Boundary Lyer Theory

McGRAW-I4ILL SERIES IN MECHANICAL ENGINEERING 

RARRON . Cryogenic System 

JACK r. IIOLMAN, Southern. Methodist University 

Co1tsu1lin.g Editor 

ISC:KERT . hzlroduclion lo Heat and M ar Tran.fer 

ECKERT AND DRAKE . Ana1y.ri.r of Jim1 and Mos,r 7i-nnsfr.r 

ECKEK.~ AND DRAKE - Ifen1 attd A4ass 7ian.fer 

HAM, CRANE, AND RODERS . Mechanics of Machinery 

HARTENRERO AND DENAVIT . Kinen~nlic Synlhesis of Ihkages 

rrmm . Turbulence 

jmonsm AND AYRE . EtlGqineering Vihralions 

~~v1NAl.i . Ettgitteering Consideraliot~.~ n/.Ylrc.~r, .ylroi~, ntzd Slretzgth 

KAYS . Co~tvecliue Heal and Mass Trcrtzsfir 

LICIIIY . fh~bt~s/ion Engine' Proce~~es 

M A R ~ N . Kil~~tnalics and Dytian~ks a/ machine.^ 

IVIELAN . I)!/~lan~ics qf Machinery 

PIIELAN . ~ll~ldfltlletll~l~,~ of n/fecharlim/ h.rigtl 

RAVEN . Aulotnnlic Corrlrol En.gineerirtg 

SOHP,N(:K . 7'hroric.r ?f Engitteering Expwir~lenlnlio~l 

noundary -layer Theory 

Dr. HERMANN SCHLICHTING 

Profresor J3rncrit.11~ nt, tlrc Ihgincrrirrg U~~ivrr~it.~ of ~ ~~IIIIR(.~Iwc~~, 

Orr~~lnrl~ 

Forrner 13ircctor of thc Arrodynnrninclre Vcr~rrclrsnnslnlt (;iittirrgcn 

Dr. J. KESTIN 

I'rofe~sor at ljrown Univrmity in Providcncr, Rliodc Ialand 

McGRAW-HILL BOOK COMPANY 

New York - St. Louis . San Francisco . Auckland . BogotL . 

Diisselilorf . Johannesburg . London . Madrid . Mexico . Montrenl . 

New Uelhi - Pa~iarno . Pnri~l . Siio I'nulo . Singtrporo Sydnoy Tokyo . Toronto

Library of Congress Gtnlogi~~g in 1'11blirntio11 Data 

Virsl p~tl~lisl~r

vi Contents 

CIIAPTEII V. Exnct ~olutiona of the Nnvier-Stokes eqnationa 

a. Parallel flow 

1. Pnrnllel flow through n straight channel and Couetto flow 

2. The Hagen-Poiseuille theory of flow through a pipe 

3. The flow between two concentric rotnting cylinders 

4. The n~~ddenly accelerated plone wall; Stokes's first problcrn 

5. I%w forn~at,ion in Couet,tc motion 

6. Flow in n pipe, start,ing fro~n rest 

7. 'The flow near nn oscillating flat plate; Stolccs's second problem 

8. A genernl class of non-steady solutions 

b. Other exact solr~t.ions 

9. Stngrdon in plane flow (FIie~nenz flo~v) 

9a. Two-tiimensiond IIOII-steady stagnntion flow 

10. Stagnntion in three-dimensional flow 

11. Flow near a rotating dink 

12. k'low in convergent nnd divergent cl~nnnels 

1:). Concl~~ding re~nnrk 

Refrr~nces 

CIIAYL'ER VI. Very slow motion 

n. The d~fircntinl eqrmtions for the rase of very slow motion 

b. I'nrallel flow pnst n sphere 

c. The I~ydrodynnrnic theory of Iubricnt,ion 

d. The llelc-Sl~aw flow 

Rcfcrrnrrs 

Fort B. Lnnninnr boundary layers 

CHAPTER Vll. l3011ntlary-lnycr equntion for tun-dirnrnaionnl inrompreusible flow; 

houndnry lnyer on n plntc 

n. Derivation of bortntlnry-lnyer equations for two-dimcnsional flow 

b. The scp~riition of a I)o~mdary layer 

c. A ren~nrlc on t,l~e integration of tl~c bortntlary-layer eqr~ntions 

d. Skin friction 

e. 'The 1)oundnry lnycr dong a flat. platc 

I. Ib~~nclnry Inyer of I~igi~er order 

11 rlrrcnrrs 

a. l)ejwn(lrncc of t,l~e cl~nrc~ctcrist.ics of n 11onnd:~ry lnycr on tltc Iley~~olds ~~n~nbcr 

b. "Sin~ilnr" solnt~ioos of the ho~n~dnry-layer ~qnnLions 

r. 'l'rnnnforn~nt~ion of t h bo~~ntlary-laycr cqontions into t,ho hcnt-conduction 

oqnal,io~~ 

(1. 'I'l~e ~non~cnt,um nnd cnorgy-int.cgrnl equations for t,lw l~o~~ndnry laycr 

I~.clorcnrcn 

' 1 1 1 1 1 I 15unrt sol~~tions of the steady-stnk bountlnry-lnyrr rquntions in twotlirnensinnnl 

n~otio~~ 

n. I%\r pnst a wrdgr 

b. Flon in n convergent cl~annel 

c. Flow pnuta nylindcr; nymmet.rical cnso (Blnsi~~s series) 

(I. Jh~ntlnry lnyer for the potential flow given by U(x) = Uo - axn 

e. Flow in the mn.lte of flat, plab at zero inridcnce 

1. 'rho t.mo-tli~nr~~sio~~nl lnn~innr jet 

g. I'arnllrl sl.~cnn~n in Inminnr llow 

t 

11 Flou in tlw inlrt Ir~~gth of n straight cl~nnnol 

i. Tlw rnctl~od or finite dillcrcncrs 

j 13oundory lnycr of second order 

Rcfrrrnrc~s 

Contcntn vii 

\ l l X. ,\pproxi~nntc ~nctl~otls for tl~r solnt,ion of t,l~e' two-tli~~~r~~nion:il, strncly 

l)o~~~~~I:~ry.l~iycr ~rpntinns 

n. ,\p!)lii.nt,ion of tlw III~III~~II~IIIII eq~~nlint~ t.o the flow pnnL n fI11t pllbt,e nt mro 

incdcnce 

b. The npproxi~natc method due to 'I%. vou I

A I l l 

1 : 

I I I. I.IIIII~II:I~ Iio1111(1nry li~yrrs in cwnprrssil)lr flow 

I . 

IZo~~~~cl:~ry-l:~yrr rontrol ill Inluit~nr flow 

n. hlrtlwcls of l)o~i~~~l:~ry-l:~yrr co~~trol 

I. hI(it,im~ of tllr solid \ d l 

2. ~\wvIrr:~l in11 or t IN* Iio1111(1ary hyvr (l)Io!vi~~g) 

3. SIIVI inn 

4. It~jcvl irm of n clilliwwt, gns 

6. I'rrvrt~l in11 t~f trn~lsit ion hy IIIC provisint~ ofs~~i(nl~lr sl~nprs. I,n~~~it~nr 11cr0fni1s 

I;. ('onli~~g of I I I wtll ~ 

I). lto~~~~(l~~ry-I~tjcr 

s~wtinn 

I . Tl~rorvl iw~l rc:s~~lln 

I. I. I~'IIIII~:II~~II~I~~ 

COII~~~OIIR 

I .2. 15xnct SOIIII~OIIS 

I.:!. ,\1qiroxi111:1lr .ml~~tio~~s 

2. lCx~ic~ri~~~vnI:~l vrsttlts oil s~~ctiou 

2.1. I wrrasr in lift. 

2.2. I)vrrrnsc? in clr:tg 

c,. 111jrr.tiri11 of ;I dill'rrcv~l g;ln (I

Y Contents 

A " I 

XI X 'I'lirorrt.irnl nsnnn~ptiona for Ilio cnlcul:~t.ion of turbr~lcnt flows 

('Il~\Pl'lCI1 XX. l'nrlwlrnt flow tl~ror~gh pipru 

a. Exprrin~cntnl restllts for 811100th piprs 

h. J

'I'i~blr 17.1 : Ikl~cwclcnrc of rrilirnl Rcyriolcln IIIIIII~I~ of vnlocil.y yrofilrn will1 nuotioll on 

din~c~~~~ionIcs$ n~~ction vol11111e f:wIor E, :~.fter UlricI~ [24:1J 

'T:ll,le 20.1: Iht.io of Inearl to 1nnxi111u111 

n 

of t,hc vclocity cliatribution, according to eqri. (20.fi) 

vc1ot.it.y ill pipe lir,w in t,ertns of tho expone~~t 

'I':hlr 21.1: T

xvi I'orcworcI 

Thc result was t.11~ l)onk or 483 pages and 206 figuros publisl~ctl in 1061 in the Gcrrnnn 

Inng~tagn. \Vhcn t,llis book bcca.mc Icnown t,o rcscarcll workers and educntors in 

t,llc Unit,oti St.ntcs, t.l~cro was nn inunctlint.c request from srvrral quarters for an lhg- 

MI t.mnslal.ion, sinro no rnrnpar:~blc book was nvnilnhlc in the 1Snglish Inngungr. 

'I'hc tcc:l~t~ical contcnt. of t h present. I':t~glisl~ etlit.io~t is dcscril~ctl in t,hc nr~thor's 

prcf:~c.e. 'l'hc c~npl~:mis is 011 t,l~o ftlntl;rmrl~l;rI pllysiral itlras rntd~cr ~,II:LII on mntl~ctn:lt.ic.:tl 

rrfinrlnrllt,. RIt:t.l~otls of t,llcort:t,icnl nn:~l~sis qrc sot forth dong with s11c:l1 

rxlwrirnont~d tln.t,n as arc pcrt.in(-nt t,n (Icfinc the regions of applic:tbilit,y of 1I1o 

I~llcwrc:l~icnl rcsult.s or t.n givr: 1i11ysic:rl it~sifillt, i~~t,o 

t,I~t: pl~c~~omcnn. 

\Vasl~ingtot~ I). C., 1)corml)cr 1064 Ilugl~ I,. I)rytlrn 

Author's Preface to the Seve~itl~ (English) Edition 

Whcn J decided in 1075 to writ,c n new rclit,ion of t.11i.q boolc I cnmc t,o t,llc conclusion 

t.l~nt, t.l~o prccrtli~~g sc:qncnrc of n (lcvlnnn rdit,ioll followc:tl I)y nn 15nglisl1 c:clit.iotl 

was no longrr prscticnltlc. 'I'hc rcnsotl for it wn.8 lllc 11cnvily incronsrtl cost, of p.int,ing. 

Conscqncntly, I suggrst.rcl 1.0 the bwo publishing cornlmnics, G. llrnun in I

xviii Ar~t,l~or's Prefncr t.o tho Seventh (Englisl~) Er1it.ion 

Along with this ncw material, I fee1 t,hat I ought, to niention the topics which I 

spcoifioally omit,t,ctl l,o include. I do not, discuss t,he effect of chemieal reactions on 

flow processes in boundary laycrs as they occur in the presence of hypersonic flow. 

The sarnc applios t.o I)onndnry Inycrs in rna.gncto-fl~~itl-clytin~~tics, low-dcnsitty flows 

and Rows of non-Nowt,onian fluids. I still t.11onght that T ought to refrain from giving 

a.n rxposit,ion of t,lir st,at,ist,ical t,heory of t,ttrl)~~lenrc in this etlit,ion, as in t,hc prcviolls 

OIICR, hrrnusc no~~~dnys t.l~crc arc avnilnblc otlrcr, good prcscnt,nt.ions in I,oolr form. 

Oncc again, t,hc lists of refcrenccs have bcen expanclcd considcrahly in many 

rhnpt,crs. The nurnl~cr of illust,rations increasctl by about G5, hut 20 old ones havc been 

omit,t.cil; the number of pages increased hy about 70. In spite of t,his, I hope that 

t,he original character of t,liis book has becn retained, and that it, still can provide 

tlie reader wit,l~ a bird'.?-e?y view of this important branch of the physics of fluids. 

As I worlrrd on the new manuscript I once more enjoyed t,hc vigorous assistance 

that I rcccivetl from scvrral of my professional collcagues. Professor K. Gersten cont.tihutctl 

sect,ions on boundary layers of second orrlcr t,o the part on laminar boundary 

lnycrs (Seas. VIIf ant1 IX j) . This is a special field which he successfully worked out 

in rccent ycnrs. l'rofcssor T. K. Fnnneloep contributed the completely reformulated 

sc-ct.ion on the nurncrical inkgration of t,hc boundary-layer equations included in 

Scc. IXi. In t.hc part on turbulent boundary layers, Professor E. Truckenbrodt 

provitlcrl me witall a new version of the largest portion of Chapter XXII on twodimensional 

and rot~ationally symmetric boundary layers Dr. 1,. M. Mack of the 

California Institute of Technology was good enough to contribute a new section on 

the stability of boundary layers in supersonic flow, Sec. XVIle. Dr. J. C. Rotta 

tliorougl~ly reviewed Part I) on turbulent boundary layers and made many additions 

to it*. For the Russian litcrxtnre I rccrivstl nlurh help from Professor Milrhailov. The 

translation was once again cnt,rustctl to Professor J. Kestin's competent pen I express 

my sincerc thnnlrs lo all tliose gcntlcmen for thcir valuable cooperation. 

I should also like to rcpcat my aclrnowlcdgemcnt of thc hclp I rcceived from 

scvernl professional friends whcn I worked on the fifth (German) edition Nat.urally, 

their contributions havc now bcen rctaincd for tlie seventh edition. This is the ex- 

tcnsivc contribution on comprcssiblc hminar bountlary layers inChapter XIIT written 

by Dr. F.W. Rirgcls, Profcssor I

From Author's Preface to the First (German) Edition 

Since :t,I~o~tt, the Ocgit~ning of 1.11~ twrrcnt ccnt,ttr.y niot1t:rn rcsr:~rclt in t,It(* lit*ltl 

of fluitl clyn:rntics has :~clticvctl grcat sut:ccsscs nntl Itas h:n able to provitlc :I Cllc:. 

oretiral clarific:ttion of obscrvc:tl pltcnonwna which tJlc scicncc of rlnssirnl Ilytlrotlyn:imics 

of t,ltc ~)rocctling c:cntnry failctl t,o (lo. 1kcnLi:~lly t.llrt:c br:tnc:ltcs of llr~itl 

tlyr~:~rnic,s 11:~vc bccomc p:~rticnlarly well clcvelopctl during t,hc last fift,y years; tllcy 

inclutlc hountlary-layer tl~cory, gas dynamics, and acrofoil Lllcory. 7'1m prcscrit t~ook 

is conccrncd with the branch knnwn as 1)ountl:~ry-layer thcory. This is the oltl(:st 

branch of modern fluitl dynamics; it w:is fou~~tlctl by 1,. I'mntltl in 1904 wllcn Ilc: 

succcedctl in showing how flows involving fluids of very s~nnll viscosity, in particular 

wntm ant1 air, the most imporl;:~nt, oncs from the point of vicw of applications, c:ln 

11c m:dc :~tncn:~blc! lo rnnthomr~t,icnI r~nnly.qix. 'l'llis wris rdliovotl by ttiking I.II(: cl1i:c:l.s 

of friction into account only in regions whcrc they arc rsscnt,i:d, namely in tho thin 

boundary layer which exists in t h irntnctliatc ncigt~bourl~oocl of a solid body. This 

concopt matlc it possible to clarify many pllcnomona wliich occur in flows and which 

Imtl prcvionsly bccn incotl~pmllcrtsit)le. Most important of dl, it, Itas bcconto possiblr 

to subject problems connected with thc occurrenor: of drag to a tllcorctical an:tlysis. 

r 

llte 

, 

scicnco of aeronautical engineering was making rapitl progress ant1 was soon 

&ble to utilize these t,l~coretical results in pract.ical applications. It tfitl, furthertnorc, 

pose many problcms which could be solvctl with the aid of the ncw bonntla.ry-layc:r 

theory. Arronautical engineers have long sinco matlc: the conccpt of a tmuntlary 

layer one of everyday use and it is now unt.hir:Icable tm do without it,. In other fieltls 

of lnaclline design in wltich problems of flow occur, in particular in the design of 

t,url~ornacl~incry, the theory of boundary layers made much slower progress, I,trt, 

in motlcrn tin:es t,hc:sc rlcw conccpt,~ Itavo come to t,hc fore in suc:I~ applic.ztions as well. 

r 7 

I he prcwnt 1)ooIt Ims hcrn writhxi principally for cnginecrs. It is thc olzt.comc: 

of a course of Icct,urcs which the Author tlclivcrctl in t,llc Winter Scmcstcr of 1941/42 

for the scinnt,ific worltcrs of tho Aoronauf.ical Itcscarch Institut,c in I3r:~11nscl1wcig. Tho 

stll)jtwt. mnttcr has bcclrt utili;r.ctl aftcr tho war in nlany spc(:i:d 1cct11rt:s 11cld at tl~c 

ISngitleering Univcrsit,y in 13munschwcig for sttdcnb of rnccl~anical engineering ant1 

physirs. Dr. IT. IIahnclnann prepared a set of locturc notes :iftcr the first sorics 

of lectures \rat1 been givcn. 'L'lrcsc were rcad mxd amplifier1 by t.hc Autlior. They wwc 

subscq~lcr~tly published in mimeograpltctl form by the Office for Scicrltific I)ocurncnt,at.ion 

(Zontmlc fiir wisscnschaft~lichcs 13cricltt~swc.scn) nntl tlist.rit~nt.ntl 1.0 :t 

lirnit,crl circle of intcrcstctl scicntifir: workors. 

Several years after the war tho autdtor tlecitlcd con~plkt~cly to re-edit, this older 

c:ompilat.ion and to publish it in the form of a book. 'l'hc tho sccrnctl ~~articularly 

propitious becausc it appeared rip for thc publication of a comprel~cnsivc I)ook, 

and because t.hc results of tltc research work carried oul, tlt~ring fhc last trn t.o twcnt,y 

yrxm rounrlctl off trltc wltolc ficld.

'She book is tlivitlctl ink four main ptrts. 'L'hc first, part contains two introtluct,ory 

ch:tpters in which t,hc funtlamcnti~ls of 1)onntlary-layer theory arc cxpoundetl 

without, the use of mathematics and then proccccls to prepare tho matl~ernatical 

and physical jllstification for the tl~cory of laminar bourulary laycrs, and includes 

the theory of thormal boundary Iaycrs. Tho t,llird part is concerned with the pllenomenon 

of transition from laminar to t,urbulent flow (origin of turbulence), arid the 

fourth pert is devoted to tnrl~ulcnt flows. It is now possible to take the vicw that 

the theory of laminar boundary laycrs is complete in its main outline. Tho physical 

relations have been complctcly clarifictl; the meifhods of calculation have been 

largely worked out and have, in many cascs, bccn simplified to such an extent, that 

they should present no difficulties to engineers. Jn discussing turbulent flows use 

has been made essentially only of t,hc scmi-empirical thcorics which derive from 

Prantltl'~ mixing length. Tt is true that according to present views these theories 

possess a number of shortcomings but nothing superior has so far been devised 

to take their plate, nothing, that is, which is useful to the engincer. No accour~t 

of the slstistical theories of tr~rbulcnce has been inclutlcd because they have 

not yet attained any pract.ical significance for engineers. 

As int,imat,cd in the t.itle, the emphasis has bccn laid on thc thcorcticnl trcatmcnt 

of problems. An attcmpl, has bccn made t.o hring thcse consiclcrations into a form 

which can he rasily graspctl by engineers. Only a small numl~cr of results has hccn 

quoted from among Ifhe very voluminous oxperimcntal material. They have bccn 

chosen for their suitability to give a clear, physical insight. int,o the phenomena and 

to proviclc direct rcrific:rtion of thc t.l~cory prcsentcd. Some examples have been 

chosen, namely those a~sociat~ctl with tur1)nlcnt flow, because they constitute the 

fonntlation of the semi-empirical theory. An attempt was made to tlcmonstrat,e 

that essential progress is not, mndc through an accum~~lation of extensivc exprrirnental 

rcsriltn but rather through a small number of fundamental cxperiment,~ hacked by 

theoretical consitlerat,ions. 

Brarmschweig, October 1050 - IIermann Schlichting 

Introduction 

Towards the end of the 19th ccntury the science of fluid mechanics began $0 

dcvclop in two tlircctions which had pmct,ically no points in common. On t,hc onc 

side therc was the science of theoretical hydrody~tamics which was evolvctl from 

Euler's equations of motion for a frictionless, non-viscous fluid and which achieved a 

high degree of completeness. Since, however, the results of this so-called classical 

science of hydrodynamics stood in glaring contradiction to experimental results - in 

particular as regards the very important problem of pressure losses in pipcs and 

channels, as well as with regard to the drag of a body which moves t,hrongh a mass 

of fluid - it had litt,lc practical importance. For this rcason, practical cngincers, 

prompted by thc need to solve the i~nport~ant prok~lcms arising from the rapid 

progress in t,echnology, developed their own highly empirical science of hydraulic^. 

The scicnce of hydranlics was based on a large number of cxperinlentd tlal,a :mtl 

difl'ercd greatly in its mct,l~ods antl in its objccts from the scicncc of t.hcorct,icnl 

hydrodynamics. 

At the beginning of the present cent.ury L. Prandtl clisti~lguished himself by 

showing how to unify thcse two divergent I~ranchcs of fluitl dynamics. He achieved 

a high degree of correlation between theory and experiment and paved the way 

to the remarkably successful development of fluid mechanics which has taken place 

over tlhe past sevent,y years. It had bcen realized even bcfore l'randtl that the discre- 

pancies between t,he results of classical hydrodynamics and experiment. were, in 

very many cases, due to the fact that the theory neglected fluid friction. Moreover, 

the complete equations of motion for flows with friction (the Navier-Stolres equa- 

tions) ha.d been known for a long time. However, owing to the great mathematical 

difficulties connected wit,ll the solution of t,llcse equations (with the exception of :L 

small uuniber of particular cascs), tho way to a thcorcticnl treatment of viscous 

fluid motion was barred. Furthermore, in the case of the two most important fluids, 

n:~mcly water antl air, the viscosity is very small and, conseqnently, tho forccs 

due to viscous friction are, generally speaking, very small compared with the 

remaining forces (gravity and pressure forces). For this reason it was very difficult 

to comprehend that t,he frictional forces omitted from thc classical theory influenced 

thc motion of a fluitl to so large an extent. 

In a pzpcr on "Fluid Motion with Very Small Friction", read bcfore the Mathc- 

maticd Congress in IIeidelberg in 1004, I,. Prandt,lt showed how i't was possible tJo 

analyze viscous flows precisely in cascs which had great pmctica.1 importance. Wit,h 

II. Schlicl~ling and II. U6rtlcr. rol I1 pp '15-584. 

Abl~anrllnngc~~ rur

the aid of thorctical considcmtiotis anti scvcrnl simplo oxperimenk, ho provcd t.hat 

the flow about n solid body can be dividod into two regions: a very thin lnycr in t,l~e 

neighbourhood of t.he body (ho~~mla.r?j lmycr) whcrc friction plays an essential part, 

and thc remaining region ontdc this laycr, where frict,ion may be ncglcctcd. On 

tho basis of Lhis Ilypot,l~esis I'mntltl succccdctl in giving a physically pct~rt~rating 

nxplnt~at~ion of tlrt: iml)ort,n.ncc of viscous flows, achicvit~g at tho samc timc n maximum 

tlegrcc of simplification of the attcntlant rnatltemntical rlifficnlties. The t,heorct,ical 

considerations werc even tJ~cri snpport,cd by simplc cxpcrimcntn pcrformcxl in a 

small water tonncl which Prn.ridL1 built, wil,h his own hands. Ilc thus took the first 

step towards a re~tnification of tl~cory and pmcticc. This boundary-layer theory proved 

cxtrcmely fruit,ful in that, it provided an cKcctive tool for the tlevelopmcnt of fluid 

tlynamivs. Since the 1)cginning of the cnrrcnt century the new theory has been tlcvcloprd 

at n vcry fast r:lta untlcr t,hc atlditionnl st~imulns obtained from the recently 

fountlctl science of aerodynamics. In a vcry short time it hecame one of thc fo~~ndat,ion 

stonrs of modern Ilnid clynamics t,ogcthcr with thct other very inlportant tlevclopmenk 

-- t h acrofoil theory and thc sciencc of gas dynamics. 

In more recent t,imcs a good deal of at,t,ent,ion has been devotctl to stdies of the 

mntlirmatirnl just.ification of boundary-layer theory. According to tllcse, boundarylayer 

theory provitlcs us wit,h a first approximation in the framework of a more 

general t,hcory designed t,o ca1culat.e ~symptot~ic expansions of t,he solutions to the 

complet,e equat,ions of motion. The l~rol~lc~n is retlucetl to :L so-called singular perturbation 

which is then solved by t.hc mct.liod of mat,chcd asymptotic expnnsions. 

I3ountlary-layer t.hrory t,hus providcs us with n cIassic example of the npplication 

of thc met,liotl or singular pcrt,nrbnt,ion. A general presentation of pert>urbation 

rnct,horls in flnid mechanics was prepared by M. Van Dykt:t. The basis of these 

rnat,hotls can be Itraced to 1,. J'raritlt.I's early co~lt~ribut~ions. 

'I'lic 11onntlar.y-layer tlicory finds its application in the nnlcnlxtion of t,he skinfriction 

dmg whic:h ac1.s on a body as it is moved t,hrongh a fluid : for example the 

rlr:lg cxpcricncctl by a flat p1n.t~ at,xcro incitlcnce, tilo t1m.g of a ship, of an aeroplane 

wing, aircraft, t~acrllr, or t,rrrl)ine I)latlc. 13o11ndnry-layer flow 11:~s t,I~c peculiar property 

t.ll:~t, untlor ccrt.airl conditions lhe flow in the imnictliat,c ncighbonrhood of a solid 

wall 1)ccomcs rcvcrscd causing the I~ountlary laycr to separate from it. This is accompnnirtl 

11y a morc or lrss pronouncctl fonnat,ion of eddies in the wake of t,hc body. 

'J'hus t.hc prcssnrc distribution is rltangcd and differs marlrctlly from that in a 

frict.ionl(\ss strcnm. ?'hc tleviation in prcssurc tlist,ribution from the ideal is the 

canse of form drag, antl its cniculat.ion is thl~s made possible with the aid of bouriclarylaycr 

t.llnory. 13ountlary-hycr t,heory gives an answer t,o the vcry irnp~rt~ant question 

of' w11n.t shape mnst, a hotly t~o given in orclrr to avoid t.llis dct.rimarital scpn.ration. 

Scpnr:rI.ion mn also oc.c:ltr in l.llc int.crn:tl flow t.hrorrg11 R (:11nnnc1 ant1 is not, confined 

to rst,rrnnl Ilows past solitl I~otlicx. I'rol~lrms conrrcct,crl with l.11~ How of fluids 

throilgl~ t,hc cl~m~ncls ftm~~ctl by t.hc blntlcs of t,urhomachines (rotary compressors 

ant1 t,url)inos) ran also he 1,rrntrtl wit,ll tho n.itl of 11ountl:~ry-hycr t,Jlcory. I'r~rt.llcrmore, 

~~lw~lon~cnn wllic:l~ occur at, t,llc point of rnn.xirnnm hft, of nn acrofoil and which arc 

assocht,t:tl with st.:~llitl~ (:;I.II 1)c 11ntlcrsI.oot1 only on thr 11n.sis of I~onntlary-layer 

theory. Ihdly, problrms of l~cat transfer I)ctwcwl n solitl hody ant1 n fluitl (ps) 

flowing past iL also bclong to thc class of problems in wltic41 l)o~~t~tl:~ry-l:~yc.r 1)11vnomrnn 

play n dccisivc pnrL. 

At, first the l~ountlary-layer theory was devclopotl rnn.inly for tl~c two of 1:~nlin:lr 

flow in an incon~prcssil)lc fluitl, RR in 1.llis c:uw t h ~)l)~t~o~nt:t~oIo~it::l.l I~j,~)oI.I~t-sis 

for shr:~ring st.rrsscs a1rr:ttly cxistctl in thc form of Sto1tt.s'~ I:\w. 'l'l~is t,t,l,it: W:IS 

sul)scqucntly tlcvclopctl in a 1:Lrgc ~lurnhcr of rcsonrclt p:Lpcrs :LII(I rt::1(:11vtl s1tt~11 a 

stagc of pcrfoct.ion Iht at prcscnt tltc problem of Intninar llow c:1.11 III: consitlt~rctl 

to h:lvc hccn solved in its main oullinc. 1,:llcr the Ll~cory w:ls cxl.ot~clt:tl 1.0 int:lurlc 

turl~ulcnt, incornprcssil)lc bountlary layers which are morc irnportzmt from (ht: poitlt, 

of vicw of practical applications. It is true that in tltc cnsc of t~trl)~~lcnt. flows 0. Iloynolds 

introduced the fundamentnlly important conccpt of nppnrcnt, or virt,~tnl tltrl)ulent 

stresses as far back as 1880. IIowevcr, this conccpt was in it,sc.lf itisuffioirnt tso 

mn.ke tltc theoretical analysis of turbulent flows possible. Great progress was acllicvecl 

with the intmtlnction of I.'randtl's mixinglcrtgt.l~ thcory (1025) which, t,ogol,hrr wit,li 

systematic cxperimcnt.s, paved the way for the thcorctical ttrcntmcnt of turl1111c1tt 

flows wit,l~ the aid of boundary-ln.yrr t.hcory. llowevcr, a rational theory of fully 

developed turbnlcnt flows is st,ill noncxist.cnt,, antl in vicw of the cxl,rtmc complexit,y 

of sucll flows it will remain so for a consitlcmhlc time. Onc cannot even be 

ccrtain that science will cvcr be successfnl in this t,aslr. Tn modern times tho phcnomena 

which occur in the boundary laycr of R cornprcssiblc flow have becomc the 

subject of intensive investigntions, the impulse having Iwcn provided by thc rapid 

incrcasc in tllc spcctl of flight of motlcrn aircmft,. In atltlition to a velocity 1)oitntlary 

laycr suc:h flows dcvclop a tllcrrnal bonntlnry hycr ant1 its cxist~cncc phys :I.U irnportant 

part in the process of heat txansfcr bctwceri the Iluitl and the solitl body 

past which it flows. At vcry high Mach numbers, the surface of Lhc solid wall bccornrs 

heatetl to a high t,cmperature owing to the protlnct.ion of frictional heat ("tllcrrnnl 

barrirr"). This phenomenon prcscnts a tliffic:nlt analytic problem whose ~ol~ttion 

is import.ant in n.ircmft tlcsign antl in the ~~ritlcrsl~anding of the motion or satellites. 

r 7 

1 he phenomenon of tmnsit,ion from liltninar to turbulcnt flow which is ~~I~~:LIIIBII- 

t.aI for t,he science of fluid tlynamirs was first investigated at thc end of tl~c I0t.11 cent,nry, 

naniely by 0. 12eynoltls. In 3914 1,. 1'm.ndtl cnrrictl out, his fnmous expcrimrnts 

with sphcrrs antl ~uccccdc~I in showing that the llow in Ihc 1)ountlnry layer car1 also I)c 

either laminar or turbulcnt and, furthermore, that, tltc problem of separnt,ion, ant1 

hence the problcm of the calculat~ion of dmg, is govcrnctl by this tran~it~ion. Y'hcoretirat 

invest,igations into t,he process of transilion from laminar to tnrbulcnt flow are 

basctl on t.110 acceptlance of Iteynoltls's 11ypot11o~is l,liat tohe lattm occurs ns a conscclucncc 

of an instability dcvolopcd by Ihc 1nminn.r 1)onntlary layer. 1'rnntlt.l ittif.int.ctl 

his thcorc1.icn.l investigntion of trnnsition in tllc ycar 1921 ; after marly v:rin cflort.~, 

succcss came in the ycar 1920 wlicn W. Tollmicn compntrd theorct,icnlly t,hc crit,ic:aI 

Reynolds numbor for transition on a flat plate at zero incidence. Ilowcvrr, nlorc 

t.lran ten years werc to pass 1)efore l'ollmicn's theory coialtl ho vtdficd throngl~ tho 

vcry carcful experin~enLs performed by 11. 1,. 1)rytlcn antl his coworltcrs. Tho stn1)ilit.y 

tltcory is capsblc of taking into account the cKcct of a nurnhcr of parnmctcrs (pmssurc 

gradient,, suction, Mach numlter, transfcr of heat,) on tmnsition. This theory has 

found m ~ny important applications, among them in tl~c dosign of scrofoils of' very 

low drag (1aminn.r ncrofoils).

Modc:rn invcstigalions in id~c ficld of fluitl dynamics in general, as well as in 

t(11c ficld of bountlary-hycr rcscarch, are characterized by a vcry close relation 

bc!twcen theory ant1 cxpcrimcnt,. The most important steps forwards have, in most 

cases, barn t,nltcn as a result of a smdi numl~cr of fi~ndamcntd cxpcrimcnt,~ bacltetl 

by t,hcorot,icnl considcrat,ions. A rcvicw of tJ~c tlcvclopmcnt of boundary-layer 

t.Ileory wllich st~rcsscs tllc rnuf,nal cross-fertilization bctwccn theory and cxpcrirncnt, 

is containctl in an n.rliclc writtrn 11y A. lktz?. Vor about, twenty years aft,er its 

inccption I)y T,. I'randtl in 1904 thc bonndnry-la.ycr tllcory was being developed 

nln~ost exclwivcly in his own institute in Goettingen. One of the reasons for this 

st.nt,c of nffnirs may well havc been root,cd in the circum~t~ancc that, J'randtl's first 

pnblionthn on boundary-layer theory which appeared in 1904 was very dimcult to 

understantl. This period can be said to have ended with I'randtl's Wilbur Wright 

Meniorial I,ect,ureo which was dclivcrcd in 1927 at a meeting of the Royal Aeronautical 

Society in 1,ontlon. In later years, roughly since 1930, other research worlters, particularly 

t,hosc in Grent nrit.ain and in tllo U.S.A., also took an active pn,rt in its 

tlevrlopmcnt. Toclay, the study of boundary-layer theory has spread all over thc 

world; together with othr branches, it constitutes one of t,he most import,ant pillars 

of fluid mechanics. 

Tho first survey of this I~mnch of science was given by 1%'. Tollmien in 1931 

in two short articles in the "llan~lbnch dcr ISxpcrirncnt~alpl~ysiIr" :. S11orl~I.v aftcrwartls 

(1936), Prnrdtl p~~l)lishcd a cotnprnl~cnsivc presentation it1 "Aerodynamic 

'J'hcory" ctlitctl I,y W. I?. Durands. lluring t.he intcrvcning four dccndcs tllc volume 

of rescarch into this subject has grown cnorrnonsly$. According to a review published 

by 11. I,. Drydcn in 195.5, t,hc rate of publication of papers on boundary-layer 

theory reached one hundred per a.nn7r.m at that time. Now, some twenty years later, 

this rate has more than tripled. Like several other fields of research, the t,heory 

of ho~rntlary layers has reachetl a volume which is so enormous that an individual 

scientist., even one working in this field, cannot be expected to master all of its 

specializctl subtlivisions. It is, tl~rrcforc, right that, the task of describing it in a 

nlotlcrr~ Ilanclboolt has been cnt.rustcd t,o several authorst. The hist,orical development, 

of bountlary-layer theory has recently been traccd by I. Tani*. 

. . 

" 1,. J'mllrlI,l, Tho goncmlion of vortiron ill fluirls ofatn.zII viscosit,y (15td1 Wilbilr Wright Memorial 

Jfir(llr% 1!J27). J. Jtoy. Aoro. Soc. 31, 721-741 (1!)27). 

: (!/. tho bildiogr.zl~hy on 11. 780. 

: I,. l'r:~n(ll,l, T11c 111ecl1a.11irs ol' vi~coun fluids. Arrodynamii~ tl1oory (W. I?. Ihrm~d, rd.), \'ol. 3, 

34 208, I%crlin, 1935. 

6 11. Schlirh~ing. So~ne tlrvcloprncn(.s of I~oundnry-layer rcsearch in the past thirty years (The 

'I%ird L~tlcl~r~lcr Metnorin.l J,rcture, I!W)). J. hy. Aero. Soc. 64, 03- 80 (l%U). 

Srr nl?lo: 11. Srlilicl~l ing, Rccrrtt progress in houndn.ry-lnycr research (The 37th Wright. Brothers 

I ~ ~ i t r 1tt11r1, 

i : ! 7 ) \ I .Jttiri:~l 1 427 - 440 (1!)74). 

* I. 'I':\t~i. Ilislory of I~o~~nrlnry-lilyor rmrnrcl~. An~~rinl Itrv. rrf Izluid Mwhnnirs 9, 87- 11 t (1977). 

Part A. Fundamental laws of motion for a viscous fluid 

CHAPTER I 

Outline of fluid motion with friction 

Most t.Ileoret.ica1 invcst,igat,ions in the ficld of fluid dynamics arc based on the 

concrpt of a perfect,, i. c. frictlionlcss antl incompressible, fluid. In the motion of 

sucl~ a perfect flnid, two cont,act.ing layers cxpcricnrc no tnngcntinl forccs (sl~caring 

st,rcssrs) b ~~l, act on tach other wit.11 normal forccs (j)rcssums) only. This is cqr~ivalcnt, 

t.o stal.i~~g tl~nf, a pcrfvct, fluitl olrcrs no inl.crria1 rc~isI.antx to a c11angc in SII:I~O. The 

tl~cory describing !,hc motion of a pcrft:cl. lluitl is ~natl~c~~~~nt.ic:~lly vcry far tlnvclopctl 

ant1 supplies in many cases a satisfactory dcscril;t,ion of real motions, such as e. g. 

tlle motion of surface waves or the formation of liquid jets in air. On the ot.her hand 

the theory of perfect fluids fails completely to account for the drag of a body. In this 

connrxion it leads to thc statement that a I~otly wllich moves uniformly t,llrongh a 

fluid which cxt.ends t,o infinity experienccv no drag (tl'Alcmbcrt.'s pamtlox). 

'Pliis unacceptable result of thc thcory of a pcrfect Iluid can be traccd to the fact 

that. t.11e inner layers of a real fluitl tmnsmit t,angent,ial as well as normal stresses, 

this lxing also the case ncar n solitl wall wetted by a fluid. Thesc tangential or frict,ion 

forccs 111 a rrxl Ilnitl arc conncctctl with a propertry which is callctl the viscosil?/ of 

thc Ilnid. 

IZccai~sc of tho almnce of t,angcnt,ial forccs, on the 1)oundary bctwccn a perfect 

llnitl :~t~tl a. solitl wnII Lhcrc cxist,s, in gcnt~rnl, :I. tlilrrrcncc in rc~l:~l.ivc t,:~ngrnl.i:il 

vrloc.it.ics, i. c. t.11crc is slip. On t,hc other hi~ntl, in r(::11 ll~~i(ls the cxi~t.cn(:t~ of int.crmolecular 

att,ractions causcs thc flnitl to adl~crc to a solitl wall antl t,his gives risc 

l,o slrraring stmsscs. 

. 1 

, 

hc exist,cncc of tangcnlial (sl~caring) s,,rcssc:s nr~l lhc condiliols 01 ,to dip n(::~.r 

solitl walls const.itut1e the essential tliffcrcnccs bctwccn a perfect and a real fluid. 

Clert,ain fluids wl~ich arc of great, practicd imporl,ance, such as water and air, havc 

vcry smnll coefficients of viscosity. In many instances. tl~c motion of such llwids ol 

sn~nll viscosity - a.grccs - vcry well wit.11 that of a perfect Iltritl, bccausc in most cases the 

shearing stressc?~ arc vcry small. For this reason the cxist,cncc of viscosit,y is corrlplctcly 

nrglcct.cd in the t,heory of perfect fluids, ma.inly bcca.11se this introdnccs a far-reacl~ing 

' 

simplificatiott of the equations of mot,ion, as a result of ext.cnsivc niathematical 

theory I~ecomcs possilh. 11 is, I~owcvcr, islpm&, 

ss the fact that,

even in fluitls wit,lt vcry srnall viscosit,ics, unliltc in pcrfwt. fluiels, t.he rontlit.ion of 

no slip near n, solill I~oundary prevails. 'l'l~is c:ot~dil~ion of no slip int,rotlures in many 

(::~sos very hrgc tliscrcpar~cics in t,hc laws of moLiorl of perfect an(\ ronl fluids. In pnrt.icular, 

t h vcry largc tliscrcpel~cy 1)ctwccn Llle vdr~o of' drag in a rral ant1 a pnrkct, 

Iltti(1 I1:w its pl~ysical origin in the contlil,ion of no slip nwr :L wall. 

'I'l~is 11oolc t1r:rls wil,l~ 1.11~ rnot,ior~ of llrlitls of'sm:~II visrosil,y, I)(-r:~llsr of t.l~c grc:~L 

I~:~ct,ical itnporl.ance of' the problcln. Ihrirtg 1,llc course of lhc st~dy it will l~cconlc 

clear how this p:trtJy consistent ant1 p:l,rl.ly tlivcrgcnt I)cl~aviour of pcrfrct and real 

fluids can be cxpl:tinotl. 

h. Viscosity 

'I%(: II:L~,II~C of' vi~rosit~y can 11cst I)c vi~rdizcd with the :lid of t,ltc following cx- 

~wrimnnt,: Consitlcr the ~not~ion of a fluid l)cl,\vccrt two very long pn.rallnl ~)latcs, one 

of wl~inh is at rrst, the other moving wit,l~ n, constant velocity pnrallcl t,o itdf, as 

sl~owu in Fig. 1 .l. 1,ct tJ1o clist.anco hctwcc~~ thc plates bc h,, the prrssnre Iwing const,nnt 

t.l~rol~gl~ot~t tllc fluid. Exprrintcnt t.c~:rcltcs t.l~:rt t.11~ fluitl atll~rrcs l.o l)ot.l~ ~valls, so 

I,II:II, it,s vclovity :rI, the lownr p1:~t.c is zero, :I,II(~ t,11:1t 3.t Lltc ltplwr ph1.c: is rt111al to 

t.11~ vcloeit,y of the plate, IJ. Ir'rtrt~l~ermor~, I.llc vclocit.y tIist,ril)r~t,ion ill t,llc fluid 

I)ct,wccn the pIat,cs is linear, so that, the fluid vclocit,y is proport,ion:ll tto t.ltc tlist,ancr ?/ 

from t 11c. lowvr platr, :~ntl we h:tvr 

In ortlnr 1.0 s~lpport t,l~e motmion it is necessary to apply a I~n~~gc.nt,ial forcn t,o thn 

tlpprr l)lnto, tho force 1)cing in cc~t~ilibriurn with tl~c f'rict~ional forces in t,l~c fluid. 

It is Icnown from expcrimont,~ t,l~at tJtis forcc (ta.l~cn per unit awn of t,l~c plal,c) 

is proprt.ion:~.I to t,hc velocity 1J of the 11l11)er plat.c, ant1 invcrsrly proport,ion:~l to 

lhc tlist,:r.nrc~ h. 'l'llc 1'ricI.ion:ll force por nit, area, tlcnotctl by t (Srict.ional shearing 

sl,rcw) is, t,licreCore, proport.ionn1 1.0 lJ/h, for which in general we may als? ssulist.itr~t,c 

tlii/tl?/. 'l'ltc: 1)ro1)01.t~io11:rIil.y far:l.or I)ct,wcnn t ant1 d71 tly, wl~iclr we sl~all dc~~ot,c I)y ,u, 

I 

tlc11~1(1s or1 tho r~al~llrc of 1.110 ll~~i(l. 11, is ~rna.ll for. "lhiri" fluids, s11c11 nk wal.cr or 

:~l(:ol~ol, I~ut I:qn in the case of vcry viscous liquids, srtclt as oil or glyccrinc. 'I'hl~s 

wc 11;tve ol)t,:~inctl t,llc ftl~~tl:rrncnl,al rclnt,ion for fluid frict.ion in t,lte form 

> 

du 

(1.2) 

= fL ~ I Y . 

Tl~r quantity p is n propertry of thc fluid and depcntls to n great cxl.cnt on it,s ternpcrnt,rlrc. 

It is n rneasuro of tho i)i.~co,qit~y OF the fl~iid. '1'11~ I:LW of' friction givrtl by 

cqn. (I .2) is 1znow11 :LS Nrwtotc's 1rr.v~ of friction. ICqn. (1.2) cnn bc rrg:~relvrl :I.R t,llc 

c1rlinil.ion of visc:osit.jy. It. is, Ilowevcr, nccxssary to st.ross that the cxnrnplc cot~siclcrc:d 

in IGg. 1.1 (:onstit~~t.rs :L p:~rt,ic~~larly simple case of fluit1 motion. A gnncr:~liz:~l,it,r~ of 

this sitn111v e:rsc is cont,:~.inc(l in Stolccs's I:IW of fridion (cf. (!II:L~. I I I). '1'11~ ~limc~~~si~ 

of visrosi1,y c:all IIC tlotl~rc:c:tl wit,hol~t, diFlicull.y from cqn. (1.2)-I-. '1'110 sl~c:nritlg s1,rcw 

is ~ncnsurcd in N/m2 =I J'n nrld tltc vcloc:it,y grntlicnt du/tl?y in ~ o I. c ~IVII(Y* 

wllcre tho square 1~r;~(:Iccts arc IISC(~ to (Icr~ot~ 11ni1.s. '1'1~ :L~)OVC is not. 1hc o~~ly, or 

even the most, witlcly, employctl unit of viscosit,y. l'riblc? 1 .I lists t,he various t~nits 

togct.lrcr with thir conversion factors. 

.15qn. (1.2) is rc1:rtctl t.o IIooltc's law for all c~l:ist,ic: solicl I)otly in w11ic:h rasc: tl~c 

shearing sCrcss is proport,ional to the strain 

Ilrrc (: denotes lhe n~oclnlus of shear, y the change in anglc bct.wc.cn tfwo linrs 

wlliclt were originally nt right anglcs, nntl 6 tlcnotcs t.110 clisplr~ccmcnt, in t11c tlircc:t.ion 

of a1)scissae. Wllcrcas in thc cnsc of an elastic solid th: sl~caring strcss is proporl.ional 

t,o the nw~gniturle of the strain,, y, expcricnrc tcacl~cs tll:~t in tl~c case of fluitls it is 

proport,ionnl t.o the vale of chnnrlc. of strain tly/tll. If' we put 

we s1r;~ll obtain, as bcforr, 

a11 

t ' fl 

?I!/ 

bccausc 5 = XI. Jlowcvcr, this analogy is not, complctc, I~cca~lsc t.llc: st,rc:ssas in :r 

flt~itl tlepcntl on one const,atlt., t,l~c viscosit.y ti, wllcw-:is tllose irl :tn iso1,ropic vI:~sLic: 

solicl tlnpcntl on two.

8 I. Ont,line of fluid lnotion with friction 

kp soc/m2 

kp hr/m2 

I'n see 

kg/m lir J 

Ibf sec/ft2 

Ihf hr/ft2 

Il,/ft scc 

m2/sec 

m2/hr 

cm2/scc (Stokes) 

ft2/sec 

ftz/hr 

I 

1 

2.7778 x 

1 X lo-" 

9.2903 x 

2.5806 x 

Table 1 .l. Visco~ity conversion factors Numerical values: In t,lrc case of liquids the vi~cosit~y, /t, is nearly indcpcndent, 

n. Aldl~te viscosity 11 

of pressurc and tlccreascs at a high raLc with increasing tcmpcrat,urc. 111 thc case of 

gascs, to n first npproximat,ion, thc vi~cosit~y cnn be talrcrr to be intlcpcntlcnt of 

prcmitrc bi~t, it irrrrcnscs wil,lr l,cmllcrnl,rtrc. 'I'Iio Itinc?~nal,ic vi~cosil,~, 11, for litl~~itl.q 

has t,hc smw type of t,cmpcrrat.i~ro tloj~otttlottc:o as p, I)ct.n.~tso 0l1r tltwsit,y, 0, (-IIJLII~~S 

only ~liglrtly with tcnrpomI,urc, Ilowcvcr, in t h caw of gn.scs, for whiclt C, tlcc:ro:t..qc~ 

consitlcrsbly with incrc:~sitig tc1npcrn1,11rc, 11 incrcascs rnpitlly willit (,cmpcmt.urc. 

Table 1.2 contains some numerical values of Q, p and v for water and air. 

Table 1.3 contains some additional lisefitl tlat,a. 

0. I

10 I. Or~llinr of lluicl rnot,ion with fricl.ir~n 

111 ortlvr 1,o i~nswrr 1,Irc ~IIICS~~~OII of wI~t:i.l~rr it is ncccssiwy 1.0 l.aI

1. Orrl.li~rr: of firtit1 motion with rrirtiotr 

ILln. (1.1 I ) statcs thi~t thc volumc r;.tc of flow is propnrtiounl to tllc first Ix)wcr 

the 1)rC"Urc (1'0p ppr unit lrngth (pl-p2)/l irnd to 1.h.: fourt,ll powor of tllc ra(jills of 

thc pipe. Lf the mran velocity over tho cross-scctioa li = 112 is intrr)~llllsrl, 

eqn. (1.1 1) can bc rrwrittcn as 

%n (1-1 1) can be ~t,ili~(:d 6'. the cxperimcnt.al dctern~jna'io~ ilf~JIC viSCmit,y, ,;. 

v 3 

Ihc nICtllo(1 corlsisl* in thc mcanartw~cnt of tlrc rate of flow ilnd of (,llc pressurn (jmp 

across a fixall portion of ~1 cn(dl:lry tube of know11 m;llur. Thus cIlollg~l dnt,a rite 

provided to dctcrtninc 11 from ecrn. 1 . I .11). 

\ --,- 

The type of flow to'which cqns. (1.10) and (1.1 I) apply exists in reality only for 

rclativcly small radii arid flow v~locit~ics. For larger vclorities and radii the character 

of tho motion changes complctcly: thc prcssurc drop ceases to bc proportional to 

the first powcr of thc rncan volocit,y as indicated by eqlt. (1.12), but becomes approximately 

proportionnl to the second power of u. The velocity distrib~lt~ion across 

a secbiori hccomcs much more ur~iforln and thc well-ordered laminar ]notion is 

replaced hy a flow in which irregular and fluctuat,ing radial and axial velocity comporlcnts 

arc supcritnposcd on thc main motion, so that, consequently, irlt,crlsivc 

mixing in a radial djrcction takes placc. In such cases Newton's law of frict.iorl, 

eqn. (1.2), ceases to be applicable. This is the case of lurbule?tt flow, to l)c tliscnsscd 

in great tl(:t,ail latcr in Chap. XX. 

1. I'rinciplc of similarity; the Reynolds a d Mach numbers 

Thc typr of fluid n~ot~ion cliseussnid in tho I preenling Scct,ion wr. very simple 

bcnasc evcry fluid part,icle ninvcd utr(lcr the infl~lcnec of friotior~al and pressure 

hrcas orrl.y, incrtia brcrs laing cvcrywhcre cqurl in zero. 111 a divergent or convergent 

ch~~nn(:l fl11i11 p:wticlrn arc n.rtcd uport by inert.ia forces in atl(lif.io11 to pressure and 

frirtiorr forrrs. 

e. 15nciple of ~irnilarit~; the Roynolda and Much nurnbern 

In the present section we shall endeavour to answcr a very fundamcntol qllcstiorr, 

~lamcly that conrcrnrd wibh the conditions under which flows of diffcrcnt fluids 

about two gcomct,ricellg sinrilnr bodics, and with identical initial How dircctions 

~lisl,lity gcomnt,rically similar strc!ntnlincs. Such mol.iorrs which havc gconrot~rirnlly 

strrcarnlincs arc cnllctl tb?l,atrm.icctbl?y sirnilrr.r, or .qimilnr /10111~9. Jkr two Ilowa 

nl)ont, grornotrici~lly aimili~r Iwtlicv (!:. y. irbout two spltorca) wiI.11 ~lill'(:r(:ttL ( Iui~ la, 

tlillilrcllt vclocitics . r~rltl . tiillkrcni; .... -. iincw tlirncrtsior~s, to bo ~irnilar, it,, is cvidcnLly 

~~ccessary ihat the folIo~v~~~g~q!~~~t,i~n 

.. shoulcl be satislic(l ;.st ?ll~g~,~~me_tr,~~~!y sirni,l.r 

Point$ thC . . f6FCCS . - . . acting on a fluid particlc must !car a fixccl ~ tio-lt cvcry instant 

. . . 

df t.iiiG, 

\Vc shall now cdnsicter the irn~~ort,nrrt casc whcn only f'rict,ional and inert,ia 

forces are prcscnt,. IClaslic forces which may bc duc to clrangcs in volrnnc will hc 

cxcllltlcd, i. c. it will bc assumed that tho flnid is incompressible. Gmvit:r.t.ior~:rl 

forces will also be cxcludccl so th:~t, conscqucntly, frcc surfaces are not adtnittctl, 

anti in the interior of thc fluid the forcc of gravity is assumed to be bal:~r~cccl 1)y 

buoyancy. Undcr thcsc assumptions the condit,ion of similarity is satisficcl only if 

at all points the ratio of incrtia ant1 friction forccs is thc satnc. In 

a mot,ion pnrallel to the x-axis thc inertia force pcr unit volume has the magnit,urlc 

of g l)lr/l)l, whcrc u ~Icnotcs tlrc componctlt of vclocity in tlrc x-dircctiorr and I)/1)1 

clcnot,cs the sribstantivc dcrivativc. In tho casc of stcady flow wc can replace it 

by e aslax - dx/dt = e v a@x, where all/ax dcnotcs thc r:hangc in vc1ocit.y with 

position. 'I111us the incrtia forcc per unit, volumc is cqui~l to C, u aulax. For thc friction 

force it is easy to deduce an cxprcssion from Newton's law of friction, cqn. (1.2). 

Considering a fluid pnrt,iclc for which tho x-direction coincides with thc dircct.ion of 

motion, Fig. 1.3, it is found that the rcsnltant of shcaring forccs is equal to 

a~ 

=-dxdyd~. 

a~ 

Hence the friction force per unit volumc is equal to atlay, or by eqn. (1.2), top a2u/ay2. 

Consequently, the condition of similarity, i. e. the condition that at all corresponding 

points the ratio of the inertia to the friction force must be constant, can be 

written as: 

-Inertin - fxcc 2 !L =,on, t. 

Friction force p a2u/aya 

13 

Fig, 1.3. Frictional forces 

acting on,a fluid particlc 

It is now necessary to investigate how these forces are changed when the magnitudes 

which determine the flow arc varied. The latter includc the density e, the viscosit,y 

p, a representative velocity, e. g. the frcc stream velocity V, and a characteristic 

linear dimension of the body, c. g. the diamctm d of the sphcrc.

The vclwil y IL at some point i11 tlrc velocit,y field is proportional to tlte free 

strrnm velocity IT, l,he vcloci0y gratlicnt au/ar is proportional to Vld, antl similarly 

a2tr/~y2 is proporlional to V/d2. Ilcnce the ratio 

Thereforc, tllc condition of ~irnilarit~y is sat,isfictl if the ql~antil~y p V d/p f~as the same 

value in bol,l~ flows. The (pntity p V d/p, which, with 11.1~ = v, can also IN wriLt,cn 

ns V d/v, is a tlimcnsiotlloss nnrnl)cr \>cen.tlsc it is the mt.io of t,l~c t,wo forces. It is 

known as t.110 Ilayitnk1.c ~slr.?ttl)ar, R. Thus t,wo flovs arc similar when the lt:lin 

three crqna(.ions : 

F : )I -4- 0 : 0 , 

the solution of wlticl~ is 

Din~ctlsint~lcss quantities: 'I'hn reasoning followctl in tho precetling drrivi~f ion 

of the Rcynoltls numl~er can be e~t~entled to inclndc the casc of diffcre~~t Itrynolrls 

numbers in the consitlerat,ion of the velocity ficltl ant1 forccs (normn.l :mtl tangont.i:rl) 

for flows wiLh geornetrica.lly sitnilar boundaries. Let thr position of :L point in (.he 

space around the gcomctrically similar bodies bc intlica1,cd by thc coortlin:tl.t~s 1, !/, 

z; t~llen tho rat,ios z/d, y/d, z/tl arc its tlinicnsiotlless coortlirt:~l,cs. Tl~c vc~loc:il.y c:otltponcnt,s 

arc lnatlc dimensionloss by relirrring tllern to the free-stream vch:iI,y V, 

thus 711 V, 111 V, w/ V, and lhc normal and st~caring strosscs, p :~ritl t, can bo mn.clcr tlirnct~sionlcss 

by reforring thorn t,o Lllc tloubfc of t,llc tlyrtatnic lieatl, i. e. to p V2 t.hus: p/p 1'" 

and t/p V2. The previously cn~~nciatcd principle of dynnmical sinlilarit,y can Im c~x1)rt~s- 

sod in :Ln alternative form by asserting t ht for the two gcornctricnlly similar sys1,cnls 

with equal Reynolds numbers the dirncnsionless quantitics 141 Y, . . ., p/p V2 i~nd 

t/e V2 depend only on the dimensionless coortlinatcs x/d, y/d, z/d. If, Ilowcvcr, the 

two systems are geometrically, but not dynamically, similar, i. c. if t.lleir Rcynoltls 

numbers are different, t,llen the tlimensionless quantit,ies under consideratlion innst, 

also depend on the chamctcristic quantities V, d, Q, 14 of the two ~ystcrns. Applying 

the principle t,llat physical laws must be independent of the systcn~ of nnit.s, it. fi~llows 

that tl~e tlimensionless quarifities u/ V, . . ., p/e V2, T/Q VZ can only depend on a 

dimcrlsionless combinatlion of V, d, Q, and 11. which is unique, being the Itcynolds 

number R = V d e/p. Thus we are led to the conclusion that for t01c two gcon~cbrically 

similar systmns which have different Rcynolds numbers antl which arc bring 

compared, the dimensionless quantities of the field of flow can only be funcI.ions of 

tlic tthree din~ensionless space coordinates z/d, y/d, z/d and of t.lw Rcynolcls 

number R. 

The precc(ling dirr~cnsinnal annlysin can bc ~~lilizctl to tu:~ltc an irnport,:r.ttt, 

assertion about the t.otal force excrtcd l)y a fluid strealn on an imrncrsotl hotly. 7'11c 

force acting on tho bocly is the surface intcgral of all normal and ~llcaring stmsst:s 

acting on it. If P denotes the component of the resultant force in any given direction, 

it is possil~le 

to write a tlirncnsionless forco coefficient of the form P/d2 Q V2, 1~11, 

stead of the a,re:b d2 it is cnstomary to clloose a diKcrcnt charactcrist.ic aro:l, A, of 

t,he immersed body, e. g. the frontal a.rea exposed by the botly to tile flow tlircct

of I,ho resultant forcc parallcl to the unciisturbctl initial vrlority is referred to as t11e 

drag I), and the component perpencliculnr to that tlircct.ion is callctl lift, 5. Hencc 

the dimensionless cocfficicnts for lift and drag become 

L I) 

C - nnd C, = - - --- - , 

, - A 

(I .Id) 

18VSA 

if the tlynnn~ic: 11cad 4 Q V2 is SCICC~C~ for rcfcrcrlce instcatl of t,hc tlunnt,ity e V2. 

Thus tho argumcnt leads to the conclusion that the tlimcnsionless lift a,nd drag 

coefficients for geometrically similar systems, i. c. for geometrically sirnilar bodies 

which have t h same orientatmion with respect to the free-8trea.m direction, are 

functions of orie variable only, nmnoly the Reynolds numhcr: 

c,,=/,(R); CD=/~(R). (1.15) 

It is ncccssary to strcss once more that this importmt conclusion from Reyr~olds's 

principle of similarity is valid only if the assumptions undcrlying it are satisfied, 

i. c. if the forces acting in the flow arc due to friction and inertia only. In the 

casc of compressible fluids, whcn elastic forccs arc important, and for motions with 

free surfaces, whcn gravitational forccs must be taken into consideration, eqrrs. (1.15) 

do not apply. In such cases it is ncccssary to deducc diKerent similarity principles in 

which the tlimensionless Froudc numlw F = v/G~ (for gravity and inertia) and 

the c1imensionless Mach number M == V/c (for compressible flows) are included. 

The importance of the similarit,y principle given in eqns. (1.14) and (1.15) is 

very great ns far as the scicnccs of th~orct~icsl and cxpcrimcntal fluid mechanics are 

concerned. First, the dimcnsior~lcss cocfficicnts, C,,, C,, and R are irlclependent of 

the system of unilm. Secondly, their use leads to a considerable sirnplificntion in 

the cxtcnt of expcrimcntal worlc. In most cases it is impossible to tlcterminc the 

func(.ions f,(R) and /,(R) throrctic:ally, antl exporimcnt,:~i ~ncthotls must be 11sot1. 

S~~pposing tl~ali it is tlcsirccl to tlrtcrrnino thc tlr:~~ cocfficicr~t ITI, for a spot,ilic:tl 

s11:q)c of hly, c. g. a sjhcrc, tllcn witl~ot~t the application of Lhc principle of sirni1:~rit.y 

it wo111tl hc? ncccssary to carry out drag mcasuremcnt.~ for four indepcntler~t variables, 

V, d, Q, and p, antl this would const,itute a trcmondous programme of work. It 

follows, however, Lhat t2he drag cocfficicnt for sphcros of diKcrcnt tlinmctors with 

different stream vclocitics antl tliffcrcnt fluicls clcpcntls solcly on onc v:~ri:~l)lc, 1.h~ 

Reynolds r1urn1)cr. Fig. 1.4 rcprcscnts thc dmg cocfficicnt of circular cplintlcrs as 

a fi~nct~ion of the Itoynolds number antl shows the exccllcnt agrccment hetwceri 

expcrimcnt antl Reynolds's principle of similarity. The cxperimentnl point,s for 

the drag cocfficicnt, of circular cylinders of widely differing diameters fall on a single 

curve. 'The same applies to points ohtnined for the drag cocfficicnt of spheres plotted 

against t,ho Iteynoltle number in Fig. 1.5. The sutltlcn decrease in the value of thc 

drag coefficient which occurs near R = 5 x lo5 in the case of circular cylinders and 

near R = 3 x 10"n the casc of spheres will be discussed, in n~ore detail, later. 

Fig. 1.6 reproduces photographs of the stream$nes about circular cylinders in oil 

taken by P. JIomann [7]. They give a good idea of the changes in the ficld of flow 

associated with various Reynolds numbers. For small Reynolds numbers the wake 

is laminar, but at increming Rcynolds numbers at first very regular vortex patterns, 

known as Khrmhn's vortcx &recta, are formed. At sLill higher Reynolds numbers, 

not shown here, tho vortex patterns become irregular and turbulent in character. 

c. Principle of si~nileril.~; 1110 Ilcynolds nnti Mach numlwxs 

2 = V'J 

Fig. 1.4. Drag coefficient for circular cylinrlcrn n, a function of tlie Jleynoltls n~~nibcr 

4 00 

700 

C~ roo 

80 

60 

10 

70 

10 

8 

G 

L 

7 

I 

08 

0 6 

0 4 

0 2 

0 1 

08 

0 a 

Fig. 1.5. Drag coefficient for spheres aa n fiulction of tho Reynolds nulnbcr 

Curve (1): Stokcs's theory, eqn. (6.10); curve (2): Oseen'a thcory, eqn. (0.13) 

17

Fig. 1.6. Firld of flow of oil nho~~t n cirr~~lnr rylintlor at wrying IZrynolcln n~c~nbnrs nltcr Homnnn 

171: Irnnnition from lnrninnr flow t,o n vortrx ntrrt-t, ill I:~n~innr fhv. Tl~r 

R = 65 t.o R - 281 I)c tnltrn from Fig. 2.9 

freqrwnry rnngr for 

e. Principle of 8irnilnrit.y; t,he Itcynolds ntld Mac11 nlt~r~bcrs 19 

\V. Jonm, J. J. Cillotta and 12. \V. \Val. 

krr [a]

20 I. Outlinc of fluid rnolim with frict.iot~ f. Comparison hrtween thc theory of pcrfcct 11r;itls anti cxpcrin~cnt 21 

f. Compnrison between tl~c theory of pcrfcct fluids ntd rxperi~nrt~t 

In the cases of t,hc motion of water ant1 air, wllich arc the most ilnport.ant ones 

in engineering applications, the Itcynoltls nurnl)crs arc vcry Inrgc l)rml~sc of thc 

very low viscositics of thcsc fluids. 1.t wor~ld, thorcforc, apl)c:tr rcasonal)lc t,o c-xpccL 

very good :tgrecrncnt 1)clwccn cxperin~cnt, and a 1,hcory in which tl~c itlllllcncc of 

viscosity is ncglcctcd alt,ogcthcr, i. c. with the thcory of pcrfcct fluitls. In any case 

it secms uscful to bcgin thc comparison with experiment by rcfcrcnce to thory 

of perfcct fluids, if only on ncconnt of tho large num1)er of cxist,ing explicit mnthematical 

solnt,ions. 

In fact, for certain clnsscs of problems, st~clr as wave formation and tidal motion, 

exccllent results werc obtained wit01 t,hc aid of this theoryt. Most problems to bc 

rliscusscd in this book consist in I,hc study of the motion of solid 9odics through fluids 

at rcst, or of lluitls flowing through pipes a.nd channels. In such cascs t,hc use of 

the theory of pcrfcct fluids is limited because it,s solutions do not satisfy thc con- 

I 

tliLion of no slip at the solid surfacc which is always the case! with rcal fluids even 

at very small viscositics. In a pcrfcct tluitl thcro is slip at a yall, and tJlis circamst,ance 

inLroduccs, cvcn for slndl viscositics, such funtl:~.mc:~t.al tliKcrcnccs that it 

I 

is rather surprising to find in somc cascs (e. g. in the case of vcry slender, stream-linc 

bodies) that thc two solutions display a good measure of agreement. The greatest 

tliscrepancy betwccn the theory of a perfcct fluid and experiment exists in the 

consitlcration of drag. The perfcct-fluid theory leads to the conclusion that when 

an n.rhit,mry solid body movcs through an infinitely extended fluid at rcst it expericnccs 

no forcc acting in the clircction of motion, i. e. that its drag is zero (dlAlembcrt's 

paradox). This rcsult is in glaring cont.radiction to observed fact, as drag is 

mcnsurod on all bodics, evcn if it can bccome vcry smaU in the case of a streamline 

body in stcady flow parallcl to its axis. 

By way of ill~~stration we now propose to make some remarks concernirlg tlhe 

flow about a circular cylinder. The arrangcmmt of streamlines for a perfcct fluid is 

given in Fig. 1.9. It follows at once from considerations of symmetry that the resultant 

forcc in the direction of motiorl (drag) is equal to zero. The pressure clistributiou 

according to the theory of frictionless motion is given in Fig. 1.10, togcther with the 

results of measurements at three values of the Reynolds numbcr. At the leading 

edge, all measured pressure distributions agree, to a certain extent, with that for a 

perfcct fluid. At, the trailing end, the discrepancy between theory and measurement 

becomcs large because of the large drag of a circular cylinder. The pressure distribution 

at, the lowest, sobcritical Reynolds numbcr R = 1.0 x 105 diffcrs most from 

that given by potential theory. The measurements corresponding to the two largest 

Rcynolds numbers, R = 6.7 x 105 and R = 8.4 x 106, are closer to the potential 

curve t,han those performed at t,hc lowest Reynolds number. The large variation of 

pressure distril)ution wit,l~ Rcynolcls numbcr will be discussed in detail in the next 

cl~apt~er. A corresponding pressure-distrih~t~ion curbe around a meridian section of o, 

spl~cre is rcproduccd in Fig. 1.11. Here, t,oo, measurements show large differences for 

the two Reynolds numbers, and, again, the smaller Reynolds numbcr lies in the range 

Fig. 1.0. Frictionlcss flow about a 

circular cylinder Pig. 1.10 

Pig. 1.10. Pre~strre distribution on a circular cylinder in the suhcrit.icnl and er~pcrcriLic~ll range of 

Reynolds nnnihers after t,he ~neasurements of 0. Flncl~sbnrt [4] and A. Roahko [13]. qm - - 1 

e 1'' 

is the stagnntion pressure of the oncoming flows 

- frlctionlerur flow Flacl,s,,nrl 

--- R = 1.9 x 10' 

. . . . . R - 8.4 x 10' Itonlikn (1001) 

Fig. 1.1 1. Pressure distribution 

around n sphere in the suhcri- 

tical and supercriticnl range of . 

Reynolds numbers, aa mea- 

sured by 0. Flachsbart [3] 

1 i:G 

of largc clrng cocfficicnl,~, whrrcas tho Irwgcr valuc lics in lhc rmgo of srnnll clrq 

coefficients, Pig. 1.5. In this case the n,czsnrcd prcssltre-cli~t~ributior~ curve for tho 

largo Reynolds number approximat,es the theorct~cai di:rvo of frictionless flow very 

well over the greatest part of the circumfcrcnce. , 

Considerably better agrcemcnt between the theo~etical and measured pressure 

distribution is obtained for a streamline body in a flow parallel bo its axis [5], 

Fig. 1.12. Good agreement exists here over almost the whole length of the body, 

with the exception of a small region near its trailing end. As will be shown later 

this circumstance is a consequence of the gradual pressurc increash in the downstream 

direction. 

Although, generally speaking, the theory of perfect fluids does not lead to 

useful results as far as drag calculations are concerned, the lift can be calculated from 

it v~ry successfully. Fig. 1.13 represents the relation between the lift cocfficicnt and 

angle of inritlcncc, as nteasurctl hy A. Bctx [2] in thc caso of a Zhukovsltii :iwofoil

22 

I. Outliric of fluid motion with friction 

Fig. I .12. Prmsr~re distrihnt,ion 

nlm~t n ~trenw-line body of 

rcvolntion: cornpnrison bctneon 

tllcory arid mrnsuremcnt. 

nftcr Fuhr~nann [5] 

Fig. 1.13. Lift nnd drag roeffi- 

cicnt of n Zliukovnkii profile in 

plnnm flow, ns ~nenaurod by 

lktz 121

Outline of boundary-layer theory 

a. Thc boundary-layer concept 

tn tho casc of fluitl motions for which the measured pressure distribution nearly 

agrcrs with the perfect-fluid thcory, such as the flow past the streamline body 

in Fig. 1.12, or the aerofoil in Fig. 1.14, the influence of viscosity at high Reynolds 

numbers is confined to a very thin layer in the immediate neighbourhoocl of the 

solid wall. If tho condition of no slip were not to be sat,isfit:d in the casc of a real 

fluitl there wollltl 1)c no appreciable tliKcrcncc between the field of flow of thc real 

fluitl as comparcd with that of a pcrfcct fluitl. The fact thaL at t,hc wnll thc fluid 

adlicres to it means, howcvcr, that frictional forces rctarcl the motion of the fluid 

in a thin laycr near the wall. In that, thin layer the velocity of the fluid increases 

from zero at thc wall (no slip) to its full value which corresponds to external frictionless 

flow. The layer under consideration is called the boundary layer, and the concept 

is duo to L. Prantltl 1263. 

Figurc 2.1 reproduces a picturc of the motihn of water along a thin flat plate 

in which the s!,rcamlincs wcrc made visible bjr the sprinkling of particles on the 

surfn.cc of thc water. The traces lcft by the particles arc proportional to the velocity 

of flow. Tt is scen that there is a very thin laycr near the wall in which the velocity 

is' considorably smallcr t,han at a 1n.rgcr distance from it.. The thickness of this 

holtntlary laycr incrc,ascs along thc plate in a downstream direction. Fig. 2.2 repre- 

~nnb tliagrammatically the vclocity distribution in such a boundary layer at the 

a. The hollndary-laycr concept 25 

plate, with t>hc tlimensiorls across it considerably cxaggcratctl. In front of the 

leading edge of the plate t,he vrlocit,y elistribrttion is rtnifornl. With increasing distattrc 

from thc leading edge in the downstrmm direrlion the thiclrness, cf, of t,lle retardetl 

layor incrrasrs continrlor~sly, nn ilicrrnsing qunnlitira of hit1 I)oc*onlo t1TTcy-lrtl. 

15vitlcr1tly tho lhiclrnrss or the 1)ountl:~ry Inycr t1wrcvw.s wit11 Oc~crrasir~~ viwosity. 

Fig. 2.2. Sketch of borlntlnry --- 

layer on a flat plate in pnr- 

allel flow at zero inciclcnce - 

On the other hand, even with very small viscosities (large Reynolds numbcrs) t.hc 

frictional shearing strcsses T = /c au/a!j in the 1)oundary laycr arc consitlcrnblc 

bccnusc of the Inrgc vclocily gr~diont, across lllo Ilow, wllcrct~s o~tl~sitlo tho I~ou~~tlttry 

layer t11cy arc very small. This physical pict~~rc suggcst~n that the field of flow in t.1~ 

casc of lluids of small viscosil.y can I)c tlivitlctl, for tho purpose or matliornnt,icnl 

annlysis, into two regions: thc t.llin boundary laycr near the wnll, in whic:h rriction 

must be taken into account, antl the region outside thr boundary layer, whcrc the 

forces due to friction are small antl may be ncglcct~cd, and where, thcrcforc, the 

perfect-fluid theory offers a very good approximation. Such a division of the field 

of flow, as we shall see in more detail It~tcr, brings about a considerable simplification 

of the ~nat,l~ematical theory of the motion of fluids of low viscosity. In fact, t,he 

t,heoretical study of such motions was only made possible by Prandt.1 whorl he 

introclucctl this concept. 

We now propose to explain the basic concepts of boundary-layer thcory wit11 

the aid of purcly physical ideas antl without the nsc of ~nat~hcmatics. The rnathcrn:~t.ical 

bor~ntlary-layer tllcory which forms the main topic of this book will bc tlisc~~sscel 

in the following chaptcrs. 

The dccrlcratctl fluid pnrticles in thc boundary laycr (lo not, in all cnscs, rrmnin 

in the thin lnycr which atlhcrcs to thr I~ody along thc whole wcttcd lc~~glh of ~ I I P 

wall. In some cases the boundary layer increases its thickness considerably in the 

downstrcarn tlirection and the flow in tho boundary laycr beconics revcrscd. 'l'his 

causes the decclcratcd fluid particles to be forced outwards, which rnmns illat 

thc boundary hycr is scpnrated from t11c wall. Wc thcn spcalr of boundniy-ltryer 

sepalation. This phenomenon is always associatrd with the formation of vortircs 

and with largc energy losses in the wake of the body. It o_ccur_sprjmarjly nrar blunt 

bodies, such %s circular cylinders ~ncl~sph_c-~~. Behind such a body thcrc exists a region 

of strongly dccrleratrtl flow (so-calletl wake), in whicl~ the pressure distribution 

deviates considerably from that in a frictionless fluid, as seen from Figs.l.10 arlcl 1 11 

in the ~rsprctiw cnscs of a rylindcr and a sphere. The large drag of such bodics can 

be explained by the existence of this large deviation in pressure distribution, which 

is, in turn, a consequence of boundary-layer separation.

2 (i 

TI. O~~tlittr of Imun~lnry-lsyw throry 

E~tin~nIin~t of houndnry-lnyer thickllr~s: 'rhc t,l~ickness ofa boundary layor whir11 

llas riot sepnrnlrtl can I)(! casily rst,irnnLrtl in thc following way. Whcrcas friction 

forccs can be ncglcctctl with rcspoct t.o incrt,ia forccs out,side tho bourltlary Ixynr, 

owing to low viscosit,y, thry arc of a comparable order of magnitrldc inside it. 'rhc 

inert,ia forcc prr nit volun~ is, as cxplninctl in Scct,ion l e, equal to Q 71 &L/~x. For 

a pIat,o of longlh 1 tho gr:ttlinnt arr/a:r is proportional to ll/l, where IJ tlrnotes thr 

velocil,y onLsitlv the! I)ountl:wy Inyrr. Ilct~rc Ihc irlnrl,in forcc is of tho ortlcr I, 1J2/1. 

On the othcr l~antl the friction forcc per nr~it volurnc is equal to at/@/, wllirll, on tho 

assurnpt~ion of lnrninnr flow, is cqunl t,o 11, a21t/i)?/2. The velocity gratliont al~/ay in a 

tlirrcLion prrl~rnrliculnr t,o t.l~c wall is of t,lm ordcr Ill6 so that thc friction forcc ])or 

~ti)il~ ~olt~tnv is i)~/&y - lI/d2. Proni the cotdit.iorl of equality of the friction :md 

inertia forcrs tho following rc.l:ll ion is obhined: 

U e UZ 

t4 82 - 1 

or, solving for I Itr Imuntl;~r~-layrr tlriclcr~rss Ot: 

The I~nlnr,ric:nl f:~rt,or wltid~ is, so f:w, st.ill untlct,crn~ined will be drduc:ctl Iatcr 

(C!l~:lp. VII) from tho exact solut,ion givcn by [I. 13lasius 141, and it will turn out 

t.llnt it is cqrlal 1.0 5, al)proxinlatcly. llrncc for lnmiarrr flow in the bountlary layer 

wn hnvo 

(2.1 a) 

'rho tlinlrt~sionlc~ss 1,our~lnr~-lnyer thirknrss, rcfcrrctf to the length of the plate, 1. 

twronles . 

wllorr R, clrnotcs tho ltcynoltls nunlber rclatod to the Icngth of the plat.c, 1. Tt is 

won from cqn. (2.1) tallat thc boundary-layer thickness is proportional in 4; and 

t,o I. If I is ropla.cetl hy the variable tlist~nce z from the leading edge of the plate, 

it is seen that d increases proporti~nxt~ely to ii. On tho other hand tho relative 

boul~(~ary-Iaycr t,I~ickncss O/i decrems with increasing Reynolds number as I I ~ R 

so that in tho limiting case of frictionless flow, with R -+ oo, tllc boundary-layer 

t.lrickness vanishes. 

We are now in a position to estimate the shearing stress zo on the wall, and 

consrq~~ontly, t.hr t,ot,ni drag. According to Newrton's law of friction (1.2) we have 

- - -- 

t A ~~lore rigororts tlrfiniliott of Im~lrtclnry-Iayrr thicknrsn in given st the end of lhia section. 

wherc sl~bscrip~ 0 tlenotes the value at the wall, i. e. for y = 0. Witll thc estimate 

(au/a~)~ - U/d we obtain 7, - ,u U/d and, inserting the value of d from cqn. (2.11, 

we have 

We cart now for~n a dirncnsionlrss sl,rcss with rcTrrnrlrc lo I, llz, ns c~xpl:~ittc~cl 

in Cltnp. I, ant1 obtain 

c,, - = 

I'q . 

1 

- - 

The numrric:ll fartor follows from 11 Blasius's cxart solution, atttl is I 328, so tll:~~, 

the drag of a ~~lntr in parallrl 1nmin:~r flow 1)rromc.s 

Tltc following nt~mrrical rxamplc will serve t,o il11tst~~rt.c: t.hr l)rec:rcling c:st,i~rt:~ t.iolt : 

Laminar flow, stipulntctl here, is obt:~it~rtl, as is known r'ronl exprritnctlt,, for Itcynolds 

numbers CJllv not cxceccling :d)outt 6 x 10Ql.o 10% lpor 1nrgc.r I

28 

TI. Ot~(,linc! of bor~ndnry-layer thoory b. Srparation antl vortex fortnn(.ion 20 

Dalinition of Imnndnry-layer thickness: Thc clefinition of lhc bountlary-laycr 

t.lrickncss is to a ccrtain extent arbitrary l)ccausc transitsion from the velocity in 

t,l~c borlndary t,o that o~~t.sitlc it t,:~.ltcs plncc asympt,olically. Tlris is, IIOW~VC~, of 

no pract,icn.l import,ancc, I~ccnusc t,hc vclocil~y in thc bor~ntlnry laycr at.t,:iins :I. vnl~lc 

whic:h is vrry c:losc t,o fho cxl,crt~n.l vcloriLy drcatly at, a small tlistancc frotn the 

wnll. 11, is Ijossil~ln to tlcfino Lhc I)o~lnd;~~.y-l:~yc:r thioltncss :IS l.l~nl rlis1,:~noo from lllc: 

wnll wllorc: t,hc vclonity tlilTcrs 11y I pcr ct:111 from the oxt,crnn,l vrlociLy. \Vil.l~ titis 

dnfinition the rtrtmcric:d f:~.ct.or in cqn. (2.2) has thc value 5. [nst,ead of t,hc bonntlarylaycr 

t.lricknc~s, anotlrcr qunnt.it$y, thc dinplr~cement thickness a, is somct.imcs used, 

Fig. 2.3. It, is dcfinetl by thc cqnntion 

(2.6) 

'Ilc displnccment tl~icltncss indicates l.llc tlistancc by which the external strcamlines 

arc shift,cd owing to tire fonnat,ion of t,l~c boundary Iaycr. In the case of a plate 

in parallel flow nntl at zcro incidcncc tlrc tlisplaccmrnt thickness is about & of the 

bountlary-layer IJ~icltncss 0 givcn in cqn. (2.1 a). 

.. 

b. Srpamlion and vortcx forrnntion 

llte bo~~ntln.ry laycr ncnr a fht plate in par:~llcl flow and al, zcro incitlencc is 

part,icrllarly sirnplc, Ijccausc the static prcssurc remains conshnt in the whole field 

of Ilow. Sincc orlt,sitlc the 1m11ntI:~ry lnyrr tho vclocily rcnmins constant t,hc samc 

qjplics to the prcss~~re l~ecausc in the frictiorrlcss flow Bcrl~orrlli's cquation remains 

vnlitl. Furthcnnorc, tlrc prcssnrc rcmnins scnsibly constnnt over thc width of t,hc 

\)o~~~rrlary layer at a givcn rlist.ancc x. 1Icncc tlrc prossurc over thc widt.11 of tlrc 

1)ountlary Iaycr has tlrc snmc mngnittrtlc ns out.sitle t.hc boundary laycr at the samc 

tlist.ancc, ant1 the same applies lo cnscs of arbit,mry body pl~n.pcs whcn tho prcssnrc 

o~rt.sitlc 1.h~ I)o~ln(l:~ry I:~yt:r vnrics along t,lrc wall wit11 t,l~c 1cngl.h of arc. 'l'his fnct 

is cxprcsscd by saying 1,h:~L t,lrc cstcrnnl prcssnrr is "i~n~rcssctl" on thc boundary 

Inycr. Ilcncc in the cnsc of the motion pst a plate l,hc prcssnrc rcmains constant. 

througIrouL t,llr: bountlnry Inycr. 

'j'lrr phrnonrrnon of 1murrtl:~ry InycrsrpnraLiot \ ~nrt~tiot~c~tlprc~viously 

--. - - isi!rtinral~ly 

c~onnrclctl wrtl~ tlrr prcssurc t1istril)ution in ti16 orintlary layrr In the boundary 

lnycr on a plate rro srpnmlion takrs phrr as no back-fldw occurs 

In ortlcr to r\plnitr t IIV very import nrrt pl~rnornrr~on of bountlary-lnycr s~paration 

let us rorrritlrr 1 hr Ilow :~ljouI n Ijlrrnt hotly, r g abont, a rirrnlar rylintlrr, as shown 

it1 IClg 2 4 111 ft ic.1 inl~lcw flow, t l ~c flu~tl par1 irlrs nrr :~rc.rlrmlrtl on tlw npstmam 

half frorn D to E, and decelerated on the downstream half from E to F. Ifcnce the 

pressure decreases frorn D to E antl increases from i' to F. Wltcrl the flow is stmtcd 

up the motion in the first inst,arlt is very nearly frict,ionlcss, ant1 rcmains so as Img as 

t h bounthry lnycr remains thin. Outsitlo lhc I~onntl:~ry lrtycr lllcro is n tprr~l~s~ornlctl.io~ 

of pressure into 1tincl.ic energy idong 11 R, 1.110 rcverso hlting pl:~c:o r~lottg IC I(', so 

IJtaL IL parlidc nrrivo~ ILL 11' with Llto HILIII~> vclocil,y 11s it, IIIL~ nl, J). A lIrci(l ~~:~rl.iclt: 

wltich lrroves in IJIC i~nmctlinlo vioi~til~y of tho wtdl in I,llc bo~lntl:r.ry I:~.yor rc:~n:iit~s 

under the influence of the same pressure field as that existing outside, I)crause the 

external pressure is imprcssctl on the boundary layer. Owing tlo tlrc large friction 

forces in the thin boundary layer such a psrtic:lc consumcs so much of its kinbtic 

Fig. 2.4. Doundary-layer scpara- 

tion ~ind vortex forrnntion on a 

circular cylinder (dingran~n~atir) 

S - point nf scl~nrnllo~~ 

energy on its pat.h from D to E that thc remaintlcr is too slnall to srlrmount t.hc 

"pressure hill" from E to F. Such a parLicle cannot move far into t,hc region of' 

increasing pressure between lC antl P antl its molion is, evcntunlly, arrcst,ed. The 

external pressure causcs it t,lrcrl t,o move in tho opposite clircction. Tlrc pl~otogra~l~s 

reproduced in Fig. 2.5 il1nstrat.e the sequence of cvent.s near the downstrcarn side of 

a round body when ,z fluid flow is started. The prcssurc increases along t,Ile I,otly 

contour from left t,o right, the flow Ilnving been ma.tlc visil)lc by sprinltlitrg nlrtminirlm 

drrst on tho surface of thc water. Tlrc boundary layer can be casily rccognizetl by 

rcfcrcncc to tlte short traces. In Fig. 2.5s, Lakcn shortly aftcr the start of lhc rnot,iorl; 

the rcvcrsc motmion has just begun. In Fig. 2.5b the rcvcrsc nrotion lrns pci~-t,r:.tctl 

a consitlcrablc distancc forward :~nd l,l~c boundary Iayor lrns tllicltcnctl n.pprcci:~l)ly. 

Fig. 2 .5~ shows how this rcvcrsc mot,ion givcs risc to a vortex, whoso sizc is incrc,iscd 

still furthx in Fig. 2.6tI. 'l'hc vorLcx bccorncs scp:~mlctl shortly afLcr~:~r~Is n.td rnovc!s 

tlow~~strearn in tho fluid. This circnn~stancc changcs complctcly blrc fiolcl of flow 

in tho waltc, and Lhc prcssnrc clisLrib~lI,ion suKcrs a rntlical change, as cornparctl 

with frictio~rlcss Ilow. 'L'llc find statc of nrotion can I)(> inrcrrctl from Wig. 2.6. In 

t,he eddying region bclrind tlic cylinder there is consitlcrable suction, as sccri fro111 

the pressure distribution curve in Fig. 1.10. This suction causes a large prcssurc drag 

on t.he body. 

1 

At a larger distance from the body it is possible to discern a rcgul:~r patt,ern 

of vorticcs which move alternately clockwise and courrt~crclocltwise, and wllich is 

known as a IGirmiin vortex strect [20], Fig. 2.7 (scc also Fig. 1.6). In Fig. 2.6 a vortex 

moving in a clockwise direction can be seen to be about to detach it,sclf from the 

body before joining the pattern. In a further pzpcr, von Kilrmhn [21] proved 

that such vorticcs are gcncrally nrrstablc with rcspcct to small tli~t~urbancrs pnrallcl

Fie. 2.511 

Fig. 2 .5~ 

Fig. 2.5b 

Fig. 2.5d 

to thr1ns14vt:s. 'I'lrc only nrmngnncnt which shows ncnt.ral cqoilil,rium is t,hat with 

- . 0.281 ([Cia. 2.8). vort,ex sl.rcet moves with n vcloc:it,y IL, which is slnallc\r 

I,II:I.II t.Ilc flow vrIorii,y II in front of t,ho body. It cnn l)c rcpdetl as a highly idealized 

pict,~~rr of t.hc mot,ion in the wake of (,hc body. The kinetic energy cont,ainetl in the 

vrlocit,y ficltl of the vortcx strect must be continually created, as the body moves 

t.llrongh tile fnitl. On the basis of this rcpresentrn.,tion it is possible t,o deduce an 

exprrssion for t.hc drng from the perfect-fluid theory. Its ~nngnit,utle per nnit lengt,h 

of tllr eYlindric:~l hotly is given hy 

Fig. 2.7. KhrmQn vortex strcct, from 

A. Tirn~nc [38] 

Fig. 2.8. Strrnmlinm in nvortrx strrrt 

(hll = 0 28). Thr fluid i8 nt rc~t, nt 

infinity, and th~ vortrx street move8 

Circdur cylittder. 'l'hc frequency wit,lr which vor1,irc~s arr shrtl in a I

32 

I I. 011t.linc of boundary-leycr theory 

Fig. 2.9. The Stroul~nl nurnher, 5, for thc I

S - point orscpnrnt.ion 

T'ig. 2.12. I)ingmnitnnt,ic represell- 

t.nf,ion of flow ill t,lw 11o1lt)tlnry 

layer near n point, of wpnrnt.ion 

Fig. 2.14. Flow with 1)ortnrlnry- 

In.yor srlc(.iott on upper wdl of 

Irighly tlivcrgetlt clln~~nrl 

Fig. 2.1.5. Flow wit,lt honndnry- 

layer ~uction on 110th wall8 of 

highly divergent channel 

1). Scp~ralion and vortex iormntion 35 

src1.s t,hc wall at a tlcfinitc angle, ant1 t,l~c point of s~p:iri~t,ion it,sclf is cl~:tern~inctl by 

tltr ro~trlitinn that t.hc velocil,y grarlicnt. normal to the wall vanisltcts t.htrc: 

Scparal.ion, as clrsc:ril)ctl for ll~c r:~: of a c:irc~~liir ctyli~~tlcr, ciin :LISO occur in 

a highly divergent rhxnncl, Fig. 2.13. In fror~t of the t.ltroat t . 1 prcssnre ~ tlccrcasrs 

in thc dirrctiol~ of flow, atltl thc flow atlhcrcs complclcly t.o thc walls, as in a fricf,ionIcs?i 

fl11id. Jlowcvcr, bcl~intl t,ho throat t.hc tlivcrgcncc of the cl~anncl is so Inrgc? t.I~:it. t.11~ 

bountlary layer becomes scparatetl from both walls, rind vorticcs arc l'nrmcd. YYIC 

stream fills now only a srnall portion of the cross-scct.iona1 area of t.11~ cl~anncl. llowever, 

separation is prevented if boundary-layer suction is npplictl n.t t,ltc wall (Ipig~. 

2.14 ant1 2.16). 

?'lm photograpl~s in Figs. 2.16 nnd 2.17t j)rovc t.hat the atlvrrsr 1)1vss1irt: 

gr:dicnt t,ogct,llcr wilh fricl.ion near t.lra wall tlctcrn~inc the proccas of sc~):~r:iLion 

which is intlcpcntlcnt of such other circumstance as c. g. tltc curvnture of thc wall. 

'Jlhc first pictme shows the mot,ion of a fluid against a wall at right angles to it (planc 

stagnnt.ion flow). Along thc streamline in t.h~-~dane of symmetry which lm,tls ho t,hc 

st,agnat,ion point tllcrc is a cot~sitlcrablc prcssllre incrcnsc in t,hc clircclion of flow. No 

separation, howcver, occurs, because no wall friction is prescnt. 'I'herc is no sepnmt,ion 

near the wall, either, because here t,he flow in thc boundary laycr takes place in the 

direction of decreasing pressure on both sides of the plnnc of symmetry. If now a tl~in 

wall i~ placed along thc planc of syrnmctry at right anglcs to thc first, wdl, Fig. 2.17, 

the ncw boundary laycr will show a pressure increase in t,hc direction of flow. 

Conscqurnt.ly, scparnt,ion now occurs nm,r 1,Ite planc wall. 'L'hc incitlcnce of scpnmt.ion 

is often rattler scnsitivc to srnall chnngc?~ in the shpc of t.he solid botly, parl.ic:~~lnrI~. 

witen thr prcssrm tlistribut,ion is strongly affcct.ct1 by this char~gc in shape. A very 

instructive exnrnplc is given in Lhc pit:t,urcs of Fig. 2.18 whicl~ show photogrnpl~s 

of the flow fioltl altout a n~otlrl of :I mot.or vehicle (t,hc Volkswa~gcrl clclivcry van), 

123, 351. Whcn t,ho nosc was Il:kt, giving it an angular shape (a), the flow past thc: 

fairly slmrp corners in front causcd largo su&ion followed by :L large pressure incrcnsc 

along the sidc walls. This led to ronlplcte scpnration and to the formati011 of a wide 

wake behind the body. Thc drag coefficient of the velricle with this angular shape 

had a valnc: of C, .= 0.76. Thc liwgc: suction nrar the front cnd i d l h scp:~ri~t.ion 

along tl~c side walls were clinlinat,c:tl when the shape wa9 chnngctl by a.rltling th: 

round nose shown at (I)). Simultm~cortsl~, tho drag cocfticienl became rna.rltrtlly 

smaller and had a value of CD = 0.42. Further rcscarch on such vchiclcs have beell 

performed by 11'. H. IIucho [In] for the rase of a non-~yrnrnct~ric strcam. 

t Fig. 2.16. and 2.17. have I)een tdten from Llte "Strom~~ngrn in I)antpfkossrln~~lnfcn" 

by TI. FocLthgcr, Mittcilltngcn tlcr Vercinip~oc! Ilr*.'IUUU:ICrsqelbenit,7.e.r, No. 73, p. Ihl (1!)39).

IGg. 2.16. Frrc stagnation flow witl~o~~tarpn. Fig. 2.17. 1)rcrlrrated 8Lag11:~tiorl flow with 

ration, au pliotogmphrtl by Fotttingrr 

scprntion, ns pllotogrnphed by Focttingcr 

I fa1 Anaubr nose 1 I 

I (b) Round nose I I 

0. %? 

- ( - z z 0 - 

I 

no separation 

IFig. 2.18. I'low n.l,orrl, n ~n~(lcl of a motor vrl~inlc (Volltsw:i.gc:n tlclivrry vrm). nftrr 15. Morller 

1231. n) Angulrrr noso wi1I1 mpnmtcd flow nlong tho whole of the aidc wall nnd lnrge drag codficicr~t 

(C,, = 0.70); h) ltord iionc with no ~cpnrntion nntl small clrng cocflhic~lt (CD = 0.42) 

b. Separation and vortcx formation 3 7 

Separation is also important for the lifting properties of nn aerofoil. At small 

incidence anglcs (up to about lo0) the flow does not separate on either side antl 

closely approximates frictionless contlitions. The prcssurc distrihntion for slleh a cnsr 

("S~IIII~" flow, Vig. 2.11)n) WILR givo11 in Vig. 1.14. Will1 inoron~ing i~tcitlo~~cn t,lrc\rc* 

is tlangcr of srparnt,ioti on t h sucI,ion side of tho nerofoil, I)cer~~~so t,l~e l)rcss~~re ill. 

crcnw bccomcs sleepcr. Por n given angle of incidenc~, which is nljout l!jO, ~cparation 

Litinlly occurs. The scpwation point is located fairly closely behind the lcading cdge. 

Thc wr-kc, Fig. 2.19b, shows a large "(lead-water" nrca. The friclionless, lift-creating 

flow patter-n has Iwcornc dislurbcd, and the drag has become very largo. The ,heginning 

of scpnrat.ion nwrly coincidcs with the occurrence of maximum lift of the 

acrofoil. 

Structural oerodynomics. Flow around land-bnsed bluff bodies, suc11 as struc- 

tures antl buildings, is consiclcral~ly more complex than flow around streamlined botlies 

and aircraft. The principal cause of complication is the presence of the ground ant1 

the shear created in the turbulent wind as a consequence. The interaction between 

the incident shcar flow and the stsruct,ure produces coexisting static and tlynamic loads 

[8, 9, 101. Tlie fluctuating forces produced by vortex formation and shedding can 

induce oscillat,ions in thc structures nt. their natural frcql~cncics. 

The flow patterns observed on a tlctachcd rectangulnr building is shown sahrmali- 

cally in Fig. 2.20. In front of the building there appears a bound vortrx whirh arises 

from the interaction of the boundary layer in t,he sheared flow (d V/dz > 0) ant1 the 

ground. There is, furthermore, strong vortex shedding from the sharp corllcrs of the 

building and a complex wake is created behind it. So far no theoretical mcthotls have 

been developed to cope with this ext,remely complicated flow pattern. It is, therefore, 

necessary to rcsort to wind-tunnel studies with the aid of adequately scalctl models.

3 8 

\ 

Y?\ 

If. 011tli11e of boundary-layer throry 

Fig. 2.20. Overall view of 

flow pat,tmn (schematic) 

around a rcctnngular st.ructure 

[MI. a) Side view with 

foreward hound vortex in 

the stagnation zonr and a 

~cperatod roof lmtntlnry 

layer; h) ~tpwitd fme and 

vortex ~hcdding from the 

t hn windward rornrr of thr 

roof 

Fig. 2.21. Acrofoil and cir- 

cular cylinder drawn in 

such relation to each other 

as to produce the same drag 

in parallel flows (parallel to 

axis of svrnmetry of awofoil) 

circuhr cr/linder: Drag 

To conclude this section, we wish t,o tlisc~iss n particr~ln.rly telling example of 

enectively it is possible to reduce the drag of a body in n st,rearn wl~et~ the srl)nrntioll 

of the boundary layer is completely elirninatrtl antl when, in ntltlit,iol~, the I~otl~ itsrlf 

is given a shape which is contlucivc to low rcsist.nncr. Pig 2.21 ill~lstrnt.cs tllr c.i~(:ct, 

R fnvvrnble sllnpe (strenndine body) on drag: it syintrteLrlc ncrofoil n~~tl a rirc-lllar 

c:ylintlcr (thin wire) have brrn drnwr~ hrrc to n relative scdo wllicl~ rtssr1rc:s c:clrlnl tIrng 

in slwnms of cqnnl velocit,~. The cylinder has a tlrag corfficicnt (:I, % 1 wit,l~ rc?spct, 

to it,s frontd arcn (scr also Fig. 1.4). 011 t.hc otllcr hnnrl, l .1~ (Irag cocfficic:t~t.oft II(, ;I(.I.ofoil, 

rcferrctl to iLs cross-seclionnl arm, has the very low vnl~lc* of f:, - 0.00(;. 'I'll!: 

cxt.romrly low tllxg of thc ncrofoil is ncl~icvetl ns n rcsctlt, of n cnrt$r~ll~ cltosc.~~ ,)l.olilc~ 

which assures llmt the boundnry Inycr rernnins laminar ovcr nlmost t,l~c \vl~olc of its 

wett.ed Irngth (Inminnr ncrofoil). Tfit,l~is conncxion, Chap. XVf l nt~tl, c!s~)rc.i:tll~, Icig. 

17.14, sl~o~~ltl Ije consult.cd. 

c. Turhulertt llnw in n pipe and in n bot~ntlnry layer 

hlensnren~cnt,s show t11n.t the t.ypc of mol,iorl tl~ro~~glr n rirwlnr pipr which was 

calculal.cd in Section ld, and in wl~ich 1.11~ vclocily tlislril)trt.ion w:~s p:wnbolic, 

exists only at low and n~odcrnte Reynolds numbers. The fact that in thc laminar 

motion tinder disoussion fluid Inminno slide over each other, and ll~i~t tllcrc: aro no 

rndial vclocit.y romponrnt.s, so t.hnt t.he prcsslrre clrop is proportiot~:~l t,o the firs1 

power of t.he lncnn flow vrlocit.y, const.itmtrs nn esscnt.in1 c:l~arnrt.rristic: of this t.ypc 

of flow. This cI~arnrt.rrist,ic of the motion can bc mntlc rlrnrly visil,lo 1,s inlrotl~lcit~g 

a dye into the st.rmm and by tliscl~nrging it tl~rougl~ a t,llin t~~l)c, Fig. 2.22. At, t,l~c 

motlernt,~ Rrynolds nunlhers associntcd wit,l~ Intnit~nr flow tl~e tlyc is visit)lr in 

lhr form oi a clearly tlefinetl t,l~read ext,cnding ovcr thr. wllolc Irngtl~ of t,hc pip., 

Fig. 2.22a. 13y increr~sing tlte flow velocity it, is pnssil~lc 1.0 rmch a stngr. .vheii t.hc 

Ruid pnrtic!les cease to move alor~g st,m.igl~t linrs antl t .1~ rcgrllnrity oC the mot.ior~ 

brrnks down. l'l~c colourcd Lltrencl bcc:o~nc~ mixed wit,\) the flltitl, its sharp out.li~tc? 

becomrs blurred ant1 nvcmt.11a.ll.y thc whole cross-srrtioll Iwrotnrs colortrrtl, Pig. 2.221). 

On t.lw n,xinl n~otion t,hcrc are now s~~pr~.i~njmotl irrc~gr1l:tr rntlial Il~~ct.rt:~t.iot~s wl~irlt 

clli.c.t the mixing. Such a flow pnttern is cnllccl l~~fiule~r!. 'l'l~r tljw cxl~crilnrnt was 

first carried out by 0. Reynolds 1291, who nscertninctl tl~nt, the taansitsic.n honl 

the laminar to Llle t~~rh!cnt t,ypc of motion ttaltcs pl:rcc at a tlcfinit.~ v:t.lnr of IIIV 

I

In t,hc tr~rl~ulont region the pressure tlrop becomes approximately pr~port~ional 

t,o the square of the mean flow velocity. In this case a consiclerably larger pressure 

tliffcrencc is requirctl in ordcr to pnss a fixed quantit,y of fluid t.hrol1g11 the pipc, 

ns corrlparocl with laminar flow. l'his follows from t,ho fact that t.ho plrcnomcnoll of 

t.url)ltlrr~t mixing dissipat,cs a largc q~t:tt~t,it,y of' enorgy which c:~~~scs the rcsist,:tnc:c? 

1.0 Ilow t.o incrcasc considcr:tl)ly. lrurl,llcr~norr, in Ihc casr? of Lurl~ulcrlt, llow t,hc volodistritlu(.ion 

over the cross-scct,ior~al arca is much tnoro cvcn thrl in hminnr 

flow. 'rhis circumst,ance is also t,o be explained by turbulent mixing which causes an 

cxc:hangc of momcntum bctwecn the layers near the axis of the tube and those near 

t,hc walls. Most pipc flows which are encountererl in engineering appliances occur at 

such high Reynolds numbcrs that turbrllcrlt motion prevails as a rule. Thc laws of 

turb~llent motion through pipes will be discrlssed in detail in Chap. XX. 

111 a way which is similar to the motlion through a pipe, the flow in a boundary 

laycr along a wall also becomes turbulent when the extcrnal velocity is sufficient,ly 

largc. ISxpcrimental investigations into the transition from laminar to turbulent 

flow in the I,ollntlnry Inyer were first carried out by J. M. Burgers [GI and I3. G. 

vnll (lcr licgge Zijncrl 1171 as wcll as by M. IIansen [lG]. The t,ransit.iorl from 

laminar to turbulent flow in the boundary layer becomes most clearly discernible 

by a sutltlcn a.nd largc increase in the boundary-layer thiclrncss ant1 in the shearing 

stress near the wall. According to eqn. (2.1), with 1 replaced by the current co- 

ortlinatc s, the dimensionless boundary-layer thickness 6/1/1'27~; becomes constant 

for laminar flow, and is, as seen from eqn. (2.la), approximately equal to 5. Fig. 2.23 

contains a plot of this tlimcrlsiorllcss boundary-layer thickness agairlst the IZcynoltls 

number IJ, z/v. At R, > 3-2 x 10" very sharp increase is clearly visil)le, and 

Fig. 2.23. Boundnry-layer thickness plob- 

tedr against the Reynolds number based 

on'the current lcngth z along a plate in 

pnrnllel flow at zero incidence, ~s mea- 

sured by llanscn [I61 

as sprn from rqn. (2 1 a). llr~lrc to thr rritiral Rrynoltls r~urnl~rr 

there corrcspontls Rg crlt = 2800. The bountlary Inyrr or1 :I plate is Inr11in:cr near t.l~t: 

leading edge and bcconles turbulent f~lrt.llcr tlowr~st,rca~n. 'I'llc nbscissn r,,,, of tl~t 

point of lrn~lsit~ion can be clctcrminctl from L11c ktlow~~ v:~lric of R, .,,,. In t.llc caso 

of n plate, as in the prcviot~sly discussed pipc flow, the nun~cricnl vaI11o of R,,,, 

dcpcntls to a ~narkctl degree on the amount of' tlist.~lrl~ancc in tho nxt,crn:tl flow, :111tj 

the value R, = 3.2 x 10%hot1lcl be regartlet1 ns a lower limit,. With oxccpt.iorl:~Ily 

(list-rrrbnncc-frcc cxt.crnal flow, valrlcs of R, , - 10%rlrltl higllrr 11:~vc been :~tt.ail~rtl. 

A 1):~rticul:trly rernarltable phcnorncnon connccld with the transit.ioll from 

laminar to trlrbrllt:nt flow occurs in tJle casc of blunt llotlics, s11cl1 as circ~~lar cylintlers 

or spheres. It will be seen from Figs. 1.4 ard 1.5 t,llaL the tlmg coef'ficierlt ofa circrtlar 

cylintlcr or a sphcro suffcrs a sutltlcn :d consitlcral~le dccrcasc Ilr:lr Itcynoltls 

n~iml~crs 1.' I)/v of bout 5 X lo5 or 3 x lo5 rcspccLive1~. This fact was first, obscrvrtl 

on sphcrcs by G. 1I:iffrl 1141. It. is a conscquerlcc of t,ransition which causes t.he 

point of separation to movc clownstacam, l)cca~rsc, in the case of a turbulcr~t 1)ountlary 

laycr, the accelerating influence of the cxt.crn:d flow extmds furlhr due t,o t.t~rbulrr~t. 

mixing. ~Tcncc the point of separation whicll lies near the equator for a laminar 

I)o~rr~tlary I:~ycr nlovcs over a cor~sitlcml~lo tlislnr~cc in the downstream tlircct.ior~. 

In t,urn, the tlcad arca decreases considcmbly, anti thc pressure di~t~ribution becomes 

more like t,hat for frictionless motion (Fig. 1.11). The decrease in thc rlcad-wat,cr 

region consitlcmbly reduces the prcssrlrc dmg, and that shows itself as a jump in 

the curve G, .= f(R). L. Pmnrltl [26] provctl tl~e corrcctncss of t,hc prrcccling 

reasoning 11y nlo~inl~ing n Ihiri wirc ring III; a ~Ilort, (li~Im(:c in fro~tt or IJIO ccl~i:ll,or 

of a sphere. This car~scs the boundary laycr to bccome art,ificially turl)~llcrlt at n lower 

Reynolds nl~mbcr and the tlccrcasc in t,hc drag cocfficicr~t taltes place carlicr Lllar~ 

would otherwise be the case. Figs. 2.24 and 2.26 reproduce photographs of flows 

which have been made visible by smoke. They reprcscnt the subcritical pattern 

with a large value of the drag coefficient and the supercritical pattern with a small 

dead-water arca and a small value of the drag coefficient. The supercritical pat,tern 

was achieved with Prandt,l's tripping wire. The preceding cxporimcnt shows in 

a convincing manncr t,hat the jump in the drag curve of a rircular cylintlcr and 

sphere can only be interprctcd as a borindary-layer phcnomcnor~. Othor bodies 

with a blunt or rounded slcrn. (c. g. elliptic cyli~~tlcrs) display :I type of relationship 

bctwcen drag coefficient and Rcynoltls number wllicl~ is s~~l)sta~~li:illy similar. \'Vit,h 

increasing slcntlcrness the jump in t h curve bccomcs ~'iro~rcssivcl~ less pronor~nccd. 

For a streamline body, such ns that shown it1 Fig. 1.12 t.h(:rc is rlo jump, I~nc:~usc 

no :lpprrci:r.l)lc scp:~.rnt,io~~ occ~lrs; t,lw wry gmtlrr:~l Iyssrlrc ir~c!rr:lso on I,l~c Il;lclt

42 

11. 011tli11e of Iw~~ntlnry-Inyrr theory 

of suc.11 I~otlics csan I x overcome by tl~c bor~ntla.ry layer witho~~t separat.ion. AS we 

sl~all also scc Int,cr in grrat,er tlrtail, t,he pressure di~tribut~ion in thc ext,ernal flow 

t~xrrt,s a clet~isivc influt:nce on t,hc positmion of t.11~ transition point. Thc bountlnry 

Ia.yrr is laminar in the region of prcssurc deereast, i. e. rol~ghly from t.l~e leading 

ntlgc? to t.hr pint of minimum pressure, ant1 becomes t~~rhulent, in most cases, 

from t.l~:~t point onward througl~o~~t, t.l~r region of prcsslrrc inrrcn.sc. In this corrnexion 

it is iml~ort,ant to statc tht, scpamt,ion can only bc nvoitletl in rcgiorrs of incrensing 

prcssnrc n h the ~ flow in thc bountlnry layer is turlrulcnt. A laminar 1)ountlary layer, 

as wc shall see Int.er, can support, only n very smnll pressure rise so t,hat. scparat,iorr 

would occur even wit.l~ very slcndcr botlics. In prt.icular, this remark also applies to 

the flow past nn aerofoil wit,li n pressure dist,rit)ut,iorl similnr to that in Fig. 1.14. In 

t.llis cnse scpamt~ion is most liltcly t,o ocrur on t.he sncI,ion side. A smoot,l~ flow pattern 

nround n.n ncrohil, contlucivc t.o ~ I I C creation of lift, is possihlr only wit.11 a t,~~rhnlent 

bountla.ry Ia.ycr. Summing up it, ma.)i be st.at,rtl that, t.hc small drag of slencler bodies 

as wrll &s t.11~ lift, of acrofoils are ma.& possible 1,111~ough thc cxist,enec of n t,url)ulent, 

t)ountla,ry Inyer. 

Bounclnry-lnyer thickness: (~cr~erally spealc~r~g, the thicknesq of a tnrbulcnt 

Imr~ntlary hycr is larger than that of n laminar boundary layer owing to grratcr 

energy losses in the former. Nenr a smooth flat plate at zero incidence the boundary 

layer incrcascs downstream in proportion to xoR (x = distance from leading edge) 

It will he ~llown Inter in Chap. XXI that the boundary-layer tl~ieknrss variation 

in (nrt)nlrnt flow is given by the rqnntion 

f 

d lJm,l -'I5 * 

= 0.37 ( ) = 0.37 (Rl)-'1' (2.9) 

1 

whic-ll c:orrcspontls 1.0 rqn. (2.2) for laminar flow. I'ahlr 2.1 givns vnlnes for thn 

I~o~~~~~l:r.ry-I:tyc:l. t11i(~Ii11ns~ o:~l~:uIal.r~I from eqn. (Z.!)) for several typical casos of air 

:~1d watl~r flows. 

c. Twhulent flow in n pipe nr~d in a hourldnry lnycr 43 

Tnhle 2.1. Thickness of bormdary Inyer, 6, at t.rniling edge oF flnt plate nt zero inridencc in 

parallel t.nrlwlent flow 

U, = rrcr ntrenlll vrloclty: I = lrnqth or p1al.e: r = kinrn>nl.le risrasily 

Air 

v = 150 x 10-e ftZ/~~v: 

100 

200 

2 0 

5 0 

750 

Methods for the prevention of separation: Sopnrnt,ion is mostly nn r~ntlcsir:~.I~lt! 

pl~rnomcnon bccnusr, it clltr~il~ lnrgo onorgy losncs. I'nr thin rcnson rnctllo~ls I~r~vo 1,cm 

tleviscd for the artificial prcvcntion of separation. Thc simplest met.hotl, from t,l~c 

physical point of view, is to move the wall with the stream in order to rcdr~ce hhc 

velocity difference between them, and hence to remove the cause of boundary-layer 

formation, but this is very difficult to nchicvc in engineering practice. Ilowcvcr, 

I'rnndtl t has shown on n rolaling circdar cyli?zP.r tllat this method is very rfrcct.ivn. 

On the side where the wall and stream move in thc same direction separnt.ion is oornpletely 

prevented. Moreover, on the side where the wall and strenn~ move in oppositc 

tlircct,ions, separation is slight so that on the whole it is possible to obtain a gootl 

experimental approximation to perfcct flow with circulation ant1 a large lift.. 

Another very effective method for tlic prcvcnt,ion of separation is hm~d

44 11. Outline of boundary-layer theory References 45 

to the aerofoil at considerably larger incidence angles than yould otllcrwisr be tlr~ 

rase. stalling is clrl:cyetl, nntl much largtr maximum-lift values are achieved [3F]. 

Aft,er having given a short out,lino of the fnntlamentd physical principles of 

fl~~id motions wit,l~ vcry snlnll friot.ion, i. c. of thc boundary-layer theory, wc shnll 

proneed to clovc!lop n mtional theory of tl~cso pl~cnorncnn froln 0110 oq111~1.ions 01' 

motion of V~SCOIIS fluids. Thf: description will be arr:~ngctl in the following way : Wt: 

shall begin in Part A by deriving Ghc general Navicr-Stjokes equations from whicl~, 

in turn, we shall derive Prandtl's boundary-layer equations with the nick of the 

sirnplificntions which can be inl,rotlucctl as a consequence of the small values of viscosit,~ 

This will be followed in Part I3 by a tlc~cript~ion of the metjhods for the integmtion 

of these cqnat.ions for the caso of laminar flow. 111 Part C we shall discuss the 

poblem of tho origin of t,nrbulcnt flow, i. o. we shall discuss tho process of transition 

from laminar t,o t,urbulent flow, treating it, as a problem in the stabiliLy of laminar 

mot,ion. Finally, Pn.rt .D will contain the bountlary-laycr theory for completely 

tlcvclopcrl turbulent motions. Whereas the theory of laminar boundary layers can 

I)c trcat,ctl as n dctlnctlive sequence Imsctl on t,hc Nnvicr-Stolres tlifTerent,i~l equationx 

for viscous fluids, tho same is not,, at prcscnt,, possible for turbulent flow, t)ccnusc thc! 

mccl~anism or turbulent flow is so complcx t.hat it cannot be mastered by purely 

t.l~rorct,icnl mct,hods. For t,his reason a t~rc;~iisc on tlnrl~nlcnt flow must, draw 11e:~vily 

on exprrimentnl result,s ant1 t,llc subjcrt mnst Ijc presented in t,hc form of a semicmpiriral 

throry. 

References 

[I] Acl~enbach, E.: J':xperilnent,s on the flow past spheres at vcry high Reynolds numbers. 

JFM 54, 505--575 (1972). 

121 Ilcrger, J':., ant1 Wille, It.: Perioclic flow phenon~c~~n. Annual Reviow of Fluid hlcch. 4, 

% > 

313--340 (1072). 

(31 nerger, 15. : Ucst.iln~nung dcr l~ydrodyn:~l~~iwI~c~~ (:riissen einer Iins. (:iiltingcn 1!)07; 

L. Math. u. I'h~.s. Mi, 1-37 (1908); Engl. trnt~sl. in SAC'\ Thl 1250. 

151 Ulenk. H.. I~urlis, I).. and Licbcm, I,.: uber die 3lcssung von \\'irbelfrequer~zcl~. 1,uftfnhrt- 

L , 

forsrh"ngl2, 38--41 (1935). 

[O] Burgers, J. M.: 'The motion of n fluid in thr houndnry lnycr nlong n plnne sn~ootl~ surfnce. 

1 Roc. First lnternationnl Congress for Applied Mcchnnics, 1)elft. 11J-- 128 (I!)24). 

171 (;h:~ng, P.K.: Sep:~rntion of flow. l'ergnn~ot~ Press. \Vnshington I>.C., 1!)70. 

[R] C,rrn~nk, J. E.: Ap~~lirntion of fluid n~rchnnics to \r-intl enginrering -- t\ Frcetnnn Scholnr 

Icrture. Trnns. AhNlC Fh~ids Engineering 97, Ser. I, 9--38 (1!)75): sre nlso: Lahor:~tory 

sin~rtlntiot~ of the ntlnosphcric houndnry Inycr. t\lA,\ .J. 9. 174(i-1754 (1!171). 

18111 Cerlnnk, .I. E.: Acrodynnn~ics of h~tildingn. r\n~~ilal He icw of Fluid Blrch. 8, 75-- 100 (1970). 

,!I] Crnnnt. JE.. and S~~CII, w.z.: \\lint~.tunnci siln6tion of wincl ionciing on structures. 

Jlrctinp: I'reprint 1417, r\SCIC Sntionnl Structural 1Sngineering Alreting. 13nltit11orr. hfnry- 

Inrd, 171-- 2j April, 1971. 

[I01 j)nvenport, ,\. G.: 'rhc rclntionship of wind structure to wind Ionding. Pror. Confercnrc on 

\\.i~~tl 15llrrts on I3uildi11~u nnd Str~wtures, Sntionnl I'hynirnl I,nhor:~tory. 'Trtldingtol~, 

Jlid(llrnrx. (:rrnt Itritnin. 26--28 ,111nr I!l(i:l. Ilcr Mnjmty's Stationnry 0flic.r. I,ontlol~. 

\'()I. I, 54 -- I I2 (I !N\5). 

[ll] Dotnm, U.: Ein Beitrag zur StabilitAtstheorie der Wirbelstmssen unbr Berucksicht,igu~~g 

endlieher und zeitlich wachsender Wirbelkerndurchmesser. Ing.-Arch. 22, 400 - 410 

i 1954) -,. 

[12] Ihhu, W.: Uber den Einfluss 1aniinn.rcr und t~~rbulcnlcr St.riimrtng nvf dnu Riinlg~nbild von 

Wnssor untl Nit,rol~cnzol. Ilclv. phys act,:^ 12. 100--228 (I!):)!)). 

II3] Ihrgin, W.\\'., nncl l

[:j2a] I?o~cilhrntl, I,.: Thr forn~atior~ of vortices horn n surface of cliscontin~iitj~. Proc. Roy. Soc. 

A 134, 170 (1931). 

[:XI] Jl,r~bact~, 11.: Uher dir 1ht.stehung und Iqortl~c\vegnt)g dcs \Yirbrlpnnres bei zylindriscl~on 

I

48 

111. Derivation of the eqrtat,ionn of motion of a compressible viucoou fluid 

The etl~lnl,ions of motiot~ ore dcrivotl from Nowton's Second Law, which shtes 

that the product of mass and accelcrntion is equal to Lhe sum of the external forces 

acting on t,hc Imdy. In fluid mot.ion iL is necessary to consider t,hc followil~g two 

classes of forcrs : forces acLing thro~~ghorrt the mass of the botly (grsvit,ational forces) 

ant1 forces arting on Lhc bounrlnry (pmssurc and friction). If F r= Q g denot,rs the 

gravit,;\t,iorl:~l force per unit volumo (g 7.7 vcrtor of accrlcmt,ion due t,o graviLy) ant1 

I' denot,es the force on the boundary per unit volume, then the equations of motion 

can I)c written in t,ho following vector form 

1' = i I', -{ j P,, + k Pz surface force . (3.4) 

'I'l~e syn~l)ol I)tr-/l)l tlcnot.c~s hcrc t,l~c sul,st.antivc :wc:cler:~t~ion which, like the sul)stant,ivc 

tlrrivat.ivc ol' tlonsit.y, ronsist,s of Lhc local cont,ribution (in non-steady flow) 

c?tv/at, nntl tho convrot.ivr cont,ribut.ion (tl~~c to t,ranslation) drtr/dl = (w-gmd) rc~t 

'I'l~n I)otly li)rcrs arc: t,o I)c rcgardctl as givcl~ extcrtwl l'nrct>s, but, t.he surfnee forcrs 

tlrpncl on t,l~e rate at, which t,hr fluirl is stwined by the vrlocit,y field present in it,. 

l'l~o system of fhrcrs dot.crmincs a slrrtc o/ stress, :mtl it is now our task to intlicat,e 

t,l~c rclat~ionsl~i~, l~ct,\vcen st.rcss and ri~tc! of st.min, noting t.l~at it can only be give11 

empirically. In our presrnt, tlcriv:ttion \vosl~:~ll rcst,rict,ntt.ention toisotropic, Newto7iinn 

/b?cirls for which it may Iw nssun~ed 1,11:1t. t Iris relnl ion is a linear one. All gases and 

manjr litl~~itls of int.crcst it] bountlnry-hyrr t.l~eory, in pnrt.icnlnr wat.er, belong t,o t.his 

clt~ss. A fluitl is snitl 1.0 I)(, isot,ropir wl~cn the rrlnt.ion 1)ct.wcrn t.he coml)oncnts of 

st,rcss ant1 t.l~ost: of 1.11~ rat,? ol strain is t.ho sxnlr in all directions; it is said t,o bo 

Newtnni:ln whcn t,l~is rclat.io~l is linrnr, t.l1:11. is whcn the Ilnid obrys Strokes's law of 

f.ricl,ion. In t,l~r cnsc of isotropic, cl;~stir solid Imlirs, cxpcriment tmches I,lmt t,l~c 

st.nt.c of strc-ss tlrpcnds on thr rnngnit~uclc of sttrain il.sclf, most engineering mat.erinls 

obeying Jlooltc's lincnr Inw whic:h is somcwl~nt analogons.l.o Stokes's law. \Vhcrens 

t,he rcl:~tion l,nt,\vccn stxcss and st.r:~in for an isotropic elastic solid involves t,wo const.nnt,s 

whicl~ cl~nrnctcrizc t.hc propcrt,ins of a given mat,crial (e. g. elirst,ic motlnl~~s:ml 

I'oisstm's rat,io), the rc,l;~t,ion bct\vccn st,rrss and rate of shin in nn isot,ropic fl~~itl 

irrvolvcs n, singlr ronst,:~nt. (1.11~ viscosity, p) ns long as rel:~xat.ion pl~cnornena do not 

occwr wit,l~in it.. ns we sl~:lll sro in Scr. Ill(>. , 

b. General stress system in e, dcfor~nablo body 

I#. General stress system in a deformable body 

In order to writo down expressions for the ~urface forces ncting on the bo~~ndarjr, 

let, 118 imngino n mnll prirrillcpipctl of volumo d V = !lx (I!/ tlz isol~cto(l 

49 

~IIU(.ILII~,ILII~~ 

from the botly of the fluid, Fig. 3.1, and let its lower loft-l~nr~tl vcrttcx coincide wit.11 

the point x, y, Z. On the two faces of nrcn dy . tlz which arc pcrpcn(lictlIar 1.0 tho 

z-axis thcre net two resr~lt~nnt stresses (vectors = surface forcc per 1111it arca): 

Fig. 3.1. 1)rrivation of the expressions 

for the strrm tensor of nn inl~ornogcneo~ln 

utrcss nystcrn and of its syrnmetry in the 

absence of n volurl~ctric distribution of 

local moments 

apz 

pz and pl -4- dx rcspcctively 

ax 

. 

(Subscript x denotes that the stress vector nets on an elementary plane which is 

perpendicular to the x-djrection.) Similar terms are obtained for the faces dz . tlz 

and ilx . dy whir11 are perpendicular to the y- and z-axes respectivrly. Ilencc the three 

net components of the surface forre are: 

"= 

plane 1 direction x: - . 

8% 

dx . dy . dz 

and t01e rcw~lt~ant srlrfacc forrc P per unit volume is, tl~crcforc, given I)y 

The quantities p,, p,, p, are vectors wl~icl~ can be rrsolvcd into components perpendicular 

to each face, i. e., into normal stressos denotetl by a with a suitable 

subscript indicating the tlirection, and into components parallel to each facc, i. o. 

into ~hcaring slrcsscs denoted by t. The symbol for n shvnring stress will I)o ~~rovitlrd

50 

111. l)crivnt,ion of thc cquntion~ of moI.ior~ of n cornprrcmihlo viscous fluid 

with two snbscripta: the first subscript indicates the axis to which the face is per- 

pendicular, and the second inclicat8cs the direction to which the shearing stress is 

parallrl. With this notation we have 

The strrss syst,on~ in soen to require nine scalar qun.nt,itics for its clcscript,ion. ?'heso 

nine quantitirs forrn a alresa tensor. The sel of nine components of the stress 

t.rl~sor is somct,imcs callrtl bhe st-rcss matrix : 

'I'llo st.ress t,cnsor and tho correspor~ding matrix are symmetric, which means 

t.llat two shearirrg stresses with subscript8 which differ only in their order are equal. 

This can be tlcmonstrated with reference to thc equations of motion of an elemcnt 

of fluid. Tn gcncml, itas motion can be sepnrntetl into an instantaneous translation and 

an inst,znt,ancorw rotat.ion, and only t,l~c Iattor needs to he consideretl for our purpose. 

Denoting t,hc inst,antaneous angular ncccleratiorl of t.11~ element by &(ti),, h,, h,), 

we cnn wrik for t,ho rot.ation about. tjho y-axis that 

where tlly is t,hc elementary momcnt of inertia about, the y-axis. Now the momcnt 

of inertia, dI, is proportional to the fifth power of the linear cli~nensions of the par~llelr~iped, 

whereas its volumc, tl IT, is proportional to their third power. On contracting 

thc clcmcnt to a point, we notice that the left-hand sitlc of t

52 

111. I)rr.ivnt.iot~ of tho rrlrtnt,ions of ~ndion of n cotnprc.ssiblc viscous fluid 

'1'110 system of the Ihrcw cq~t:~t,ions (3.1 1 ) aotrt.air~s the six st,rrsscs a,, a,, a,, 

T,,, t , s,,. Tho next tdc is t80 tlct,crlninc the relation between them and the 

strair~s so as to enable 11s t,n introduc~ the vrlocity components u, v, w into eqn. (3.1 1). 

Before giving this rclat,ion in See. 111 tl wo shall ir~vcstigat.~ t,l~c syst,om of st,mins 

in great,cr detail. 

c. The rntc nt which a fluid element i~ stmined in flow 

Whrn a cont,inl~o~ls l)odq. of fluitl is rnatlc to flow, every rlcmcnt in it is, gcncrally 

sptvdting, clis~,ln.cctl t.o a new posit,ion in t1he course of time. 1)uring t.his motion 

rlcmpnts of flrlitl l)ccon~c st,minctl, ant1 since the mot,ion of t.hn flnicl is completely 

tlet.rrminrt1 when the vclocit,y vect,or rrr is given as a funct,ion of t,ime and positpion, 

tr, = ru(z,?/,z,t), them cxist Itincrnat,ic rrlnt,ions l)et,wcen the components of t,hc 

r:~.tc of st.min and t,ltis function. 'Vhe rntc nt which an clement or fluid is strained 

tlcprt~ls on t.11~ ~~1.lbz.t: n~otion of t,wo poin1.s wit.hin it. We, therefore, consitlor the 

t,wo neigl~l~onring points A ant1 B whirh arc st~own in Fig. 3.2. Owing to t.he presence 

of t,ho vrlority field, point A will be tlisplncctl to A' in t,ime dl by a distat~ce s = ro dt ; 

sinrr, I~owevor, tho vclocity at B, imngincd at a dist,ancc dr from A, is different, 

point H will move t,o B' displ:md from 1% by s -1- (1.9 = (ui -1-tlrri) dt. More explicitly, 

if t,hr componcnt.~ 

l)oint, I 0. 'i'l~c rclat,ive velocity ol' 

any point B with respect t,o A is now 

antl tho field consists of planes x =- const which displace thcmsclvcs nniformly 

wit,ll a velocit,y which is proportional Lo the clistancc tlx away from the plane x = 0. 

An elementary parallelepiped with A anti R :~t its vertices placed in such a vclocity 

field will be distorted in extension, its face BC receding from AD wil.11 nn inorcasing

Fig. 3.3. Tmcd clintor1,ion of fluid rlenlc.111 

whcn &/ill: > 0 wil.11 nll olhr lrrnin bring 

eqnnl to zrro; rrniforrn exlcnnion in the z- 

direction 

vr1ocait.y. Thus 2, reprcscnt~ t.he rate of rhn,gdion in the x-dircct,ion sufli?retl by the 

clcmctit. Similarly, the atlclit,ivc terms C, = a11/'11/ ant1 C, = aio/i)z drscri1)e the rate 

of rlorig:rl.io~l ir~ t,llc y- xntl z-tlircctions, rcspcctively. 

(~r I az lr I, dy dy

56 11 I . 1)t.rivntion of t.lm cquntionrr of n~olinn of n rotnprcasibln viscous tlnitl 

arltl r?n/8z 11avc posit,ive nonvanishing vnlnrs, the right, angle at A will distort owing 

t,o t,l~e sl~pcrposit,ion of t.wo n~ot,ions, t,llc st.:~tc of affairs bcing illr~st,ral,etl in Fig. 3.5. 

1L is clr:~r that, 1.hr right n.tiglr at. A now distorts at t,wiac the mtc 

tlcscrilwtl by I wo of the orf-tliagon;~l . . t,rr~ns of matrix (3.15%). In general, t,hc thrcc 

~ff-dii~gon~l t.rrms Ex!, - F,/,, F,, = d,,, :LII~ E,, = Fyr tlcsrxibc the rate of dist,ort,ion 

of a right, nnglc locatrd in ;L plane nornmnl t,o the axis the index of which does not 

nppt'ar ns n srll)script.. 'l'hr tlistort.ion is volume-preserving and affects only the 

shape of t,hc rlcmcnt . 

(lirrr~mstanrrs nro ilgain tli!fcrcnt in the pzrticulxr case when au/ay = - av/az 

illrrwt.r;ct,ctl in Kg. 3.6. k'roni t.11~ preceding considerations and from the fact that. 

t~ow 2,, - t) \\.e ran infsr n.t oncc tllat, the right angle at A remains undistorted. 

'I'his is also rlrar from thr diagram which shows that the fluid element rotates with 

rcsprrt, t,o t.llr rcfkrencr point A. Insla~rtnneo~~sly, this rotdon occurs without 

dist.ortion ant1 call Iw dcsoril)rtl as a rigitl-l)otly rot,ntion. The instant,ancons nngulnr 

vrlorit,y of this rot,at,ion in 

It, is now rasy t.o see that the component. < of curl in from cqn. (3.15b), known as 

t,hc vort.irit,y oft.11~ vclocit,y ficld, reproscnt,~ t.lw angular velocity of this instant.nneo~~s 

rigitl-lmlj~ rrfi:.:t ion, and that, 

c. The! rate ~t whir11 a Ihid elcnwnt is ~Lrainctl in flow 57 

(a) A pure tmnslation t1escril)ctl by the vclocit.y components w, v, 1il of it,. 

(b) A rigid-body rotation described by the con~poncnt,~ 5, 7, 5' of c~~rl icr. 

(e) A volrrmctric dilatation tlcscrilmi by e -- tliv in, the iinmr dil:it,:~tions in 

the tlircctior~ or the axes bcing described I)y d,, i, :tnd E,, rrs~)e(:t.ivoly. 

((I) A tlist,ort.ion in shapc drscribctl by t.hr cornponcnt.~ i,, (,I,(: wit11 rnixt:tl 

inclicrs. 

Only tho last, two motions produce an intxinsic tleformation of n llr~itl olcme~lt, 

surrour~tling tho rrfercncc point A, lhc first two causing a mere, general, tlisplacerncnt, 

of its location. 

T11c el~ment~s of matrix (3.15a) constit,rrte the componcnt.s of ;t symrnet~ric 

tensor known :IS t,he rate-of-slmitt lensor; it,s mat.l~ematical propertics arc analogous 

to those of the cq~rally symmct.ric st,ress t,cnsor. It is known from the theory of 

elasticit$y 13, 71 or from general c:onsidcrations of hnsor algebra [I I ] tht wit,l~ 

every symmetric tensor it is possil)lc t.o associate three rnrlt.rrally orthogond pritt,cipnl 

axes which tlctormine tlrrce mut,nnlly ortllogonal principnl plar~cs t,l~:~t is a privilcgctl 

Cart,esian syst,crn of coordinat,cn. In t.llin syst,cm of coortlinatlcs, t,he stlrcss vert,or 

or the inst~wt.t~ncor~s nrolion in tiny ono of the prinoip:~l planes is nornr:d lo it., LhaL 

is, pnrallel t,o one of the axes. IVlrcn sr~cl~ a special system of c~ordinat~cs is used, 

the n~at~rices (3.10) or (3.15a) retain their diagonal tmms onlv. DcnoLirra the valrrcs 

of tlw respective romponer~ts by symbols with I);ir 

matrircs 

It slrould, finally, bc remembered that, such :L t,rnrrsli)r~i~:~liot~ of c~oortlin:~t.vs tlo~ 

not affect, the sum of the diagonal terms, so that 

Iz'ig. 3.7. I'rincipial axes for 

st.rrss ;ind ral.c: of sl.r:~in

of flltitl is strcsscd in thrcc nlutually pcrpcndicirlar tlire~t~ions, and its faccs arc 

displaactl instantaneortsl~ also in tlrrec niut.ually perpendicular directions, as suggested 

by Pigs. 3.7a antl I). This tlocs not., of course, moan tlmt bllcre exist no shearing 

strrssrs in ot.Ilrr pl:~nrs or t,llat. t,l~o sltapc of t.lw clcmrnt rcmr~ins ~lntlistortrd. 

11. Rclntion between stress and rote of deformation 

It sl~nuld, 1)crhaps, 1~ st.rcsscd once more that the cql~at~ions which relate tho 

surface forces to the flow ficld must, be ol)tainetl by a pcrc~pt~ivc interpretation of' 

experimental resulB and that our intcrcst is restricted to isotropic and Newtonian 

fluids. l'hc consitlcmt,ions of the precctling section provided 11s with a useful mathcmatical 

franlcwork which allows us now to statc thc rcquircmcnt~s suggcstcd by 

experimcnt.~ in a somewhat, morc prerisc form. 

When the fluit1 is at rcst, it dcvclops a uniform ficld of hydroslatic st,rcss 

(nrgat.ivc prrssurc - p) which is idcnt.ica1 witli the thermodynamic pressure. 

When the fluid is in mot.ion, t,l~c equation of statc still tlcternlines a pressure at, 

ovcry point ("principlc of local st,atc" 141), and it is rnnvcnicnt to consider t,he 

tlcviat,oric normal strcssrs 

togrtl~cr wit,lt the nnc:lrangotl shearing strosscs. The six q~~ant.itirs so obtjainctl 

cor~st,itrct,c a symnlct,rit: strcss tensor thc cxistcnrc of which is tluc to the tnotiorl 

hccnusc at rcst. :ill it.s componcnls vanislt irlcnt.irally. l'rom what Ilns bcen snit1 

brfnrc it follows that tllc components or this tlcviat,oric tcnsor arc creatctl solcly 

by t,hc componcnls of the ra1.c-or-shin tensor, t.11at is to thc exclusion of the cornponcnt.~ 

v, 17, m of vc~1ocit.y as wc:ll as of the componcnts (, ?I, 5 of vort,icit,y. This 

is rclnivnlent. to s:r\,ing I,l~:rt the inst.nnt.:lncolts t.r:tnslat.ion [component motion (a)] 

as wrll as tho inst,:~.rtt,ancou rigitl-hotly rot,:~tion Icompor~c~~t motion (I))] of' an 

t,lrtnrnt of llrtitl protlucc no surface fnrccs on it in atltlit.ion to the exi~t~ing cotn- 

~wmrnt.s of l~ytlrost.:tl~ic prrsstlm. 'l'hc procctling st.atcmcnL, cvidcnl.ly, rnorrly rcprcscnt,~ 

x prcciso 1oc::d formnlnt.ion of wltat we cxpoct Lo observe whcn :I Gnitc I)otly 

of Ilt~id performs n gcncral mot,ion wl~icll is ir~tlist~ing~rishablc from tltnt of an 

~:q,~~ivalcnt rigill hn(ly. We thus cmncl~dr th:it the erprrssions for tho components 

a, , 0,'. . . ., T ~, of blrc tlcviatoric st.rcss t.cwsor can ront,airr in lhcm only the velocit,y 

gratlicnt,~ aupx, . . ., alll/az in approprintc combinations which we now procccd to 

clet,crmine. 'I'ltcsr: rclntions are postnlnt.ct1 t,o Oc lincar; thcy must rcmain unchangccl 

by a rotation of the syst.crn of coortlirlalcs or by an intcrct~ange of nxcs 1.0 ensure 

isot,rolry. Isotaopy also rctluirc-s that at every point, in t,hc continuum, tfhc principal 

a.xcs of t,llo strrss t,cnsor must roinridc with the principal axes of the rate-of-strain 

t,crisor, for, ot.I~(~rwisr, :I. prrfcrrcd diroct,ion wollld 1)c introduced. rllltc simplest 

way to arllicvc otlr aim is to select an arlrilmrf point, in the cont,inunrn antl t,o 

itn:~girin Llr:~.t, the locnl syst.crn of roordir~at~cs ?, ?7,'2 has been provisionally so clloscn 

as t.o coincitlc with the t,llrcc common princ:ipal axes of tltc two trcnsor~. 'l'hc corn- 

I)ot~i.ttt.s of t,l~v vo1ocit.v lic*ltl in t,his syst,c:m of coordinntcs are dcnototl by ?(., ii, 111. 

of which conicides with it and on the sum of tho three, each with a different factor 

of ~moportionalit,y. Thus we record, dircctJy in terms of tho spacc-clcrivat,ives, tJ~at, 

'l'!.,? rlmlt,iti?s ?I,, 0, :~nd 4, r], C do not appcnr in ti~csc cxprcssiorts for t.11~ ~.r:~.sotls 

jtlst explained. In each expression, the lnst term represents thc appropriate rate 

of lincar dilatat,ion. that is, in essence, n change in allape, and tho first, terrn rcllrescnb 

the vollrmetric clilatfation, that is the rate of change in volume, in csscnct., 

a change in density. Thc factors 2 kl the last terms are not essential, beirrg mcrely 

convenient to facilit,ate thc interpretation, as we shall see 1at.er. Tlrc fact

80 111. l)rrivnt.ior~ of the cquationu of lnot,ion of a con~presaiblc viscouu fluid 

where div rrt has been used for hrrvity. 'J'hc rcatlrr may notice the regularity with 

whir11 the indices x, y, z, the componrnfs n, v, in, antl tl~r coortlinntrs x, y, z arc 

permutcdt. 

Applying t,l~ese equations t,o the si~nplc casc rcprescnt.ctl in Fig. I .I, we rccovcr 

eqn. (1.2) and so confirm that t,hc precctling more gcnernl rrlat,ion rccluces to 

Newt,on's law of friction in t01r casc of simple shear ant1 docs, t,l~orcfore, const.itxtc 

it,s proper gcnoralimtion. At the samr timr, we identify tshe factor 11. with the viscosity 

of t,he fluid, amply disc~~ssetl in Scc 1 h, antl, incid~nt~ally, justify the factor 2 previously 

inserted int,o eqns. (3.21). The physical significance of the second factor, 1, requires 

furt.Iicr tliscilsaion, t~ut we 11ot.c that, it, plays no part in an incompressible fluid when 

div 119 = 0; it then disappears from the equat,ions a.lt.oget,hcr, ant1 so is seen to be 

in~port.nnt for r~ompressible Ruitls only. 

e. Stokes's hypothesis 

Althougl~ the problem l,l~at we arc about to discuss has arise11 more than a 

ccntury and a half ago, the physical intcrpretatiort of the second fact,or, 1, in 

eqns. (3.21) or (8.22a, b) and for flows in which tliv rcJ does not vanish ident,ically, 

is still being disputed, even though the vabe which should be given to it in the 

ioorkirtg eq~u~fio?~~ is not. l'his numerical VRIIIC is determined with the aid of a. hypothsis 

:~tlvancod by G. G. St,oltcs in 1845 11.71. Without, for Lhe nlomcnt,, concerning 

o~~rsclvcs with the physical reasons which just.ify Stokes's h?yjvath~s~:s, we first st.ate 

that according t.o it,, it is neerssary to assume 

This rclatrs the value of the fartor 1 to the visrosity, 14, of thr romprrssible fluid 

and redures thr number of propertics whic41 rhamct~erize the field of stresscs in 

a flowing romprcssiblc fluid from t,wo to onr, that is to thr same num1)cr as is 

rrq~~irctl for rcn incomprrssihlr f111itl 

Subst,it.nting t,l~is v:duc ir~t,o eqr~s. (3.22a), we ol~tairl the normal corni)oncnt,s 

of tlevin.t.orio stmss : 

aw 

a,' = - /L div IJ 2 ,u az , 

3 

t 'hc aboyc ncL of six cqnnl.iona can be oontrac:tcd to a single one in Cartesian-hnsor notation 

(wit.l~ Einflkin's .sn~nrnnt,ion convention): 

u'IIc~(. tlw Kronrrkrr tlrlta dl, - 0 for i + j nntl dij - I for i -- J . 

the ~l~raring stresse~ remairhg unrhangrtl. Malting usr of eqrls. (3.20), wr obtain 

tl~r so-railed conrtitutioe eqlantion for an isotropir, Newtonian fluid 

in it,s final form, 11ot,ing that p reprcser~t.~ the local t,l~errnotlynarnic: prrssurrl- 

Regartled as a pure hypothesis, or ever1 guess, eqn. (3.23) can certainly be 

:~cceptctl on tho ground that the working eqr~at.ions which result from the substitut.ion 

of cqns. (3.26a,b) into (3.11) have been si~bject~cd to an unusually 1;trge number 

of cxpcritnentnl verifications, even ~~ntlcr quite cxt,remc conditions, as t,he reader 

will cor~crtlc after having studied this book. Thus, even if it should not rrprescnt, 

thr state of affairs exact.ly, it certainly constitut.cs an rxcellent approximation. 

Since the deviatoric components are the only ones which arise in motion, 

t,l~cy rcprrscr~t those components of stmss which produce dissipation in all isothertnnl 

flow, t,l~crc bcit~g further dissipatior~ in a t,cmperature field ~ IIC t,o thermal cor~tIuct,ion, 

(%:L~I. XI I . Fl~rt.hcrmore, since t,hc S:~cbor 1 occurs only in tho normal cornpo~~rnt,~ 

cr,', a,', a,' wl~ich also cont~i~in the thcrnmdynaniic pressure, cqrls. (3.20), it I)ccomcs 

vlvar t,l~at. t,hc p11ysic:d significnncc of 1 is connectctl with t.he nicchanism of tlissip:\t,ion 

\\.IIPI~ t.he volume of t,hc fluid clcrncnt is changcyl at a finite rate as well :as 

\\.it,h t.11~ r.rl:~t.ion I)rt,wrrn the tohl st.rcss tensor :wtl t.l~c:rnmotlynnmic: II~OHSII~O. 

f. Bulk viscosity nrd tl~errnodynamic pressure 

\Ye now ~rvcrt to the genrral tlisc~~ssion, wiL11orrt ncccssarily arcaptir~g th(~ 

\aldiI y of Stokrs's hgpothcsis, but, confine it to the casc wl~erl no shearing str~ssrs 

arr irivolvctl, 11cmusc their physical signifiranre arid origin is rlcar Conseq~lrntly, 

6 

In the compact tcnaorial notation wr would write

1:i~ 3.8. Qltnsintzt.ia cotnprrssion and orroill:rlory mobion of n upllorical maw ol fluid 

Wllcr1 tho syst.om is con~prcssed qrl;~sist~ntinally and reversibly, we again mcovtv 

1.11~ prcvions casc becauso then div rrt -> 0 :~s~mptoI.icall~. Wo note Lhnt in such 

cascs t,ho rate at which work is performed in a t.hcrn~oclyn:~rnicdly rcversihle process 

per 11ni1, vnlume hccomc:~ 

rP - ptliv rrr (3.2Ga) 

which is t,l~r same as 

in the not,at.ion cr~sl,omary in thermodynamics. 

tl v 

Jk 7 p -- - (3.26 b) 

dl 

When div rrv is finite, and the fluid is compressed, cxpar~tlcd or made to oscillat.a,' 

at, n fuita d o , qualily hotwesn and -- P pcmist-9 ot~ly if tho coefficirnt 

11' " 1 I 

2 

3 lL 

(3.27) 

valrishm itlrntically (Stokes's I~~poLhsis); otl~or isc it docs not. If p' -l=O, the: 

oscillatory rnot,ion of a spl~ericat system, Pig. i .8b, would produce dissipation, 

oven if tht! tcnipcmture remained constant throughout the bulk of the gas. The 

snmp would be true in t31ie casc of cxpnnsion or compression at a finite rate. Por 

this reason, tlw coefficient 14' is cnllcrl the bulk viscosity of the fluid: it represenh 

tllnt. properLy, like t,lw shear visconit,y {L for deformation in shape, which is responsihlc 

for energy clisaipntion in a fluid of uniform tcmpornt~~ro during 8 cllnnge in volumo

g. The Nnvier-Stokes equations 

\Vith thc aid of rqns (3 20) the non-viscous pressrirr terms can I)r srparatcd 

in the equation of motion (3 11) so that thry bcrolnc 

Jntroducing the ronstitntive relntiotl from cqns. (3.24) we o1)tain the resultant 

surface force in tornis of thc velocity components, c. g, for the 2-direction we obtain 

with the aid of eqn. (3.10a): 

and corresponding exprcssior~s for the y- and z-cornponent,~. 111 the general case of 

a compressiblc flow, the viscosity /A nlust be regarded as dependent on the spacc 

coordinates, bccause p varies considerably with temperature (Tables 1.2 and 12.1), 

and the changes in velocit,y and pressure t,ogethcr with the heat due to friction 

bring about considcral~lc tetnpcraturc variations. The temperatllre dependence of 

viscosiLy p(T) must. bc obtainetl from expcrimcnts (cf. See. XTIla). 

If thcse expressions nrc introtlucrtl into the funtlamelltal eclunliol~s (R.11), we 

oljtain 

Tl~mx vwy wcll known tliffcrcntial equations forb the Itasis of tllc whole science 

of fluid mechanics. They are usu:dly rcferrcd td as the Navier-Stokes equations. 

t Iq indicinl notation: 

g. 7'11~ Nnvier-Stokes equatiotls 65 

It is necessary to include here the equation of continuity which, ns seen frorn ctp. (:j.]), 

assumes t,ho following form for cornprcssible flow: 

'I'lle :tltovc ctpat,ions tlo no1 givc n cornplcln tIoscript,ion of t.110 ~no(.io~r ol' :I, cwn- 

;mssible lluid bcca.usc changcs in pressure and dcnsit.y clfcct tcrnpcrature varint,ions, 

and principles of tl~crmodynamics must, t,Ilcreforr:, oncc morc enkr into the consit1cr:~tions. 

From thermotlynarnics we obtain, in the first, placc, the cllaractcristio 

equation (equation of state) which combines pressure, cIcnsit.y, :inti t,crnpemt,urc, 

:lnd which for a perfect gas has ttlle form 

with 12 drnoting the gas constant autl 7' denoting the nholutc tmnpcmt,urc. Srcontlly, 

if the process is not isothcmial, it is fnrthcr neccssary to makc IISC of tho cnrrgy 

cquat.ion which draws up a tdancc htwccn Itcat, and tnccl~ar~ic::tl cncrgy (First 1,:lw 

of'I'l~cr~notlynamir.s), and which furnishes a dilTerenLial equation for the tempcmturo 

tli~t~riln~tion. The energy equation will be tliscussctl it1 greater tlctail in C~I:L~). XI I. 

The final equation of the system is given by the empirical viscosity law p(Z7), it.8 

tlepentlencc on pressure being, normally, neglected. In all, if the forces X, J', Z are 

considered given, thcrc are seven eqnations for the seven v:wiablcs u, v, 70, p, p, T, p. 

For isothermal proccssos tltcsc rctlucc to five cqlmtions (3.29n,b,c), (3.30) nntl 

(3.31) for the five rtnknowns u, v, W, p, p. 

Ir~com~ressible flow: The above system of equations beconles further simplified 

in tl~r case of incompressible fluids (e = const) even if the temperature is not 

constant. First, as already shown in cqn. (3.la), we have tliv iu r: 0. Secondly, 

since tVcmpcr:~ture variations are, generally speaking, small in this case, the viscosity 

may be taken to be constantt. 

The cquation of state as wcll as the energy equation bccome superfluous as far 

a8 t he cnlculatioo of the field of flow is concerned. The field of flow can now be considered 

intleprntlmtly horn tl~c cqrtntions of tllcrmodynnniirs Tllc cquntions of motion 

(3 29n,l),c) ant1 (3.30) can be simplified and, if the accclcration terms arc wr~ttcn 

out fully, they assume the following form:

66 

111. 1)wivntion of the eqnntions of motion of a conlprr~sible vimwrr fluid 

With known body forcrs there arc four equations for the four unknowns u., v, tu, p. 

If vechr not,ation is nsrtl thc simplifird Navicr-Stokrs equations for incompres- 

~iblc flow, cqns. (3.32a,b.c), can bc shortcncd to 

whrre the sym1)ol V2 denotes the J,nplnce oprrator, V2 -- a2/i)x2 -1- a2/&y2 -1- a2/az2. 

Tho nlwvc Navier-Stokrs rquntio~~s diKer from Euler's equations of motion by 

the viscous terms ,IL V2 (11. 

'r'hc solutions of the above eqnations herome fully clrterminrtl physically whcn 

thr 1)ourltlary and initial rontlitions arc sprcifietl In the case of viscous fluids the 

rontlition of no slip on sold boundaries must, be satisfird, i. e., on a wall both the 

normal and tangential components of the velocity must, vanish: 

v, -- 0 , v, = 0 on solid walls . (3.35) 

. lhc 

, 

equations under discussion wcrr first tferivcd by M. Navier [9] in 1827 and 

by S. 1). I'nisson [lo] in 1831, on the basis of an argument which involved the 

ro~~sidrmtion of int,ermoleci~lar forces. Later the same equations were derived 

without the use of any such hypotheses by 13. de Saint Venant [14] in 1843 and 

by G. G. Stoltcs [13] in 1845. 'l'heir tlcrirations were based on the same assumption 

as made here, narnrly that the normal and shearing stresses are linear functions 

of thc rate of deformation, in conformity with the older law of friction, due to 

Newton, and that the thermodynamic pressure is equal to one-third of the sum of 

the normal stresses taken with an opposite sign. 

Since the hypothesis of linearity is evidently completely arbitrary, it is not 

a priori cerhain that the Navier-Stokes equations give a true description of the 

rnotio~~ of a fluid. Jt is, therefore, necessary to verify them, and that can only be 

arhicvecl by experiment,. In t.l~is connrxion it should, in any case, be noted that 

the enormous nla.tI~ernatical difficulties encountered when solving the Navier-Stokes 

cq~iations have so far prevented us from o1)taining a single ~nalytic solution in which the 

aonvcrtiva t,crn~s int,eract in a gcr~cral way with the friction terms. However, known 

solnt.ions, snch RS Inminnr flow throngh a circular pipe, as well as boundary-layer 

flows, to bc discussed later, a.gree so well wit,h experiment that the general validity 

of the Navior-Stolrcs cqnations can Iiarclly be doubted. 

CYliliclricnl coorclinntcs: We shall now transform the Navier-Stolccs equations 

t,o cylindrical coordinates for future reference. Jf r, +, z donot,e the radial, azimuthal, 

and axial coordinates, respectively, of a three-tlimc~lsional system of c~ortlinat~es, 

and v,, v,, v, donote the velocity components in the respective directions, then 

the transfor~nntion of varinbles [3, 111 for the rase of incompressible Huid flow, 

eqns, (3.33) and (3.34), leads to the following system of equations: 

Curvilinear coordinatee: Tt is often usrful to employ a aurvilinrar ~ysicm of 

coordinates which is adapted to the shape of the body. In t h rnsc of two-dirr~ensior~nl 

flow along a curved wall, it is possiblr to srlect a coordinate syst,cm whose 

abscissa, x, is measured along the wnl;, the ordinate, y, being rncn,snrrrl at right angles 

to it, ld'ig. 3.9. Thus the curvilinear net consistsof curv~s whicl~ are parallel to tlrr wall 

Fig. 3.!). Two-tlinlennionnl honntlary layer along n cnrvetl wall 

and of stmight, lines perpendicular to tlhem. The corresponding velocity compo~~er~ts are 

denotcd by 7~ and v, respectively. The radius of curvature at positmion R: is derlotetl 

by R(x); it is positive for walls which are convex outwartls, ant1 nrgat,ivc whon the 

wall is concave. Tho appropriate form of t,he comp1et.e Navicr-Stokes equation~1 has 

been derived Ity W. Tollmien [lFi]. They are:

111. 1)crivntion of tlir eql~ntions oC motion of a rornprcssiblo viscous fluid 

r 1 

lhe stless components we 

R au at, 

-- - 0 

1, , !/ ar -I- I -7& - 

and t,hr vortiritg [see cqn. (4.5)] bcrorncs 

[I] tlr (:root, S.lt., nnd Mwur, T.: Non-cqnilibriuni t,hermotlynnniics. Nort,l-llollantl J'ubl. 

Cn.. 19ii2. 

(21 I'iil)pl, A,: \'orlesl;ngcr~ iiber tcchnisclw hleol~nnilc. \'ol. 5, 'J'cul)r~er. Lcipzig, 1922. 

1.11 Hol~f, I,.: Zilw I"1iisniglteilcri. Cont,ribul,io~l to: lln~~tll~ncl~ tlcr Pl~ysik, Vol. VII (H. Geiger 

nrlcl J(. St.llrrl, rd.), lirrlin, 1027. 

[4] Kestiti, J.: A COIII.~~ in t.l~crt~iody~~n~~~it'~. 

VoI. I, J3lniplell, 19(M. 

[T,] I!cs(.i~~, , J.: Il:(~rtlo tllcr~nodynanliqr~c drs ph6no1niwe~ ~rrdversihlcs. Ilnp. No. 66---7, Lab. 

d Aht.I~t-r~niqr~c, hlcutlo~\, \!IN;. 

I(;] I,:LIII~. 11. : IIytIrodyr~~~~~ii(.~. 6th nd., (hnl~ridgc. I!lT,7; also ])over, 1045. 

(7 1 I,ovr. A. 1':. I I.: 'l'lw ~~~nt,l~e~~~:~t,icnI 

t,Ilcory of c~lnsticity. 41.11 cd.. Cnnlhridgc Uliiv. Press, 

1052. 

[HI hlc.iurlrr. .I.. nr~l Tlrik. 11. C.: 7~lic~rn~otly~~nn~ilc tlrr irrc~rrsil~lrn I'rozessc. Co~iI,rihut.iorl to 

I~:III(II,IIcII (11.1. I'I~y~ilc. Vol. 111/2 (S. I+'liiggr, ctl.), Springrr, 1959, 11p. 413 -523. 

[!I] Nnvicr. 174.: hI6rnoirc srlr Ics lois tlu ~~iortvc~nont clos flr~itlcs. MCm, cle I'Aci~rl. tlc Sri. 6, 

380---410 (1827). 

1101 I'oisson, S.ll.: hl6111oiro sur les 6qrlntions g611~rnlon tlr I1(.qnilibrc rt drt riln~~vetiicmt, (Ips 

c~orpmolitlos 6l11ntiq11os ct tics Iluiclcs. $1. do I'lkolo ~lol,y(yc~h~~. 1.1. I3!1-- l8li (IH:!l). 

[Ill I'rugrr, CV.: J~~t,rotl~tc(.iot~ to IIICOIIILIIICR OF co~~t,inun. C:imJ RI. CO., I!)(;]. 

1121 l'rigoginn, I.: I

CHAPTER I V 

General properties of the Navier-Stakes equations 

Reforc pssing on to thc int,rgrat,inn of the Navitx-Stokrs cqunl.ions in the 

following ch:lpt,ers, it now sncms pcrtincnt, to discnss some of their general properties. 

In doing so wc shsll restrict ollrsclvcs to irrcornprcssiblc viscous fluitls. 

R. J)c.rivntion nf Reynolds's principle of sindnrity from the 

Nnvicr-Stokes cquntiorla 

TJr~til I,Ilc prrscnt day no gcncd a.nn.lyt,ic n~rthotl.s 11:tvr I~rcotnc availnblc for the 

intc-gmtion of t,hc Navirr-Sl.okcs~cl~~at~ions. I~urtl~crmorc, soluLion~ wl~inh arr vnlitl 

for all values of viscosity are Irnown only for some particular cases, c. g. for Poiseuille 

flow through a circular pipe, or for Couctte flow bet,ween two parallcl walls, 

onc of which is at mst,, the other moving along its own plane with a constant 

velocity (set: Fig. 1.1). For this reason tl~c problcm of calculatir~g the motion of 

a viscous fluitl was attaclrctl by first tackling limiting cases, that is, by solving prohlcrns 

for very large viscosit,ics, on the one hand, and for very small viscosities on 

the other, I)cmusr in t,ltis manner thc matllcmatical problem is considerably simpli- 

Actl. liowevrr, tho casr of modcratc viscosit,ics cannot be intmpolatd I~ct~ween 

thsc two rxtrornes 

1l:ven the limit,ing cases of vcry largc antl very small viscosities present great 

mat,hemntical tlifficulties so that rescarch into viscous fluid motion proceedetl 

to a largc cxtcnt. by experiment. In this conncxion t,hc Navier-Stnlrcs equations 

furnish vcry uscSul hints which point to a considerable rcduct,ion in the qnantity 

of cxperimcntal work required. It is ofhn possible to carry out. expcrimcnts on 

models, which means that in the experimental arrangement a geometrically similar 

model of tho aot11a1 body, but reduced in scale, is investigated in a wid tunnel, 

or other s~ritahlc arrangement. This always raises the question of the dynamic 

sim~ilnril?y of fluid mot.ions which is, evidcnt.ly, intimately connectmi with the question 

of how far rcsult.~ obta.inod wit,h motlcls can Jlc ntilizcd for the prediction of 

tho Id~aviour of the full-scale body. 

As alrr:dy oxpl~incrl in Chap. I, two fluitl nibtions are dynamically similar if, 

with gc?ornct,rit:ally sirnilnr k)oundn,ries, the velocity ficltls are geometrically similar, 

i. e., .if t1tc.v have gromctricnlly similar strcnrnlincs. 

This question was answcrrd in Chap. 1 for thr caw in which only inertia and 

visrntts fnrt~s t:~Itc pitrl. in the process. It was found there that for the two motions 

I 

the RrynoItIs IIII~I~JC~S mnst be rqunl (lirynolcls's pri~~~i~)lcb of sirnil:~rit.y). 'I'his 

roncllrsion was drawn by astimating thc forces in the strewn; wr now propose to 

tlctlr~ce it again directly from thc Navicr-Stolrcs equations. 

'rlrc Navicr-Stokes cqr1a1,ions cxpress tho condition of cqt~ilil>ri~~~~t, 

II:IIIICI~ 

that for cnc11 pa.rticle thrc is eqrrilibriurn betwccn hly forcrs (woigI~(.), SII~~;LC~ 

for~cs a~ttl jncrti:~. forcrs. 'J'hc sr~rfac!c forc:c:s co11sist. of prcwurr for(*c.s (IIO~~II:~~ Ii)r(:(:s) 

and frictiotl forrcs (sl1ea.r forccs). TZotljr forccs n.rc in~port,nt~t, only irl c::t.sc>s ~IIC:II 

tlicro is a free s~~rfncc or whcrl l,lto tlrtlsily clisl.ril~trl.ion is it~l~orno~c:~~c:o~ts. III 111,: 

(:xs~ of a hornogcnrol~s fl~titl in tltc :l.hscnc:t? oS n Srrc wtrfi~cc tltrrc is c:(l~tiIiI)t.i~tt~~ 

l~ctwc(~t tho wcigltb of'c:at:l~ p;~.rl.ivlc at111 it,u I~.yrlrwld,iv I )IIO.~IIII~:~ l'orc;~!, in tl~c S:I,IIIC 

w:~y 3.8 at rost. Ilc:nco in 1.11~ rnot,iorl of a I~o~nogcncons Illticl, ir~ thc nt~sct~c:c of:^ I'rrc? 

snrf:icc, body forces can 11r canrcllctl if prcssttrc is t,dt~n to IIIC:II~ tho (Iillcr~~ncc 

I~ctwccn that in n~ot,ion a.nd at rcst. In t h following arpttnc~tt, wc sl~nll rc.st,ric:t. our 

at,tc~~tion to cases for whic:h this assttn~ption is trtrc bccalisc they arc t,ltc: tnost imporhnt 

oncs in n.pplicntions. Tltr~s bltc Nnvicr-Stoltcs rqnations will now c:ortt,air~ 

only forces clue to pressure, viscosity, and inertia. 

Unclor thwc assumptions and ronvcntions ihc N~~vicr-Stolccs rqn:ttions for 

:In inromprcssiblc fluid, rcstrick:tl 10 stci~dy llow nncl in vcclor fnrttt, sinl1~lil:y to 

This clifl'crential equation must he indrpcnclent d the clloicc of the utli(.s for t.lrc 

various physical quantities, suc:h as velocity, prcssnrc, clc., which appe:lr in it. 

We now consider flows about two gcomctrically similar boclics of diKcrcnt lincar 

tlimcnsions in streams of different velocitics, c. g., flows past two spt~cms in wllictl 

the densitics and viscosities may also bc different. Wc shall invcstigatc the condition 

for dynamic similaritfly with the aid of tho Navier-Stokcs cquat,iot~s. Evidently, 

dynamic similarity will prevail if with a suitablc choice of the units of Icngf.h, 

tirnc, antl force, the Navicr-Stokes cqn. (4.1) is so tmnsforn~ctl that it, I)ccomcs 

identical for the two flows with geomctric:ally similar botir~tlarics. Now, it is [~ossiblc: 

to free oneself from the fortuitously selechcl units if clirncnsiorllcss q~~ntltitics n.rc 

introduced into cqn. (4.1). This is achievctl by snlcct.ing ccrt,:~in suitnhlc c:har:rt:taristic 

mxgnitudcs in thc flow as our ~rrtil,s, antl by refixring all otlwrs t,o t11c:nr. 

., .Ll~us c. g., thc frcc-slrcarn vcloci1,y anrltl tllc tlianlcl.cr of Ll~r sphrrc: cnlt IJC srl(:c:t.t:tl 

as the rcspcctivc 11ni1.s of vcloc:it.y and Icngth. 

1~r.t V, 1, and pl clcnotc tl~csc characteristic rcfcrcnco magt:itrltlrs. II' we now 

introtltlcc into thc Navicr-Sl.okcs oqn. (4.1) thc tlirrrrnsionlrss ri~tios

72 I\'. C:rncr:tl prol,e~lirs of thc Navier-Stokrs rquat.ions c. The Navicr-Stoltes equations intcrprctd as vortirit,~ t.rnnsport eqr~ntio~~s 73 

'J'his princildo was tlisc:ovrrctl I)y Osbornc Iteynoltls when he invrst,igxt.ctl fluitl 

~nolio~l thro~~gh ~'ipcs ""(1 is, t IlcrcSorc, ltnown ns the Reynolds priucipla o/ similnriby. 

'I'hc rli~nrt~sio~~lrss ratio 

e." = v z _ R 

Cc 

v 

(4.3) 

is cnllctl the Itcynoltls nrlni\)cr. JTere tho ratio of the dynamic viscosit,y 11, tm the 

clcr~si(,y e, tlcr~otctl by v = ,I./@, is the Itincmatic viscosity of the fl~~itl, int.rotl~lccd 

cn,rlicr. S~~niming np we can state that, flows nhout geon~ctrirally sirrlilnr bodies 

are tly~~n.miaally similar whcn the Rcynoltls numbers for the flows arc equal. 

Thus Itcynoltls's similarity principle has been deducctl once nlorc, t,his t,imc 

from t,he Navicr-Strokes cq~mtions, having I~ccn previously derived first from an 

c:st.irnnt,ion or Sorccs :in(] sccontlly from dimensional analysis. 

b. I.'ricliordenn flow as LL801u1io~~n" of the Navicr-Stokes equations 

It nay bo worth not,ing, prent.hrt.ically, that, the .solutions for incomprcssil~lc /riclionless 

flown may also bc regarded as exact solutionn of tho Nnvier-Stokes cquat,ionn, bcca~~sc in such 

rases tho frictional tcrnls vanish itlont.irnlly. In the case of incomprcssiblc, fricl.ion~csn flows tho 

vr1oc.il.y vector can he rrprrscntn?tl an tho grntlicr~t of a potcnt.ial: 

w = grad di , 

whrrr t.he potential @ RR~~S~IOS t,hc L:lplacc cquat.ion ,' 

V2@=0. 

We thn nl~o have grad (V2 @) - V2 (grad @) = 0, that is, V2 w = 0 . 

t See foot.nota on 1). 48. 

Tl~us the frictional terms in eqn. (4.1) vanish identically for potential flows, but generally 

speaking both boundary conditione (3.36) for the velocity cannot thcn be satisfied sin~ulta~~cot~sly. 

If the normal con~ponent must ccsmtmu prencribed vnlucs along n bouncinry, thn, in potential 

flow, l.l~o t,iw ont.inl oon~ponont i~ tl~oroby tlolorn~i~rnd no 1,Ilnt I,lm 110 dip oo11c1iI.io11 IVII~IIOI~ IN) 

sdislicd nt Lf~o mtnm l,i~no. Jd'or Lhis reason ow cnnnok regnril pohntinl ilowe a" pl~ysidl~ 

moaningfill nolutiona of 1.110 Nnvicr-Stokon cquntionn, bocnuno tlmy do not nnt.inry thc ~w~:scril,rd 

boundary conditions. l'hcro exist^, howcver, an important cxccption to tho prccccling ~tx~cmcnt 

which occurn whon tho solid wall is in motion and when this condition docs not apply. 

The shylest parlicular case is that of flow pant a rotating cylinder wl~cn the pofential ROIIItion 

does constit,utc a meaningfnl solution to the Navicr-Stokcs cquntions, as explainctf ill 

grcatcr detail on p. 80. The rcadcr may rcfor t,o two papers, one by G. 1InnieI [4] and onc by. 

J. Aclteret [I], for fnrt.ller details. 

The following sect,ions will be rest,ricted to the consideration of plane (two-din\rnsional) 

flows because for such caocs only is it possible t,o inclicato son~e gcncral properties of Lhc Navicr- 

Stokes equations, and, on Mia oClrcr hand, plane flows ronstituh by fir tho lnrgrst clans of 

prohlcrns of prartirnl i~nportance. 

c. The Navier-Stokes equations interpreted as vorticity transport equatinns 

In t,he case of two-dimensional nori-stcatly flow in the x, y-pla~lc t,l~o vcloc:it.y 

vector bcco~ncs 

and the system of rquat,ions (3.32) and (3.33) trnnsforms into 

whicli furnishes three equations for u, v, and p. 

We now introduce the vector of vorticity, curl W, wl~ich rctluccs to t.hc one 

component about the z-axis for two-dimensional flow: 

I~rict,ionless motions are irrotat.ionn1 so that curl cct = 0 in s~lcll cascs. Eli- 

minating pressure from eqns. (4.4a, b) we obtain 

or, in short,hand form 

This equation is referred to as the vorlicity transport, or transfer, equatzor~ It stalvs 

that the subskmtive variation of vorticity, which consists of tlw lord ant1 rot~vrcl,~ct~

74 

TV. Gencrol proprtic~ of tho Novior-Stoke8 eqrtnlionrr 

terms, is cqrtal t,o thc rate of clissipntion of vorticity t,l~rough friction. Eqn. (4.6), 

togclher with thc equation of contir~uit~y (4.4c), form n system of two equations 

for thc two v~lorit~y components ?I, and V. 

Finally, it, is possible to transform t.hcsc two equations with two unknowns 

ir1t.o one eq~iat.ior~ with one unknown by introtl~tcing the &,ream fnnc,tion t(r(x. y). 

Put,l.it~p 

we see t,llaL tho cor~tinnil.~~ equation is s:tt,islictl aut~orn:rt.irally. In ncltlit,ion l.lle vorLici1,y 

from eqn. (4.5) I~t:conrcs 

w=- +v2y,t (4.9) 

C. The Nnvirr-Stoke3 rqwl.ion8 intcrprrtccl IW vorlicit,~ Lrnnsport rqunt ion3 76 

In this form the vorticit,y trnnspo~t cquntion contains only one unknown, 11). 'C11c 

left-hard side of cqn. (4.10) contains, as was the casc with the Navicr-Stokcs 

equations, the inertin term#, whcrcas kho right-hhrd siclo cont,ains tho frictional 

tcrms. It is a fourth-order partid dilfercntial cquatiorl in the strcarn functior~ 7,' 

Its solution in gcneral terms is, agnin, vcry clifficult, owing ho its bcing non-linr;rr. 

V. G1. Jenson 151 found a solution to thc vorticity transport cquatio~~ (4.10) 

for the case of a sphere by numcricnl integration. The resulting pattrcrns of s(.rcainlines 

for different Reynolds numbers arc seen plotted in Fig. 4.1 which also contnirls 

clingrams of the distribution of vorticity in thr flow fioltl. Tltc srnnllod Itc:ynol(l~ 

number included, R = 5 in Figa. 4.1s anti 4.1~1, corresponds to thc casc w11c11 thc 

viscous forces by far outweigh the inertia forccs and the resulting flow can bc described 

nR crecping motion, Scc. IVd ar~tl(~11nptr.r VI. III this casc the wholc flow fit-It1 

is rotatior~al and tho pattcrns of strcnmlirlcs forward and aft are nenrly identicnl. As 

thc ltcynolds number is incrcnscd the sphere dcvclops on its rcar a scparatcd region 

with back-flow and the intensity of vorticity is progressively more concentrated near 

the downstream portion of the sphere, wherca.. in thc forward portion t11c flow becomes 

nearly irrot,ationnl. The flow patterns undcr consideration which have been 

deduced from the Nnvier-Stokes equation, allow us to rccognizc thc chnractcristic 

changes whieh take place in the stream as the Reynolds number is made to increase, 

even if at t,he highest Reynolds number rcachcd, R =. 40 in Figa. 4.1 c and 4.1 f, the 

boundary layer pattern has not yet had a chance to develop fully. 

Very extensive experimental inve~tigations of the wake behind a circular cylinder 

in the range of Reynolds numbers 5 < R < 40 nre described in two papers by M. Coutanceau 

nnd R. Bouard [lc, ld] who covered both steady and unsteady flows. 

The development of very efficient clcctmnic computers in modern times has made 

it possible to solve the Navicr-Stokes equations for flow past geometrically simple 

bodies by purely numerical methods. In order to do this, the differentid equations 

are replnccd by difference equations. The numerical techniques uscd for this purpose 

will be explained in Sec. 1x1. Without discussing this matter here in any ilcpt,h, we 

quote one interesting result. Figure 4.2 shows the flow past a rectangular plate placed 

at right anglcs to the stream calculated by J. E. Fromrn and 1;". H. Harlow 131. At 

the back of the plate there forms a vort,cx street similar to that bchintl a circular 

cylinder shown in Figs. 1.6 and 2.7. Figure 4.2ashowsan expcrirnentnlly detcrrnined 

pnttmn of slreamlincs, where*! Fig. 4.2b rcprcscnts thc calculatcd ficld, both for n 

Reynolds number Vdlv = 6000. Thc agrconcnt bctwccn tjhc two pttcrr~s is rcrnitrkably 

good, in spitc of thc fact that in this rnngo of 1tcyr1oldanrtml)crsl.l1c flow aequircs 

an oscillatmy character, Fig. 1.6. Tltc earliest attempts t,o obtain such nl~rncrical 

solut,iol~s t,o t,lm N:~virr-St,okw cyt~:ltions can hc t.rnt:c~tl 1.0 A. 'l'ltotn 101 ~111, ~,crformed 

such calculations for a circular cylindcr at the low Rcyr~olds nurnl~crs R -- 10 

to 20. I,ater, the calculations wcre carried to R = 100 [2]. As the R.cynoltls nurnbrr 

increases, the degree of difficulty of such numcrical int.egrat,ions increases st,ncply. 

In this conncxtion it is worth consulLing the comprchensivc sr~mmnry by A. l'horn 

and C. ,I. Apclt [7], as well the work of C. J. Apclt [I n] and I). N. tlc (2. Allen and 

R. V. Southwcll [I I)) nnd of If. B. Kcllcr nnd 11. Takami L5nl.

76 1V. Genrrnl proprrtir~ of thr Nnvirr-Stokm rqustio~~n r. 'I'llr litr~iting cnw of wry small visoo~~s hrrr~ 7 7 

Fig. 4.2n 

Fig. 4.2. T'nt,tcrn of st,rcnmlinra 1)cIiiutl 

a rect.nr~gulnr flat. plnto (lf/rt = 1.6) plnctrtl 

nt. right, nnglr t.o th flow at. n Itry~~olds 

nu~~iI)rr R = I' ![/I) = 6000, nfkr J. I m). 111 this case the [lroccss of m;~t,h(~lrl:~l,ic::~I 

sin~plificntion of the tliffcrcntial cqn. (4.10) requircs a consitlcrablc amount of rart:. 

It is not pcrlnissiblc simply t,o olnit (,lie viscous tcrrns, i. c., t,lle right,-h:~ntl side of 

(:(in. (4.10). This woultl rntlnco the ortlcr of t,llc oqu:~t,iot~ from four to two, :LII(I t.110 

solution of the simplifictl cquat.iou could not be made to salisfy the full bountl:lry 

condit.iol~s of t.Iw originn.1 cquat.ion. The problem wl~icll was ontlincd in tllc prccctling 

scr~t~cnccs belongs essentially to tl~e rcdrn of hou~cdnr?/-kr!/er lheory. We now proposc: 

to tliscuss briefly the genrral st.at,r~nc~lt,s which can t)c made about the solutions 

of the Nnvicr-St,oltes cqnnl,ions for t,ho special mse of small viscous forccs as cornpared 

wit.ll t,hc incrt,in forccs, thnt is in t,11r limiting case of very Inrgr: 1tt:ynolds 

n11m1)crs. 

The following analogy rnny scrvc to illust,rate tl16 c:llsr:~cter of the solut.ions 

of t,hc Navier-Stokcs cquat~ions for the litnit.ing c;~se or vt:ry small viscosil,y, i. c., 

of vrry small friction terms, as compared with t,llc inertia terms Tl~e tcrnprrat1lrcb 

distribution O(r, y) aboul, a hot, I)otly in n fluid strrarn is clcscrihtl 1)y the following 

tlilTrr'rrrntial rqrlation, Chap. XI 1.

Ilcre v, c, :tntl k tlrnol,c 1.I1t: tlrnsity, sl>rc:ilic Itc::rt, :mi contlucl,ivit.y of Ihc lluitl 

rc!spwlivrly; 0 in tltc! tlilli~rcnc:c? I)ct,wco~~ Ihc loonl t,t:llll)onrt,r~rc: ant1 (,hat at :t vory 

1;rrgc tlisI.:~t~(:c fron~ t11c I)otly, wl~orc: IJlc: l.c:tnpcr:~l,~~rc:, 7', is c:onsl,ant ant1 cq~~:rl lso 

'/, i. c . 0 - - '/ - '/',,,. 'I'llt: vcloc(ity lic:ltl w(z, y) :rt1!1 ~(z, y) in oqn. (4.12) is ;~.ssurnctl 

to I)c known. 'L'hc t,ernpnrat~~rc: distribution on the I~ountlarics of Lhc botly tlcfinetl 

b?~ '/I,, 3 7', is prrsc:ril)ctl nrttl in the sirnplcst cnsc it is constant wit11 respect Lo 

sl)nct: and t.imc 1)111., gcncrnlly spcalting, it varies wit11 both. I'rom the pl~ysioal poinl. 

of view cqn. (4.12) roprosents the 11c:rt 1):~lanc:c Ihr an clcn~cnt,ary volut~~e. 'l'hc Icfh- 

Ilantl sitlc represents t.11~ qu:mt,it,y of heat, c:xcl~:~~~gotl I)y convcc:tiorl, whereas the 

rigl~t-ll:rnd side is the ~~~;~n(.it.v of 11r:lt t:xt:I~:~n~cd 11y con(I~t(:tion. T11c frit:l.ion:~l hcatl 

gcneratcd in tile fluid is ncglcetctl. Tf 7',, > T,, tho prol)lom is that of detcrrnining 

1.ltc tcmperatl~rc field around a hot body which is cooled. By inspection it is scrn 

that cqn. (4.12) is of the same form as eqn. (4.6) for the vorticity w. In fact thcy 

hccomc itlentiral if the vorticity is replaced by thc tcmpcraturc tliffercncc and t.hc 

kincmatic viscosity v by t,hc ratio kip c known as thc thcrmal diffusivity. 'l'hc bountlary 

conclit,ion 0 - 0 at a largc distance from thc body corresponds to the condition 

tr, = 0 for the undisturbed p,nrnllcl stmam also at a largc distance from the body. 

llcncc we may expect that thc solutions of the two equations, i. e. the dislribntion 

of vorticity antl that of tcmpcraturc around the body will be similar in chnrnctcr. 

Now, tllc tcmpcratlrre dist,riI)ution around the body may be pcrccivcd intnitivrly, 

to a ccrtnin cxlcnt. In the limiting ca.sc of zero velocity (fluid at rosl) the inflncncc of 

tilo I~eatccl 11otly will extend ~~niforrnly on all ~itics. With very small velocit,ics tho 

fluid arountl the hody will still he affectcd by it in all directions. With incrcnsing 

velocity of flow, howcvcr, it is clcarly seen that the rcgion affected by the higher 

tempcreturc of the body shrinks more and more into a narrow zone in the immetlint,c 

vicini1.y of the body, antl into a tail of hcatcd fluid bchind it., 1Pig. 4.3. 

. - 

-.__ -- - 

--__ 

Fig. 4.3. Annlogy betfween trnlperntuw 

and vorticity di~tribution ill tho neigh- 

bol~rhd of R body plnml in a strerrrn 

of fl\lid 

a), b) IAndCq of rrgion or iw.rrhsrd trmprrsture 

n) for ~rnnll vclucitlrs 

---------__..______ 

Ir) fur Inrge vrlucitirn uf flow 

'rllc so111l.ion of eqn. (4.1 2) ni~rst-, a.s mcnt,ionkd, be of a chn.ra.cter similar t.o that 

for vorticit,y. At snlall velocities (viscous forces hrge compared with inertia forces) 

t.hcro is vorticity in ihc whole region of llow around the body. On the other hand 

for' largc vcloeitics (V~SCOIIS forccs smnll compnrctl wibh ir~ctl~in forccs), we may 

rx-prct, i field of flnw in which vorticity is confined to a small Iaycr along the surfacc 

of the I)otly antl in a wake behind thc boily, whereas thc rest of the fcld of flow 

c. 'l'ho limiting caw nf vory ~nlnll V~RCOIIR li~rtm 79 

remains, practically speaking, free from vortioity (scc Vig. 4.1). 11, is, I.l~ercforc, 

to be cxpected that in the limiting case of very small viscons forces, i. e. nt 1;rrgc: 

Itcynolds numbers, the solutions of the Nevicr-S(.okcs cq~~:~t~ions arc SO ('otlsti(.~tt.rtl 

as 1.0 permit a suldivision of the ficld of flow inl,o an cxtcrn:~I rrgion wlti(*ll is fr~o 

from vorticity, and a thin layer near the I)otly togcthcr with a wakc I~(:llit~tl it.. I11 

1.11~: first, region tho Ilow mny Im oxpnctctl 1.0 sntisfy OIO ctl~~~rtions of I'ri(:(,iot~ltw 

flow, the potcr~t~ial llow theory bcing uscd for ih cvalnation, whcrcas in tllc sc~-otr(l 

region vorticity is inherent, and, thcrcfore, the Navicr-Stoltcs cq~iations m~~st. hn 

uscd for its evaluation. Viscous forccs are importm~t, i. c. of thc santc ortlcr 91' 

mngnitt~tlc n.9 inertia forces, only in tl~c scc:ontl region known :is thc bou~tdrtr?y Irr?yrr. 

This concept of a boundary layer was introduced into the scicncc of fluid mechanics 

by L. Prantitl at the beginning of thc present ccntury: it has proved t,o he vcry 

fruitful. The subdivision of the field of flow into tho frictionless oxtcrnnl Ilow iwtl 

the cssentinlly viscous boundary-laycr flow pcrmitkd thc reduction of the rnnt,llcmatical 

difficnlties inllorent in the Nnvicr-Stokes cqnatior~s to snch an extent, that 

it, lmmne possible to integrate them for a large numbcr of cascs. The tloscript.ion 

of t,l~csc methods of integration forms t.hc subject of the boundary-laycr thnory prcscntctl 

in the following chapters. 

From a nt~mcrical analysis of the available soluteions of the Navicr-Stokc~s 

cq~~ations it is also poasiblo to show directly that in tho limiting case of vcry lnrgc 

Reynolds numbers there exists a thin boundary laycr in which the influcncc of viscosit,y 

is conccntratcd. We shall rcvcrt to this topic in Chnp. V. 

The previously discussed limiting case in which viscous forcrs heavily outweigh 

inertia force3 ((creeping motion, i. e., very small Reynolds number) results in a considerable 

mathematical simplification of the Navier-Stokes equations. By omitting 

the inertia terms their order is not rcduced, but they become linear. 'J'hc second 

limiting case, when inertia forces outweigh viscous forces (boundary layer, i e. very 

large Reynolds numbem) present8 greatrr mathematical difficulties than creeping 

motion For, if we simply substitute v = 0 in the Navior-Stokcs equations (3.32), 

or in the stream-function equation (4.10), wc thereby suppress the derivatives of Lltr 

highest order and with the simpler equation of lowcr order it is impossible to satisfy 

sirr~ultancously all boundary conditions of the cornpleto tliKrrcntial equnt~ous. 

However, this does not signify that the solutions of sucll an equation, sin~ldificd by 

t.he elimination of viscous terms, lose their physical meaning. Moreover, it is possil~lc 

to prove that this solut,ion agrees with the &mplete solutionof the full ~ svic~:-~toke~ 

cq11nt.ions nlmost. everywhere in t,he limiting case of vrry large Reynoltls nrtmb~rs. 

Tho exception is confincd to n thin lnycr near the wall - the bountlnry In.yc;r. l'h~ls, 

thr complete nolution of t.hc Nnvicr-Stmkcs cqtralions c:nn I)c tl~orrgl~l of nrc t:onrcisI,ing 

of two sointions, thc so-cnllctl "outcr" solution which is ohtninctl with the nid of 

Eulor's equations of motion, and a so-callcd "inner" or bonndnry-1n.yc.r solnt.ion 

which is valid only in the thin layer adjacent to the wall. The "inner" solut,ion 

satisfies thc so-called houndary-layer eqmtions which are dctlncctl from tho Navicr- 

Stokes equations by ~oortlinat~e stretching nnti pwqsagc to tho limit R + m, n.s will 

be shown in Chnp. VII. The outer and inncr solutions must he malchcd t,o ench other 

by exploiting the condition that thcrc must exist nn overlapping rrgion in which 

bbth s&tions are valid.

80 

I\'. Crnrtnl proprrtirs of the Nnvier Stnlzcs rqr~ntions 

f. Mntlwn~nticnl illctntrntion of tltc process of goirlg to the limit R 4 oo t 

Let IU rotinitlrr t.lie tlnmpr(l vihrntious of n point-mass tlmrrihctl hy t,hc tlifirrnlinl cr(~~nt,ion 

Herc irr donolrn the vihrnling mnsn, c (.lie spring c:or~ntnnt., k I.l)c. tlnniping f:wto~.. r t.I~t: Irng(.ll 

roordinnt.~ nlcrlmcrccl from t,lw jiosit,ion of ril~~ilibrin~n. nnrl I t.lw ti~nr. '1'11r initial ron(lilions arc 

ILRRIIII~~C~ t,o be 

r-O at 1-0. (4.14) 

In ruinlogy with (.tie Nnvier-Stnkon eqr~ntions for t.lie cnse \rhcn thr lrinrnintic visronity, I*, is very 

sninll, we condcr hcrr tlw limitsing rnsc of vrry smnll mnss nr, hrrn~~nr this loo rnrlnrs 1.11~ lerm 

of thc Iiigllest ordrr in cqn. (4.13) to brromt: very small. 

l'lir complrtr solrition of cqn. (4.13) s~~hjrrt tn the initinl rondit.ion (4.14) hns the form 

x = A {exp ( I ) 

-- cxp (-k 11iri)): irr -t 0, (4.15) 

where A in n frre constnnt \vl~osc v11111e rnn hr (Irtrrn~inr(l with rrfcrct~cr to 11 srron(l initinl contlit,ion. 

If we put, in - 0 in eqn. (4.13), we nrc lrtl to t.lw simplified rqr~t~t.ion 

wiiirli is of first orckr, nnrl whose solrclion is 

d x 

k- f e:r =0, 

tlt 

TO(/) = A rxp ( - ctlk). (4.17) 

This solrrtion is idcnt,irnl wit,h the first term of the aomplctc solr~tion dur to the feliritous choice 

of t.lw ndjuntnhle co~~utnnt,. However. this solution rnnnot he ~nntlc t.o satisfy t,lie init,iol coridit.ion 

(4.14); it thus reprc~entR a eolut,ion for 1n.rge values of thr time, t ( Lco~~l~r" so111t.ion). 'l'hr8oIntion 

for smnll vnlrtcs of time ("inner" solt~tion) snlisfies n.noLhrr diflerentinl equation \rliirlt can also 

be dnrivwl from eqn. (4.13). 111 order to nchicw this, t.hr in~lopcntlrnt vnrinl~lr: t is "stret.cllcd" 

in t,hnt a now "inner" vnrinhle 

iv int,roduccd. 111 this manner, cqn. (4.13) is Ira~~sformetl t,o 

wliicl~ p)vrrnn ll~r "innrr" ~ol~~t~ion. WII: soI111io11 now is 

I 

1.1 (I*) = A, rxp ( - kt*) 1 A,. 

t* = t/m (4.18) 

t 1 nln i~~dcl~f.cvl 1.0 Profrssor Klnns Grrnten for (I1c rosisrtl vrrnion of t,liin section. 

* 1,. I'r~l.wlt.l, Annrhnr~liche 1t11t1 ~~wtzlirhc hlnt,henintik. I,rrt,ures drlivrrrd nt. (;oet,t.ingrn Univrrnil.y 

ill t.hr \Yint.rr-Srmcnt,rr of 1!):11/:12. 

f. Mnt,lremnticnl illust,ration of the procens of going t.o the limit R -t m 8 1 

In sl>il.c of thn sirnl~lificatiorl, Il~e diflixentinl rqrrnt,ion (4.20) is onr of noco~irl ~lcgrrr: it c.nn 1)r 

mntlc? t.o e:rt.idy LIIO initial rondit,iot~ (4.14) hy t.1~ clmirr 

P 7 

Il~c vnll~c of const.nnt. Az folloaw froin t.110 tnnl.ol~i~ip; to t11c "o~~lrr" nnl~~t,io~~, rt111. (4.17). 111 1111 

ovrrlnpping rnngr, tht. is for ~noclcr:i(c vnlt~rn of tiinr, t.hc nol~~lionn in nqnn. (4.17) i ~nd (4.21) 

nit~nl. ngrr:r. 'l'lit~s \VO tnr~sl, II~v~~ 

or, in wortln: 'l'h "or~t,nr" limit of t.lic "innrr" solntion IR~, 

"outer" solnlion. Condition (4.23) Icntls nt, oncc to 

he C~IIRI lo the "innt:r" lin~il 01 thr 

nntl no to the innrr solrltion 

.rr(t*) = A (1 - cxp (-- kt*)}. (4.25) 

'I'l~c snme form rnn be obt.ninrtl from Lllr ron~plclr soI11t icm fro111 rqn. (4.15) by r*x~~:~n(ling (It(. 

tirut t,rrni for small vnlws of I nnd rctni~~i~lg the GrnL tern1 only, Ihnt is by p~t,ling 

7'11~ t\vo iic!ution~, tlic onter so111tion from eqn. (4.17) nrid Ll~c inner ~olution from rqn (4.26). 

togct,l~er form the m!!iplcte solution on condition tlint cnrh is 118rtl ill its projwr I.III~~C of vnlidity. 

ht finite 1, cqu. (4.15) tends (c the outer solut~ion for nt + 0. whcrens at constant t* eqn. (4.15) 

tol~tl~ t,o the inner nolution. 'l'lle pnrtinl solut,io~is give I llc cornplrtc, cotnposit.~ nol~~t.ion which is 

vnlitl in the cnl,ire rnnge of vni~cs of t i)y ridding thrnl togrtl~rr, rcmemhrring that Ihr ronlmon 

tcr111 from eqn (4.23) ~nnst. he included only once, tlint, in sul~t,r:rclrrl from the R I nr.rorrling to 

tho prcsrription 

x(1) = ~"(1) + rt(t*) - Iim x: (l*) = TO (t) I rt(t*) -- lim xn(1). (4.27) 

I* -. m 1 -. tJ 

A graphical roprencntation of the complete .soIr~t,ion from eqn. (4.15) i~ nhown in Fig. 4.4 

for the cnse when A > 0. Curve (a) corresponds to t,l~e outer solution (4.17). Cnrvcs (I)), (r) nntl 

(d) represent solutions of the cotnplct,~ tlifirrntinl equation (4.13) \vitlt vr clrrrmsir~g from (h) 

to ((I). 

If wc now cor~ipnrc this rxamplo with thr Navier-St,okcs cq~mt,ions, we COIICIU~O 

l.liat. t.Iie 

r.o~nplctc cqrt:~t ion (4.13) is nn:~Iogonn 1.0 thr Nnvicr-Stokes cq11a1.ions for n vi~oonn Iluicl. wlwrms 

Ilw sirnpIiliv(1 tqwtt,io~~ (4.1(;), t!orrcs~~on~ls In lh~lcr's rquntiom for nu i(lral ll~tid. WIP i11iti:11 

Fig. 4.4. SOIIII~OIIS of thr viOr:itio~~ rq~~:iti(~n 

( t . I:!). (a) Sol111io11 of tl~e sin~plifitd rqn:~th~ 

(s!. 14). 111 -- 0: (11). (c), ((I) rcprrsent so111tions 

of 111r vo111111rt(: tlil'li~rcnt in1 cquntion (4.13) 

\I it11 vnriol~s V:IIIIIY of i11. JVhcn irl is wry 

s~nnll. soI111io11 ((I) arq~~irrn I~nun(l:tr,y- layer 

141:1rnrtrr

conclil.ion (4.14) plays n part wl~ich ia ein~ilnr tn 1.11~ no-slip condit.ion of n red fl~~id. 'Chc latter 

cnn be saLislic4 by Ihc solutions of 1.11~ Nnvicr-Stokcs eq~~etions I~ut not by those of Euler'a 

cquntio~~s. 'I'lw slowly-varying solnt,ion is trnnlngrr~~s In) tile frictionlcsn solution (ptrntinl flow) 

whicl~ f:rils to satisfy the no-slip contlil.ion. 7'11~ f~rst-vnrying solntion rcprcscnts Lhe counlr?rpart 

of tho bonndnry-lnycr ~oIuLion whicl~ ia delcrn~incd by t.ho prcscnm of viscosit.y; it clin'cn fron~ 

zero only in n narrow zone near tho wall (boundary Inyer). It is to bo nokd that the second 

bonndnry condition (no slip at tho wall) can only be sal,inficd if this bountlnry-layer solution is 

a.dclwl, t,l111s mnking tho whole sol~~bion phy~icnlly red. 

This simple rxarnplc cxl~ibit,s thc sarnc matbcmstical Aaturcs M t.l~osc ch?usscd in 1,110 

prcrcding cl~u.plcr. It is, nrrrnrly, not pcrn~iasil~lc si~n rly In onlit tl~c viscou~ tern18 ill tlw 

Nnvicr-Stokm equation, wlmn performing the process or going over to t.he limit Tor very small 

viscosit.y (vrry I;irge llcynolrln n~~rnbrr). This wn only bc: clonc: in tile intrgrnl solnl.ion itxlr. 

We sha.ll tlcnionstratc latm in grcatcr c1cl:iil tht if, is not t1cc:c:ssary to rctain lhc 

Cull Navirr-St.olrcs eqnnt,ions for the process of finding the limit for R -+m. For 

lhc salte of n~athrmatical simplification il will provc possible to omit certain t.rrni~ 

in it, pnrticnlnrly certain small viscous tcrnls. It is, however, important to note that, 

not all viscous trrma can bc ncglrctrtl. ns this woril~l depress tllc ordrr of tho Navier- 

Stolrrs rqnntions 

[l] Ackcrct, J.: Ubcr cxnkte J5sungen dcr Stokes-Navi~r- Glcicl~ungen inkomprrmihler 1Pliiimigkciton 

bci vcriin~lcrbn (:rr~~r,l~c~li~~~rtngc?~~. 

%1\;\11' 3, 259--271 (1952). 

[In] Apeelt. C. ,i.: 'l'hc ~trnrly Ilrtiv of a viscous flnid pat n circulnr cylindcr at Reynolds numbers 

40 and 44. Ikitisl~ AltC ItM 3175 (IWil). 

(Lh] Allen, D.N. 1)c G., m~ci So~~t.hwcll, 1t.V.: ltclaxation methods npplicd to deternline the 

mot,ion, in t,wo di~nm~iona, of R viscons flnid pnat n Bxetl cylinder. Q~mrt. J. Mecb. Appl. 

MnLIl. 8, 12!)-145 (1!)55). 

[lo] Coutnnccau, M., nnd Uo~lnrd, R.: 15xpcrirncntnl dckr~ninnt.ion of t.lw main fcnt,nrrs of the 

vinrn~~n flow in ...~. tho wakn of R circular cvlinder in 11uifor111 tra~~slation. Par1 I. Stendy now. 

~ 

JFM 78, 231 -256 (1977j. 

[Id] (huta~~cenu. M., ~1r1 Ih~nrd, It.: RxprritnrnLal detcnnination of thc mnin fcnCuren of thc 

visco~~s flr~w in tbe wake of R circulilr cylinder in uniform trnnslation. Part 2. Unsbndy flow. 

.11W 79, 257- 272 (15377). 

[2] Ihmnis, S.C.K.. and GRII-ZII Chang: Nn~ncrical soI~~t,it)ns for stcarly flo~ past x circ~~lnr 

cylintlcr nt, I

84 

V. Exact. solul.ions of t.he Nnvirr-Stokrs equn1,ions 

1. Parallel flow tltrough a straight channel nnd Col~eltc flow. A very simple 

solut,ion of eqnat,ion (5.2) is obtained for the case of stm~tly flow ir~ ;I channel with 

t,wo parallel flat walls, Vig. 5.1. 1,et. t,hc tlist.ancc bctwcen the :valls be denoted by 2 h, 

so t.liat cqtl. (5.2) can Iw writ9tcn 

Vig. 5. I . i'nrellnl flow with p:~rnholir 

vclocit.y disl.ril~~~t.i~tl 

Another simple solution of eqn (5.3) is obtained for tho so-mlled Couette 

flow 1)ctween two pnrnllcl flat walls, one of which is at rest, the other moving in 

its own plan? wit11 ;I velocity rJ, Fig. 6 2 With the boundary conditions 

we obt.n.in the solution 

y=O: u=O; y=h: u=U 

\vhic-11 is shown in Fig. 5.2. Tn parLicular for a vanishing pressure gratlicrlt we have 

'I'his p:l.rt.inllar case is known ns simple Couct,tc flow, or si~nplc? sllcar Ilow. The 

gcrlrral casc of (:ouct,tc flow is i supcrposit,io~~ of this simple casc over the flow 

between two fht, wn.IIs. 'I'l~c sllapc of t.I~c vclocit.y~profilc is tlctcm~incd by t,hc dirnoltsionlrss 

pressure p~.:~tlicr~t, I 

For 7' > 0, i. e., for n prrssurc tlccreasing in t01c tiircctior~ of mot,ion, the velocity 

is posit.ivc over the whole witll,ll of tl~o channel. For ncget.ive values of P the velocity 

over a porkion of the cl~:~nncl width can I,ccomc ncgntivo, t,l~at is, back-//OW may occur 

near the wall wllich is at rest, and it is seen from Fig. 6.2 that t,his Irappens ~llcn 

I' < - 1. Tn this case the dragging act.ion of the fast,cr layers exertetl on fluitl lmrt,iclcs 

in the ncigl~bourl~ootl of the wall is insufficient to ovcrcomc thr, influence of 

t,hc adverse pressure gradient. This type of Coucttc flow with a Iwcssurc gratlicnt 

has some importance in the hgtlrodynamic theory of luhricatio~t. 'J'hc flow in tltc 

narrow clcnrar~ce bctwcen journal and l~enrirlg is, by a.nd large, identical with Couct,tc 

flow with a pressure gradient (c/. Scc. VTc). 

2. The IIngen-Poiseuille theory or flow tl~rorrgh a pipe. Tl~e flow t.llrougl~ a 

stmight, tl~bc of ciroular nross-scc:l.ion is the casc with rol.:~t~ionn.l sylnlnct.ry wllic*h 

rorr(~+~)onds to tho prcc:otli~~g casc: of lawn-tli~~lcnsior~:~l flow t.llrol~gl~ :I oll:~.r~r~t!l. 1,~t. 

tho z-axis be solcct.c:tl along thn axis oL' t,l~n pipc, Pig. 1.2, :md Ict, y t1onot.o 1,llo I.IL,I~:LI 

eoortlinate mcasurcd from the axis outwards. The vclocit,y componcnt.~ in t.ltc 

tnr~gt?nl,inI and radial directions arc zero; the velocity component pnrallcl to the 

axis, denoted by 11, depcntls on y alone, a,nd the pressure is const#ant in every crosssection. 

Of the thrcc Navicr-Stolres equations in cylint1ricn.l coordinates, cqns. (3.:16), 

only the one for the axial tlircct,ion remains, nntl it, simplifies to 

tlw boundary condition being u = 0 for y = R. The solution of cqn. (5.6) gives the 

velocity distribution 

1 dp 

IL (y) = - - - - - (R2-- y2) 

411 dz

86 V. 1':xact solul ions of tlir N:ivirr-Shkm rqnn.l.ir~~is n. l':irn1Ic~~ flow 87 

where - -tlp/tlz : - (p, pa)/l! = CWSL is the pressure gr:dic~rt,, t,o be rognrclctl as 

given. Solut,ion (5.7), which was obtdincd hcre as an oxact solution of thc Navicr- 

Stokns equations, agrees with thc solution in cqn. (1.10) whiclr was oht,;~iricd in on 

c-l(:~ncnt,ary w:~y. 'l'lic vchcity ovor tlic cross-section is tlist,rihutctl in the form of a 

pnr:tI~oloicl of rovolut.ion. 'I'llc mnxirnurn ~c:loc:il.~ on t.hn axis is 

1 lie nirali vch.il.y 17. -- v,,, that is, 

r . 

:~ncl t.ho vol~r~nc. mtc of flow I~ccorncs 

'I'ho larninar flow elc~scribcrtl 11y tho al~ovc? solution occurs in practice only ; l long ~ as 

t.he Itcynolcls nurnhcr R .-- .li d/v ((1 =- pip cliamc?tc:r) has a vnhc which is less than 

t.hc so-callccl critic:d Jtcynoltls number, in spite of the f:~ct that t h above formulae 

coristitutc :m cx:tnt solution of thc Nnvic:r-Stokes equations for arbitrary values of 

ctp/cln:, R, :~rrtl p, or llcnc:c, of IT, R, and fr. Acsorcling Lo cxpcrirncnk 

('p) = Rcri, = 2300 

Cril 

approximatdy. For R > R,,,, the flow pnttcrn is entirely tliPFcrenl, and bccomca lur- 

hulc~tt. Wc shall disc~~ss 

this type of flow in greater detail in Chap. XX. 

'I'hc rcl:rtion bchwecn thc pressure graclicnt and thc mean velocity of flow is 

~~or~nally rcI~roscntcd in cnginccring applicat,ions by introducing a resistanc~ coe//icient 

o/ pipr /low, l. 'l'his coc:fticicnt is tlofincd by setting tho prcwlrc gr:ulicnt proportional 

t,o the clyn:~.rnic Iicatl, i. n., 1.0 tlic square of thc moan vcloc:itsy of flow, aocorcling to tho 

lr~trotlr~cing the cxprcssion for dp/dz from cqn. (5.9) we ol)tain 

with 

- - - 

t This qndrntic Iaw which nnmrnra dp/dz - 12' fitn t.urbulcnt flow vcry well. It is rebind 

fin Irmminnr flow, although in Lhnt rnngo dp/dz - 12. Thus lor Innliner flow A was to bo 

a mnnbnt. 

Fig. 5.3. Immitinr flqw t,l~rongh pip; 

resi~t.nnrc cocfficicnt, A, plottcd ngninst 

Itcynold~ number (rncasured by Hngcn), 

from Prnntltl-l'ictjrns 

R = (id 

1' 

Jlcro R tlcnotes Lhc Roynolds numlm calculated for thc pipe dinmcbr and moan 

vclocity of flow. The laminar equation for prewuro loss in pipcs, cqn. (5.1 I), is 

in cxcellcnt ~grccmcrlt with experimental rcsvlts for thc laminar range, as SCCII 

from Fig. 6.3 which rcj~rotfuccs cxpcrimentnl pinh moas~~rcd by (2. 1I:i.gc:n [I()]. 

From this it is possible to infer that the Ilagcn-I'oiscuillc parabolic vclocity 

distribution represents a solution of the Navicr-Stokes equations which is in agrcemcnt 

with expcrimental results [22]. It is also possiblc to indicatc an exact solution 

of the Navier-Stokes equations for thc case of a pipe with a circular annular crosssection 

1201. The problem of laminar and turbulent flow through pipcs with cxcentric 

annular crow-scctions was discusscd theoretically in ref. [38] which also contains 

experimental results. 

3. The flow between two concentric rotating cylinders. A f~~rtlicr cxamplc wliieli 

leads to a simple exact solution of the Navier-Stokes equations is affortlcd by the 

flow between two concentric rotating cylinders, both of which move at tlifTvrcnt 

but steady rotational spccds. Wc shall dcnotc thc inner and outcr radii by r,, and r2 

rrspcctivcly, and similarly, the two angular velocities by w,, and w,. Thc Navicr- 

Stokes equations (3.36) for plane polar coordinates redwe to 

with dcnobing the circi~nrfcrcntial 

velocity. The lmundary contlit,ions arc: u - rl r11~ 

for r = rl and u = r, (0, for r = r2. The solution of (5.14) which satisfies tticsc rr- 

quirements is

88 V. Rxnct uolutions of tltc Nnvinr-Stokcs rqr~ntions n. Pnrallrl flow H!) 

x 1-1.2 

(1) A,-.--- (~nner rotntinp, outcr at ITS^). (5.lGn) 

11, I -x2 x 

TI is ~tol.t.worIhy IItnt, t,ltr vclocit.,y vnrirs strtmgl~r wil.l~ t,hc rnt,io x - rl/rz of t,hc I~WO 

radii ill Cnsr I, whrrras for (hsr I1 it is almost intlcprnclc~~t of' it. MThcn x = rl/rz + 1 , 

Iwt.l~ c~tscvi tclttl t,o the linrar vrloeity tlistril)l~t.ion of (!oucstt.(, flo~, as it, ocrurrcd 

1)cI.wc~vn l,wo flat plat,cs in thr rase rrprc~srntccl in Pig. 1.1 . The cc~nnt~ion of' Cnsr J 

yicltls tho satrtr linlif fiw r1 -- 0, i. C. fnv x = O \vhcn 110 in~ier rylintlcl is prrsc~lt. In 

(.Itis c,nsr, IJIt(~ Il~~itl 1~)(;11t>s insitlt: IIIC out,cr eylintlrr as n rigit1 I)otiy. Ilcncc il. is seen 

lIln.1 ('nso I1 yicsltls n lineal vcloril~y tlislril)~~t,iotl POI. llir t,\\~ sy111pI.ot.i~ C~SCS x -- 0 

n11t1 x -- 1. 'l'ltis I)rhnvio~ ~rtaltcs it rn.sy to ~intirrst,e.rltl why t,hc vclocit,y tlist.ribut8ions 

for lhr. ot.lter, inl.crmrtliatc valurs of x tlilTc,r so liklc from n stmight line. 

- x' - - r-r, 

5 - 5 

Fig. 5.4. Vclorily dist~ributinl~ it1 thr nnn~~lr~n l)c~t\r.c~r~~ 

I:tlrtl with tl~c i~itl of cqtm (5.l5n, b). 

n) Cnsr I: irinrr cyli~ltler rot.nt.ing; orct,rr cylitlticr at, rcst, ro2 - 0 

h) (he I I : inner cylitlclrr at rcst, tol = 0; o~tt,rr cylitlclrr rotati~l~ 

r, - r;uIius or i1111l.r 1.~1i11dcr. r, = r:uIim oI'o1111,r cyliwlpr 

t,\\.o.c-o~lc.c.~ltri~.. ~.ot;tti~~~ ~~Jlill~l~~rs 

>In ~~;lll~ll- 

r 

?J, = --.I. . 

2nr 

It, is sccn, therefore, t,ll,zt, t,he case of fric:l.iot~lrss flow ill t,hc r~eiglll)o~~rhootl of a 

vort,cx line constitput.cs a. solut,iorl of t.hc Navirr-Stokes cquntior~s (c/. Scc. IVb). 

In t,llis connexion it may be instn~ctive t,o n~cnt,iotl a11 cxnrnple of an cxnct nmslendy 

solut.ion of the Navier-Stokcs cclunt,iotls, rlnmcly that which tlescribcs the 

process of t1cca.y of n vortex t,hrough bhc acthl of viscosity. The distribr~t~ioll of t.l~e 

t,angct~t~ial vrlority component 7~ wit,lr respect to t.l~e radial tlista~lce r and tirnc t 

is give11 by 

Pig. 5.5. Vclocit.y distribution at varying 

times in tho ncigltbo~~rhood of n vortex 

filament cnl~setl by tho action of viscosity 

1; - circulnlinn or llw vortox nnrncttl nt 11 nio 1 =. 0 

w1:c.n vircoslly Iwylnr lo ncl: ti.- I;/? n r.

90 

V. ICxnct sol~ttion~ of tho Nnvior-StOkcs cqi~:~liot~~ a. Parallcl flow 9 I 

as derived by C. W. Osecn [21] and G. 1Ia1nel [I]]. This velocity distribution is 

represented graphically in Pig. 5.5 Here 16 dcnotcs t,he circulation of the vortex 

filamolt, at time 1 =1 0, i. c. at tho moment whcn viscosity is nssumed to bcgit~ it* 

actiol~. An cxperimenLal investigation of this procoss was ~~ntlcrt~nlta~~ 11y A. Tirnmo 

[40]. K. Kirdc 1171 mndc an nnnlytio study of the caso when the velocity distribution 

in t,ho vortcx tlilTcr~ from I.hnt irnposctl hy pot,cnt,inl theory. 

4. The suddenly necelernted plane wall; Stokes's first problem. We now procccd 

to calcuhtc somo non-steady par;rllcl flows. Since the convcctivc acceleration terms 

vanish itlcr~tic:aIly, the frict.ior~ forcos intrmct with tho local ncce1crnt.ion. Tho si~nplcst 

flows of this clam occur when motion is stnrtcd irnpr~lsival~ from rest. We sl~all 

begin with the c,wc of thc flow near a flat plntc which is s~tldcr~ly acce1cr:~tcd from 

rest and n~ovcs in it,s own plane with a constant vclocitty [lo. This is onc of the pro- 

I~lcms which wcro solvctl by (2. Stokes in his colcbr:rtccl memoir on per~tl~~lurr~s 

[3ri]t. Selecting tho z-axis along the wall in the dirnction of U,, we obtain the 

simplifiotl Navicr-SOnlccs oqmt.ion 

'rho prrssuro in tho wh01o space is constant, and Lhe bol~nclirry conclit,iol~s are: 

The cliIT(:rcnt.ial equation (5.17) is icIcnt.ical with the equation of hest conduction 

which clcscribcs the propngatrinrl of Itoat, in tho space y > 0, whcn at time 1 = 0 the 

wall y = 0 is sudtlcnly hcatcd to a t,cmpcr;~t,nre which oxccecls that in the surroundings. 

'l'lle pnrl,i:~l tliffcrcntial oquation (5.17) can be retlucctl t.o an ortlinary diTcrt:ntial 

cqu:~t.ior~ 11s tho sul)st.il,nt.ion 

Y 

(5.19) " --3' 2 1/ 

If wn, further, n.ssumc 

x = Uoj(r]), (5.20) 

wc o11I.air1 the followi~~g ordinary tliITorcntial equation for / (q): 

t.hc complemenhry error /um%ion, erfc q, 11.w been tabulatedt. The velocity distribution 

is rcpresontcd in Pig. 6.0, and it may bo notctl that tho vclocity profiles for 

varying tinies arc 'similar', i. e., they can bc rctl~lccd to the same curve by changing t,hc 

scalc ttlong the axis of ordinates. Thc cornplcmcntary error f~~nctior~ whicl~ appcnrs 

iu eqn. (5.22) has a valuo of about 0.01 at 7 -- 2.0. %.king into accor~r~t tho tlcfir~ition 

of t,l~c: t~hic:ltnoss of the: I~ot~ntl~try Inyor, 0, wc: ol)t.r~ir~ 

6=2qaJZx4 JZ. (5.23) 

It is seen to be proportionnl to the sqnnrc root of tho protll~ot, of kir~ornnLio visc:osiOy 

silt1 time. 

This problem was gcncralized by E. J?ccker [3] to ir~clr~dc: more genrml rat.rs 

of nccclnratio~~ as well as the cqses involving suctior~ or blowing or tho cfict of 

compressil)ility. 

Fig. 5.6. Vclocity dishibution 

above a snddenly accelerated wall 

5. Flow forn~ntion in Cmuette motion. The s~~bstiLuI.ion (5.10) which Imds to eqn. (5.21) 

dm not, in general, lend to a sol~ttion of 1.hc so-cnllcd lwnt conduction cquntion (5.17) ir morc 

complicntod boundnry contlilions aro itn~~osctl, Sincc cqn. (6.17) i~ linear, solution^ (i)r il, OILII 

be obtained by the use or the 1,nplncc t,mnsfor~nation nnd by tnoro direct nlcl.hods dcvclopc:tl 

in conncxion with tho study of the conduction of hcnt in solids. Mnny rc~~lkt obtni~~ctl, c. g., 

for the tcmperaturc vnriation in nn infinite or semi-infinite solid, cnn be tlircctly transposed 

and uacd for the ~oIut,ion of problems in viscons flow. Thw the prcccding problem in which the 

formation of tho boundary layer noar a suddenly accclcrakl wall has bwn invwtigntrcf can 

also be nolvcd for tlw CDSC when the wall movur in a direction parallel to ar~otl~or flat w:dl at. 

n ~ and t at a distantx, h from it. This is the problcm of flow forn~ation in Couettc motion, i. c., 

t Soe c. g. Shoppard. "The Probability Tnbgrnl", Rritish Atwoe. Adv. Sci.: Math. Tsblea 

vol. vii (3039) and Works Project Administration "Tables of the Probability Function", New 

York, 1041.

the of how the velocity profilc varion with tirne tcnding nsyn~ptotically to the linear 

diutribtrtion nlrown in Fig. 1.1. The diITcrcntinl cqriation is the same en before, cqn. (5.17), 

lmt with modified I)o~~r~clary conditions which now are: 

'rllr solutio~~ of eqn. (5.17) which sntinficn tho bor~ndary :d initial ro~~ditior~s ran Iw 

oht.:~inrd in t.l~c form of a ucrio~ of con~plcn~c~~tnry error fun~l~ions 

7x1 

11 'y' 

- = x erfc r2 n 4- 711 - x crfc [2 (71 -1 I) ?I# - ?I] 

('0 ,,-I, , -.I3 

rrfc - rrfc (2 q1 - tl) -1- crfc (2 -1- 71) - rrfc (4 11, - - 11) 1- rrfc (4 71, .1- 71) - . . . 4- . . . 

wllerc 71, := h,/2 1/ F i (lot~~t.cn 

(5.24) 

the cli~nenniol~lcsn tlistancc between t,l~c two wnlllr. 'Tho solut,ion 

is represellted in Iiig. 6.7. 'rlw corly profiles nre &ill aplwoxi~nntely similar and rc~nain so, an 

long nn t,llr bolllldary layer l~ns not sprcad to the stationary wall. The s~lcceeding vcloc:ity r)rofilcn 

:).re no lo~~grr "similer" a~~tl tc~~cl nsynptotirnlly to t,lre linrar distrihrrt~ion of tile skndy st&?. 

Exact solrtt,ions for non-stratly Coric.l.t,c flow werc rlcrivcd I)g .I. St.rinl~c~irr (331 

for 1I1r (xsr WIII~II OIW ol' 1,110 wdls is ILI, ITS^, in ;I, shn(ly flow II,II(~ is 111v11 SII(I(I~II~.S 

wc:r~lv~.at,c:cl to R givc.11, c:onstnut, vcloc:it,,y. 'l'o (lo t,his, il, is Iicbvc:ssal,y 1.0 solve! ['(lit. 

(5.17), whirli is itlcnt,ical with tho one-dirnrnsior~nl Iicat conrluci.ion cqtlat,ion, l)y 

lncnrls of n l~otiric~r. srrirs. A spccic'll CR,SC it, t.llis class of soluf.iotls is t.hd whrn t'l1~ 

moving wnll is sutltlrnly st.oppctl so t,l~al, it rcprrscnt,s tho decay of (h~ot,t,c flo~. 

a. I';L~IIVI flow n:! 

lnyor ncnr tho wall. 'The influonce of vi~cosity rcnrhcs the pipe ccnf.rr only in the 1:rt.c.r st,:~grn 

of motion, antl tho velocity profile tonds asy~npLoLically to tho pt~roldic tlistribt~l.io~~ for ste:rtly 

flow. The corresponding solut,ion for an nnn~tlar circul~rr cross-section was given 113. W. Murller 1201. 

,, 1 IIC nccrlcr~~th of 11 I111id ovrr ~,IIc wl~nlc ICII~I,II of pipe di~c~~sst%(l 11~re IIIIIH~, 

din~ir~g~~inl~ctl from t.lic acrcIornt.io~~ of n fluid in tl~c illlet j~ortio~~a or 

,.41rcsrlllly 

a pipe in ~IJ*:L~I,~ IIOW. 'I& 

rechngnlar ve1ocit.y profile whicl~ exists in the entrance ucct.iol~ is grncl~lnlly transfor~~~ed as 

t.he fluid progresses through the pipe with x increasing, antl tends, ~~ndcr the influence of viscosity, 

to nssnine the Hngen-Poiscnillc parabolic diaI.ribntion. Since I~c?rc a@z :t 0 tho flo\rs is not 

onc-rli~nensiond, nncl the vdocity depends on x, nu \vrll ns on t.ho rndi~rs. Thin proh~n wak 

rlisrusricd by 11. Srl~lichl.ing [DO), who gave t,l~o solrlliolr for L\vo-tlin~c~~sio~~nl Ilo~ tl1ro11~11 

n st.r:~igl~t. rhannel, antl by I,. Srhiller 1291, ;~nd B. 1'1111nin 1241 for nxinlly symrr~rt.rir,nl Ilow (rirc~~lar 

pipr): srr nlno Sew. IX i nnd X 111. 

Fig. 5.8. Vclocit,y profilc in n rircrrlnr pipe d~~ri~~g 

ncc~rlrration, 

art given by 1'. Szgtnnnski [87]; T .- v //I12

94 

V. Exnct sohltiono ol' tho Nnvicr-Slnltcq oqr~nt,ions 

Tho velocity profile u (!y,t) thus has thc form of a damped harmonic oscillaLion, thc 

amplitude of which is I/, c w i ? ; , in which a fluid layer at s distance y has a phase 

lag y l/;t% with respect to the motion of the wall. Fig. 5.9 rcprcscnts - this -. motion 

for scvcral instants of time. Two fluid layers, a clistanco 2 n/k = 2 n d2 v/n apart, 

oscillate in pli~c. This distancc can be regarded as a kind of wave length of the 

motion: it is somctimcs called the depth o/ penetration of tho viscous wave. The 

layer which is carried by tho wall has a thidrncss of t11c order d - Jq and dccrcasos 

for decreasing kinematic viscosity and increasing frequcncyt. 

8. A rlnm of non-steady solutions. A general c1:rss of no11-stcntly sol111ions of the 

Nnvinr-Stnltw ,scq~latio~ls which possran bor~ndnry-lcycr o11arnctr:r is ol~tainrd in the sr)ccinl mm 

when tho velocity componcnta arc indopcndcnt of Lho longitudin:~l coordinnl,c, a. 'rhc systcn~ of 

rrlr~nt.ions (8.02). writlnw for plnno flow. nasun)cs 1.11~ form 

----. .- 

t Tltc ROIIIL~OII in cqn. (5.2(in) roprcscr~t.s also t,l~c tcmporat~~rn c1intril)ution in Lhc rarth which is 

muwd by t.hc pcricxlio Iluc*t.r~at.ion of I.ho k~npcraturc on tho surfncc, my, from clay 14) d:ly or over 

t,hc scnfu~ns in a yc::w. 

b. Other oxnct solulionn 95 

If we now prescribe a cnnutRnt vclocity v, < 0 at thc wall (suction), wo notice that cqn. (5.27~) 

is satisfied in~mediately hy a flow for whicl~ o = v, and that the prc.wuro p bcaorncs indcpnndcnt, 

of uirnultrmco~~sly. Accordingly, we put - (l/e) (+/ax) = tI(J/cll,, whom 11(t) donotes bile frwstrrnm 

vrlocity nt jr very largc dishnc:~ from t.1~ w:rll, nncl I~cncc obtain 1l1c followir~g clilTc.rcl~l.itrl 

cqtmtion for u(y, 1): 

au b 3t 1 

dU azu 

l,,, -. ..-- 1 

ag dl ay2 ' (5.28) 

According to .I. 'r. Stuart m2] thorecxista an oxnct soluI,ion ofccln. (5.28) for tllo arld,r:rry oxkrr~al 

vclocity 

'lll~is so1116ion is 

whcro 

Sllh~tituLing thc I.wt thrw cqu~tions ink cqn. (529, we am led ID n psrtinl diffcrrntial oq11st.ion 

for the unknown function g(!/. 1) = g(7. 1); thin hnn 1110 forrn 

Tllc following non-di~ncnsionnl varinhlcs hnvo been irltrodud in the prccocling: 

I'ie. 5.9. Vrlocit,y rlistrihution in 

the neighbourhood of an oscillating 

wall (Stokes's second problem) Solutions of (5.32) hnve hccn ohtaincd by J. Wnhn (411 who crnploycri Lnplaoo transformations 

and who restricted hirnuclf to severnl apecinl forms of the functior~ /(1). (:cncrally 

speaking, the following cxternnl flows, U(1). hnve been incIudw1: 

a) dnrnped nnrl undampcd oscillations, 

h) stop-likc chnngc from one vnluo of vclocif.y to xnot.lwr, 

c) linear incre.nuc from ono vnltlc to anoll~cr. 

In the upncial c.wc whcn the cxlcrnnl flow is indcpcnclcnt, of time, /(t) - 0, cq~~ation (5.32) 

I-~ds to the uirnple solution '(7, 7') = 0. This CDIIRP* v01oi~il.y prolilo from oqn. (5.30) to 

I~oromc iclont.ic:rrl wiI.11 Llw nuyrnptoLic s11c1io11 prolilo givcw IILIAW ill WIII. (14.l;). 

The preccding examples on one-tlimcnsional flows were very simplc, I)cca~~se tho 

convective acceleration which renders thc equations non-linear vnnishcd idontically 

everywhere. WG shall now proceed to examine sorno exact solutions in which thcsc 

terms are retained, so that non-linear equations will havo to t)o considcrcd. We shall, 

however, restrict oursclves to steady flows. 

9. Stagnation in plane flow (Hiemenz flow). Tho first simple examplc of this 

t,ype of flow, represented in Fig. 6.10, is that lending lo a shgnc~tion point in plane,

whcrc n tlcnoks :L cot~st,nnl. This is an cxa.~nplc of a. plane polcnt,ial flow wl~ich arrives 

from thc !/-axis and impinges on a flat wall placed at y = 0, dividrs into two 

streams on the wall and Lenvcs in bot,h directions. The visco~~s flow mwt adhere lo 

t,he wall, wl~crcas tho potential flow slides along it. In pot.entia.1 flow the pressure is 

given by Rcrnoulli's cqr~nt.ion. Tf pa, dcnotcs the stagnat,ior~ pressure, and p is t.11~ 

prcss~~rc nt. a.n arbitrary point., wc have in pot,cnt.inl flow 

For visco~~s Ilow, wc: now ninkc t,hc nss~lmpt~ion~ 

?I = x /' (71) ; ?I = - 1 (?I) , 

Po - p = 

Q (L ":r2 -1- F (y) 1 . 

111 this way t,hc cquat,ion of cont,inuit,y (4.4~) is snt,isfietl idcnt,icnlly, :~nd thc t.wo 

Navicr-Slnltcs cqr~at.ions of plane flow (4.4n,l)) n.re snfliciont to dctmminc ll~c f~~nclions 

i(y) and F(y) Substituting cqns. (5.34) an(\ (5.35) i1it.o eqtl. (4.4a.,b) wc 01)tain 

t,wo ordin9.r~ tlifTercnt.inl eqantions for / and F: 

aqtl 

i'z - i y' cf,2 1- 3, j"' (5.36) 

/ /' =

1). 0t.lwr exact volr~tio~~n 97 

'I'hc lm~nrlary eonclit~ions for / and F arc obt,i~ined from 11 -2 v -- 0 at. tl~e wall, wl~rrc. 

?/ =-. 0, n.nd 2) : po :tt the stagnation point, as wrll as froin 11. == (J = n. x at a Inrgt: 

tlisl,ancv: Sroni t,ho wall. '~'IIIIS 

'I'hc solution of' the? diITerrntial cquntion (5.3!)) w:ts first givcn in a thcsis I)y I

Table 5.1. Functions occrtrring in thr .solution of plnnc nnd axinlly ~ymmct.rionI flow with 

atngnntion point. Plnne cam from L. Hownrth 1141; nxinlly symmetrical cnnc from 

plnnc 

nxinlly uyrnmot,rical 

ITence again, as Idore, thc laycr which is influencr.d by viscosity is small at low 

kinematic visrositics and proportional to 6 l'hc pressure gradient ap/ay becomes 

proportional to Q n i ia and is also very small for small kinematic viscosities. 

It is, furthcr, wort.11 noting that tho dimcnsionlcns velocity distribution u/fJ 

and thc bo~~nclar~-la~cr thicltncss fron~ cqn. (5.40) arc indepcntlent of x, i. e., they 

do not vary along tho wnll. 

I 

'rho t,yp of flow under consideration does nbt occur near a plane wall only, but 

also in two-climcnsional flow prst any cylincirical I)ocly, provided that it has a blunt 

noso near tho stagnabion point. In SIICII cnsrs tho solution is valid for a mall neigh- 

bourliootf of tho stsgnntion point, if portion of the curved surface can hc replaccd 

by it4 tangcr~t planc war t1w sL:~gn:Aion point. 

Tho nnn-steady flow pattern which results qwu tho sl~pcrposit.ion of :rn nrbitrary, 

timc-dependent transverso motion of thc pl:~nc was st~ldicd by .l. W:~kotl 1.121 

'Chc spcrial cam of a harmonic tran~vrrsc niotior~ was solved carlicr by M. 1%. (:l:rt~rrt 

(1143 in Cllap. XV). 

9.. Two-dimensional noo-steady ntngnation flow. The cnm of non-stnndy, t3wo-tlinlrnsionnl 

flow sturlicd by N. ltott 128.1 conut.itutm n gcncrnli~atiot~ of Lho prcrcdit~g cnnc:. W

10. Stagststinn in thx-dimensional flow. In :I similar way it is possible to ol~tain 

an cx:~c:t, sol~~t.ion of the Navicr-Stoltcs cqnations for the three-clirncnsiorlal case 

of flow with st,a.gnat,ion, i. c., for the axisy~nmct~rical casc. A fluid stream irr~l)i~lges 

011 a wall at, right, nnglcs t,o it and flows away mdially in all directions. Such :I casc 

occrirs in t.11~ ~lri~l~l)orlrl~oo(l of a st.agn:~t.ion l,oi~il. of a 1)otly of rcvo111t.ion in :I flow 

I)arallrI to it,s :ixis. 

7'0 solve the problem we shall use cylindrical coordinates r, 4, z, and we shall 

assume tlrat, the wall is at z = 0, the stagnatiorl point is at the origin and that the 

flow is in the direct,ion of the negative z-axis. We shall denote the radial and axial 

components in frictionless flow by IJ and 11' respectively, whereas those in viscous 

flow will be tlcnc~ted by u :.= v(r,z), and w = w(r,z). In accordance wit11 eqn. (3.36) 

t,hc N:~virr-St,okcs rqrrntion for rotnt,ional symmetry can be written as 

111 1.h~ r:wc of viwor~s flow we assllmc t . 1 ~ following for111 of t.11c. sol~~t~io~~~ Sor (.IlP 

vrloc.i(.,v and prcssllrc clist~ril)ut.ions 

It, can be easily verifird that :L solution of Lhc form (5.4:1) s:itisfic~s the: cq~~atio~~ of 

fi,llowing t\ro 

c-ontinuity idcnticnlly, wl~crcas the cqr~ations of niot,ion Iv:~rl 1.0 t.l~n 

cy~~ations for /(z) ant1 F(z): 

p- 2jJ" = .2+ "Jl1l1 (5.45) 

2jJ' = ) a 2 P f - "J". (5.46) 

As I)cd'nrr, t.lw first of the t,\vo equations for / awl F c:ln l)o frc:ctl ol' Ill(- c:o~~st.;r~~t.s 

rr" 

and 11 l).v a sirnilarit,y transforrnat.ion, wl~irll is idrr~tical \!,it11 tl~:~t. i r ~ t,hr III:III~. ,.;IS(', 

t hns

11. Flow taenr n rotnting disk. A furlhor cx:~tnl)lo or :tn cxact soluLion of the 

Navicr-St,okcs cqu:~t,ions is furnishctl I)y t.11c: flow around a flat clisl; which rotates 

at~out an axis pcrpcntlicrtlnr to it,s planc: with n nnik,rm :tng~~lar vclocit.y, cr), in a fluid 

ot.hcrwise at, rest. Thc layer near the disk is carried by it througll friction and is 

thrown outwa.rds owing to tllc :xction of ccntrif~~gnl forces. This is comprnsnkd by 

part,iclcs w11it:h flow in an axial direction towarrls tho disk to be in turn carried 

and c:joetrcl centrifugally. 'I'hus tho east: is seen to bo ono of f111ly three-dimcnsionsl 

flow, i. c., tl~cre exist volocit,y components in the racli:d dircction, r, the ciro~rm- 

[rrrtrtial clircct,ion, 4, and the axial direction, z, which we shall denotc rcspcctively 

1)s 7s. v, and tt,. An axonornctric: rcprcsent.:~t.ion nT this flow ficld is shown in I'ig. A. 12. 

At. first. t .1~ cnlcnl:xt,ion will bc pcrlomed for thc case of :MI infinite rot.atir~~ pla~~c. 

11, will t,11e11 I)c easy to cxtcnd tl~c~rcs~tlt to inc:lndc :L disk or finito cli:trnct.cr I1 -- 2 11, 

on contlition that the edge rlTcc:t is ncglcctctl. 

'raking int,o acco~lnt rotational symmetry as wcll :m t.hc not.at,ion for i.ho prol)lcnl 

wr car1 write down the Navicr-Stokes cq~~ations (3.36) as: 

Fig. 5.12. Flow in I,ho nnighbo~rrhod 

of s disk rotating in s fluid 

st rest 

Velocity compon~nk: u-radial, s-rircllrnlerwilinl, 

ro-axial. A lngar nf fluid in rarricd 

hy the disk nwing Lo the ncliun of 

viscnua lorcrs. Tho eenlrilupal lorccs in 1hC 

thin layer givs ria. to ~cconrlnry flow wllicll 

is dircelcd rndinlly oatw*rrl 

., I ho no-slip condition at the wall gives the following bountlary conditions: 

z=O: u=O , u=rw, w=O, 

z=w: w=o, v-0. 1 (6.49) 

We shall bcgin by cst.im:~tirtg tho ttlic:knc:ss, A, or tho In.ycr of fluitl 'c::l.rric:tl' l)y t.llo 

disk 1,231. It, is clear that Lhc t11icknt:ss of the Iaycr of fluid which rot;dras wit11 thc 

disk owing to friction tlncrcmrs with th: viscosity r~nd this view is c:onfirtrtrtl wht:t~ 

c:om pared with the msnlb of the prccoding cxamplcs. l'ttc ecrl tri fugnl li)rc:c per 11 ",it 

volumo which neb on a fluid p:~rticlc in tho rolatirlg Inyor at a tlisf.ancc r lon~ (.It(: 

axis is cqu:~l to p r (3. lrencc for a volumc of :ma clr . tls ant1 I~right, (1, the rrntri- 

fugnl forcc I)ccomr,s: p r cuz 6 tlr c1.v. The same olcrnc:r~t of fluid is nctctl upon I)y :I 

sl1c:tring stress t,, pointing in thc dircction in which the fluitl is slipping, and forming 

an angle, say 0, with the circumfcrcnt.ial velocity. The radial comportent of the 

sl~carirtg stress must now IN cqual to thc centrifugal forco, ant1 hrncc 

or 

T~ sin 0 dr (1s = p r co2 6 dr (1s 

T, sin 0 -- e r (oZ fi . 

On the other hand tho circumfemrttial componcnt of the sltcarir~g stress must I)c 

prop~rt~ional to thc vclocit,y gradicnt or tho circumfcrcntial vc1ocit.y at thc wall. This 

condition givos 

T, eos 0 N (14 r co/O . 

Rliminating tw from these two equations we obtain 

a2- I) 

m tan 0. 

If it is assumed that thc dircction of slip in the flow near tllc wall is indcpenclent of 

thc mdius, tho thickness of thc layer carried by the disk bccomcs 

which is idcnticiil with tho rrsult obtziinctl in the case of tho oscillating wall on 1). 94. 

Ihrther, we ran write for the sharing stros.9 at the wall 

- 

t,-eru~~d-erw fvw. 

'rho lmquc, which is cqual to thc prcnlucl of shcaring strcss at tllc ~1111, arva :LIIO 

arm 1)ccomcs 

R denoting the rnclius of the disk. 

In order to integrate the system of equntions (5.48) it is convcnicnt to introtluco 

a dimensionlrm distance from the wall, 5 - z/d, thus putting 

I

104 V. Jq:.tacL solutions oi tlrc Navirr-Stdtrs rqrtations b. Othrr rxnct, solutions I05 

I'rrrtl~er, the follo\viy assun~ptions arc ~nntle for t,l~r vrlocity romporrents nntl prcs- 

Inserting tllcse cq~laLior~s into eqns. (5.48) we obtnir~ n syst,rln of four sin~nllanrous 

ordinary diflerrntial eq~lat~ions for the funotions F, G, 11, ant1 P: 

'I'he boundary rontlitions can be calculat,rtl from cqn. (5.49) and are: 

r 7 

1 Ilr first solution of t,hc syste~n of eqns. (5.53) by an approxi~nnt~c methotl was given 

I)y a method of numerical intcgmtiont. They are plotted in Fig. 5.13. The starting 

values of the solut,ion indicated in Table 5.2 were given by E. RZ. Sparrow and 

J. 1, Gregg 1321. 

Fig. 5.13. VolociLy tlinlril)~~lion 

nvar n disk rot.nt,ing in n fluid at rrsl, 

TII t,he cnsc nntlnr discussion, just ns in td\c exn.mple involvir~g R stlagtlatrion 

[)oinl,, t,lle vclocit,y Geld is the first, t,o I)o cvnluatcd from the cqnnt.iol~ of corlt,inr~it~y 

ant1 the ccpat,ions of motion parallel to the wnll. 'J2fc prcssurc distribution is form1 

s~~l)seqwmt.ly from the equat.ion of motion perpendicular to the wall. 

. - 

t '1.his ~olut,ion wns ohhind in tho form of n power series near 1 = 0 and nn anymplotic uories 

for largo values of C which were then joined togehher for moderato values of 1. 

Table 5.2. Vnloos of t8he functions nerdetl for the drsc*ription or thr flow of n clisk rotntirlg in a 

Iluict nt rest, cnlrulatrrl at. the wnll antl nt n Inrge rlistn~~rc froru Ilrr wnll, 11s rnlc~~ln(ctl IIJ' 15. A]. 

Slmrroa nnd .I. I,. Grcgg 1321 

It is sccn I'rom Fig. 6.13 tht tho tli~tnnc:n lion^ thn w:ill ovcr wl~irl~ t.11~ 11rrip11rr:rl 

wlocit.y is rctlucecl t.o half the disk vclocity is do., = d&/~ . It is to I)c r~ot-cd I'IXIIII 

the solution that when h = Jv/ij is sm:lll, t-he velocit,y components 11, antl v l~nvc 

;~.pprecint)lc values only in a t,hin layer of f,hi(:kncs~ l/;/0) . 'J'h v~locit~~ compot~(:t~t W, 

normti1 t.o t.he tlislc is, nl. :it~y ral,r, srnnll a1111 or the or(1cr 1/1~ ,I). '1'11~ ~ I I ( ! ~ ~ I I I L ~ . ~ ~ I I 

of I h rdat.ivc st.re;rmlincs Ilc:lr the wnll \rrit.ll rcspcct tto Lhc circutnlcrc~~t.i:\l clirc.c:(.iott, 

if the wall is imagined :it rest. nntl the hid is tnlten t,o rot,nt.c at, a I:lr~c tlist:~t~c,c: 

frotn t,hs wall, becomes 

Alt.l~o~tpl~ the calculation is, strictly spcalcing, applic.able t.o an infi nitc disk or~ly, 

we may utilize t,hc same rcsults fbr a finit.e disk, provitled that, its ratfius R is largo 

cotnparcd wit.l~ the thiclrness tY of t,hc layer carried with ttlc disk. We shall now 

evnlnnlo t,llc turning moment of such a disk. The cont.ribnt.ior1 of an annular disk 

clcmrnt. of widt,h dr on mdius r is dM = - 2 n r tlr r t,+, and hcncc the moment 

for a dislr wetked on one side becon~es 

llcrr tr+ p(av/az), tlcnotes the rircumfercntial comporlrt~t of tl~n shqarir~g stress 

Iirom rcln (5 52) we obtain 

11. is cust,omnry to ir~trotlucc the following tlil~lc~~~sio~~lt~ss ~non~t:~it; cocfficicr~t., 

Ch, -- - - . 

2 1)l 

- . . 

: e wZ Rr ' 

(5.55)

106 V. JCxact solutions of the Navicr-Stokm cq~~ntiona b. Othrr exact, solutions 107 

This gives 

or, tlcfining a Reynolds number based on thc radius and tip vclocity, 

R'o 

R = - - 

and int.ro~lttcing thc nnmerical vnll~c - 2 zG'(0) = 3.87, wc obtain finally 

Fig. 6.14 shows n plot of this equation, curve (I), and compares it with mcasuremcnta 

1391. For RcYnolcls numbcrs up to about R = 3 x LOS there is cxcellcnt 

agreement hnt,vecn tltoory nnd exporimcnt. At highor Raynolds numbers the flow 

bccorncs turbulent, an11 tho respective case is considered in Chap. XXI. 

Curves (2) and (1) in Fig. 6.14 arc ohtainnl from thc turbulent flow thcory. Oldor 

mcasuremcnts, carried out hy G. Kernpf [lG] and W. Schmidb [31], show tolerable 

ugrrwnrnt with throrotiral resalts. Prior tn Lrsr aol~t~ions, I). Riahoachinsky [2Gj. 

1271 estaldiahcd cmpiricnl fonnulac for the turning mon~cet of rotating disks wllich 

werc hmcd on vcry carcful mcasurrments. Those formulae showed very good 

sorrcmcnt with the thcorctical equations discovcred suhscqucntly. 

-0- - 

The quantity of liquid which is pumpcd outwards N a result of thc centrifuging 

nctdon on tho one sidc of a disk of radins R is 

o NACA Report No. 7g.3 

v 0.48 lo 1.69 

Kernpf 

0 NSchmidt Fig. 5.14. Turning mo- 

ment on a rohting dink; 

crlrvc (1) from eqn. 

(LM), hmimr; eurvea 

(2) and (3) from eqns. 

(21.30) and (21.33). tw- 

bulenl 

Calculation shows t,hnt 

Q = 0-885 n R2 (11 = 0.885 n Rn (1, R-lI2 . (5.67) 

,I I he q~tanf.ity of fluid flowing towards thc disk in the axial dircct,iorl is of cqrlsl 

~napnitutlc. Jt is, filrther, wortlly or no(,c tht tJtc pressure tlini:rrt~cc ovcr t,hn 1:~yrr 

carried by the (lislr is of tho orrior e r1 a), i. c., vcry small for s~unll vi~co~it~ics. TIIC 

prcssnm (Iist,ril~ntion tlcpcntls only on tho tlist,nrlc:o from th(1 wall, rinrl (.II(w is r~o 

riuli:r,l ~~rcss~trc grntlicwt,. 

A generalisctl fnrnl of tl~e prccetling problem has becn stutlictl 11y M. G, Itogers 

and G. N. Lance [28] who assumed that the hid moves with an nnnllIar Vl,IO(*i(,V 

antl thc sccond boilntlnry contlition for tho function G(() mttat, Ito rrplncwl t)y 

C(m) = s . In this conncxion a comparison should hc mndo with thr caso ofrotating 

flow ovrr n fixrd disk given in Scc. XTn. Nnmcrical ~olutions for rotatio~~ ilk tl\c 

s:nnc srnsc (s > 0) can be found in [20]. Whcn the rotations arc in opposite scnsps 

(s < 0). physically meaningful solutions can bc obtained for s < - 0 2 only iTunifc)rrn 

suction :it right, :i~~gIrs to the dislr is n(lniittc(1. 

The prol~lem of a rotating dislr in a housing is discussed in Chap. XXl. 

It, is particularly tiotcwor(hy that the solutior~ for tlrc rotating disk as wcll as 

1.llc solutions obtained for the flow with stagnation are, in the first place, exact 

solutions of the Nnvicr-Stokes cquntions anti, in the sccond, that thcy are of a 

houi~drcry-la?/rr 

(me of vcry small viscosity t,hese solntions show that tho influence of viscosit.y 

rxl.rntls over a vcry small lnycr in tllc ~~cighl~ot~rl~ootl of. the solid wnll, ~ 1 1 c ~ t . c : : ~ ~ 

ill 1,llc wl~olo of 1.l1c rcmnining region t.hc flow is, j)rnct,ic;llly spcalring, i(lrnt.i(~:ll 

\vit,l~ (.he corrcspontling itlcal (potcnti:~l) cnsc. '~hcsc cxamplcs show Surthor l.l~;~t 

the b~~nnilary hyer has a thickness of the order iv . The one-dimensional examples 

of flow discussed previonsly display tho snmc bonntlnry-layer character. In this 

conr~cx-ior~ the rcatlcr may wish to conwit a pnpcr by G. I

The periphcrnl vclocit,y vanishes evcrywhcrc. Int,rodnring (his form into the Navier-St,oltes 

cqllnLiolls writ.lnl~ ill I~oln,r rc~nrtlinnlcs, cqn. (:l.:l(i), nntl rlin~in;~l.i~~g prmsllrc from t.hc cql~nt,iollu 

ill t,llo r nlltl $ clircrtions, we obtain t.hc following ordinary tlilhrrnl.i:ll cq11a1.io11 for I('(4): 

'I'lle co~~st,:rnt, I( clrnotrs the radinl prrssurc gr:~.clic.nt, ;tl, 1.11~ ~~alls, Ii =- --(I/@) (i)l~/ar) (r"v2), 

where \ve llave 17 0 fore - n anti+ - -a. ns wc:ll as E" - 0 ford) -7 0. The so111lio11 ofrq11. (5.58) 

,,.as givrn by (;. lIalncl [ll],'l'l~c f11nrtion 1.' ran be cxprrssrtl rxplicit.ly ns an ellil)l.ic fnnction of$. 

\Vc sllnll now briefly sltrt,ch t.l~c ch:~rnrt.cr, of the solution refrailling from cliscussi11g the 

(let,nils of t.lre derivation. The grqh in Fig. 5.15 shows n family of vclocit,y profiles for a tollvcrgc~~t, 

and a clivrrgcnt chnnnrl for diITerent. IZ.cynoltls nnn~bers plot,tcd on the I)asis of the 

Il~~r~lerinaI raIor~l:itionn pcrrorrncd by I

cnu bo npplirtl nt mont, nn fnr RS 1.l1r point, of nrpnrnthn ot~ly. F~~rt.hormore, t.hc now t.hcory Rw- 

ccodn in nome canen evrn \vil,h t,hc cvnIunt.ion of tho cotnplox flow pntterns which exist. in the 

region of hnck-flow behind t.lm point, of sqmrntion ng woll RR t,l~nt in the rrgion of re-ntl.arl~~netlt. 

111 Al,rnn~owitz, M.: 011 Iinrkflo\v of n visrn~~n ll~~itl in n tlivrrgi~~g chnnrl. .J. M:IIII. I'hyn. 28, 

152 (~!i~o). 

141 I!orker, It.: IntCgrntion tlcs i%pmtionn ~ I ~nouvcrnent 

I 

(1'1111 fluidc V~S~IICIIX i~~co~~~prrssihl~. 

(,ontribution to: Hnndbuch dcr Physik (8. I'liiggc, ed.) 1'11//2, 1-384, Rcrlin, 1D63. 

[51 . . Rlaniuu, 11. : I,nrni~lnro St.riimr~nji in K~l.nA\en wecl~nclnrler Ihile. Z. Mnth. 11. I'l~ysili 68, 

226 (1!)10). 

(61 (:nt,l~ernII, I)., nntl hltinglrr, l

CHAPTER VI 

Very slow motion 

a. The clifirential equntiono for the case of very elow motion 

lIl {,lliq 1~~1~111~,,~~ jvl. llr,l~ll~~p 10 1Iisv11ss somv :1~1proxit11:11.(? sol~~l.ions oft,hr xavierril~,lta~~ 

OI~IIILI,IOIIH WIII~III t~rt, vttliti 111 IJII, Ii~~~ll.lng (:IISO \~IICII 1.111, viscous furccs :we 

consiclerably grcatcr tlmn the incrtia forws. Since tltc incrt,ia forces arc proportional 

to tho square of the velocity whercas thc viscous forces are only proportional to its 

first power, it is easy to appreciat,e that a flow for which viscor~s forcrs arc dominant. 

is obtained when t,l~c vclocit.y is very small, or, sl~rnlring morr gcncndl~~, WIIPII trhtr 

Reyr~olds number is very small. Whcn t,hc inertia terms are simply omitted from the 

equations of motion the resulting solut,ions are valid approximately for R 6 1. This 

fact can also be deduced from the dimensionless form of t.11~ Navier-Stokes equations, 

eqns. (4.2). whcrc the inertia terms arc secn to be multiplied by a factor R = e V 1/p 

compared with the viscous terms. In t,liis connexion we may remark that in each 

particular case it is necessary to examine in detail the quantities with which this 

Reynolds number is to be formed. However, apart from some special cases, motions 

at very low Reynolds nllmbers, sometimes also called oeepittg mntiotl,.~, (lo not 

occur too often in practical applicationsi 

It is seen from eqns. (3 34) that when the inert,ia terms are ncglrrtctl the incom- 

pressible Navirr-St,oltrs equations assume the form 

or, in cxtnntlrd form 

div 111 = 0 , (6.2) 

az I 

t In tho canc oC a ~pltcre 

when t.11~ dinlnetet d - 0.04 in (: 0.00333 ft.) and t8he velocity V = 0.048 ft/sec. 

falling in air (v 1- 160 x 10 ft2/soc) wc obtain C. g. R = lid/v -- 1, 

This systern of equations must be supplemcntcd with the same boundary cotltlit,io,ls 

:IS the fill1 Navier-Stokes equations, namely those expressing the ~bscr~cc of slip i r ~ 

the fluid at the walls, i. e. the vanishing of tlw normal and tangent,iaI con~poncnt,~ 

of velocity : 

ll,, = 0 , ?!, = -- 0 :I t< \v:1 I Is . (6.5) 

An irnportnnt c:l~n.ractcristic of creeping motion can bc obtair~ctl at once fro111 ~:(III. 

(6.1), when the divergence of both sidcs is formed and when it is ~~ot~icctl t,lrat t,lle 

oper:tt8ions tliv and V2 011 the rigI~t-l~n.rd side may 1~ pc:rforr~irtl in the rcvt5rso ortl(%t;. 

,. I bus, wit,Ir cqn. (0.2) we have 

'I'l~c pressure fioltl in crccping motion s:~tisfics the potcnt,ial cquatior~ :tr~tl the Iwcssurc 

p(x,?/,z) is a poLent.ia1 function. 

The equations for lwo-dimensioned crceping motion become parl,ic~~ln.rly sirnljlc in 

form urit,ll the introtlnction of the stream f~~nction 71) tlnfirictl Ity ?L =+/if!/ at111 

?I = - ay~/r3x. As cxplainctl in Cl1n.p. I V, :tnd as sccn from cqns. ((i.:%), wllcr~ I,rt,ssllt.c: 

is t:Iiminl~ImI l'ron~ l,l~t? first, I,IVO ctp~t~ioll~, stret~~n 

III,IS(, s:~l.isl:~ (.It(! 

t h 

J'IIIIC~.~~II 

rquation 

V",l =1 0 . 

., I he strcam funct,ion of plane creeping mot.ion is t,hus a bipot,rnt.inl (Ijil~nrnlollic,) 

function. 

In t,lic remaining scct,ions of this eltapt,er we propose t,o discuss tllrcc ex:~tn~)lcs 

of creeping motion: 1. Parallel flow past a sphere; 2. The l~ydrotlynnmic theory of 

I~tt~rirnt,ion; 3. Thc Iiele-Shaw flow. 

b. I'arnllel flow past a sphere 

- - - - . - -. 

'I'hc oltlcst known solrlLion for a creeping nrotion was given by (:. (:. 

who in~csl~igatctl 

St.olrcs 

t,hc rcsnlt of his calrlllations witlmlt going into the ~n:~lhcrnat~icnI tlct,:~.ils 01' t.lto 

tl1cor.y. Wc shall Imc our tlcscription nn thnt given Ijy 1,. 1'rrtntlt.l (121. '1'11t. sol~~t.ion 

Lhc? case of pamllol flow past, a splrcre [17]. Wc shall IIOW tl(~sc-ril)t~ 

incitlcs with tltc origin, ant1 w11icl1 is 11l:~cctl in a pamllcl 

I/,, Fig. 6.1, along the 3:-axis can t)c rcprcsrntctl by t,hc 

pressttre arlcl vcloc:it,y componcnl,s: 

3 11 Urn Rz 

p - p,= -- 

2 r3

114 VI. Vcry slow motion 

where r2 = z2 + y2 4- z2 has been introdneed for the sake of brevit,y. Tt, is easy to 

verify that these expressions satisfy eqns. (6.3) and (6.4) and that t.ho velocity va- 

ninhes at all pointa on the surfnco of t,ho sphere. The pressure on t,he surfnre becomes 

'rho rnnximllm ant1 n~inimurn of prrssllrc occurs nt points P, and I'2, respectively, 

thrir valnrs bring 

3 11 uw 

1)1.2- pcn - -1- --ji - 

(G 7 1)) 

Tile prcssnrc distribr~t,ion along a 1ncridia.11 of t,hr sphere as well as alor~g the axis 

of al)scissar, r, is shown in Fig. 6.1. '1'11~ shrnring-stress distribution over the sphere 

can also be cnln~libtctl from the n.lmvo formulae. If, is found that the shearing st,ress 

has it,s largoat value nt poirll /I whcro t = ij ,IL fJ,/I1 :m(I is r~(11al to the pressllre 

riso nt PI or prrssurc tlccrrase nt /',. Tntrgmting tho pressure distribut.ion ant1 the 

shrnring sl,rrss over the surfacr of tho sphrre we obt,nin t,ho t,ot,nl tlrng 

'This is f,ltc vcry wcll known ,Wko.~ cr/udion for thc: tlrag of a spl~rrc. It, can I1v shown 

t,l~at. ol~c t.llirtl of ihc t1r:t.g is tlrro t,o the prossure ,list,ril)~ition n ~ ~.II:L~, ~ d tho ron~ni~~ir~g 

t.wo t,l~irtls nrr t111o t.o tho cxistctico of shcar. It is fr~rlhar rcrnark:~l)lc t,h:lt t,hc ctr:~g 

is ~xo~)~rt,io~~nl to the first, powcr of vclocity. If a t1ra.g coefficient is fornled hy 

rc.l;.rrillg f,llc tlr;la 1.0 t,hc tlyu:~mic hc:r.rl a Q 11,2,nntl t.he rrontd arca., :IS is dotlc: 

in tllc c.:rsct of highrr Ib~~noI(Is r~n~nlwrs, or if we p~t. 

b. Parallel flow pnst 8 spl~rrc 116 

A coniparisot~ het,ween Stolzcs's equation nnd rxpcritnont was givan in Vig. 1.6 

from which it is seen that is applies only to rases when R < 1. The pnt,tern of 

strcamlinos in front of and behind the sphere must be the same, as by rcvvrsing 

t,he direction of free flow, i. e., by changing the sign of vclocity con~ponents in cqns. 

(6.3) and (6.4) t.he syston~ is transformed into it,self. The st,reamlincs in viscons 

flow past. a sphcro are sl~own in Fig. 0.2. Thy were tlrnwn ns they woultl nl)pear 

to an observer in front of whom the sphere is dragged with n constnnt, vclocity U,. 

The sltrt,ch contains also velocit,y prolilcs at scvcral cross-s~ct~ions. It is scon f,l~nt 

tho sphere drags with it a vrry witlr layer of flnitl wl~id~ rxtr~~tls over :iI~out, one 

tliitrnclor on I)oth sitlns. At, vory high Itryrtoltls nurnl)ors tl~is I~o~ln~liiry li~y~r 

I)ccornes very thin. 

Ipig. 6.2. Sl.rcnnilincs nnd vrloci1.y di.st.ri- 

brttinr~ in Stokm' snlut.ior1 for n spllcrr ill 

pa r;dlrl flow 

[Pig. 6.3. S1.rc;itnlirlr.s in llir flow 

ORCVII'.~ improvcrnt~~~t: An ~ITI~)~~VI!ITI(-II~, of' St,ok~s's srv is', 11' nntl 711' nrr t,l~o pcrl,url):l (.ion f.cwns, : rr~tl :IS SIIV~I, stn;r ll wit 11 ~.c~sl~~~,, 

t.0 

(.he f'rcr st,rca~n vclocity (1,. It is to noted, I~owevcr, that, this is r~ot, tr~r ill 1,11c 

irnnlrrliatr neighl~onrhootl of t,hc spherr. With tho ns~nnl~~i~iot~ (0.1 I ) 1.111- illrrt.ia 

t.crms it) t,hr Na,vier-Stdokt,s rqns. (3.32) arc tlrcv~nl)osed in two ,pro~~l)s, r. g. : 

allr av' 

(loo , U, -- 

ax 

, . . . and 

a ~ ' , avr 

"Iaz , IL ax 

--, . . .

116 VI. Vrry slow motion 

, 3 I 11c sccontl group is ~lrglnctrtl as it, is small ot' the sccontl ortlrr rotnparell wit.ll the 

first group. Thus we obt,nin tho following cqllat.ions of rnot,ion from the Nxvier- 

St~oltcs cquat.ions : 

P 1 

I he pat,t,rrn of sl,rr:wnlir~rs is now no lo11gc:r t.11~ sanlcx ill front, ol' :LII(~ 11ehi11ll 

t,hr sphere. 'I'his can be recognizcd if' rcfcrerlcc is tnatle to eqns. ((i.12), I)rrn.ust: if wr 

(:11:1ngc 1.11~ sign of t,llr vclorit.ios and of thc: pressure, t,hc cq~~nt,ions do not t,mnsforn~ 

i111,o thclnsclves, wl~crcas tl~c Shlics cqu:~t.ions (6.3) tlitl. 'l'hc st.rc:a.mlitlt:s ol' 

t.11~ Osrcn cquat,ions arc plot,tcd in Fig. 6.3, and t,lle observer is again assunlctl to 

I)c a,t rrst wit,ll respect. t,o Che flow at a large disttance frorn t,he sphere; it is itnagi~~rtl 

tO~at, f,l~c sphere is dra.ggctl wiLh a constant, vclocit,y 11,. 'rhc Row in front of thct 

spltrrc. is vcry similar t,o that given by Shkcs, but, behind the sphere the st.rearnli~~rs 

arc closer tqq?t,llcr which mc:lns that tho vclocit,y is larger f,han in t.he forrncr case. 

I'ttrt.ltcrrnorr, 1)ohintl t,l~c sphere some pnrticlcs follow it.s mot.ion as is, in Ikvl., 

ol)srrvc:tl rxprrimrntally at, large T

1 I8 VT. Very slow motion r. 'l'l~o Ilytlrorlynn~~~in tllmry of I~~brivntirm I I!) 

The tliffrrcnt,ial cq~lntinns of crrrping motmion, cqns. (6.1), can be fitrLhrr simplifier1 

for the case ulltlcr consitlcration. T11c cqu:~tion for tllc y-tlirechn can be ornittctl 

altogether bccnusc the component v is very small with respect to u. lhrthcr, in tl~c 

cqnnt.io11 l'nr 1,110 x-(lirrct.ion i)21c/ih:2 ca.n I)c ncglcct,ctl with rcspocl, 1.0 a21~/r?y2, be+ 

calrsc the forrnrr is sn~allcr t.Il;l.n the lnt,t,cr by n hetor ol the ortlcr (h/1)2. '1'he prcssnrc 

tlist,ribution must, snl.isfy 1,llc contlition t,hnt p .r= p,, at bot.ll entls of tho slipper. 

Comp:trrrl wit.lt 1,hc rase or flow bct,weon pnmllcl slitlin~ walls, thc pressurc grwlienlf 

in I,llc tlirrc1,ion of ~nol~ion, ap/ax, is no longer constant, but the very small prcssurc 

grntlic-nt, in 1.11~ ?/-direction can I)c nrglrrt.rt1. With 1.11csc sitnplificntions the tliffrrrnt,inl 

rq~tal~ions (6.3) rt:tlucc to 

nntl i,hc eqn:~t.inn of cont.innii.y in tliffcrrnt~ial form can bc rcplnccd by the contlith~ 

that t.11~ voln~nc of flow in evcry scct,ion must be constnnt: 

The sol~ltion of rqn (6.15) whirh satisfirs the bountlnry conditions (6.17) is 

similnr 1.0 Pqn. (5 5), namrly 

or, solving Cot. 71': 

'I'IIIIS tho mass flow is known wl~cn t.11~ shpo of the wctlgc is given as t.llo f~lnrl.iol~ It (:t:). 

Eqn. (0.19) gives tlle prcssnrc gmdient., ant1 eqn. (6.20) ives the prcssurc tlist,ril~~lt,iol~ 

ovrr tllc slipper. 

wllit:11 n.ppcar in cqn. (0.20) tlcpcntl only on t,llc gcornct,ricnl sl~al)c of' the gt1.1) Iwl.\vr~t~ 

t,ltc slitlrr nntl I'Iln plane. 'l'llrir rn.t,io 

c (2) = hl (x)/b2 (x) (6.23) 

wllicl~ Ilns t,lm tlimcnsiorl of n 1cngl.lr plays nn imporl,a~~t, pnrt in lhc bllrory of 

Iubricnt,ion; it.8 value for the wholc cl~ntrncl, 

is somctimcs callctl the clw~rrrclerislic Ihicknes.3. 1Vit.h it.% ni(l, the crluatiot~ of' c w - 

tinuity (0.21) cnn Iw contractccl to 

q =; UlI, (6.25) 

from wllich it,s pllysirnl int,crprct.i~t,ion is cvitlcnt. Tlre pressure can now be n.ri1.t.cn 

and the pressarc gradient. 1)ecomcs 

It (z) > II for 0 < z < x,,, implyilll: 2)' > 0 

h(r)

120 

VI. Vcry ulow motion 

and for the pressure distribution 

2 (I-z) 

p(x) = po + 6pU-----. h2(2 a-1) 

(6.29) 

The relations hecome somewhat simpler if the shape of the channel is described 

by t,he gap widths hl and h, at inlet and exit, respect,ively, see Fig. 6.4. 'The c1lnr:lc- 

t,erist,ic witlt,h now becomes equal to the harmonic mean 

and the condition for positive pressure excess, eqn. (6.28), now requires that t,he 

channel should be convergent. In this notation, the pressure tlistribution is given by 

and the result.ant of the pressure forces can be con~putecl by int,egration, when we 

obtain 

with k .= h,/h,. The resr~lt~ant of the shezring stresses can be calc~~latctl in a similar 

manner: 

1 

It is interesting to note [el that the resultant pressure force possesses a maximum 

for k = 2.2 approximately, when its value is 

and whcn 

Tl~c corfficicnt of frict,ion F/P is propor16onal to hz/Z and can be made very small. 

The coordi~~ates of the centre of pressure, x,, can be shown t,o be equal to 

For small angles of inclinat.ion between block and . h c (k w I), tile pressure distri- 

hution from cqn. (6.29) is nearly parabolic, the charact,erist.ic thicltness and cent,re 

of prcsssnre being very nearly at z = 1 t. Pni,t,ing hm = h(4 1) we cnn find that the 

pressure tli[l:ronce 1)ccomcs 

If we compare t,his rcs~llt with t.hat for crccping nlotion past, n sphcro in cqn. (6.71)), 

we not,ice that in the case of t.he slipper the pressure tlifTcrrnec is grcalrr I I n ~ f:rct,or 

(lll~,,,)~. Since Ilh,,, is of the order of 500 t.o 1000 (1 = 4, A,, =x 0.004 to 0.00s ill). 

t,he prevailing prcssurcs are seen to assume vcry large val~~es-1. 'l'hc occll~~c?t~c~: of 

sucl~ high pressures in slow viscous motion is a pocwlinr proprrt,y of 1.11~ (,,yp(: of flow 

(~nrot~t~t~~~rotl 

ill I~~lriwdon. At, l h tmmo tirn~: itf is tw!op~iz~!(l I II:I~, I,IIc :III~CI~~ li~~.ttt

122 VI. Very rrlow motion 

ext.endcd to include the case of bearings with finite width [I, $1, when it. was found 

that the decrease in t,llrust supported by swh a hearing is very considernble due to 

the sidewise decrease in t,he pressure. Most theoretical calculations have been conduct,cd 

under t,hc xssumpt,ion of constant viscosity. Tn reality heat is evolved tshroogh 

friction and the temperature of t,hc luhricating oil is increased. Since the viscosiLy 

of oil dccrcascs rapitlly with incrri~sing t,c:tnpcrat,ure (Tahlc 1.2), the t,hrust also 

drcrmsrs grc:nt,ly. 111 rnorr rcwnt. t,in~cs 1'. Nahme [I01 extcntlcrl t,hc I~~drotlynarnic 

throry of' lubrication to inclutfe t,hc cffrct, of t.hc varintion of viscosit,y wit,ll t,?rnperaturc 

(cf. Chap. XI1). 

d. The Hrle-Shnw flow 123 

Here R, and Uc donote, respectively, the rndit~s and the peripheral relocit,y of the concentric 

journal (e = 0) and d ia the width of the gap. 

After the onset of inatnbility, the flow in tho gnp developn rrgulnrly spaced, ccll~rlnr vortices 

which n.ltcrnntcrly rotate in opposite dirertionrr. 'l'l~e nxcs of t,hesc vortireu coincide \vitl~ the 

circumferential direction, ns shown achemnt.irnlly in Pigu. 17.32 nnd in the photogrnpl~ of Fig. 

17.30. In n certain rnnge of Taylor nrnnbers, the flow in the Tnylor vortice~ remainn Iaminnr. 

l'rnnuition t,o turbulent flow ocrnrs nt vnlue~ of the Taylor nuntbcr which c,onrriderahly oxrcrcl 

the limit of ~t,ahility. Tho tho rcgi~ncw of (low (ns will be rrprnl.rrl in See. XVIIf nnil in I'ig. 

17.04) nre chnracterized as follows: 

T < 41.3 Inrninnr Cor~ot,to flow; 

41.3 < T < 400 Inininnr Ilow with cell~~lnr 'l'nylor vort.irm; 

T > 400 tmrb~~lcnl Ilow. 

\Vllcn the flow becomea nnat,nblc, the torqne nrting On tho rot,nt.ing cylinder inrrrancu sI.rrply, 

t~rcn~tsc? the kinetic energy nhrerl in tlw uccontlnry flow ~trt~st he c.on~pen~nt.etl by work. 

The snnie flow phenomena, generally speaking, occur when tlw henring is londed nnd 1,l1e 

gap witlt,h vnrie~ circumferentinlly, bnt, t .1~ dct.nila of t .1~ flow bcro~nc more romplrx. At,ten~pts 

hnvc lwon rnndc t,o cnlct~lnta tho tttrb~~lcnt llow in n gnp of n bonring with t.hr rritl of 1'rn11tIt,I'~ 

mixing length [Chap. XIX, eqn. (19.7)1. The set of these problems hnu at,t.mcked n wide circle of 

invr,stigntorn, utrch a8 I). P. Wilroek [19]. V. N. Con~l~nntineu~n 12, 3, 41. E. A. Snit)el nntl N. A. 

hlnckrn 114. 151 have writton two gcncrnl a.cror~nte t,hnt ront.nin ntlrnwotls litcrnt,ttrr rrfrrcnrr~. 

d. The Ilcle-Shnw flow 

At~ot~hcr remnrkn.l)lc sol~ttion of tho tfl~rcc-tli~ncnsio~~nl cilunl,ions of crrrping 

mol.ion, eqns. (0.3) and (6.4), can bc obt,ninctl for the case of flow botr\vcc~~ two 

parnllrl flat walls separated by a small tlistance 2h. If a cylindrical body of n.rbitx:~ry 

cross-section is inserted brtwccn the two plates at rightf nnglcs so that it conlplclcly 

fills tllc space bctwecn tlwn, the resulting pilttcrn of stm:~~nlines is idcnticnl wit811 

1,ltnt in potential flow about t,he anme sl~apo. 11. S. Jlclo-Shnw [7] nscd this tncthoil 

to obtain expcrimcntnl pnttcrns of strenn~lincs in potential flow about. :~rl~it,rary 

botlies. It is easy to provc that the solut,ion for crccping motion from O~~IIS. (6.3) 

nntl (6.4) possosscs the same st,rm.mlincs n.s the corrt:spording pot.crttial flow.

124 VI. Very slow motion 

of tho t,wo-tli~nc~~~sionnl potential flow past the givrn body. Tl111s ?I,, v, and p, satisfy 

tho equations 

J'irst wr nol,ioc all O~CO from 1.11~ soIut,ion (6.39) that 1.11~ cqr~at~ion of continuity and 

the cquntion of motion in the z-tlirection are satsisfietf. The fact, that the equations of 

motion in the z- and y-directions are also satisfied follows frorn thc potential character 

of ?I,, and v,,. Tho functions ?I,, and v, satisfy the condition of irrotationality 

so that the pot.cntin1 equations V2 71, = 0 and V2 v, = 0, where V2 = a2/i3z2 + 

iI2/B?p2, are sal,isfied. 

The first t\vo rqnnt.ions (6.3) reduce lo ap/az - /L a2u/az2 and ap/a?j = /L a2v/az2; 

t.11r.y nrc, howcvcr, uat,isfied, as seen from C~IIS. (0.39). Thus eqns. (0.39) rcprcsent 

a solt~t.ion of t.hn equations for creoping mol.ion. On tho othcr Irantl Ll~e flow rcprcsentctl 

by rqns (6.39) has the same streamlines as potential flow al)out the botly, n.nd the 

st.rcamlines for all parnllcl layers z = const arc congruent. The condition of no 

slip at tho pla.1.c~ z = f h is seen t,o be satisfied by eqn. (6.39), but the condition 

of no slip at the surfacc: of the body is not sat,isfied. 

'rhr ml.io of incrt,i:~ t,o viscous forces in JTele-Shaw motion, just as in the casc 

of i,l~c mot.ion of Irll)ricat,ing oil, is givc:~~ 1)y t.11~ reduced Itnynolds number 

whc-rr 1, tlcnoks a n11amat.cxistit: lincar tlimc.nsion of l.hc botly in the R., ?/-plane. 

If R* c:scwtls unit,y the inertia tmms Iwco~ne considrmlllc nntl the motion tlcvin1.r~ 

from 1.11~ sin~plc sol~~l~ion (6.39). 

'1'11~ solt~t.ion given by oqn. (6.30) can bc improvotl in the same mannor as 

Stolcc~s's solu1,ion for a sphcro or t.hc solution for very slow flow. The inertia t,crms arc 

cnlcnl:~.l.otl from t11c first approsimni.ion and introd~lce;l into the cq~tat,ions ns 

c!xt,rrn:rl forws, :~ntl an improvcd solution results. This was carricd out I)y F. Riegels 

1181 for t-he casc of Tfclc-Shnw flow past, a circular rylindsr. 

For R* > 1 fl~c st.rt:nmlinos in t.11~ various layers pnrallcl to the ~valls cease 

to l)o congn~cnl.. Tho slow p;~rt,icles near t,ho t~vo plates are tlefleclcd more by 

t,l~o 1)rcsmc:r of (.he hotly t11n.n i,hc fast,rr particlcfi near tho ccnt.rc. This causes t'hc 

st,rcan~li~~rs t.o n.pprn.r somcwhnt blurrrtl a.rd i,hc phcnomcnon is more pronounced 

at, tJtc rcnr of lhc botly than in fronl, of it,, Fig. G.6. 

Solutiol~s in thr case of cwcping motion are inherently restricted to very small 

Rrynoltls n1tni1)rrs 111 prineiplr it is possiblr to extend tho ficltl of applicat,ion 

to I:~rger Reynolds numbers by successive approximnt,ion, as mentioned prcvionsly. 

IIowevcr, in all cases the calculations become so complicated that it is not practicable 

to carry out more than one step in the approximation. For this reason it is not 

possilh to reach t11o region of motlcrntc Rcynolds numbors frorn this tlircctiol~. 

, .lo , all intmts and purposes the region of moderate Re~nolds numher~ in which 

1,110 in(:rh tincl viwo~~s forc~s nrn of ~~oIII~~I~~:LI~Ic m~~~~t~ilwlo 

I~II~OII~IIOIII~ IJIC lia>l~l 

of flow 11a.s not been cxtcnsivcly investigntcd by analytic means. 

It is, therefore, the more useful to have the possibility of intograting thc 

Navicr-Stokes cq~~ation for t,he othcr limiting casc of very large Rcynolds numbers. , 

'I'hns we arc lctl to the boundary-layer theory which will form the subjcc.1 of tho 

lollowing chapters. 

Fig. 6.6. IJcle-Shew flo~v 

past circulnr cylinder nt 

R* - 4, shr Iticpln [I:$] 

Referencer 

[I] Bauer, K.: Einfluss der endlichen Breite des Gleitlngcrs nuf Trngfiihigkeit uncl IIeibr~ng. 

Forschg. 1ng.-Wes. 14, 48-02 (1943). 

121 Constnntinescu, V.N.: Analynis of bearings oprmting in turl)ulrnt rcgin~e. 'l'rn~~s. i\SI\lE, 

Serb D, J. Ilnsic Eng. 84, 130-151 (l!)(i2). 

[3] Constnntinescu, V.N.: On the influence of inertin forces in tnrbulent and Inminnr selfacting 

films. Trans. ASME, Series F, J. 1,llbricntion Technolo~y 92, 47:1--481 (1970). 

[4] Constantinescu, V.N.: On gun lubrication in turbulent regin~e. Trans. ASMI':, Series 11, 

J. Basic Eng. 86, 475-482 (1964). 

[R] Frossel, W.: lteibl~ngs~viderntnl~d unrl Trngkrnft cin~s Gleitnch1111cs endlichrr Brrile. Po~.scIlg. 

1ng.-Wcs. 13, 65--75 (1042). 

[GI Giimbel, L., and Everling, 13.: Ileibung und Schn~ierung in1 Mnscl~incnbnu. 13crli11, 1025. 

[7] Hole-Shnw, H.S.: Inve.st,igntio~~ of thc nnturc of surfncc renist.nncc of wntar nntl of st,ron~n 

motion nndcr ccrtnin ox~icrin~cntnl conditions. T~OIIA. Innt. Nnv. Arch. XI, 25 (IA!)H); ~ co 

nlso Nnture 58, 34 (1898) tint1 Proc. Roy. Innl. 16. 40 (1899). 

(81 Knhlert, W.: Dcr Einllusx cler l'rlgheit.~ltrlfk bei der Irydrotly~~a~~~iscl~etl Schtniertnittolthcorie. 

Uius. Brnunschweig 1947; 1ng.-Arrh. 16, 321 -342 (1948). 

[9] Michell, A.G.M.: Z. Mnth. 11. Phys. 52, S. 123 (1905); mealso Ostwald's IClnssiker No. 218. 

(101 Nnhtne, F.: Beitrrigc zur l~ydrodynnn~incl~e~~ l'hcorio der 1,agcrrcibung. 1ng.-Arch. 11, 

191 -200 - (1040). \- 

[I I] Oseen, C. W.: Uber die Stokcs'sche Forrnel und iiher einc verwnndte Aufgnbc in der Hydrodynamik. 

Ark. f. Math. Astron. och Fys. 6, No. 29 (1!110). 

[I21 Pmndtl, I,.: The mechnnics of viscous fluids. In W. F. Ihrand: Aerodynnn~ic Theory Ill, 

34-208 (1035). 

[I31 R,iegels, F.: Zur Kritik des Hele-Shnw-Vcrsucl~es. Diss. Gnttingen 1038; ZAMM 18, 05- 106 

(1933). 

1141 $nib&, E.A., and Mncken, N.A.: The fluid mechanics of Inbricntion. Annual Review of 

Fluid Mech. (M. Van Dyke, ed.) 5, 185-212 (1973). 

[IR] Snibel, E.A., nnd Mnoken. N.A.: Non-lnmirmr bchnvior in bcnrings. Critic-nl review of the 

li(.rrnt,urr. l'mnx. ASMIC, Scric~ F, .I. I,~lliricntion 'I'ccl~nolo~y 96, 174---I81 (1974).

126 

VT. \'cry slow rnot.ion 

Part B. Laminar boundary layers 

Boundary-layer equations for two-dimensio~~d incon~l~rcssil,le 

flow; boundary layer on a plate 

iVc rww ]wot:t:rtl Lo c:x:~~nir~c 1.11~ sccontl 1irnit.ing cnsr, tr:~rncly t.l1:1I, of very srnnll 

visc:osil.y or very large Iteynoltls nrlm1)cr. An iml)orln.nt corrt~ril)ut~ion to (.Ire scit:ncc: 

of ll~ritl motion was mndc by I,. 1'mrrtlt.l [21] in 1904 wl~cr~ Irc clarifitxl t,l~t: cssrrlhl 

i~illr~r~~w of vis~osit~y in fIo\v~ i1.L Irigh Ibc~rnoltls 1111n11wrs nr~tl sl~owrtl I~ow tlrr 

Nnvicr-St,oltrs cqrrn(iorrs coulcl bo simplifictl 1.0 yinltl npproxirnnt,r sol~~t,inns for 

(Iris rnsc. \Ye shall cxplnin thrsc sirnplilicat.ions wit,h t,ho nit1 of an :crgnnlcnt wl~irlr 

[wrsrrves tlrc: physical pict,urc of the pl~rnorncrron, nntl it, will be rccnllctl t,l~:~t ill 

Ilrc 1)11Ilt of t.lw fluid ir~cxtia forrrs prctlorriitr:~t.r, tlrn i~ifl~lr~~t:o of viseo~~s forws being 

vn ~~isl~i~i~ly srnall. 

Il'or I,II(: S;II

128 VII. Ihr~nclary-layer rq~lnt,ions for t\vo.ditnrnuiotd flow; ho~~ndary lnyer on n plate 

layer, t.hc so-cnllctl bounclary layer. In this manncr there are two regions to consitlcr, 

rvrn if t,l~o tlivision t~et,wrcn t,hcm is not very sharp: 

1. A vcry thin lnyrr in the immct1i:~tc ncighbourl~ood of the body in which thc 

vrlocit,y grn.tlicnt normal to t.hc wall, a?~/r?y, is very largc (hnimrlrr~y hjer). 

111 t.l~is rrgiolr t,ho vrry small viscosiLy ,u of Ihc flr~itl cxcr1,s :rn rssc:nt,i:~l infl~rnnc.~ 

ill so f:~.r :LS 1.11~ sl~caring sI.rcss t -7 l~(i)~/i)y) tn:~y :ISSIIIIIO Inrgt: V:L~IICH. 

In gcncml it, is possible to st,at.c l.hat the Ll~icltncss of thc bonndary layer intmvtscs 

wit.11 viseosit,y, or, more generally, Lhat, it decreases as the Reynolds numbcr 

incrensrs. It. W:LS SCC~ from scvcrnl cx:lct solutions of t,he Navicr-Stolzcs cqrtntiorls 

I)rrser~tcd in C11n.p. V that t,ltc bonntlary-lnycr thickncss is prop~rt~ional to the sqliarc 

rool, ol' Izinc:m:tl,ic visoosit,y : 

6-fi. 

\\'llcrr malci~~g t.lrc si~nplific:~Lions t,o I)c inl.roducrd into I.he Nnvicr-St.ol;ns eqnnt.ior~s 

it, is nssr~lnrcl t.l~at, this thickncss is vvry small cornpnretl with a still ~~nspeciliotl 

linrnr tlimc?nsiot~, T,, of thc hly: 

0 < 1, . 

\Va shall now proccctl to discuss the simplification of the Navicr-Stokes 

rq~tnt,ions, awl in ortlcr t.o achieve it, we shall make an estimate of the order of 

rn:~gnit,ntlc of c:~,cl~ tnrn~. 111 the two-dimensional problem shown in Fig. 7.1 we 

sl~:l.li Itcgin I)y RSSI~III~~I~ Lhc wall tio be flat :~nd coinciding with tho x-dircctrion, t h 

1,-:txis being I~crl)cndic~llar to it. Wo now rcwritc the Navier-Stokcs equations ill 

tli~r~r~lsionlrss I;)r.m by rehrring dl .;clonit,ies to tl~c free-st.rcnm vclocit,y, V, and 

I)jr rcfrrring all lilwtr tlimnnsions 1.0 a rhar:~etcrist.ic Icrrgt,h, L, of t h I)otly, wltioh 

is so srlccl,cd ns t.o rnsnrc that the tlimcnsionlcss ricri~at~ive, au/i)z, clocs not, cxcocd 

rrt~it,~~ in tlrr rrgiot~ r~ntlcr consitlcmt,ion. 'rl~c pressrlre is made dirncnsinnlnss with 

p 1f2, nntl I irnr is rrfcrrctl to r,/ If. lPurt,I~cr, the cxprcssiorl 

With the :~ssnmpI~iotrs mndo ~~rwiously t.llc: tlirnrtrsionlrss l)o~~t~~l:~ry-l:~~ 

t,hiclzncss, dll,, for which \vc slra.ll retain t,hc syml)ol (\, is very small wit,ll rrsprct. 

1,o ~tnity, (0 < I). 

\\.'c sl~all, further, assume that the nolr-stcndy :~ccclcr:~t,ion i)~/al is of the! sn.tnct 

ordcr n.s (.he convective term 11. aupr which mcnns t.llat vory s~ttltlcn nccolcrnt.iotrs, 

s~lcl~ as occur in vcry lnrgc prcssnrc waves, arc cxclr~ctctl. Zn accort1:~rlcc wit,ll orlr 

previous n.rgtlrncnt sornc of the viscous terms must be of thr samc ordcr of rnngnit,ntlr 

:IS t.hc incr1,i:t t,crms, at lcnst in t,lrc immctlintc ncighl)ourl~ood of the wall, nntl ill 

spi1.c of t,lw srn:~llt~css of (,he fact,or 1/R. JIcncc sornc of t,hc second deriv:it.ivcs of 

vrloci1,y must t~ccornr vcry Ixrgo nnar tJtc wall. In nt:cortl:tncc with what w:~s wit1 

Iwforc: this can only :~pply to a27r./ay2 ant1 i)2~~/ay2. Since t.ho component of vc:loci~,~ 

p:~,rallrl t80 tl~c w:dl increases from zero :tt. t,ltc wall 10 t,ltt: vnluc: I in t211(. frwsl.rc::~rn 

arross 1.l1r 1:Lycr of t,hic:lrncsn (?, we 11;tvc: 

wl~c~rens i)v/if?j ,- 016 - I ant1 iPv/if?~~ - l/d. If tltcsc v:drtes :~ro inscrLctl ittt.o c:qirs. 

(7.2) at~d (7.3), it. follows from the first cq~~nt,ion of molion Il~at the viscous f'orrrs ill 

t,lrc honlltl:wy layer can bccomc of t.hc samc ortlcr of rnngl~ihtlc :ts thr inrrt,i:t ii)rc.c:s 

only if tho Itrynoltls nurnl)cr is of the ortlcr 1/02: 

. 

ll~c . first, cquat,ion, tlt:~t. of cont,inuit,y, rcrnnins unnltcretl for vcry lnrgc Ilcynoltls 

ntlmbcrs. The srcontl cquathm can now be sirnj)lifirtl Ily ttcgl(:ct.it~g iPi~/r?.c.2 

wit,h respect to a2i~/ay2. From t,he t,hirtl eqnation wc may infrr t,hnt i)p/al/ iu of t,Ire 

ordcr d. The pressure incrcnse ncross the bonndnry Inycr wl~icl~ woultl he ot~t,:r,ined 

I)y int.cgrn,t,iny! 

'l'lt~ls 1.11~ I)I.~.SSIII.~\ 

l.lw third tqnirt,ion, is of 1.11c otdrr 02, i. V. very st~~n.ll.

130 VII. Boundnry-layor cquntionzl for two-dimensional flow: bormtlnry lnyer on a plnte 

in a diroct,ion normal to t,he boundnry layer is pra.ct,ically con~t~ant.; it mo.y he assumed 

equal t,o Lhat at the ontm tdge of the 1)oundary layer whore its value is determined 

by Lire frict,ionlcss flow. 'Vho pressure is anid t,o be "impressed" on the boundary layer 

by the out.er flow. It, may, therefore, be regarded as a lrnown futlction as far as 

houndary-layer flow is concerned, and it depends only on the coordinate z, and on 

time t. 

At the outm ctlge of the bountlary layer the parallel component 7r becomes 

ccl~~a.l lo that in t,lre outer flow, U(x,t). Sinco there is no large velocity gradient 

hnrc, the viscous t.rrms in eqn. (7.2) vanish for largo vn.luca of R, and ronscrlnent,ly, 

for the o~rt~rr flow we obtain 

whore again tho symbols denote dimensional quant,ities. 

In the case of stently flow the equation is simplifictl still f~~rlhcr it1 thnl the 

pressure dopentis only on s. We shall rmphasize this cirrumstnnrc by writing ttir 

tlrrivntivc ns dp/tl~, so that 

This rnny also he writ,t.cn in tl~c usllnl form of TZrrnor~lli's cyw~tion 

1, -t 4 p 1J2 = eonst, . (7.6) 

r 3 

I IIC hoi~ttcl:u.y oontlil.ions for t.11~ cxt.rrnnl flow :Ire nrnrly tho snnic :w. for frict.ionlcss 

flow. 'l'ho I~onntlnry-lnycr t.hicIzncss is vcry stnnll :tntl t.hr trr~nsvrrsc velocif,y component 

v is very smnll at, the edge of t,hc boirntlary hyer (a/ I' - cT/L). 'i'1111s potcnthl 

non-viscous flow ahont Lhc hdy nndcr considerntion in wl~ich the prrpcntlicular 

vrlocit,y component, is vanishingly smnll nrnr the wall offers n very good npproxin~nt~ion 

1.0 tho nct,nnl cxtcrnnl flow. The pressure gmdicnt, in t.he 2-tlireet,ion in the boundnry 

k~ycr can I)c oht.ninod by simply npplying t,ho Bcrnor~lli erjl~at.iorr (7.5%) to the st,rcnniline 

at t,l~c wall in t.hc known po(.ent,inl flow. 

witslt tho I~onntlnry condit,ion.s 

1). The ucparaLiotl of n hor~ndnry lnyer 131 

!/-=0: u--0, a - - . O ; ?I:-m: 71:-(J(:r). (7.121 

11, is necessary t,o prescribe, in ntltiit.ion, n vclocit.y prolilo nt the init.ial sc,cl.iot~, 

1 : J,, say, by intlicnting t,he fi~nct~ion ?r(a,,,y). Tho problcrrl is t,I~lts scbrtl t,o ~.t~tll~c-r 

it,sclf tto the cn1t:ulstion of tho furtller change of a given vc1ocil.y profile wit.11 n ~ ivc,~~ 

pot~cntial motion. 

: 

. 1 , Ile mathemnt,ical simplificntion acllievcd on the prccctling i)n,Aos is col~sitl~:v:~l,lc 

it, is (,rue t,lint,, as distinct from the rase of orcoping inot,ior~, tho tro11-lit1car c:l~nmrtrr 

of t,ho Nnvirr-Stolrcs oqu~tion 11n.s been prrscwetl, hut of thc t.11rre origir~nl c~tl~rnt~io~ls 

for 11, I,, nntl p of t.11~ lwo-tlirnensioniLI Ilow proble~rr, ono, I h cqurlt,iot~ of rnot,iott 

normal Lo tho wall, has been clroppcd con~plct,ely. Thus the number of 11rllinc)wrrs 

has 1)rrn rrd~~cctl Ijy one. Tl~crr rcrnnins n s?jst,cni of t.wo si~rt~tll~:~.~~col~s t.cll~:~.l.io~~s 

Sor tjl~(* LWO II~I~IIOWIIS I& a,n(l 11. '1'11~ pr(:ssllro cc:~sctl lo IN- :III IIII~

Fig. 7.2. Sepnration of tlin 11ot111dnry layer. 

;I) I'low past :i body wit11 sopamtion (S = point 

of soparation). b) Shape of st.rrnnllines near 

r 7 

I l~r. ~inint of sc.lt:~r:~.tio~~ i:; tlofit~ctl :IS 1,I1r limit I)et,wccn forwar(l and rcvcrsc llow in 

thc. I:I yvr in tht. immctli:lt.o ncigh1)ourhootl of t,he wall, or 

1 

In order t.o nnswcr the question of whether and where scpnrat,ion _~~ru_rg,~ it 

is nc~ccssn.iy, in gc~lcr;d, first to intcgratc tho boundary-lnycr eqrtntions: Chcmlly 

spc:tlting, the I)orlntl;~ry-lxyrr equations are only valid as far as t11c point of scpnrnt.ion. 

A short (list,:~ncc tlownst,renrn from thc poinl, of scpnmtion lhc bountlnry-l:lyrr bec:otnvs 

so t,l~ic:lc th:lt, I,hc nss~~rnpt,io~~s whic:lt wrrc tn:~tlc in t.I~t! clt:riv:l.l,ion of t,h 1)01111tlary-ln.ynr 

rqu:lt,ions no longer apply. 111 tJ~c c?sc of Lotlies with blunt. stcrns_t,ho 

sopnr;~.(.etl I)or~ntl:~ry Ia.yor displaces t,hc potcr~t~ial How from t h body by an npprccin1)lr 

tlist,;~ric:o xntl t h prrssnro disteribnt,ion in~pressctl on thc bountlary lnycr nus st 

I)c tlrt,crmi~~ctl I)y cxpcrimcnt,, I~eoarrsc the cxt,crnn.l flow tlcprntls on tho phcnomcn:~ 

cwttncc:t.c:tl with scp:lrntion. 

'I'hc fact t,l~nt, sepnmt,ion in stcntly flow occnrs orly in tlccclcratcd flow (tlp/tl.~: > 0) 

can I)(? rnsily inft:rrctl from a. consitlcmthn of the relation t~et~wecn the prcssurc 

gr:ltlicmt, tlp/tl.r and lfhc vclocil,y tlisl.ril)nl,ion II.(?J) with tllc aid of the boundary-layer 

t 'I'll~ velonit,y profilc n.t, the point, of urpnrnt,ion is seen to I~ave a perpendicrllnr t.nngent at the 

wall. 'Cho ve1ocit.y profiles clownutrennl from tho point of ~eparation will sl~ow regions of reversotl 

llow near tho wall, Vig. 7.2~. 

c. A remark on the inBgrat.ion of the boundary-layer equntions 183 

rquations From eqn. (7 11) wit.11 the bounclnry contlitiorls 71 T v =- 0 wc 11;lvr at 

?/ = 0 

In the irumediat,e ncighbourhood of the wall the cnrvatnre of t,he volocit,y profile 

depends only on the pressure gradient, and t.he curvature of the ve1ocit.y profile 

at the wnll changes its sign with the prcssure gradient. Por flow with dccrcnsing 

prcssure (accelerated flow, dpldx < 0) we have from eqn. (7.15) that (a2u/ay2),,,, T: 0 

and, therefore, a2u/ay2 < 0 over the whole width of the 1)oundary layer, Fig. 7.3. 

Jn the region of pressure incroase (dccelcrntctl flow, dp/dx > 0) we fi nd (a2u/ay2) 1 0. 

Since, however, in any case a2u./ay2 < 0 at, a largc distance from the wall, Lherc 

must exist a point for which a2~~/ay2 = 0. This is a point of inflexiont of the v~locit~y 

profile in the bollntlnry Inycr, Fig. 7.4. 

Fig. 7.0. Velocity distribut,ion in a borrndnry Fig. 7.4. Vclocil,y dislribution in a borlt~dar~ 

layer \vit,h pressure derrease layer with pressurc increase; 1'1 - point, of 

inflexion 

c. A remark on the integration of the bnundnry-lnyer rquntinns 

In order to integr:lto t.11~ boundary-layer eq~mtions, whethr in thc non-st.oady case, cqns. 

(7.7) and (7.8), or in the shady case, cqna. (7.10) and (7.11), it is ofkn convenirnl to int,rotl~lcr 

n stream function yt(x, y, 1) defined by 

U= av .=-a~ 

ay ' ax ' 

(7.17) 

t Tho exisknce of a poi111 of inllcxion in tho vclocil,y profiln in tllr boundiiry Ixyer i4 inlport.ant 

Tor its stability (trnrleitiot~ 

from laminar to turhtllent. flow), ueo Chnp. XVI.

'1. Skin friction 

\VlIat1 t.11~ I)ollntl;lry-Iayw rqust,ions arc int~grnt~ctl, t.hc ~rloc:il.~ tlislribut.ion 

(:an I)c tledr~ood, ant1 t,lrr position of the point of srpnmt.ion can be dctcrrninotl. 'I'lris, 

in t,urn, perrnit,s us t.o m.lculxt,c lhc visrous tlrng (skin frirt.ion) nrorlrltl t,llo surface 

I,y a silnplc process of int,carnl,ing tho sllmring st,rrss nt t.hc wnll over thr surface 

of t.lio 1)otly. 'I'l~c sllraring stmss at, t.hc \v:lll is 

I 

L), = 1) to cos 4 ds , 

1-0 

r. The boundary lnyrr along a flnt plnte 

e. Tlw bowldnry layer nlong n flnt plnte 

il. was cliscussetl hy 11. Bli~si~ts 

[2] ill his doclor's t,l~csis at, (hct.t.ingrn. llrt 1.11~ 

lending edge of the plate L)c at x -= 0, the plate being parnllcl to the r-nxis and 

infinit.cly long tlownstmam, Fig. 7.6. Wc shall considor sleatly flow wit,h ;I frcc- 

sLroa.m volocitty, [I,, whiclr is pnmllrl to the x-axis. Tlrc vclooity of potmlinl flow 

is corlst,xnt in this case, and, thcrdorc, dp/& z 0. Tho boundary-lnper oqnntiorls 

(7.10) t,o (7.1 2) I~cco~nc 

3 36

136 VII. lloundnry-layer equations for two-dimensionnl flow; boundary laycr on n plnte e. The houndnry layrr along a flat, p1at.e 137 

ces x can be made identiral by selectling suitable scale factors for u and yt. The 

scale fact,ors for u and y appear quite naturally as the free-stream velocity, U, 

ant1 the bountlary-layer thickness, S(x), rcspcctivcly. It will be noted that the latter 

increases with tho current distance x. Ilcnce the principle of similarity of velocity 

profilrs in the boundary layer can be written as u/lJw = 4(?//6), where the func- 

tion 6) must be thc same at all clistanccs x from the lcatling rtlgc. 

We can now estimatc the thickncss of the boundary layer. From the exact 

solut,ions of tho Navier-Stokes equations considered previously (Chap. V) it was 

found, c. g. in t,hc case of a suddenly accelerated plat2c, that (1 - I/yE , where t 

clcnotctl tho time from the start of the motion. In relation to the problem under 

consideration wc may sub~t~itute for 1 the time which a fluid particle consumes while 

travelling from the Icading edge to the point x. For a parbiclc outeide the boundary 

layer this is t - x/lJ,, so that we may put S - 1/ v x/lJ, . We now introduce the 

new tlimcnsionless coordinate 77 - y/S so that 

'I'hc equation of continuity, as already tliscusscd in S~L. VIId, can be integrated 

by introducing a stream function y~ (x, y). We put 

v = I/VZ~/(T), (7.25) 

where J(7) tlcnotcs 

poncnLs become : 

the dimensionless stream function. Thus the velocity corn- 

.,‘=" = 3!?!=Um~~(,,), (7.26) 

ay a! ay 

the primc denoting differentiation with respect to q. Similarly, the transverse 

vclocitv com t~onent is 

Writing down t.hc further tcrms of eqn. (7.22), and inserting, wc have 

Afkr simplification, the following ordinary differential equation is obtained: 

J J" + 2 /"' = 0 (Blaaius's equation). (7.28) 

As seen from eqns. (7.23), as well ns (7.26) and (7,.27), the boundary conditions are: 

71-0: /=O, /'=O; T=W: /'==I. (7.29) 

t Tho prohlem of a//inity or similarity of velocity proflr~ will be considered from n more general 

po~nt of view in Chnp. VJII. The more exnct theory shows that the region immediately behind 

tho lending eclgo miwt bo excluded; ROC p. 141. 

In this cxamplc both partial clifferential equations (7.21) and (7.22) have bccn 

transformed into an ordinary different,ial cquation for thc stream fnnclrion by the 

~imilarit~y transformation, eqns. (7.24) and (7.25). The resulting diffcrcnLial equation 

is non-lincar and of the third ordcr. Thc Llrrce 1)orrnd:wy conditions (7.29) arc, 

thcrcforc, sufficient to ~let~crminc the so111tion complctdy. 

'I'ht? nnalyl,ic: cvrdlr:kl,ion of Iho sol:rl,iort of lho tlifi:ro~lLinl c!tllrr~l,ior~ (7.28) is 

quite t,cdious. 11. Ulssius obtained this solution in thc form OK a series expansion 

:wound 71 = 0 and an asymptotic expansion for 71 very large, the two forms being 

matchcd at a suitablc valuc of 7. The resulting proccdurc was described in detail, 

1)y 1,. Prandtl [22]. Subscqucnt to t,hal,, I,. Bairstow [I] and S. Coldstcin I131 solvcd 

thc sanlc cquation but with the aid of a slightly modified procedure. Somewhat, 

rarlicr, C. Tocpfer [27] solvcd the Rlasius equation (7.28) numerically by thc 

:ipplic:ation of thr mcthod of Runge and I

138 VII. no~~ndnry-layer cqnntions for two-dirncnaionnl flow; houndary layer on a pink 

This means that at the outcr edge there is a flow outward which is due to the fact 

that the increasing boundary-layer thickness causes tho fluid to be displaced from 

the wnll as it flows along it. There is no boundary-layer separation in tho present 

case, as t,he pressure grndient is equal to zero. 

J. St,cinhcuer [25] pr~hlisllctl a systctnatic rcvicw of t,hc solut.ions fro TJln.sius's 

equation. 111 part,ic:ulnr, hc providcd a tli~n~~ssion of t~llr chnrnct.cr of the sol~rtions ill 

the intcgrntion rnngc where r] < 0 in the presence of a varict,y of bountlnry conditions. 

It turns out. that t,llcrc exist, three set* of solr~tions which differ from each other by 

their nsyrnpt.otio ldmvior atf 7 + -m. Apart from t01r larninnr hountlnry layer on 

a flat platme, the solutions which can I I givcn ~ a physically mcaningrul intcrprctnt,ion 

include Inminar flow between t,wo parallel streams of which the two-dimensional 

hnlf-jet. is a special ca.se (scc See. IXII), larninnr flow with suction or blowing nt right 

angles (see Src. XIVb), as well ns tho laminar bonntla.ry Iaycr formed over a wn.ll 

moving parallel to thc stream in the same or in the opposite direction. 

Skin friction: Thcskin friction can be easily clctertninrtl from the precotling tlnta. 

From qn. (7.19) we obtain for one side of the plate 

wllcrr h is t h width and 1 is the Iengt,h of the plate. Now tho hlcal shearing stress 

at the wall is given by 

wiLh /" (0) -- a - 0.332 from Table 7.1. llence the ditncnsio~~lrss shearing st,rcss 

1)ccornra : 

Co~~sccjrmttly, from cqn. (7.30), t,lw ski11 friction of one sitlc I)ct:olnt.s 

nntl for a. plnt,r wrttrd on I)of.h sitlrs: 

1 

It, is rrtn:l~kn.l)lr tI1:tt. fhn ski11 l'ric+iotl is 1rroport.ionnl t.o t 1 1 ~ powrr # of vclocit,y 

whcrcns in rrcx?l)ing rnot.io~l t.hcrc was ~)roport,ionali(.y t,o the first, Imwrr of vclocit,y. 

I~i~rt~l~rr, f,llr: t1m.g incrca.sos wil,h I.llo sclrtwc roof, of t,hc Icngt.l~ of the 1,la.t~. This 

(.:I.II l)c i~~t.c~rprotc-tl :IS showing t,ha.t. t h (lownst.re:~rrl ~ 

1)ort.ions of the pI:~t.(: cont.ril)~~Le 

11rol)ort.io11:1t.c-ly Irss to t.ho t,ot,nI tlr:lp t0wn t.hc portio~ls nra.r t.11~ Irntling rtlgr, 

c. The bountlnry laycr along n flnt, plah 1 3!) 

Tahle 7.1. The function /(v) for tho boundary layer along a flnt plate at zero incidence, after 

L. Mowarth 1101

140 VJ1. Bounclnry-layrr rquat.inns for t,~rn-tli~iic.~ixio~~nl flow; I~oundnry layer o~i n plnte e. Thc boundary lnyer dong n flat, plate 141 

because they lie in the region where thc boundary laycr is thicker and where, consequently, 

the shearing stmss at thc wall is smnllcr. Introtlurinp, as usual, a dimensionless 

tlrng coefficient by the definiCion 

2 1) 

C r-la;lu,F, 

-- 

whrrr A = 2 1) 1 clsnolcs thc wcttcd surface aroa, we obtain from cqn. (7.33) thc 

formula : 

I 

Ircrc R, = 11, I/v denotes thc RcynoItIs number forrnctl with the Icngth of the 

platc: and the frcc-strcnm velocity. This law of friction on a plate first dcdnccd by 

IT. Blasius, is valid only in the region of laminar flow, i. e. for R, = IJ, l/v < 5 x 10" 

to I06. It is rcprcscntctl in Fig. 21.2 as cwvc (I). 111 t,lw region of t.urbulcrit, motion, 

R, > loG, the drag bccomcs considerably grcatcr than ti,i~t givcn in cqrt. (7.34). 

Rou~dnry-lnyer thickncss: 11, is impossible to int1ic:~t.c a hor~ntlary-layer l.l~ic:lzncss 

it1 an ~lnamhiguous way, because tlic influence of vi~cosit~y in the bonndary laycr 

clccrcascs asymptot,ically out,wards. 7'110 parallel component,, u, tends asymptotically 

to the valuc [Im of thc potcnLiaI flow (thc function /'(?I) tends asymptotically to 1). 

If it is tlcsircd to define thc boundary-layer thickness as that distance for which 

IL --- 0.99 [I,, thcn, as scon from l'ahlc 7.1, q 5.0. ITcnec t01c bonnt1:~ry-laycr 

t,lliclrness, as tlcfinctl Ilcrc, becomes 

A physically meaningful nrcnsurc for t.hc 1)ound:wy layer t.hiclcness is tJro rlisplnrxmnt 

lhickmxs (TI, whit:li was dreatly i~~trotlucntl in eqn. (2.0), JGg. 2.3. 'l'llc tlisplaccnlcnt 

thickncss is that distance by which thc external pohntial field of flow 

is displaced ouLwards as a conscquencc of thc decrease in vclocily in tho 1)ountlxry 

m 

layer. Tlic dccrcasc in volumc flow due to tlie influence or fricl.ion is j ((I,, 

- 0 

so t,hnt for 0, wc havc thc definition 

--I&) cly, 

Wilh 1r./17, from cqn. (7.26) we obtain 

where q, denotes a point outside the boundary layer. Using tlic value f(q) from 

Tablc 7.1 we obhin q, - / (Q) = 1-7208 and hencc 

. lltc . clislnncc y =; dl is sl~own in Kg. 7.7. '1'11is is t11c distnncc by wlucl~ tl~c strcarnlines 

of the external potential flow are displaced owing to the effect of friction near 

the wall. The boundary-layer thickness, 6, givcn in eqn. (7.36), over which the 

potential velocity is attained to within 1 pcr ccnt. is, in round figures, three times ' 

larger than the displacement thickness. 

Wc may at this point cvaluate the momcitli~m thicknms a2 which will be used 

latcr. The loss of morncntum in the boundary layer, as comparcd wilh potential flow, 

m 

is givcn by ,g J IL(TJ, - u) dy, so that a new thickness can be defined by 

0 

m 

e ~ ~ ~ b , = ~ ~ u ( ~ ~ - - ~ ) d y , 

u-0 

aZ =I & (1 - &) dy. 

Numerical evaluation for the plate at zero incidence gives: 

Y-0 

- 

4 = 0 . 6 1 ~ / (momentum ~ thickncss). (7.39) 

It is necessary to remark hcre that near the leading edge of the plnte tbc boundmy-hycr 

theory acascs to apply, sincc thcrc thc assumption 1 a2u/8x2 1 

is not satisfied. Tho boundary-laycr theory applics only from a ccrlain valuo of 

the Rrynolcls numbcr R = lJ, x/v onwards. Thc rclntionship near tho Icatli~rg 

edge can only be found from the full Navier-Stokes equations becnusc it involves 

a singularity at the leading edge itself. An attempt to carry out such a calculation 

was made by G. F. Carrier and C. C. Lin [5] as well as by B. A. Bolcy and M. B. 

Fricdman [3]. 

Experimental inveatigationa: Measuremenk to test the theory given on the 

preceding pagcs were carried out first by J. M. Burgers [4] and B. G. van dcr lleggc 

Zijnen [16], and subsequently by M. Hamen [14]. Particularly carcful and com- 

prehensive measurements were reported later by J. Nikuradse [20]. It was found 

that the formation of the boundary layer is greatly influenccd by thc shape of tho 

leading edgc na well as by thc very small prcssure gradient which may exist in tho

142 VII. Hor~nclnry-lnyrr rq~~ntii~t~s for t\vo-di~i~rr~nio~l~~l Ilow; I)o~~nilr~r,y Inyrr on n plelo 

Fig. 7.9. Vclocity rlist.ril)~~lion in tho Inn~i~inr hounrlnry lnyrr on n IlnL plntr nt, xrrn i~~rirlrncr. 

nn ~ncasr~rcd by Nikr~rndso [20] 

I:ij!. 7.10. Lord rocffiricnt 

nf 6Izi11 frirtion 011 a flnt, 

])l:iln at ZPITJ iriridr~lco in 

0.001 

i~~con~prcssi blo fln\v, dclcrminc~l 

from tlirocl. tncnnnrc- 

0 0005 

twnt of shearing strcss by 0.0067 

O hd~recl skm fiiclioon measurement 

fmm velonfy profile 

D/recl skin frctlin measuremen/, x - 28 6 cm 

0 t r ,x-56cm 

- ! I l l l l l l l I 1 I 

,. 

I ltc laminar law of friclion on :L flat phtc was also subjectctl to careful expcrirncntal 

verification. The local shearing stress at the wall can be determined 

intiirecthj from the slope of the velocit,y profilc at the wall together with eqn. (7.31). 

In rccrnt t,imcs IT. W. Liepmann and S. Dhawan [18] measured the shearing stdress 

tlirocI.ly from the forco acting on a small porl,ion of t,ho plnto which wns nrrnngctl 

so t,l~at it could move slightly with respccl to the main plate. The results of tl~eir 

wry careful measurements arc seen reprotluccd in Fig. 7.10, which shows a plot 

of the local coefficient of skin friction cf' -- to/k Q 1Jm2, against, thc R.cynolds 

number R, = 11, z/v. In the range of R, = 2 x 10"o 6 x 10"both laminar and 

t,urbrrlcnt, flows arc possible. It can be sccn that direct and indircct mcas~~rcmcrtt~s 

nrc in cxcrllont agrrorncnt, with each ol,l~or. Mcns~~rcrnonl,~ in tho lnn~innr rnnga give 

a strilcing (!o11fit311:kt,io11 of 1Hwius's oqn. (7.X) frm~ which cff .?: 04Wi/d~~. 111 

the turl~r~lcnf~ range there is al~o goo(1 ngrecmcnl with I'rntdt.1'~ thcorctif:nl forrnr~l:~ 

which will bc deduced in Chap. XXI, cqn. (21.12). 

r 7 

Jllc conlplctc ngrccmcnt bctflwcen t~hrorcticnl and expcrin~cntd rcsr~lt.~ wlliol~ 

cxist,s for the velocity distribut.ion nncl t,hc shcnring stxcss in a 1aniinn.r honndn.ry 

lnyer on s flat plate at zero incidence that, has hcen hrougl~t inta evidcncr in Figs. 

7.9 antl 7.10 for the rmge R, > lo5 nnequivocally dcmonstratcs thc valitlity of t,he 

bountlnry-ln.yer n.pproximntions from the physicnl point, of vicw. TII spit.(! t,I~is,

144 VTI. Bonndary lnycr equations for two-dimensional flow; boundary layer on a plate f. Ronndary layer of higher order 146 

cartrain matl~emnticjnns have axpenclod much effort to create R. "mnthemnf,icel proof" 

for t,ho validity of theso simplifications; in thiw connexion consult the work of 

11. Schmidt and I

146 VII. I~o~~ntlnr~ Inyrr cq~~ntions for t~vo-tli~~~r~isio~~nl 

flow; bo~~ndnr? lngrr on n pink 

'rllrsr are rxnctlg I'rnt~cltl's bo~tridnry-Inye1 rcl~~ntions, rrlris (7 10) nnrl (7 11) trn~lsformcd to 

coordinntrn x, N. In ntldition p1(1) - I'I(J. 0). 

'rhr nnlt~tion 7cr (T, A') nllons us to cotnptte the rlinplncctnetlt tl~irknc~ is!, drfitirtl as 

'Shr rcl~~ntions of first ordrr, rqns (7.4!)), do not rontnin lhr Ht*ynolcl~ 11111i1l)rr explic.itly. It 

folhvn 111111 uI (.r, 8) nnd cl(.r, ,V) nit~st nlso I)(! indq)en(lrnt of lhc I~C~IIO~~R 

n~~n~lrrr. 'I'his 

prvvrs thnt t hr 1ot.nt ion oft IIP point of lnminnr sepnrnt.ion is inrlependcnt oft hr Ilrynolrls nrl~nher. 

wit 11 t hr Iminrhry rond~tions 

N = 0: 1(2 -- 0.19 T 0, 

AT -+ m: 1 , ~ = llz(r. 0) - 1i lrl(3, 0) N, 

112 = I'z(T. 0) 4 1i IJ; ( ~ ~ A'. 0 ) 

'Shr o11t.rr I)o~~ntInry ronrlitionn (i. r. for h' -+ m) of tlir inner solutionn nn u.cll ns the inner houndnry 

c~onditio~~s of the outer sol~~~iorln (c. g. rqn. (7.45) for l'~(r, 0)) follo\v fro111 t.he matrlril~g of 

t.hc inner nnd orlt.nr solntions; ner nlso 171. 

'I'hr systc~ii of rqr~ntions (7.52). (7.53) for ll~r! ~erond-nrdrr 1)01111dnry Iny(-r loo doru not, 

ronl,nin t.l~c: Iltyolcls IIIII~IO~T t*~pli(*illy. I10wrvvr. it (:onlninu solntio~~s of lil.st, orb nrd is nlorv 

t~~tr11~iVr Ihn I.!Ic Iirsb-order nynIt.ni. but it w~~sistn of linrnr di~li!rcnIinl (.tltllltioll~. 1'01. Illis 

ronson, it. is ~~ossible, in t,~~rn, In wpnrntc the n.l~olr so111tion into n sum of pnrtinl solutions. 11. 

has l~rco~iir r11s101nnry 1.0 split tihe soll~t,ion illto rr r~~rvnt,ure t.crn~ nnd into n tliuplnre~nent term, 

IIIII wt. shrill nol, IIII~RII~ this rlinn~~~sion any f~~rther I~ere. 

Uur tn t.l~c In(-t t,l~nt tho 011rvnt,11re of t.1~ wall is nrro~~ntcd for in t,lic sroond-order theory, 

thrrr nppcnrs n prrssurc grntlicnl in Ihc dirrrtion normal lo Il~r wnll. For this rrnson, the prmsurr 

nl. Il~r \~IIII I~txwnwn cliktw~t fro111 t,lrnt. whirl^ is i~~~~~rrssrtl 011 t.11~ 1)011ndnry I~iyrr Ily tlie outrr 

flow. Inlrgr~~ting ncrnss Ih

148 VTI. Boundnry lnycr eqnations for two-dimensionnl flow; boundary lnyer on n plate 

Fig. 7.1 1. Skin-friction coefficicnt, 

of n flat, plat^ of finite 

length at zero incitlcnco 

(I) 'f'llrory aftrr 11. Illnsius,oqn.(7.34) 

(2) l'l~cnry nncr A. 1'. Mrlrsitrr I 1RI)I. 

rqn. (7.00) 

A Tlirnry nrter Ih~lain (nolullor% of 

Nnvier-Stnkrs cqoaLio~~a) 

Here, the trailing edge has been ncco~rnted for, bnt not the displacerncnt effect. 

The dingrnrn in trig. 7.11, rcprotlnccd from the work of It. E. Melnik nnd It. Chow [18a], 

shows t,l~at t h vnlucs of c, computed with t h aid of eqn. (7.60) ngrec very well with tlie results 

obtained frorn the complck: Navicr-Stokes equntions as well as with those of ~nensnrements down 

to RI = 10. At Rl = 40 eqn. (7.60) leads to c, = 0.316 which is less than 2% in excess of the exact 

vnlnc cl = 0.31 1. 

Sertion 1Xj will ret.urn to the discnssion of exact soltltions of houndnry-layer equations of 

srcoritl order. 

References 

111 13airntow, I,.: Skin friction. J. ltoy. Acro. Soc. 19, 3 (1025). 

121 I%lnsius, M.: Grenzuchichtct~ in Fliimigkeiten mit kleiner JEcibnng. Z. Mnlh. J'hyn. .SF, 1-37 

(1008). Engl. transl. in NACA TM 1256. 

131 Bolcy, U.A., and Friedman, M.U.: On the viscons flow aro~rnd the lcading edge of a flat 

plntc. JASS 26, 453-454 (1059). 

141 Ihrgcrs, J.M.: The motion of n fluid in the borlndnry lnyer along a plane smooth surface. 

l'roc. First Intcrn. Congr. of Appl. Meell., Delft 1924 (C.B. Biezeno and J. M. Burgers, ed.) 

Delft, 1925, pp. 113-128. 

[R] Carrier, G. I?., and I,in, C.C.: On t,tie nnturc of t,ll& bonndnry layer near t,lic leading edge 

of a flnt plate. Qnnrt. Appl. Mnth. VI, 63-68 (lp48). 

[ti] I)linwnn, S.: Direct n~crcqurcmenta of skin friction. NACA Rep. 1121 (1953). 

[7] Van Dyke, M.: Higher npproxi~nntiona in boundnry layer theory. Pnrt 1: General analysis. 

JI'M I4, lti1- 177 (1962). I'nrt 2: Application tm lending edges. JFM 14, 481-495 (1!)62). 

I'nrt 3: l'nrrrboln in uniform streani. JI'M 1.7, 145-IR!) (1964). 

[R] Van Dyke, M.: I'crtnrbntion rnct,hodu in fluid mechanicu. Acnde~nic Pre-%,New York, 1964. 

I91 Van 1)ykc. M.: Higher-order boundary Inyer theory. Annonl Iteview of F'luicl Mech. I, 

2tiR 2!)2 (I!)(;!)). 

[lo] Geraten, K.: Grenzschichteflkkte hiiherer Ordnung. Anniversnry volume com~ncmoratit~g 

Professor H. Schlichting's 05th anniveranry (Sept. 30, 1972). lbp. 7215 Inst. f. Stromungumech. 

Techn. Univ. at Brnunschweig, 29--53 (1972). 

1111 Gersten, I

CIIAFTER VIII 

Gencral propertiee of the boundary-layer equatione 

12cforc: passing to t.lw ca.lcr~l:ll~ion of furtl~cr cxarnplcs of bountlary-layer llow 

in t.ha next, chnpt,rr, we prol)os': first, t.o tlisc~lss some grncral propertics of the bound- 

:~ry-l:rycr t:quatiorls. 111 tloing so wo shall ronfinc our atttlention to steady, twotlimension:tl, 

ant1 ir~c-o~r~~)rt~ssiI)l(~ l)o~~n(lar~ I:ly~rs. 

Alt,hougl~ t.Iw ~)o~~ntl;trj--l:i.yt~r rcl~~:ttions have hen simplified to a great axtmt., 

as coml)arctl \vit.l~ t,hr Navir~.-St.oltcs rclr~at,ions. thoy arc still so tlifficult from t'he 

matJrrn~at.ical point of vicw tht. not vdry marly gcncml ~t~atcn~rnts :rbout tlicrn 

ran I,c matle. 'I'o I~c:gin wit.ll, it. is import-antf to not.ice that t,he Navier-Slolrcs 

aqlla.t,iotls :trc or t,I~t? rllipt.ic. typa wit,h rrspcct to tllc c:oordin:~l,cs, whcrms Pranrltl's 

l~o~~t~tI:~.ry-l;t~~t~r cq~~:~,I~iot~s :tro p:ir;th~lic, It, is :L cowo~~~~t~rwr or lhc sin~plifying 

nss~rrnpt~iol~s in Imuntl:rry-layer t,hcory that tho prcssuro can be assumeti constant 

in R clirrction n.t right :~nglcs to the hountlwy Inycr, whereas along tho wall the 

1wess11rc can be rcprdetl as bcing "imprcsscd" I)g the external flow so that it bccwncs 

a givrn f~lnc:I.ion. The rcsr~lt~ing omission of t,hc arlnntion of motion porpcntlicul:ir 

t.o the tlirccliott of flow can be i~~tcrprctctl physically I)y stat,ing that a fluitl 

~);trt.ic.la in tha l)our~tl:~ry Iaycr has zcro mass, and sulTcrs no frictional drag, as far 

;rs it,s motlion in t.11r t.mnsvcrsc ttirecLion is conccrnrcl. It is, tl~crcforc, clear t,ha.t8 

with sr~t:lr f~lnrl;trnrt~t,al cl~angcs introtl~~ccd int,o the cqtlat,ions of n~ot~ion we mnsb 

nnt.ic.ipatc t.ll:~t, tllrir solut,ions will exhibit certain rn:ltI~cmatical singnlarities, 

nn(1 t.ll:lt, :tgrrrrnrnt I,c:t,wrcr~ ol)scrved :t11(1 ~illt:ulat,ed phrrlon~ona cannot always 

'I'ho assu~npt~ions which warc rnatlc irt tho tlcrivation of t,hc tmuritlary-layer 

rq~tntions are s:~tisfictl with an increasing tlcgrce of accuracy as the Itaynolds number 

ir~c:rrnses. 

,, 

l hils hountl:~ry-layer thcory can bc regardcd as a process of nsymplolic 

i~itrgmtiol~ of t,llr Nn.vicr-Bt,olrrs rqnn.t,ions at wry In.rgc Itcynoltls nurnl~c~~s*. 'rhis 

sl.:~trmrnt, Irntls 11s now to R tlisc~ission of the yclnt,iortship bet.wcen t(11c Itcynoltls 

nirmhcr and t.he chn.rnctt~ris(.ics of a t~~indary 1h.yer on our individrlal body-under 

consitlcrat,ion. It, will 1)a reanllctl t,hat in thctlcrivat~ion of the boundary-layer equations 

--- - - - - -- 

t C/. Set-R. Vllf nncl IXj. 

* 'I'llo srg,~n~rnt, t~ont,ninctl in tl~in nwtion wan nlronrly tlinc~ls.st?d in Sec. Vllf on high-order 

II~)~~~OX~III:~~~OIIR. 

'I'IIv itll~l~lilir~~tio~~ is givm I~c.rr for t.lw mltc of Iwl.l.rr ~l~~tlrrnt~it~tlil~g. 

a. Drpel~denrc of the rhnmcteristicn of n. boundary lnyer on the llry1101dn IIIIOIIIPT 151 

tlinler~sionlcss quantities were used; all velocities were referred to the free-stream 

velocity IT,,, all lengths having been retfuced with thr aid of n cl~aractcristic length 

of thr botly, 11. 1)cnoting all tlirnensionless magnitutfes I I a ~ prime, thus v/fJm, =u', 

. . . , x/L = z', . . . , wc obtain the following equations for the steady, two-tlimrnsionnl 

CRSR : 

scc nlso cqs. (7.10) t,o (7.12). Itere R dcl~otcs t.lro ltcynolds nurnbrr for~ntyl wit.11 t,)lc 

nit1 of 1.11~ rcfcrencc qunntitics 

It is seen from eqns. (8.1) and (8.2) that, the boundary-layer solution dcpcnds on 

ow parameter, the Iteynolds number R, if the shape of the botly, and, hcnc:c, t,hc 

potential motion U1(x') are given. By the use of a further transformation it is 

possible to clirninnt.~ the? Rcynoltls number also from cqns. (8.1) nnd (8.2). If wt: p111. 

eqns. (8.1) and (8.2) transform into: 

with the boundary conditions: v' = O and v" =O at y" -0 and 71' .= U' at y" =a. 

, J , hese equations do not now contain the R.rynolrls numl)cr, so that the solutions 

of this system, i. e. the functions u1(z', y") and v" (sf, y"), are also independent of the 

Reynolds number. A variation in the Reynolds numbcr cnnscs an nffinc t,rnns- 

formation of tho boundary lnynr during which tho ordinn.t,o nntl the vclonily in 1,11(. 

transverse dircction arc mult,iplictl by R-'I2. In othcr words, for n given botly tho 

tlimcn~ionless velocity components M/U, ant1 (v/U,) . (U, L/V)'/~ n.ro fur~cl.ions 

of the dimensionless coortlinates z/L and (?//I,) . ((I, I,/V)'~~; the functions, marc,. 

over, do not depend on the Reynolds nun~bcr any longer. 

The practical importance of this principle o/ nim.ilat.il?y wifl~ resp-1 lo Ilrynold.~ 

nirmher consists in thc fact that for a given body shape it suffir:cs to find the solrtt,iot~ 

to the l~oundary-layer problem only once in terms of the above tlimcnsionless varia1)lcs.

162 VIII. General properties of the boundary-layer equations b. 'Sin~ilnr' solutions or the boundnry-lnycr cquntions 

Such a solution is valid for any Reynolds number, provided that the boundary 

layer is laminar. In particular, it follows further that the position of the point of 

separation is independent of the Reynolds number. The angle wl~ich is formed between 

the streamline through the point of separation and the body, Fig. 7.2, simply decreases 

in the ratio 1/R1I2 as t,he Reynolds number increases. 

Morrovrr, tl~c far!, lht, srpar:ll ion tlors ldtc phcr is prcsrrvrtl wl~c*n tlic- proccss 

of passing to the limit R + co is carried out. Tl~us, in the case of body shapes which 

cxhibit separation, the boundary-layer theory presents a totally different picture 

of the flow pattern than the frictionlcss potential theory, even in the limit of R 400. 

This argument confirms the conclusion which was already emphatically stressed 

in Chap TV, namely that the proccss of passing to the limit of frictionlcss flow must 

not be performed in the differential equations themselves; it may only be undertalren 

in the integral solution, if physically meaningful rcsults are to be obtained. 

11. 'Similnr* soletions of the boundary-lnyer equations 

A sccond, and very important, question arising out of the sol~~t~ion of boundarylayer 

equations, is the investigation of the conditions untlcr which two solutions 

arc 'similar'. We shall define here 'similar' ~olut~ions as those for which the component 

u of the velocity has the propcrty that two velocity profiles u(z, y) locat.ed 

at different coordinates x differ only by a scale factor in u and y. Therefore, in the 

rase of such 'similar' solutions the velocity profiles u(x, y) at all values of x can 

be madr congrnent if they are plotted in coordinates which have been made dimensionless 

with reference to the scale factors. Such velocity profiles will also sometimes 

be e:llled mifine. The local potential velocity U(x) at section x is an obvious scale 

factor for u, because the dimensionless u(x) varies with y from zero to unity at all 

sect*ions. The scalc factor for y denoted by g(x), must be made proportional to the 

local boundary-layer tl~ickncss. The requirement of 'similarity' is seen to reduce 

itself to the requirement that for two arbitrary sections, x, and x,, the components 

~(x, y) must satisfy the following equation 

't'hc boundary layer along a flat platc at zero incidence considered in the preceding 

rl~apter possessed this property of 'similarity'. The free-streani velocity U, was 

the scalc factor for u, and the scale factor Sol y was equal to the quantity g = 1/ v x/U, 

which was propor(,ionnl to the boundary-layer thickness. All velocity profiles became 

- - 

it1ent.ica.l in a ~lot of u/IJ,, against y/g = y )/ U,/v x = T] , IFig. 7.7. Similarly, 

the rases of t,wo- and threc-clirnerisiorlal stagnation flow, Chap. V, afforded examples 

of solutions w11id1 proved to be 'similar' in the present sense. 

r 3 

I he quest, for 'similar' sol~lt~ions is particulyly imporbant with respect to 

t.he mnthomnticnl cl~trrnctnr of the solut.iorl. In cnses when 'similar' soldions exist 

it. is pwsiblr, 11s we sl~nll sre in ~norc? drtnil later, to reducc the system of partial 

dilt'rrent.in1 equations to onc involving ordinary differential equations, which, evidcntly, 

cot-~stit.ntcs a considerable mathematical simplification of the problem. 

'i'he ho~~nclary layer along a flat platc can serve as an example in this respect also. 

-- -- 

It will be recallad that with the similarity transformdon T] = y 1 / -/v ~ r,cqn. 

(7.24), we ohtained an ordinary differential cquation, eqn. (7.28), for tho strcan~ 

function /(q), instead of the original partial diKercntial equatior~s. 

We shall now concern ourselves with the ty~~os of potential flows for 

wl~ich .such 'similar' sol~~l.ions exist. l'11is pro1)lom WILH (IIN(:IIHH(:CI it1 ~ron(, tI(,(.l~.il 

fir~l~ by S. (h~ltl~l,oi~~ 1.4j, m t l I I L ~ : by ~ W. Mangler [!)J. ,'1'11~ point or d(:pt~r!,~~rt> is 

to consider the boundary-layer equations for plane stdady flow, cqns. (7.10) and 

(7.11) together with eqn. (7.5a), which can be written as 

au av I 

& -t -=o, 

ay 

the boundary conditions bcirig ?r. =-7 a -- O for y = 0, : ~ r d u - I/ for ?/ --. oo. 'l'ho 

cqu:ltion of c:ontinuit1y is it~tc:gratctl by 1,110 introc1uc:tion of the tilrc:r~n func:Ition 

y(x, y) wibh 

Thus the equation of motion bccon~cs 

with the boundary conditions ay/az = 0 and appy = 0 for y = 0, and aypy = IJ 

for y = oo. In order to discuss the question of 'similarity', dimensionless quantities 

are introduced, as was done in See. VIIIa. All lengths are reduced with the aid 

of a suitable reference length, L, and all velocities arc made dimcnsionlcss with 

rdference to a suitable velocity, I/,. As a result the Reynolds number 

appears in the equation. Simultaneo~~sly the y-coordinate is reforred to the climonsionles~ 

scale factor q(x), so that we put 

proposed by F. Schultz-Grunow [Gn, 15a], ninkes it poasiblc to rcduce uevcrnl problems involving 

self-similar solutions to that of bl~e flnt plate at zero incidence. If A = 612 R is chosen 

as the curvature parametor, the trnnaformntions can be npplicd to flows nlong longitudinnlly 

curved walls with blunt or shnrp lending edges as well ns wit,h blowing or suction (Chnpt. XIV). 

The preceding trnnsfnrrnation is exnct to second ordcr in curvnt,urc which mcnns tbnt all t,crms 

of the ordcr A hnvr been inclded. 

163

164 VIII. Ccnernl propertic8 of Lhe boundary-layer equntiotis b. 'similar' solution of the boundary-hycr equntioris 

The fact,or I/~-for the ordinabe already appcarod in cqn. (8.4). The stream fnnct,ion 

is mde di~ncrisionloss by t.ho suhst,it,~ltkm 

where the prime in /' clcnot.cs difircnt,iat,ion wit,ll respect, to 71, and wit,h rf:spc?ot, 

to z in g'. It. is now seen directly from cqn. (8.12) tlhat the vclocit.y profile-s ~s(x, 11) 

nre similar in t.lro previonsly tlcfincd scnso, when t,hc st,rc:lm firnc:l.ion / tlel)t:ntls only 

on the one vnri:tblc 7, eqn. (8.10), so t,I~:it, tho clcpct~tlcnc:c of j on [ i~ c.anccllctl. 

Iri tlri~ (:we, moreover, the p:~rti:tl t1iffcrenli:~l equ:lt,ion For tllc st,ream Functioir, 

eqn. (s.!)), must retlrlce itsclf 1.0 nrr orc1in:wy tliffcrcr~li:rl equation for j(?). If we now 

proceed to investigate the conrljtions untlcr which this retluction~oi eqn. (8.9) takes 

placr, we sllidl obtlain the condition w1iic;h must be sat,isfictl 1)y the potential flow 

IJ (2) for such 'similar' solut,iot~s to exist.. 

If we intmducc now t.hc tlimcnsionless variables from eqris. (8.10) and (8.1 1) 

inl,o cqn. (8.9), we obt,n.in the following tliffcrenlial cquation for /((, q): 

'Sitni1:tr' soIt~t.ions t.~isl only IVIIO~I / :in(l /' (lo not, (lc:pw~(l on 6, i. c. when tl~v 

right,-l~:~nrl ~itlv of ocln. (8.13) vanisl~cs. Sitr~~rll,:~.ttro~~sIy 1 1 cocffi~ic?nt,s 

~ 

a ant1 P 

01; t.hc Irft,-ltnncl side of cyn (4.13) rn~tst* IIV itrtlo~,cntlcn~ of x, i. c., lhcy must, IN: 

~~~tistmit.. This l:ilicr co~~tlit,ion, rotnbit~d with cqn. (8.14), furnishes l,wo qu:~t,ions 

for Ilic polcnl,i:l,l vc~lovil.~. I:(R-) ant1 t,l~c scnlo f:tc:t,or q(z) for 1.110 ortlinal.r, so thl, 

they wn INS I~\*:LIII:~I,c~I. ilcncx:, il' silnihr soIut,io~~s or 1~o11n~I:~ry-l:~ycr flow arc lo 

(,xis1 , t IIV st.r(~~ni f~tncl~ion /(?I) rn~tst, 

This cquat,iorl was first given by V. M. Falkner and S. W. Sknr~ [2], and its solutions 

were latm studied in detail by I). R. 1l:trtroe 101. We sllall revert to this poi11t; i r ~ 

the surceeding chapt~r. 

10 remains now t,o dctermino from nqn. (8.14) Clw nondit.iorls for lJ(z) and 

~(z). 

From (8.14) we olhin first 

1"urthc.r from (8.14) we llavc 

ant1 Itn~~cc 

so t,llnt upon integration 

a-Dr. L 99' u 

where I< is a constant. The elimination of g from cqnc (8.17) and (8.18) yic+ls t.he 

velority distribution of the potential flow 

As srcn from cqn. (8.14) the result, is intlrpentlrnt of any comnloll f:lct,or of 

a nntl p, ns it ran I)c ittcludcd in g. Therefore ns long as a + 0 it is perrnissiblc t,o 

pnt a =- -1- 1 wit.llout, loss of gcneralit,y. It is, furthcrrnorc, c.onvrnient t,o int.rot1t1c.c 

:I. now c.ot~st.:~.nt~ 111 t.o roplacc p l)y puI,t.ing 

s:~l,isfy t,llc following or(lin:try (lilTcrcnt~i:~l :IS it) l,ltis wt1.y lho plty&:~.l TII(::LII~II~ 01' 1 I I ~ 

sol~l iott will I)(~(:oIII~ 

155 

&x:~r(tr. I Ic~ttc.(- 

so t11:1l., wiI.11 a = 1, the vc1orit.y clisf.rilnlt~ion of t.llc% ~)otc,t~l.inl flow :~nrl t.11~: sc::~ltr 

Iac:bor !/ for t.llc ordinnt,c Iwcomc

VI11. Gcncrnl propertties of the boundary-layer cquations 

and tho tmnsformation rcluntion (8.10) for the ordinatc is 

It is thus concludccl l.h:rt, siniilw solul.ior~s of Lhc bou~ttlary-layer cquat.ions arc 

ol)taincd when thc vclocit,y tiistributior~ of thc potcnt.ial flow is proportional to a 

power of thc lcngth of arc, rncnsurcd along 1.11~ wall from the stagnation point. 

Such pot.cntial flows occur, in fact, in the ncighbourhood of thc stagnation point 

of a wedge whose inclutfcd anglc is cqnal to n /?, as shown in Fig. 8.1. It is easy to 

verify with thc aid of potcr~tial theory tht we havc hcrc 

whcrc C is a constant. The rclntionship k)rtwccn t,hc wedge angle factor /? and thc 

cxponrrit, m is cxactly that givcn in cqn. (8.21). 

Fig. 8.1. Flow pas1 a \vcdge. In the neighhour- 

I~r~orl of tho leading cdgc Ilm pobnlid vrlocit,y 

rli~l.ribul.ion is lJ(z) - Crm Particular case8 for n =: I: (a) For =- I we have n = I, ant1 cqn. (8.22) hccorncs 

U(z) = rc 2. 'l'his is thc case of two-din~rnsional strqn.al~:on /loin, which was considered 

irl Snc. Vh 9, and which locl t,o an exact, solut.ion of thc: Navier-Sl.oltcs cqu:rt,ior~s. 

Wilh a -- I, nncl /? =-: I, the di&:rcnt,ial equation (8.15) transforms irlt,o cqri (539) 

which was already considcrcd carlicr. 'l'hc transformat,iori equation for the ordinate, 

ccln. (8.24). hccorncs identical with thc alrcaciy familiar oquation (5.38), if we put, 

IJ/z -- a. 

(b) For /? =- 0 wc havc nh -- 0, hwcc IJ(z) is const,ant and equal to U,. This is 

t.licc:ascof :~/kcl plde d zero incirhnce. ltfollowsfrop cqn. (8.24) that r] = y 1/ U,/2 v z. 

'I'his value tlifli:rs only by a faclor 1/2 from that idtmduccd in cqn. (7.24). Correspondingly 

Lhc clifTc:rcntial cq~~xtion /"' $-//" =0 which follows from cqn. (8.15) differs by 

a fidm 2 in Ohc soconcl term from rqn. (7.28) which was solved cerlier. The two 

equaLions hrcomo idcnlical whrn tsransformcd to identical definitions of r]. 

Solut,ion for diiTcrer~t, valuos of m will be corisiclercd latcr in Chap. IX. 

cl. Trnnsformation of the boundary-layer equations into the hcat-conduction equntion 157 

The case a = 0: The case a = 0 which has, so far, bcen left out of account, 

leads, as is easily inferred from eqn. (8.19), to potentinl flows U(z) which arc proportional 

to l/z for a11 values of /?. Depending on the sign of U this is the case of 

a two-climensional sink or source, and can also be intarprctcd ns flow in a divcrgrnt, 

or convorgcnt dlanncl with flat walls. This type of flow will also be con~itlcretl in 

grratcr tlctiiil in Chap. 1X. 

Thc second casc excluded earlier, namely that when 2 a - /? -. 0, leads to 

'similar' solutions with U(x) pr~port~ional to ep2, where p is a positivc or negativc 

constant.. We shall, howcver, rcfrain from discussing this casc. 

, Lhc . problem of the cxistcncc of similar solutions i~lvolving non-stcatly bountlary 

layers was discusscd hy 11. Schuh [l!j]; thc same problcm in rclation to con~prcssil~lo 

boundary layers will I)c tliscusscd in Scc. XIIId. 

d. Transformation of the boundary-layer cquations into the heat-conduction equation 

It. von Miscs [lo] published in 1927 a rcmarkat~lc transfornation of t.hc 

boundary-layer cquations. This transformation cxhibita thc mathematical chnract.cr 

of the equations even more clcarly than the original form. Inslcad of tho C:~rtasi:rn 

coordinates z and y, von Miscs introduced the stream function y~, together with the 

lcngth coordinate z as indcpcntlcnt variables. Substituting 

into eqns. (7.10) and (7.11), as wcll as introducing the ricw coordinatcs [ = x and 

r] = tp instcad of z and y, we obtain 

J-Ience, from eqn. (7.10), it follows thnt 

Introducing, Furthcr, the 'total head' 

wherc the small quantity 4 p v2 can bc ncglcctcd, wc obtain, reverting to Lhc syml~ol 

z for l: 

We may also put

158 

VIII. General ppropcrtics of the boundary-layer eqr~nLions 

l'lcluation (8.27) is a tliffrrent,ial equation for tho totd prrssuro g(x, vi), and its 

I)outrtlary rontlit.ions arc 

g = p(x) for rl, = 0 and g = .p (2) -1- 

Q U2 -- const for )I) = GO . 

2 

JSq~mtion (8.27) is relat,ed to t,hc hcat,-conduction equation. Tile differcnt~id 

rqnn.t.ion for t,he one-dimct~sional case, e. g. for a bar, is given by 

whrrc 7' tlcnot,cs the t.cmpernl.t~re, t tlcnoLcs 1.11~ t,in~c, n.nd rc is t,he t,l~rrmal tliKusivily, 

scc Chap. XII. Jlowevcr, the transformed 1)oundary-layer cqnation, unlike eqn. (8.28), 

is non-linear, ~CCSIIS~ tho thermal tliffusivity is rrplaced by v .u, which tlopentls on 

the indepentlent variable x, as well as on the tlepcndet~t~ 

variable g. 

At the wall, VJ = 0, 14 = 0, q -- I), eqn. (8.27) exhibits an unpleasant singularit.y. 

Thr Irft.-hn.ntl side becomes ag/ax = dp/dx + 0. On thc right,-hand side we have 

16 = 0, and, therefore, @g/avi2 = oo. This circumst,xnce is dist.tlrbing whrn numerical 

methods are used, and is intimately conncct,ctl with the singular belraviour of the 

velocity profilc near the wall. A detailed tliseussiorr of eqn. (8.27) was given by I,. 

I'mndtJ [I I], who had dctlnccd the tmnsfornration a long time before t,he paper by 

It. von Misen appcnmd, wit.hout,, however, publishing it?, cI. [I, 12, 161. 

11. ,J. 1,11oltcrt [8] applied eqn. (8.27) to tlre example of t.lw boundary laycr 

on a flat plat>e in order to test its pm~ticnbilit~y. 1,. Rosenhead and H. Simpson [I31 

ga.vc a. rrit.icnl cliscnssion of the preceding pul)lirntion. 

e. Tl~c niomcnttlm and energy-integral eqrrntions for the boundary layer 

A complete calculation ol the houndary layer for a given body with the aid 

of the differ~rit~ial equations is, in many cases, as will 60 seen in more detail in the 

next chapter, so cumhersome and time-consuming that it can only be carried out 

with t.he n.id of an elcct,ronic computer (sec also See. 1X i). It is, tlicreforc, desirable 

1.0 possess nt Im,st approxi~natc methotls of solution, to be applied in cases when an 

exact so111I.ion of t,hc bo~lndnry-hycr cqr~at.ions cannot be obtained with a rcasormble 

an~oltnt, of work, cvctl if thoir :iccumcy is only limited. Such approximate ~nethotls 

can he tlevisctl if we do not insist on satisfying t.he tlifferential equations for every 

fluid part.icle. Irtst~catl, t.1~ boundary-layer eqr~ation is ~at~isfietl in a st,ratnm near the 

wall nntl nmr t h region of transitior~ t.0 the external flow by satisfying the boundary 

rordit.ions, togct.l~er with cert,ain compat.ibilit,y ~ontlit~ions. In t.ho remaining rqion 

of flrtitl in the boundary layer only a mean over the tliffcrcrrlial ~quat~iotr is satisfietl, 

tlie wcnn heing hken over the whole tlliclrncss of the boundary layer. Such :I mean 

vnl~re is oht.ai~red from t,he momentum equation wltich is, in tmrn, tlerivetl from t,he 

rr111:ition of niol.iorr I)jr it~t~cgmt~ion over tlrc bor~ndary-1:~ycr t.hicknc:ss. Sinw 1.lris 

c-tl~t:~,tiott will Ipc oll,c.rr 11wt1 itr t.110 ~~~)proxitrr~il.(: ~trt~~.l~o~I~, to I)(> ~I~H(:IINS(~~ litt~~r. \v(. 

slr~II ~IC~UCC it now, writing it down in it,s motlcrtr Irm. Thc oqrt:ition is know~l :LS t,Ir(: 

nlo~ttentunt-integwl equation of boundary-laycr theory, or as von Kiirm;in's irrtcyr:il 

cqnntion (7 J 

\\'c sltnll rcsl,rict ourselves 1.0 t,lrc cnsc of slcwly, t.\vo-tlitnct~sit~tti~l, :wtl irlcv)tn- 

~)ressiblc flow, i. c., we shall refer to cqns. (7.10) tso (7.12). Upon intcgr:ttit~g t,lle 

rqu:it.ion of motion (7.10) with rcspert to y, from y = 0 (wall) t,o ?I =- 11, wl~crc 

the layer ?/ 1- IL is c?verywhLrc out,sitle t.lrc bo~~ntlnry Iayrr, we obtain: 

h 

'rhr shenring stress at tho wall, T,, 118s l~rcn substituted for p(au/ay),, so tht 

rqrr (8 21)) is sern to br valid both for laminar and turb~~lent flows, on condition 

that, in the latter case u and 7~ deuotr the time averages of the respechive velocity 

romponents. The normal velocity ron~ponrnt,, v, can be rcplacrd by v -. - J (iIu/r?z)d y, 

as sren from the equnt.ion of continuity, and, conscqncttt.ly, we have 

1nt.rgrxting hy part,s, we obt,ain for the second t,erm 

so that 

j~W-wY 

0 0 

h 

1- (Ir' & J(u -U)CIY -= zn e . 

(8 mi) 

Sincr in both int,rgmls the irrtegmnd vanishes outsitle 1,hc boundary Inyrr, it is 

prrmissiblc to put h + oo . 

We now introduce the displacement thicknrss, a, and the momcrrtr~nl tlrirl~tr~ss, 

d,, which have nlrcady bren liwd in Chap. VIJ. They arc dc4ncd I)y 

Y

160 VITT. General propertirs of the boundary-layer equations d. The rnolncnti~rn and energy-inkgrnl equations for the bounclary layer 

and 

m 

6, U = 1 (U-~)dy (displacement thickness) , (8.30) 

y=o 

a, 

6, U2 = u(U-U) dy (morncntum thickness) . (8.3 1) 

It will be not& that in the first tcrm of the eqn. (8.29a), differentiation with respect 

to x, and integration with respect to y, may bo interchanged as the upper limit h 

is independcnt of z. IIence 

This is t,hc momenlum-integml eq&ion lor two-dimemional, incompressible boundary 

lmyers. As long m no statement is madc concerning T ~, eqn. (8.32) applies to laminar 

and turbulent boundary layers nlike. This form of the momentum int,egral equation 

was first given by 11. Gruschwita [5]. It finds its application in the approximate 

thcories for laminar and turbulent boundary layers (Chaps. X, XI and XXII). 

Using a sirnilnr approach, K. Wicghnrtlt [17] dcduced an energy-inlcgral eqdion 

for laminar boundary layers. This cquation is obtained by multiplying the equation 

of motion by u and then inkgrating from y = 0 to y = h > a(%). Substituting, 

again, v from thc equation of continuity we obtain 

The second term can bc trarlsformcd by integrating by parts: 

whercas by combining the first with the third tcrm we haw 

0 

Finally, upon integrating thc right-hand side by pnrts, we obtain 

'l'hc upper h it, of irltegrat,ion could here, too, be rcplaced by y = 00, becausc the 

intcgrantls become cqual to zero outaide the boundary layer. The quantity p (&I*)' 

represent8 the energy, per unit volumc and time, which is transformed into heat 

by friction (dissipation, cf. Chap. XTI). Tho term & e (U2-u2) on the 1~"-l~ad 

h 

1 

sidc rcpresenta the loss in mechanical encrgy (kinetic and pressure encrgy) taking 

place in the boundary layer as compared with the potential flow. IIcnce the tcrm 

m 

4 p / u(U2 -u2) dy T C ~ O S C ~ ~ ~ 

161 

the flux of clissipntcd cncrgy, ant1 tho Icfl.-l~r~~ltl side 

n 

rrprescnts the rate of chnngc of the flux of rlissipatctl cnrrgy prr unit, lc:ngt.11 ill I.h(. 

x-direction. 

If, in addition to the displacement, and momentum thickncss from eqns. (8.30) 

and (8.31) ~wpxtivcly, we introduce the r1issip.ation-energy thickness, d,, from the 

definition 

m 

U3 a3 = [ u(U2-u2) dy (cncrgy thickness), (8.34) 

0 

we can rewrite thc crtcrgy-inbgral equation (8.33) in the following sirnplifictl form: 

which rcpresents the energy-integral eqmtion for two-dimnsionnl, lnminnr boundary 

lu yers in ineom.pre.wible flmu t. 

In onlcr to visualize thc displacement thickness, the momentum thickness, 

and the cncrgy-dissipation thickness, it is convenient to calculate thcm for thc 

simplc case of linear velocity distribution, as shown in Fig. 8.2. In this casc we find: 

displacement thickness dl = ) 6 

momentum thickness 6,=+d 

cncrgy thickncss d, -= 1 d. 

The extension of the preceding approximatc method to axially symmetrical 

boundary layers will be discussed in Chap. XI. Approximate mot hod^ for thermal 

boundary layers are trcatcd in Sec. XIIg; those for compressible and non-steady 

boundary Iaycrs will bc given in SCC. XIIId and Chap. XV, rcspcctivcly. 

Fig. 8.2. Boundary layer with lineor vclo- 

city distribution 

d - boanrlary-lnycr thickness 

6, - clisplaccment thickness 

d, - momentsm thickness 

4. - Energy lhicknesa 

t In the ease of turbulont flowtr, the energy-inbgral equation wsurnes tho form

VIII. General propertien of the boundary-layer equations 

Hct,z, A. : Zur Bcrccl~nung des Uhcrgnnges Intninnrer ~renzschichtm in die Auxsen~trijmrrng. 

Ii'ifty yorrs of boundnry-lnyer rcscarch (CV. Tolltnicn and 11. Giirtler, ed.). Brnunscl~wcig, 

1955, 03-70. 

Fnlkncr, V.M., and Skan, S.W.: Some npproxitnntc solutiono of the houndnry Inyer equntiono. 

I'hil. Mag. 12, 865-896 (1031); AltC RM. 1314 (11130). 

(his, Th.: Kl~n~icho Crenzscl~ichten an Jtot.etio~~nlriirporn. Fifty years of bountlnry lnyer 

resenrcl~ (W. Tolln~icn nnd II. Cvrtler, cd.), Urnunschweig. 1955, 294-303. 

Goldstcin, S.: A note on the boundnry - lnyer cquntions. Yroc. Cnrnlr. Phil. Soc. 35, 338-340 

~ 

(1039). 

Grusohwit,z, I

1 64 IX. Exact solutions of the shady-state boundary-layer equations a. Flow past n wedge 165 

a. Flow paat a wedge 

Thc: 'sirnilar' solutions discussed in Chap. VlIl consLit.utc a particularly sirnplo 

class of solutions u(x, y) which have the property that the velocity profiles at different 

distnnccs, x, can be made congruent with suitablc scale factors for u and y. The systcr.~ 

of p;mt,ial differential equations (9.1) and (9.2) is now rednced to onc ordinary 

rliffcrcntial cquation. It was proved in Chap. VlIl that such similar solutions exist 

when the velocity of t,hc potential flow is proportional to a power of tho length 

coordinate, 2, rneasurcd from the stagnetiot~ point,, i. e. for 

Jrrorn cqn. (8.24) it rollows that thc transformat.ion of thc int1ol)endcnt. v;l.riablc ?I, 

which lends to an ordinary tlifl'crcnt,inl equation, is: 

'J'hr r~uation of ~ontinuit~y is intrgratcd by the introduction of a stream function, 

as S(~PII from cqns. (8.1 I) ant1 (8.23). 'l'hus the vrlocity romponel~ts become 

u = u1 2" /'(r]) = u / '(r]), 1 

1 nt.rotluc:ing t,l~osc vnluos into tt~c 

nnrl put,t,ing, as in cqn. (8.21), 

we ol)Inin the following differential equallion for /(I)) 

/"' -t / /" -1- p (1 - 1'2) = 0 

equation of motion (9.l), dividing by ni. IL~ 

zZ"'--I, 

It. will IN: roc::tllrtl t.li:tt it, was ;drcady given as eilrl. (8.15), antl that its I)OIIII(IR~Y 

f 

contlil ions a.re 

Y] 7 0 : / = 0 , 1' --= 0 ; /l=1. 

the velocity profiles have no point of inflexion, whereas in tho case of decolcrat.rti 

flow (m < 0, p < 0) they exhibit a point of inflexion. Sepxrat,ion occurs for 

= - 0.199, i. e. for nt = - 0.091. This result sl~ows tht the laminar horl~\tlnry 

layer is able t.o support only a very small dccelcration witl~o~lt separat,iorl occurir~n. 

by IJxrtmr. The additional solution leads to a velocity profile with baclz-flow (cl. 

Chap. Xf). 

Tl~c potential flow given by U(Z) = 1 ~ xm , exists in thc ricigllbourllootl of the 

stagnation poil~t on a wedge, Fig. 8.1, whosc included anglc 8, is given by eqn. 

(0.7). Two-dimensional stagnation flow, as well as thc boundary layer on n llat. 

plate at zero incidencc, constitut,~ particular cases of the present solutions, the former 

for p = 1 and in = 1, the latter for = 0 antl m = 0. 

Fig. 9. I. Velocit.y distri- 

bution in the 1:~tninar 

boundary layer in tile 

flow past a wedge given 

by U (x) = a, zm. Tllc 

exponent m and the 

wedge anglc P (Fig. 8.1) 

arc connecbd tlirongll 

cqn. (9.7) 

,, lhc o:~.sc fl :- &, m 2- .j is worL11y of att.c~nI.io~r. 111 I,llis cnac I.llo tlillrc!t~l,i:~l 

equation for /(q) hccomcs: /"' .I- / /" 1- 4 (1 = 0; it, t,rnnsforn~s irlt,o ~.II(* 

tlifi:rcnt.inl c!qunt.ion ofroL:~(.iordIy symrnct.ric~~l flow with slngn:~l.io~t poit~l,, ocltl. (5.47), 

i. e., 4"' -1- 2 4 4" + 1 - 4" = 0 for $(C), if we put r] = 5 1/2 and d//dsj = d+/d

'I'his c.quat,ion Lmnsforms into t1in.L for n flat plat^, cqn. (7.28), in (.he special case 

whcn m = 0. The solut.ions of t,hc Falltncr-Slran eclun.tion (9.8) have been discussed 

in tleta.il in 1611. 

According t,o J. Sl~cinllcr~cr [631, nn interesting cxtensiot~ of 6hr solr~t.int~ of t,llr r 'nlltner-Sltnn 

eq~lntion (9.8) which in vnlid for ret.nrded flows (P < 0) in cases when velocity dintrihtct.ions possest+ 

ing n velocity cxccns (I'(i1) > 1) with n~naxin~utn near the wall arc ndrnittcd. In RIIC~I cnscs, the 

limit /'('I) = 1 for 11 -+ 00 is nttnincd nsympLot,icnlly "from abovr" rnthcr thnn "from hclow", 

as was t,lle cnw RO far. SIICII uo1uI.ions can he interpreted pl~ysirnlly as corrrspo~lding to a laminar 

wall-jet prod~~cctl in nn oxtnrnal strcnrn wit.11 n positive pressu~e grndient,. dplda: > 0. Ileferenro 

[G3] drrnonnt,rntn~ t,hnt t.lw limiting cnne of ll~mo uolut,ions, oI)tnin~d W ~ I I,IIc I tnnxirnun~ velocity 

cxccns tend^ 10 in fin it,^, trnnnforms illlo tllr wr-ll-known dl-sitnilnr nolttl.ion of n plro wnll-jet in 

t,lm absence of nn cxtrrnnl vclociLy -- n cnnc trcr~tccl hy RI. 11. (:lartt:rl (ucc 1401 in (Illr~p. XI)-~when 

we put, p = -2. 

A pnrtirulnrly drt,niletl n~onogrnpll on exnrt., self-nirnilnr solt~t.ions for lnminnr Imlndary 

lnyeru in two-din~cnsional nnd rot,ntionnlly symmetric nrrangemcnt,~, inrl~lsive of the nssocintrtl 

thcrinnl boundnry lnycrn (am Chnp. XTl),wns prlhlinl~cd hyC. 1'. J>cwey nntl J. F. Grosn [141. 

Their consitlcrnt.ionn inclntlc t,lle elTt:ct.s of con~presaibilit~ (nee Chnp. XIJI) wil.11 and mitl~out, hcn,t 

tmnnfer, relate Lo vnryitlg vnlnes of t.he Prar~tlt,l number, and incJ~tde some rases of suction and 

blowing. 

K. 1(. Clien nnd P. A. Libby 191 cnrried out nn cxtx?nsivc invcst,ignlion of bo~~~~rlnry lnycrs 

which are el~ornctorizcd by ~mnll clcpnrtnrcs from t.11~ nelf-ui~nilnr \vctlge-flow boutltlnry lnycrs 

of tho I'nlknrr-Sltan type. Rvidcnt,ly, RII~II 1)ounrlnry Inyerrr nre no longer nolf-~in1iln.r. 

b. Flow in n convergent channel 

The case of potmt,ial flow given by thc eqlrnthn 

U(s) = -2L 

x 

is related to flows pt~t a wedge, and also leads to 'similar' solutions. With > 0 

it rcprcscnt,s two-dimengional mot,ion in n convergent ohnnncl with flat, walls (sink). 

The volume of flow for a frill opening angle 2n and for a strnt,~~nl of ttnit 

I~cight is ($ = 2 n ?I,, (Fig. 9.2). Int,rodncing t.he simi1nrit.y t,ransformat.ioti 

Fig. 9.2. 1 % ~ in n ronvrrgrnt rhnnnrl 

, I . ltr I)~IIIII~*L~.~ ~on~li1,ion~ rollow Prom c(ln. (V.3) nl~tl nrc?: /' : 0 nl. o, 0, I / I 

1tt1(1 /" = 0 a(* 17 == w . 'I'lris is nlso :I j)nrl,icrrlar caso of I,llo clasa of 'similar' sol~lt~i~tt~ 

consitlcred in Chap. V111. ISquntiot~ (9.12) is obtnincd from 1.11~ more gcncral tlifli~rc~tltial 

equation (8.15) for the case of 'similar' boundary layers, if we put a - 0, ntld .-. 

4- 1. The example under consideration is one of the rare cases whcn the sol~ttior~ of' 

tllc botrndary-layer equation can be 01)tdncd analytically in closcd form. 

First,, upon ~nult~iplying cqn. (9.12) by 1" and integrating ol~ce, \vc? I1:1vc 

where n is a ronstnnt of intrgmtiou. 1t.s value is zero, as /' .- 1 ant1 /" -- 0 lor 

7 

v; 

f 00. '1'1111s 

-- - -- -- 

T = (,I - 112 (I* + 2) 

d 71 

whrre the additive constmit of intrgration is seen to bc cq~lnl to zero in virw of 

tile Im~ndary condition /' = I at 17 = oo . The int,egral ran be rxprcssctl ill closrtl 

form as follows: 

or, solving for 1' = w/11: 

/' = = 3 t8anh2

168 

IX. Exnot, soI~rt,ionn of tho ntrady-~t~atc boundary-layer equations 

Fig. 9.3. V~lority distribution in t h~ 

laminar Iio~~ndnry Inyrr of tho flow ill a 

convcrgcrlt cllanncl 

The prrcetling solution was first obtained by I

170 TX. 1Sxact uolutions of t.ho steady-state boudary-layer eqrrntionu 

cirnts nnd Lhus obt,ain n ~ystc~n of ortlinnry di~crenl~inl equat,ions for t.he funct,ions 

/3, . . . . The first two equntions turn out to be 

JII t,Iipse, ~lill'(;rc:~il~inl~ior~ wil,l~ rrs~)ccl, l,o r1 is ~lrt~ol.r~l hy pritnrs. '1'11~ ~~.ssoci~~t,ccl 

bountlnry contlit.ions are 

All difTercntinl cquatsions for the functionnl coefficients are of the third orrlcr, nnd 

only the first, one, t,haI, for fr, is non-linear; it is itlct~tical with the equation for twodimensional 

stagnnt,ion flow, eqn. (5.39), discussed in Chap. V. All rcmnining equntions 

arc linenr and their cocfficientps nrc expressed in t,erms of the f'unct~ions associat,etl 

wil)h the preceding t,crms. The frtnctions and hnvc been ralculatctl already 

by I

172 

1X. Exact m111ti0118 of tlw st.rady-state 1)oundnry-layer equations 

If. tJle power wcrc t,erminated at ~ 9 t1he , point of separat,ion would tllrn ollt LO 

be at, +s .-= Iog.oo. Iktt,er accuracy can nowadays bc obtainetl with numerical 

mct.l~ods, sco Sccs. JXi antl Xc3. 

'rhc nccllracy of t,his r.ale~~lnt.ion I)nsod on 11 powcr scrirs can I)n t,cst,od for spced 

of convcrgcncc of t,hc omit,t,ctl Imrtlion of t,hc serics by invoking t,hc co?adilions o/ 

com.pnbiOility at, the wall. I\ccortling 1.0 ctln. (7.15), wc ~nt~st, Ilnvc: 

Fig. 0.7. Verification of Ihe first coni- 

pntibility condition from eqn. (9.21) 

for the laminar boundary layer on a 

circular cylintlcr from Pig. 9.5. Thn 

first compatibility condition is satis- 

fied approximately as far as some point 

beyond separation 

FigIlrc 9.7 compares t,l~c curvat,urc of the velocit,y profiles mcasuretl at. t,he wall wit.l~ 

its exact value rcprescnt,ed by UcllJ/tlx. 'I%c agrrcrnent is good for a distance fi' 

lIryond t,llc point of seprat,ion. We may, Lherrforc, conrlutlc that t,l~t, Ulxsius series 

terminat,ing at t11c t,ertn ~ 1satisfies 1 t,l~c compatibilit,y conclit,ion on a circular cylindcr 

up t,o a point. which lies bryod 1.l~ point of scp~rat~ion. It does not,, howover, 

~~ccessnriljr (allow t,hat. 1,lrc: Ir11ncnlrtl srrics rrprrsc~lt.~ t.11~ velocity profile with good 

nc,rltrnc*y. 

As nlrently mcntionetl, in t.hc cnsc of more slender body-sllnp~s cor~siclernbly 

morc t.rrrns of the J3lasius serics are roquircd, if it is tlcsirctl t,o obtain t,hc velonit,y 

profiles as fa.r as the point of scparatior~. Ilowcver, t,he evalunt.ion of furl,l~cr fnnc~iond 

coefficients is hinclcrctl by considerable difficult.ies. These are tluc not orily to t,lw 

f.:~ct that for every atltlit,ional t,crrn in t,he series the numbcr of cliffercnt,inl equations 

lo I)& solvetl incrcnscs, but also, antl even morc forcibly, the difficult,ics are tlnc t,o 

t,lw nerd to r.va.lnnt,e t,l\e funct-ions for the lower power ternls with ever increasing 

nrcurary, if 1.11~ funct,ions for the higher power terms are to be sufficicl~t~ly n.ccnrat,e. 

1,. IlowartJ~ 1401 rxtentlrtl t.lic prrscnt nuet,l~od tn inclrldc the asymmet,~.ical 

casts, l)nt t.hc t~a.ltulnt~ion of t,l~c fi~nct.ional coefficients was not carried bcyo~lcl those 

c.orrcspontlina t,o the powcr z2. N. lhessli~lg (231 carrictl out an estension of this 

rnc:t,liotl h the rot.at,ionally sym~nrtricnl cnsc which will be considered in Chap. XI. 

illrnsnrc~nrnt~s of t.11~ prcssnrc dist.ril)~~tion nronntl a circular cylind?r wcrr 

rcport,rd I)y I

174 

TX. Exnct ~olut,ionu of tho stondy-state boundnry-lnyrr eq~latioris e. Flow in tho wake of flat plxtc nt zero incidoicc 175 

in r* for the ~trcnm fitortion in n mnnncr ~irnilnr to thc enao of the cylinder, Scc. TXr, the cocfficirnta 

being functions of y: 

1I~ncc the vrlocit,y of flow becomes 

Tnl.rothring t,I~rse vnlnes inh t.hr rqunt.ions of motlion (9.2) and comparing coefficients we obtain 

a ~yutcni 01 ordinary dilTcrontia1 equ:tt.ionu for t.ilo FIIIIC~~OIIR fg(rl), lI(11), . . . . Tho first tho 

of Ll~cuc are: 

lof" -1- I 1 " - 0 , 

0 0 

/['" -1- 1, - 2 lo' -k 3 /,>" = - 1 , 

fz"' -1- 1, 12" - 4 1,' 1,' -1- 5 1,' 1, = - 4 + 2 1,'" 3 I, I,", 

Only tho first cqnnt,io~~ is non-linmr, nntl it in idcnt.ical wit.11 tlint for n flat plats nt zero incidence:. 

All rornnining equations are lincnr nnrl contain only t,he function f, in the homogeneotls 

portion, wherons t,he non-liomogeneous brnw arc for~nrd wit,lt t,Iie nid of the remaining funct~ions 

1. I,. flowarlh solved trho first. scven tliKcrcntinl eqnations (up tO and including I,), and calctllatod 

t,zblcs for Llicm. 

'rim ucricn (9.25) converges wrll with t.hcso valnrs of I,, in t.he rnngr - 0.1 _< x* 5 -1- 0.1. 

Jn tllc casc of decclorntctl flow (x* > 0) t,l~c point. of scpration is at z* = 0.12 npproxi~nntrly, 

I)ut for thc sliglit.ly cxhndetl rangc of valnns t.ho convcrgor1r:o of the scries (9.25) is no longer 

wsured. 111 ordor to roach t.lrc pinL of separnt.ion,-I,. 1Iownrlh used a nninericnl proccdum for 

tho ronl.innnt.ior~ of the no111t.ion. V~1~rit.y profilrs for sevcrnl vn111cs of r* for hot,h ac~~lerntrd 

t r . J ho in&ycntlrnt varin1)lo in Lhr nhovc rqunths difkrs from that in Chap. VIT by R factor 1. 

and tlorolcrnt~ed flow are uenn plot8t~cd in Fig. 9.8. 11. ~htmld be noted t,Iint. nll profih in tlccclcrntcd 

flow have n, point of inRexion. D. It. IInrtroo [38] repcntcd tl~cso caloulntior~s nrid obtninctl good 

ngrcemc~lt, wit11 L. Hownrth. The case for a/iJ, - 0.125 wns rnloolnted more ncc~tvnt.cly by 

1). C. F. Ileigll 1441 who ~ 8 nn ~ clecl.ronic d digit.al computer for t,hc purpose nnd who pnitl ~l~ccial 

nL,tentio~~ to tlic region of scpnrntion. TIIC valuo of the form fnctor at l.110 point of sepnrat,ion il.srlf 

wns founrl t,o ho x* = 0.1198. 

, llic . nictliod ornploycd by L. JIownrtl~ was cxkt~tlnrl by I. 'l'ntli 1001 t.o ir~cluclc I.lw caws 

corrcspontling to n 2 1 (with a > 0). tiowcvcr, I. 'l'nni did uot publisl~ nny t.nbles of the furw- 

Lionnl roeflicicnts but confined liitnsclf to reporling lho lid rosnIL for n = 2. 4 nnd 8. 111 Iiis 

cnsc, (no, MIC poor ronvcrgcnro of tlic ~crics did not pcr~nit him 1.0 dotcr~ni~ic the poiut of sqi:wit- 

tion wil,l~ unfficic:nf, ncrurncy and 110 formrl himnrli rowpell~cl Lo IIUC I,. Il(~wnrl.l~'s IIIIWT~V:~ , 

fw~~li~~~~tilion RCIIWII~~. 

e. Flow in the wakc of flat plate nt zero iucidence 

The application of the boundary-layer equations is not rcstrictml to rcgions 

nmr a solid wall. They can also he applied when a stratum in which thc infltwncc 

of frict,ion is rlominating cxists in the interior of a fluid. Such a case occurs, among 

ot.l~crs, whcn two laycrs of fluid with tliffcrcnt vclocitics mcct, for instnncc, iri tho 

wake bcliind a body, or whcn a fluid is tlischarged through an orifice. We shall 

consider three examplcs of t,his typo in the prcsent ant1 in the succccding scctions, 

and wc sl~all return to them whcn considcririg turbulent flow. 

As our first examplc we shall discuss the case of flow in the wake of a flat plate 

at zero incidence, Fig. 9.9. Behind the trailing edge the two vclocity profiles coalesce 

int,o one profilc in the wake. Its widt,h increases with increasing distancc, and its 

mean velocity decreases. Tlie magnitude of the dcprcssion in the vclocity curvc is 

dircctly conncct,cd with tho drag on tAc bocty. On thc wholc, howcvcr, a.s wc shall 

see later, the velocity profile in the wake, at a large distancc from thc body, is 

intlrpenrlent of thc shape of the body, cxccpt for a scale factor. On the other hand 

thc vclocity profile very closc to thc body is, evidently, detcrmiricct by the boundary 

layer on tho hody, and its shape dcpct~ds on whcther or not thc flow has separated. 

The momcntum equation can be used to c.alculatc thc drag from the vclocity 

~wofilc in t.hc wnlro. For this jn~rl~osc wc draw a rcrtarigr~lar control snrfacc AA, 13113, 

Fig. 9.9. ~\pplirnt.ion of the niomcn- 

tun1 equation in tho calculation of the 

drag on a flat plak nt zero incidence 

from thc velocit,y profilo in the wake

176 

IX, Nxnrt ~oi~~liot~s of ~IIC ~~cndy-state boundary-Inyer rqrmtiona 

as shown in Fig. 9.9. The bonndary AIBl, parallel to the plate, is placed at such 

e distancc from the body that it lies ovcrywhere in the region of undisturbed velocity, 

I/,. Purthorrnorc, t,hc pressnrc is constant over the whole of t,he control surface, 

so t01at j~rcssurc forces (lo not contribute to the mornenturn. When calc~lat~ing 

the flux of momontunl across the contml surfacc it is necessary to remcmber that, 

owing to ront,innity, fluid nu st loxvc t,l~rongh tho hountlary AIBl; tho q~lantit~y of 

fluid leaving Ll~rongl~ A1lll is cqu;rl t.o tho tliffcrent:c I)clwccn tht ontcring Lhro~rglt 

AIA and loaving through BIR. 'rho boundary AT3 contribntcs no term to t.hc 

nom men tam in the x-diraction becanso, owing to symmetry, the transverse velocity 

vanishes along it,. The momentnm balancc is given in tabular form on the next page, 

and in it the convc~ltion is followed t.11:bt inflowing masscs are considcrcd positive, 

and ontgoing masscs arc taken t,o bc negative. The width of the plate is denoted 

by b. 'l'hc tot,al flux or morncntnm is cqnal to the drag D on a flat plate wetted on 

orlo sitlc. 'l'hus we have 

03 

D =be/u(~,-u)dy. 

v-0 

Intrgration may bo prrformctl from y = 0 to y = oo instcad of to 2/ = It, because 

for ?/ > h thc intcgrantl in eqn. (9.26) vanishes Ilrnce thc drag on a plate wetted 

on both sides bcromrs 

+ 

2 D = b e / u(u,-u) dy . (9 27) 

- m 

This cqnat,ion applies to any symrnet,rical cylintlrical body ant1 not only to a flat 

plat,o. Tt is t,o bo rcrncmbcrctl that in the more general case thc intcgral over the 

profile in t,he wake must be t,aken at a sufficiently distant sect.ion, and one across 

whirh t,ho st.at.ic pressure has it.s undisturbed value. Since near a plate there are 

no pressure tlill'crrnccs cit,hcr in t,l~e longit~ldinal or in the transverse direction, 

ccln. (9.27) npplins t,o any tlist.ancc brhintl the platc. Furthermore, eqn. (9.27) may 

11c: nppltc(i t.n any section x of tlhc I)o~lntlary layer, when it gives the drag on the 

portion of t-l~c plate between the leading ctlgc and tlltat sect,ion. The physical meaning 

of tho ir~t~cgml in eqn. (9.20) or (9.27) is that it rcprcscnts tho loss of momentum 

due to frict,ion. It is itlcntical with the intcgral in eqn. (8.31) which dcfir~ed the 

mome?ltum thickness a, so that eqn. (9.26) can he givcn tllc alternative fbrm 

Wc shall now proccrd to calculate tthc velocity profile in the waltc, in particular, 

9.1, a. large dist.ance x t)ehintl the trailing edge of the flat plate. The calculation must 

bn p(:rformcd in t,wo sLcps: 1. Through an expansion in thc downstream direction 

from I.he Irntling t.o t,hr tmiling ctlgr, i. c. I)y n ~:~lculation which inv?lvc:s thc cont.inu:~.t.ion 

of tJ~o Illilsius profile on thc plalo near d.hc tmiling cclgo, antl 2. Through an 

expansion in t,hc nl)st,rrarn direction. 'fhe lattw'is a kind of asymptot,ic'int,egration 

for x Inrgc tlistancc behind thr plate and is valid irrespective of the shpe of the 

1)orlp. It. will 1)c nrrrssnry hrrc 1.0 n~nkc lhc nssrrmpt,ion t,llat t.he vc1orit.y difference 

in t.11~ wn kc 

711 (", !/) ' U, - - u(z, y) 

(0.29) 

e. Flow ill ll~e wake of flnt plntc at mro incitlrncc 177 

Croswxxtion I Rnte of flow I Dlonient~~ni in dircclion r 

C -- Control srlrfnrc 2 Rnte of flow = 0 ::Mornctit~~m flus -= Drng 

is stnall rotnparctl wit.11 Urn, so thnt q~~n.tlrn,t ic nntl highrr t~crtns in 711 IIIIIY hr 11t~gltv~t.rt1. 

, , 

I l~c ~~occ~luw mnltrs 11xc of n nict,l~otl ol' c:o~~l,inuir~g n. Iznown solul.ioii. 'l'ltc~ (:ILI(:u- 

Int,ion st,arts with t.11~ p~viile at the t.miling ctlge, calculnt.ct1 with 1.11~ aid ol' Jllnsius's 

~ncthotl, and we sha.11 refrain from furthrr disrussing it hrre. 'I'hc asympt,ot,ic cxpmsion 

in t.he upst,rraln direction was calcnlatcd by W. Tollrnicm 1091. Sinrt: it, is 

t,ypical for problems oF flow in t,hc wake, antl since we shall mdte nse of it in t,hc more 

ilnport,nnt, tmbulcnt case, we propose to devot,c some t,itnc t,o an account, of it. 

As thr prrssnre trrm is rqr~al to zero, the bonntlary-layrr cynntiot~ (9 2)rombinetl 

wit11 rqn (9 29) gives 

'I'he partial tlilli:rr~~t.inl cqunt,ion call, here 1.00, be tmnsformctl into an or(li11iir.y 

tliffcrcnlinl ecpat,ion by n snit,a,blc? t,mnsrormnt,ion. Sirl~ilnrly to 1.11~ assuml)tion (7.24) 

in 13lasirrs's mct.l~od for t,hc 11x1 plate wr put. 

antl, in adtlit.iot~, wr assnme t.hxt( u, is of' the forin 

tl1 = U-c (-;)-kg(,]), 

whew 1 is the lrngt,ll of thr platc, Fig. 9.9. 

Tho power -- .j for 1: in eqn. (9.31) is just.ifict1 on the ground that the ~no~nent.urn 

int,cgrnl whicll givrs t,hc drag on tho plnt,c ill oqn. (!1.27) I~IIS~, I)r intlrpondrnk of r.

178 

IX, Exact solutions of tlrc steady-statc bonndary-layer equations 

Hence, omit,t.ing quadm.t.io terms in 15, the drag on a platc wetted on ht,h sidca, 

as givrn in eqn. (9.27), is transformed t.o 

+m 

2n=beCJ,/u,dy. 

lnt,rodurir~g, fttrt.llcr, t . 1 ~ assumpt,ion (9.31) into (9.30), ant1 dividing t,hrough by 

C (I,2 . (x/l)--lIz z-1, we obt,ain the following tliffcrenti:tl cquation for g(t1): 

with the lmnntlnry conditlions 

Integrating onre, we have 

y--m 

JI" 1- 4 71 JI' -1- h q =7 0 (0.33) 

0' = 0 at 11 = 0 and JI -- 0 at 71 = co . 

0' I :: 71g --0, 

\ 

whoro flrc rorlstnnt of integration vanrshes on account of tho tw~ndnry condition 

at q = 0. Rcpcatcd integration gives the solution 

g = exp (- '1 ?12). (9.34) 

llerc the constn.nt of int.cgrat,ion n.ppcaw in Lllc form of a cocffcicnt and can be 

ma& cqr~nl to unit,y without loss of generalit,y, as the velocity distxibution function u, 

from eqn. (9.31) st.ill contains n free coefficient G. This constant C is determined 

from the condition t.hat thc drag calculated from the loss of morncr~t~um, eqn. (9.32), 

,. (7.33). 

011 t.11r: ot.llnr I~nntl, from cqn. (7.33) we cnn whc tlown tho skin fric+t,ion on n. plntc 

I 

wct,t,otl on I)oll~ sitlcs in the form: 

difference in tl~c wakc of a flat platc at zero incidonrr becomes 

, I . Ito volocit.y clist.ril)~tt.ion given Iby this n.syrnplotio cclllnLion is rr:prrsc:nt,otl itt I'ig. !I. 10. 

It is remnrkablr that the vclocitty distxil)nt.ion is identical with (::~uss'.s c:rror-tlistribntion 

function. As assumed at the boginning, cqn. (0.35) is valitl only at grcnt, 

distances from the platc. W. Tollmicn verified that. it may bo nscd at about z -- 1. 

]pig. 9.1 1 corlt,nins n plot, from wllirh tho wliolr vc:locit,y.lit*ltl rnn IN! ittliv.r.t~tl. 

Thc: flow in tllc \dto of n platc as wc-ll as in tl~at bc:l~intl any othrr body is, 

in most cases, turbulent J5ven in the case of small Itcynoltls nnn~hrrs, say R, < 106, 

w11en the bountlnry laycr rcrnains laminar as far as tho tmiling cdgc, the flow iri 

t,Ile waltc still bccomes furb~~lcnt, because the vclocity prolilcs in the wnltr, all of 

which posscss a point of inflexion, arc c~t~rcmcly ~~nstnI)lc. In othrr wortls, cvcn 

with c~mparat~ively small Rcynolds numbers tho wakc 1)ecomes turbulent.. 'l'ur\)ulent 

wakes will be discussed in Chap. XXIV. 

f. The two-dirnensionnl larnir~nr jet 

The efflux of a jot from nn orifice affords a furtllrr oxample of motion in tho 

abscnco of solid boundaries to wliich it is possible to apply the boundary-layer 

theory. We proposc to discuss the two-dimensional problem so that we shall assume 

Fig. 9.10. Anyn1pLot.i~ vclocitydistribrttion 

in tho laminar wake bohind s flat plate, 

from erp. (9.35) 

Fig. 9.11. Velorily distribution in tl~c 

la- t 

minar wake l)cl~intl a flat platc at zcro 

innidenco

180 IX. Exact solrltions of tllc stcntly-stato boundary-layer equations 

that t,l~e jet cmcrgcs from a long, narrow slit and mixes with the surrounding fluid. 

This 1rol)lom was solved by 11. Scl~lichl.ing [60] and W. Biclrley [3]. In practicn, 

in this case, ns in the previous ones, tho flow becomes tl~rbulent,. We slinll, howevcr, 

discuss hero the laminar c:we in some tlet,nil, since the turbulent jet, wlticll will be 

oonsidcretl later, can be analyzed mntllcmaLically in an identical way. 

Thc emerging jet carries with iL some of the surroutttlit~g Iluitl whicli wns 

originally at rest becauso of the fridion developed on its periphery. The resulting 

patt.ern of strcarnlines is shown in Fig. 9.12. We shall adopt a system of coordinates 

wit.11 i1.s origin in Lhe slit and wit,l~ ita axis of abscissae coinciding with the jet axis. 

The jet spreads outwa.rtls in t.hc tlowr~stmam diroct.ion owing t,o the influence of 

frict,ion, whcrc:w its vc?locit,y in t,l~e cetrtm decrcascs in the same direction. For the 

saltc of simplicit.y we sllall assume that the slit is infinitely small, but in order to 

rt!t.:lin a finite volnrnc of flow as well as a finite motno~tum, it is necessary to nssumo 

an infir~itc fluitl vel6oit.y in tl~c slit. 'l'l~c prcssurc gratlicnt tlpltlx in ~JIC direction 

can Iicrc, as in t.he previous cxan~plc, be neglected, bwnose the constartt pressure 

in 1.11~ surrounding fluid irnprcsscs itsclfon the jet Consequel~tly, the total nioment~~m 

in t.he r-tlirrct,iou, clcnot,c:tl I)y J, must, remain const8ant arid intlcpo~~dnnl of 1,he 

distance r from tho orifice. Ilcnco 

11, is ~)ossil)lc t,o tnnlzc a snit.;~l)lc assumption regnr(ling blic velocity distribution if 

if, is ror~sitlcrctl t11:tL the velocity profilcs ~i(r,y), jnst :IS in the oasc of a flat plate 

at ZWO inci(lcnce, arc most prol)al)ly sinli1a.r. 1)ecnuse the problem as a wltole possesses 

IIO ch:~mct,nrist,ic! li~~enr tlimcr~sion. \Yo shall ass~!mc:, t,hercfore, that tho velocity u 

is a fi~nntiot~ of ylh, where h is the \vitlt,li of t,he jet, suitably defined. We shall also 

nssumc t,l~:tt. h is proportional 1.0 x*. Aacortlingly we can write the strcam fut~ction 

it1 t,hr lorm 

I. 'Chc flux of tnon~cnt,t~r~~ in fhc z-tlirwt,io~l is i~ltlop~tlcnt, of r, at:c.orcli~~g to rqu. 

(936). 

2. HI^ :~ccclrrnl,iot~ t~(~rttis :IIIII LIIC I.rivl,iott Ltvm in CIIII. (U.2) nrt: of' 1,111: WIIIII* 0r111.r 

01' n~:~gnitu(lc. 

(:or~sr:cl~~c~~t.ly, the assumptions for the iritlrpcr~tlcr~t~ vari:ll)lt: at~tl for the st,rcntn 

func.t,ion can be writtcri as 

if s~~it;tblc c:onstxtlt fa,c:tors arc it~t:lu~lr~l. 'I'l~rrrlnrc., t,lt(. vc,l1)c.it.3r t~om~~o~~t~r~ts 

arr 

given I)y 1,llc f'ollowing expressions: 

I 

whcrc OI is n lice constant,, l,o be clot~crmittctl Inter. 'l'hus t01c a.lmvc cq~~:~t.iot~ 1,r:lnsfolms

and t,llc? clnsh now clet~otos tlilTrrcntia.tion with rcspcct, to (. Thc boundary contlitions 

art? 

( ~ 0 F=O; : t=oo: r=O (0.41) 

whcro t.1~ consl;~rtl~ of inl.rgr:it,iotr was m:itle cqunl to I. 'l'llis li)llows if we p t 

Ff(0) - 1, wlticlt is prrnlissihlo wil.ltottt losx ol'gc~lcrnlil,y Imnnsc of I,llo frcc cotlst.:inl. 

a in t,l~c rrlat.icin Iictwoen f ~ n P. d 1Cq11atio11 (9.42) is n clill:rrnt.ial cqn:tt,ion of 1tic:t::~t.i'~ 

typc ancl can Iir int.rgrat.ctl in closctl t,rrms. \Ye oli1.ni11 

I 11vr1 t ing this rqnnt ion wr obtain 

Since, furt,llcr, tlP/tlE - I - 1:in11~ 

qn. (9.37) and is 

F 

1 - exp(-BE) 

I. =t,anh E= - - - 

1 4- cxp ( 3s) ' 

E, the vc1ocit.y (li~lril~~tl~ion (:all I I tloclucctl ~ from 

1 - r (I t.an11~ 6) . (9.44) 

.I 

1.he vrlorily tlisl.rilwtiorl from cqn. (9.37) is soon plott.ctl in Pig. !).Is. 

1L now rcn1:tins t,o dc(.crtninc. t81rc const:tn(. a, :LWI this ciln be (lone wit.11 the 

aid of condition (!).:3R) wl~ich shtcs that t,l~c rnomcnl,um in 1.11~ x-tlirrcl ion is ronst,nnt.. 

(hnbining rqns. (9.44) :111(1 (0.36) we obhin 

we shall assume that tho flux of momontum, J, for thc jet is given. It is proportional 

to Lhr excess in pressure with which the jet leaves the slit. lrrtrodricing the kinematic 

mo~nenlmt .I/@ = K, we have from eqn. (9.45) 

Fig. 9.13. ,VrIcirit,y dist,ril~ulio~~ in x t,\\o-rJimrt~. 

sion111 nn(J cire~ll~w frcc jcL fro111 cqns. (9.44) 

:md (11.16) icspect~ivcly. For tho two-tlirner~. 

xionnl jct [ = 0.275 KIP y/(v~)~/~, and for the 

circnlar jct. C - 0.244 y/vz. I< and K' 

t1twot.c: Ilir kincrnat.ic monwnt.um J/e 

and, hencc, for tlio volocit,y distribution 

K. Pnrnllcl &reams in hminar flow 

r 7 

Ihc transvcrsc: vclooil,y at thc bountlnry of Iht. jet is 

-1 00 

ant1 the volume-mtc of discltxrgc per unit height of slit bocorncs Q = e J v (I!/, or 

- m 

Q = 3.3010 (I< VX)"~. 

(!).48) 

Tlic volumc-rate of tlisclmrgo increases in the tlownstrcam direction, bccai~sc: flnid 

particles are carried away with the jet owing Lo friction on its boundnrics. It also 

increases with increasing momcnt~um. 

The corre,sponcling rotationally symmct.rica1 casc in which the jet cmcrgcs from 

n small circ~~lar orificc will be tliscussed in Chap. XI. The problem of t,hc twodirne~~sional 

laminar compressible jet cmcrging from narrow slit was solvctl Iiy 

S. 1. P.zi [4!)] nntl M. Z. JZrzywo1)locki [42]. 

Moasurcmrnts performed I)y TI:. N. Antlrntlo [I] for tho t,wo-tli~ncnsiot~:~I 1n.rnina.r 

jct confirm t.he preceding thcorct~icd argurncnt vory well. 'l'llo jct rcn~ail~s laminar 

np t,o R - 30 appro~irnat~rly, where the Ibynoltls number is rcfcrrctl to thc cfflrrx 

vclority and to the widL11 ol' tho slit. Tho casc of a Lwo-tlinlensional ant1 t.llat of .z 

circular trtrl~ulent jct is discusscd in Chap. XXIV. A comprchensivc review of all 

probloms involving jets can be found in S. I. Pai's book [49]. 

g. Pnrnllel streoms in laminnr h w 

Wo shall now 1)rirfly cxnminc the laycr 1)ctwccn two pnrallcl, Inminnr sl,rcnms 

which move at tlifTercnt vclocitics, xntl so provitlc a htrtl~cr cxnrnplc of the npplicability 

of the bountlnry-laycr equations. Thc forrn~~liition of thc problctn is scot1 

il111sLraLctl in Fig. !).14: Two it~il~ially scp:ir:~Lc(l, ut~disO~~rI~~xl, prnllcl HL~~!ILIIIS whith 

move with the vclocit.ics TJ1 nncl (I,, rcspcctivcly, l~cgin tm intcrc& thro11g11 frit:l.iorr. 

It is possi1)lo Lo assurnc thnt the transition from the vclociLy U, to vclocity (I, talccs 

in n narrow zone of mixing and that the transvcrsc vcl&ty component, v, is 

everywhere smalc oomp,zrcd with the longitudinal velocity, 11. Consequently, the 

boundnry-layer equation (9.1) can be usctl to describe the flow in thc zoncs I and 11, 

and the pressure t1crm may be omitted. 

In n manner analogous to that employed for thc boundary layer on a flnt platme 

(Scc. VIIe), it is possible to obtain the ordinary tliffcrentinl equation

184 IX. Exact solutions of the steady-skate boundary-layer equations 

by int'roduring t,hc dimensionless transverse coordinat,e 9 = y 1/ lJl/v z and tlte 

stream fur~ct.ios y~ = 1/ v V1 z /. Assuming t,l~nt IL/U = /I, we are led to t,Iic boundnry 

contlit ions 

IICCXIISC Y) =- 0 t,l~cre. The sol~lt~ion of the dilTerential equation (9.49) subject to the 

boundary contlitior~s (9.50) and (9.51) cannot be obtained in closed form, and a 

numerical mcthd nimt be employed. It is possible to obtain exact numerical solut 

h s I)y the IISC of asymptotic expansions for 77 + - co and 17 -+ -1- cro togetfher 

wit.11 a series expansion about r] = 0; several such solutions were provid~d by R. C. 

1,oc:Iz 1451. 'f'hc prthlcrn was first, solved by n~lmerical integration by M. 1,essen [44a] 

st,art.ing with an nsymptot.ic expansion for r] -+ -00. 

. Jlw . tlia.gr:~rn in Pig. 9.14 prc.scnt,s volo~il~y profiles for I = U,/U1 = 0 and 0.5. 

An irnprovcd ~~umerical solution was p~~hlishctl by W. J. Christian [lo]. This special 

cnsc of the int.eract,ion l~et~ween n wide, l~olnogericous jet ancl an adjoining mass of 

quiescent, air is oftm tlescribcd by tho term "plane half-jet". 

Fig. 9.14. Velocity distrihut,ion in tl~c 

zone hetrvecn two int,crnct.ing parallel 

streams, after R. C. Lock [45] 

11. Flow in the irlet lengt.h of n strnigl~t chnn~~rl 185 

R. C. J.oclr [45l studied, in atltlit.ion, the case wl1e11 t,hr t.wo half-jets differ in 

their clensit.ics ancl viscosit,ies, and riot only in tllcir velocities. An exanlplc of stlch :I 

case is t,lrc flow of air over a wnt,cr srlrf:~.cc. The solution now tlrpcntls on t,l~c p:lr:Ltnctcr 

x -- I,, p2/p1 p1 in atltlit.ion t.o I. Lock provided sevcrnl cxnct solut.iotls ns wcll 

as solutions which were l)ascd 011 the rnoment.um int.cgral rquat~ion. An approsim;~lc 

mc.t.llotl was also conccivc.tl 1)y 0. I

1 86 

10 

0.8 

0.6 

0.4 

0.2 

0 

-0.2 

-0.4 

-0.6 

- 0.8 

- 1.0 

IX. Exact, aolrtt,ionn of the nt.rnnuntlary lnycr rlcvrlops in the same way as on a flat plate at zero 

inritlrnce it1 unnccrlcratad Ilow, so that from cqn. (7.37) we Iiavo 

is tllr cl~nriirlrri~tir dinirnsin~~lrs~ inlct Icng111. 1':qnntion (9.53) ran dso be written as 

Y 

uo 

U(x) = V, {I + Ir', a + I

\\+ere j = 0 (plane flow) or j = 1 (flow with axial symmetry). The boundary condit,ions 

arc 16 -- I) = 0 at y = 0 n.ntl 11. =- 11 (n.) at ?/ - 0. For turbulent flows u-and v 

a.rc t.he n~pprop~.ink lnrnn vclocit,ics n,tltl ~1 rrprescnt.~ a suitably defined cdtly viscosit,y, 

scc for inst,nncc A. M. 0. Smit.11 n.nd 'l'. Cc1)coi [Dl]. I'or Inminer flows €1 .= 0. 

7'11~ l,rnl~sft,~.r~~al,iot~ of rqt~s. (9.50) ancl (9.57) to din~en~iot~l~s~ vnria1)Ies incorporaLes 

11otJ1 t11(, l$ln.sir~s ant1 lhc Mn~lglrr l.rnnsfot~tnn.titr~~ st^ nlso II. (:orrl.lcr I:13, 341) nntl 

is tldined n.s ft)llows : 

Tllc cont.it1ui1.y equn.t.io~~ is sat.isfictl I)y the st,rcarn function 

nntl E, is t,llc rdtly visvosit,y from ecln. (39.2). The s~~l)script,s tlrnotx- part.inl ctifkrctit.ia.Iion, 

anrl the qunnt,it,y 

5 

77~0; /=O; / =On.r~tlq=cm; I,-- I. (9.63) 

Fi~~it~e-tlifl'c?rcncc: cqnnt,ions of sccontl order can I>(: SOIVC~ (by mnt,rix inversion 

ront.inrs) rn11r11 nlorc cfficiont.ly trllan t.llirtl (or higher) ordcr equations. It is of intcr- 

?st,. t,hct~eforo, t,o rrclurc equations (9.01) to sccond ordcr. To this end the variable 

I;' -= /, is int,~ducrtl and eqn. (9.01) is rewritken as 

INF'v],l I /FII --F~)-T-~((FE'~--~~F',~]. (9.64) 

'J'llis rqua(iot~ now conlains two unknown functions, f and 1'1, hut tllrse ale related 

by thr sirnplr rxprcwion 

In the absence of srlctior~ or t)lowing the boundnry eonditions nrr 

This strip is completely covered by a grid with lines drawn parallel to t,he ( and 

coordinates as illustro.t,ctl in Fig. 9.16. Tho stq sizr A[ rcl)rcsc~~t.s t.11~ tlist,nnrc. 

bet,wcer~ t,wo snr.crssivc grid lincs 5 = const,a~~t; it is prost~~nctl i.o I)c stn:~ll IIII~, is 

ot,hcrwise rrnspccifictl. 'I'hc corresponding step sizes in t.hc q-tlirrctiou nrr spc:c.ilictt 

t,o vary in geometric progression. The rnl,io Octwecn t,wo s~~ccrssive grid lincs, TI, 

and qn+l, is denoted by I< = I -1 k where 1 kJ varies from 0 t,n 0.05 in l.ypical cases. 

Each notlal point is itlcntified by a dou1)lc intlcx m, ?L which tlclinrs it.s posit,ion 

Fin, 7, according tso 

111 writing t>he Anitc-tlilTerence quol.icnt,s it is corivcnlent t,o int.rocluce t,he moan of 

two successive Aq-values 

In the step-by-step calculations the solution is considcretl known at 5,n ancl ell 

preceding grid lines, and the variables F nntl / are sougl~t at. [,,,, 1. 

Fig. 9.16. \'ari.zble stcp size finit.c-rliffrronce grid 

for th rnlcr~lnt,ion of laminnr and turbuletlt' 

Iiounrlnry Inycrs 

x knon.~~ vnlucs, 

O r~nknnwn rnll~rs

ccnt,rred nt, (m -1- 1, n). 1'110 two t,xptwsions are t.hertwpon cornt)ined in such a way 

thnt ternls of ortlcr Aq2 are elitnina1.rtl. The corrcsponclir~g difference qnot,icnks can 

I,c given the, form (index we 1 I omitted) : 

a',, 1 

{ I I I 2 T I I 1 7 I] I 0 ( A . 1 ) (9 69) 

il,l 2 /I 11,' 

where 

p1 - 1 

l2 (I -+- K), rPz - L ~, 

.-- 2 r1 J'~, I>,, -= I . 

Eq~tn.tions (9.69) nncl (9.70) rcrlr~cc t.o t,hc st.nntlartl form for cent,ral difl'crenccs when 

K = I. 

For the (-tlcrivnt,ivrs in cqnnt.inn (9.64) a simple bnclrwnrtl tlini.rrner formule is 

used 

y - Fsr 11. n -- Fnr, n 

E - 

-I om. (9.7 I ) 

At -- 

The 1nrgc.r I,rtlnc.at,ion error which appcn.t,s here is balancctl by t,hc it.crat.ivc scltclne 

proposc(1 for solving thr tlilkrcncc c-qt~at,ion. 'l'llc non-lincn.r t,ernis in r(lnnt,ion (9.04) 

Imvc to be rcplaced by lincarizccl diflkrencc quoticnt.s. Tho tcrlns fFIl and FFg may 

serve as exntnples and thcy are writ,t,cn as 

l'hc lincnrizrtl Iini1,r-clifl'ercnw qnotionts given nhove are su11st~it.ut~erl into the 

tlifl'crentinl ty~tnt.ion (9.G4) nnd Lhe result is multiplied througlt by A E to give n 

tlilYcrcnce equa.t,iotl. 'J'his is writden ns follows 

i. The n~ctl~otl of finite dilTwcnces 191 

111 cqun1,ions (9.75), 6 and 0 nre cvnlunlctl nt (111. 1- I), mtl ot~ly the vn.rir~l,lcx wit.11 

sttp(wcripll i ntt 11~1cInlc~1 through st~cccssivt: it,crntions. To s~~xxl-III) t,Itv il~r~~:t,l,ion 

proces.s t,he tcrrns (/S)t can be licJ)t constnrlt. (equnl to t . 1 ~ &luc nt t.hc prrvious 

shtion) unt,il initial convergence is nchicved. 

Method of nolution: Equations (9.74) rcprc~ont~ n ~ oof t N-1 si~ntlll.nncwus r~.lgc:- 

I~rnic equntionrr for the unlrnown k;ntl,n (n = 2, 3, . . ., N). At, cnch levcl IL t.l~rcc 

unknown quantities nppenr, namely Fnajl, .-I, Fm.kl,n and Fniit, ,,+I, but sincc 

F,+I,~ and Fm+l,~ nre known from the bonndnry conilitions, 1.11~ totnl nurnbcr of 

cquntions equals the nunlber of unlrnowns. The set of nlgclmic cqunt,ions rnn be 

writt.cn in so-callod tllrcc-tlingonnl matrix form. MnLriccs of Illis ttypcwhcro oK-tlir~go~tnl 

elements vnnish outdc the three-tlingonal band can bc inverted bg n sirnplc: and 

direct nlothod well suit.cd for digital con~putcrs.l'o end tlriseqna.tion (0.74) is rcwrit,toll 

in "stantlard form" (subscripts (m -1 I) ornittcd) 

Thc botlnclary conditions arc 

F1 = 0 nntl PN = 1, (9.70) 

wllere IL = I tlenotcs tbc wall and ?t = N thc edge of the bountlary Inycr. J1, is asst~tn- 

etl now t,llat a solution existst in tllc form 

The boundary condit,ion F1 = 0 nnd t,he rcyuirernent t,llnt rquation (9.77) sl~ould 

rcrnnin valid indcpcntlcnt,ly of the sl,cp size /Iq leads to 

A direct, colisrquencc of rqnntion (9.77) is that 

When the preceding expressions are substitutml int20 oqn. (!).741,), 1.11~ following 

relohion is obt.aincd 

By tncnns of equalion (9.81) and the condition (9.78), it becomes possil~lc t.o cotnpot,~

192 

JX. 1Cxact sol~tLion~ of tho stmdy-state l~ountlary-layer equat,iona i. Tho method of finite diffcfcrrnoes 103 

R, and G',, for sucoessive values of n startling wit,h ?z = 2 for all grid point,s between 

the wall and the edge of the 1)onndary layer. 

Sinco 17,,.,l for 12 = N-1 is known from rqnat,ion (!).70), it lmx~mcs possible to 

evaluate all nnlznowns F, by means of equation (9.77) whilc t,mversing td~o boundary 

layer from t,ho edge t,o t,he wall tJlrongll (Icrreasing va111cs of n, i. c. for 17. -- N-l , 

N--2, . . ., 2. '1'11is cornplr1,rs lhr cdc1lln1,ion of Il',, (7.- F,, ll.n) in ono it.c~.nl,ion. On(:(: 

I{',,,, 1111s I)c~w tlr~r~~nii~~r(l. thr cot.~.c.sj)ondi~ig solntion for /0141,n ca.n be found by 

rlireot. nun~wirnl inhcgrxt,ion of equat'ion (9.05). The t,rapezoidal rule snfficcs for t,his 

purpose. 

The calculat.et1 vniues P7n4.1,n a.nd /,,+l,. are used t,o dntmnline new and improved 

,. 

valrtcs of t,he coc,fficients A,,, I?,,, C,, which in t,urn leads t,o new and improved values 

of F,+I,,, antl f,, , I,,. 111~ ~WOCCSS is t.~rmin:~tcd when t,hc rcsnlt,~ of two s~~cccssivc 

it,rrnt,ions ngrco t,o within a specified tdcrancc, typically of order 10-5. 'l'ho convergrnce 

is nsnallp rapid, t01ree t,o four iterations being adequate in most cases with 

st,rp sizrs A.r in l,he range 0.01 t,o 0.05. 

In crrt,t~in pro11lc:nis it, I~ecomrs n~ccssnr~ t,o nllow for bonnda.ry-li~y(~ growt,l~ 

1)y inrrc,asing N (or ve) as t,hr calcnlatrions proceed tlownst,reant. The houndarylayer 

edge is rlcfinctl by thr rcquircmcnt t,hat tho difference FN-Fnr~l should be 

Iws t,lian a sprrificd value, t,.vpically of ordrr 10-4. 7'hc growth, in t,crrns of the presrnt 

variablcs, is usually very modest even for cases involving separation. 

A vnrial~lc of primary intcrcst. in the calcnlat,ion is the s1.rrs.s at t,hc wall; it,s 

vnluo can bc tl~t~erminetl with good accuracy from the five point formula 

Iuirinl vnlr~cs: \Vlic~n using hnl111lrr1.c.tl similar solrrl.ions as s1,arling vdrlcs, ext,c:nsivc 

int~c~t,j)olat,ioti is rcclnirecl whrncver variable step sizes Ay,, are nsctl. It is 

rnorcx convcnirnt, antl efficient also t,tr gcnerate t,hc sin~ilarit~y solut.ion by finite 

tlifi~rr-rcs t.hroupl1 surcrssivc iterat,ions. The equat,ion t>o be solved is oht,aincd from 

cclna,li~m (!).64). and can I)c writtam in 1inoar.ized form as 

guessing a so la ti or^ which salisfics the boundary condit,ions), whereas those wit,ll 

index i arc to be found in the 1:-th or cnrrcnt itcmt,ion. 'L'lte tlifTcrc:nc:c quol,irnf,s 

(9.69) and (0.70) are now snl)stitut,cd ink) equa0ion (!).84). 'I'hc rrsult. is a tlilli~~~~ncc 

equat,ion which can be writt,en in the standard form of eqnat,ion (9.74), with coeffi - 

A linear variation in F suffices as an initid guess, Fo, and the corrcspontling value 

off is detmmincd from equation (9.86). The coefficients A,, /I,,, C,, and I),, nro (XIoulntcd 

next, and tltc corresponcling vnlncs of /?,, and (r,, arc tlct~errninctl ~~eross thtbonntlary 

layer. The recurrence rclnlion (9.77) and t,he bountlnrg contlit,iorls (!).78) 

are then used to determine the new it,crat.e, FI, across thr bounclnry lager. 'Yhr 

process is repeated until the difference bct,wcen successive it,eratrs becomes smaller 

than the specified t,olerance. The number of it.emtions required is typically of order 

8 to 12. The method is simpler t,llan t,he usl~al W~ooting'' ~ncthod used for two-point 

lmnntlary-vnlnc problem$ arid it converges in many rases WIICIT t,lrc In,t.ter mct,l~otl 

fails, for inst,ance for very large blowing mt.cs. 

Applications: The finite-difference method prescnt.ctl hrrc is in1,cntlctl as n prnctical 

engineering t,ool. Great,rr accuracy cnn be achicvcrl with a more clalwrntr procedure, 

1,111, t,his in turn leads t,o greater cotnplexit,y in fornlnlat,ion mtl progrnmrning 

and to an increased demnlid for computm t,ime and cnpacit.y. The conipnting time 

nnd accnmey tlrpcnd for all tlifTcrcncc ntet,lrotls on the skp sizr nsrtl in thr rnlrw 

Iations. It, is of int.crest, to exa,mine the accuracy of the present, mct,hotl in a few 

cases for which very accurat3e solut,ions are known. The cases considrretl are 11owa1~t.h'~ 

linrnrly retarded flow (cf. Scc. IXd) a,nd the circular cylinder with a pressure clistrihut.ion 

ncrortling 1,o pot.cnt,inl 1.hrory and nccotding t,o t.hc cspcriments of I[ic.n~r~lz 

(c/. See. X c). '1'11~ rrsult,s for n "normnl" step sizc and a "srnnll" step size arc tabnlntcd 

11clow. lhtn t.11~ C:IICIII:L~~ rcs1111~s only the locat.ion of the scl~n.rt~.l.ion 11oin1s IIII! 

sl~own. 

Case ('o~kIerrd 1 Present redts 1Sxnct 

1,inenrly set.artlrtl Ilow 

Circular ryli~~tlcr 

(l'okntinl flow) 

-. 

Circular cylit)tlrr 

(Ilic~ncne prms. tlntn) 

I 

(1) x,' = 0.1227 

(2) x,* = 0.1210 

(1) 4, = 106.13" 

(2) 4, = 105.01 O 

r8* = 0.1 I!)!) (Ilownrtli) 

or r,* = 0.1 198 (l,eig11)[44] 

I or T,* - 0.1203 (Sc4~ortinr~~) 

4a - 104.5' (Srl~ocnni~er) 

(rf. Scc. Xc) 

-- 

(I) $, = 80.98" 

(2) 4, = 80.08" 

#. -= 80WC (.lnlli: nnd Stnitl1)(.42/ 

(interl)olntrd) 

.-

194 

IX. Jqxnct eolutions of thc stcdy-ntnto I)or~ntio.ry-layer cqr~ationn 

The computing time with t.11~ "normnl" strep sizc is tyltically Ci to 10 scconds on 

t.hc UNIVAC 1108 compntcr.'l'he accuracy wit,l~ the smnll step sizc is seen to Ire bet,tcr 

11ut at t.he expense of a twcnt,y-fold incrcww in comtrut,c.r time. For engirlenring calcn- 

Iations Lhe conlwr grid shoultl suflicc; il, rc(luires running times of t,hc ortlcr 10 

sccor~tlrr in cr~sc of pr:~otinnl int.cw-st suclt as Ll~c In.tnir~nr lmr~ntlnry Inyrr ol' nn ncrofoil. 

11nlm)vcd econotny cnn bc nc:hiovc:tl Iiy vnrying the step size ns Lhc cnlculntion 

proccetls, tht is wing thC fine mrsh only in the critical region near separation. 

A summary account of nurncricnl methods in fluid mechanics is give11 in thc 

lecture notes of Smoldc,ren [G5]. 

j. Uoui~dnry layer of second order? 

The secontl-orclrr rquntions, cqns. (7 52) nnd (7.53) for flow in n hountlary layer 

were dcrivd in Ser. Vllf. This system of linear partial differential equations ran be 

solvcd if the first-order solutiorls ul (z, N) nnti vl (x, N) are known, and if the func- 

tions K(T), IJz(x, 0) nntl I'z(x, 0) nre suitably prescribed. 

It follows that the calculation of n second-order bounclary hyer on a given body 

in a strcnm requires that the following steps should be taken: 

(a) Cnlculation of the potential flow (external flow of first order) about the body 

with the boundary conditions IT1 (~,0) 

= 0. The solution yields Ul(s, 0). 

(b) C~lculat~ion of the first-order boundary layer for given Ul(x, 0), that is, determination 

of the solution of the oystem of cquations (7.49). In pn.rticular, from the 

uolution irl(r, N), vl(~, N) we calculate the function Vz(x, 0) with the aid of equ. 

(7.45). 

(c) Calculation of the second-order external flow for the boundary conditions Vz(x, 0) 

and zero velocity at large diotance from the body in accordance with eqn. (7.45). The 

solution provides us with Uz(x, 0) and Pz(x, 0). 

In what follows, we shall assume that t.hese steps have already been taken. We 

shall concentrate on more detailed second-order calculation for several particular cases. 

Symmetric atngnation flow: This type of flow wau analyzed in detail by M. Van 

Dyke (see also Chap. VII, [7]). It is assumed that. the expressions for the external 

flow of first antl second ordrr on a convex wall at the stagllation point (K = 1 at 

x = 0) have bcen found and yield 

U(2, 0) = Ull x -t F UZ1 2 + 0 (c2), 

(9.87) 

whrrc IJu nntl TIzl nrr conslsnts which dqwntl on thr shnpc of the lmdy. According 

to eqn. (7 48), wc make the following assumption for the inner solution: 

t I ~III i~idnhtrd t,o Profc~sor I

l!Ni IX. Ihnrt solutions of thc .stt*n(ly-stntr 1)onndnry-layer equntioll~ j. 15oundnry layer of second orclcr 1!)7 

Pnrnl~nln in n ~ymmetric strclnm: 'r'h sccond-o~dcr l)out~tlary layer on a parabola 

in n ~yrnrnrl.rit: st.rrn.111 was rnlcnlnt,cstl I)y M. Van I)yltc (see c~luo ('lhal). VII, 171). 111 

t.110 ~~cigl~l)o~~~.l~ootl of st,n.gn:~t,ion, wr II:LVC 

111 t.hc rnsr of the pnrnl~ola wc haw nt onr tlisposid a r1111ncricn.l solut,ion of t,hc co111pbt 

r Nn.vit:r-Stokrs cc]llnt,ions drrc to It. '1'. I):Lv~s I I I I nnd c:Ln use it. for a tlirc!ck 

cval~lnl ion of thr irnpr.ovc:n~rnt mntlr by t.hc sccontl-order t,l~oory. Pignre 0.17 sl~ows 

a plot. of the skin-fricl.ior1 cocfficicnt from (9.!)7) at a st,ngnat,ion point, of a parabola. 

in t,rrrns of 1.11~ Reynolds nurnhcr forlnrtl wit.11 the radius of cnrvature nt, thc vertex. 

It follows I'ron~ cqn. .(9.!)7) Lhat 

Cnrvc 2 in Fig. 9.17 is a plot, of this relation, wl~crcas Curve 1 dcpictss the first-order 

solut,ion. Curve 3 hi14 IJCCII plo1,tctl with t.hc rcsult.~ of It. 'I?. I)nvis's n~~mcrical solution. 

'l'11r ronsiclcrablc i~nlwovcrncnt~ cflkct~ctl by the sccontl-order tl~cory in the lower 

rango of Jtcynolds n~lmlms is clcnrly visible. In wtltlition, t,he dingrams give an unslnl)ignous 

intlicnt,ion that t,he sccontl-ordrr t.htory allows us to itlent.if.v the range 

of vnlitlity of first,-order lhcory. Jf an c~~or of up to 2% is to be tolcmtcd, it follows 

tl~at first-ortlcr thcory applies at J

O~l:cr ehnpw: Second-order cITcetls fi~r hdf-borlics havo Ixrn invrstigntetl 1)y 

Id. Devan 11 21. 'I'hc rcsultrs nrc similar to thosc for t.hc parnholn.. 'l'hc cocffi cimk for 

cqns. (9.!)7) and (9.98) arc 

1JI1 :- 1.5; 1JZ1 =r--0.02. 

l~ut.l.l~rr sol~ll.iorw or t.l~c: I)onntlary-lnyrr cyun.l,iol~s (7.52) ant1 (7.53) of scco11t1 ortlcr 

arc availitblc, as rnigl~l, 11n.ve 1)cr.n cxl~ccl.c:tl, for oases which 1en.d Lo sclf-sin~ilitr solutions 

in first order, Sec. VIII b. In the case of flows whose first-order external flows 

arc of the forn~ IJl(x, 0) - zm tht: scc:ontl-ordcr thcwry n.lso Irarls to sdf-similar 

solnt.ions if 

K(x) - x(*n-l)lz; U2(:x, 0) - xn. (9. 101;) 

l~urtl~cr tlchils conccrl~ing lhc cfrefr.ot*s of srcontl ortlrr can bo Sountl in CIhnlx Vl I n.s 

well as in V1lI [Gal, [IFia]. Tflc htlter cor~t,nin indi~nt~ions about second-ordcr effects 

in t4he prcscncc of suction, blowing, hcnt. tmnsfer and compressibility. Secot~cl-order 

clfccts ncqniro increasing impori.ance for high Mach nnrnl)crs antl in the presencc of 

blowing. In this connexion consult [24, 25,47,48,59]. 

[I] An(lrado, 15.N.: 'I'hc vnlocit.y dist.ribnt.ion in :I liqr~itl-i~~to-liq~~i~l jet. The plnnr jet. Proc. 

Phys. Soc. 1,nndo11 51. 784 - 7!)3 (I!):!9). 

121 Ihxtcr, I).(>., nrd I'liiggc-1,ot.z. I.: 'l'hr sol~ltinn of romprc~ssihlc laminar bo~~ndnry layer 

prd~lcnin by a finitr: tlilkrcncc rnrtd~otl. h r 11: ~ l~nrkl~er dist:nssion of the rr~ct,l~od and 

con~pnt;lt.inti of exnwplcs. Tcchn. Jl.cp. 110, IXv. 15ng. hfec:h. Stanford Univ. (l!)57); short8 

version: ZAh11' gh, 81 !IT, (1!)5X). 

[3] 13icklcy, Mr.: 'J'lw pl~tno jrt.. I'hil. Mag. Scr. 7, 26. 727731 (1939). 

141 I!Ianius. 11.:. (:r(~~~zst-l~i(:l~(r~l in 1~liissigkcit.cn n~it ltlcincr Itcibnng. Z. Mnt.11. u. T'hys. 56, 

I ~-37 (1!)08); I311gl. t,rnnsI. in NACA 'I'M 1256. 

I5J I3lott~rcr. F. (:. : I"initc difli:rcncc n~rthoda of solntioli of t.hr bor~l~dnry-lnyrr cqnalions. 

AlAA ,I. 8, 193 - 205 (1970). 

[f,:l] 13lott,ner, I?. ( 2.: 1nvent~igat.ion of some 1init.t: rlifircn~:~ tcclrniq~~cs for solving the houndnry 

1nyt.r w{~~:~tinns. (!OIII~. ill:tl,l~. Ajq11. MI!(:II. l4:11g. 6. I -- 30 (1075). 

161 (k+cri, 'I'., and Sn~ith, 11.M.0.: A Iinitc dilTcrc~~cc 1nct.l1011 for vnlr~~lating ronipressihle 

Iarnin:~r nnd t.r~rh~llcnl bonndnry Iayrru. 'l'r;ins. ASM 15, ,I. Ihsic I':t~g. 92, 52:)--535 (1970). 

171 Chnpn~nn, 11. It.: Imr~innr nixing of n wn~prcssil~lc llnid. NACA TN 1800 (1949). 

181 Ch:~l~~i~an, J). It..: 'I'h~!orrt.ic:rl :in:ilysis of hen(. 1.rnnsfcr in regions of scparat,cd flow. NACA 

TX 37\12 (1!l!it;). 

[!)I Chcn, I

200 1X. Ixxnrt nolntiona of t.lin ntmtly-nt-ah bounrlnry-lnyor equations 

1441 Leigh, I).(!. IT.: 'I'Iw 111111innr honnclxry layer equation: A lnetl~orl of solnt,ion hy tnnnnn of 

an nuton~ntic con~ptor. l'ron. Cnmbr. I'hil. Son. ,51, 320-332 (1956). 

r44nl 1,rsnen. M. : 011 t.11r &tl)ilit.y of t,he Inwir~:tr frcc bnunhry layer hrt.wt!en pnm.llcl ntrcnrnn. 

NA(:I\ Jkp. !)7!t (I!t50); SCC :dso Sc. I), 'I'l~nsin, MIT (l!M8). 

[46] I.ork, I

202 X. Approximate rncl.hoda for steady cqrlntionu n. Applicnth~ of the morncntrlm rqr~ntion to Lho flow pnut n flnt plnLo at zcro incidcr~cc 203 

the c:ont.rol s~lrfacc, consitlcrrd fixrd in spacr, is cq~lal to the skin friction on the 

plate D(s) from the leading cclge (s =0) to the current section at x. The application 

of thc momentum equation to this particular case has already been cliscussecl in 

See. IXt It was then found, cqn. (9.26), that the drag of a plate wetted on one 

side is given by 

m 

D(4 =be/ u(u,--u)dy, (10.1) 

u-0 

where the integral is to he taken at scrtion s. On tho other hantl tho tlmg mn bo 

expresscd as an intrgral of the shearing stpress to nt thr wall, lnltrn nlong tl~c plntr: 

X 

1) (s) = b 1 r0 (x) dx . 

Upon comparing eqns. (10.1) and (10.2) we obtain 

0 

This equation cnn bc also dcclucccl in n purrly formal way from t,hc 11onntlnr.y-layer 

equntion (7.22) by first integrating the equation of motion in the x-direction with 

respect to y from y ---- 0 to y = m. Equation (10.3) is, finally, obtained without difficulty 

if the vclocit,y component v is eliminated with the aid of the equalion of 

continuity, and if it is noticed t.hat p(au/a~),,,~ =to. 

&'Iu:$ 

mtmI surlace 

Fig. 10.1. Application of tho momcntun~ equa- 

-5 u @.Y) tion to the flow pmt n ant plnto nt zero incidencc 

-x 

Introdncing thc morncnturn thiclir~css, a,, defined by rqn. (8.31), wc have 

Tllc momcntom cqmtion in ils form (10.4) rcprosrnb n particular cnso of the gcncrnl 

momentum eqnntion 01' bountlary-laycr Lhcory as given in eqn. (8.32), heing valid 

for the cnse of n llat plntc nt zcro ir~ciclcncc. 1t.s phpical meaning expresses thc fact 

thats Iht! shearing stmss at the wall is cqunl to thc lhss of momentum in tho bonntlary 

I:lycr, because in tho cxarnple under consiclcrnlion t,hcre is no conl.ril~ution from t,llc 

prcssure gmtliont. 

'So far rqn. (10.4) int,roclucncl no ntlclit.iona1 :~ssnmpLions, as will be the case 

wit,l~ the ajq)roximntc method, bul, 1)c:forr tliscussing this matter it might be nscful 

to nolc x ~cI:LI.~oII I~CLWCCI~ to nnd S2, WII~CII is obtaincti from cqn. (10.4) by int,rotlucing 

the exact value for to from eqn. (7.32). Putling tu/p Urnz =a iy/urn2 with a =0.332, 

we have 

E .- 

' 

With rcfc:rcncc to qn. (10.3) or (10.4) wc con now porfortn nn npproximnto 

calcnlnt.ion of the l~ountlnry laycr nlong n Il:bt, plnlo at zcro incitlcncc. '1'11t: CRWIICO 

of the npproximatc method consists in assuming a suitable exprrssion for the vclo~it~y 

tlist,ribution u (y) in the boundary laycr, taking cnrc thnt it sntisfics the importnnt~ 

boundary conditions for u(y), and that it contains, in addition, one free parameter, 

such ns a ~nitddy choscn boundary-layer thicltncss which is finnlly dctcrmincd wit.11 

t,he aid of the momenlum equation (10.3). 

In the particular case of n flat plate at zero incidenco now being considered 

it is possible to t,ake advantage of the fact that the velocity profiles arc similar. 

IIencc we put 

where r] == 2/16 (s) is the dimensionless distance from the wall referred to the boundnrylayer 

thicltness. The sin~ilarity of velocity profilcs is here acconnt.ed for by assu~ning 

that /(?I) is a function of 7 only, and contains no additional free parameter. The 

function / must vanish at the wall (7 = 0) and tend to the value 1 for large values 

of 17, in view of the boundary conditions for u. When using the approximate method, 

it is expedient to plnce the point. at which this transition occurs at a finite distance 

from the wall, or in olher words, to assume a finitc boundary-layer thickness 6(x), 

in spit.c of the fact that all cxnct solutions of t.hc houndnry-layer equations t.cntl 

asympt~otically to the ptential flow associated with the particular problcnl 'l'hc 

boundary-lnyer t.llislrncss has no physical significance in this conncxion, being only 

a quant.ity wl~ich it is convenient to use in thc computation. 

Having assrimcd tl~c vclocity profilc in cqn. (10.0), we c:~n now proceed to 

rvnl~~atc tho momentum intcgml (10 3), arid we obtain 

for short,, we have 

ru(uW- u) dy=um2~2 = a, 8 urn2, 

v-0 

or d, = a, 0 .

204 X. ApproximnLc rnetl~ods lor steady equations 

'I'hc value of the displacement thickness O1 from cqn. (8.30) will now also be calcn- 

laktl as it will be required later. Putting 

wc: oltt,n.in 

1 

a2 = J (1 - l) dtl, 

0 

(10.10) 

0, --a, d . (10.1 1) 

I?'l~rt,hrrniorr, thc visrous shearing stress at the wall is given by 

Introtlllcing thcsc valnrs into the niomcntum equation (10.4), wc obt,ain 

Int.cgrat,ion from 0 0 at z -= 0 givrs t.hc first, result. for tho approximato thcory 

in t.11~ form 

ITrnc~ tllo shearing strrss at the wall from cqn. (10.12) beronics 

Finally, the l.otd drag on a plato wrttetl on both sides mri be written as 

I 

2 I1 -- 2 h J to 

0 

tlx, i. c. 

:1n(1 fro111 r(lns. (10.1 1) and (10.14) we obtain the tlisplaccmerit t,hicknrss 

A comparison of t.11~ approximaf,c oxpressions for the Itonndary-laycr thickrwss, 

li)r the shraring st.ross at tho wall, ant1 for drag with thc respcctivc formulae of 

t.11~ :lrc:ur:~t,c throry, rqns. (7.37), (7.31) ntd (7.:13), shows that Lhc use of tho iritcgr:d 

rnorner~l~um cqui~tion lcatls in all cn.ses to a peufcctly correct fornlulation of the 

cqmtions. In other words, the dcpcnrlcnccof tliese'quantitics on the current length, x, 

the frcc-stmxm vclocit,y, Urn, antl the coeffioiont of kinematic viscosity, v, is correctly 

tlctll~twl. li'urt,I~crniore, the rclation 0ct.wecn momentum thickness and shearing 

strrss nt, ttw wall givrn by rqn. (10 5) ran also be dcducrd from the approximate 

rnlrulation, as is rnsily vcrifird. The still-unknown coefficients a,, a, and P, can only 

o. Application of tlir mo~nrnt~~rn rqr~ation to Lllr flow past a flat plr~lr at mro incidrnrr 205 

hc calculated if a specific assumption regarding the vclocit,y profilc is matlr, i. r. 

if t,lte function I(??) from eqn. (10.6) i~ given explicitly. 

Whcn writ.ing down an expression for f (q), it is ncrrssary 1.0 sat,isljr cc.14nin 

boundary condit,ions for z~(y), i. c. for /(?I). At lcast the no-slip t:onclit.io~~ IL -- 0 

:~t y -- 0 antl tho condit.ion of continuity whorl passing frorn t,ltc hottnd:tr~y-laycr 

l)rolilo l,o I.hc ~ml,(:~)l.i~rl vl:loc*il.y, TI . (1 III, - , 0, tnr~sl~ lw s~~lisli(vl. I~III~IIVI~ VOII. 

(litlions might inclutlc t.hc continuity of thc tangent ant1 curvalurc :IL tl~o 11oirlL, 

wlicrc t,lic twr~ solutions aro joined. Tn othrr words, wc may scrlr to satisfy tha con- 

ditions a~l./i)?l =: 0 and 321~/a?/~ 

= 0 at y = 8. In tho case of :L plate tho cot~tlit~io~t , 

that a2u/tJy2 = 0 at y = 0 is also of importancr, and it ran I)o scon frorn rqn. (7.15) 

tl1a.t it is satisfied by tho exact solution. 

Numerical cxamplcs : 

Wc now propose to test the usefulness of the prccctling approxirnak mct.hotl 

wiLh the nit1 ofscvrrnl rxnmplcs. Tho q~~alit:y of thc rcsnlt tlcpcntls to a grcat cxI.cn1, 

OII t,hc assurnpI,ion which is matlc for thc volocity f~lndion (10.6). 111 ally c::~sv, 

as already mcnt,ionecl, the funct,ion /(q) must vanish at 17 = 0 in view of the noslip 

condit,ion at the wall. Moroovcr, for large values of 17 we tilust havc /('/) = 1. 

Tf only a rough approximation is tlcsirccl, the transition to the valuc /(q) = 1 may 

occur with a discontinuous first, tlcrivativc. For a bettcr approximation, corrLinnit,y 

in dj/dfl may bc postjulatcd. lndcpcntlcnt~ly of the pnrticular assumyt,ion for l(q) 

the cruant,itks 

must Itc pnrc numI)crs. Thry can bc easily calculated from cqns. (10.8) to (10.17) 

Fig. 10.2. Vclocity tlislrilmhn in t.hc boundary 

layer on n flat plntr nt xrro i~icitlanrc! 

(1) Lincrr aplrroxirnntion 

(2) Cubic npl~rrrxirnntiou Irom Tablr 10.1 

Tablo 10.1 contains results of scvcral calcrtlations wil.11 a.lt,crnativc veloc:it,ydistribution 

functions. Tho first two fun~t~ions nrc illuslr:l.tod with tlrc aid of I'ig. 10.2. 

'I'hc linear funct,ion sat,isfics only the conditions f(0) -- 0 antl /(I) -= I, wllcrcas tho 

cubic function satisfies in addition tho conditions /'(I) - 0 :~nd /"(0) :x 0; finally, 

a fonrth tlcgrcc polj~nornial can be made to satisfy the atldjtional contlition /" (1) =-- 0. 

Thc sinc function satisfies the same I~oundary conditions as the polynomial of 

folirtli dtgrcc, except for /"(I) = 0. The polynoniials of third antl fourth tlrgree 

and the sine-function lead to values of shearing slrcss at Iho wall which arc in 

error by loss than 3 per cent and may bc considrrcd ent,ircly atlcquatc. 'The valnrs 

of the djsplaccmerit thickncss 6, show acccptablc agrecmont wiLh thc corrcsponditlg 

cxect values.

206 X. Approxitnnte rnct.l~otls for steady equations 

Table 10.1. Rrsultn of the calcrllation of the bolnltlary layer for a flat plate at zero incidence 

baaed on approximab thcory 

Vclonity 

dint,ribtrtion 

ll/U = f (11) i 

It is seen that t.hc npprosinmtc mct.liocl Icacls to sa.t,isfnctory rcsult,~ in the case 

of a flat plate at zero inciclcncc, and the extraordinary simp1icit.y of the calcnl:~tior~ 

is cluite remarkable, compared with the complcxit,y of thc exact solution. 

We now propose to clcvelop thc approximate method of thc preceding section 

so t,llnt it can l)c applied to t.hc general problrm of a two-tlirncr~sional hotrntlnry 

layer with prcssurc gradient. The tnct,l~od in its original for~n was first intlicatctl I)y 

1C. l'ohlhar~scn [Is]. The succeccling tlcscriptiotl of thc method is based on iLs mnrc 

motlcrn form as developcrl by tT. TIolstcin and T. 13011len [GI. \Vc now choose, as 

before, a system of coortlinat.cs in which x c1enot.c~ t,hc n.ro mcasured along the wcttetl 

wall and whcrc y tlcnotos the tlisLancc fronl t,hc wall. 'rhc hsic crlnat,ion of thc monrent,nm 

theory is ol)t,ninctl by intcgrnthg the eqr~:~l,ior~ of motion wit#h rcspcct tm y 

from t.hn wall atf ?/ -=- 0 t.0 a ccrt8i~.in tlistanca h(x) which is a~sntn(:d to be outside 

t.11~ I)o~tndary layer for all val~ros of x. With this r~otat~ior~ the momentum cqttat.ior~ 

' 

11a.s the form nlrcntly givcri in (8.32), namely 

This eq~~ationgives an ordinary diffrrrni ial rqnation for the ho~~t~tlnry-la~pr thiclrncss, 

as was tire rase with thc flat plntc in lJic prccctling scc.lion, provitlccl that a 

I 

b. The approxirnnto nirthod duo to TIN. von Jchrnlhn and K. Polllha~~ncn 

form is assumed for the vclocitty profile. This allows us to calculate the momentum 

tl~ickncss, the displacement thickness, and the shearing stress at thc wall. In choosing 

a suitablc velocity fi~riction it is necessary to talrc into account the same considerat,ions 

ns beforc, nnmnly thosc regarding the no-slip condition at t,hc wall, as wcll as thc 

reqi~ircment,~ of cont.inclit,y at, the point whcrc this sol~tt,ion is joinctl to tho poLcnti:d 

soIut,ion. I~t~rI.l~t~rn~orr, 

i11 I,hc prcsonco of IL pr(-ssuro grmlic~tt~ tho f~~nc:I,ion n~~tst 

atln~it the cxisbc~icc of profilcs with and without a point of inllcxion corrcspontling 

t.o t,hcir occnrrencc in regions of nsg,zt,ive or positive pressure gradients. In ortlcr 

to kc in a posit,ion to cn.lcr~latc tho point of scpamtion with tho aid of thc npproxin~at~c 

n~etl~otl thc existence of a profile with zero gratlicnt at thc ~ ~ (au/ay),-,=0 1 1 must 

also be possible. On t,hc ot.hcr l~ancl functions postulating similarity of vclocity 

profiles for various valws of x may no lorigcr be prcscribctl. Following I

208 

where 

X. Approximate methods for steady equations 

It is easily recognized that thc velocit,y profiles cxprcssed in terms of g = y/b(z) 

constitute a onc-paran~ctcr family of curves, the tlirnensior~less quarlt,it,y A being 

a shape factor. The tlin~cnsio~iless qu:lnt,it,y A which may also IIC written as 

. PdU dv 6 

ran be intrrprcted pllysically as thr ratio of prrssuro forces to viscous forces. In 

order to obtain a quantity to whirh real physical significance can be ascribed, it 

would be nerrssary to replnce 6 in the above definition by a linear quantity which 

itself posscsscs pl~ysical significance, such as the momentrim thickness. This will 

be done Ialcr in this section. 

Fig. 10.3. Tho functions F(91) and G(77) for 

the velociLy rli~t~ribntion ill Lhe boundary 

layer from rqns. (10.22) and (10.23) 

'1'11e two ~IIIIC~~~OIIS F(77) and (:(q) 

Fig. 10.4. Tho ot~c-parntncter family of vclocity 

profiles from eqn. (10.22) 

tlcfinctl by nqrr. (1'0.23), which togt.l,hcr compose 

tthc vc1ocil.y-rlist,riI)r~t~ior~ function givnn in cqn. (10.22), a.rc serrl ploltcd in Fig. 10.3. 

Vrloc:it.y profilrs for v:lrions vnluns of A arc shown in J'ig. 10.4. The profile which 

crmcspontls t.n A -: 0 is ol)f.ainc?rl whcn tllJ/tlx'= 0, i. r. for ihc bourrda.ry Iilycr 

wil.l~ no prrssltrc gr:~tlirnt (Iht, glnt,e nt zero i~vcitlrncc), or for a point whrre the 

vrlocity or t,hc pot.rrrtinl flow pnssrs thlongl~ ~nilrim~~~n or a ~nn,xinln~n. In this case 

lhc: vclorit.y profilo l~cconws itlenticnl with the fourth-drgrcc polynon~ixl uscd for thn 

IlnL plate in t,ho prcrrtling snct.ion. 7'11c prolilo at separation wit.l~ (itu/?i/),, -- 0, 

i. e. nib11 (1 =- 0, occnrs for A == - 12. It will he shown later that 1.11~ profile :LL 

tlhr st~agnat.ion point corresl~ol~ds t,o A - 7.062. For A > 12 vnlnrs ?r/U > 1 occrlr 

b. The approximate rnethocl dl~o to Th. von Kkrnldn ant1 K. Pohll~nnsen 209 

in the bolrntlary layer, but this must be exrlntled in stcady flow. Since behind the 

point of separation thr, present cxlculalion bxsrd, xs it is, on the boundary-layer 

concept,, loses significance, the shape fnctor is secn to be rcstrictrtl to thr rangr 

- 12 1 A -k 12 , src Pig 10.4. 

J3cforc proceeding to cnlculat,c the bountlary-layer thiclrncss S(x) from the 

mon~cnl,~lnl I.llrorcm~, it, is now cnnvrnicnt, 1.0 (:ILIIYII~LI,O t h ~IOIIIO~I~.~IIII ll~idcllrss, 

S2, the tlisplaccment thickness, dl, nntl the viscous sliearing strcss aL llrc wall, t,,, 

with thc aid of the approximate velocity profile in the same way as was done for 

the flat platc at zero incidence in the preceding section. 'J'hus we obtain froin cqns. . 

(8.83) and (8.31), t,oget,her with eqn. (10.22), 

Computing the definite integrals with the air1 of the values of F(q) and (:(I/) from 

eqn. (10.23), we havc 

Silnilarly, tho viscous stress at the wall, to = !~(il~~/ay),_,, is given by 

In ortlvr t.o tlct.rrmii~r the s(.ill-unknown s11n.p~ factor A (z) and, hrncc, t11c fun(:tion 

O(x) from cqn. (10.21), it. is now necessary t,o rcfrr to the momcnlum cqunt.ion (10.18). 

h111lt.il)lying by d,/v IJ we can rcprcwnt it in t,hc following tlitnc~lsionlcss form:

210 X. Approximate mcthod~ for shady rqi~etions 

which is connortctl with thc momentum thickness in thc samc way as t.lle first 

shape factor, A, was connected with t,hc boundary-layer t,hickness, 0, in cqn. (10.21). 

Tn atltlit,ion wc sldl put 

= %" 

v ' (10.28) 

so that 

It is wrn from crjns. (10.21), (10.27) nntl (10.24) that (.hc slmpc fa.ctors ,,I anti li 

satisfy Lhe universal relation 

for thc saltc of brcvity, mtl suI)st,ituti~lg, I< ant1 Z from eqns. (10.27) and (10.28) 

rcspecl,ively, togothor wil,l~ fl (K) ard /,(I() from eqns. (10.31) and (10.32), we 

obtain, furthcr, from Lhr momcntum cqu:~t,io~l (10.20) togcthcr with a2 6,'/v = 4 ~IX/tlx, 

thc rrl;~t,iotl 

2 /2(1

212 X. Approximntr 11iet11odg for stc~rly cquntions 

'bhlr 10.2. 1211xili:iry fr~nrtio~in for the npproximntr rnlr~~letiorl of lnlliitier boundary I:~yera, 

b. The npproxitnste mct,hod clue to 'rh. von Kbrmhn nntl I

214 X. Approximato mothotls for shady equations 

or, u~ing the ~lumerical values for a and b given earlier: 

I'hus the solution of eqn. (10.36) is secn to reduce lo a simple quadrature. An analogous 

quadrature will br uscd in C!hnp. XXIl for thr ~olution of thc cquntions of 

turbulent. flow. 

-0 R -am -OM -am m 602 o am onc m om 

K 

Fig. 10.6. Tho auxilinry funchn F(K) for the col- 

culstion of lnminnr boundnry lnyer by thc method 

of JIolntcin and Bohlen [5] 

(1) train* eqn. (10.35): 

(2) lincar nlrproximntioll F(R) -- 0-470 - 6 K; 

S = ahgnntion point: 

lf -- vclori1.y milximum 

c. Con~pnri~on between the npproximntc nnd cxnct solutions 

I. Flat plnte at zero incidence. It is easy to see from cqn. (10.22) that the 

Pohl11a11ncn i~pproximation becomes cquivalcnt tjo 1Sxn.mplc 3 in 'Ihblc 10.1 for the 

case of a fI:~l. piat,c at zero incidence. l'his ease can also t;c obtainctl directly from 

eqn. (10.36), whcre U(z) = U, U' = 0 and hence R = A = 0, so tlmt ejn. (10.36) 

givcs dZ/tlz = F(O)/U, = 0.4698/U,. Taking into account t,hat Z = 0 at s = 0 

it follows that R =0.4098 z/fJ,, or a, - 04XH3 I/l.&/ii, in ngrccmcnt wi0h Tal)lc 

10.1. Table 10.1 contains cx.zct and approxi~nat~c values of the boundary-laycr pnrarnct.crs 

for tho purpose of comparison. Tt is sccn tb'nt agreement is very satrisfactory. 

2. Two-clin~cn~io~~al stngnntion flow. The cxart solntion of 1,he problem of 

I wo~dimrnsional stagnation flow for wllirll U (r) = U' . r, was given irl Sec. V 9. 

r 7 

Ihe exart vnlnrs of displacrmcnt ll~iclzness, momentum thickness and shcnring 

strrss at tho wall, calrulalcd with the aid of that theory, arc given ill 'I'aljlc 10 3. 

c. Compnrison lwtwccn tho upproximutc and oxnct uolrlth~ 215 

Tablo 10 3. Compnri~on of exnrt nnd npproximnto vnlr~rs of Lhr honndury-lnyer pnrurnctors for 

tho caso of tim-dimen~ionnl stngnntion /lout 

oxnct, solul.iori 

In tho approxi~nnt~c mcthotl wc havc Z0 = R,/U1, and from cqn. (103) it follows 

- -. - 

that t11r momentum tl~ickness is given by (T2 iVfi = dif; = i0.0770 = 0.278 . 

It is seen from cqn. (10 81) that the displacement thickness is approximatctl by 

6, JiF/v = /,(~i,,) 1/ii, = 0.641 anti cqn. (10.32)gives t0/p u - 47~' = /z(Ko)liR, - 

0 332/0.278 = 1.1!) for tllc sllcnring slrcss at the wall. 'L'h ngrccnlcr~l bclwccn 

the approxirnnte ant1 exact values is here also complet,cly satisfactory. 

3. Flow past n circulnr cylinder. A comparison of the result of the approximate 

calculation for a circular cylindcr with the mlution due to Ilicmcnz (See. IXc) 

was given by I

216 X. Approximato n~ethods for steady eqllations d. Further examplcs 217 

well as in the shearing stress are concerned, and predict an earlier point of separation. 

W. Schocnauer found that thc separation angle is at$a = 1045Oas = 109TP' 

ol~t.ainat1 with t11c nit1 of the l'ohll~auscn approximation and $s = 108.8O suggested by 

tllc series cxp:ansion cont,inuctl up to the term z". A comparison between t,hc vclocity 

tli~t~ribut~ions, IGg. 10.8, leads to the conclusion t,l~at there cxist,s almost perfect 

agrwmrnt. hrf,wcrn t,Im exn.ct solut.ion and the npproxiniation in t.hc m.ngc of angles 

0 < 4 < !)On, t,lrat. is in t,l~c range of acc:clcrat,ctl c*xt.crnal flow. I 12 (K > O.O!IR), because the plot of X against A !.urns at, this poit~l 

(Table 10.2) and cantlot, tl~rmlore, be contirlucd 1)oyond K -- 0.0!15. Moreovor. for A > 12 

the vrloriby prnfilns l~rco~no r~nnrrcpt,al)le as t.liry rontnir~ point,^ for \vl~irl~ ti/II -.- I (Fig. 10.4). 

'I'l~rar cliffioid!.ira arc obviatcd wl~cr~ cqli. (10.37) is IIRC~.

218 X. Approximntn met,hods for shndy equntions d. Further oxnrnplea 219 

-4 f~~rther example is shown in Fig. 10.12 which contains results for a symmetrical 

Zhukovskii nerofoil at zero incidence. The point of minimum pressure is at x/l' = 

= 0.141 which is very far forward on the aerofoil. The pressure rise at the rear is 

very gradual so that the point of srpamt,ion lies very far downstream of the point 

of minimum pressure, i. e. zll' = 0.470. Since the Zhukovskii aerofoil has a cusped 

trailing edge t,hc potential velocity at tho trailing edge is djffcrent from zero. For 

details of additional systcmntic boundary-layer calculations concerning nn extensive 

serics of Zhukovskii aerofoils with different thickness and camber ratios and at 

cliffrrcnt angles of incitlcnre, refcrcnce may he made to a piper by K. nussmnnn 

and A. Ulrich [2]. 

Pig. 10.0. Potcntinl velocity 

distribution function on ellip- 

tical cylinders of slenderness 

o/h = 1, 2, 4, 8, tlic direction 

of the stream being parallel to 

tho major axis 

R ;-; position or point of srpnrrtion 

Fig. 10.10. Ilrsulls of Lhc? cnIculnt,ion of hor~ndnry In.yers on ollipt,icd cylinders of ~lcndorncss 

n/b t 1, 2, 4, 8, Jcig. 10.9. n) displnrernnnt~ t,liickrirsa of the boundary layer, h) shape factor 

r) chraring st.rrw nt the? ~d. 2 1' -- rircllllifcrr~~~~c of blic rllipsc; n/b -. 1 rirct~lnr cylintlcr; n/b = m 

flnt plate 

A review of the very numerous approxi~nate methods which have been proposed 

so far is contained in the collective book entil.lod "Laminar Boundary Layers" 

1181 - . nnd edited by 1,. Rosenhcad. 

In an effort to improve t,he accuracy of the calculation of laminar boundary 

layers, many authors replaced the preceding single-pnrameter methods by oric employing 

tim pramelers. This is acl~icvcd whon t,hc encrgy intcgrnl cquntion is sntisfed 

in addition to the momentum intcgrnl equation (ace e. g. I

220 

X. Approximate n~etlrods for steady equations e. Laminar flow wii.lr ndveruc presnure gmtlient; sepnrclt.ion 22 1 

e. I,nmit~nr flow with nclverse pressure gradient ; sepnrntion 

Flows with :dvcrso pressure gr:itlicni.s (rrtmdrd Ilows) arc of great practical inrportancc. 

111 I.l~in connoxion it is always desired to nvoid ncpcr.m/ion from t,lw wall, hccanse thin phcnomenon 

in associatccl with large energy IO~RCS. '1111~ flow al~wt an nrrofoil in a case in point,. Owing b 

t.hc hct that on the nucLion side the pressure must, increase to it# free-stream val~~c at the txailing 

edge, the flow is always likely 1.0 sc:parxtc. 'l'l~c flow in a divrrgcnt channcl (din'~~scr) nfTords 

anollrcr cxan~plc. 'l'l~c objcd in using this sl~n.pc: of cl~nnncl is lo convcrt. kincl.ic cnrrgy i11t.o 

prr:ssurc rncrgy, and if Lire angle of tlivcrgrnco is ~nndc t,oo large, srpamt,ion mrry oocnr. 

l'hcoret~icd i~~vrst.igat.ions on the l)cl~nvio~~r of the I)ountlary Ia.ycr in the vieini1.y of I h 

point of nrparntion hvr been carried out, hy S. Goldsbir~ pi] and 13. S. Strni.ford 121 :rl. C/. talm 

rcvirw I)y S. N. Ibown a ~ 1C. d Stfiuvrrtnon I I). 

Ol~scrvat.ions slrow thnt a Inlninnr 1)ounelnry Inyor whirlr separates from a wall frcqr~ent,ly 

I)reolnes rmt.t.:drc~d lo it, having first hrron~c. t.~~rl)r~lcni.. Thin Irads t,o tho crent,ion of a laminar 

separ:~t,ion bnbl)le. Fig. 10.13b, wl~iclr pl:~ccn it.sclf bet,\rrerr the separation point S ~hnd 1,110 rcaLt,nchtnent 

~~oint, R. 'l'h flr~itl in the bul)l)le 1)erfornrs a rircrllntory motion. According t,o 1%. 

10.1:h, the prrssure tlistrib~lt.ion nlong the wall can be represrrrtctl, in ~iml)lified fashion, by fi 

ronstant, vnlne brtwcen thr point of separation S and point I' of largest thickncns followed by a 

litrcnr inrroasc from I' t.o the point of reattachment n. Phenomena of this kind have been denrrilml 

in tlchil l)g I. 'rani [2:3]. More recent experimental investigations into tl~e nnturc of 

I;uninnr scpnrat.ion I~nhhIrs I I R V ~ I)rcn pcrfort~~rtl by A. 1). Young rL nl. [2R] as well an hy hf. 

Gnstar 1481 and J. L. Van Ingen [GI. For theoret,ical contribr~tions see [Zb, 3a, 5~1.3. 

It, will now bn shown wil,l~ the aid of srvrr:d cxntnplrs that, a laminar flow can only support 

vrry sn~:~ll arlvrrsc: prcssorr gradin~~t.~ wiI.lro~~t srp:lrnt.ion. Adverse pressure gradients wlriclr exist 

in practiral npplicxtions wonld, tlrorcrorc, nlmont nln.ays Ictul Lo separation if the flow were 

laminar. 'l'hc: circ~~msl.ancc that real flows c:~n support consitlcr:~l)le rates of pressure increase 

in n large nr~tnl)cr of rnsrs without scpnr:~t,ion is duo t.o the fact that the flow is mostly turbulent. 

It will hc srrn later t.l~a.t. t.w+ulcnL flows arc r:~pal,le of overc~otning n~uclr larger adverse pressure 

gr:~rlicnts \vil.l)out scpral.ion. 'l'lrc Iwst known rx:~mplcs inclutlc the cases of flow past circular 

c~yIindrrs and s~II(.I-os, WIIOII srpnrat,ion ocr:urs ~nnc:lr f~~rt.lrcr r~psLrcam in laminar than in tur- 

1)11lrnt. flow. In pri~cti(:t> wl~en a~lv~rsc prcss~~rc gr:dicnf.s exist, the flow is aln~ost always turbulent 

l)rcn~~sc, in atlclilion, fl~o cxistcncr: of an nrlvorsc prrssure gradient favours the transition from 

laminar 1.0 L11rh111c:nt. llow. 1 t is, ~~cvcrtlrcl~~~s, 11sef111 to clarify some of t,he f~mdamcntal relations 

:wso~~i:~lcd o.it.11 tho provo~tlio~~ of srp;;rntin~~ on fl~c cxarnplc of I:m~itrnr flow, in particular, 

0cw11sr I:IIII~II:I~ l10ns i~rc 11111~11 Inore rrarlily :I~II~II:II)IC 1.0 n~atlrcmai,icnl treatment than is tho 

wsc wit11 I ~~rhlrnt, Ilnws. 

'I'l~vrr arc. svvrr:tl ~nrt.l~ocls of prcvcnt,ing scpr:tl.ion. The simplcst of t,lrenr consists in 

:rrr:~ngi~~g for IIIV ;~~lvrrsv 1)rrss11r(: gri~dir~~i.~ lo rrn~i~in hrlow the limit for wlrirl~ ncpnrat,i~n 

I'j 

s V R -- 5 

I 

Fig. 10.13. Scpar:~t,ion bul)l)le in a laminar bountlsry 

li~yor nfiar I. 'l'nni 123). a) Shape of bubble (nchcnratir): 

b) l'rcssl~ro distrib~~tion in hnbhle along tl~e wall (w!w 

matic). 'l'hc nrc'snurc hetwoen S and V in tlro I)r~hhlo 

docs occnr. 12 numcricnl example will serve to make thin idra clonr. Another possibility consisln 

in ontrolling the bountlnry Iayer, e. g. by suction or by injecting fluid into it, or by addition 

of -11 arrofoil at a poit~t where ils presence favoural)ly afictn t.lrc I~oundary Iayer in critical 

regions. Thrsc mct,lrods will l)c discrlsscd rnorc fnlly in Chap. XI\'. 

I'olloaing 1,. 1'randt.l [I61 we sllall show how it is possil)lo 1.0 cni.imaLc t.lw pcrn~issil,lc 

rl~ngnilntlc of the :~dversc prcssure grn.dicnt for wl~irlr scjmrat.ion is jnst. prcvcnt.crl. 'l'llc arg~~mrnt 

will he I):~.srd OII ~,III? von I ~&~~II~~II-I'~IIIII;LII~~II a~~~~roxi~~~:~tiot~ 

disc~~ssrd inSco. XI). 11, will hc:~w~~nir~l 

ll~:it. tl~r I ~ o I I I I liiy~~ 

~ : ~ ~ in iwlwl 1111nt1 I1.y t.11~: 111~.xwrc: didril~ution (Irt(.r~ni~~c.(l I),y t110 11.(~.51 ~I.:IIII 

poLr111i:rI flow 111) 1.0 :h point, which lirs wry clclsr in t,lie point, of scp:wation, sue11 as point8 0 in 

Fig. 10.14. St:rr.Ling with this point, it will I)c assumed that i.hc pressure gratlicnt is srtoh that t.110 

s11:1pc of IIIC vnlnrit,,y profiln TOIII:I~IIR IIIICIIRII~(~ ~)rocrc~ling (I~WIIR~~~:LIII, or trIu~t,, in oI,11rr \v~rds, 

tho filq": C:~f:lor A rc:mainn ronst:~nt.; fiincc aL sc:lr:~ral.ion A .- - 12 a valnc of A -7 - 10 will 

be cl~osrn. As seen from 'I':tl)le 10.2 this lcadn t.o a definite value for the second slrnpr factor, 

narncly I( = - O.l:169, so L11:lt Il'(K) = 1.Tr23. Using tl~cse valurs it is sccn from rqns. (10.28) 

and (10.29) that prevention of separation i~nplies the following relationsliip between the vclocity 

U(z) of potential flow and the momentum thickness d,(x): 

It follows that dZ/dz = 0,1369 U"/U'2, or 

8,' 0.1369 

- =z= - 

v - V(x) . 

Fig. 10.14. Devclopnrent of boundary Fig. 10.15. I'otcntial velocity fi~nction 

I:~yer in thr! case when laminar separation for n laminar boundary layer with and 

is prevented without separation 

On the otlw hand the succeeding vclocity proflcn are given by tho ~non~entutn rquation (10.36) 

for 3. =. 0, or 

(le 

U --- = F(I

222 X. Approximate methods for steady eqnations 

Qnalitnt.ivcly it is at once possil)lc to mnkc the following nt~atcmcnt regarding the shape 

of [,he potential velocity function U(x) which Icatls to no xcpnrntion. In viow of cqn. (10.41) 

U" > 0 

is a nrccnmry condition for n rctnrtlrd flow (IJ' < 0) to xrlhcro t,o the wall. In other words, t.11~ 

~nn.gnitude of the advcrsr pressure grntlicnt, n~r~st tlccrrnsr in t,lie flow direction. Ii'ig. 10.15. 

, % 

J hns scpnrntion will nlwiiys occllr il' the f~~r~t:l,ion I/(%) iu cx~rvctl tlownwnrtln 1)chinrl its mnxi~nnm 

(11" < 0). In the opposit.e rase, whim tho vc1ocit.y function ci~rvrs i~pwnrds (U" > O), srjmration 

tnny he ol>viatetl. 15vcn the limiting c,we of IJ" = 0,i. e. the rase of a velocity which tlccreanea 

lir~carly with the length of arc, always Icatls to separation. Thin latter remark agrees with the 

rmitlt fonnd in Src. IXd; it was conccrnrd with the boundary lnyer aaaoriatcd with a potentinl 

flow vc1ocit.y which dccrcnwd linearly, and the solution of thc tlil~rret~tial cquntiona wm quokd 

from a pnpcr by I,. Ilowarth. The su//icient condition for the absence of aepnration iu givcn by 

Wo RIIRII now procrcd to cnlcnlntc tho potential flow and the varintion of hountlnry-lnycr 

thickncus which are wnorinktl wiLh t,I~r 1imit.ing cane of o - I I, whrn tho bonntlnry lnycr re~nninu 

on t,ho verge of sepraLiot~. Ikwr cqn. (10.41) we Imvo 

. U" . .. - U' 

u, -- I1 x - 

or, npon intrgrathg: In U' -. 1 I In 11 -I- In (- C,'), i. r. IJ'/CJ1I = - C,', whero 6,' denotes 

tho constant of integrat.ion. ltcpcntccl intagrnth~ ~ivcs 

u 

- 1 

U-lo = C,' z + C, . 

For z - 0 wo uhould hnve lJ(r) .- IJ,. no that C, = $6Nn-'O. Putting furthcr C,' UOl0 = C,, 

we obtnin from cqn. (10.41) 

w. 

Eqnat,ion (10.43) reprcsrn1.s the pot.cnlia1 vslocit,y for wl~ich cloparation can jnst be nvoidwl. 

Thr constm~t C, can IIC tlctnrminccl from the vnlnc of the bountlnry-layer I.hickness do at the 

origin z = 0. We hnvc A - U' P/v = - 10 or d - 1/10 ;q/(--D7). From eqn. (10.43) we ohtain 

and hrnrc 

From 6 - r!, nt x = 0 we hxvr C, - 10 rl/lJ, doZ, which gives the final solution for thr potcntinl 

flow nntl thr vnrinl,ion of bo~~ntlnry-lnyrr thirknrsu 

It, in srrn that, t.hc n,ngnil.~~tlr of I.hr prrmissil)lr dcrrlornt.ion (tlcrrmsc in vclorit,~) is very small, 

Irring ~)roporlion:il to .I: (1 1. Its vnl~~c is vrry nearly rcnlizctl for t,hc cnsc of constant vc1ocit.y 

nlr~na tho IInh p1:ilr at. zrro iwitlr~~rr. Jn the prcsrnt. cnnr the incrrnsc in hound:wy-lnyrr thirltncsn, 

0, is ~~roporlionnl to 3:I'.5" :his vnlnc also tli1h.m hnt lit,t,lc from lhe rase of n fln.1. plate nt zrro 

in~~i~lr~~~~t- 

I'cw whi~,Ii 

0 - 2+5. 

lly way of n further cxam~)le of retarded flow we shall ronnider the flow t.hrorrgh a 

divergent chnnncl whose walls nro straight. This ca.w in corollary to the cum of the houndn.ry 

layer in a divcrgmt chnnnol trentcd in Sec. IX b. The flow is nccn sketched in Fig. 10.16, where 

x tlcnotrs tIw rndial tlislnncc frorn t.11~ nourrc a1 0. The wall is nsann~ctl to Iqin nt, x .- n \vhcrc 

the entrnncc vcloril,y of the potrntinl strrarn is put cqnnl to U,. The poknlinl flow in givcn by 

Computing thc vnhc of the qnantity a from cqn. (10.41), which is decisive for separation, we 

obtnin here o = 2. Applying the criterion given in eqn. (10.4111) we cor~clude thnt, scpnration 

occurs in :ill cnscn irrrnpcctivc of the megnit~~de of t.ho nnglo of divrrgence. This oxnmplc RIIOWR 

very clrnrly thnt c lnminnr nt.rm~n has only n vcry limitctl cnpncity for nnppwting nn ntfvrrsn 

prcrrsnrc grntlirnt without ncpration. 

Acrording t,o a c:alculation pcrforrnctl hy K. Pohlhnuscn [In] tho point of ~cpnrnt.ion occurs 

nt xr/rl = 1.21 nntl is sccn to be indcpentlcnt of tho anglc of divcrgence. 

Fig. 10.16. J,nniinnr honnclnry layer in n tlivrrgent 

chnnnc4. SrpnmLion occnrs at r,/n = 1.21 intlcpen- 

dcnt,ly of the nnglc of tlivcrgence 

-x : 

p-- ,-, -I 

rc----,ys ): 

The prcrrding concl~~sions npply only n.s long as t,he displaccnwnt clTect of the Iioimdary 

lu,yrr n~ny be nrglcrlccl. Ilo\vevcr, this is not the cnsc uhcn the angle of divcrgcnco in small. 

Whcn thin nnglc is small, the boundary laycrs fill the whole channel cross-scction aflcr a certain 

inlet length (r/. Scc. XI i) and the flow gorn over nsytnptoticnlly to that discussed in Scc. V 12 

undrr the heatling of channel flow. When the included angle does not excced u certain valnc 

which drprnds o~? the Reynolds number, there is no separation. 

Ilcccntlg, S. N. l3rown nnd I(. Stewnrtnon [I] ~)~~l,linhrtl n nuninlnry rrvirw on nrpnrntior~ in 

whirh the rnathematical qucst.ion renbrcd on thr ning~~larity which occllrn in 1.hr tlifl'~:r.rlll.ial 

eqmt.ionn at tho critical point has been ernphnnized. Sccilso tho work of S. (:oldstein 141. 11 Inore 

physicnlly inspircd review of thin ]~rob~cln nrm h ; rccc~~lly ~ been puh)inhrd by J. C. \villi:lln. 

11 1 (291, n.nd by P. IC. Chang [2c]. 

45-72 (1969). 

121 I3ussn1ann, I

224 

S. appro xi mat,^ III~I.IIO~R for alcady equation4 

14111 C:~slr:r, hl.: 'l'llc sIr~rct.~~rn :ind I~el~a~iour of Inn~innr srl):lration I~ul)l~lrs. A(:,\l

220 XI. Axinlly symniobricnl nnd thren-di~~~el~sionnl boundnry lnyers 

dist>ance from the wall are in equilibrium under the influence of the centrifugal 

force which is balanced by a radial pressure gradient. The peripheral vclocity of 

the particles near the wall is reduced, thus decreasing materially the centrifugal 

force, whereas the radial pressure grndicnt directed towards the axis remains the 

samc. This set of circumstances causes the pnrkiclcs ncar the wall to flow radially 

inwards, and for rcasons of cont,inuity that motion must he oompcnsntcd 11y an 

axial flow upwards, as shown in Fig. 11 . I. A supcrimposctl firld of flow of this nature 

which occurs in the boundary layer and whose direction deviates from that in 

the external flow is quite generally referred to as a secondary flow. It was first discovered 

by E. Grusc11wit.z 1451 whrn he nnnlyzcd the flow in a curved cl~nnnel, scc 

also E. Becker [R]. 

Fig. 11.1. Itotnt.ion of llow near 

the ground 

Velocity component^: u - radial; 

a - tan~ential: 10 - axial. Owlng to 

rrlctlon. the tangentid velocity sulTera 

deceloratlon in tho ncfghbourhood of the 

dlak at rest. Thls glves rise to a aceon- 

dary Pow which la directed radlslly In- 

wards 

The secondary flow which accompanies rotation near a solid wall and which 

hna bcen described it1 the preceding paragraph can bo elearly observed in a teacup: 

after the rotation has bcen generated by vigorous stirring and again after the flow 

has been left to itself for a short while, the radial inward flow field near the bottom 

will he formed. Its existence can be inferred from the fact that tea leaves settle in a little 

heap near the centre at the bottom. 

In order to formulat,e the mathematical problem, we shall assume cylindrical 

polar coordinates r, +, z, the stationary wall being at z = 0, see Fig. 11.1. The fluid 

at a large distance from the wall will be assumed to rotate like a rigid body, with a 

constnnt angular vclocity w. We shall denote the velocity componenh in the radial 

direction by u, that in the tangential direction by v, the axial component being 

derloted by W. For reasons of axial symmetry the derivatives with respect to 4 may be 

dropped from thc Navicr-Stokcs cquntions. The solution which we are about to find 

will be an exact solution of thc Navier-Stokes equations, just as wae that for the 

a. Exnct solr~tion~ for nxinlly syrnmntrical borlndnry lnyers 227 

rotating disk, bccause the terms which are neglectcd in thc bo~~ndary-laycr cquntions 

vanish here on their own accord. 13y eqn. (3.86) we can write down the Navier-Stokcs 

equations as 

The boundary cor~ditions arc 

It is convcnicnt to introtlucc thc dimensionless coordinate 

in place of z, as in tile case of the rotating disk (Sec. V 11). Wc assume that the 

vclocity components havc t,he form 

Thc radial prcswre grnclicnt can bc con~pntctl for t.lia frictionless llow al a large 

distancc from tho wall from tho conditiorl: (I/@) . (aplar) = V2/r, or, with V = r CO, 

In the framework of the boundary-layer theory it is assumed that the same pressure 

gradient acta in the viseoua layor ncar the wall. lntroclucing cqns. (1 1.4) and (1 1.5) 

into eqns. (ll.la, b, d), we obtain a system of ordinary dilrcrcntial equations which 

is analogous to that in Sec. V 11 : 

with the boundary conditions

228 XI. Axially symrnrlrir:d rind Il~rrc.dimrn~ionaI bountlary lnyer~ 

l'nhlc 11 1. 'l'hr funct.inns for thr velocity distribution for the cxsc of rotation nvpr x 

slxtiot~xry w:t11, nftcr J . E. Nytlahl [81a] 

> 

a. Exact solutions for axially symmetrical houndary layorn 

Fig. 11.3. Rotation near a uolid mall, 

after Boedewadt. Vector representation 

oC the horizontal velocity component 

4 

Fig. 11.2. Rotatior1 near a solid walh 

aftrr Soedewadt [Dl. Velocity rlistribn- 

tion in thc bounctnry layer from 

eqn. (I 1.4); RCO a180 Table 1 I .I 

difference, i. e. that between the ground and that at ( --= 4.63, is 58". It is further 

remarkable that tllc axial velocity component w does not depend on the dist,nnce r 

from t.llo axis bltt only on the distance from the grountl. Tllc motion at all l>oir~t,s is 

llpwartls with la > 0. As alrcatly mcntiorlctl, this is ca~lsctl by the inwnrtl Ilow near 

the ground, consequent upon tho tlecrcase in the rcntrifrigal forces. In any anse, as 

seen from Pig. 1 1.2, this is compensated by n mdial flow o~ltward~ at, n groat,or I~eight~, 

1)1it or1 t,lic wl~ole, i.11~ rntliiil flow inwr~rtls 1)rt:(lo1ilir1t~t,cs. 'I'IIc 1.ot.11I VOIIIIII(> IIowi~~g 

towartls (.he a.xis taken over a cylinder of radius R around the z-axis is 

Tnscrtiug thc numerical value of II(m) from Table 11.1 we obtain 

Q = - 1.387 ~ r R2 , in) 11 . 

(1 1.8) 

Tile volume of flow in t,he positive z-direction is of ~-. equal mngnitudc. The 1:lrgcst 

II~)w:I~(~ motlion oc(wrs at [ == 3.1, wl~~rc III = 1.85 $0) 1) . 1 t. is nlso wort,l~ not,it~g tht,

230 

XI. Axinlly symmctricnl and three-dinicnsionnl bonndnry layers 

the boundary layrr cxtends considorably higher than in the example with the disk 

rotating in a fluid at rest (Scc. Vb). If thc boi~&r?/-hyer thickness 8 is defined as the 

height for which thr drviation of thc pcriphrral vclocity is cq~~al to 2 prr ccnt , wc 

for the stationary fluid. 

shall obtain 8 = 8 1/ v/w as against 8 = 4 d q 

The cxa~njtlc of t,11c mot.ion of a vor1.c~ sourcc hl.wccr~ t,wo pzrallcl walls consiclcretl 

by U. Vogclpohl [I201 is rel:it,ctl 10 somo cxtcnt to thc prcscnt cam. For 

very small ltcynolds numbers thc velocity distribution deviates little from the 

parabolic curve of Poisc~~illc flow. For large Rcynoltls numhcm thc velocity profile 

approachcs a rectangular tlistribntion, and a boundary laycr is sccn to bc forming. 

The corrcsponding case of turl)ulcnt flow was discussed by C. I'fleidercr [%I. In this 

conncxion t,hc papcr hy R. Bccker [F1 may also be consulted. 

Similnr phnno~ncnn cnn bc found in swirling flow through n conical funncl-like channel 

investigated by I

232 

XI. Axially synimet.rica1 and thrcc-dimensional boundary laycrs 

Jnscrting these values int,o cqn. (11.10a), we obtn.in the following equation for the 

strcam function 

FF' F.' FF" - 3- (F,, - :) 

tlz tl tl dtl 

wIric41 can be int,egratetl once to give 

FF' =F1- qF". (11.13) 

Thc I)onn(Iary condit,jons arc IL = IL,,, and v - 0 for y = 0. It follows that F' = 0 and 

F = 0 for "1 = 0. Sincc TL is an even funct,ion of r], F'lr] must be even, F' odd and E' 

evcn. I3ccausc of F(O) =. 0 the constant tcrrn in the expansion of Fin powers of 17 must 

vanish, which tletmmincs ono constant of intcgrat,ion. The sccond constant of inte- 

grat.ion,whicl~ will be dcnot,erl by y, can bc evaluated as follows: If F(r]) is a solution 

of eqn. (11.13), t11cn F(y v) = F(() is also a solution. A particular solnt,ion of the 

tlill'crcnt,inl cq~mt.ion 

dF dF dZF 

F . - = - - - 

dt dt d ~ ' 

which sat,isfics tl~c hountlnry condiLion ( - 0: F -1 0, F' -= 0, is givcn by 

F 5' 

1+:t2. 

(11.14) 

Ilcncc wc obtain from aqn. (11.12) 

JJere [ = y y/x, and the constant of inlegmtion y can now be determined from the 

givm value of momcntum. 

From cqn. (1 1.9) we obtain for t11c momenttlm of tl~c jet 

l'inally, t.11~; at)i,vc rcst~lts can bc cxprcssctl in n. form t.o chtain only the 1ti11cmnt.ic 

viscosit.y, v, ant1 the kin.ematic momentu.m, Ii' == Jlq. 'l'l~ns 

n. Exact .sohttionn for nxinlly ~ylntnrt,rical botnldary layers 233 

Figurc 11.5 rcprescnts a streamline pattcrn calculated from thc prccccling equations. 

The longitntlinal velocity IL is shown plottctl togcthcr with t.l~~t ror,t,llc? two-tlirncnsionr11 

jcl, in Fig. 9.13. 

m 

r 7 

1110 volumc of flow Q = 272 / u y tly (volnmc per sccontl), which incrcasrs 

n 

with the tlist~ancc from the orificc owing to t,ltc flow from the surro~~ntlings, is rcprcscntetl 

by the simplc cqunlion 

Q=8nvx. (11.18) 

Fig. 11.5. Strranlline pattern for a circulnr 

laminar jct 

This equat,ion should be comparccl with eqn. (0.48) for thc two-dimensional jct,. It is 

secn t.hn.t,, nncxpcctrtlly, the volume of flow at a givcn tlist.ancc from tllc orificc is 

intlcpcn(lcnt of thc morncnturn of the jot,, i. e., inclepcntlcnl, oT t,l~c csccss of prcssurc 

undcr wl~icll the jet leaves tlrc orificc. A jet wl~icl\ lcnvcs under a large prcssure 

tliffcrcnce (large velocit,y) rcrnrtins narrower t,han onc leaving wit,lt a smnllcr prcssurc 

difference (small veloci6y). The latter carries with it comparatively more st,ationary 

fluid, namely in a manner to make the volume of flow at a givcn distancc from the 

orifi cc equal to that, in a faster jet, provided tlrat t,hc kincn~:~,tic viscosit,y is t,llc same 

in lmt.11 cnsrs. 

The corresponding cnsc of a cornprcssiblc circular 1:mirmr jet was cvalu:~t.cd 1)y 

M. Z. JCrzywoblocki 1611 arid U. C. Pack 1831. In the subsonic rcgimc, tllc tlcnsit,y 

on the axis of thc jet is larger, and the tempcraturc is smnllcr tltan on its 1)ountlary. 

These differences arc inversely proporti~nal to the square of the distance from tl~e

23.1- XI. Axially uynimet.rira1 nncl three-dimrnuionnl honnrlnrg laycrs n. Exnct solution4 for nxially symmctricnl boundary layers 236 

orifice. According to 11. Goertler [4317 the case when a wcak swirl is superimposed 

on the jet can also bc trcated mathematically, and the effrct of the swirling motion 

present in tho orifirc can bo tmcrd in the downstream direction. Jt turns out that t h 

swirl decreases fastcr wit,l~ the dist,ancc from the orifice th:m the jet velocity on the 

axis. 

3. The axinlly symmrtric wake. 'rhc flow in an axially syn~mctric wakc, s~ch 

as occurs downstream of an axially symmetric body ~laccd in a stream parallcl to 

its axis, can also be tlcscribctl with tho air1 of the system of equations (ll.IQa, b). 

Tho solution is quite analogons to that for the two-tlimcnsionnl case whicl~ was tlcscribed 

in dctail in Sec. IXf. Let U, denote the oncoming vclocity and let ~ (r, y) 

be the flow vclocity in the woke. We assume, as was done in eqn. (9.20), that, the 

vclocity differcncc in the wake, 

U,(X,Y) = urn - ~ (x,Y) 

(11.19) 

is very small compared with U, far downstream. Consequently, we shall neglect 

quadratic terms in u,. With this simplification it is possible to deduce from eqns. 

(11.10a) and (11.19) the following differcntinl equation for 11,: 

Thc analytic form to be assumed for the dependence of the velocity differcncc 

ul(x, y) on thc axial coordinato, x, and on thc radial coordinat.e, y, can be discovered 

from the condition that the drag evaluated from the momentum of the wake must 

become independent of z at large distances downstream of the body. This leads to 

the relation 

03 

D = 2 n~ U, / u, . y dy = const, (11.21) 

which is satisfied by the form 

where 

1 (rl) 

U, = ClJ, -- 

Z ' 

This form is analogous to that in cqn. (9.31) for the two-dimensional problem. 

Substituting eqns. (1 1.22) and (11.23) into eqn. (1 1.20), we obtain a differential 

equation for /(q). This is 

(?I/')' -1- 2 q2 i' -1- 4 q / = 0 , (11.24) 

and tho boirndary conditions arc 

I Jfi 

4 i, 

It is easy to vrrify that thc solnt,ion of eqn. (11.24) has the form of an exponential, 

/(7) = exp (- $1 9 (1 1 .25) 

this form, too, being analogons to that in eqn. (9.34) for tllc two-dimrnsional case. 

IIcnce, thc velocity difference turns out to be 

The value of thc ronstnnt C must be tletermincd 

rqn. (11.21); its value is 

from thc drag with the aid of 

wl~rro c, tlcnotcs the drag cocficicnL rcforrrtl 1.0 tho frontal arra of t.11~ botly, ant1 

R = 11, d/v . IIcncc we obtain 

The plot of the velocity difference from eqn. (11.26) is the same as that in Fig 

9.10. Experimcnt.al data can be found in F. R. 1Inmn's work [4An]. 

4. Boundary layer on a body of revolution. The flow of a viscous fluid past 

a body of revolut,ion when the stream is parallel to its axis is of grrat practical importance. 

The bounrlary-layer equations have I~ecn adaptcd to this case by E. Boltzc 

[lo]. Assuming a curvilinear system of coordinates (Fig. 11.0), we dcnotc by z the 

current length measured along a meridian from the stagnatron point, y denoting the 

roorrlinate at right angles to the surface. The contour of the body of revolution will 

bc specified by the radii r (x) of the sections takcn at right angles to tl~c axis. We 

assume that there are no sharp corners so that d2r/dx2 does not assume extrcmcly 

large values. The velocity components parallel and normal t,o the wall will be denoted 

by u and v, respectively, and the potent.ial flow will be given by U(x). According 

to Uolt,ze t,he boundary-layer equations will I hen assume the form : 

with thc boundary conditions : 

Fig. 11.6. Ilounrlnry lnyrr nrnr a body 

of revolntio~~. Syatcrn of coordinates 

y=0: u=v=O; y=w: u=U(a,t).

236 

XI. Axially symmetrical and three-dimensional boundary layera 

The cq~~ation of motion in the x-direction is seen to remain unchanged compared 

with two-tlimcnsional flow. An order-of-magnitude estimate of terms in the equation 

of motion in the y-direction shows that the pressure gradient normal to tho well 

ap& - u2/r - 1. Con~equent~ly the pressure difference across the bounda.ry layer 

is of t01c ortlcr of the boundary layer tJ~ickncss S, and it is again possible to assume 

t,llnt, t,l~r ~~rcss~ire gr:dicnt of the potential stream, ap/ax, is imprcsscd on thc bounciary 

hycr. 

Wc shall limit the consitlemtions of this chaptcr to the case of steady flow. 

111 order to intrgratc cqns. (11.27a, b) for the axially symmctricd casc it is oncc 

rnorc possible to introduce a stream function y(x,y) given by: 

This tmnsforms eqn. (1 I.27a) into 

with the boundary conditions 

Wc now procced to give a brief account of the methods used to calculate the 

bountlary layer on a body of rcvolut,ion. A tnore det,ailecl account can be found in an 

earlier ctlition of this book [loll. The numerical results for a sphere, however, will 

be discossctl in more complete detail. The t~orrntlary layer on a bod!/ of revolulion of 

nrbitrrtr?/ .. ~hape - can be determined by the same method as that. used in See. [X c for 

the caso of a cylinder of arbitrary cross-section (two-dimensional problem). The 

velocity of the potential flow, U(z), is expanded into a power series in z and the 

st,reatn-fut~ction,~~, is assumed to be represented by a similar series in N, with coefficienta 

depending on the wnll distance (Blasius series). Following N. Froessling [29] 

it, is found that here also the coefficient-functions of y can be so arranged as to become 

independent of t,he parameters of any particular problem. In this manner the functions 

can bc calculated once and applied universnlly. 

- 

t The equation of continuity can also ba snlisfied by an'alternative stream function @, such that 

'Tliix form of bhc st.rcnrn function was 11scd by E. Boltzo when he calculated non-steady 

axinlly symnictricnl bonndary layem, as tlnscribcxl in Scc. XVb2. 

a. Exact solutions for axially symmotrical boundary layers 

The body contour is given by the series 

the potential flow being defined by the series 

The diatenco from thc wnll is rcprcucntetl hy 1.110 tlirncnsionlra~ coorclinnlo 

and in analogy with eqn. (11.32), the stream-function is represented by the Blasius 

series 

Substituting eqns. (11.31), (11.32) and (11.35) with (11.36) into eqn. (11.30) and 

comparing terms, we obtain a set of differential equations for the f~rnct~ions /3, . . . . 

The first equation is 

where differentiation with respect to 77 is denoted by primea. The boundary conditions 

are : 

The first equation of the set is non-linear and identical with that for three- 

dimensional stagnation flow which was considered in See. VlOt. A plot of /; is 

shown in Fig. 5.10, where /; = #'. The equations for the terms in 13 and z5 havc been 

solved by N. Froessling [29]. The succeeding ten functions of the term 27 havc been 

evaluated by F. W. Scholkemeyer [102]. 

Example : Sphere. 

In a manner analogous to that employcd for a circular cylinder in Scc. IXo, wc 

can use the preceding scheme to solvc the casc of the sphcrc. Thc cnrrcnt, rntlius for 

a sphere of radius R is given by 

r (N) = I1 sin x/R , (1 1.37) 

and the velocity distribution at the surface of the sphere we have 

3 3 

U(x) = - Urn sin x/R = - U, sin $, 

2 2 

(1 1.38) 

where $ denotes the central angle measurccl from the sta.gnation point,. Comparing

the two wries expnnsicws for sin (%In) in eqns. (11.37) antl (1 1.38), we det,ermille t,he 

cocfficicnt~s of cqn. (1 1.3) as follows 

Tlie resulting vclocit,y dist.rit)rtt,ions for various val~rcs of the n.nglc $ nre seen 

in IGg. 11.7; for t,l~eso graphs t,hc vclonity ?L has been con~prlted up to t,l~e term 27. 

Tho vclocitfy profiles for > 90° exhil)it a point of inflexion bcrausc they are associated 

with the rnnge of prcssnrc incrcnsc:. 

In connexion wit,h t,hc prot~lcrn at hand, we can repeat our previous rernarlrs 

concerning the gcnrral pract.icahilitay of applying n Blasit~s series. 'J'l~c cnlc~rlnt.iol~ of 

the fundnment,al cocffiaicnt.~ beyond t,lrc t,t:t.rn r7 involves an unaccept>able arnogrnt 

of con~put,ntion, md furt,hertnorc, the calculation of slender bodics rcqr~ires consid~rd)ly 

more t,c:rms. All t.11is pr~t.s n very severe 1irnit.ntinn on this method. For frrrther 

resnlts concerning spl~cres, r~fercnce sl10111d b0 rna.tle t,o the succcctling section. 

Trnusverse curvnlure. We hnvc! statrd rcprnlcdly t.llxt the rqunt,ion of nlotion (11.27n) 

of an nsinlly syn~tnrt,ric: flow 11ns t.hc snrnc for111 as tliat for t.110 t,wo-di~~~r:~~~io~~:iI mse o~ily 011 

condition t,lr:d bl~c l)oun~lnr,y-ln.ytrr tl~irknms is cvcrywl~r!rc much stnnllcr t11nn 1 . l rntli~w ~ of tl~o 

conl.our of tlir I~ody (R< r). '('his contlilion is nol. ~ ~hi~fi~d in t.11~ case of n long but thin cylintlcr 

or, for thnt ~nnttm, in t.he c:m of nny long and slcnder hly of rcvolul~ion. 'I'l~c bor~nrlnry Inycr 

on surlt R I)otly grow (IOWII~L~P~III and iLq I,~I~~II~I~ss ~ C ~ J I Icornp&nl)lc I C ~ wit11 tl~c rrltli~~s cvc~lt.unlly. 

'l'ltis I~rings intn rvi~lw~co tho rsnc:nl.i:~ll.v I.l~rco-tli~~~c~lnio~~nl nntnro r)f ISllo I)o~~~ltl:lry Inyrr 

on n I)otly of rovolr~I.ion \rrl~iedl rrs111I8 from t h con~~~:rrnt,ivcly large cr~rvnb~rrc of (he surfnro 

ol tho body in tho transvrrsc direction. 

R,. A. Srhnn ~ntl R. Jk)ntl [95] t,rmt,rtl I.lir mso of n nlcntlcr rylinrlcr, of rnrli~m r, -- a = ronst, 

plnccd in a nrriforrn axial stIrc:t~n. . 1l1c . RRIIIO J)~OI)ICIII was st~~dird Oy 11. 11. Kelly [Go] ~ h o 

introtlncetl ccrtain nnn~cricnl rorrrrtions. M. D. G1:iurrt nnd M. J. LigI~Lhill [41] ol~tninrrl 

~0111Lionq hy tl~c npplication of J'ohlliausrn's approxl~nnta n~cthod (scc See. Xlh) antl of nn 

cwyt~ipt~ntic ueries axpnn~ion. Tho flow dong the generators of o cylinder of arbitrary cross-srction 

anr, worked out hy .J. C. Caoko [IR] wllo employed a Blnsius seriw nn well ns I'ol~lhnusen'n npproxitnnle 

procedure. 

The nlnrc gc~~crnl msc of n con~prmsil,lr, nsinlly syn~n~rI.ric I~o~tnclnry I;~yar on ;I I~ody 

of rcvolr~tio~~ whosc ronto~~r is a f~lnction or t.l~c lonp,ilr~rlit~nl c:oortlinnlc, a:, ill ~~wlir:~l:ar, 

tlm cnsos of n circulnr cylintlrr nntl n spllcrc, wcro sl.otliccl by It. I". I'robstcin nnrl I). 1Slliot. [RR]. 

I1 turnrtl ont t,l~nt thc trnnsvcrsc curvnturc has the S~IIIO cni\ct on RIICII Ilo~s wit11 n. prcssurc 

grnelicnl 11s n ~II~I~I~~IIII:I~~~~~, 

fiivonr:ibIo prrssttrt! gn~dirnt. As 11 r~wtlt,, 1110 sl~wring dr~w is 

inrrtvwrcl C I I sc:l~rrrnl,ion 

~ is tlclayctl. 

b. Apprnxirnnte solutin~~s for nxially crytnmc~ric bo~a~~tlnry lnycrs

240 

XI. Axially symmetrical and three-dimensional boundary lnyera 

The significance of r(x) may be inferred from Fig. 11.6. Retracing the steps of 

Sec. X b we obtain the following differential equation for the quantity Z = cJ,~/v: 

'l'he quantitirs Ii, fl(R), f2(K) have the same moaning as in tho two-dimensional 

case, eqns. (10.27), (10.31) nnd (10.32). Introducing F(K) as before, cqn. (10.34), 

we have 

1 dr U 

F(K)-2K--,I; K=ZUt. (1 1.40) 

dz U r dz U 

It is casy to see that the substitution 

g=r2Z 

transforms the prcrcding equation to the form 

This form is preferable to that in cqn. (11.40) because it does not contain the 

derivative drldx. 

The point of separation is again at A = - 12, i. e. at Ii = - 04567, but 

at the stagnation point the values of tfhe shape factors A and K are now different. 

If the body of revolution has a blunt nose, we have at x = 0, i. e. at the upstreanl 

stagnation point, 

With this value the terms in the bracket in cqn. (11.40) reduce to F(K) - 2 Ii. 

By following the same argnment as in tho two-dimensional case it is found that the 

initial valw of Ii at the stagnation point is determined by the condition F(Ii) - 2 R = 

-- 0 , or, explicitly 

A, = -t 4.716 ; R, = 0.05708 . 

Ilrnrr thr initial valurs of thc intcgml rurvc (11.40) at thc stagnation point l)cromc 

K 0.05708 

7 - -.!. -1 

'0 - ur,, u', I 

The initrial slope is zero for a body of revol~t~ion, because for reasons of symmetry 

we must have (I,,'' = 0 at t.hc st:~grrat,ion pojnt. 'l'hc mcthotl of tlircct integration 

tlrscribetl in Scc. X b can hc cxtendcd t,o the case of axially symnictrical bodies, 

as shown by N. ltott and 1,. F. Crabtree [931. Equation (10.37) for the momcntuln 

~t.hiclrness is now rcplacctl by 

Some nunleriral examples have been calrulatrd by F. W. Sc!loll~emcier [I021 

7--.- -8- .. 

in llis pr;scrlkcd - - -.-- -- 

t-o tlij T-E"-----'--

242 XI. Axially nymmct.riral and tl~rrr-tli~r~rt~siot~al 1)onn~l:~ry Inyrra b. Approximate aolutiona for axinlly apmmctric boundary lnyer~ 243 

Fig. 11.8. Velocity distribr~bion in thc inlct port.ion of n pipe for the lnn~inar ca.se; mennurements 

perfornicrl by Nikuradac and quotrd from Pranc1t.l-Tiet.jen vol. TI. Theory dl10 to 

Scliillrr (901 

ltns prarf irally tlrcayctl al, :I clistancc of 40 pipe radii whcn thc Rcy~rol& nuntber 

has a value of R = 10:' This is in good agreement with experimental results. 

3. Rour~clnry layrra on rotating bodies of revolution. The simplest cxarnplc of 

e bounclary laycr on a rotaling hotly is tIi:~t considerod in See. Vb 11, namely the 

problem of a disk rotating in a fluid at, rest,. The fluid prticlcs which rotate with the 

boundmy laycr arc thrown outwards owing to the existence of ccntrifugal forces 

('centrifuging') anti are rcplaccd by part,icles flowing towards the boundary layer 

in an axial direction. Tlic casc of a disk of mtlius I< rot.nting with an angular vclocity 

o in an axi:~l sl.rc:~m of velocity U, :~lTords a simplc cxtcnsion of the previous 

problem. In thc lat,t.cr case the flow is govcrnctl by two parameters: thc Rcynoltls 

number and the rot,af ion pammetcr, U,/Rw, which is given by the ratio of frecst.rcam 

to tip vclocity. An cxact solution to the problem under consiclcration was 

given 11y Mi* D. M. Ilannah [46]t and A. N. Tiffortl 1.1 131 for tho case of laminar 

flow; IT. Sal~licl~ting and R. Truckcnbrotlt [98] providcd an approximate solution. 

E. Truclrcnbrotlt 11 191 investigated the case of turbulent flow. .Figure 11.9 cont,ains 

a plot of the torqnc coefficient,, C, = ilf/g e (2 R" in terms of the Reynolds 

numbcr and rotation parameter, U,/ll(u, obtained from such calculations. Here M 

clrnotcs the t,orque on thc leading side of the dislz only. When the disk rotatcs we 

may stmill assumc Ifhat separation occurs at the edge of the disk. 'l'he 'stn.gnant,' 

fluid Itchintl the clislr part,ly rot,n.tcs will1 thc,clislz and contributrs lit,l,le to the 

torcpc. Any such contribution has been lcft nt of account in (7, in Fig. 11.9. 

It is seen that Llie torque increases rapidly wi P 11 U, at constant angular velocity. 

t Arl.~tnlly rrf. 1381 solvm n rrl:rlr~l pro1,lrrn in wltirh the! cxtcrnal ficltl in t.l~:ct, rluc to '& source 

at infinity. 

Pig. 11.9. 

Morncnt cocllicient on 

a rotating disk in axial 

flow, aftor Gchlichting 

and Truckenbrotlt 

[98, 1 191 

Cnr - Mlf Q w' R'; 

bf - torque on lcnrling sidc 

or dirk 

W4 2 4 6 WS 2 4 6 106 2 4 6 lo7 

Reynolds number R =g$ 

Thc flow in a circular box provided with a rotating lid shows a markcd rcscmblance 

to that between two rotating dislts mentioned in Scc. V b 11. 7'11~ cnse 

of the flow inside the box was investigated in deLail by 1). Grohne [44] who discovered 

two peculiar features in it: First, the flow in the friction-free core in tlie 

interior of the box can only be determined by taking into account the inllncnrc of 

tlie boundary layers which form on the wall, in contrast to normal cascs whcn onc 

naturally nssymes that t,hc influence of tho flow in a bountlary layer resu1t.s at, most 

in a d.isplaccment. Secondly, the boundary laycrs arc unusual in that they join car11 

other. Siniilarly, in the arrangement oonsis1.ing of R rota1,ing channrl irivc?stligat,etl 

by IT. 1,udwieg [68], it is possiblc to discern two regions of flow when the spcxd of 

robation is sufficic?nt.ly high, ttamcly a fricd.ionlcss corc and bottndnry layrrs which 

form on the side walls and which givc risc t.o a secondary flow. 'l'hc t.hcory lcads 

to a large increasc of thc drag cocfficicnt which is dnc to rotation, ant1 this fact has 

been confirmed by experiment. 

Blunt bodies, sncli as o. a. a sphere or a slcrdcr body of revolution, placctl in 

axial streams, show a marked influcncc of rotation on dmg, as cvitlrncwl I)y tho 

measurements performed by C. Wicsclsbergcr 11231, ant1 S. 1~1t.h:~ndcr and 

A. Rydberg [69]. Fig. 11.10 contains a plot of thc drag cocfficicnt of n rotating 

sphcre in terms of the Rcynolds numbcr. It is secn that Lhc critical Rcynoltls 

number, for which the drag coefficient dcrrcascs abrnpt,ly, depends strongly on tlie 

rot,at.ion paramcler U,/Rw, and the same is true of the position of tltc point, of snp:~ration. 

The effect, of rotary motion on bl~c posilhn of the linc of 1aniirtn.r sc.pnr:~l.io~~ on 

a spltcrr is (l(;s(~il~cd by lhc grc~pli in ltig. I I .I I ; 1,Itc (IILIJL ror it IIILVO INWI ~ X ) I I I ~ ~ I I I ~ I ~ I I 

by N. E. lloskin [50]. When the rotatmion para.mctcr 11:~s nl,tdinrtl t,hc vn111c: 

Q = w R/[J, = 5, the line of sepnm.t,ion will have moved by about lo0 in 1,hc upst,rean~ 

direction, as compared with a sphere at rcst. 'l.'hc physicnl ren.son for this 

bef~aviour is connected with the centrifugal forces &ding on t,hc fluit1 parLiclcs rolat,ing 

wit,]i the body in its bour;tlary layer. Thc crt~trifngal forces have tlic sn,mc rni~t n.s an 

atltlit.ionnl pressure gratlirnt dircctd towards t,hc plnne of I,ltc erlun.t,or.

244 XI. Axially symmrtrical nnd tl~rcc-dirnertsionnl boundary lnyers 

Fig. 11.10. Ihg coefficirntr, 

oi n rotnt.ing spl~crr in axinl 

flow in trww of tl~c ltcJrnolds 

number R and rotntion 

Imramrtrr .Q - ioR/lI, 

h t.l~corct.iwl cxpl;~nntiou of t,hc vrry cwnplcx thrrc-dimrnsionnl cll'ccb in the boundary 

Iayrrof rotating I~odics of rcvolut.ion in axid flow is contained in the papers by H. Schlichting [00], 

IC. 'I'rnckrnhrotll~ [I181 antl 0. I'arr 1841; thcse authors onployed the approximato method 

!:xplainod earlior. It is t.rne that the boundary layer of a rotating body of revolution 

In axial flow still rctains it^ axial syn~mctry, hnt owing to the rotation there appears a peripheral 

vc1ocit.y cotnponent in addition to that in the mcridional direction. For this reason, the calculation 

for such a I)o~lntlsr,y layor must int,roduce a ~norncntom eqnat.ion in the circumferential direction 

(11-direr:t,ion) in atlclit.ion Lo that in tho n~crirlional direction (x-direction). Assuming that the 

a~~gulnr vclocit,y of t.11~ I~ody is io, antl ilcnoting t,he coordinate at right angles to the wall by y, 

wr ran writ.(: 1.11~ 1.~0 erluat.ions of n~otn~nl~un~ in tho form 

r. 

I hr component,^ or the shearing stress at thc wall are then given by 

~ig. 11.11. Position of line of laminar 

separation on a spl~ero rotating in axinl 

stream, after N. IF. Hosltin [SO] 

c. Iblation hel:ween axially ~y~ntnetrical and t\\o-cli~~~c~~aio~~;rl I,onntl;rry I;ryr~s 245 

and tho displacement and momentum thicknesses arc defined as 

m m 

In the procctling equations, the local pcriphernl velocity w, - r u) hm been cl~onen ,w n rofcmnrx: 

veloc~ity for the a7.imutal con~poncnt, w,(x, 2). 'I'ho preceding equations ~nnke it possi1)lo Lo pcrforrn 

cnlculntions for Inminar as well as for turbulent flows, it being necessary to introduce difircnt 

expression8 for the shearing stress at the wall in the latter cme (see ref. [R4] and Sec. XXllc). 

In some of the cases, it proved possible to evaluate the drag coefficient in addiLion to t,he t,nrning 

tnonwnt,, the former decreasing as the parameter mR/Um is increased. In this connexion, the papers 

I)y C. It. Illir~gwortl~ [54] and S. T. Clru and A. N. TilTbrcl [13] may nlso hc stdied. The approxilnate 

procedure conceived by H. Schlichting [98] was extenrlcd to compressible flows by .I. Y;rnlnga 

[125]. The preceding investigntions have bcen extcnded for laminar as well ns for t.nrl)ulent. 

tlows by theoretical and experin~ental investigation^ described in ucveral papers by ,Japanese 

authors [29n, 10, 01, 79, 801. 

l'rohlcn~s connected with laminar flow nbout a uphere rotating in a flnid at, resL IIGVO IICCII 

discussed by I.. Ilowarth [51] and S. I). Nigam [All. An extension to the case involving ellipsoids 

of revolut.ion wns provided by B. S. Fadnis p6]. Near tho poles, the flow is the same as 

on a rotating disk and near the equator it is like the one on a rotatin cylinder. The aecornpanyi~~g 

secondnry st re an^ causes fluid particles to flow into tho boundary yaycr near the poles, nntl out. 

of it at the equator. The rate of this secondary flow increases with increasing slenderness, the 

cquabrial area and peed of rotation remaining constant. However, the phenomena in the 

plnne of tho equator where the two boundary layers impinge on each other and are thrown 

outwards can no longer be analyzed with the aid of boundary-lnyer theory, el. W. 11. If. Banks [5a]. 

Further theoretical and experimental investigations of t.his problem have been later under- 

taken by 0. Sawatzki [94] and by P. Dumsrgue et al. [21a]. Reference [94] describes n~edsure- 

rnenls d the torque exerted on a rotating sphere in the rango of Ibynolds number 2.105 < R < 

1.5 x 106 which goes far beyond the laminar regime. Tho invwtigntion of Ref. [21 a] included 

the vi~unlizntion of the spiral strenmlines near the wnll on n sphere nnd on cones of various in- 

cluded angles as they occur'in laminar flow. 

It has been observed that in axial turbomachines there may, under certain circumstances, 

appear an extended zone of dead fluid in tho whirl behind the row of stationary blades antl 

ncnr the hub. This phenomenon was described in great detnil by K. J3antmert and H. Klaeukens 

[5]. The origin of this dead-water area is conneckd wiLh the radial increase in prcssurr in Iho 

ontwnrtl direction which i~ due to the whirl. Owing to tho whirl the axinl pressure inerrme nrnr 

lhe huh in the bladelorn annulus behind the guides is much greater than at the outer wall. The 

influence of tho houndory layer is here only ciecontlnry. ALLonLion rn!ly, further, ho drawn Lo 

an invesbigotion duo Lo I

in cross-flow, tlrprntls only on lhn potat~l,i:d vclo~it~y tlistril)ution. IJ(z). By ront.r:~st,, 

whcn an axially symmctrical I)oundary laycr is stutlictl, for rxamplc that on a 

rotating 1)ody of rcvolntion, it is found that the contour r(a) of the body entcrs 

explicitly into thc corresponding rquations. Tile prcsont scction is clcvotctl lo x 

more tlotailctl invcst,ignt,io~~ inl,o thc rolntion 1)ctwc~cn two-tli~nr~~sit)t~:iI nlicl axi:illy 

symmntric l~our~rlary Inycrs. 

In st,~n.tly flow the Oountlary-layor rqr~:rt.ions for Lwo-tlitncnsiond flow :~ntl fnr 

axially symmot~rical flow are given I)y cqns. (7. lo), (7.1 I) ant1 ( I 1.278, h), rospectivrly. 

l'hc Int,tcr rcfcr to a curvilincnr systc~n of c:oortlin:~.l.c:s with z tlcnot,ing t,l~r cttrrrnl, 

arc: Icr~gt~h nntl y tlrnot,ing {,l~o tlist,nncc from t,hr wall in :L tlirrt-tion normal t,u if.. 

The rcspcctivc vc1ocit.y components n.rc tlcnotwl I)y IL nntl v, and IJw mn.gnit,ntlcs 

wit.11 a bar rcfrr t,o tho two-tlinicnsiond cnsr. Wit.11 these syml)ols, wo Itavo for 

tho two-tlinirnsional msc: 

for t.hc axially symn~ct~ricnl vnsr 

Ilrre ~(z) dcnot.cs tJto (list,ancc of a point, on t11c wa.ll from 1.11~ axis of symmntq. 

Thr first eqnnfions of l)ot,l~ systems nro iclrnt.icn1, the tliffrrrncr Ixing onl,y in t.llc 

npprnrnncy of t,hr rntlirls ~(n.) in t.11~ rqun.l,ion of conI,innit,g. 

It, sc~ms 1.1111s rcasnnnl)lc to inqnire wlteLllrr it, is possil)lo 1.0 intlic.at.c a transformal,ion 

wl~icll woultl permitf t.hc nsn of t,l~n solt~t.ions of Lltc two-tlirnrnsionnl cnsc 

1.0 tlrrivc solr~f,ions of t,l~c n.xinlly syn~rnrt,ricnl cnsr. Such n gvncml rc1at,ionsI1il) 

bctwocn t,wo-dimcnsiond nntl n.xially symmctrical I~ounrlary laycrs Itas bccn cliscovcrctl 

by ITT. RTn.nglrr [72]. It rr~lrlccs tho calcnlation of thc hminn.r 11ountla.ry Inyor 

for am n.xially s.ymmct,ric.:~l botly t,o tl~nf, on a cylintlricnl I)otly. 'l'he givcn body of 

rrvolut,ion is nssocin.18rtl wit.11 n.n itlcnl pot,cnt,inl vclocit.y dist,ril)ntion for n rylint1ric:ll 

body, the f~lncl~ior~ Lcing rnsily calcnlatfctl from the conhur ant1 the potcnt,inl vcloci1.y 

tlist.ril)tlt,int~ or t.ho botly of rovol~tkn. Mnnglrr's tfmnsl;mnation is also valid for 

comprcssil)lt: Imtlntlnry In.ycrs, n.s well ns for tllcrmnl boundnry 1:iycrs in In.tnil~:ir 

flow. Wr sh:1.11, I~owcvrr, consitlcr il, here only in rclat,ion to incomprrssi1)lo flow. 

According to Manglrr, l.hc cqrin.t.ions whic:l~ t.m.nsform tJle coortlin:~.t.es ant1 111~ 

velocit.ics of t.hc xxinlly symmct.ricnl pro1)lcm to t,hosc of t.he eqiiivalcnt two-tlimcns- 

ional problrm n.rc as follows: 

Z 

w11orr 1, (Icnotrs a const~arit Iw~gth. Itcn~cnibcring that 

it, is rasy to verify tht thc syst,rm of c.q~t:itions (1 1.60) l,mnsforms intm oqns. (1 1 .4!)) 

by t,hn wc: of LIto sul~sl~itulions (Il .GI). 

WIC hountla,ry layer on a 1)otly of revolution r(z) having tho itlml pol,rnt.ial 

vclocit.y tlisI,ril~clf.ion IJ(z) nnn l)c cv:dt~al.c:tl by con~pnting t h t,\vo-(litllr~~sio~~:~I 

I,ot~n(l:~,ry l:tycr for tt, vcloc:it,.y tli~l,rih~rI,ion o(:?), wltc:ro /J r-: ~ I NZ I ti~~l :I: tire rcl~iI3wl , 

I)y oqns. (1 I .GI). Il:~vi~tg c~alcrtli~t,rxl I.hc voloc.il.,y oornpot~ol~l.~ ii, nntl 6 for l.11~ l.wotlimctl~iottal 

I)ortntl:~ry Inycr it is possible tlo tlctcrrnine tho con~poncntn I* nntl IT or 

tho n.xinlly symmct.rical bountlnry laycr $1 ith tho nit1 of thc t,mnsforlnnt,ion rquations 

(11.51). 

Iloncc, from rqn. (11.51), we Itavo 

l'hc pol.rnt.ial flow of tllc associatd two-tlimct~sional flow bccornrs 

J - --- 

U(2) = u, 113 L2 2, 

so that 0 ( ~ = ) C 5' , wllcrc (: dcnotc~ a constant. 'rhc associat.~d two-tlitncnsio~la.l 

flow bclongs to thc class of w~tlgc flows disoussotl in Sce. 1Xa ant1 is givcn 

by I1 = C an', with m = + for the present example. l'rom cqn. (9.7) wc find the 

wc~lgc nnglp P = 2 m/(m -1-1) = 4. Thc associntctl two-tlimcnsiond flow is t.ltat 

past a wcdgc wit,h an anglc n P .= n/2. '1'11~ fact that nxinlly symmct.rical stagnn,l.ion 

flow ran be rcduced to the case of flow past, a wcdgc whosc angle is n/2 wa.s st,nt,etl 

in Scc. 1Xa and is now confirmed. 

11. Three-clin~ensio~~nl Lountlnry lnyers 

IJtttil now wc have restricted o~~rsrlvcs nln~ost cxc:l~lsivcly Lo the consitlcrat,ion 

of two-tlimrnsionnI :mi axially sylnmnt.rical prol~lcrns. 1'rol)lcms of t.wo-tlinlct~siot~al 

nncl of nsinlly syrntncI,ricnl flow havo this in common l.ltat t,ho prcscril)otl 1)oI,cnt.in.l 

flow tlrprntls ot~ly on onr sp~cr: coortlil~:il.o, :l.ntl tho l,wo vc:loc:it,y con~l)c~~ottls ill Ih(: 

I~o~tntlnry I:~ycr tlc:pcntl on t8wo space roordin:~tcs 

sional 1)orrntl:iry lnycr thc cxhcrnal potcnt.ial llow clcpcntls on two coortlin:~.l.cs in 

thc w:~ll srlrfaco and t.ltc llow willtin tllc Imuntlnry laycr posscsscs all tl~rcc vcloci1.y 

componrnts which, moreover, tlcpcnd on all three spxco coort1in;~tcs in thc gcncr:~l 

cmc. 'l'hc flow abont a disk rot,nl,ing in a fluid at rest (Scc. Vb) and rotntion in thc 

nc~ig11l)ourllootl of a fixed wall (Scc. Xla) const,it3utr cxarnl)lrs of t,l~rcn-tlimcnsionnl 

I)ol~ntl:~ry I:~yors, rrpnrt from Ijcing cxnol sol~~l~ions of tllc Nlbvior-Stokes cqttn.t,io~~s. 

(\:LCII. [II I,IIc cnsc ol' n t ,I~t~(~(:-tli~~~t:t~--

248 

XI. Axially symmetrical and three-dimensional boundary layers 

If the streamlines of the potential motion are straight lines which either converge or 

diverge then, essentially, the flow differs from a two-dimensional pattern only in that 

there is a change in the boundary-layer thickness. On the other hand, if the potential 

motion is curved the pressure gradient across the streamlines of the potential flow 

impressing itself upon the boundary layer gives rise to additional influences, such as 

secondary flow: out,sidc the boundary layer the transverse pressure gradient is 

baln.nced with t,he centrifugal force, but within it the centrifugal forces are clccrcasccl 

because of the decreased velocities and, consequer~tly, the pressure gradicnt causes 

mn.ss to flow inwards, i. e. towards the concave side of the potential streamlines. 

The rotation of air over a fixed wall affords an example of this belmviour antl illustrates 

the existence of a flow inwards. 

A further example of sccondary flow is affordcd by the mot.ion on the sidewall 

of the channel formed by t,urbino or compressor blades or by a deflector. The bound- 

ary laycr which forms on the wall dcvclops a sccondary flow from the pressure 

side of one blade to thc suction side of the next one owing to the curvature of the 

streamlines in the external flow ficld. The secondary flow caused by the sidewall 

is further affected by the boundary laycr on the blades themselves causing the flow 

pat,tcrn through a turbine or compressor stage to become vcry complex. This prcsent.~ 

a vcry difficult problem to 1)ourldary-layer theory bccausc the three-dimensional 

nature of Lhe llow is essential to it. For a long time problems of this kind hat1 been 

stutlicd by cxpcritncnt,al means only [471. 

1. The Boundary layer on a yawed cylinder. Another important case of a three- 

clirnensional boundary laycr is that of an aeroplane wing, whose leading edge 

is not pcrpcntlicular to thc stream, as in the case of swept-back wings and ynwccl 

wings. It is lrnown from cxpericncc that on the suction side considerable quantities 

of the fluid move t,owartls t,hc recctling end, the phenomenon having a very tlet,riment.al 

elfcct. on t.hc aerodynamic behaviour of the wing. 

111 two-tlirncnsionnl motion t,ltrough a 1)ountlary laycr, the geometrical shape 

of t,hc I)otly inlluenrcs the ficltl of flow only ir~lirect~ly, i. e. through the vclocil,y 

dist.l.il)ut.ion of t.he potcnhl flow which alone ent.crs the cnlculation. By cont.rast,, 

1,ltrro-tli1nensio11nl t~ountlary layers arc affcctctl by both: by the extcrnal vclorit,y 

tlist.ril~ution ant1 by t,hc gcornct,rirnl shapc directly. For example, in the case of 

a I~otly of rcvolulion t,lrc variat,ion of the ratlius with distance cxpressctl by tho 

funct.ion n(r) nl'pcars explicitly in tJtc dilTcrent,ial equations, see eqn. (11.27 11). 

For tJtc purpose of rst.al)lislling the I~ourltlary-hyer equations we shall confino 

o~~rsrlvc~ 1.n l.11~ simplrst, rase of a plane w:dl or t,o a curvccl wall which is tlrvrlol~:tl~ln 

into :I pln.nc (Pig. 11.12). T,ct 3: and z drnot,e t,hc coortlinat,cs in the wall surface, 

1, (Irnoting (:IS p~~cviousl~) 1,110 coor(1in:~t.t: which is pcrl>endicular to t,he wall. 'l'hr, 

vrloc~il,y vrc!t.or of pot,cnt.i:ll llow 1' will be assumetl lo hnve the cmnponcnts 11 (x,t) 

nntl II'(r.z), so 1.11:lt in thc st.catly-st,al.c msc t,l~c pr??surc di~tribut~ion in t.he potcnlinl 

If wr now prrfortn lhr snrno cst,iniat,ion, untlcr the assumpt.ion of vcry large Iteynolds 

n~~ml)c~rs, rvlalivc; t,o 1.11~ t.ltrcc-tlimrnsional Navicr-St,okr.s cqttnt,ior~s (:1.32), as 

?xphinctl it, tlol,:~il in Soc. VII a in rclat,ion to the two-tlimcrrsiona~ casc:, wc sirall reach 

the corlclusion t,lln(, in bllc frictional terms of the equat,ions for the z- and z-tlircctior~s, 

re~pcct~ively, it is possible to ncglect'thc tlcrivat,ivcs with respect to the coordinaLes 

which are parallel to the wall as against the derivative with rcsprct to t,he coortlinat.r 

at right nnglcs to it,. Itcgartling the equation in the y-tlirecliorr wc again obl,nin t.lrc 

result t,liat ap/i)?y is very small and may be neglected. Thus the lmxsure is secn to 

depend on x and z alone, and is impresscd on the borrnda.ry Iaycr hy the pot.ct~t,i:ll 

flow. 'll~o rst.irnr~.l.iot~ furIJ~rr sl~ows that,, gcx~crnlly spc~~king, nom: 01. t.11~ ( ~ ~ I I v ~ ivv Y . ~ 

terms may be ornitLetl. 'l'lle trllrec-tlilnensio~~aI liountlary-layer cqunt.iotls arc, t,ltrn, as 

follows: 

with the following boundary conditions: 

Fig. 11.12. Sy~tem of coordinates for 

n t,l~ree-clinirnsiond boundnry layer 

7'he pressure gradicnt,~ i)p/ax and aplilz arc known from the potmtial flow in accordance 

with eqn. (1 1.52). 'l'llis is a system of t-hrec equations for qi,, v, and lo. For 1Y 0 

and lo - 0 t,he system transforms int,o the familiar systcm of equations (7.10) a~ltl 

(7.1 1) for two-tlimcnsionnl boundary-layer flow. 

Up to the prcscnt time no exact ~olut~ions of this gcncml systcm of cqnatiotts for 

t,hrec-tlimcnsiond floiv have brcn found, apart from tho cxamplcs wl~icl~ \vc 11:lvc 

mcnt,ionctl prcviorrsly. 7'11. Gcis [33, 341 invcstigatcd tho spcrid class of flows whirl^ 

lead to similar solutions. In analogy with wedge flows, the velocity profiles arc now 

similar in the direction of each of the two axes of coonlin:~tes,\and this :~llows us 

to transform the systcm (11.53) into a set of ordinary diKcrenti/l equations. 

A prticular case of three-dimensional boundary-layer flow wjlicll is consitlrr:~l,ly 

more amenable to numerical calculation is that where the potcnthl flow depends on n: 

but not on z, i. e. when 

U=U(x); W=W(x). (1 1.55) 

These conditions apply in bile case of a yawed cylinder and npproxirnatcly, in 

trhc case of a yawctl wing at zero lift,. 'L'hc systmn of ccluntrions (1 1.5:1a, I), c) is simpli-

250 XI. Axinlly nyrnmet,rirnl nntl t,liree-dirnenaiorid honnrlnr.~ lnynrrr 

fied in that, t.hrre is no clepc~ntlcncc on z. With W = IV, = const and taking into 

account thnt - (I/@). (ap/ax) = U . (dU/dx), we obtain 

with tho same boundary condit.ions as 1)efore. In this partic~~lar case the system is 

rcduciblc in t,he sense that it is possible to cnlculate IL and v from tho firsL and last 

cquntion, the solution bring iclcr~t~ical wiLh that for a two-clirnensional case, and subscq~~enl,ly, 

to cotnpl.ctc lhc c:~.lcitlnl.ion of 111 from the secontl equation, which is, 

moreover, linear in lo. This rendors such cases really simple. Tncider~tally, it, might be 

not,ctl that the equation for the velocity component u) is identical with t,hat for 

the tmipcratnre distribution in a two-c\itncr~sional boundary Inyer when the I'mntltsl 

nmnber is cqc~n.l to unitmy (soe Chap. XTJ). 

Specializing tho syst,crn (1 1.66) still further for the case when TJ(z) = IJ, = 

ronst, we obtain the example ofthc flat plaLc in yaw hut at zero incitlcnce. 111 Lhis case 

tho pressure term in the first, cqnnt,ion vanishes, and t,hc secorltl oquat.ion becomes 

itlcnt,ical wit,li the first when lo is rcplncctl hy u. Thus t,he solutions ~ (x, y) and ~ ( z y) , 

brcome proport,ional, w(x, y) = const. . 71 (x, y), or 

Tltis means that in t,hc cnsc of n yawrtl flat plate t.hc ~ ~ R I I I ~ of I I L Lhc vcIocit,y 

in t,hc bo~~nclnry layer whic:lr is parallel to t.hc wall is also pnr:~llcl to thc poLential flow 

at :ill ~minta. 'l'l~o fact 1,liaI. trhe plate is ynwcd is seen to have no influor~no or1 Lhc 

f'ormat.ion of thc bountlrwy 1a.yer (intlcpcnrlcnce principle). 

Whrn t.\ic llow in the honntlnry Iiryor on a yawed flat plate I)ccomcs t~~rhvlott, 

the right,-hn~~tl sitlcs of t.ho first two cqu:~tions (1 1.56) must be sripplcmcntcd wit.11 

t*l~c~ I,crlns tluc t.o t,l~rl)ulcnt 1Lcynoltls ~t~rcssrs (C11;lp. XIX). 'I'llrn, t,hc two cqunt,ions 

can no longer be trnnsforrnctl into rn.rh ot.l~cr by the substit,ution of IL for w nntl vice 

vrrsn. Conscql~cntly, t,l~c st.rr:~ntlil~cs in 1hc boundnry Iiyer ccnsc to bo p:~r:~lIcI 

t,o 1,llc llow tlirct:l,ioti it: (,Ilc I'rcc sLrc-:~~n, :ln C:LII bt: vwifictl Og tlircclj c:xpt:t~irncnl, [Dl. 

111 :ttltlil.io~~, ref. [3J Jys rstnhlisllctl Ll1:~t. Iho tlisplnccrncnt thidtrlcss of' n Inrblllcnt, 

I)ountlnry lnger on n y+wc-(1 plnfc grows somcwhnt fn.st,cr in the tlowrlsl,rrn.m tlircct,io~l 

t.11n11 is I,hc c:~sc wit,I~ ~III IIIIJ~:IWC~~ plalc. This ag:tin tl(~mo~istrat.csc thc i~~:~~~~~licaI~ilil~y 

ol' IIIV intlrpantlcncc l)rincipla t,o t.nrl)~tlrl~l, I~o~~nthry Inyrrs. 

'I'l~c c:~I~~~l:it.ion of Iho I,I~rc~r-tlitnt~~tsio~l:~.l botintl;~.ry lnycr on :I, yawed cylintlcr, 

rqns. (11.5(i), ran I~crnrric~l olit, I)y n. ~nct,hotl simil:~r t.o thnt wrtl in t,ho cnsc of 

two-~tlimrr~sionnl llowlnl~out a cylit~tlw \vl~osc a.xis is n.t right nnplcs t,o 1.11~ sl,rmtn 

(Sco. TXe), i. c. I)y asbnming n scrios expansion with respcct to the lengt,h of arc, X, 

tncnsl~rcd from tlic .st.ngnn.t.ion point,. For n syrnnictricn.l cylintlcr we may put 

cl. Thrcc-dimensionnl boundary Inyers 25 1 

Tt is frtrt,hcr asslrmetl that the vclocity componct~t~s ~(z, y) and ~(z, y) of 1.his flow 

(in which the stagnation points lie on a dcfinito lim) may also be oxprossod wiLh tho 

nit1 of n scrics ill z with cocfficionta dcpcntling on y (13lasius ~crios), tho flow pntt,crn 

bcing i~ltlrpcv~tlont, of tho coordinntc z rncasr~rcd along tho gcncralrix of tho cylinc1t:r. 

'ilhns, putting 

(1 1.57) 

'J'hr fnnct.ions IL. 13, . . . satisfy the diRcrrnt,ial cqunt,ions (9.18). 'I'hc rornpnt,n.t.ion of 

the con~po~~c~it I& vras first given by Mr. R. Scars [IOB]. It was Inter consitlcrenbly ext.cndctl 

I)y 11. Coer(.lrr [42]. The funrt,ions go, gz, . . . snt.isf.y the differential cclunt,ions 

wl~osc botintlary corditions are 

As intliralrtl by L. Prandtl [861 the equa(ion for go can bc solvccl by dircct inte- 

gmtion, the result bring 

J!'{ exp (- f j,dv)j dv 

go ( v) = OZ - - -- O - - (I I 60) 

j - j'~,d~)) dl 

0 0 

Fig. I I. 13. 1,nniinnr honndary Inyer on n ynwcd 

rglintlrr. The functions ge nnd gz for the vclority 

ron~porirnt. 111 nlorig t 11e axis of the rylindcr, cqn. 

(11.58~). At the ntngnntion lilw wc have w/ll', 

= go (11).

252 XI. .\xi:rlly ~yrnrnctriral and t,llrce-ditncnsiot~al bonntlnry layer3 

Approxi~~lntr ~rtrtl~ntl. 1,. l'r:~.t~rlt.l 1721 laic1 tlowtr a. progr:~~r~tnc: for ot)t,airlillg 

so111lio11s wilh I,IIc nifl of IIIV tnonrct~tum I~I~wr(~rn, i. r. ill a wajr which 

:I.II~II,~X~III:I~(* 

is siniihr 1.0 I hat. r~scvl ill Sro. X'1 11. In l)art,icular, t-hc set of rqunt,ions (1 1 .45) to (1 1.48) 

tr:~nsli)rtns illto lh;~,, Iiw :L y:~wcttl cylintlcr when it. is assurnccl formally that - const 

:111tl when t,llc a.zitnr~ll~:ll IIIOI~~II~~IIIII thiclctlcss rj~?.~ is rrprrscntcd 11.v t)l~c formula. 

=7 

A siti1il:tr :tpl)roxil~lat~(~ rncthod was usrtl by J. M. Wild 11241 for the solution of 

thc prol)l(trn of the ynwocl cylirdcr. Figure 11 .I4 reprcscnt.~ the pn.t.tern of st.rcamlines 

for :I y:~wctl rllij)t.ic cylirdrr of slrndcrnrss ratio 6 : 1, placed at, an angle of incitlcnce 

to the sI.mnm. 'i'hc lift corfficirnt has a valuc of 0.47. The arrows shown ill the sketch, 

intlic:at.c~ the ~lirrct~io~i of flow of thn vclocity conlponent pan.llcl to the wall in its 

immctlint~c ~lcighl~or~rllootl, i. e. thc value 

A 

Il'ig. 11.14. Ihnd:~~y-l:~yrr flow abor~L a 

y:rwrd rlliptical ryli~drr with hfL, altar 

.I. M. Wild ( 1241 

Vig. ,I 1.15. JSxplnnation of origin of crous- 

flow, on a yawcd wing at an angle of inci- 

dcnce. Curves of constant pressure (isobars) 

on the ~nction side of the wing. Near the 

leading edge on the uppcr snrface of the wing 

there is a harp pressure gradient at right 

angles to the main stream and towards the 

receding end causing cross-flow 

d. Three-dimensional boundary laycrs 253 

The respective streamline is shown as a broken line, and the potential streamline 

is seen plotted for comparison It is noticeable that thc flow dircction in the boundary 

layer is turned by a large angle towards the rrrecling end of tho cylinder. This rirrum- 

stanre is very important when flow patterns on yawed wings are obscrvcd with the 

aid of tufts 

Swept wings. The cxistencc of cross-flow which occurs in the boundary laycr of 

a yawed cylinder is important for the aerodynamic properties of swept wings. When 

yawed or swept-back wings operate at higher lift values the pressure on the suctiot~ 

side near theleading edge shows a considerable gradient towards the receding tip, 

the effect being due to the rearward shift of the acrofoil sections of the wing. This 

phenomenon can be inferrcd from Fig. 11.15 which shows the isobars on the suctibn side 

of a yawcd wing. The fluid particles which become dc~clcrat~cd in the boundary layer 

have a tendency to travel in the direction of this gradient, and s cross-flow in khe 

dircction of the rccctling tip results. As dc~nonsLr~~.tod by in011~11romotit.s p~rror~no(l 

by R. T. Jones [58] and W. Jacobs [55], thc boundary layer on t h receding portion 

thickens, the effect leading to prcmaturc scpnration. In aircraft cq~~ippcd with sweptback 

wings separation begins at the receding portion, i.e. ncar the ailerons, nntl causes 

the dreaded one-winged staU to occur. It is possible to avoid this kind of sepamt.ion, 

and hence to prevent one-winged stalling, by equipping the wing with a 'boundarylaycr 

fence' which consists of a sheet-metal wall placed on the suction sidc in the 

forward portion of the wing, thus prevent,ing cross-flow. An aircraft with swept-back 

wing.? and x boundary-layer fence on each half of the wing is shown in Fig. 11.16. 

W. Liebe [66] reported on the improvement in wing charactmistics which can be 

attained by these means. A paper by M. J. Queijo, B. M. Jaquet and W. 1). Wolhart 

[90] t1cscril)cs extensive mcnsurcment,s on models providctl with 'houridnry-layer 

fences'. The papers by ,J. Black [8] and I). ICucchemann (641 contain morc details 

Fig. 11.16. Jet fighter De ITavilland D. 11. 110 wil.ll rrwcpt-back wings and a I~wndnry-layer 

fence at cdge of each ailcron; from W. J,icl)c [66]

XI. Axinlly ~,vn~n~et.rir;~l nnrl Il~r~.e-cli~ne~~sior~nl I~o~r~~clnry Inyera 254 

Si~~rc: s11c.11 rx1~~rn:iI flows nrt. 1\01. irrul,:~licn~:~I. l11c vcIot:ilry in l.11~ l)ound:~ry I:~yrr ran Ircconlc 

Inrgrr III:LII I.lln1 in t.lw frw sl,rc;m. 'l'l~c cxc:css ill vclocity is rltrr? to the swo~rtlnry flow in I.lic 

I~nu~~tl:rry hyrr \vl~irl~ lr:~t~sli.rs inln) if, lluitl p:~rl.iclrs from rrgionn of higher energy. It. so~ncti~nrs 

also II:I~I)~IIS ~,II:L~, I.II(~ inili:tl veloril,y prolilw in 1 . l ~ l)ri~~cirnl llow dirrrtion slrow rcgiot~s of 

l);~d-Ilow which, ~~~~vcrll~rlcss, do not, signify scp;~r~ilio~~; they 11n11d1y tlis:~ppc:~r f~trl~lwr tlownsLrtw11. 

'l'l~is type of brh:lviour can also bc cxplni~~ctl an bcing due to a trar~sfer of rncrgy by 

t,llc srrontlnry flow. 'The rrndcr will recognize from the preceding exarnplc that t,lrc definition 

of sr~):~.rnI.ion is hcscl. \vit.h cliffic~rltics w11cn three-rlin~cnniorr:~l boundary layers arc being consitlrrrd. 

'l'his in d ~ 1.0 ~ the e fact I,lwL t.110 rclnl.ion brlween I~nclt-flow atd ~hcaring strrss has ceasctl 

trr hr :IS sitnplc nn in t.lw t.~r~o-tli~~~c~~sior~:~l 

rax [4!), 771. A scp:\rot.ion of tcrms itlcnt.icnl will1 

th: ow twro~~~~tc:rcti in cwnnt!xion wiLh 1.11~ free st~rcn~n (1c~cril)cd by cqn. (1 1.55) call bc srrcccssft~lly 

nrhit?vrtl, :~c.rortling 1.0 1,. 16. 1~og:~rly [24], mlwn considering an infiuitely 1o11g wing wl~iclr 

is III:I~C 10 rrrl,ale nl)ouL a vcrl.ical axis (I~cliropter rotor). It is found that, the rotary motion 

c1oc.s not nll'rct, ll~r cl~ortln.isc velocity co~n~~oncnt nntl so I.hc incitlencc of scparnt.ion rclnnitin 

~ln:l(li.c~trd. I101.:1lio11 ~nnrcly cntlscs 1.h~ nppr:lr:ulrc of slight r:idiaI vclocit,y roniponcntr4. 

A ft~rl,l~rr sprri:tI casr of I IN: gr11m11 ~iroI)lcr~~ r1cscriI)cd l)y cqtrs. (11.53) and (11.54) which is 

atnrtl:1I)I1' 111 r:~lw~lnl.int~ orwcrs B.II~II 1111: cxlcrnnl flow consisls of a tlvo-tlin~rnsionnl basic patlrr~~ OII nl~ii.l~ tl~rre- is SII~I~~III~~SI'~ n wr:tlz tlist,~~rl~nnce of 1.11~ kind tlrsrrihrtl I)y 

(1 (.r,z) = [l,,(x) I lJl (y.2) , Uc1 , 

ll'(.r,z) = I\'# (r,:) , lrl < [lo . 

2. Ilow~cl:rry Iayrrs 011 d ~ c Idies. r 'I'l~rr~t!-tli~~~c~~~ion:~l 

l)~t~t~tl;~r~'-In.yrr flr)ws hcronrc even 

III~II,~ (.o~~~l)li,.:~lr~I ill r:is~s \VIWII Ih t!~I(~rn:~l flow 1::111110l, I)P r(:l~rt:siwl~~~~I xi~~~ply Ipy 01c s11l1t-r- 

~~tsilio~~ (PI' I.\CO VOIII~)OIIC.IIIS. '1'110 Iiil kr IYISI: orrrtrs, li)r ~:X:I.IIIII~T. 011 n y:~n.cvl I~r)(ly of ~r.vol~il.i

256 XI. Axially ~yn~motricnl and three-dimcneionnl boundary layers 

tho oxperi~nent,nl pnt,tnrr~ in Pic 11.17 b. It is, tllcreforo, not at all eaay to establish a critorion 

for scparat,ion in a thrco-clirnc~~siol~nl boundary layer, if proper weight is given to this type of 

bel~aviour. At this point, we wish to draw tho reader's ~t~tention to the investigations on yawed 

cmwn he to W. J. Itninbird, It. S. Crnbbe and L. S. Jurewicz [91]. 

It, ap~)rars Lo bo possible to attempt, a theoret,ical analysis of t,hree-dimenaional boundary 

layers wilh t.ho aid of ir scheme snggcsted by L. Prandtl [RBI who proposed to introduce a 

cwrviiincnr systrrn of coordinatm in which the potential lines and streamlines of the free stream 

would play t.lm part of coordit~atos. This progmmme wns cnrrictl out by E. A. Eichelbrenner 

and A. Owlnrt [22] whon Lhcy calmllnlad tho laminar cmo ment,ioncd earlier. It hna already 

Iwrn mcnt.ioncd l.l~nt good qualit,ativc agreerncnt rrsultetl. as shown in Fig. 11.17c. See also 

It. 'I'imtnnu [I 141. 

'I'he mcthotl of c:nlculnt,ion proposed by L. I'randtl r86] was recently developed 

numerically by W. Gcisslcr 135, 36, 371. Figure 11.18 illnstrntes the result.s referring 

t,o tlrc t,hrcc-tlilnensional l~oundary layer on a yawed ellipsoid of revolution. In 

acldition t80 t1hc potmtial lines and st.reamlines of Llre external flow, Figure 11.18~ 

shows the separation line A'; the latter has a course ~imilar to that in Fig. 11.17. Figures 

ll. 18 11 and 1 1.18 c represent t,he v~locit~y distribution in the boundary layer nt 

various st,alions on a particular potc,nt.ial line. 

The lamir~ar I~oundnry layer on a yawad rotating circular cone in a supersonic 

stream was earlier invcst,ignted by R. Sedncy 11041, whereas ,J. C. Martin 17.71 

irivrstigat.ecl the Mngrrns eKect8 on bodies of revol~tt~ion at, R, small angle of incidence. 

Fig. 11 .lR. \'rloc.il.y tli~trilmlion in t.11~ t,hrco-di~r~cnsio~~ai bor~ndary-layer on an ellipsoid of mvo- 

Iution ornxis mtio LII) - 4 nt an angle of inci~lcnco ? = is0, after W. Geisslcr 136, 871. a) SysLcm 

of potrnt,inl linrfi ~ ~nd st,rrn~nlinm in outer flow; S = sopaintion line. b) Primary flow velocity profilrs. 

~r/lJ,, in t,hr tlircvt.ion of the outer flow strcaniliue~. c) Secondary flow velocity proBles,iu/Um, 

nt ripht anplr~ to thr dircrtion of t,lw outer flom strcatnlines. 'rhc velocit,y profiles arc given for 

pot.rnt.inl line 1 - (13) nt, di(Torrnt st,ntions 111, wit.h a7,i~n~it,h nnplc 4 and st.nt,ion x as pcr table above 

(6 - 0'' - wir~dwarrl sytnn~rt,ry) 

Another irnport,ant, example of a t,hree-dimensional bountlary layer can 11e fount1 

in the corner formed by two mutually pcrprndicular planes in a slrcarn prallrl t,o 

their line of intersection. This flow config~rat~ion was invrst,igat,ed t.llrorrt~irnlly 1)y 

V~~sant,n 1dn.m 1921. 'I'lin est.rrnnl rcloc:it.y at, fnr t1ist.ntlc.c I1as hen nssunrc~l lo Iw of 

Ilnrt~.co's l,ypc, i. c!. givcn by 

It, is recalled from See. TXa that this type of external strrnln leads to sirnil:w vclocity 

pofilcs in the boundary layer. This feat,ure continues t,o hold in the case of flow in a 

corner. Some of [.he results of these studics arc givcn in Fig. 11.19; t,llis shows the 

vclooity distril)utions in the corner for three cliflkrcnt vnl~~cs of the prcssnrc parallletcr 

nt. A comparison between the distribulions for different values of ns demonstrates 

that the boundary layer in s corner thiclrc~~s apprrcia1)ly in t,hc prcsrnce of a 

pressure increase in the external flow. 

Expcrimcnt,al ol)scrvat.ions [82, 391 suggest that t,he flow in t,lle corncr s(:p.:~t,rs 

carlicr than th:~l on llic portions of t01e walls at a larger dislnncc from it, cvv11 in thc 

prcscncc of sn~nll ntlversc pressure grntlicnts. 'J'lris pliysic:nlly r~r~tlcrst~r~.~~tl~~~I~I~ 

111011(: 

of I)clinvior is fnlly conlirrnetl by Lllesc thoorctionl r~sult~. 011 a flnt plr~l,c: H(-~):LI.:I(~ 

occurs at m = -0.091 (see Fig. 9.1), separ:rtion in a right-nnglcd corncr occ~rs as 

cnrly ag for m = -0.05. At na = -0.08, Fig. 11.19, the flow in the nciglll)ou~.lrootl 

of t,he corner displays a separation region with revrrse flom (IL < 0). By oontrnsl, at 

a large distance no reverse flow occurs. M. Za.mir and A. 1). Young [120, 1271 carrid 

out extensive experiments on the laminar bomndary layer xlot~g a right,-anglctl corner 

a.t zero incidence. See also S. G. Rubin [93a].

258 XI. AxhIly syn~rnrtrirnl nncl tl~rrr-diri~msionr\i borlntlary layers 

AII rxI.(v~sinn of l'~~I11l1au~rn's nictl~od 1.0 rot11.ting hodieu \vns given by G. J~rngclaus [40]; 

lie :~pplir(l it t.o the: invrsLignt.ion of rclnlivo ti~olion throng11 a curved chnnnel which is important 

in the Clrror,y of rrrilriftll~.nl 11111nps. 1.110 LI~rory Icnils 1.0 ~~rcdirl,ionu regarding scpamtion which 

nrr: in good :igrcrn~r:~~t, with n~ras~~renirnl..r. 

111 conrlr~sion, attention nlny I>r: drnwr~ to t.lir mlr~tl:~tion of t.lie 11o1111dnry layer on two 

~nnt~~~nlly 1)crl)rf1dic1llnr flnl. plalcs at. znro incitlt:ric:c pc:rforn~ctl hy (:. IT. Carrier 1121 and I

260 

XI. Axinlly syrnmrtrird nntl Il~rrr-tli~~~c~t~sio~~al bonndnry Inyrrs 

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[I81 Cookc, J.C.: Tho flow of fluids nlong cylinders. Qrmrt. J. Mech. Appl. Mnth. 10, 312-331 

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242 - .- Il9R!II. I-. .. ,- 

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konatnntrr nnd orlnvi-riintlerlicllrr Wanclten~l)rr~~tur. Ihs. I3raunschwcia - 1951 ; ZAhlhl 

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References 261 

,/ 

1231 Eichelhrcnner, E.A.: D6collement laminnire en troia dirncnsions aur un obstaclc firti. 

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5, 339-360 (1973). 

[25] Eldcr, J. W.: Tho flow poet a flat pin& of finite width. JFM 9, 133-153 (IDGO). 

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[27] Fagc, A.: Expcrimcnta on a sphere at critical Iteynolds-numbers. ARC 1tM 1760 (1036). 

[28] pgnrty, L. E.: The larninnr boundary layer on a rotating blade. JAS 18, 247-252 (1951). 

[29] triissling, N.: Verdunstung, Wiirn~ciibcrgang und Geacl~windigkcitavcrkiinng bci zweidimc~lsionalcr 

und rotationusymrnetrischer lnrninarer (2rcn7ficl1ichtnt.riimung. 1,nncln. Univ. 

Areakr. N. F. Avd. 2, 35, No. 4 (1940). 

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rotating bodies in axial flow. J. Appl. Mcch. 37, 17-24 (1970). 

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layer on a rotating body. Bullet.in of JSME 9, 702-710 (1966). 

[31] Furuyn, Y., and Nnkemnra, I.: An cxpcrin~ontnl invcatigntion of the skowod bonndnry 

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[32] Garbscl~, Ii.: Ubcr dic Grenzschicht an dcr Wnnd cines Trichtars mit inncmr Wir1)cl- nnd 

Radialstriimung. Fifty ycnrs of boundnry-lnycr research (W. Tollmien and H. Giirtlcr. cd.), 

Brannschweig, 1955, 471 -486; six also: ZAMM-Sondcrhcft 11 - 17 (1956). 

[33] Ccis, TI).: Ahnlichc Crcn7~chichtcn nn Rotationskorpcrn. Fifty ycnrs of bonnilnry-lnycr 

rcscnrch, (W. Tollmicn, and H. Oortlcr, ed.), 13munschwcig, 1955, 204-303. 

1341 . - &is, Th.: ,,Khnlichc" drcidi~rre~~sionnle Grenzsclricl~ten. J. Rnt. Mcch. Annlysis 5, 643 --- 

686 (1056). 

[35] (:cisslcr, W.: Rcrcchnung dcr I'oknlinlstriimung unl rotntion~~yrnn~ctriscl~c Itiitnpfc, 

1Ungprofilc nnd '~ricbwcrkscinlii~~fc. ZFW 20. 457-462 (1072). 

[36] Geiealcr, W.: Uercchnung dcr drridimcnsionnlcn Inminarcn (:rr?nzechicht an nngwkllkn 

Rotntionskorpern mit Abloanng. AVA-Bericht 74 11 I0 (1074); Ing.-Arch. 43, 413-425 

(1974). 

[37] Ckiwler, W.: The throe-dirncnaior~nl Inminar boundary lnycr ovcr a body of rcvolution st incidence and with separation. AVA-Bcricht 74 A 08 (1974); AIAA .J. 12, 1743--1745 

(1974). 

[38] (:ersten, K.: Corncr interference cfich. r\GARD Rep. 290 (1959). 

[39] Ger~ten, K.: Die Crcnzsc11iclltatron1ung in cincr rccl~twinkligcr~ Eckc. Zi\MM 39,428--429 

(1 959). 

[40] Glnucrt, M.B.: The wall jet. JFM 1, 625-043 (1956). 

[41] Glnuert, M. B., and Lighthill, M. J.: TIIC nxisymrnctric bonnclnry Iaycr on n long tl~in 

cylinder. J'roc. Roy. Soc. London A 230, 188- 203 (1955). 

[42] (Xrtlcr, 11.: Dic laminnre Grcnzsrlricht nnl schicbcnclcn Zylincler. Arch. Math. d, Fwc. 3. 

21(i-231 (1952). 

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Mnth. Hisp.-A~ncr. 11'. Scr. 14, 14:)-178 (1054). 

[44] Grohnc, I).: Zur lnrninarcn Strvmung in cincr krciszylindrischen Dose mit rot~icrendcn~ 

Ihckcl. ZAMM-Sondcrhcft 17-20 (1956). 

1451 (>ru~rl~witz, I(.: Turb~~lcnte Rcibungsschichte~~ mit Scknndiirstriimung. 1ng.-hclr. 6, 

355-365 (1935). 

Han~n, F. It., nnd Peterson, L. F.: AxisymmcLric laminar wnkc behind n slcndcr body of 

revolution. JFM 76, 1 - 15 (1976). 

Hannnlr, I1.M.: Forcrd flow agninsh n robling disc. AltC JtM 2772 (I!)52). 

1Iansrn. A. G.. Hcrzie. 1I.Z.. nnd Costello, G.R.: A visunlizalion stndy of sccond:iry flons 

in cnscddcs. NI\CA 'FI; 2947 (3953). 

Ilnnsrn . . . . . A C.. and Ijrrzie. 1I.Z.: Cross flows in lan~ioar incon~pressiblc boundary Inyrrs. 

. . 

lit 1 h1:lgt.r. '1.: 'l'l~idc I:IIII~II:I~ h~l~chry l:~yvr 1111iIer SINI(I(VI II(TIII~J):I~~OII. Il'ifl,y ~(.;II.s 

IIIIIIII(~~~ lilyrr rrs(~:~rvh (\I1. r~'O~~llli('ll :11111 11. (:iirl,l~:l.. (.(I.), I

264 

XI. Axially symmetrical and thrm-dimensional boundary layera 

[107] Steinhcucr, T.: Three-dimensional boundary layers on rotating bodies and in cornera. 

AOARlJograph No. 97, Part 2, 567-611 (1965). 

[I081 StewartRon, K., and Howarth, L.: On the flow past a quarter infinite plate using Oscen's 

equations. JFM 7, 1-21 (1960). 

[I091 Stewarteon, K.: Viscous flow pnst a qunrtcr infinite platc. .JAS 28. 1- 10 (1961). 

[I101 'J'nlbot, I,.: I~minar swirlinl?. pipe flow. J. Appl. Mcch. 21, 1 -7 (1954). 

[1]():r] 'Jh, S.: On lnminirr bo~mdary lnyer over a rotnt~ng blnrle. .JAS 20, 780 -781 (1953). 

11 I I j 'J'nylor, (:. 1.: 'J'ho 1)oundery laycr in the ronvcrging nozzle of n swirl ~rlornizvr. (2urtrt. -1. 

Mcch. Appl. MaLh. 3, 129- 139 (1950). 

11 121 Tetervin. N. : Boundary-layer momentum equations for three-dirncnsional flow. NACA 

> 

. . 

TN 1479 (1!)47). 

11131 l'iiford. A.N.. and Chu, S.T.: On the flow around a rotnting disc in a uniform stream. 

L , 

JAS 19. 284-285 (1952). 

[I 141 Timn~nn, R.: Tho theory of three-dimensional boundnry layers. J3oundary layer eiTecle in 

acrodynamirs. Proc. of a Sympouium held at Nl'l,, landon, 195.5. 

[I 151 'I'imman. It., and ZturL. J.A.: Eine Rcrhcnmet.hode fiir dreidin~ensior~alc laminare Grenzschichkn. 

IWtv vears of boundnry-layer - " research (W. 'l'ollmien and II. GBrl.ler, cd.), 

Ur~unschweig, 1655, 432-445. 

1, [I101 Tomotika, S.: Ihrninar bonndary layer on the surface of n sphcro in n uniform stream. 

ARC 1tM. I678 (1935). 

, [I171 I'oniotika, S., and Imai, I.: On the transition from laminar to turbulent flow in the 

bounclary lnycr of a sphere. Rep. Aero. Res. lnst. Tokyo Id, 389-423 (I9:SR); and Tomo- 

tika, S.: l'roc. Phya. Math. Soc. Japan 20 (1938). 

[I 181 l'ruckcnbrotlt., E.: ]$in Quadmturvcrfnhren zur Bcrcchnung der Reib~~ngsscl~icllt an axinl 

angestriin~kn rotiercndon I)rohkiirycrn. In&-Arch. 22. 21 -35 (1964). 

[II!)] Truckcnhrodt, E.: Die turbulente Striimung an eirier angeblancnen rotiercndcn Scheibe. 

ZAMM 34, 150-162 (1954). 

[I201 Vogelpohl, G.: Die Stromung dcr Wirbclquelle zwischen ebeuen Wiinden mit Reriicksichtigung 

dcr Wandreibung. ZAMM 24, 280-294 (1944). 

11211 Weher, 11. E.: The boundary lnycr inside a conical surface due to swirl. J. Appl. Mech. 23, 

587 --592 (1950). 

1' 11221 Wieghardt, I(.:- ISinigc Grcnzscl~icht~ilr~~s~~tigc~~ nn S

206 XIT. l'l~rr~nnl bo1111r11iry lnycra in Inminnr flow 

'1'111- t.vr111 tl lC7./tlt rc:prrsrnls n st~l)st.ar~t id tlcrivativc which consist,^ of a local ant1 

:I cwr~vc.c.l.ivr c.o~~l.ril)l~t.io~~. 

Ilrrrs lr I.l/n~ sc~ t l q l tlcr~ot,cs the 1h0r1n:~l (:o~~(l~~t:I.ivit,y of 1.11~ (Il~id. 'l'he rtcg:~l.ivc 

. . 

slgn s~g~~ifics ~,II:L(, thc hrnt Ilux is rccltol~cvl :IS positivr in the tliroc1,ion of the t,cmpc- 

r 3 

I Ilr rl~nt~gc ill t.l~r f.ot.nl rl~c~fiy. tllCT, (:on!iist,s of a chnngo tlE : p,,l l'tlt. in t.hc intcrn:d 

c~~crgy :tntl n c.l~:wgc in Iti~~rl.ic: oi~crgy by at1 ntnount, (1 { 6 0 LI I ' ( ~ L ~ 1- v2 4 w2)}, if 

I IIV ~ I : I I I ~ ill V tl~r l)oi~v~~tial cwrgy (Iuc to a displ:~ccrn~~~t it1 Ihc gr:~vit:~t,im:~l Iirhl 

is ~~rglrcl(~l. I lvncc 

Tho nrgst,ivc sign is a.tltletl in order to follow t,hc sign convcnt,ion of cqn. (1 2.1) accortling 

to wl~ich work adtlcd to tho fluid from t,l~r! oui,sitlc is rlcgativc. Thc tol.:d worlc 

j)erforlnrcl by t.hc normal :rnd shearing shrssrs prr unit! time rsrl now l)c writ.t,c:n :IS 

JIrrc a,, n,), . . . , s,, rlcuok tho rlorlnn.1 ant1 sl~rari~~g si.rcmt:s itit.rot111rrtl r:~rlivr ill 

eqns. (3.20) and (3.26). Substituting eqtls. (12.3), (1 2.4) ant1 (I 2.0) into cqn. (1 8.1 ), 

nntl pcrfnrming s numl)cr of obvious simplificst.ions, i~~cl~ltling those inI.rotl~lcrtl by 

cqn. (3.1 I), wc ol~t,ain, after some calculnt,ion, the: followi~~g oncrg.y o(j~~:~I.iot~s 01' 1.l1c 

flow: 

Jlcrr @ rrpresrnts the tlissipatiorl firnct,ion give11 I)y 

Rquat.ion (1 2.7) enjoys gcnrral valitlit,y, I)ut in most pr:1,c4ir:d rnsos it is possil~lc to sirnplify 

it still furt01cr. Jrl doing so, it, is ncccssnry carcfrtlly l.o(list,il~g~~isl~ 1)ctwcc11 l.hc rnsr 

of a pcrfcc:t, gas n.nd t.l~st of :LII incom~)rrssil~lc Iluitl. 'I'll(- f.l~c~r~notlyt~n~~~ic? proprrt,ic,s 

of t,l~c Int,t,rr do wol ronsl.itnt.e n 1irnit.ing m.sc of 1.11~ prnl)c.~.l irs of the f'orl~lrr. 

tJic va.riat,ion in iJ~o int,crn:d energy of :L prrfrc:t gas is clc I-- c,, ti'/', wllcrrns l,l1:11~ 01' 

it,s ontl~allly is tlh. c,,dT. Tlin corrcspo~~tling v:~r.iat.iot~s for nn inc-olr~prrssil)l(~ fl~litl 

:1ro (if? = c (17' a,ll(I dl& .r c (17' -1- (l/@)dp . 

M'itlr the nit1 of' Illis rqu:~t,io~t :III~~ of 

c, (IT - c,, d 7' 1 (1 

111 li~~t.,

268 XII. 'I'hcrrnal borrnrlary lnycrs in laminar flow b. Temperature incren.w through ndinbntic cornprcssion; stagnation tempcratme 269 

Here c,[d/kg (leg] represents the specific: heat at const,nnt, pressure per unit mass. 

In general, c, clepcnds on t,ctnpcmtnre. In the casc of a constant thrmal contluo- 

tivit,y, we obtain the simpler form 

In thc ca.se of an incomprcssil)lc fluid, wc havc tliv rct = 0, antl cqll. (12.7) togrthcr 

wit.11 dn .- c tb7' yields 

r 7 

I he tan~pnmtnre changes brought, about by thc dynamic: pressure variation in 

a comprc:ssil)le flow arc important for its heat bala~ice. In particular, it appears useful 

to compare t.11~ tc~nperaturc diffcrcnccs which result from the hcat due to friction 

wit,ll those cattsctl by comprcssion. For this reason wo shall first cvnluatc the tcrnpcra.t.trrc 

increaso due to compression in a frictionless fluid stream : 1 f the velocity 

varies along a st.rm.tnlitlc t,l~o tcmpcr:~t.urc must vary also. In order to simplify - thi: .. 

argumcnt it, is perrnissi1)lo to assumc that the process is adiabatic and rcvcrsiblc 

bccaltsc the small value of conductivit.~ and the high rate of change in the thcrmodynamic 

propcrtics of state will, in gcncral, prevctit, ally appreciablc cxchange of 

hmt with the surroundings. In particrtlar wc propose to calculate the temperature 

increase (AT),, - T, - 7', which occurs at the stagnation point of a body in a 

stream anti wltich is due t,o compression from p,, t>o p,, Fig. 12.2. 

l'ig. 12.2. Calc~~lation of bl~c tctnpcraturc incrmno 

at stagnation point due to adiabatic comprc3sion 

(A7'),,, = To - 7.m 

For the casc of zero hcat conduc%ion in frictionless flow the energy equation 

(12.11) givcs the following relation between temperature and pressure along a strcam- 

liric (coordinate s) 

I 

where w(s) dcnotcs the vclocity along a streamline. Dividing by euy and integr,zting 

along a strcamlinc we obtain 

" 

Mercury 

1,uhr. 

oil 

Air 

(ntrnooph.) 

Temperature 

6 T 

Table 12.1. Physical constants 

(1 d = 1 Nm; 1 IrJ/kg dcg - lo3 m2/sec2 dog) 

Specific 

hent 

Cv 

[lzJ/kg K] 

Thernrnl Ther~nnl 

condnrtivity difl~mivit.y 

k n x 10' 

[Jim noc K] [ni2/mc] 

Vincoaity 

/1 X lom 

[kg/~n ncc 

= r~ R] 

Kinernntir 

vinronity 

v x 108 

[rn2/uec] 

Prnndtl 

nurnber 

In an analogous manner, thc complete Navicr-Stokcs r(~na1ions (3 26) lead to the 

I%ernoalli equation when viscosity is neglected in them and whcn an intagrnl along a 

streamlinc is talren: 

so that the tcmpcrature increase 

1 

T - T, = -- (wW2 - w2), (12.14a) 

% 

antl, in particular, the temperature increase at the stngnat~ion point (w - 0) t111c to 

adiabatic contprcssion becomes 

Here w, dcnotes the free-stream vclocity (Fig. 12.2). The temperature T, assumrd 1)y 

the fluid when the velocity is reduccd to zcro is known as the slagnntio~t te~npernlurr, 

sometimes also referred to as the total lernperature. The difference (AT),, = To - l', 

brtween the stagnation and the free-stmam temperature will hcre be called the 

ndiabtrlic trmprralitre incrrosc 

P 

I-]

270 XIT. 'I'l~rrtnal boundary layers in leniinar flow 

Fig. 12.8. Atli:rl>nf ic tcnlp.-ratrlro 

inrrmwo at stngnntion 

point for air frorrl qn. (12.1411) 

(r,, = 0 24 l%t~~/ll)i Iot,os t,llr rorffi(:icn{, of I,llrr~n:~l cxpnnsiot~ nl. l.r~ml)c~r:r.l.~~~~c 'I'm, 1' is t,llc die 

"1' tJlc t,wo sprcific Iir:r.t,s, ant1 cm is t.hr s1)rrtl 01' sout~tl ol' 1I1v llilitl. 

r > 

I IlC Inst t.isrm (:an 1)c nrglrrt,ctl in flows wliirlr n1.c nfl'cv+c.(l by a~.n\rita.tio~~. 'I'l~is 

nlrnns, getlrl~ally sL)ral

272 XII. Tl~ern~al boundary layers in laminar flow 

In the general case of a compressible medium, eqns. (12.17) to (12.20) form a system 

of six sim~~ltmwous equations for the six variables: u, v, w, p, e, Tt. For incompressible 

media (liquids) the last equation as well as the tferms u ap/& etc. which represent, 

compression work vanish. In this case there are five equations for u, v, w, p, 1'. 

1t is noccsnnry to ernphnsizo t.l~nt tho sytnl)ol p does not donotc the sumo physical quontity 

in eqnu. (12.IR), (12.19) and (12.20). Whorens in tho lnut two cqunlionn p stnntls for tho thcrrnodynamic 

property, the symbol p in eqns. (12.18) represents the difference between the actual prcusure 

and the static pressnre of the mcdinm at rest when its density is em (cf. remark concerning 

fluid8 without free surfaces in Sec. IV a). In the cn.908 treated in detail in the litmature so far, tho 

pressnre term has been included either only in eqns. (12.18) - the case of free flows - or in the 

pnir of equations (12.10) and (12.20) for compressible flows. 

Before procceding to indicate solut,ions of the above equations, which we shall 

discuss in tho sl~ccccding sections, we propose, first, to examine them from the poir~t 

of view of the principle of similar it?^ [10B]. In this way we shall discover the dimension- 

less groups on which the solutions must depend. We begin by introducing dimension- 

less quantil,ies iuto eqns. (12.18) antl (12.19) in the same manner as in Sec. IVa, 

when Itcynolds's similariLy principle was deduced from the Navier-Stoltcs equations. 

All lengt,l~s will be referred t,o a representative length I, the velocities will he made 

dimcnsionlcss with rcfercnce to the free-stream velocity U, the density with respect 

to e,, antl the pressure will be rcferrctl to e, Urn2. The temperature in the energy 

equakion will be madc dimcnsionlcss with reference to ttpe tempcrat,ure dircrence 

(Al'), = T,,, -- TW between the wall and t01e fluid at a large distance from the body; 

thus O* -- (7' -- Tc,)/(A 7')". hnoting all climcnsionlcss quantities by a star we obtain 

from eqns. (12.18) anti (12.1 0) for the equation of motion in the x-direction and for 

the energy equation in the two-dimensional case with g, = -g* cos a: 

" 

ao * ao* av* a20* 

Q* (u* h* -i- V* ay*) - e,r, * 1- 1 - 

The tlimcnsionlem dissipalio~~ function is here givcn by 

. . . - . - .- . . . - 

t sine(? the viu~onity /I wn.8 aflsllnled constant the above system is valid only for moderatc changes 

in tcni rrnbure. In the rmc of large temperature tlilTerences in gnscs (over 50" C or 00" F), 

or mocfcmtn ones (over 10" C or 18' F) in liqnids, /I mu~t be hken to vary with tcmpcmtllre. 

In thin c:rse t.he equation of motion robins the form (3.29). The six equations under consideration 

mnst he snpplenwnbri by the empirical viscosity law /c (l'), eqn. (13.3), and, in all, we 

hnvo n syutrm of nevcn uirnnlt.nncoun cqnnl.ioncl for tho scvrn functions It, 11, tfl, p, e, 7'. 11. 

I 

c. Theory of sitnil:~rit,y in 11c:tt tranafrr 273 

It is recognized that the solutions of eqns. (12 21) nntl (12.22) tlcl)cnti on t h following 

five dimensionless groups: 

'l'hc /irst group is tl~c nlro:uly f:tmilinr Ilcynoltls nunlbcr. The fourlh nntl fifI11 groups 

differ only Ijy {,he factor R, so that, in all, tShcrc arc only /our inrlrpenrIe111 dintcit.?iol~lc~s 

qua,i~lilies. The second group call be rcprcscnlctl as 

This gives tl~c Grashof 1111m1)cr 

q /? 1:' (A 

G 

7') 

. .- -- --- 0- 

v2 

'i'hc third quantity m.n bo wriltcn as 

whcrc 

is the tl~erinul di//uaivity [m2/sec or fL2/soc] and 

is the dimensionless Prandtl number. It will be noted that it depends only on tho 

proprrtirs of the medium. For air P = 0.7 npproxirnatcly and for watcr at 20°C P = 7 

approximately, whereas for oils it is of the order of 1000t owing to thir large viscosity 

(see also Table 12.1). The Jourlh dimcnsior~lcss quantity leads directly to the tem- 

perature inrrease througli atliabatic compression as calculat.cd in cqn. (12.14b). We 

1 1 % ~ ~ 

wl~cre E is Irnown as tllc dirncnsionlcss Eclrcrt number. Tl~c quantiLy E = lJm2/c1,(A71)0 

is sorncLirncs used. It in related to the Prandtl number by thr equation P, = P R 

l'hc ratio of thc two tempcrat~~rc dilTcrcncrs has, 80 far, not received a separnk nalne. l+dowing 

n snggcution by Professor E. Srhmidb it hnrr bern proposed in nn rarlinr edition to call it, after 

I'rofcu~or IF. It. C:. Eeltort,, antl to give il t.l~e namc of t.hc 1Srkrrt. nnmbrr, E.

can I)r rct.ni~rrvl in inrotnl~rrssil~lt~ flow :IISO. I)ut, the inf~cr~~rctnt~ion wit11 reference to 

atli:~l):~t.ic: t:omprcwion vcnsr>s l,o I)(: vnlitl. It is now possil)lc 1.0 conclutle that frictiot~al 

hc~11, nncl IICRI, (IIIC 1.0 (:ompr~~sion :Ire in~port~nnt. for the c:n.lcnlnt.iorr of the temperature 

fivltl whrn tltr I'rw-st,rrn.~n vc4oc:ity 11, is so lnrgc t.l~n(, t,lro ntlinhntic t,crnl,er;~ture 

int:rtx:~sr is ol' l,ltc S:IIII(~ orb nf m;~p~iI~t~lt~ 

:IS I,lrc prt?srril)c~l I~~:~r~pt>rnt,~trc (lifT

276 XtI. Thermal boundary layers in laniinar flow 

Whrn sprcial solr~tions are consitlerctl thcn, in most cases, one or more of the 

tlimensionlrss groups will disappear as the problem will only seldom be of this most 

general nature. As srrn from eqn. (12 27) the temperature field and, hence, the 

coefficicnt of heat transfer tlcpcnd on the Eckert number only when the tcmperature 

cliKcrences arc large (50 to 100° C or 100 to 200° F) and whcn, simultaneously, 

the vclocitics arc very large and of the order of the velocity of sound. With moderate 

vclocitirs the ten~peraturc and vclocity fields depend on the Eckert number when 

temperaturc differences arc small (several degrees). Further, even with moderate 

vrlocitics, the buoyancy forccs in eqn. (12.21) caused by temperature differences 

arc small comparcd with the inertia and friction forces. In such cases the problem 

ceases to dcpend on thc Grashof number. Such flows are called forced flows. Iience, 

for forced wnvrdion 

N, = f (R, P) (forced convection) . 

The Gmshof nwnbcr becomes important only at very small velocities of flow, 

particularly if the motion is caused by buoyancy forces, such as in the stream 

which riscs along a heatcd vert.ica1 plate. Such flows are callctl natural, and we refer 

to thc prohlcrn as one in durn1 co?~vection. In wch cases thc flow becomes indepcntlrnt 

of the Rcynoltls numbcr, and 

N, = f (G, P) (natural convection) . 

Examples of problcms in forced flow are given in Sccs. c to g of the present chapter; 

cxan~ples of problems in natural convection are contained in Sec. h. 

(I. Exnct solr~tions for the problem of temperature clistributinn in a viscow flow 

We shall now proceed to solve several particular problems of temperature 

distribution. The examples to be discussed will be sclected from the Iarge number 

of possible cases on the ground of mathematical simplicity. We shall begin by dis- 

cussing several cases of exact solutions, as given by H. Schlichting [loll, just as we 

have begun with the discussion of examples of exact solutions of the equations of 

flow with friction in Chap. V. For the case of incompressible two-dimensional flow 

with constant properties the system of equations for the velocity and temperature 

distxibution in steady flow along a horizontal z, z-plane we obtain from eqns. (12.17) 

to (12.19): 

d. Exact aolutiona for the problenl of temperature distribution in a viscoua flow 

Fig. 12.4. Velocity and tcmpcraturo T-T, 

distribuLion in Couette flow. a) Velocity 

distribution. b) Temperature distribution 

with heat generated by 

friction when the temperatures of 

both walls arc equal. c) Tcmperatrlre 

distribaLion with hc~t gcncnted by 

Friction for tho cast when Lhc lower 

wall iu non-conducting 

where 

1. Coucttc flow. A particularly simple exact solution of t,llis systcm is obtained 

for Couette pow, i. e. for the case of flow between two parallel flat walls of which 

one is at rest, the othcr moving with a constant vclocity U1 in its own plnnc, l'ig. 12.4. 

The solution of the equations of motion in the absence of a prcssurc gratlic~~t in the 

x-direction is 

A very simple solution for the temperature distribution is obtained when it is pos- 

tulated that the temperature is constant dong the wall, the boundary conditions being 

277 

y=O: T=T,; y=h: l'=T,. (12.35a) 

In this case thc dissipation function reduces to the simplc expression @ = (a~/ay)~, 

and the equation for temperature distribution becomes consequently 

With thc boundary conditions (12.35a), thc above equation has a solution which is 

independent of x. Since, with v = 0, the term v aT/ay on thc Icft-hand sitlc also 

vanishes, all the convective terms on the left-hand side of eqn. (12.34) become cqual 

to zero. The resulting temperature distribution is, thercfore, due solely to the gencration 

of heat through friction and to conduction in the transverse direction. Prom 

eqn. (12.35b) we obtain 

d2T 

k-=-p dy2 (:)', (12.35~) 

and, substituting dultly, we have 

'UIC solution of this cquntion which sntisfics conditions (12.36n) is

if wc: pillf - - 'I1, - (A7'),. 11, is sccn t,llnt it cat1 1)o oxprcssctl it1 t,orms of 1.11~ 

1'rnntll.l 

1111in1)t:r 1111(1 l.lw 16c:ltcrt. numt)cr from cqn. (1 2.28). In lhc cnso urlclcr consitlorntiot~, 

i. c. wlwn thrc is no convection of I~ei~t, the temperature distribution is seen to 

clrpond on the protluot P x E. If, finally, t,hc ilbbrevintion rl = ?//it is introtluced, 

tho followin,n wry sitnple equntion for temperature distribution is obtaincd: 

'l'l~is si111111v rx:11111llf\ shonx I,II:II. tilt, gwt~:tI ifw of' IIC:II, IIIIC lo r'rict~io11 rxcrts n 1:1rgc 

vl1i~c.t. 011 IItr ~)r.oc:rss of' rooling :tl~tl l,11:1.t. :I(, lliglt 1-rloc.il.irs t h w:trlncr wall Inny 

I)IYYIIIII% 11(~:1l(~l inslt':~tI 01' I)rin!: (:ooIr~I. 'l'llis t-lh't. is of (.IIII(I:UII~~III:II ill~porl,:lnc.c lor 

t IIV ro~~si~lt.t.;~t ion ol'c~ooli~~g :11. 11igl1 vt:loc.il iw. It. \viJI r.c:c-~r~. in t,ltc. ~)~.ol)lt>tns c~ol~tlc:ci,t.tl 

~villt t11t~1n:11 1~~11ntI:try I:tjrrrs :111tl will I)(, ~1isc11ssr~l I:~.t,c:r. 

d. Exact ~olutions for tho problem of tmnpornture di~t~ribrltion it1 a viscou~ flow 279 

Fig. 12.5. Telnpern1,rlre distribution in 

Couettc flow for vnrions temperaturea of 

both walls with Iteat generated by friction 

(7'" = tempcrnturc of the lower wall, 

l', - kmpcmlrlrc of t,he upper wall) 

0.8 

v 0.6 

Tltis tlistributior~ is seen plotted in Fig. 12.4b. Tlic Iligllrst tcni[)ernturc T',,, c:rrnt,ccl 

by frirt,ional hcat occurs in thc ccntrc ant1 Itas a va111e givcn by 

In the case of coinprcs.ribic! flow for wllicll tho al)ovc solutiot~ rcmains v:did provitltrtl 

that t,he vis~osit~y may I)o assumrtl to be intlcpcndcnt of tcrn~)cmturc, eqn. (12.38) 

cnn be put, in the following tlimcnsionless form 

cvhcro M =F [J,/C~ clcnol.cs the M:~ch nulnl)c~. :~ntl c,, is 1,110 vclocil,y of so~l~ltl :kt 

le~nl)cral~r~rc l',. It is rcrnarltal~lo t.llnt Lhc mnximum t.crnpcrat.r~ro docs not clcpcntl 

on the tlistanro bct,\vccll t,llo wn.lls. 'l'ho ql~:~ut.it.y of 11t::~t grnrrat,otl 11.v fric:l,ioll is 

disl~ril~uted cvcnly bctwcen t.hc st,at.ionary ant1 thc moving \v:~ll. 

Tlie tcmpcrnturc distrilmtion in t.11~ 1)rcsctlt c:xa~~l~l,lc is import,:lrll. for I.llc 

flow in the clearance bctwccn a journal and it,s bcnring a.ntl was tliscussctl itt t1ct;ril 

by G. Vogclpol~l [143]. The flow in thc clcnmncc is 1anlinn.r in vicw of t.hc s~nall 

tlitnonsions of the latter and of thc Iligll viswsit.y of t1I1c nil. '1'11~ t1rn~1)c.r:t.t.~rrc riso 

tluc t,o frict.ion becomes consitlcmblc cven :tt, nlotlrrat,c: vclocitics, as sllo\v11 by 

t,hc following cxanlplc: Viscosity of oil at, motlcmtc t,crnprrnt~t~rc (my ROO (1) from 

Table 12.1 : ,L = 0.4 I

280 X11. Thermal honndary layers in laminar flow 

conrlitions for tcm pcraturc bccomc : 

The solution of eqn. (12.34) with the above boundary conditions is 

it, is seen plotted in Fig. 12.312. Thus the temperature increase of the lower wall 

is given by 

T (0) - To = T, - !Po = ,u UI2/2 k . (12.41) 

The value T, is callctl the adiabatic wa.21 temperature as already mentioned; it is 

cqual to the reading on a thermometer in thc form of a flat plate. Upon comparing 

cqns. (12.41) and (12.38) it is seen that the highest temperature rise in the centre of 

the channel for the case of equal wall tcmpcratures is equal to one quarter of tho 

adiabatic wall temperature rise 

The criterion for cooling in the case of different wall tempcratnres given in eqn. (12.37) 

can be simplified if the adiabatic wall temperature T, is introduced. We then have 

11. M. de Groff [48] generalized the preceding solution for Couctte motion to 

incluclc the case when thc viscosity of the fluid depends on temperature. The further 

extension to a compressible fluid was given by C.R. Illingworth [68] and A. J.A. 

Morgan 1871. 

2. Poiseuille flow thror1~11 a channel wit11 flat walls. A furthcr and very simple 

cxnot solution for tempcralme tlistribution is obtained in the casc of two-dimensional 

flow through a channcl with parallel flat walls. Using thc symbols explaincd in 

Fig. 12.6 we notc with I'oisc~~iUc that the velocity distribution is parabolic: 

Fig. 12.6. Vc1ocit.y and tcrnpcra- 

turc distrihution in a channel with 

flat walls with frictional Ilmt. fx7.lwt1 

int,o anconnt 

d. IPsncb solrttio~in for thc problrtn of totnpcraturc tlisLrilmbion in a visoons flow 281 

Assuming, again, equal tmnpernt.urrs of the walls, i. c. 7' -- l',, for y = ,I h, we 

obtain from cqn. (12.35~) 

the sohition of which is 

The t,cmpcrat,uro distribution is reprcscntcd by a parabola of thc fourtd~ dcgree, 

Fig. 12.6, and t h mnximnm tcmpcraturc rise in Iho ccnl,rc of t h chi~nnrl is 

An extension of the solution to the case of tempcrat,~trc-clcpnnclcnt viscosity w:~s 

given by IT. liausenblas 1631. The corresponding solut,ion for a circular pipe was 

given by U. Grigull [47]. 

A further exact solution for the thermal bounclary layer mn bc ol~txincd for 

the flow in a ronvergrnt and a divergent channrl alrcatly cortsitlrrrtl in Sro. V 12. 

The solution for the velocity field due to 0. Jeffery and 0. Iiamel quoted in that 

section was utilized by I(. Millsaps and K. Pohlhausen [86] in order to solvc thc 

thermal problem. The temperature distribution across the channcl is seen plotted 

in Fig. 12.7 for different Prandtl numbers. Owing to the dissipation of energy which 

is particularly large near the wall, the resulting temperature profiles acquire a 

pronounced "boundary-layer appearance". In fact, boundary-layer-like appearance 

becomes more pronounced as the Prandtl number increases. The velocity distribution 

u/us from Fig. 6.15 has been plotted in Fig. 12.7 to provide a comparison. 

Fig. 12.7. Tcmpcrnluro dis- 

tributions in a convergent 

channel of included angle 

2 a = 10" at varying Prandtl 

numben P, afhr I

Rotntirig rli~k: (:II:I~I. V, in pnrtkular Pigs. 5.12 and 5.13, ~ont~aitlrd a solrtt.ion 

1.0 t>lw flnw prol)lr~n nrountl an infinit.cly large disk rotat,ing in a fluid at rest. l'l~in 

wns n solnt,ion of' t.lw syst,cm of eqns. (5.88). In order to tletermine t,he t.ernpern,t,ure 

field in thc noiglrl)o~trlrt,otl of n hcni,cvl. rot,at,ing disk, it is necessary to expand the 

prwcvlirrg syst.vm of'rqr~xt.iotrs by inclrrding t3hc eqnat,ion for t,Ire t,emi)eratr~rc distribut.ion 

it.sc.lf (cwcsrgy c-clunt,ion). Sntah cn.lctt~ln.t.ions have I)ccn prrforrnctl by I

284 

XIJ. Thcrninl bormtiary layers in laminar flow 

In view of t,hc 

obtained estimation for the thickness of thc vclocity boundary 

layer 8, - l /l/~ , wc obtain 

It follow^ that the ratio of the thickncsses of the two boundary layers is independent 

of the lteynolds number. If energy dissipation through friction and the buoyancy 

forces are omitted, the ratio of the two boundary-layer thickncsses becomcsdcpende~~t. 

on a sin& characteristic number - the Prandtl numbcr. In this case it is possible 

to givc a vcry good physical interpretation of the Prandtl number, as will be shown 

in Sec. XI1 f 4 in more detail. 

Rstimnting the rrmnining tcnns in thc: cnrrgy rql~ation it is concl~rtlcd that, 

in the cxprcssion for tlrc tli~sipat~ion function only the tcrm (i?u/L~y)~ rcmains signifiranL, 

and 

l'hc hrat clnc to friction is sccn to be important only if 

Jn the case of gases thc hcat gcncmtcd I)y friction bccomcs important only if thc 

temperature risc due to adiabatic compression is of the same order of magnitude 

as the difference in tempcraturc bctwccn the body and the fluid. The same rcmark 

applies to the work of compression. 

Reverting to dimensional quantities and taking into account the dependence 

of viscosity on temperature, wc obtain tho following simplified equations for two- 

dimensional compressible fluid flow: 

Since in thc fr:rmcworlc of boundary-laycr theory thc pressurc may be rcgardctl 

ns a given, impressed force, we havc hare a system of five simultaneous equations 

for tho five unknowns e, IL, v, T, p. 

f. General properties of thermal boundary layer 285 

Regarding the differences in the significance of p in eqn. (12.50b) on the onc hand 

end in eqn. (12.GOd) n thc other, we rcfer the reader to thcremark made in Scc. 

XI1 c just after eqns. (12.17) to (12.20). 

For the incompressible case (Q = em = const) and for constant viscosity these 

equations reduce to 

giving three equations for u, v, and T. 

f. General properties of thermal boundary layers 

1. Forced and natural flows. Thc diffcrcntial equations for the velocity and 

thermal boundary layer, eqns. (12.51b) and (12.51c), are very similar in structure 

They differ only in the last two terms in the equation of motion and in the last 

term in the temperature equation. In the general case the velocity ficld and the 

temperature field mutually interact which means that tfho tcmperaturc distribution 

depends on the velocity distribution and, convcrscly, thc velocity distribution 

depends on the temperature distribution. In the special casc when buoyancy forces 

may be disrcgnrdccl, and when thc propcrtics of thc fluid may bc assumcd to be 

independent of tcmperaturc, mutual interaction ceascs, and the velocity ficlcl no longcr 

depends on the temperature ficld, although thc convcrsc depcndencc of the tcmperature 

field on the vclocity ficld still persists. This happcns at large vclocitics 

(large Reynolds numbers) and small tempcrature tliffcrcnccs, such flows being tcrmed 

forced (cf. p. 276). Thc process of heat transfcr in such flows is doscribcd as forced 

convection. Rows in which buoyancy forces are dominant are called natural, t,hc 

rcspcctivc heat transfcr bcing known as natwal convrction. 'l'his casc occurs at, vcry 

small vclocitics of motion in the prescncc of largc tcmpcrnLwc tlifrercnccs. 'l'ho sI,atc 

of motion which accompanics natural convection is evokctl by buoyancy forces 

in the gmvitational field of the earth, the latter bcing duc to tlcnsity dil'fcmt~ccs 

and gradicnt,~. For cxamplc, the ficld of motion which cxistsoiitsidc a vertical l~ot 

plate belongs to this class. Porcctf flows can be subdivided into t,l~osc with rnotlrr:tt.c 

and those with high vclocities depending on whcthcr thc hcat tluc to friction and 

comprcssiorr nccd or necd not bc taken into account. In bot.11 cascs thc tcmpcl-aturc: 

ficld depcnds on the ficld of flow. At modcratc vclocitics, wl~cn thc hcat cluc Co 

friction and comprcssion may be neglected, the depcndencc f the trmpcratrrrc ficld 

on the vclocit,y finld is govcrncd solely by thc Prantltl num1)cr. To rael~ ,~in!/le 

vclocit,y field thcrc corrcspontls a singly infinitc family of tonlprrntl~rtr tlisl.ril)~~l,io~ls 

with thc I'rantltl rruuibor as it8 paramct,cr. At lrigl~ vc1ot:ilics work duc I)ol,l~ to 

friction and comprcssion must be included. Whether this is ncccssary or not dcpcnds 

on tho Eclrcrt nurnher E - 2(A7'),,/(A7'),,, i.c. on wlrct,lrc-r it, is compn.r:ildo with

286 X11. 'IY~rrrr~al bo~~ndnry layers in laminar flow 

11ni1.y. Tn ot,her words, t .1~ \vorIz due to frict.ion and compression must he taken into 

accol~nt, when the t,crnpcrnf.ure increase drlc to friction nnd cornprcssion is comparable 

with f h tmnpcr:~t.urc tlilTcrcncc prcscri0ccl as a bourdary condition (tnmpcrature 

tliffrrenco I)ct,\r;rc:~~ I)otly and fluid). If t h prescribed tenlperatlire difference is of 

t.ho ortlcr of t.hc mmn at)solrrt2c t.rmpcral,nrc, t h work duo to frict,ion and cornprcssion 

1)rcorncs ilnlwrtallt, only if t,hc vrloc.il.y of llow is compnr:ll)lc wit,ll t.hat, of sol~ntl. 

2. Atlinlmtic wall. I~in:~lI,y it, is IICC~-SSRI.~ to mrnti~n t.l~i~t. the vi~rict~y of possiI)Ic 

sds of 1)ounclary contlitions is much grontcr for the ttcrnpcrat~~re field t,Ilan for 

t,hc: vrlocit,y fieltl. Tllc tcmpcmtnre on the surface of the body may bc constant 

or w~riable but., moreover, it is also possil)lo to encountm pro1)lcrns for wl~ich tho 

hrat flux is prescribetl. In view of eqn. (12.30), this means that t,lle t,crnperaturc 

gmtlirnt at the wall may appear as a boundary condit,ion. The so-callctl rtrlinbalic 

wr/l constiLutos a pnrticulnr rxn.nlplo of the Iatt.er class of cases, since it rnnst be 

post,ulat,cd thn.t there is no heat flux from the wall to the fluid, i. e., the borlr~clary 

t:ontlit.ion at, t,he wall is 

r 1 

1 his case c:Ln 1)c visrrnlizotl Ily imagini~lg 1,hat the wall of the body is perfectly 

it~sul.ztctl against I~cst, Row. 'I'lic heat generated l)y the fluid through friction serves 

1.0 l1cn.t 1,ho wall 11nt.il tho contlit,iorl (a7'/an),,. , = 0 is reachctl. Thus the t.crnperat~rrc 

of t.llt, wnll wl~ic:li we may :~lso call the rdirtbcclic unll tempernlure Imxmes higher 

t,l~:~.tl thaL of the lluitl at, some clist,nnco from it. Srtcll conditions are satisfied in 

pr:~ct,ic*c \vhrn a so-called plate t.hcrmornct,c:r is usctl, i. (:. when t.llc ternperabnre of 

:I, fll~itl st.rrnm is rnc~as~~rctl with t.hc aid of a flat plat,c which is placed parallel to 

t,lit. sl.rc~nm 1. 'l'hc PXC~SS l.wq)crats~~rc 011 1.110 plnt,r const~it,ut,c*s t.hc rrror of t.hc pink 

t.lwrmon~ct.c~r. 'I'llc error mns0 I)c tlrtl~~c:t.t-tl ill ortlrr t.o obtain t.hc 1,rnc t,empcrat.urc 

of t.Iit: tnovirl~ Il~~id. 'l'llis tlifkrcnc:c is somctimrs cnllctl the kinetic. lemperntwe. 

It, hns i)c~n showt~ ill Stw. VI 11 :I t.llnt nlk solni.ions of t.llcr two-tlirncnsional 

I)ol~ntlnry-l:tycr cq~~;l.t.iorts for an inron~prc~ssil)lc flnitl I~xvc the form 

If t.1~ work of compression as well :IS t,hc cvolut.ion of Ilo:~t. tltro~~gll tlissipat,ion 

c : be ~ neglect,etl, tile same reasoning shows that, rtll solut,ions of oqr~:~t.ions (12.51 P) 

wl~id~ tlcsrril~t: th: i,l~rrm;tl 1m1ntIa.r~~ l;~,yt:r, nii~st, I)(: of the form: 

Iloncc, the heat flux from rqn. (12.30) can I)(; writ,t.cn 

1

288 

and 

XII. Thrrn~al boundary layers in lnminar flow f. Gencrnl propcrI.ien or thcrmal bor~ntlnry lnyrr 289 

It follows immediately from the temperature equation that 

In analogy with eqn. (12.53), t!~e local Nussclt number formed with the coordinate 

z assumes the form 

where 

(11 5 

N,=-r = 1 /~,-F(~,P), (12.55) 

The function F(m, P) will be discussed in more detail in Sec. XI1 g 2 (see eqn. (12.87) 

and Fig. 12.14). Thus between the local skin-friction coefficient 

and thc Nusselt number there exists the relation 

The simplest type of flow, that on a flat plate at zero incidence, is characterized 

by the value m = 0 and by the fact that eqns. (12.51 b) and (12.51 c) for the velocity 

ficld and the temperature field, respectively, become completely analogous if the 

Prandtl number has the value of unity. In this c~ase, the solutions themselves acquire 

identical algebraic forms, and we have 

Consequently, 

and eqn. (12.56) simplifies to 

Z(0,l) = 1 , 

N, = 4 cf,' R, (m = 01, P = 1) , (12.56 b) 

when applied to a flat plate. This is the simplest form of the Reynolds analogy; 

it was, as elready stated, first discovered by 0. Reynolds himself. 

The preceding argument is applicable, so far, only to laminar, incomprcssiblc 

flow^ at constant wall tcmpcratura and on condition that energy dissipation may 

be neglected. Nevertheless, thc preceding rcsnlts can bc cxt,o~tlctl t,o inclutlc other 

cases, RUC~ as that of a flat plnto with frictionnl heat (sect oqn. (12.81 ) ntd ioot.11o1.c on 

p. 2!)9) or that with compression work (sce Sec. Xlllc). It is p:rrtioulnrly nof.c?\vorthy 

that thc Itcynoltls analogy can bc rccovcrctl in turbulcn t flows whcrc it.plnys:u~ rsscv~,,i:~l 

part in the calculation of heat-transfcr rates (c/. Chap. XXI [I). 

4. Effect of I'randtl numher. The cor~sidcrationa of this c.haj)trr c~mvint~c 11s 

that thc J'randtl number conatitutcs that parameter wl~osc vnluc i~ dc.c*isivc Tor t h 

extent of the thermal boundary layer and, thcreforc, for thc rate at which hrat is , 

transferred in forcctl or free convection. According to its tlrfinition 

thc Prandtl number is equal to the ratio of two quantities: one of t,l~ctn (viscosity) 

charactcrizcs the hid's transport propcrtics with rcspoct to IJIC tmnsport of rnorncntum, 

the other (thermal diKusivity) doing the same for the transport of hcat.. If 

the fluid possesses a particularly large viscosity, it can bc stated looscly that its 

ability to transport momcnLurn is Inrgc. Const:qucnt.ly, thc tln~t~ruction of ~notncf~t.~~n~ 

introduced by the presence of a wall (no-slip condition) extends far into the fluid 

and the velocity boundary layer is comparatively large. Similar statcmcnts can bc: 

made with respect to thc thcrmal boundary layer. It is, thcrcforc, undcrstantlal~lc 

that the Prandtl number serves as a direct measure for the ratio of the thicltnesscs 

of the two layers in forced flow, as already demonstrated in eqn. (12.49). The special 

cwe when P = 1 (already discussed) corresponds to flows for which thc two boundary 

layers are approximately equal in extent; they are exactly equal along a flat plate 

at zero incidence whcn its temperature is uniform. In addition to this, thc two 

limiting cases whcn the Prandtl numbcr is either very large or very small arc also 

worthy of attention; they are representcd schematically in Fig. 12.8 

Very small Prandtl numbers: It is clear from Fig. 12.8 that in the case of very 

small Prnndtl numbers, such as occur in molten metals (for example in mercury), it 

8) P-U(l~uid metals) 6) P -01 liquids, oils) 

Fig. 12.8. Comparison between the tcmpcmture and velocity fields for bonnclnry lnyrrzl wit.11 

vrry amall and with very large valr~es of Prandtl number

290 XlI. Tl~rrnmal boundary layers in laminar flow 

is possil)lc to disregard the vc1ocit.y boundary layer in the calculation of the thermal 

bountlary layer. Conscqucntly, the velocitty components ?L(x, IJ) and v(x, y) can be 

replaced by U(x) and V(s, y) = - (dlJldr) IJ, respe~t~ively, the approximation for 

V stemming from the continuity cquat,ion applied at the wall. The energy equat,ion 

(12.51~) then assumes t.ho particnlarly simple form 

we can transform t,lw partial diffvrentinl cc(nn.tion for temperntr~rr, ~listribut~ion into 

an ordinary ow. 'L'l~is, in tarn, Icads to the following univcr~al expression for tile 

Nus.selt, nutn1)cr 

15qnations (12.59a) and (12.5!3b) are special cases of this general equation. 

In t,I~c 

7',,,, we obt,ain thf! satno tlini:rt!nt,inl cq~l:&t,ion a8 that encountered in anot,her corlnexion 

in (:hap. V, cqn. (5.17). Its solution is 

casc of a flat plate (II(3:) :.- [I, T. const) with a uniforrn wall ternpcrature 

Aecw-tling to rqn. (12 31). thr cormspontling Nussclt numbcr is 

In 111(, cast. of st~agr~nl.iorl-l~oi~tl. flnw (11 (3:) L-= u', r), it follows that, 

f. General proportics of tl~crmal bonnclnry Inycr 2!) 1 

Very large frandtl numbers: The second limiting casc when P + oo was solvccl 

for the first time many years ago by M.A. 1,evi:que 1701. Jlc inLrotlucotl t#he wry 

reasonnble assumpt,ion t,hat the wholc of the ttcrnperaturc field is confi net1 inside t,hat. 

zone of the velocit,y field where t,he longitudinal velocit,y wtnponent,, u, is st.ill pro- 

port.ional t,o the trnnsversc distance y. The samo circumstanc:cs can also oc:c.ur at 

int,errnccliat,c valucs of the I'rantlt.1 number in caws whrn t,ho tllcrninl 1)ound:try I:tycsr 

starts wibh a temperature jump at t,he wall at a: = To (cf. Vig. 12.17) irlsitlo n clcvc-l~pctf 

veloaitly bountlnry layer. Accordingly, in t,llc enrrgy cqnctl~ion, iyn. (12.51 t*). wt3 SIIIIposc 

that the velocit,y tlist,ribution in tile velocity bount1nr.y layer is rcprcsc-11t)ccl 1 ) ~ 

11 = (to/p) IJ. It can then be verified t,l~at, in accordance u.it,h refs. [76J and I(iYa] 

(see also refs. (1 111 and [I 121). the substit~~bion 

tmnsforms t h 

cnrrgy cqnation into the following ordinary cliffervnt.inl tyn:~lion: 

Ilcrc 2, dcnntcs tl~a coordinate at whicll thc t.cm~~c?r;ttnrc ju~nl~ at. tlto w:~ll has 

bccn placed, il I)cirrg rcmembcrcd t,liat the cffcct of fr.rictiot~:~l Iwat has hen 

ncglcct.ct1. l'ho solution of this ortlinary differential cql~:ttion c:~n 1)c cxprcssetl in 

closed form in t,crms of t11o incomplete gamma functions. I'orfor~ning 1,111: rrql~irrtl 

calculation, we would obtain t,l~e Nussclt numbor 

It is shown in Fig. 12.14 Llmt t.11is cquat.ion rrpwscnfs n very gootl approsin~alion

292 XIT. Tl~er~nnl Imn~dary layers in laminar flow g. Thermal boundary layers in forced flow 293 

evcn in the case of moderately-valued Prandtl numbers. At the stagnation point, 

the corresponding cquation is 

N, = 0.661 P1I3 1/% (stagnation point, P -+ oo) . (12.6213) 

Analogous, simple asymptotic formulae can also be established for the case of frcc 

convect,ion on n verlical flat plate, [73], see also eqns. (12.118a) and (12.1181)). 

g. Thermnl bounilnry layers in forced Row 

In the present section we shall consider several examples of thermal boundary 

layors in forced flow. In solving thcso problems, uso will bo made of tho simplified 

thermal boundary-layer equations. Just as in the case of a velocity boundary layer, 

the general problem of evaluating tho thermal boundary layer for a body of arbitrary 

shape proves to be extremely difficult, so that we shall begin with the simpler 

example of the flat plate at zero incidence. 

1. Parallel Row paet a Bat platc at zero incidence. We shall assume that the 

x-axis is placed in the plane of the plate in the direction of flow, the y-axis being 

at right ar~glcs to it and to the flow, with the origin at the leading cdgc. The boundarylayer 

equations for incompressible flow and constant properties (i. e. independent 

of temperature) have been given in eqns. (12.61 a, b, c): assuming that the buoyancy 

forces are equal to zero as well as that dpldz = 0 [18, 941, we obtain 

'I'hc: 1)ountlary ronclitions arc: 

11 = 0 : u = v = 0 ; T == T,,, or aT/8g =O 

'I'he vrlocit8y ficld is it~tlcpcndrnt, of t,hc tcmprraturc firltl so that tlrc two Ilow 

equations (12.03a, b) can be solved first and the result can be employed to evaluatc 

the tscmpcmtnrc field. An important rclatior~ship between the velocity distribution 

and thc temperature distribution can bc obtained immediately from eqns. (12.63 b) 

:md (I 2.fR c). Jf lhc hcat of friction p may be neglected in eqn. (12.63~). 

the two rquat.ions, (12.03b) and (12.63c), become identical if T is rrplaced by 76 in 

the sccond cqr~at~ion a.id if, in addition, the properties of the fluid satisfy the equation 

If the frictional heat is neglected then a temperature field exists only if there is a 

difference in temperature between thc wall and the extcrnal flow, e.g., if Tw - T, > 0 

(cooling). Hence it follows that for a flat plate at zero incidence in psrallcl Row 

and at small velocities the temperature arid velocity distributions arc idcr~tical 

provided that the Prandtl number is equal to unity: 

Thi~ result corresponds to eqn. (12.52) which Icd us lo thc f~rrnulat~ion of Llw 

important Iteynolds analogy between heat transfer and skin friction. 

11. Blasius introduced new variables for thc solution of the flow rquat.ions, 

sce oqns. (7.24) and (7.26). (y) is 1110 slrcnm fnnclion): 

'rhe diffcrcntial cquation for /(q), cqn. (7.28) bccornos 

f f" + 21"' =0, 

with thc boundary conditions: rl = 0 : f = f' = 0 ; 11 =- cm : 1' L- I . 'I'l~e solution 

of these equations was given in Chap. VII, Table 7.1. 

Including the eflect of frictional hcat, as seon from eqn. (12.63c), the temperature 

distribution T(7) is given by the equation 

It is convenient to represent thc general solution of eqn. (12.65) by tho super- 

position of two solutions of the form: 

ITcrc O1(7) dcnotes the grncml solution of thc hornogcncous cqu:~tion and 02(t7) 

denotes a particular solution of the non-homogeneous equation. It is, further, 

convenient to choose the boundary conditions for 01(7) and O,(q) so as to rnakc 

01(7) the solution of the cooling problem with a prescribed temperature diKcrcnce 

betwecn the wall and thc external stream, T, - T, with 02(7) giving the solution 

for the adiabatic wall. Thus 01(7) and Oz(v) satisfy the following equations: 

with 0, -1 1 nt 7 -1 0 arid O1 -- 0 nt 11 == oc> , r ~ l

with 0,' -. () at, r] - 0 and 0, = 0 at q -= co . 'l'lic value 0,(0) pcrmit,s us 

t.o rvalnntc tJie constnnt C from cqn. (12.66) in n manncr to satisfy t.lie boontfnry 

nondition 1' - T,,, for 7 - 0 . This yidtls 

(hding prohlcnt: The solution of cqn. (12.67) was first given by 13. l'ohlhnnsen 

[94]. It rnn Iw writtcn as 

IIct~co for P = I : 0,(q) = 1 - /'(q) = 1 - u/U,, and for P = 1 the temperature 

dint.riltc~t,io~~ bccomcs itlont.icn1 with tJlc vclociLy tlistrib~~tion in accordn.nce with 

cqn. (12.64). The t.cmpcmt,ure grntlicnt at, t.hc w;dl, as calculated from eqn. (12.60), 

wil.11 /"(O) = 0.332, bccomcs: 

- ("'I) = a, (P) = (0.332)' 

dtl 0 

Tltr coristnnt rrI is scol t,o tlcpcntl solcly on thc PrantltJ numbcr, a, (P). Some valucs 

c:clct~l:~t.rd hy 1:. I'ol~ll~nr~srn nrc rcproclnccd in 'l'ablc 12.2. They can bc interpolated 

with goofl nrrltracy from t,hc formula 

For very sm:tll I'rantll,l numbcrs, cqn. (12.59~1) givcs 

rs, - 0.564 ;/-P (P -- 0) , 

/Irlinhnfic ~twlb: '1'11~ so11tl.ion ol' w~n. 

'v:lri:lt,io~~ ol' t,lle 1)nrnrnrt.rr'. Itz is 

(12.(i8) cmi I)c ol)t.:einctl by t.hc mcthotl of 

g. Thcrrnnl bonntlary layrrs in forcrtl flow 2!15 

Table 12.2. Dinicnnionlcss coefficient of heat trensfcr, a,, nntl din~cnnionlcns ndi:hnt,ie wall 

ternpcmturc, b, for R flat plate nt zero incidence, from eqnn. (12.70) and (12.76) 

Fig. 12.9. Tempc.ratnre ~listril~ntiot~ on :I I~mtrtl flnt plate at zcro iw~tlrnrt~, 

plotted for various Prantltl nnn~hcrs P (frictional 1lr:it ~~rglrrtctl) 

I\ tt 11 sn1:i11 vrlo(.tty 

'I'llc t.cnil)crnt.urc wllicll is assumctl by t,llc w;111 owing to fri(:l.ionnl Ilml. 1.11~ rrrlirthrtlic 

~c~nll f~ntpern/?ur. 'l',, is thus, by cqns. (12.00) nntl (12.72): 

l'?,,, 

r l 

- = I - T,, -- Urn? b (P) 

rP 

(12.74) 

from cqn. (12.72). I'or a const.ant 1'rantll.l nnml)cr t,ho :~tli:~,l~nl.ic: \v:tll l,c-tril,rr:\.I,t~r(~ is 

prol)orI,ionnl to l,l~c n(lin~l)ntic t.ctnp(~r:~tnr~ I ~ 1!,v,2/2 C rl, \vltit;l~ W:I.S ~)lol,t.i~~l ill 

Icig. 12.3. Some n~lrnc.ric:al valucs of IJIC I;wI.or h(P) ;we givc.11 ill 'I':IIIIv 12.2; 1'01. 

motlrmt,c: Prnr~tlt~l ~iumbcrs them vn.lucs may 1~: int,cl.pol:rt,od wit,l~ sr~lIinionb ;I(:(;IIracy 

from thr. forniul:~, 11 - 1 /P. The valncs for Iargvr l'r:~t~tlt,l n111111)t\rs ~:III II(> inf'crrrd 

fronl l'ig. 12.10. In t.11~ bmit.ing msr, wc! I1:1vc (841

296 

XI. Thcrtnnl boundt~ry layrn in lnminnr flow 

It is remarliablc that for P = 1 wc havc cxactly b = 1. Thus, for a gas with P = 1 

flowing in a parallel ~t~rcam with velocity U, past a flat platme at zero incidence 

the ternperaturc rise due to frictional heat is equal to the adiabatic tempersture, 

i. e. to that which occurs from velocity U, to zero. The adiabatic wall temperature 

[16, 201 measured st various Reynolds numl)ers U, x/v is seen plottcd in Fig. 12.1 1. 

Thc agreement, is vcry good in thc laminar region. At the point of transition from 

laminar to turbulent flow in thc boundary layer the temperature increases suddenly. 

The temperature distribution for an adiabatic wall represented non-tlimcnsionally is 

and is seen plotted in Fig. 12.12 for various values of the Prandtl number. From 

eqns. (12.74) and (12.75), we obtain that the constant C from eqn. (12.68a) is 

c = (T, - T,) - (T, - T,) = T, - T, . 

The general solution for a pxcribcd tempcraturc difference between the wall and 

the free stream, T,#, - II',, eqn. (12.66), is thus 

uz 

T(7) - T, = [(T, - T,) - (T, - T,)] 01(7, P) + ---- 02(7, P) (12.76) 

cP 

with T, -- II', from eqn. (12.74). Thc dimensionless temperature distribution be- 

comes 

It is shown plotted in Fig. 12.13 for various values of the Eckert number 

E = Um2/c,(TW - T,), from eqn. (12.28). For b x E > 2 the boundary layer 

near the wall is warmer than the wall itself owing to the generation of frictional 

heat. In such cases the wall will not be cooled by the stream of air flowing past it. 

Heat transfer. As scon from eqn. (12.2) the 11c11.t flux from plate to fluid at, 

station x has the value q(x) = - k(t3T/ay)v-o or 

Thc rate of heat transfcr per unit time for both sides of a plate (length 1, width h) is 

1 

Q = 2b / q(x) dx, so that I 

a) Neqkding /rictioml heat : In this case T (q) - Tm = ( T, - T,) o1 (7) by eqn. 

(12.69) with (~ll'/dq)~ = - a, (T, -- T,). With nl from cqn. (12.71 a) we have 

I'ig. 13.10. Adiabatic 

wall tctnpcrnturc 7'" of 

a flat plate nt zero incidrnce 

with velocity Ua 

for v:irioun vnlur~ of tho 

I'rmrltl 1111ni1)cr; ahr 

1':. 15ckcrtanclO. llrcwitz 

[IB] as wcll RR D. Moksyn 

[84]. Vor large I'rnndtl 

n~ttnl)rrn, according to 

D. Meksyn [84], we 

havc 0 = 1-0 PI13 

Fig. 12.1 1. Measurenlcnt of ndiabatic 

wall ternpcrnturo on n flnt plate in n 

parallel air strenm at zero incidence in a 

lntnittar nntl trtrbnlcnt boundnry Iayor, 

nfter Eclzert and Weise [20]; theory for 

laminar flow and P = 0.7 

g. lrl~crtd bountinry lnycrs in rorcctl flow 

Fig. 12.12. Temperature excess in the laminar boundary laycr on n flirt plate at zero in- 

cidence in s parallel stream with high velocity in the absence 01 hrolinq for vnrions l'rnndtl 

numbcra (ndinbntic wnll) 

207

2!)H 

XI[. Thcrn~al bor~nrlnry Inyrrs in Inn~innr flow 

IGg. 12.13. 'rrn~l)cr:~t~lrc. elinlri~~ntion in a I:in~innr 1)ounrlary Iaycr on a Iicnbrl (E > 0) and 

cooled (E < 0) llat plntc: nL zcro incidcncc in a parallol stmnn~ for the case of a laminar hoi~tidnry 

layer and wil.l~ frict.ionnl IIC:I.L accorrntctl for nn calc~thbd frow rqn. (12.70). I'mntlt,l nntnber 

P 0.7 (air). 'l'llr lcn~pc!rat,nre of the wrdl is n~aint.air~ctl constnnt at y,. (hrvc: h x E = 0 for 

zrro rrict,ictn:~I Ilr-nt,; CII~VC h x E = 2 corrcspo~id~ to an ntliabalir wall; E = 1Jm2/c,,(7',, - 7>,); 

b .- 0.835. I'or h x E > 2 (.he hot wall ccb:tsrs I*, I?c cooled by t,hc strf:n~~i of coo1er air, RIII(.C 

tJ~o 'hral, c:nnliion' providctl I)y frichnal I~rnl. prc!vrtit.s cooling 

I~~[rotl~rcirrg tlitnr.nsionlcss coefficients in the form of thc Nlrssclt ~lurntxr from 

WIII. (12.31) instoatl of t,l~c? locnl and total Iicat flux, rcspcctively 

The cnsc of turbulcnt flow can bc approximatcd by lhc equations 

N, = 0.0296 'fi. R,O'~turbulcnt) , (12.79~) 

N, = 0.037 ;/F . R,O.R (turbulent) , (12.7!)d) 

which we quotc here for complctencss, but without proof. 'l'hc preceding forrnulim 

for the rate of Iicat transler arc in good ngrccmcnt with thc n~rasnrcmontn tho to 

1'. Elias [31], A. Rrlwarcls nrd B. N. I'urbor 1271 nntl .J. J(c:slin, ID. 1'. M~wclc*r ILIHI 

1%. E. Wang [66J. 

b) With frictional heal: In this case with T(q) from cqn. (12.76) we obtain 

wherc T, is the adiabatic wall temperature. It is identical with the wall tcmpcrature 

in the thermometer problem and follows from the equation 

T,-T,=b(P) 

Urn2 - Urn2 

- ----zip ---. 

(12.80) 

2 c, 2 c 

liere b(P) can be takcn from Table 12.2. Iritrotlucing the Mach number M = U,/cw 

from (12.27), T, may also bc taken from 

Thus we obhin the following expressions for the local and total heat flux from cqns. 

(12.77) and (12.78) respectively 

It now ceascs to bc useful to basc the cocfficicnt of hcat transfer a(x) on thc t.cmpc- 

raturc diffcrcncc (Tw - T,) from eqn. (12.29) or to clcfinc thc Nnssclt numl)trr :rx i ~ t 

eqn. (12.31) because the heat flux is no longcr proportional to that tcmpcmturc 

diffcrencet. 

t E. Eckcrt and W. Wcisc [17] hnvc, thcrcfore, ~t~ggrnLed to introclr~co a NIIRRC~~ nnrnl)cr N* 

based on the difference (T, - T,). Wc mighL tlicn cxpcct to obtain rrs a 8rnt approxiin a t.' Ion, 

even in compre~ible flow, the mmc forrnu1.w for N* a9 in eqn. (12.79a. b). If, on tho otllcr 

hand, wc retain the Nu~clt number basd on (T,- T,) thcn eqn. (12.81) Icnds to tlic following 

cxprcasions instead of (12.79a):

300 

XI1. Thrrrnnl borrndary layers in lan~inar flow 

The cooling action of a stream of fluitl on a wall is considcm1)ly rctluccrl because 

of t,hc hrat gencmtrd by friction. In thr nbsrncc of frirtionnl heat, heat will flow from 

the platc to thc fluid (q>0) as long as T, > II', but in actual fact,, if frictional 

hcat is prcsont, a flow of hcat persists only if T, > T,, eqn. (12.81). Taking into 

acronnt thc valuc tlcducctl for T,, we obtain the condition that heat flows from wall 

to fluitl (nplwr sign) or in tho reverse tlircction (lower sign), if 

A numcricnl cxamplc may serve to illust~mtc the signifirancc of cqn. (12.82): 111 

a stream of air flowing at TJm = 200 m/scc, P = 0.7, cp = 1.006 k.J/kg dcg wr 

obtain 1/ P 1JW2/2 c, = IF tlcg C. The wall will begin Lo be cooled whcn 

If tltc tenipcrat,urc difference bctwren wall and stream is snlallcr than this value 

the wall will pick up a port,ion of thc hcat generated by friction. In particular this 

is tho case whcn thc tcmpcraturc of the wall and stream arc equal. 

An equation for thc rate of hcat transferred from n flat platc at zcro incitIcnce 

but with variable material properties was derived by H. Schuh [110]. The tempe- 

rature field on a platc placed in a stream with a linear temperature distribution w54 

studicd in ref. [128]. 

2. Additional sinlilar sol~~tions of the equations for thermal boundary laycrs. 

In the casc of a flat platc at zcro incidence, the velocity and the temperature profiles 

t~lrnctl out to bc similar among themselves. This means that the distributions at 

tliffcrcnt clistanccs z along thc platc coulcl hc mn.tlc congruent by a sniLahlc stretching 

in the y-direction. Since it is lcnown that there cxist velocity boundary layers other 

i.han those on a flat platc for which this is true (e. g. the wedge profiles discussed 

in Chap. IX), it, appcnrs useful to stucly the possibility of tho cxistcncc of additional 

similar solutions of the energy equation. This problem was investigated in detail 

in ref. 11351. At the prcscnt time, we sha,ll start with the class of velocity boundary 

leycrs on wedges and will awnme that t,hc cxtcrnal flow is of the form U(z) = tc1 x"'. 

111 an analogous manner, we stipulate that tho wall-tcmpcraturc distribution also 

x". Walls 

sa1,isfies n power law, say one of thc form T,(x) - Y', = = TI 

of constant t,rmpcrat,ure are inclutlctl as thc casc n == 0, and t11c valuc 12. = (1 --711,)/2 

corrr~ponds to :I, crn~stc~nl, hrat flux q. 1nt.rotlucing tlic sitnilarit,y variable 

wr ol~tnin tlic f'ami1i:ir rquations (0.8~1) for the bdocity u = iJ(.z) . /'(q), or 

g. Thcrmal boundary layers in forced flow 

ant1 thc solution must satisfy thc boundary contlitions 

17~0: a:=]; 17=03: O-:() 

lIcrc E =. 71,2/c,, '/I1 rrprcwnts the nppropriaLc form of 1I1r I':ckorl. t~ttrnl)cr li)r illc 

prol~lcm. 

It is clear from cqn. (12.84) that its right-ltantl sitlc vanisltcs in tltc al)scncc 

of frictional heat and that all solutions arc thcn of tlrc similar typc. IIowcvcr, if 

frictional hcat is includcd, similar solutions arc rcstrictctl to tltat combinatio~~ of 

pnramctms for which thc right-hand sido becomes intlcpcntlcnt of z. 'rl~is occ:urs 

whcn 2 ~n - 7t = 0 , tltat is, whcn thcrc cxists a firm cor~pling 1)ctwcon tltc vclocity 

distribution in thc cxtcrnal flow ant1 tho tcmperatarc tlistril~ution along t.110 w:ill. 

According to this result., thc casc of a co~~st~ant tcmpcmt~ure lratls to similar solut,ions 

only on a flat plate (1i1=1t =0). 011 thc olhcr hnntl, if tho contlil,ion 2 111, -- 11. - 0 

is satisfied, thcn for every pair of values of m and P thcrc cxists one tlcfinitc valur: 

E, for which there is no flow of ltcat (O'(0) = 0). Jn this rasc, the tcmpc:mt,~~rc 

distribution along the wa.11, once again lcnown as tfhc atliabalic wall-tcmparat,urc: 

distribution T,, is given by 

Numcricnl valucs for thc function b(m,P) havc bcnn romput,rtl by 1% A. llrun 171. 

In the particular case whcn m = 0, the numerical valucs of l'ablc 12.2 arc rccovcrcd. 

Wlicn thc cffcct of dissipative hcat is ncglcctcd, wc obtain the simpler cquation 

whose solutions for different valucs of thc parameters m, n, and P have bccn 

published by a number of authors [79, 121, 32, 33, 89, 1401. E.1E.G. Eclrcrt [I91 

has dcmonst,ratcd that for n = 0, the local Nusselt number is given by thc equation 

Eerc 

ax U (x) . 1: 

N =-=- 

% k v---y O' (0) = - id, 0' (0) . ( 12.88) 

The function F(m, P) is seen plotted in Fig. 12.14 on thc basis of the numcriral 

data provided by 11. 1,. Evans [33]. Jn addition, thc asymptotes for very small 

and vary large l'randtl numbers, cqns. (12.57) arid (12.01 n), rrspcctivrly, have also

302 XJI. Therninl boundary layers in laminar flow g. Tliernial bonndary Inycrs in forced flow 303 

F~R. 12.14. 1,ocnl Nu.sselt nwnher as a fnnction of tho Prandtl number and of the flow parameter 

m for flows wliosr fire-stroam velocity is distributed according to the law U(z) = u, zm = 

= v, zPl(2-0) (wwlge Ilow) hut for a constant wnll kmperature and in the absence of dissipation 

Asymphtic approximations for P -t 0 

I\riylilpt.r)tin approximations for P .+ ca and P + - 0,100 according to eqn. (12.61 a), and for 

P + u, and p = - O.I!)O: 

Approximation for inbrmcdinto PrnntitI nurnbcrs and /i' = 0, according to eqn. (12.71a). 

Fig. 12.15. Tornpcrntnre distributioli , along a 

lienkd wnll (T,,, > T,) in a right-angled corner 

in n laniinar boundary layer with a constant external 

velocity Urn (inclusive ofdi~sipntion), aftnr 

~n.qnnd Rani (1441. 

1,irictn of const.nrit bmper:~.tnrc: for P = 0.7 arid E = 2.4. The local temperature exceeds the wall 

t~rnlwraf.nre ( 7' ;- 'l',.) in the hntcl~r:d rcgion; conseqnent,ly, in that region heat flows fluid + wall 

in spite of the fncl, t,liat the wall tmnperatnm exceeds the free-stream t.aniporntnre. Tlie reason for 

t.I~in proccm lic~ in tlis~ipntion. Rckert nnnibcr E -- IJ&/r, (T,,, 7',) 

been indicated (see also [119]). For thc Rat plate (m =0) the earlier rclntions from 

eqns. (12.69a) and (12.62a) are, naturally, recovered. The caso of stagnation flow 

(m= 1) leads to eqns. (12.69b) and (12.02b). In tho special cnsc of a separation 

profile (m = - 0.091) it becomes necessary to adopt a different asymptotic 

approximation for P -t oo, as shown in 1321. 

The thermal boundary laycr associntcd with tho thrcc-dirnc:nsional vc:locity 

boundary layer on a rectangular corner at zero incidence is also of thc sclf-similar 

type when the external velocity distribution is of thc IIartrcc class given by U(r) -- 

Cxm. The velocities as well as the temperature distributions for this case havc bccn 

worked out in a thesis by Vasanta Ram (ref. 1921 in Chap. XI). Figure 11 .I!) givcs 

an idea of thc vclocity distribution for dilTcrcnt vnlrics of tlic ~)rr..ssurc-grntlicnt paramctcr 

m. I'hc diagram in Fig. 12.16 supplcmcnk tlic procctling om in t,l~rtt, il, c:c~~~t.riitix 

an example of the associated temperature distribution. For a uniform cxtcrnal Ilow 

with U(x) = Um = const and in the case of a hotter (i. e. cooled) wall for which 

T, > Tm the solut,ion nevertheless exhibits a zone near the corner itsclf, shadcd in 

the figure, in which (T - Tm)/(Tw - Tm) > 0, that is in which T > T,,,. 'l'his zonc 

occurs when dissipation is included and corresponds to a condition where thc local 

fluid temperature exceeds the wall tempcraturc. Thus, locdly, the heat flux is reversed 

and proceeds from the fluid to the wall in spite of the fact that at a largc distancc 

from the wall the temperature of the fluid is lower than that of the wall, Tm < T,. 

The physical reason for this seemingly anomalous behavior is rootad in the increased 

local ratc of heating due to dissipation which occurs near thc corner. Phcnorncnn of 

this kind are important in the hypersonic flow rogimc. 'l'hc rcsulling Inrgc inc:rcnscs 

in temperature which occur in such cases can cause burning of the surface of the body 

in the stream (ci. Sec. XI11 e). 

3. Thetmal boundary layere on i~othermal bodies of nrbitrary shape. N. I' ' rocssling 

[39] carried out calculations on the tcmpcraturc distribution in thc laminar 

boundary layer about a body of arbitrary shapc for tlic two-climcnsionnl and axially 

symmetrical cases. In his calculations, in which friction and cornprcssion work wcrc 

neglectccl throughout, he assumed a powcr serics for the potential vclocity clistribution 

around the body expanded in terms of the length of arc (Blasius serics), similar to 

Sec. IXc, i. e. of the form: 

U = u, x 4- u, 2" -1- u, x"k . . . . 

The velocity distribution in the boundary laycr is ass~~mctl to have the form: 

Correspondingly, the assumption for thc tcmpcrature distributio~i was of tlic form : 

In a manner similar to that for the velocity boundary laycr in Scr. TX c it, is found that 

the functions TI (y), T3(y), . . . satisfy ordinary diffcrcntial cqnations which include 

thc functions fl, i,, . . . of the vclocity distribution. In this case, howcvcr, thc functions 

T,, T,, . . . also depend on the Prandtl number. The first auxiljary functions TI(?/)

304 X11. Tl~ortnal boundary Iaycra in laminar flow 

for t,hc two-tlitncnsional and axially syramotricnl case were evnluntcd numerically 

for n Prantltl number of 0.7. 'L'h met,hocl under consideration is somewhat cumber- 

some by its natnrc, ns was the case with the velocity boundary layer, particularly 

for slcntlrr htly forms when a large number of terms in the power expansion is 

rrqnirrd, as shown ill 1281. 

Nurnrrous solut,ions for self-similar thermal boundary layers inclusive of the 

elTwt,s of blowing and suction can be found in [34, 44, 134, 101. 

In t,lw sprcinl case when P = 1, antl when the heatf due to friction is neglected, 

the tfiKcrcnt.inl cquat,ion for the temperature distribution in the boundary layer 

around an xrbit,r:sry cylinder is itlcnt.ica1 with that for the transverse vclocity 

component (vdocity component in the direction of the generatrix of the yawed 

rylintler). This ran be seen upon comparing cqns. (12.63~) and (11.58). The relation, 

which has already t)ecn discusscd in Scc. XId, was ut,ilized by I,. Golancl [46] for 

t,ho eval~~nt~ion of the temperatlure distribution in the boundary layer around a 

aylintlcr of sprcinl form. 

In the ~~eigltl~ourhootl of a stagnation point, where the velocity distribution 

is r~~~rcsont~~d by IJ (z) == x with nz =.. /? = 1 , thc Nusselt number dcfincd in 

eqn. (12.87) a n be rcprescntad by thc cquation 

on contlition thatf oncrgy dissipation is neglected. The character of the function A (P) 

emerges from Fig. 12.14 and Table 12.3. In thc former, the curve for /? = 1 corrcsporlt1.s 

to tl~c function A. For .z circulnr cylinder we put U(z) = U, sin (x/R), so 

that. 91 I 4 fJ,/D. lrcncc 

The above rxprc.ssion agrcrs reasonably wcll with the measurements performed by 

R. St.llnliclt antl I

306 XII. Tllcrrnal hor~ndary layers in laminar flow 

I = U ( ) 2 -- 2 11"1- ?i4] == IJ (x) F(7) , (12.92~~) 

,. 

I htr vt:loc.it,y tlist,riln~t,ion st.il)~~lnl.ctl Ilcrc c:orrcspontls to t,lw I'ohlhauscn assumpti 

or^ in oqn. (10.23) :~.t~cl t,I~t: liwm of t.l~e t.cmpcr:~t.~~rc clistribt~tion func:tion is so 

srlrc:l.ctl as tm cnsurc: itltwl,ic::tl vclocity :m(l t(:n~~~~ra.l,urr (listril~~~tion~ for BT ---- 0, 

as rcrquircrtl 11.y t,hc Jtc~ynolcls analogy for n flnt pI:~t.c at P = I, cqn. (12.64). 011 

sul~stitat~ing cqns. (12.!)2:1, 1)) into cqn. (12.91), wo obhin 

Performing the intlicatcd integrations, we obtain 

2 

II(A)= -A- 

15 

? /In"+ 

140 ' A4 

180 

for A < 1 

and 

3 3 1 

~I(A)= lo---+ 

10 A 

2 1 

15 A2 

3 1 1 1 orA>, 

. +- - -- f 

140 A' 180 A6 

Some numerical valucs of the function [[(A), calculated by W. Dienemann[ll], 

havc been listed in Table 12.4. 

Tablo 12.4. Nurnoricnl vnlwa of tilo function H(A) 

The integration of cqn. (12.93) yields 

'I'lle vclority l~oundary-layer thickness (J can bc evaluated with thc aid of cqn. (10.37) 

wl~cw it is rcrncml~crt:cl from cqn. (10.24)t thnt ?~/d, = 316/37. Thus 

I 

I . 

t Iptw IIir ankv of~in~plidy tlic rdalci~lalion is ba.wd throughout on the flat-plate relations ( A = 0). 

I 

Upon dividing eqn. (12.95) by eqn. (12.96), we obtain 

U~~UII.(I~ 

4 1 0 

A2. H(A) = i-4 - (12.97) 

z 

H JUQI x 

O 

Since H (A) is a known function, Table 12.4, the preceding equation can bc used 

to doterrnine A (x). The calculation is best perrornicd by uurccssivc npproximntio~w, 

starting with the initial assumption that A -= rot~sl~. Jltwrc wc oMain 

The resulting value of A is now int,rvtlucctl into thc 1t:ft.-11:~ntl sitlc of ac111. (12.!)7) 

thus leading to an improved value of A. In general, two steps in the itcration nrc 

found to be sufficient. 

The local rate of heat transfer becomes 

and hence the local Nusselt number referred to a characteristic length I is 

The steps to be taken to evaluate the thcrmnl boundary layer, and in partficular, 

to determine the variation of the Nusselt number along a body of preseribcd shape 

are thus the following ones: 

1. evaluate A (2) from eqns. (12.97) and (12.978) 

2. evaluate d (x) from eqn. (12.96) 

3. steps 1 and 2 give dT(x); finally, the local Nusselt number follows from 

eqn. (12.98). 

Flat plate at zero incidence: The preccding approximato method will now be 

compared with the exact solution in tthc case of a flnt plnto at zero incidcncc. Insert- 

ing U (z) = U, into eqn. (12.97), wc obtain 

The expression A = P-ID constitutes an approximation to the solntion of this equation 

which is in error by no more than 5 per ccnt. as compared with the exact solution. 

The boundary-layer thickness from cqn. (12.06) is 

I

308 X11. 'I'hrr~nnl hotrnclnry I:ryrrs in lnn~iunr flow g. Tl~cr~nal boundary layers in forcrtl flow 30!) 

w11rrrn.s t,l~c c\xa.rt sol~~l,io~~, cqn. (12.79a), showed the numerical coefficient to b~ 

eq11:1l lo 0.:1:12. 

Alt.rrn:~l ivc :~pproxirnat,c proccdores for the calculation of-the tjhcrmalq!o~ndary 

1j1,ycr on 1)otlit:s of n.rl)itmry sl~apcs have baeq intlicnlc;i_Ly...E ..lj;ckert [l-)j and Gy 

JC. I~~C:I.~ worlc, b111 thir ancur:~ey is improved. 111 this connexion the 

p a l ~ I)y ~ s \V. I)ic~nrniann 11 I], 11. J. Merlc 1851, M.B. Skopek [I181 and A. G. Smit,ll 

n.1~1 1). 1%. Spnltlit~g 1.1 In] rnny I)? usrful t,o t h rcadcr. In ronf,ra.st. with 11. 13. Squit.~'s, 

t.l~c: I:~t.Ivr prorwl~~rrs rn:~.lco us(: of t.lw rcsulI,s of t,hc t,hcory of similar lhcrn~:il Iwur~tl:rry 

I:~.yrrs 011 l.liwt l in t.11~ prrnctling scc:tion. 'I'l~is improvcs the accuracy of the calculation. 

r 

I . hn v:irio~~s :~l)lrroxirnat,c nlct.l~otls have Imw examined crit.ically and comparctl with 

rxch ol hor in n pnprr hy 1). 13. Spnlding and W. M. Pun [I221 : their accumcy has been 

jutlgrtl I)y pc:rlimning c:ornpn.risons with the exact solut,ion for the circular cylintlcr 

proritlvtl Ily N. 1~rorssling. According t.o ~JICSI: studies, the methods due t,o IT. J. Mcrlc 

/85] nntl A.G. Sn1it.h and D. R. Spnlding [lln] t,ulm onl, to I)c rcht~ivcly t,hc most, 

~~cIII.:~I(: in spiIlc of l.l~rir simpli(:ily. 'I~IC Iattcr relerencc shows that at a PrantU1 

nurnl)cr of P :-= 0.7, t.hc similar wcdgc profiles satisfy with good accuracy thc relation 

'I'his c.ql~at.io~~ is rx:ic% for /3 - 0 (plnlr) :~ntl /3 -- 1 (stfaanat,ion point). If it is supposctl 

(I~nl rqn (12 100) vnjoys ~~r~ivrrs:d vnlitlily, it, is possil)lc immetlintcly to writ.? down 

I lrrc IT,, and I tlonok c.onsl,:~nt, rcfcrcltcc values. This equation corresponds to eqn. 

(10.37) which W:IS tl(vived I)y A. Wnlz for t,hc momcnt~nm thiclrncss. The local NussclL 

~~u~nl)cr is n.gait~ tlcl,cmni~~otl I)y cqn. (12.98). At, the st.ngnalion point we ol)t.:~it~ 

,. , 

-. 

i . ~,~~i~~j~l:,ij 

310 

XJI. 'I'hcrmnl boundnry layers in laminar flow 

In n simihr matrnrr. thr hrrit flux q(r) mn IIV ronlp~~trtl from thr known distribution 

~(~~rt,) = q* (r.rfl) (7's - 7'm) 

for lhr stnntlnrtl ~~rohlrlrl of PI^. 12.17. 111 this case 

0 

(12.105) 

Ilr*rr, P ~IIR. (12.101) nnd (I.L.lO5) cnntai~~ Sticll.jr*n inlrgrnln. When tho dintrihul,ions 7'.

312 SII. 'I'hrr~nnl I~ountlary laycrs in Inntinar flow 

pert prrforrncd measurements on circular cylintlers in a cross-flow of air covcring 

avcry witlcmngcof Itcynoltls numbers. ITigure 12.18co11tninsaplot ofthe mc:kn Nussclt 

numbcr N,,, tnlrcn for the wholc circumfcronce of the cylinder against the Itcynolds 

number R. 150th N,, and R arc based on the diameter of t,he cylintler. As n first 

crutlc a.pprnxiniation it can be a.ssumccl that N, is pr~port~ional t,o RU2 as confirmed 

by tl~c tllrorctird calculations for the f1:i.t plntc at zero incitlcncr, orp. (12.79a, b), 

and for t,he flow nonr a sbagnation point, eqn. (12.90), in laminar flow. 

The loral cocfficiolt of hcat transfer varies considerably over the surface of 

cylintlers nntl ot.hcr bodies; mrn.surcmrn1.s on circrllnr oylinrlcrs tlnc to R. Schmidt 

nntl I

314 X11. Thermal boundary laycrs in laminar flow h. Thermnl boundnry Inyrrn in nnbitral flow 315 

It iu rccognizcd that, there exiutr, a difficulty in providing nn rrnequivocal description of 

nucb Iloct.uetirig ntrenma. Since turbulence involvca atochnstic fluctuations, ntrictly epcnking, 

110 two turhttlctit utrcnrns can ever bc nimilnr. However, it is found by oxperinlent that certain 

nvcr:rgc propcrlicu of tho osrilhtiodu arc adequate to dmtcribe them. Tlirse nre: the intensity 

of titrl~uloncr, T, clcfinctl in Scc; XVI d 1, and I.ho ncnlc of b~trhulcticc, L, defined in See. XVIIltl. 

It, iu found, furtltcr, that in cnaca when ttic scale of turbulenm is small comparcd with the din~elinionn 

of 1.11~ I)orly. whinh occurs in most cmm in prnctice, ttic degree of turbirlence alone sr~fficca 

lo ~-l~artkrtd~rizc t.hc Ilow. It in, t.hcn:fore, b bc cx]xcted t.hnL lhc NIIRRC~~ number for gconlctrically 

uitnilar, inot~licrni:~l I~odicn wlticli arc placcd in fluct~unti~~g, parallel, isolhcrtnal streanis, depc~iclu 

on blir trtrlntl~:ncc intenniy, T, in addition to its clc~~cntlcncc on the l'randt,l and l

316 

XI1. Thermal boundary layers in laminar flow 

In the casc of a verlical hot plate, tho pressure in each horizontal plane is equal 

to the gravitational pressure and is thus const+ant. The only cause of motion is 

furnisl~etl by the differcncc? between weight and buoyancy in the gravitational field 

of thr earth. The cquntfion of motion is obtained from eqns. (12.61 a, b, c) with 

tlp/tlx ,: 0 ant1 = 1/11',. Nrglccting frirtional heat we have 

IJrre n 2 k / c, ~ i3 the thrrmal difTusivit,y and 0 -- (7'- T,)/(I', - Z',) is the dimt~t~sionlrss 

lorn1 tc.rnprmttrre In n thcorcticxl invcstigntion conecring the cxperiinrnidly 

drtrrrninrtl trmpcraturc and velocit,y fieltl of a casc involving natural 

convection on a vcrt~cal hot plate, due to E. Schmidt and W. Beckmann [104J, 

E. Pohll~ausen tlrrnorlstrntctl thaL if a strcani function is introduced by putting 

IL - &play ant1 v = -- atppr, then the resulting partial differential equation for y~ can 

be mtlucrcl to an ortlinnry diffcrent,inl rquation by the similarity transformation 

Fig. 12.28. 1'rmlirmt.nro rlistrit111t.ion in tho 

laminar bonnrlary hycr on a hot vcrtiral flat 

plate in natnrni convcolion. Tllroroticnl 

curves, for P - 0.73, nhr J':. I'ol~II~u~~cn 

[!MI and $. Ostrnrh I!)3] 

Fig. 12.24. Velocity distribution in tho I:m- 

inar boundary laycr on n hot vcrtirnl 

flat plate in nataral convection (RCO also 

Fig. 12.23) 

'rhe vc1ocit.y component,s now become 

U = ~ V X ' ~ ~ C V=VCZ-~~~(~C'--~~), 

~ ~ ' ; 

and the tc~tipernt~urc tlistribution is tlct,rrminccl by the function O(7). Equnt.ions 

(12.1 12), (12.1 13) a.ntl (12.1 14) lcatl to Lhc following clifiro~tial cqunt,ionn 

wit.h the bound:~ry contlit.ions 5 = 5' = 0 and 0 = 1 at 77 -- 0 and I;' =- 0, 0 = 0 

at q = 00. I'igures 12.23 and 12.24 illrrstratc the solr~tions of thsc cqunt.ions for 

various values of P. Figures 12.25 antl 12.26 conL:~in a c:omp:rrison I~t,wc:or~ t,I~t: t:dculatccl 

velocily and t.crnpcraturc dist.ribrrtion and those nicnsr~rctl by 11:. Srhmitlt. 

antl W. J3cckmann 1104].'l'he agreement is sccn to be very good. It, is sccn, fnrtl~cr, 

that the velocit,y and tJ1crma1 boundary-laycr thickness arc proporlional to r'lJ. 

Fig. 12.25. Tempcmturo dislribution 

in the laminar botlndnrj laycr 

on n hot vdicnl ht plntn in Imtnml 

convcction in air, mcasnrrtl 

by I

318 XIT. Tl~crnial bonndnry layers in laminar flow 

lcrigtli I anti wit1t.h 0 is Q == 0 / q (~) (Iz, and hence 

0 

1 

'rho mran Nnssrlt nnrnbrr tlcfinctl by Q -- h k Nm(T, - T,) thus l~ecomes N,, = 

= 0.677 c PI4, or, inserting the vnlue of c from cqn. (12.1 14): 

N,, = 0.478 (G)'14 , (12.110) 

is t,hc Grnsliof n~~tnhcr. Tt mn also hr writhn as G = g I"(T,-T,)/v2 in tho 

casc of liquitls. 

'l'lie tli:tgmm in Pig. 12.27 gives a comparison between tlieoreticnl resultR on 

free convcction wiLli measurements on heatad vertical cylinders and flat platcs 

pcrformcd by E. R. G. Eckert and T. W. Jackson [22]. When the product GP < 10R, 

1.11~ flow is laminar, and for GP > 101° the flow is turbulent. The agreement hetween 

theory and experiment is exccllcnt. 

E. Pohlhausen's ralculations have been extended by IT. Schuh [I091 to the case 

of Iwge I'mncttl numbers such as exist in oils. 

The casc of very small Prandtl numbers is treated in a paper by F. M. Sparrow 

ard .T. I,. Gregg 11261. The limiting cascn whcn P + 0 and P -zoo were exambled 

by E. .I. Lo Fcvre 1731, according to whom we may write 

Fig 12.27 Average Nimelt nom- 

brr for frre convcction on vertical 

plntes and cylinderti, aftcr E. R. 

G. 15cknrt and T. W. Jnrknon 1221 

f'urvc (I) lnniinnr . 

Nm - 0 556 (GP)'I'; GP < 10a 

VII~VQ (2) 111rl1111rnt. 

Nm - 0 0210 (GP)'l'; GP > 10' 

n P 4 ' m7 ma W' on mn w* mn 

GxP 

h. Thermal bornidary Iaycrs in nnturnl flow 3 1 $1 

Tnblo 12.6. Coefficients of heat transfer on a heatod vertical plate in nntnrnl convection (laminar), 

according to refs. [XI, 94, 109, 1201 

Some numerical values for intermediate Prandtl numbers are contained in Table 12.6. 

Calrulntions with n hmpemt~~rc-clcpcn(lctitt viscosity were performa11 1)y 'l'. Tllirn 

[50]. The olI'ccL of suction or blowing on tho rate of llrnt Lrnnshr from n vcr1,irnl 

plat,e in naturul cortvrction is tlcscribctf in refs. [29, 1241. Atltlitionnl rlassrs of similar 

solutions in natural {lows were discussed by I

320 

X11. Thrrn~nl bonnclnry layers in Inminsr now 

gradient at (.he surface, i. e. to the local coefficierlt of heat transfer. Figure 12.28 represents 

a Schlieren photogmph taken on n heated vertical flat plnto. The contour of 

the plate is shown by a broken white line. It is easy to recognize on the shadow that 

t,hc boundary-layer thickness increases as d4. The edge of the zone of light sllows 

(,It& the local roaflicicnt of heat bmnsfcr is proportional to z-'I? The picture in 

Fig. 12.2!) gives an intcrferogram for the samc type of bounclary layer; it was obtained 

hy E. R. G. Rckert antl E. Soehngen (13J. 

Fig. 12.29. Intcrferogra~n of a thermal boundnry layer 

or1 n vcrt,ical I~cnted flat platc, nftcr R. 1%. C. Iklzrrl 

nnd E. Soehngen [10] 

Otl~er ttlmpes: 'I'IIC mot,ion (111c to nat,rlral chnvcction sror~n(I a l~orizorlt.:~l IIC:LL(Y~ 

circular ogli~~tlcr was t,rcn.t,cd in an n.na.logous way by R. Ilcrmann 1551. tic 

F~IIII~ for P := 0-7 a Inem 11cat tmnsfcr coefficient N,, = 0.372 G'I~, where G is 

I~asnd on t,l~o clinmcter. blcasc~rcmcnts in air performed by I

322 XII. Tl~or~nnl I)onndnry lnyers in lnminnr flow 

[22] I5ckrrt.. 15.11. (>., anel Jnckson. T.W.: ~\nnlysis of tnrbalent frec convection boundnry lnycr 

on n flnt. plate. XI\(:;\ I

Xll. 'I'l~crn~nl 1)onntlnry Inyrrs in Iarninar flow References 325 

1.e IPur, 13.: Convrction do In chnleur en r6gin1e laminnire dar~s le cas d'un grndient, de pression 

et, d'nnc ten~pi.rnLure tIe pnroi quckponques, Ic lluitle 6t.nnt A propriet6~ physiqurs 

constanteo. In(,. .l. Iqeat Mass 'I'rnnsfer I, G8-80 (10Ii0). 

Lcvi.qtrc, M. I\.: lacs lois do In t.rnn~nlission dc chnlel~r par convection. Ann. Mines 13. 

201 -- 2311 ( 1928). 

1,cvy. S.: Hrnt trnnsfer to const.nnt propert,y lnn~innr bonntinry Inyer flows \\it11 powerfunct.ion 

frro-strmtn vclocit.y and wall t~wpcrat~~trc vnriation. ,I,\S 19, 341 --348 (1952). 

Lieptnnt~n. H.\V.: A uin~ple derivation of l,ightl~ill'n Iwat trnnsfc~ form~~l:~ .Il:M 3, 357.- 

360 (1958). 

Lictzkc. A.F.: l'l~corct.icnl and ex]icrin~rntal invrstigntion of hmt. t,rnnsfrr lip Intninnr 

nntural convcr1.ion bct\r~cen parallel platcs. NAG\ Jlep. 122:) (IO55). 

I,igl~t.l~ill. M. J.: Ch~lribul.ion~ t,o the throry of hat trnnafrr t,llrougl~ n Ix~ninar houndary 

Inyrr. I'ror. Roy. Soc. London A 202, X!)-377 (1850). 

Lorenz. 11.11.: 1)ir \Viirineiibertrngung an einer ebetien sc~~lcrrchten Plntte nn 01 hei nnliirliclirr 

J

326 

X l I. 'l'l~crn~nl hounrlnry lnycrs in Inniinnr flow 

(12131 Spnrrow, 1C.M.. and (:re g, J L Ijent, trnnsfer from n rot,nting disk to fluids of any PrandlI 

nutnbcr. .J. Hd, l'ronskr 8i, i49-251 (l!)s!,). 

(1301 Sparrow, K.M., and Gregg, J.L.: Mnsa t,rnnrrfer, flow, and hent tmnsfer nhout. n rotnting 

disk. .I. Hoat. 'l'rnnsier 82, 21)4-302 (IOOO). 

[131] Squire. 11. 11.: Scct.iori of: Modern I~cvtrloptnmt~ in Fluids Dy~~nn~ics (S. C:old~tein, ctl.), 

Oxford, 11, li23- (727 (l!)RR). 

11921 Squire. 11. I{.: Ilrnt Lrnnsfrr cnlr~ilat~ior~ for ncrofoils. AIEC ItM 1986 (1942). 

(133) Squire, 11.11.: Notc. on the effect of variable wall ten~pcratr~re on Imnt trn~iefer. ARC lthl 

2753 (1!)53). 

[I341 Stewnrt., W. 15., niicl Proher, It.: Hcnt. trnnsfcr and dirnnion in wedge flows with rapid liinnn 

trnndcr. InL. J. llcnt, 'I'rniisfer

328 X I1 1. J,nn~i~~nr bo~~ntl:~ry Inyrrs in comprcusihle flow 

or1)itnl velocity of a satellite of w, = 8 lzmlsec, the temperature rise even in a real 

gas is still of t,l~c order of 10,000 tlcg C. Tho mngc of Mach numbers M, 1 6, in which 

thrrc cxisl, Inrgo cIiKcrrnncs bctwcon tho bchavionr of a real as opposed to a perfect 

,gas, is givrn t.Itt: tmme of hypersoltic flow. 'l'llc occtlrrence of chemical reactions 

(ioniznl.io~l, tlissocint.ion) wl~ich sc:t in behind a shock wave or in t.he Imundary layer 

OII :I. solitl Imly ill :I Itypt?rsonic sl.rt::~m by virt,no of tho cxistcncc: ol:t high f,c:tnj)c:r:~.t,trrc, 

c.otwitlcr;tl)ly c.otnldic:xtcs t,llc I.;~slr of annlyzing t,ho flow. For this reason, wc sl~i~ll 

rt,s(.rit:t, our consitlcrat.ions to that range of Mach numbers in which tho fluid can 

still 1)c :~ssl~mctl t.o obry 1.11~ perfect-gas law; ir~ air, this corrcspontls to a range 

of M.,, -r: 6. In motlcrn t~itncs ni11(:11 nttcnt.ion has I)ccn giver: t,o tllcstntly of l)onntJary- 

Inyrr Ilows at hyprrsonic vclocit.irs ant1 in 1,hc presence of c1iemic;tl reactions. For 

drtails, tllc rrntlrr is rcfcrrctl to t.he hole by 1%'. 11. J)ormnce [20]. 

Fig. 13. I. Tcmpcrnt,uro rise in air in ternis 

ol thc flight velocit.y, w,, and t,lle Mach 

w,Rm/secJ 

n~~nil~cr, M,. 'Vhc curve Iabcllcd "l)erI'ect 

gas" mas calculst~d wit.11 the nit1 of eqns. 

(13.1) nntl (13.2). Thc velocity 111s = 7.0 

km/sec in that of nn nrtificinl satellite in 

orbit,, and lo,< = 11.2 km/scc roprcsenh 

the e-9cnpc ve~ocit~y of n satalIite from 

I 

0 

~ 

6 I2 

I 

8 

I 

24 

I 

36 

I 

42 

, the earth ~ 

Him 

Even in I.llc mngc of snpcrsonic Mach numbers ( M, < 6 in air), the t,cmperature 

rise irt thr gascww stream is high enough to force us to talrc int,o a.ccount the effect 

of t,cmpcrehrc: on the proportics of tJllc gas, in particular, on ils viscosity. The lrinematic 

viscosit,y of most gases, and of air wnong Lhem, incrcascs cor~sitlcrably as the 

t.ernl)craturc is incronsctl. 

In t,llc caso of air, as sltown I)y E. R. van Driest [30], it is possible to use an interpoI:~t.ion 

fbrmuln l):~.sctl on I). M. Sntdlcrland's theory of viscosi1,y. This can be written 

wllrrc /I,, clcnotcs the viscosity at the reference Lrnjperaturc To, and Sl is a constant 

whit:l~ for air assumes the value 

S1=llOK. ' 

l'ltc lm?twli~tg rel:~t.ion I)ct~weon tlle viscosiby /I of air and the temperature, T, is 

scerl plottd a.s curve (I) in Fig. 13.2. Sinco t,hc relation (13.3) is still too complicated, 

it is c:nst.omnry l,o npproxirn:~.l,c i(( in thcorc~t.icnl calculat.ions by tho simpler power law 

where 1,llc constant h sc3rvcs to nchicve a better apl)rosim:tl,ion 1.0 thc more cx:~c% 

Sut,hrrIn.nd formt~la (13.3) in 1.h~ nciglll)onrl~ootl of a tlcsirotl l,rt~ll)c~r:l.l.lr~~c r:111p 

(cf. Scc. XIlTtl). 

Fig. 13.2. The dynamic vis- 

coaity, 11, of air in tcr~ns 

of the temperaturc T 

Curvc(1) ?dras~~rc~ncnls 

and inler- 

pnlalion forlaula (13.3) hased on 

Sntherlnntl's rquntlnn. Ourvcn (2). 

(3). rind (4) pow~r lacs. cqn. 

(13.0, wit11 difirnnt values of 

thc exponent ro

330 X[[1. lmninar bor~ndnry lnyers in compressible flow b. IEclation between the velocity and the temprratl~re firlcls 

The pllonomcna wielcr consitlcrat,ion hccome, naturally, very complicat,ctl 

becnuso of t.110 intcraction hetwccn thc velocity and the thxmal hountlary Inycm. 

Conrparccl wiLh incornprcssil~lc flow there are at least four atldit,ional quantit,ics 

which must I)o tdtcn into account, in tJie calculation of comprcssiblc boundary layers: 

1. thc Mach nilrnbcr 

2. th! l'r:tn(IlJ n~~mbcr 

3. tho viscosity function p (7') 

4. hountlary condition for tcmperatm-c clist,ribut.ion 

(hcal, tmnsfcr or aclia1)atic wall). 

It is clear that tho large numbcr of additional pnmmctms, compared with incomprcssiblc 

flow, causes the numbcr of csscs likely to occur in practice to becomc almost 

int.mctablo as a consequence. 

Comprchcnsivc roviows of tllc numcrous papers concerned with comprcssil)lc 

boundary laycrs were givcn by G. 1Zucrt.i 1571 and 8.11. Young [106]. Details of 

spccial m:~t.hcmat.iral mcthods cmployctl by various nuthors have hccn discussed 

by N. Curle [26) and I

332 XI I I. I,:trni~~:w l)o111111r1ry liiywx in co~nprrssiblr flow c. Thc flnt plntc? nt zero incidcnrc :$:%:I 

of tho heat flow is rlrt,erminctl by the gradient (dT/tlu,), at tho wall. In fact, wo ran 

tlednce from eqn. (13.13) that 

so that, for (tIZ'/tl~,), < 0 t.hrrc is a flow of hcat from tho wall to tho lluitl, :tnd (,on- 

versely, for (dT/tlu), > 0 hat flows from tho fluid to tho wall. In this ~nannor 

Ilwr 7', (a) (ICIIOI,I,S 1.11~ t~crnprm1.11rc :I.(* 1,llc: mtlw rtlgr of the I)onn(l:~.ry Inycr, :III(I 

1,hc- sol~~l.iol~ l)cw)111rs urnz 

T," - T, 3 - or 

2 C, 

ll\v - l', ,Y-1 

- 

7'- 

Irlt.rotl~~c*i~~g 1.11~ Rlacl~ n~lrul)cr M = U/c, whcrc cI2 = (y -- I) cp !ltl we ciu~ rcwrit.r 

r'lm (1 3.12%) in the form 

Heat flux wall ;'_ fluid, valid for P = 1 

Fig. 13.3. Rclationship bct,wcen vclocity and 

temperature clistribution for the compressible 

laminar boundary layer on a Rat plate including 

frictional heat,, from eqn. (13.13) 

Pmndt.l nt~mhrr P = I. TI,, = wall tcmpernlllrr; 

!Ir, = lrw.sLr?a~n t,rn~p~nIurr, lpnr 

$ (Y-1) Ms > (7',,>- Tm)/Tm 

wc h%vc (i?7'/i?!,),,,0,1 > 0, nltd l#wL Is trntt4errcd to the 

rnll owing Lo the inrpo qu:tntity or lwrt eenrrshvl hy 

c. The flat plntc nt zero incitlcl~ce 

2 

Mm2 : 

The boundary layer on a flat plate at zero incitlcnce has been studiccl cxl,er~sively 

in numerous publications, and we propose to begin with n more tlctailctl cliscussiotl 

of this case. First we shn.ll deduce the ralatio~l bctwccl~ tho vclocil,y and tclnpcral~~~ro 

tlist,ribution on a flat plate from the prccctling grnrrnl proposit.ion. 

Tn t,l~c case or an rcrlirhlic wtll (flat,-plat,c tl~ernlornclcr) wo s~tl)stitmt,c! -: '/I,., 

nncl [J == (I*, i~tt,o cqn. (l3.12), SO t,h:tt t,ho t,crn~wr;~t,~~rc (lislxil)~~t,iot~ it1 I,IIO IIOIIIII~:I~.V 

layer on a llat pla,l,c bcconlcs 

and the ntlialmtio mall trmpcrat,nrc, rqrls. (13.128, I)), is 

wl~irh follows with M, = U,/c,, ant1 c,2 = (y - 1) I-,, l', . [t is worth noting 

that the t~mprrnt~urc of a wall in comprcssiblc flow give11 by eqn. (13.17) is itlcrlticnl

334 XIIT. Lnrninnr 1)oundary laycra in nompressihlc flow 

with that for an irlromprcssiblc fluitl from eqn. (12.80) provicled that in thc former 

rase P = I. IT. W. I'Gnmons and J. G. Brained [34) have shown t,hat, it1 the rasc 

of T'mntltl 11ntn11crs which differ from unity the deviations in wall tempcraturr 

caused by comprrssibility effects, as compared with the incon~pressihlc cqt~ation 

(12 SO), arc- only very slight2. TIIIIS Ihc atIinbat,ic-\vnll trmpcmtnre cqnatiorl 

remains vnlitl for rompressil~lo flows with a vrry gootl tlcgrrc of npproxirn:lfio~l 

For nir, with y -- 1.4 :mi P -- 0.7 1, wt. ol)t,nill 

Thv rrs~~lling tlrprntlrnce of thr ntlinbatic-wall trmpcraturc on thc Mar11 nun~ber has 

Iwrn rrl~rrsrr~trti gmpl~icwlly by tho plot in Pig. 13.4. For rxamplc, at, a Mnrli 

nl~nil)rr M,, -= I thr wall 1)ccornrs Ilc~atrtl by 4.5O C (or 80° F) in roirnrl fig~rrrs. 

A[. M,, - - 3, t llc tr~nprratl~rr inorc:rso brc~ornca ns l~igll as 400' C (or 720° P), ;i~ltl 

nl M,, = 5, it is as rnlrrll as 1200° C (or 2200° 1'). 

c. Tllc flnt plntc at. zrro incitlrnct! 

The recovery /actor, r, then represcths tl~c ratio of the frictional lnmpcraturr inc.rcnsr 

of tllc plntc, (T, - T,), to that due to adiabat,ic con~prrssion, 

urnz 

AT, = -- - > 

2 c,, 

from cqn. (12.14). 011 cornprrirtg rqns. (13.lH) ant1 (13.19) il. is scwl l.I~:,t, t,l~o mcovc:r,y 

factor has the val~rc 

Ilcncc: for air 

:wi 

r = dF- (Inminnr) , (lXl!):~) 

r -- d0.71 = 0.84 (Inniinnr) . (l:!.I!)l)) 

Fig. 13.5. Mrnunrrrl rrcovrry Z't 10 IZ 

factors, r, for laminar boundary 

layen on conra nt sqwsonic 

veloriticq lor difkrcnt 

Mnrh n11111brm nnd Ilrynoltln 

v %I0 

A 60' 

0 Boo 

br? TO 33 

019 to 25 

%I lo 18 

numbcra, al1r.r C. 11. I ~ rJ2]; Y r 

ronipnrison \\ it11 theorel icnl 

vnlncs lrotn rqn. (I 3.19%) 

'.rhc diagrams in Fig. 13.5 rcprcscnt the rcsults of ~nnn.suronirnls on t,llc rccovcry 

factor in the cast of laminar boundary layers on conos in supcrsonie strca.rns, porformed 

by G.R. Eber !32J. The numcrirnl valnc r = is swn to be ~onfi~~n~t:~I 

lly t,llcse men.surernents. Similar results follow from ~nc:~surcnlcr~l,s p(di)rtllt:~l on 

various cones and a paraboloid pcrformcd by B. dcs Clcrs ant1 J. St,ert~l~crg 1271 :d 

It. Scl~crrcr [89]. 

Velneity nncl tctnpernturc distributions in thc nbnetlce nl lwnt trmslcr: 'I'wo 

ppt:rs by W. Ilant.zscht? ant1 11. Wcntlt. [44, 461 ant1 :L p:l.prr by 1,. (Irorxo 121 1 

coll(,nill cxplici(, formll]ac for the c:rlc~rln.l.ion or Ll~c vvle~cil~y :1,11tl I,c:ln~)c~l~:li.llt~t* tlish'i- 

~)llt,iorl ill Illlmbcr of spceific cnscs. J'ignrc 13.0 conI.:iilw 11lot,s of lSllt: vtdor.il,y rlisl.l.ihlt.ion 

in t,lle l~ol~ntlary 1;~ycr for sovcral M ;~I numlmrs. It, reprrst!~lt,s Cl-oc:c:o's c:~l(:i~- 

Intions for a boulldary hycr on an arlirrhcttic /kt plnlc on tho :~,ssnn~l)l ion of ;I. vise-11sil.y 

law = 1 and for P = 1. The distance, y, from the wall has 11cc11 rnntle (litncnsiolllcss 

\vit,ll rcfcrcnco 1.0 ~GTu, where 11, tlrnotcs 1,lle Itinrrn:rl,ic visc80sit,y in 

t,llo oxt,crrln,l flow. It is seen l,Il:~t for incrcasirlg I\l:rcl~ tl~tml)rw l,llc:ro is :I. c:r~l~sitl~:~.:~.l)le: 

t~lickcrlillg of tllo Ilollntln.ry laycr and tll;~I, for very ~:LI.~I: hl:11.11 n11t11111.t.s (.)I(: v~.lor,i(y 

clist,rihllt,ion is approximately lincar over ib W~IO~C thicloless.

,, 

Ilie t.c!rnprmtt~rc tlist,rib~~tion is also shown in Fig. 13.6, and it is seen that 

tlic fric:l.io~~nl ir~crcasc in thc tcmpcrature in tho boundary layer assi~mes large 

valurs for Iargc Mach nwnlms. 'Clic pnpcr by W. I~antzsc:l~e and JI. Wcnclt [44], quotccl 

cnrlicr, contnins calc~ilat,io~~s for P = 0.7 (air) for the case of a Iicat-conducting plat,e. 

It is SII~~VII t.l~:~t, the velocity tlist,ril)~~lion u/rJ, plot,tcd in tcrms of y 1/ U,/Z i," 

drviatcs c:onsitlc:r:~l~ly Srom thnt for P -= 1 when 1.11~ Mnclr nurnlm- :i.ss~~nics largcr 

v:rlrlrs. The vchc.it.y . lwofilrs shown in Fig. 13.G can bc mstlc ncarlv t.o coincitlc 

A 

. . . . . . 

when thc dist,a~~cc from t h wall, y, is matlo tlinicnsionlcss with rrfcrcncc to l/a,,, z/cI,, 

Fig. 13.7, wl~c:rc v,, tltnotcs the Irinomat~ic viscosity of the air at the wall. This 

circn~nstancc dcnot,cs pl~psicnlly that t,llc incrcasc ill I~oundary-laycr tl~iclrncss with 

Rlach number (at constant Rcynoltls number) is mainly due to the increase in 

volnmc which is nssociated with thc incrcasc in tlrc temperature of the air ncar the 

wall. This fact was first noticcd by A. N. Tifford [98]. 

Jiig. 13.6. Vvl~wit,,~ lind t~cn~prr:~t,~irc (Iistx- 

1r11tiori in (;o11111rc,%qil1lc:, 1a111iri:ir l)o~~ncl:~ry 

layrr on adinlrnlic flat plate, nfkr Crocco 

P1.1 

l'rx~idil I I I I I ~ P ~ - ~ I , m = i, y = 1.4. 1)istanr~ 

rrwn wnll rrfcrrvd lo I-, ~111 I' 

In this method of plotling. lhc curvrs fcsr 

dillcrcnt Marh nunlbcrn hnvc lwcn rnrcle 

nearly lo coincide. It is possible to conclude 

from ll~in that tl~c Iargc incrcnsc in llw bovndary-layer 

tl~lcknclis will1 nlnch n~~wbcr is 

mainly duo to tl~c incrcnac in volomc shlrl~ is 

associnted with tho increase in tcmpcrat~~rc of 

tllc air ncar tllc wall 

7 

Jqig. 13.7. Vc-looit,y lint^ il~~~l~ion~ iu t l ~o 

lr~ininnr I~ounclnry layor on an acliabatic 

Rat plate at zero incidence; data 

identical with those in Fig. 13.6. The 

dist~nce from tho wall is referred to 

I/v,,, Z/U;. For w = 1 , we have 

1/ II~,,/V~ = T,/Tm 

Jiig. 13.8. (hllicient oldtin frictio~i on dia- 

Oolic flat plate with rornprcnniblc, Ian~innr 

i)ot~ncl;~ry 

ll:~nt~.sd~c :111ql \f1c!n~1ta [44] 

a. 'Lh: flat plnto nt nrro inriclci~cc: 337 

Fig. IX!). Corfficic~~t, olsliin lric,t,io~i for ediabrrlie 

11:1(. plaLc at zero inoiclrnrt: with coin- 

layer. P =. I, ), = 1.4 (air), nfl,cr prrssil~lr, laminar borlndnry 1:1yc.r, : ~ h r 

IHY] 

It111wsi11 1inc1 .1111111son 

Adinbntic coefficient of skin friction: 'I'llc rocSfic.ic:i~l, of skill f~.ic:t.ioi~ Sor :II~ 

adiabatic wall, as cnlculnt.ct1 by W. Ilnntzschc nntl 11. \Vr~~tlt, 11as 1)crn plot.t,ctl in 

tcrms of tho Mach nnmbcr in Pig. 13.8. For co -.. 1 t,hc protluct c, R is intlcpr~ltlcnt 

of thc Mach number, but Sor tlifircnL val~~cs of rr) t-l~c c:ocKicicwt of slii~~ S~.ic.l.io~~ 

decrcascs with increasing Mach nurnbcr, the ratc of ~Iccrtasc I~cing largcr for srn:dlcr 

va111es of o. Figurc 13.9 contains a comparison bctwccn Lhc valucs of tllc cocl'ficicnt 

of skin friction for an adiabatic flat plat,^ obtai~~ctl by scvcm.l aut,l~ors. i. e. for 

different valucs of t,hc Pmntltl numbcr, P, ant1 of the cxponcnt in t.1~ viscosity 

Fig. 13.10. Bfeasl~ren~mtrr of t,he 

velocity diatrihution in nnadiabulic, 

Inminnr I~oi~n~lriry 

Inyrr in nl1pc.r- 

sonic Llow, al'lcr 11. M. O'l)ont~olI 

[28]. Mach number Mm = 2.4. 

Theory from ref. [I31

338 XI I I. Im~linnr I>orirtclt~ry Inytm in con~prcnnil)lr flow 

funrtiot~ 'I'hr plot sltows t11:11 the I'rantlfl numbrr exert*^ a much smaller influcncr 

on the rorffic.ienlf of skin friction than the cxponcnt to. 

Vrlority nid trniprrnlr~rc tlistril~~~lianq in lhr prcsrnrr of lwnt 1rn1tnft.r: In ~ I I V 

grnrml ca:lqr, 1111th hrrrl trtr?i+r p~rsrnl, thr rrhtiort I)ctwcen tho vclocity and t,oml)cratt~rr 

tlist,ril)ution (mi I)r tlctlucrtl from rqn. (13 13a). Wl~eri P = 1, it can br 

wril t t.11 

whtw 'I1,, is givc.11 I)y rqn. (13.17). '1'11r prcccding rtluation can bc cxt.c~rltlctl 1,o 

I'r:~ntltl 1111n1l)t~rs tlifYrring froni 1111it~y I)y the int~rt~tlucl~ioti of trhc rcrovrry factor, 

whctl wc: ol)tn.in 

In 1.llis cq~rntion, thr acli:~.l)n.t,ic wn.11 tcniprmt,~~re, l',,, shoultl be ~%lcuhted from eqn. 

(13.18). 1)11t, il, ni~~st, IIC rr;clizrtl 1.l1n.t~ this is only an approximation. Thn direction 

in which 11(-:1.1, is tr:lnsft:rrcxl c:i.n I)c tlctlncotl from eqn. (13.21) n.nd written 

Since for tt) :- 1 1,111: corfficicnt of slzin fricfion is intlcpcntlcrlt of the Mach 

tlumlw.r (IG% 1:3.8), thc r:cl.c :el, which 11td is tr:~nsk~rrt:tl brcornrs equal t,o that in 

nil inc:omprcssil~lt: strmrn. cvlri. (12.81). A survry of lirnt.-l.r:lt~sfc-r cocffirirtlt,s ant1 

rct.ovc:ry Ij~t.tt~rs liw I:IIII~II:I,~ :1.11(1 t~ur1~11~11~ IIow :I.I. I~igl~ Mi1t41 IIIII~I~)(:I.S t:wt I)e (o1111(1 

ill :I ~mpr I)y .I. IC:tyo 1551. In t,l~is rpr~nc!xion rrf. (1051 may also I)(% mcrit,ionetl. 

Fig. 13.11. Vcloci1.y and tc?mpcroturc 

clistrihrtt.ion in oorn~)maqihlo Iiitninnr 

bountlnry lnyer on flat plate nL zcro 

incidence with hrnt tranrler, aftor 

IIantzaclre anti Wendt [44] 

Wall tempernlllrf! free rtrrrm letllpernt~ltre. 

T,. - T,: P - 0.7. = 1 ; y = 1-4 

Calculations conccrnjng compressible boundary layers on flat plntcs which are 

based on the momentturn-inhgml eqnat,ion (Chap. X) have been pcrfornictl by 

Th.. vorl ICBrmrin and 11. S. Tsie~l [S]; see also Pig. 13.9. Approxiniate solutions 

for the flat plate were also published by F. Ro~~niol and 15. A. Ric!l~cll)rrtlricr (71, 

1). Colrs [I 71, 1,. Crocco 1221 ant1 11. .I. Monngltrcn 1751. S~l~~tions for 1,110 tyu~lt.iow 

of lnmir~nr I)ountlary lrrycr~l with vnrialh pro1)crtiw WC~C givt:~~ l)y I,. I,. Moort: (77 1 

and G. B. W. Young and E. Janssen [1081.

340 XI 11. Iml~innr 1)oundnry Inycm in co~nprcwil>lo flow (1. noundxry layrr with non-zero prrRunrr grndirnt 3-1 1 

d. Ilor~ritlnry layer with no~~-zero pressure grnclient 

1. Exnrt solutions. The ralculaiions conccrning boundary layers with non-zrro 

pressure gratlicnt s are more difficult I Imn t hose concerning flat plates, owing to 

thr Iargr nrlmbrr of intlrprntlrnt variables 1,. Crorco [21] tlisrovered quite early 

a t~mnsformat.io~t which simldifics the task of int,egra.ting the equations for tl~e 

cases when rithrr (1) P = 1, ant1 t.hc viscosil,y function /A(?") is arbit,rary, or (2) whcn 

thn 1'mntll.l numlm has an arltitrary value but, p/T' = const (i.e. when w .I 1). 

111 1.11~ spcri:rl cases of ;MI atlinl~nt~ic wnll wi1.h P -- 1 and to = 1, I,. Ilowarth [481, 

C. It. Illingwort,h 1701 a.~~tl I

342 XI11. Tmninnr Imlndary layeru in rotilprenaiblc flow 

The viscous trrm in tho cquntion of motion can bc transforrncrl with thc aid 

of cqn. (13.4%) nntl 1,Itc pcrfect-gns law p == p, = p R T to yield 

or. by i~~lrotluning t h tli~ncrmsionless Lcmprrxtt~rc f~lrlction (rrlat,ivc: stngnal.iollcr~tl~nlpy 

diffcrcncc), tlcfincd by 

Jlt-rc IL tlwwtrs tlw local, ns tlistil~ot from thc stng~~nlion r~~t~llalpy. Introducing llmc 

This tmnsfort~~cd cquatior~ cliffc~cw from the corrcspondi~~g I~oundnry-hyrr equation 

of irmcornpressiblc flow merely by thc factor (I -1-8) wl~ich rnultiplics tlmc prcssuro 

trrnm. 

111 order to transform the energy equntion, we multiply rqn. (13 0) 11y ?L ard 

add ccp. (13.7) Rc~ncmbrring that tl~c Prandtl nun~bcr is 

As WXR clonc! for oqn. (13.28), we cxpross the pnrt,i:d tlorivativc:s with rrspcct to z 

cxprc&ot~ ill rqns. (14 XI), (I3 34). nt~d (13.36) into rqn. (13.0) imngincrl diviclrtl nnd 21 by tllosc wi+h res1)cet to j: nrmtl $, lmtc that .- h 11, p, p,/ptl p :&I makc 

use of the definitions (13.40) to obtain 

by Q, we tlcrivc : 

Ilrrr, M, -- II~/C, is tJ~c Mnrh nurnl)cr of t,lic cst,rrn;ml flow. Sitit:(: 

thc fact.or of (li)~~)~ in cqn. (13.46) can 11c put in fror~l, of Llle operator P/r??y2 in 

cqn. (13.44), so t,I~nt tltc trnrlsformcd cncrgy ccjlmt.io~~ :~c:qt~ircs t,Ile form :

344 X111. Imninnr bo1ttrr1:~ry Inyrrs ill comprcsrriblc flow 

Eqr~nt.ions (13.41) ant1 (13.47) t.ogcthcr with the continuity equation 

wlticl~ is n tlirwl, consrqncnao ofotjn. (I:!:lO), now (:ot~sI.il~~ll,o L11c IIOW ~otoFI~o~~ntlar~- 

I:lyc:r t:clr~:~l,iorts. 

The syst.cm of equations (13.6), (13.G), (13.7) was subject to the bountlary 

contli thns 

thc latt,cr dopentling on whothcr tho wall is adiabatic or isot,hcrmal, t.ogether with 

It is easy to sco that these 11011ntl:wy cor~tlit.ions t.mnsforrn as follows: 

I/irnilirtq crrsrs: If P := 1 Iltrn S - 0 is a spccid solut,ion of the cr1crg.y cquaI.ion 

(13.47). 'I'ogcl~l~rr wil,l~ rqn. (13.:%0), it. Ict~tls to t.lta rcl:~~tion between t,cmperat,l~ro and 

volorit,y for nn ntli:~li:~.l.ic \vall tlisrovcratl r:~rlirr as cqn. (13.12). In this case, cqn. 

(I 3.11 ) assumrs 1,hc "incotnprrssil)lc" form of rqn. (9. I) rnnctly. 

t.r:~t~sf(>r~~~nt,ior~ 11a.s l~r(~n 

usrtl t.o rlvrivc cs:wt, sol~tf.ions nntl l,o formulntc n I:~.rgc nntnl)cr of al~proxin~al,c 

proac~tlnrt~s. Srlf-sirniln.r solrttions piny an important, part, wit,l~in t>he class of exact, 

solnt~ions. 111 tlrc sonLrxL of incomprrssil)lc Ilows, we consitlerctl that n solntiot~ 1)nlongvtl 

to this group if I.lm vchcit.y 1)rolilrs 11 (R:, y) atf two clill'crcr~t st,at,ions n: cor~ltl IN: 

n~:~.tlc t~ongrncnl. by I.hr npplicat,ion of :t singlc scale l;l.t:t,or ritc:I~ for IL and y (Scc. VI 1 I 1)). 

It was t.11c.n sht)wn t.l~nt surh sin~ilnr solut,ions existrsd in t,l~~ prosrnco of a dcfinit.e 

gronp of t~slnrnnl Ilows II,(:I:). In cnscs of t,l~is Itir~tl. I,ltr pnrli:~l tliffrrant.inl orlr~ntion 

for t.Iw st rmtn frlttc~l~ion rrtl~trctl t.o nn ortlinary ililrrrrnt.i:ll aqrtnt.ion wlrich is consitlrr:~l~ly 

r:~sirr lo solve I,II:III 111s I'orn~rr. 

I 

Rl:~liillg IISC of :L 111ln11wr of sl.t~(lit~s, l'nr (:x:~tnl)lt: 148, 40, 50J, 'I.'. Y. I i nnct 11. 'l'. 

N~~:IIII:I(.SII [(XI. 611 (I~~IIIOIIS~.~;L~~~:(I in a n~tlnl~cr of pr:liscwort,l~y invrst,igntions t.l~at. 

S I I ~ I sin~il:rr sol~tl.ions rxist it1 1.11~ c-nsr of comprossiitlc bo~tntl:rry layrrs ns wcll. 

As liw as 1I1c vrlorit.y Ito~tr~tl;~ry hyrr is cor~ccrr~rcl, l~crc t.oo, ~irnilarit~y rsl.cntls t,o 

1.11~ longil.~ttlirt:~l vrlovily rotnponrnt~, 76: wi1.h rcsl~oct to the t,l~armal la.ycr, similarity 

1.2. Srlf-similnr SOI~I~~OIIU. 'I'ltt- Illi~~g~vtt~~tl~-Sl~t~~v:~rl~so~t 

Sin~iln.r solr~I.ions for comprnssil~lo I~onntl:r.ry l:~..yc:rs twnsl,it.~tlo rx:cct. sol~rl.ions 

of t31to sysI,t~trr 01' w~~t:r~I.ions w~tl :trc, l,I~t>t~~l'orc~, ittlxi~tsit~:rIl~y vt*r.y itrt~~orh~rl.. l'vrI~:ri)s 

cvcn ~norc itnport;lntIy, solutions of this Itintl art: cntployctl as t.011c41st.ont~s :~g:r.inst, 

whir11 t.hc n.ronmay of n.pproximal.c prorrtl~tras ran 11a j~itlgotl. For l.l~rsc: rrnsons. 

we 11ow proposo roughly to slteld~ the line of rcnsoning which It::uls l,o sin~ihr solut.iot~s 

starl.ing wit,l~ the lllingwortl~-SLc~vart~so~~ t,mnsforn~nt,im. We sh:tll ror~c:lutls t,l~i.s 

topic with n number of r~~lrncricnl results. Wc shall postulate tl~c valitlilsy of t.l~e 

viscosit,y Inw from eqn. (13.4n) so that ro = I ant1 P - I n.ro implic:tl. In t,l~o t::~sc of 

bountlary layers wilh herrt lrn?is/er, an nrl~ilr:~ry, 1,111, c:onst,:lnt, w:~ll I~:III~)(~~:I.~.III.~:, 

II',,, will be assnrncd, so that A', will 11ccornc a oonst:~nf,. In prol)lwns i~tvolving ;HI 

nrlirrbnlic wall, t,ltc stagnation ent,l~nlpy is given by cqn. (13.12): 

ant1 rentains cot~st,nnt over t.hc 1)ountlnry-lnycr t.l~ic~ltt~c~ss, itnplying S : 7 0 (c/. also 

end of preceding section). In this cnsc, the sirnila.ril,y of tho st,agl~:~(,iort-cr~tl~:~l~ty 

prof les assumes a trivial form. 

Employing the stream funct.ion 111, we rrwrik cclns. (13.41) iltttl (14.-t'i) in f,ltr 

form : 

Thr similarity vnriablr is int,rotluccd wit11 fl~c aid of IIIC following assttrnl~t,io~ts. 

where A, 11, r, s, t pl:~y the part,s of rtntlcl.rrrninotl COIIS~.:III~.S, 

stream funct,ion, ant1 S(q) is the tcrnpcr:tt,~trc S~lnct,ion tlclinctl in oqn. (13.35), now 

cor~ccivcd t,o be n functi& of 77 alone. 

1Squat.ions (13.50) and (13.51) arc now bmnsfomlcd to tlto coortlin:~tcs 2 an.ntl 17, 

and in the result.ing cxprcssiorls it is clcmantlctl that t,ltc terms in 3 must tlisapprar. 

In this manner we obtain ordinary tliffcrcnt.inl cqrinl,ions for t.110 fnncl.ions /(q) :r.nti 

S(17). Snch c:~lcnl:~t,ions 11:~vc 11ecn pcrforn~ncl by 'I'. Y. I,i :wcl 11. 'I'. N:~~irrnir.t.su JOOJ 

who found that there exist,ctl four clnsscs of solut,ions for 7i., (Z). I~ollowing this work, 

C. U. Colten [I61 dcmonst~rht,cd that t,hree of t.11csc classr,~ can be rcd~lcctl to thc 

/(?I) is an II~I~~IIOIVII

346 

XI 11. T,ntninar I~or~nrjary hycrn in cornpressihlo now 

(Ic' nntl VL arc c:onst,:~rlls). '1'111, fourth case 

?it 7-- I

XI 11. I,nrninnr bo~~ntl:~ry Inycrs in rornprrsnil~lc flow 

and wrik tlo\vn (IIP t.ransformcrl l)o~~ntlnry-layer rquations (13.50) and (13.51) in 

tlrr form of tl~r following two ordinary tliiTercntial eqr~at~ions: 

in wl~ich primrs tlrrrot,c tliKcrcnliat,ion with respect to 11. 'rhc pnratnetcr 8, in the samr 

way r7.s oarlicr in cqn. (9.7), is tlcfinctl by 

it, rlr:lr:~vIrrixt.s tl~r prrssltrc gmtlicnt of t,lrc cxtcrnal stream. 

we corrclr~clo wit,h tJrc aitl ofeqn. (13.60) tl~:~t J' constitntcs a tlimcrisionloss form of tlrr 

1ongit.11tlinal vrlorit,y component in t . 1 ~ bo~~ntl:~ry layer, because 

Sinrr ?/ -- 0, or ?/ --r oo implirs 71 = 0 arid 71 = ro, rrspect,ivcly, the boundary contlitions 

for t11t- systrm (13.til) rnr~st he wril,t,rn 

In the case of an crditrhrrtic loall, the srcond equation (10.01) is sn.t,isfictl itlcnticnll~~, 

:r~itl it. is ncrcssnry t,o solve t,he singlc c.qn;~t,ion 

/"' -1- //" = P(/'" 1) , 

\\~11rn IIIC \vnll pcrrnits thr /rrrirn/pr o/ hccrl, it brcomcs necessary to solvr thr 

system of rquntions (13 01). Sincr thr wnll tcmperatnre, T,, can he prescribed in an 

arbitrary manner, it will be fol~ntl that thc solutions depcntl on the paratnrter 

in addition to their dcpcndcncc on P. Solutions for a largr nrmbcr of valws of thsc 

two par;~mrt,rrs have brcn workcd out by 'l'. Y. Ii tintl 11. '1'. Naf;nmatsrl [(ill ris well 

as by C. J3. C!ohen and E. Reslrotko [16n]. 

Pig. 13.13. Vrlocity arid rntl~nlpy dislril~~~lions in 

ro~nprrssiblr, 1:trninar honntlnry lngrrn with prwsuro 

grndirnt, and hrnt tr:lnsl'rr, nftcr C. 13. Colrrrr and 

E. hshotko [lGn]. and in conforrr~ity with cqtls. (13.62) 

and (13.35) 

I'rn~~(l tl N111n1)cr: P -- 1 ; OI - 1 . rJ(x) = (z) tlcnolm 

t.111, vdor.il.,y of I.lm t:sl.cr~tr~l flow. n), I,), c) vc:locily 

distsil)ut,ions; tl), e) cnthlpy dint,rihrttions; n) IS,,, = 0; 

T,,, = 7'0; ntlinlmtic wnll; h), d) 8, = -0.8; T,, - 0.2 

7'" cooled wll; c), e) S,, 1.0; T,, = 2 TO; hcnt.ctl \vall

350 XIII. 1,mninnr honnclxry lnycrn in comprrrwihlo flow d. l%onnclsry lnycr with non-zero prrnwlro grndicnf. 35 1 

It is wort,hy of notc that the system (13.61) subject to thc boundary conrlitions 

(13.63) yirlcls t,wo pllysically sonsil)lc solutions wl~cn Jl < 0 (this is also lrue in the 

cnsc of an atli:lbnt,io wnll. c/. Sca. IXn). Aceorcling to the vicws expressccl by C. B. 

Cohen and 15. Itrshotlzo [lea], Lhc one ofthe two solutions which scls in in an expcriment 

is clctnrnlinrtl by t.llc initsid ~ondit~ions which cstnblish t,he prcssrrrc field acting 

1iig1trc.s l:%.l:~~l, c rvl~rwct~l, 1.11~ 

,Y, in lht: 11o111t~l:~r.y I:~,yc:r 

ct~thlpy dis~d~~~~ior~, 

it1 

accorcJn.nc-o with cqn. (1:!.:!5) for '/I,, - 0.2 7',, nntl 7', 2 'I1,, rcspcc:t.ivcly. It is 

socn tllnt, t,hc: prrssuro gr:l,clicvlt, nxnrl~ n consit1or:~l~ly stronger infIrlc:nna on the vr1ociI.y 

prolilvs ~.II:I,II 011 lhc: (:t~l~I~:t.l~~y prolilvs. 

Tlw figures c:orll,:ai~l I)lols of /"(71) for clifli:rrnt val~rc:s of tllo pnramcltrrs fl and 

Whrn t,l~c cxt,c:rnn.l flow is nc:cclnrat.ecl (/I >O), thc lnrgest. shearing slrcss occurs at, 

tho wall iLsdf (1, - - 0); wlwn the flow is tlccclern.t,ctl (P (O), t,his rnaxim~rm niovcf 

away from the w:dl awl plnccs itself furlhcr from it ,as t,hc pressure rise is increased, 

tllnl, is for 1:wgc:r nl)solutc v:~lucs of the: ncgntivc val~ro of Jl. lr~trotlrlcing the k ~al 

skin-frirtion c:orffic:innt, 

C, =- 

b e,,, 16, 

lG6. 13.14. l)ixlvilml,iw~ of dwnri~~g nl~rrw~:n i11 ~ I I I I - 

prmsildo, I:tn~innr h~nclnry lnyorn wiLh prrnwrc 

gradient and heat tmnsfer, after C. 1%. (hhen nntl 

E. Rwhotko [IBa], and in conformity with eqns. (13.64) 

I'rn~tcll.l n~~lnl~crr P -- I : ro -- 1 

a: A', Y- 0; 'I*,,, 7 T,,; 1~1Ii1ilm1,io w~ll. 

b: S,

352 

X'I I I. I,:ur~i~~ar Imt~t~clary hyrrs in corr~p~r~riil~lr flow 

l h vn111c.s of/,,," lor tlifTornnt vn.lrlcs of A', are seen plol,tcd in t,crms of P in Fig. 13.15. 

It is rcvognizrtl t,llot n chnngo ill pmssrrrn grntlicnt cxcrt,s n n~uch sbronger it~flucncc 

on I,,,", nntl 11c:llc:c: on the sho:~ring stxoss n.t the w:~.ll, when t,l~e wall is I~cntocl (Xu, >0) 

1,11:111 WII~II t l ~c I:~l,l,c,r is aoolrtl (A', ( 0). In t,I~e mngo of ncgat,ive vnluos of P there 

vxisl, l,\vo v:~l~ir.s of r,,, fnr (YI(:II v:l.ltlo of /I. 'l'his is n consoqllonoo of t.ho rxistcnco of 

Lwo soI~~l,iot~s in l,l~is r;tngc, ns n~r~~l,ion(xI wrlivr. WIICII l,l~o w:tll is n,(li:~l)n,l,io (AS,,, ==O), 

I,II(: low(-r I )~;IIIC~I of' t,l~(: CII~VC yi(,ltls 11cp1,ivc V:I~II(~S of sltcnring s1,rcss wl~id~ j)oinLs 

t,o rtwcrsc Ilow. When t,hc wall is hcnt,cd (A',, > 0) il, is possible to find sufficient,ly small 

valrtc~s of p -- P,,,,,, for w11ic:h I)ot,l~ vxlncs of I," arc ncgnt,ivc, that is for wl~ioh the 

flow 11:~ rcvorsc~tl it,s dirccLion. 1 n the c:~so of :I coolctl wall (8, < O), 11oIh valrles of 

I,," ran Im positive, 1.11al. is 1)0(,11 can r~prcscnt~ non-separated flow patrlmns. It is 

swn, fin:~.lly, Ifhat, snp:~m.t.ior~ (I,,," =O) rnovcs in the tlirect,ion of smnllcr pressure 

risw :IS t11c l.cmprr:~.l,~~rn of 1,Iw wall is incrrn.sctl. 

III ortlvr t,o t,r:~nsform fro111 1,11o vn.ri:~l)lc 11 f,o the pl~ydoal t1isLnnc:c y, it is ncccssnry 

1.0 111 ili.xv vqns. (I :1.H), (1 3. IO), (1:1.24), (I R.25) nntl (13.62). It is then found tht 

Y'ho f:rct,or nl~oatl of t,hc ir~togml is comrrrlt,cd from cqn. (13.53), and the func1,ionnl 

rcl:tI.ion b(:t,woon z nntl 2 IIIIIS~, bc tdic11 from cqn. (13.5G). According to eqns. (13.46) 

:LII(~ (l3,(;2), I,IIc int,~gr:bncl is 

I

354 XI 11. 1,nminnr h~nrlnry 1:ryrm in rornprranil~lc flow 

we can rewrite t,hc nncrgy c-quation (13.7) in tho form: 

The borintlary contlit.ions arc 

a) with hcat t,c~r~sfw: 

h) for an aclinl)nt.ic w:dl 

cla 

Rtpiations (13.6), (IR.fi), (13.8) and (13.71) togclhcr with thc boundary conrlitions 

(13.72) constitute a systcrn of four equations for the variables u, v, e and A. The 

pressure p(z) is known from Bernoulli's cquation and is given by eqn. (13.9); it 

remains constant over the thickricss of thc bounilnry Iaycr, i. c. ap/atJ = 0. Since 

the prcssure remains constant across the layer, wc havc at every point 

where h,], TI, el tlcnotc tho vnlurs of cnthalpy, ternpcrature, and dcnsity, respectivcly, 

at tho orttcr ctlge of the boundary laycr. 

We now introduce a displaccmcnt thickness, a momentum thickness and an 

energy-dissipxtion t,hickness in the samc way .M in incompressible flow and sevcral 

adtlit.ional quantities clcfi nccl with thc aid of cnthalpy. In this connexion the formcr 

paranirtcrs arc so dcfincd as to reducc to the respective quantities for incomprcssiblc 

flow, cqns. (8.30). (8.31), arid (8.34), whcn p = const is sobstituted in the definitions. 

Ihioting Lhc bo~~nclnry-Iaycr thicknem of the velocity laycr by d, we int,rodiicc the 

dc:finitions : 

dl : J 

A 

- P-'! ) (ly (displnccment thiclrness) , 

el (1 

0 

0 - - [ Yl (I - ) I (rnomontum tliickncss) , 

6 

U 

(1. Tlountlnry lnycr wit,h non-acro prrssrtro grnclirnt 355 

(~elocit~y t,I~iclzncss) . (13.78) 

It is easy t.o vcrify from cqns (13.73), (I3.74), (13.77) a.ritl (13.78) t,l~nt t.hc paranvhrs 

nl, b,, ant1 d,, satisfy tlic rclntion 

lnkgr:1ling t,hc tnon~cnt,~~n~ c:q~~nIhi (13.0) ant1 tPhc cwcrgy tyitn.I.ie)n (13.71) 

ovcr y, in the snmc way ILR was (lono for in~o~nprcs~il)lc flow, wo t::~n oI~l.:~i~i l,ltc 1110- 

~nrnt.~rt~~-ititcgr;~l :mcl energy-integral eqr~:ktion for c:ouiprcssil~lc flow. 'J'aiti~~g int,o 

nccount. that 

I ~ (Ix ~ -u - dz ~ e . 8, ~ ( ~ + _ ~ iLv ~ w ~ ~ ) = (lxRO) ~ : ~ ~ 

The equation for n~cc/mnicn.l encrqy is olI.airicc1 by first multiplying cqn. (1B.G) by t,hc 

vrlocity component TL ant1 then intcgrnting with respect to y. Making use of thc 

continuity cqi~ntion and performing a ni~nibcr of sirnplificnt,ions, we obtain 

On t.lic left-hand sidc of this cqi~ntiori wo tliscovcr t,lic mcc.l~anicnl work of t.hc flow, 

tho trrm on tho right-hand sitlc rcprescnt,ing the tlissip:~tion. 111 incornprc~~il~l~ flo~, 

the sccoricl term on thc Icft-liarid side vanishes bccaiisc then, with e r= const*, wc linrl 

that a,, = 0. As n residt, eqn. (13.81) tmnsforms irito cqn. (8.35). 

Thc cquation for thc increase in cnlhnlpy - Iia1)itnnlly lrrlown as tho energy 

equation for short - is obtninctl as a rcsult of tlic intcgr:~tion of eqn. (13 71) over y. 

Thus 

6 

d tl u 

(el hi U a,,) -/- el Uz-&-. 8,, - - - -- ----clz 

( ) - ( I . (13.82) 

Thc loft-h.zntl sidc of this cqunl~ion reprrscnt.~ the OIIRII~C in f,hc rnt.linll)y of tho 

strcnrn, whcrcas 1.11e tmms on t.l~c right,-hnntl sitlc clcscrilw it,s cll:~ngc-s tlt~t? In ~IIO 

tr:~nsl't:r ol' llt!at, :kt, Ihc w11,11 (sul~scrjpt, W) :~ntl Lo ilhv.gt-~~t:r:~l,io~~ 1Jtre111gh cIissi~j:~l,io~t. 

Noting that cqn. (13.81) describes the loss in meclixn~cnl energy, whereas cqn. (13.82) 

describes the gain in enthdpy, we cnn obtain an cq~lnt~ion which describe-a the incrcnsc 

in toCd etzlhnlp?l in the x-direction by forming t.heir difference. This yields 

0

it, vicw of cqn. (13.85). whcrc M = TJlc, tlcnotcs thc local Mach n~tnlbcr at the outer 

(:tlge sf t,hr hott~alnry layer. Taking inta accnant the relations (13.79). (13.81) md (13.86) we ol)t,nin t,hc final form of t,he energy-i?ztegrn,l eqlc.ntio?z: 

I

Fig.. l3.16. In 13.18. lain~innr honnclary layrr in co~nprrwihlc subsonic flow for the s~~ction 

nick? c,T t.llc NACA H4 10 nrrol'oil on I,lm nsnu~npt.ion of :in ncli;r\iat,ic wall. Angle of incirlcnre a - 0". 

h1:1~11 11111nlwr M,., -- Il.rr/~,.,; I'r:\ncll.l nl~nilwr P -- 0.725. Chlwliltpion b:wcd on the approxi- 

e. Intcractioa between shock wave and bountlary layer 

Wlicn a solid hotly is placcrl in a stream whose velocity is high, or when it flies 

through air with a high velocity, local rcgions of supersonic velocity can be 

forri~ctl in il,s nc~iglil~ourliootl. The transition from snpcrsonic velocity to subsonic 

vclocit.y against, the adjoinir~g adverse pr~wn~re gradient will u~nally take place 

I hrough :L s110ck WRVC. 011 crossing the very thin shock wave, the pressure, density, 

:rntl t.rm~)or:~l.urc of L11c Iluid chnngc at cxt,rcrnely high mt.os. l'lio rates of changc 

arc so high t,liat, thc t.ransit,ion can hc rcgdcd as heing discontinuous, except for 

the irnmetliat,~ ncighbourhood of the wall. The existence of shock waves is of functarn.cnt,al 

itnport,ancc for the drag of the body ,as they often cause the boundary 

I:~ycr 1.0 scp:lrnt,e. 'l'ho t,licomtical calcalatio~~ of shock waves and associated flow 

lit-lais is wry tlil'lin~rlt,, n.nd wc do not propose to discuss this topic here. Experiments 

sliow tl~nt, the processes of shook and boundary-layer formation intteract strongly 

Fig. 13.18. n) Velocity distributions and 

b) temperature distrib~~tions in tho boundary 

lnycr nt dilTcrant Mar11 numlmm 

e. Inl~rnolion lwtwccn sl~ock wave and I~OIIII~II~~ lnycr :m 

6, 

with each othcr. This leads to plienotncna of great cornplcxity I)cca~isc tlhc I)chavio~~r 

of the 1)orrndary layer clcpentls mainly on the Reynolds nurnhcr, whereas t,hc conditiorrs 

in a wavo arc primarily tlcpendor~t on t,hc I\Iac:Il numI)rr. I'hr c.arIicst. syst(xmatic 

investigations in which tlicsc two influences urcrc clcarly scparatrtl Iin.vr been 

putt to hand a long t,irnc ago. .I. Acltcwt.. IF. Fcltlnin~it~ alltl N. 1tot.t [I], 11. \,V. Lir11- 

mann 16x1, G.R. Qndtl, W. lloltlcr and J. I). Rcg:~.n 

l38] varictl ill t.llcir cslwitncnki 

t.he Reynolds and Mach numbers inrlcpendcntly of cach other ard so s~tcc:ccdcd in 

providing some clarification of this complex interaction. The most import,ant rcs111&q 

obtninotl in t.hc ahovc t.tircc invcstignt.ions nrc tlr:xc:ril)c.tl in t.llis wc:t.ion. \Vc IIIIIS~,, 

however, add that a cornplctc ~~ntlcrstantling of tthcsc complcx l)hc:nonirn:~ 11:~s 1411tlctl 

us to this clay. 

The pressure incmasc along the l~ounclary laycr must ultin~:~tcly l)c t,hc same 

as that in the cxLcrnal flow because the streamline which sepamt?es tho two rrgions 

must, 1)c:oomc pnr:cllol to tho c:oritn)ur of tfho body :~f't,c.r I,llo shoc:k. 111 1,111: I~otrrttl:wy 

hycr, by its n:bturc, lhe parliclcs rlcnr the wall rnovc with subsol~ic vc1oc:itics 1)ut. 

shock waves can only occur in supersonic stmarns. It is, thcrcforc, clear t,l~at a shock 

wavc which origirratcs in thc extcrrlal st,rcarn cannot rcach right 111) to the wall, 

and it follows that tho pmssurc gmtlicnt. prnllcl to t.1~: wnll musk On much rnorc: 

grwl~rd in the ~lcigIil)ourl~ood of lhc wall thn in tho cxt,crrl:~l sl,ro:~n~. N~ir 1.110 ~toirtt 

whcrc the shock wavc reaches t,owards the wall, the rahq of changc of al~laz and

360 

XIII. Laminar bonndory Inyers in compressible flow 

l'tg. 1:1.1!). Srl~licren photogrnpl~ of shock wave; direction of flow front left to right, aftcr 

Adtrrrt, Ifrldtnnnn nnd Ilott [I]: rr) 1:lrninnr bonntlery layer; tnult.iplo I-sliock, M = 1.92. 

R,r, - :!!)(I; I)) turl)ulrnt 1)011ndwy lnyrr; normal ul~ork, M - 128, Rn, - 1159 

I'ig. 18.20. lsohars in a shock re- 

gion in Intninnr flow (I-sl~ock), 

i~flrr Arkc-rrt. Iprldrnnn~~ and 

I(ot,t I I ] 

au& t)econie of the same order of magnitude, and tmnsvrrsc prrssurc grntlirrlts ciLn 

also occur tllcrc. Both conditions rcntler tho well-known nss~~mptions of hnrttltl:uylayer 

theory invalid. 

The a,ppenrancc of tho shock wave is funtlamcrltnlly tliFfcrctlt tlrl)c.t~tlillg 011 

whrthcr the boutttlary lnycr is laminar or t,url)rtlcnt,, Fig. 133.19. A sllor(, tlist:~iic-p 

allcntl of tltc point wltcrc the csscnt,ially pcrpctdicrtl:~r sltovlc wave itnpit~gcs "11 :I 

laminar boundary layer, there nppea.rs a short Icg forming a so-cnllcct I-s~Io&. 

Fig. 13.19n.. Tn gcncml, wllcn the boundary lnyer is turbulcnt,, the rtor~nnl sltoc:li tloc.~ 

not split and no I-shocks a.rc formed, Fig. 13.19b. An obliv~ce shock wlticlt ilnpit~gek 

on a laminar boundary laycr from the outside becomcs rcllectcd from it in t,llc for111 

of n fan of expansion waves, Fig. 13.30a. Ilowever, whcn t.he bountlary lnyrr is 

trurbnlcnt, the rcflcxion nppmrs in the form of n mow concc?ntmt,cd cspnt~aiotl wn.vcx 

(Fig. 13.30b). 

The plot of isobaric curves in Fig. 13.20 ant1 the prrssrtrc curvcs in lcig. 1:j.21 

how t,hat t.lw rat,c of prcssnrc incrcnsc along a Iaminnr or :I tnrl)~rlcnL Im~titl:~r~. 

lnycr is more gratlnal than in tltc cxtcrnal strcam. 'l'his llrtttcnirlg ol' Lltc prcssltrc 

gradient in the boundary layer is described by stating t,llat the prcssurc dist.ribrrtion 

"diffuses" near the wall. It is observed that diflusion is much lnorc prono~tncctl 

for a laminar tphan for a turbulent boundary Iaycr. The tlifkrcncc bcLwccn 1nniinn.r 

and turbulent shock diffusions can also be recognized from Fig. 13.22 which roprcscr~t,~ 

tho pressure variation along n flat platc placed parallcl to n supersonic st.rmtn. 'L'hc 

mcasnrcmcnts were pcrforrnctl by 11. MT. Ilicpmnnn, A. lto~ltl~o attd S. I)h:r\viw 

[64]. The pressurc plob llavc been tnlren mar thc point on tllc platc whcrc t,hc 

oblique shock produced by a wedge interacts with thc boundary laycr. Tltc prcssurc 

gradient is co~;sitlcrnbly stccper for the turbulcnt t,hnn for the 1:rrninnr I)oltntlary 

Iaycr. The witltl~ of diffusion is cqunl to about 100 d in t,hc case of int.nr:wfiotl with 

Fig. 18.22. Prrsqnro tlistrihution along n 

flat plate at supersonic velocity in the 

016 

ncigl~bonrl~ood of the region of reflcxion 

of n shock wave from laminar and tur- 

8 7 

hrllcrtt borrntlnry layers, xftrr Lirpmann, 

008

a Intninn.r l)out~rlary I:~.ycr, but; rlccrcasrs to about 10 0 for n turh~llent hountlary 

Iayor; t,l~r! syn~l)ol 8 tlcne>lcs hrrc t,l~c: I)o~tntlary-layer t,lliolrness in i,hc si~oolr region. 

The liigliar tlcgrcc of tliffusior~ wl~icli is cli:tmcte:rislic of laminar bouticI:~ry layers 

can IN untltrsi,ootl if it, is not,ctl that tllc subsonic mgion of flow extm~ds furfhr 

away from the wall in a laminar than in a trirbulcnt boundary layer. 

Irrrspcclivc of wl~ntllcr sepxmt,ior~ elocs or clocs not occur, the bountlary-layer 

t.l~ic:kncss incrtascs alicatl of the poinl, of :~rrival of the shock wave. The pressure 

increme at Lhc out,cr ctlgc: ol the Iiountl:iry I:~yer, ant1 hcncc also insitlc the I~ountlnry 

layer, corrcspondn to the c~lrvrtl st~rcan~linc wllic:h is convex in the direction of 

t.11~ wall ant1 which scp:~,mtc~s the: exl,t:r~~:il from the I)oirntlary-hycr flow. lhwn in 

the clomain of influcncc or i,l~c cxpnnsion waves which appear in the rcflcxion of 

an ol)liquc shock wave, the sligl~t, tlccrc:asc in pressure in the bountlary hycr, l'ig 13.22, 

corresponds to the fact that the curvature of the dividing streamline is concave 

lowards tho wall. A laminar boundary layer which has not sepsratetl can support 

only very small pressure rises because tlw external flow imprcsscs on it the prcssurc 

cxclasively through viscous forces. A non-sepnmt,ecl tnrl)rllcr~t ho~tndnry 

layer can take up much larger pressure padicnts because now the turbulent mixing 

motion aids the process. Both laminar and tirrbulent boundary layers nre in a position 

to snpport the large pressure increasaq of strong shocks if they separate. Tn particular. 

Fig. 13.23. Ilrllrsion oi n sl~ock wnvc from a f.nrldoril bonndnry lnyor on a flat wall, after 

S. M. Ih)grlot~oll' nntl C. IF. I

364 

XIII. Laminar boundary lnyrrs in comprcssihle flow 

butions along thc wall arc shown plotted in Fig. 13.23~ for different deflexion angles 

(and hence ctiffcrcnt shock strengths). Separation occurs for O > go. The pressure 

rise which lratls to scpzmtion is independent of the deflexion angle and has a value 

of ahont p /p, = 2. 

The incidence of transition and srparation in the nciglhourhood of an impinging 

shock wave are governed principally by the Reynolds number of the boundary 

layrr and by the Mach numbcr of the extcrnal stream. When the shock is weak 

and the lteynoltls nnmbcr is very small, thc boundary layer remains laminar thronghout. 

Tncrrasing the Reynolds numbcr at a fixed,.'small Mach number, causes transition 

towccur at tbc point of impingement,. When thc shock is strong (largo Mach number) 

and thc Itc.ynolds number is small, tho laminar boundary layer will scparatc 

ahratl of thr shock front owing to prcssnrc diffusion; it may also undergo transition 

ahrad of the shock front.. When the lteynolds numbor is large enough, transition 

in thc t)onr~tlary lnyrr occurs ahcad of the shock, w!~cther tho boundary layer has 

Fig. 13.26. Schlieren phot,ograph of the flow past an acrofoil. Shock-wave and boundnry-layer 

interaction. Cnm (4): Boundary layer trlrbulent ahead of shock, no uepnration. M = 0.85, 

R = 1.69 X 10" after Ii~prnann 1631 

separated or not. According to observations made by A. Fage and R. Sargcnt [RBI, 

turbulent boundary layers do not separate when the pressure ratio pJpl is smaller 

than 1-8, which corresponds to a Mach number M, < 1.3 for a normal shock wave. 

Thtl~cr cxperimcnt.al rcw~lts on t h it~l.c:racIio~l I~c!l.wcr?n sl~oc:lc wirves itr~tl I~oilnclrl.r.y 

lagcrs car1 bc fonnd in the pu1,licntions by W. A. Mair [G9], N. Il. .Johanncscn 1521, 

0. Itartisley and W. A. Mair [R], and .J. Lulcasicwicz and J. I

1 I Itc vnriolts c4Twl.s ol'sl~oclts impinging on a 1)ortndary In.yrr will now be illustmkt1 

\vit,l~ roSct.rnce to Schlicrcn pl~ol,ograplis. As point.rtl out by A. TI. Young 1. IOB I, 

it is possil~lc lo tlisI.itlgnish tl~c following mscs: 

(2) 'I'lic nppronc:l~ing I)o~trltl:rry lager is laminar, but scpamtcs n.head of the shock 

Iwmlsc of 1.h~ ntlvcrsc pressure gratliont antl lhon retnrns to the surface in 

c:il.hnr a Ianlinar or lt~rlntlcnt stmate, Fig. 13.24t. 

(3) 'I'lic nppro:whing I)outitl:lry lnycr is Iaminnr, srpwatcs cornl)lol.rly from the 

s~irf:rnc: RIIC:L(I of I,hc sho(4z, and (10~s not, r(:-attacli itr9rIf to the surface, Fig. 13.25; 

t,lw shorlc is normal arttl sprorrt.~ a A-limb. 

(4) 'i'llo :tppronc.hing Imtntlnry layer is I~trl)~~lrnl. and clocs not sepnmto from the 

s~~rf:wc, I'ig. 13.26. 

,. L llr c.o~~sitlt~t.:~l,iot~s vonwrning 1.11~ Ix~havioltr of' I,ol~n(l:~ry Iayrrs on acrofoils in 

Ill( I1.;111sonir ~.~~gini(: t.l1:1,1. follow ~vScr (wrnt.ially t,t) t,nrl~ttlrnt, 0011ndnry Iajwrs whic.li 

\\ill I)(. sttttlictl 1n.tc.r in (%:~ps. XSll nnd XX11I. Sinrc, Iiowcvrr. t,tmlsit,ion pl:l.ys 

:I 11art in IIIw(~ ])t~o~:c~ssrs, wc: shn.ll it1st.14 I.lt(w1 II(w. P\.PII t.l~o~tgll t,hc trn~nsil~ion 

pru(.tw itsc~lf' will also I)(, tlisc:ltsst~tl l:~.I.t:r. nn~noly in (bps. XVl nntl XVI I. 

t 'I'~I:III~

308 XIII. Laminnr boundary layer8 in compressible flow c. Inkroction bctwccn sllock wnvo nnd 1)ounclnry lnycr 

1 . 2 Scldicren plrotogmph of the flow past nn aerofoil. Sllock-wave and boundary-layer 

interaction. Case (5): Turbrllcnt boundary layer with atrong separation behind shock. M = 0.90, 

R - 1.75 x 10'. after Liep~nann [W] 

the inll~loncc of Ilcynoltls nnnlhcr on tdlc I)onr~tlnry layer - and hence also on the 

shook wave as well as on the associated point of se,paration - is quite considerable 

ill transonic flow. As a result,, tho value of t,hc Reynolds number has a much great,er 

cll;v:t, OII all acw)tlynnrnir. cllar:tct,crist,ics of an aerofoil in the tra~~sonic range o/ Mach 

nu~n.hrrs f,han cit,llcr in subsonic or in tho purcly supersonic rrgime. For this reason 

it is ncccwary t.o exercise ntmost c:ultion when tcst results fronl wind tunnels in 

t.Iw t,r:~nmnic t.:&ngc7 arc uectl to predict, I)cl~nvionr in flight. Further experimental 

rcw~lln on Ihis t.ol)ic:

370 XIIJ. I~tminar boundary layers in compressible flow 

Yet nnot.hcr important problem of interaction between boundary layer and shock 

wave occurs in h?ypemonic corner /lou~ at zero incidence. 'rhe flow is accompanied by 

int,cn?ie 11mt.ing in thc cornrr cnt~xccl by l h very m~ich lnrgcr mt.c ol'clis..ipat~iorr in the 

corner compnrctl t.o t,hc clissipation in the ncighbouring two-dimensional flow. A hint 

in that direction is visible in Fig. 12.16. It was sliown there that even in incompressible 

flow along a rectangular corner with the wall being at a temperature exceeding that 

of the free st,rcam there exists a heat flux transferring hat from the fluid to the wall. 

By contmst, at a large distance from the corner, t,he flow of heat takes place in t,he 

reverse direction. 

Snirnt,ist.s lwcnmc: aware of t.hc almvc prohlrm only rcccnt.ly, nnmcly in con- 

nexion wit,li t.hc flight t.csts in the range of hlnch numbers M - 3 to 6 on the American 

experimental aeroplane X-15. Reports on this phenomenon were puk~lislicd by R. I). 

Neumann [82, !In]. Figure 13.28 reminds t,he reader that such corner confignrations 

exist at t.he root of the wing, at the side fins, at the engine pods or at thc air inlet in 

air-brcnt,liing engines. 

More rcccnt. expcrimcrltnl invest,igat.ions on hyj)crsonic corner flows wcrc pcrforn~rtl 

hy I

XTII. I.nntinnr l)o~~rldnry 1:iyrrs in romprmsihlr flow 

'J'lrc sul)sc.ript wp rcfrrs to the point of srp:~mtion, thr subscri1)l 0 ~~~~~~~ibrs the 

shte upstream of the sl1oc4c wavc, and subscript 1 clenotes the state at the edge 

of thr I)ou~itlary Inycr. 

'I'lrc pressure coeffirirnt at separat,ion turns out to 11,zve the form 

whcrc R lT a./v, ant1 TI,,, Mtr do~~ot.o tho pressure and Mach numbor, respect-ivcly, 

~tpst,rr:r~n of 1.11~ shodc w:~vc. 

Numerical solut.ions which contain tl~e zone of interaction between n separated 

laminar bountlary lnycr and n frictionless suprrsonic &ream were perforrned by V. N. 

Vaka arlcl S. I). Rrrtke [I 011, as well xs by 0. R. Burggraf [9J, G. S. SetdJes, S. M. 

12ogtlot1olT and 1. E. Vns 193a.J. 

111 Ackcrct,, J., I~cldnrnnn, F., and Rott, N.: IJr~tcrs~~chrlngcn an Verdichk~ngsstiissen nnd 

~~~II~.~CII~~III.~II 

in nc:l~ncll hcwcgtcn Cnsen. Itcport No. 10 of L~IC Inst. of Aerotlynamics 

I:'IxII Ziirich 1!)4(i: nee also NACA '1'M 11 13 (1!)47). 

121 Applrt,on, .I. I'., uncl I)nvics, fl. J.: A note on tho intcmot.ion of n nortnal slroclc wavc with n 

tl~crtnnl hotr~~tlnry Inycr. JAS 25, 722---723 (1958). 

[:$I J

\'all I)rirst., 15.11..: 'l'hr prol~lenr of nnrocly~rnn~ir Irmting. t\ero. 1h~. R.cvicw 15, 2fiL 41 

(1!15(i). 

I4:l,rr. (:. I

376 

XI 11. Jfiminnr ho~~~tdnry 1:rycrs in compressible flow 

[Bla] Murphy, .I.]).: A critical evnh~ntion of analylhl mcthods for predicting laminnr boundary 

Inyer, ahock-wave internction. NASA TN 1)-7044 (1871). 

I81111 Murphy, ,I. I).. l'rcsley, LL., and Roue, W.C.: On the calculation of supcraonic scparnting 

and rrntt~aolti~tg flows. ACAltl) Cortf. I'roc. Flow Sepnrntion, No. 168, 22-1 to 22-12 

(1975). 

I821 Neumnnn, R.I).: Special topics in hypersonic flow. AGARD I~cture Series No. 42, 1, 7-1 

to 7- 64 (1972). 

[831 I'ni, S. I., al1t1 Sl~cn, S. IF.: Hypersonic viucoun flow over an inclined wedge with heat trnnsfcr. 

Fift.y ycnrs of bout~dnry-layer rcsenrch (W. Tollmien and H. Gortler, ed.), Branttscl~weig, 

1955, 112- 121. 

1841 Prarccy, 11. ll., OnImr~tc, .I., n~td llninr~, A.I%.: The interaclion between local rtTccta at the 

shock and rcnr scy~nrnbio~~ - rr uotlrcc of sigt~ificant soale clkcb in wintlt~l~~t~el tent8 on airfoils 

n ~ wings. d A(:AItI) Conf. lJroc. No. 35, I I -- I to I 1-23 (1968). 

1851 l'ooln, (:.: A noluLion of the comprcsaihlc ln~ninnr hou~~dnry lnycr cq~intionrr wit,h Iloat 

Lrn.nsfrr nntl :idversc prcmwrc gradient. Quart. J. Mech. Appl. Math. 13, 67-84 (1860). 

[HC,] Itcd~ot.ko, I

CIIAPTER XIV 

Boundary-layer control in laminar flow t 

a. Mntliod~ of boundnry-layer control 

,. 

l iicrc art. in existence several ineLliods which have been devcloped for the 

purpose of nrt.iGcinlly conbrolling the Imliaviour of the bonntlary layer. The: piirposc 

of fhcsc mctliods is to affect the wliolc flow in a clcsircd dircction by infl~rcricing 

the strtrctmrc of the boundary layer. As early as in his first paper published in 1904, 

1,. I'rantltl rlrscril~cd sevrrnl cxperirnents in which tlie borl~ldary layer was cont,rolled. 

Ire: inl,cndctl to prove t h validity of his funtfamonta.l ideas by suitahly designed 

ex~)crilncnt,s n.nd arliicvctl quite: rcmarltn.l)lc rcsr~lts in this WRY. Fiprc 14.1 shows the 

flow p;mt n c:ircr~l:\r c:ylititlor witli s~ict,ion :~pplit:d on one sitlc of it t,hrotrgh a srnall 

slit.. 011 t.hc s~~ct,ion sitlo tho llow ndhnrrs t.o I.lic cylintlcr over a consitlcrably larger 

~,ort.ion of its srirf;lce: nntl scpnr:rt.ion is nvoitlctl; 1.11~ tlrng is rcduoctl npprcci:~ldy, 

anti si~nnlt~niiconsly n Inrgc cross-force: is intlt~cctl owing to the lack of syrnnictry 

in the flow pnt,tcrii. 

1 I'roft~swr I),.. \IT. \VIWS(, t~sxixl~~l ill l,lw lm~1x1r:tI.itm 

b'illll IC11ilio11 of I h i ~ lmk 

of t,Ilt: ~ ~tw vorsitm of lhi8 rIlnplt*r for l h 

in nct,r~al npplio:~tioris it is oftcri norcssary to prevcrit scp:~ml.io~i in ortlor to rc-clnc:t: 

tlr:ig ant1 t.o athin high lift. Several niethods of cont,rolling the botrntlnry lnyor 

I~nvc \)con clcvclol~cc~ expcriniontnlly, nnrl nlso on the h ~ix of I,lioorotic::~l norrsidc:rntioiis 

16, 76, 701. Tlicsc can t~c t:lnssifictl as follows: 

hlot.ion of tlie solitl wall 

Accclcrat,ion of the boundary hycr (blowing) 

Suct.ion 

1njac:l.iori of :I. cliffcrerit gas (biriiwy Imnndary Inyc:r~) 

I'rovt:nt.ior~ or t,rnnsit,iori to L~~rlmlonl flow I)y t11t: provi.siori of utliL:~I~It! 

(I:i.minnr :~t~ohiIs) 

('ooli~ig or Llio wrdl. 

NII:L~;(:H 

Mrthotls 1 to 4 will be discussetl in (.his chnptrr. Methods 5 antl fi will be 

tlcscri1)rtl in (:II:L~. SVII in conriexion with tho rorisitlrration of I.lw t.liwry of 

t rnnsil ion from I:ui~in;w to ti~rl)tiIriit~ flow. 

r 

I . lio t.rcat.isc: e~it.it,lt:cI "Bo11ncl:wy-1,i~ycr antl I'low (hl.rol" 1441 Ily (:. V. I,;L~:IImann 

~ont.:~ins a summary of the sirl~jcct of boi1nd;~ry-layer control accorcti~ip to 

the state of rcscnrcli at t.lic time; compare also I.'. I

380 X IV. llorlntlnry-layrr control 

placed in a strcam nt right anglcs to its axis. On thc upper side, where the flow 

and t,he cylinder move in tlic snmc direction, separation is completely eliminated. 

I~ttrt,llcrmorc, on tho lowcr sitlc wl~cre the tlircotion of Hr~itl motion is opposite to 

t,llnL of t.hc solitl wall, scpnration is clcvclopcd only incotnpletcly. 011 the whole, 

t,lw flow pnt.t.ern wllich exists in this case npproxinintcs vcry closely the pnt,t,ern of 

frit:l.ionlt:ss Ilow psi, x circular cylintlor with oircnla.tion. 'rhc: sl,rram exerts n consitlcmbl(: 

force on t,hc cylintlcr at right :~nglcs t,o the mean flow tlircction, ;lncl t,liis 

is somctimcs referred to ns t,he Magnus clTcct,. This effect can be seen, o. g., when a. 

tennis 11dl is 'sliced' in 1,I;~y. Attctnpts wcrc also mntlc to ~lt~ilizc the occilrrcncc of 

lift, on rotating cylintlcrs for the prol)l~lsion of ships (I'lettner's rotor 111). With 

the cxcept.ion of rotating cylinders, tho itlc:l of moving thc solid wall wit,l~ the stronm 

can I)(: realized only at tho cost, of vcry grcat complic:~tions as far as sl~apcs 

othrr t.l~nn ryli~~tlrionl are conoornccl, ant1 conscqt~ently, this nictl~otl has not fonntl 

mnalr practical application. Nevorthc:loss, A. Ihvre (261 mntle a thorougl~ experimc%nt.nl 

invest.ignt.ion of the inll~~cncc of a moving bountlary on an nerofoil. A port,ion 

of l.I~t~ upper sttrfacc of the norofoil was limnetl into an c:ntllcss Idt which n~ovatl 

ovvr two rollcrs so t,li:~t tltc: rcLttrn tnol,ion occurred in the interior of tlic motlcl. 

'J'lie arrangcrncnt, proved vcry cffcctivc for the avoidance of separakion, nntl yicltlcd 

vcry high maximum lift, cocfficicnt,~ (C,,,,,, = 3.5) at high angles of incitlcnce 

(a. z 55'). The Inrninar boundary laycr for a flat plat,c moving in its rear part with 

t.hc shmn 11n.s I~ccn c:rlculatctl by ]I:. 'I'rncltcnhrotlt. [IOO]. 

2. Accclcrnhn olthc bnu~~tlnr~ layer (blowing). An nllnrna.t,ive tnc~t.ltotl of l)rcvc:nt.il~g 

scp:~.rat.ion consists in supplying atltlit.ional energy to the part,iclcs of fluid wl~irh 

arc bring rct.nrtlrd in the boundary laycr. This result can be at-hicvetl by tliscl~arging 

fluill from t,hc itlt,c?rior of t,l~c I~ocly with the aicl of n special blower (I'ig. 14.:ln), or 

by tlrriving t.hc tcquirctl energy directly from the main sham. This Iathx effect. 

pa11 IIP protlricctl by connect.ing the rctmdect rcgion t.o a rcgion of l~iglier pressure 

t.l~rot~gh n slot in the wing (slottcd wing, Pig. 14.3b). In citl~cr ca.sc atltlit.ionnl 

crirrgy is intp:~rkd 1.0 the pnrticles of lluitl in tho boundary layer .near the wall. 

When fluitl is tlischnrgcd, say in the manner shown in IGg. 14.3a. it is mnntlntory 

t,o pay mrcfitl at,tcnt.ion to the sllnpc of t,hc slit in order to prevent the jet from 

dimc,lving into at 8 short* distnncc behind the exit section. 1,atcr expcri- 

~ncnt,s pc~rli)rmc~tl in France [04] Iinvc rnatlc it, vcry ntI.racl.ivt: t,o n.pply blowing :I.{, 

t,hc (,railing ctlge of n.rl acrofoil in ortlcr to incrcnsc its mnximum lift. Att,cnipt.s 

consid(.ra.l)ly to incrmsc the masimum lift of a Rnp wing tl~rongh blowing in the 

slot. Imve dso mrt with success (c/. Sec. X l l b 6). 

111 1.11~ c.;~sr of t .1~ slottctl wing [7], shown in Ipig. 14.311, the cfict is prot1uc:ctl 

:IS follows: 'l'l~c I)o~tntl:~ry laycr formed on the forward slnL A - 13 is c:arrictl inlo the 

tn:tit~ sl.rc-n.tn brfore sepnml,ion occurs, :&ntl from point (: OIIWR~~S a t~ew bot~ll(la.~.y 

I:~yt-r is liwnc.tl. IJnJcr favourablc contlitions this new boundary layer will rcnclt 

1.11t: t.miling rtlgc 1) without. sc?pamtion. In this way it is possiblc to relegate ~rparnt~ioti 

t,o consitlem.bly larger a.nglcs nf incidnnce, nrtd to achieve much larger lifts. Fig. 14.4 

shows n polar tlia.grnni (lift, cocfficicnt~ plotted against drag ~oeffi'licirnt~) for a wir~g 

~(dion \vit,l~ and wil~l~ot~t, forwnr

5. J'rrvrntion of trauaitinn by the proviriwt oC a~~itnl~lr almpcs. Ln~r~inar nerofoiln. 

'I'riinsit.ion l'ronr hmint~r to tnrl)lllc~rt, Row car1 also 1)o tlt>lnyctl l).v 1.11~ nsa of suitably 

sl~:~p~l 11otlic.s. 'l'llo ol)jct:t,. :LS in t.hc cnso of suc:l.ion, is 1.0 rctlncc frictionn.l tlrng Ity 

c.:~~lsing tho point of t.r:l.nsil,ior~ 1.0 movo tlowrrst,rcnni. It, hns bccn es1,nI)lislrc:tl IhaL 

thr, loc.:rt.ion of t,lw point or t,rn.nsit.ion in tho bonntln.ry layor is st*rongly inlltrrnrct1 

by the I)rossurc: gr:~tliorrt in Lllc cxt.crnal strewn. With a decrease in pressure, t.mnsition 

oc:c:tirs at, much I~igI~cr ItcynoItIs rllrrnl~crs tllnn with prossure increase. A decrease 

in prc~ssurc 11n.s R Itiglrly st,nl~ilizing cfic:t. on tho borrrlclary layer, and tho oppositt: 

is t.1.11t: of' an incrc:~.sc in prcssuro nlollg tire stream. 'l'llis circumstn,ncc is nt~ilizt~tl 

in niotlcrrl low-tlrag ncrofoils. 'l'hc dcsirctl mst~lt. is acliicvrtl by displacing the sectpion 

of n~:~xirnnn~ t.l1ic4crwss far rc:~rwnrtls. In this manner a largo porLion of the ncrofoil 

rcvn:~ins tlntlrr t.11~ inflncnor of' n prcssurc: wllit41 tlecreasos tlowrist,rcam nntl a Inniinar 

I)o~rntl:~ry Iaytv is mnint:~inctl. We sll:~ll rovcrt to t,tlis q~~cst,ion in (:hap. XVII. 

, 

I 

, 

Ilc rnol.llotl of 1)oundary-layer nont,rol Ijy suct,ion, togctl~cr with the prevention 

of t.r:lrisil,ion on I:l.minn.r acrofoils, Imvc the grc::~tcsl pmc:t.io:d iniport:~.rlac :Lrnong all 

t h rrlc:tl~otls tlis(:nsscd prcviot~sly. For this rcason vnrions ~~iat,l~crnnf.it:nl ~ncllrods 

for ti~t: c~:~l~:uI:~l.ion of t,hc inllnorcc of surtion on bonr~tl:~ry-hycr llow have been 

tlcvolo~~ctl, i~nt1 wc now propose to mvicw Llrom lwiclly. 

b. Doundnry-layer suction 

1.1. Fundamental equations. It is sirn~~lcst to Iqin tl~c rn:~l~l~r~nnt~ic:nl stntly 

of the laminar bounc1ar.y laycr with s11cLio11 by first co~~sitlcri~lg t,ht: case wifh (:ontin~tol~s 

suntion which may be irna,ginctl rc.nlizcd with ishc :tit1 of a 1)oro~s ~:l.lJ. 'l7he 

usual system of coordinatcs will bc atloptetl, the z-axis I~oing along tlrc wall, ant1 

tho y-axis Iicing at right angles to it., Fig. 14.5. S~~c:t.iot~ will Ijc ac:c:or~r~l~t:tl li)r b,vT 

pmsnribing n non-mro normal vclocit.y compo~tcr~t v,(z) :I.(, t.I~c wnll; in tho msc: of 

sut:fhl SJIRII put v,, < 0, making v,, > 0 for discl~argo. It, will IIC assr~n~ctl tlrnt, 

the rjrlanlily of fluid rernovcd from tho strcam is so sn~nll tht only fluitl p:trt.icl(:s 

in the immediate neiglrl~ourhootl of the well arc? suolrctl away. This is ctlt~ivaI(:rlt 

to saying tht. I,trc rat,io of suction vclocity v,(z) to frce-slfro:~tn vcloc:ity IT,,, is vt~y 

small, say n,,/lJ,, -- 0.0001 to 0.01 t. 'I'llr oontlitior~ of no slip nL tl~o ~1.11 is rc~l.:~i~rrcl 

with suct,iorr prescnt, as wc:ll as the cxprrssioli to = 11 (au,/&~),~ for thn shc:~r-il~g stross 

at, t h wall. 'l'l~t: q~~nr~tit,y of fluid rctnovotl, (J, will bc cxprcssccl l.llro1rg11 a tli~l~rtrsionlcss 

volume corfficicrit by pnt,t,ing 

~ ---- -. 

t Jn orclrr t,t> cns~~rc 

that z flow wit,l~ s11ctio11, or bIo\ving, nt l,l~c wtll snt.isfivs t,llt\ sin~l~lif~itty, 

cotdit.io~rn wliinh for111 the Imsin of bo~~ndary-):byor t.l~cory, iL is noc.e.ss:try tx) li111it l,I~c: \~c.Iot:itY 

no at, Lhc wnll to n ~naguitutlc of Lhc ortlcr of I/,, R-112, wllt~rr: R -. (I,,, 1/18 atltl 1 c1c~t~1t.c.s :\ 

clmrantrrist.ic dinrc~lsion of 1.I1c: solid 1)otly pl:~rt-tl in I.IIo Ilo\tr. At. R loR this c.o~~clil.io~~ t:iyc.s 

v, - 0.Ol)l 11,. Ll'l~ot~ 1.110 s~lnI.icm vcloc:il.y is of HWII n m1:tII orclt*r 01' ~~~:~.pt~il.~t~lc~, 

ill is 11ossiltlc. 

to IIP~IWL the Icms of rnoss or c'~i~~k-cI~t!f:L" 011 1110 rxlrr~i~tl 11ol~wlid 110\~. 111 ol.11c:r \101.114, 

the potcnt,inl flow 111ny be :tasulncd to rcnlair~ unnrfc:ctcd by scd~ I~lotving or srtolion al~plicvl 

at Lhc nurfacc of Llw solid body.

384 

XIV. Boundary-layer control 

Assunring incompressible two-dimensional flow we have the following differential 

ccinnt,ions 

with the 1)otindary conditions 

lSvitlcnt,ly, the integration of the above system of equations for the gcncral case 

of arl~it~mry body shape, implying an arbitrary velocity function U(z), present-s 

no frwcr clifficulties thnn does the case with no suction. 

Nrvcrtltrless, the qnalitative effert of sr~ction on separation can bc rstimated 

with the :lid of t,lrr preceding equations cvcn without, intrgration. Along the stream- 

lirtr at thr wall (?/ -0). eqnnt.iotis (14.3) antl (144) yield 

It is seen that in a rcgion of adverse prossure gmdient (dpldx > O), the superpositiori 

of sunhion (vo < 0) rctluccs the curvature of the velocit,y profile at the wall. According 

t,o the :irgumcnt,s adva~iccd in Chap. V11, this signifies that the point of scpnratiot~ 

is tlisplaced rearwards. Now, in accordance with t,he theory which will be giver1 in 

Chap. XVIT, t.l~is has the additional effect of stabilizing the laminar houndary 

layer. 'rhcse t,wo clTcct.q produced by suction, namely avoitlance of separation and 

t,he relrgat.ion of t,hc point of Iami~~ar-t~~rb~rlcnt transit.ion to higher Reynolds ~rurnbcrs, 

hn.w bcen c:onfirmctl hy thc results of experiments. 

A snmmary of mcthocls used fvr the: c:~lct~lat.ion of Imnntlary layers with sncLiorl 

\\':is pnl)lishc~I I)y \V. Wl~cst [108]. 

1.2. Exnct ~nlu~inns. 'I'IIc mct.11otl of l~sing n power-soric.s cxpi~.trxion in t.rrrns of 

t,lrc! longt,II of arc for t.ltc: I,ot.cr~tinl vclocity (Illasins series) dcscribctl in Scc. IXc 

(::In, in principla, l ~c al)plictl in this case as well. IIowevcr, just as in the case wi~hout 

snct,ion, the result.ing comput,ations become very laborious [75]. Reasonable sirnplc 

sol~tt,i~ns can bc ot)t,aincd only in t,hc case of a flnt at zero incidence. 

FInt A snrprisingIy sitnPIe soInt,ion C:HI IN 0bhi110d in the case of a JEnt 

7,!,11c tr( zo.0 itlc:it/e,tcr. wit,lr 7tu,i/o~.m, ~wlio,~, Fig. 14.5. '1'11~ systcm oT tlilfcrcnt.ial 

rquat.iorls now rctl~rccs to I 

with the boundary conditions: u = 0, v = v, = const < 0 for = 0, and ZL = U, 

for y =- co. It can be secn at once that t,llis system posscssrs a part.ic&r solu(,ion 

for which the vclority is intlcpcntlcnt of t h ci~rrctrt IctlgL,I~ x [52, 781. l'nl,t,it~g 

aup~ zs 0 we sec from the equation of conLiriuity that v(z,?l) = vo = eotrst. Ilcnce 

witA thc solntion 

the cquation of motion bccorncs v, au/ay =. v a27~/r??yz, 

7r(y) = U, [I - exp (voy/v)] ; v(x,?y) - vn < 0 . (14.6) 

It is wort.ll noting that this sirnplc solut,ion is cvcn an cxnct solution of tho coml~lctc , 

Navicr-St.oltcs cc~rrat.iorrs. 'I'l~c tlisplaccmctrt thic:l~t~css antl t.hc ~nomcntntn tI~it:l~~lrss 

arc 

and the sl~earirig strcss at tl~c wall T, 7 p (a@y), bccornrs simply 

and is indcperirlenb of viscosity. The vclocit,y clistribution is seen plottnd in Fig. I.t.(i, 

cwvc 1. Curve 11, dmwn for the pilrposc of comparison, rcprcscntn the I5lasins 

velocity distribution witllout suction. It slroultl be noted that the suction profile 

is fuller. The solution thus discovered can be realizcd on n flnt plat.e at zcro ir~cidcncc 

with uniform suction only at some distance from the loading ctlge, even if suct,ion 

is applied from the lcatling ctlge onwartls. 'rho bount1:~ry I:l.ycr, cviclont,ly, I~c-~it~s 

to grow from zero tl~icltness nt the lcading cdgc and conLinucs downsham t.rntlit~g 

asymptotically to the valuc given in eqn. (14.7). The vclocit,y profile attai~is thc 

simple form given by cqn. (14.6) only asymptotically, i. c. from tlw practical point, 

of view after a ccrt,ain initial length. For thcsc reasons the preceding particular 

solut~ion may bc regarded as the asymptotic suction pl-ojile. 

Fig. 14.6. Velocity distrihut.ion in the boun- 

dary layrr on a flat plntc at, zcro incidence 

A more tlctailetl investigatiorr irkto the flow in the initin1 kngth, i. e., brforc 

the asymptotic stmate has I~een reachcd, was carried out by R. Iglisch [40] who 

has shown that the asymptotic state is reached after a IcngtJl of about 

'The vclocity profiles in the inilial length are not similar among tlienisrlvcs 'rhey 

arc practically itlrntical will1 thosc for Cho cnso with no surl.ior~ nt shorl dist:~nwa

frorii th; 1c;ulitig rxlge! (l3l:wi11s profile:, Wg. 7.7). 'l'hc: ~~a~l~tcrti or st,rcnn~lit~c~s in t,hc 

it~il,i:~l Icngt,ll is UC~II tlr:lwn in Vig. 13.7, :LII~~ the: vc:locilfy ~)rolilcs arc scon ple)tAc~l 

in IGg. 14.8. 'l'lie way in which tho I,outicl:~ry-layer tliick~icss incrcsscs from 

n.t, the Icnding lctlgc? to it8 asym~)tot,ic value given in oqri. (14.7) is clo~cribotl hy 

tdic values in 'I'nblc 14.1 which have bcon taken from It. Jgliscli's pepcr. 

1'artic.ular int,crrst, is st.t:wlirtl to thr nnvi*g in drug ceusc.d by preserving 

I:imit~ar flow with fhr aid of si1c4ion, an(l, thcrrforc, to the Inw of frirtiorl for the 

'r:t1,l1: 14.1. I)iti~rttsio~ilrsq ~,ol~llc~:rr~-~:r~~~r 

Ll~irlz~irss :rnd sh:rpo fnofnr R,/d, for t,hc vdorit.y profilc-s 

ill t.lir* itlil,i:~l lc~t~~t.li 011 n I1:rt. plirl~ :kt, mro i~rciqlc~i(:e? \viCh rtttili~rtn ~ucl.iott, idler It. Iglisd~ [40J 

ht -- ~lis~~lt~~~rtti~~trt. 

t.liirlzt~css; 

v ,?, - rnotiictil.it~~~ t.liinknr,sn 

Fig. 14.8. Flnt plate wit,h 

~l~iifortn liuctio~i ; velocity 

profilcn ovcr init,inl length, 

aftw lgliscli [40] 

Tllr rurve 6 - a, corrrspowls 10 

llw 'ruy~nplnlir s~tcllnn pronle' of 

rqn (14.0) 

,. I hin is the* (1~1.c clt~c to siukit~g, i.c. thf) (1r:~g oxlivrir~~cwl l~y ,I. II~IIJ lvlti(+ is ~I:I~,I>,! it1 :L 1.1.ic.. 

tionlrss st.ro:~tn of vrlociLy lJm ntid whirl^ 's~nllown' :I. q~tnnl.il.y (2 oI' IIt~icl. 'l'lrv :rl~~\~c. VY~I~(.X. 

aioti call bc: tletl~toctl vcry nit~iply by 1.11~ npplie:ation of I.ho I~IOIII~~I~.IIIII 1.l1~ort.111 (TI. I'r:tt~~lt~l- 

'I'ieljor~n, Ilyclro- 11. Aerorncchnnik, vol. I I, l!):lI, p. 140, Ihgl. t.rnllul. 1,~' .I. 1'. tlct~ l ln.rl,og, 1934.

388 XIV. 13ountl:~r,y-laycr control 

r, 

l he tlritg c:ocilit.icnt, is Inrgor for small Ileynoltls ~,u~tnl~crs, becn.r~se the shearing 

stress is groa1,t.r t)vt:r the front portion of thc plate, i. c. lhat which falls within 

I.hc ittitiill rcgiol~ rwtl whoro the 1)ountlnry l:~yor is thinner than furLhcr tlownst.rcam. 

The tlr:~g on ;L plate with a turbulent boundary layer with no sucbion is 

shown plotted in l'ip. 14.9 for the purpose of comparison. It will be tliscussetl more 

f~~lly in Chap. X X I. 'I'llc s:~.vi~~g in drag can be dcdnc:ed from this diagmm only if the 

v:rlue of the stn:~ll~:st voll~ntc: cocflicicnt ofsut:tion whit:h is capable ofcnsuring Iarnini~r 

conditions in the bor1ntlar.y layer at large Reynolds numbers is Itnown. This prohlcm 

will I)c invcst.ignl.c:tl in Clln,p.'XVIT, togcrthor with thc phcnorncnon of transil,ion. It will then I)(: shown t,lt:~t thcrc exists a curve of 'most favoum1)lc suction'; il can be 

sc:c:rt plottctl in Fig. 17.10. It will be noticctl that the rctluction in drag through srrction 

is vcry consitlcral)lc antl t,l~at the required intensity of suction is very small, as it 

cwrrospontls to values of the order cQ = 10-% A soll~tion for the flat plate with nniforni 

swt,ion in a c:ornprc~ssit)lc st,rcam was fonnd l)y 11. G. Lew antl .J. B. Fanuoci [47]; the 

s:mc prol)lt:nl for cylindrical lmdit:~ of arldxary cross-scction was solved by 

W. WIIOSL [107]. 

J.M. Kay [41a] undcrtoolt to verify thcse theoretical results for the flat plate 

at zero incit1cnc:c with t h aitl of cxpcrimcnts. The assnn~ption that uniform snction 

btyjns at the leading ctlgc, which formed thc basis of Iglisch's theoretical calculations, 

was not satisfied in the Ccst platc. The latter, moreover, had a portion near the 

lcatlir~g edge complctcly devoid of suction. Fignre 14.10 shows a comparison betweell 

the measured and calculated displacement thickness and momentum t.hicltness 

rcspnctively. The asymptotic valucs from eqns. (14.7) ard (14.8) are seen t-o have 

been confirmctl by tho measnremc.nts. Fignro 14.1 1 shows a comparison bctween 

tl~oory and mcasurerncnt for various values of (; the mensl~remcnts havc been 

performed by M. It. Ilcad [%I. Again, the agreement is very sat.isfactory. Measuremonfs 

pcrforrncd by P. A. Iitjby, I,. 1Canf:nann and It. P. 1I:~rrington [48] confirm, 

in adtlition, tho strong stal)ilizing ellixt cansed by suction (increase in the 

nrit.ion.l Ileynoltls number), as will be reported more fully in Sec. XVITc. The 

large decrease in l,hc skin friction which results from the preservation of laminar 

flow when suction is applictl, and which is shown in Fig. 14.9, was confirmed by 

nicnsun:mcnts performed by M. ,Jones antl M. R. Ileatl [41], and A. Raspct [70]. 

Fig. 14.10. Laminar bortntlary I:~ycr on 

a flat plate at zrro inciclrncc with 

uniform suct,ion. Displaccmcr~t thickness 

6, anil motncntnn~ thicknws (7, havc been 

meeured by J. M. Kay [41 a]. Theoretical 

curves afkr R. Iglisch 1401, 'hblo 14.1 

a = necLion at which suclion begins 

Pig. 14.11. Velocity distribution in the laminar 1wund:lry layrr on an acrofoil with srtct.ir)n 

applied t.llrough its porous surface. Mea.wccn~cmb pcrforrnccl by M. It. Ilcacl [XI: ccrrnpnrison 

with the theory due Lo It. Iglisch [40] 

Boundary layer with pressure gradient1 Acltlitional czncl solutiona of 1.11~ bot~r~tlary-l:lyc!rcq~~n. 

tions (14.3) and (14.4) are known only for flow patterns which can bo nwociaLt?cl'wit.l~ sintilar 

velocit,y profiles. The class of similar sol~rtions discussed in Clr:lp. VIlI can be cxknclrnl 10 inclutlc 

boundary layers with suction and blowing. When the vclocity in the external strcxn~ can bc tlcscribed 

by the function IT(%) = 76, zm and whcn the sr~ction vclocity v,,(a) is proportinnnl to 

z(ll3(~~~-l), werecover from the boundary-layer equations the already fa~uiliarortlinary clilti:rt~r~i.i:rl 

equation for the ~Lrcam function /(11), hst clrrivecl hy Fnlkncr and SIZ:~, narnrly tho r;l~nili:tr 

equation (!).8) : 

in which 71 has been clcfincd in eqn. (9.8). That this is so can \I(: infcrrctl by inspection frnnt rqn. 

(9.b. In t.lw present case, t.he slrcanl functio~~ /('I) 11.25 a VLLIIIR w11icI1 is dill'twnt fro111 mr11 :LL 

tl~c wall WIIRII 11 :-- 0. This value is positive in the tasc of snol.ion and ncpll.ivc Tor I~lo\\.i~~g. 

Tlw part,irular case for 111 =: 0 wluch corresponds to a flaL with a nucf.ion vrlot:it.y 

-- 

6 > 0 : s~~ction 

z (' < 0 : blowing 

was invcstignkd by II. Sol~licht~ing and K. 1111ssn1;~nn [7!), 801. The rcs~~ltitlp vc!locily prt,liilc.s 

for scvrrnl values of the vol~~ntc cocrficicnt I~nvc I)ccn ploLt,ctl in Fig. 14.12. 11, is worth not.i~~g 

thnl all relocil.y profilcrr for the c..zsc: of tlisclt:~r~c hrlvo poin1.n of inllvxion 1r.il.11 i)2i~/i'!y:! -- 0. 

'I'l~is fnrt is important for the sl.udy of tmnsit.ion (Chap. XV1). Sintil:tr vrlorit.y prolilc.s nrc. also 

obtninrd in the case of I.\\-o-di~~~c~tsio~~al stngt~ation flow with a vc:locit.y fi~nc*t.ion O(x) -: 71, r 

wiClt suc1.io11, provided that 11" - consh. This caw urns also inv(vd,ig:it~d in 1.110 p:lll(:r IIY I I. S(:ltli,.l~ting 

: ~ t d I

390 XIV. Ilo~~n~lnry-lnycr contml 

Fig. 14.12. Vr1orit.y ~listrilwt~ion in t.hr 11ouncl:rry 1:ryl.r on n hl. plnkr nt. 7.0ro inritlrnrc with snetion 

and t1lsrh:rrgc nc:cordinp 1.0 t,l~c I:rw a,(z) - 1/1/ i from cqn. (14.1 I), :rflcr Jl. Scl11icht.ingnncI 

I 0 dcnotm s~~ct~ion, I(0) .: 0 

clcwh~n I~lowing, and 

n) K. D. P. Sinhnr [86] stutlietl the ewe of nn infinitely long, ynwccl cylinclnr with surt.ion. Tho 

velocity distribution n.long tho strcnn~ wns nsaun~od lo lm proporlionnl to zm. 'l'he inv(~n(igntion 

lmu ROIIW I*onring on t010 cvnt,rol of tho 00~1dnry lnycr on R W C wingu. ~ ~ ~ 

IJ) When t.hc Lc:rnpc:rnt~iro of Lhc fluid being 1,lown out, is diKerenL froln t,l~nL in CIIC cxtcrnrrl 

flow, tho bonndary layer will develop o tcmpcrnturc profile; Lhc rwnlt.ing thorn~nl I)ountlnry 

layer was cniclllntcd in refs. 1551 and [Ill]. 'rhc knowledge of the tcn~peral~lrc distrihrltion 

in t h bonnclnry layer is of particnlnr importance for the ~)robletn of cooling. It tnrnn ont 

that cooling by means of Ihwing tho coolant, through a porons wall, so-cnlletl t.rannl~ir:rtion 

cooling, is much more eKcctivc than cooling tho wall on tl~c insiclo. Jn this connc?xion the 

pnprs by I). Jhown [I 1, 121, p.1,. I)ono~lghe and .J. N. B. Iivingoocl [I!)] nnd W. WII(:S(~ I 10!)( 

nrny I)o conn~iltntl. 

C) 'I'hc cooling prol)leni I~ccon~cs wry itnportntit nl, high vel~~cit.i~:n of flow. (:. hl. ls)w 151 1 forr;~cl 

uoIut.iot!s for ( 1 rnsc ~ of r.~rnprc~~i\~le (low ewer nn isot.l~rrn~:~l IIIL~~ IIIR~; RCO nlm 145, 1101. 

. > 

*3. " 

cly 

p " -- ' - 

cly '" dy ( I' -&j j . 

tlu - 

-. 

71 - (Irn /L 

(14.12) 

1 1 1 ~ p~wwli~ig rrl:l.l.iotts :ire valicl for :LII :~rl)il.r:~ry V:I.~III~ of' I.Iw I'I~:IIIIII,I 11111111~1.. 

I I,(: sl~c*:tring sl.r~,ss :,I. I.IIc w:ill is now 

7 , 

T,, =-

392 XJV. I%outldnry-layer control 

(P .- 1; adiabatic wall). When thc flow is incompressible, we have T, = T, , and 

cqn. (14.18) rrc111ccs to cqn. (14.6). 

1.3. Approximate noltikiona. In t11c general case of an arbit,mry body shapc 

and ILII nrl~itmry law of ~uction we must resort to approximate methods based 

on t.110 morncntum cqt~atiotl ; tJlcy wcro clcsc:ribetl in Chap. X. The momcntom ocluat.ion 

for (.he case with s~~(:tion is ol)t,ninccl in ex:~c:Lly tho snrnc wny as bcfore, cxccpt that 

it is now ncccssary to take into account the fact that the normal component of the 

velocity at the wall tlilTcrs from zero. I'crforming t.he same calculation as in Scc. 

VI IT c, we firttl that, the equation for the normal component of velocit,y at a distance 

y = h from the wnll now becomes 

Vh = lJO -- j 

-- 8% ( I . ~ 

ijz 

0 

'She ralct11ati011 is ~:onl,inucd in exactly thc same way as in Sec. VTTTe, and leads 

finnlly to the following rnornrntt~m rqit:~tion for thc bouttdary I:tycr with suction 

the- rnrrgy-intcgral rqtml ion, areording to I

. - v, (x) =cons/ 

Fig. 14.14. 1,antinar l)ou~lcl:try 

layer on n ~yt~~~ncLrie:~l Zltukovskii 

norofoil with uttiforttt s~~cl,iott; 

II,,(x) -- COIIRI~, t~ngle or it~c:idc~~w 

a - 0, ns cnlct~lah:d I)y I':.'l'ruckc~tbrodt 

[loll 

II'ltrr~ srtc.t,iort is :~pplirtl t,o n wing, it is ncccssary to discern two distinct problcn~s 

whir11 ntipltt~ : I I : ~ 

1. 11. ~rl;ly I)(% (l(:sirrrI t.o incrcnsc t11c maximum lift hy dolaying scparnth~. 

2. 11, nlny I)o (lnsil;rblo t,o m:~inl.nin 1:rminar flow and to avoid tmnsition in order 

lo 1.(~t1~~(~(~ sl

XIV. Boundary-lnycr control b. Boundary-lnyer suction 397 

Fig. 14.16. Incrrascin thr ~nnxitnnrn 

lift, of n swept-I~n~~I~ wing 

by s~~ctio~~.Cornpariso~~ brtwern 

rontinuous suction and suction 

applied tl~rough slib, ns Inca- 

uured 11y 15. D. Popplct,on 1661 

Ileynolds nurnbcr R =. 1-3 x 10'; 

rrlrlivc width ol slila rll = 0-004 

:~c:rolhils. Sint-t: :11. high a.nglvs of inc:itlonc:c: thin acrofoils tlcvciop n s11:~rp ncg;~.I.ivc.prcss~~rc 

p(dc nc::~r I,hc IIOSV on t11c 11p1)t:r sitlt:, it, is nw~:ss:~.ry 1,o :~pply S I I ~ I ~ l,l~crt-. 

~ I I 

In this t:onncxion it is important tlo know whctl~cr to apply suction t,hrough a porous 

wall (~~nifnrm surt.ion) or t.hrough a systcrn of slits. The diagram in Fig. 14.16 shows 

;I, comprison l~t:l,wccn t,he results of cont,inuous suction and suction applied through 

slits on n swr,pf,-hack wing as measured by E. 1). Poppleton [66]; see also ref. [38]. 

11, is dc:w that the snmc incmasc in tho lift coefficient can be ohtaincd with n much 

rccl~~cod mass flow when c:onl.inuous suc:tion is usctl. l'hc diagram in Fig. 14.17 

c~on1,:iins informnl.ion on the most favonral~lc position of t,he suction zone at the 

nos(:. 'l'hc mcasummcnt.s carriccl out on an 8% thick symmet,rical aerofoil seem 

t.o intlio:~.t,c th:~.t continuous suction is most cfTcctivc when it is confined t.o the npper 

sitlt: or t,hc wing and when it cxtcnds over a region of 0.15 1 approximat,ely. 'I'he 

tni~~irn~~rn mass flow rtyuirrtl t,o avoid scpnrntion depends on the position antl thc 

I'ig. 14.17. MTcct, on in- 

c:rmsc ill lift, coefficient of 

c11:uiging tho position of the 

porons si~ction snrfacc for 

nn S :(, thick acrofoil at. an 

nnglc of incitlcncc ofa - 15" 

extent of the porous surface and, even more significantly, on the Ileynolds nunher. 

This, of course, is a very important consideration when results of model experiments 

are applied to full-scale arrangements. Some data on thc depentlcncc of t1he mass 

flow on the Reynolds number are shown in Fig. 14.18. 'J'hcy arc based on mensurcmcnts 

performed by N. Crcgory and W. S. Walker [32] on a thin symmetrical 

arrofoil. 'l'hc graph shows thc minimum volumc flow of suction rcqnirotl to avoid 

separation for a f xed anglc of incidence or a = 14O plotted in terms of tl~c Rcynoltis 

number. Several curves of cQ 0 = const, whicll were obtained from the theory 

of purely laminar flow, have also been plot,tcd for comparison. 

2.2. Decrease in dmg. An exprrimcntal proof of the fact that it is possible to 

maintain lnrninar eontlitions in the bonntlary lnycr with the aid of sn(+or~ wns 

firsl, givcn by 11. Ilolst,cin [37], and shortly afterwards by .J. Acltcrct, M. ltas 

Fig. 14.18. Minimum suction volume 

required for the prevention 

of sc*pnrnt,ion ns n fnncbion of ll~o 

Reynolds 1111m1)er for an angle of 

incidence of u = 14", after Gregory 

antl Walltrr [32] 

and \IT. Pfenninger [3]. W. I'fenninger [el] cardcd out oxtensivc cxperimcnts on 

the problem of reducing dmg by the application of suc:t,ion tllrough which 1nn1in:~r 

Ilow is maintained. Figure 14.19 reproclnccs some of his results, ol)l.ninctl with a thin 

:wrofoil which was provitld with a largo n~~mbcr of sliction slit,s. 'I'hc gr:~1)11 in Vig. 

14.19a sllows thc optimum vnlucs or Lhc skin-fricljion c:ooflicic:nt plothl ill I.c:r~ns ol 

tl~c Ilcynolds number. It is sccn that t.hcrc is a largo saving in tlr:rg, cvcn if the 

power consumption of the suction pump is debited against it. 'l'hc graph shows, 

I'urtl~cr, that, at moderatc values of the lift cocfficicnl,, cvcn at largc Itr~n01~Is 

~~u~nl)c:rs, the values of the skin-friction coefficient arc not muc:h l~ighcr t.l~n~~ I.llosc 

for a flat ~)lat.c at zero incitlcncc. Moreover, Fig. 14.1!11) ticmonsl,rnl~cs t11:~t thcsc 

low vnlurs persist, over a ~onsi(lcr:~1)1c rangc of vnl~cs of t11c lift (:n(:fli(:ic~li,, c,,. 

J'III~IIC~, the expcrimc:nl.s tlo~~~o~rsi,r:~l.c:tl 1.11:~t l.I~t: tlcc:rc::rsc ill l,l~c tltxg t~fT~:ctc.tl I)y 

~~~:lirlt:~ining a Iatni~~nr I)ol~ntl:~ry I:~,yc:r wiLh thc :~.itl of sucl,ion tlcpc~~tls I:trgrly on 

n rnrrfnl shaping of slits. If this proc:~nt,ion is not t,al

invc?st.igai.ctl. In this msc too, sltl~st~:~.nti:rl rcc1uc:tiorm in tlrn,g were achicvccl, allowing 

for lhc mnclrnriic::t,l work roqrtirctl to maintain it. 

W11(?tt an n.t.t,cmpt is matlc t,o prcscrve a laminar I~olinclary laycr citllcr by 

suction, or, as already rncnt,ionctl, merely by proper slln.ping, it is vcry important, 

t,o have a gootl Irnowlctlgc of bllc potcntti:d velocitcy clistrib~~tion. In cither case it 

is ncccssn.r.y tt.o arrangc for t,Ilc prcssltrr 1.0 t1c:crcnsc ovcr as Iargr. a portmion of the 

section as possible. Very oxt.cllsivc cxpcrimcrrts on this sul)ject were carried out, 

by S. C;loltlst,cin 1311 n.ntl Iiis coll:tl)or:tl,ors. 'I%(: cnlc:ulxtions lccl to the tlct.crnlinat,ion 

of the* sl~n.pc of I.llc? scc:l,ion of t,lto :~.c:rofoil which wo~~lcl procltlcc: n ~)rcsc:rilml pot.cnl.inl 

vvloc.it.y clislril,ttl.io~~. III ortlvr to ol)t.:r.itl :tc~roli)ils which nlnintnitl n. Inminar 1)ountl:l.ry 

l:~.yc:r :IS hr :IS lilt- tr:lilillg c~ljy: it. W:I.S s~tgg(~st.ecI 1.0 ltsn slinpcs sliowing a 

dcerc;:lsc: ill prcwttrc: (:it1 itlcw:lsc: in vchc-il.,y) ovvr l.11~ wholo Iotiglh, n.nd otrly 

(lisl)l:~.yi~~g :IWI :I.IwII~~, prwsltrc in(wxsc :~t, on(: posiI,ion, n.s sl~own in Vig. 14.20. 

If t.Iw slils arc nrr:~.t~grtl nt. 1.11~ ~)oint~ of prcssltro jump, n.s snggcst,ctl by (:riflil,li 1731, 

it is possible t,o secure a laminar bonndn.ry Iaycr otl thick nerofids as far as the 

lit and separat,ion is prevcntctl 1)chirid it. B. Itcgcnscllcit 171, 721, and B. 'l'l~wait.os 

[Mj proposcd Lo 'regulate' tho lift on vrry t,lliclc norofoils by varying the 

intcnsit,y of sl~ctdon ant1 so to obtain a lift which is intlcpcntfcnt of the angle of 

incitlrncc. In morc recent times there were many proposals to use tho air srtclretl 

aurn.y from the bountlnry laycr for the purpose of irlcreasirtg the thrust of :t jat 

a.ircraft [87]. 

'1'11~ papers 1j.y F.X. Wortmnnn [I051 and W. I'frnningrr 102, (31 rrport, on more 

rc~c-nt~ r~s~tlt,~ (:onc:orning tlic design of latninar acrofoils and of the tlcl:ty of t,r:tnsit.io11 

o~t xwc:pt-l):tcrlc wings. 

,, I ho ~WO~:CSS 01' t.rn.risition frorn hninnr to trrrl)~~l~:i~l~ Ilow in tho Im~~rcli~ry 

layer witl~ s~i(:t.ion will 1)r. stltdicd in (lrt.:til in Sc:n. XVl l c:. 

1. Theore~ical results. 

U 

.- 

urn 

c. Injcctinn of a diflcrent gns (Ihnry boendnry layers) 

1.1. The fundamental equations. Whrn R sp,zco-vn11ic:lo robr~r~~s to the tlcnaor layers of the 

nt~nospl~rro, t.lw ~l,zjinnt.ion cfTccct whir11 is protl~~ortl nt t.l~c? t~ose or in 1,110 I ~ o I I I I ~ Inycrs : ~ ~ ~ along 

1,110 wdls givc8 risc l,o vcry high t.c!t~~l~er~il,~~rt*.q. 

111 ortlvr Lo rcx111w 1.11~ q~~:inLit,y of 11c1at Lr:u~sfwwd 

to tile vchicle to s~~tell proportions, it is possil~lc t,o i~~jcct, n light. g:ts or a fluid tl~ror~glr a porous 

well. l'hc light gns or thr v:rponrizing fluid thus rrcnls n thin filw nlong tho wdls. A si~nil~r ctl'rct 

ran :also IN: procl~~nc~l il 1.1~: rnn1nri:al of t.lw will (r.g. grnphitn. gl:wn, or n ~ynthct.ic ~l~:al~.rinl) 

i~ nllowctl Co sublit~wtc thtm rrdl~cing it8 tl~ick~lcss (n.blat.ion). III :dl S I I ~ onnrs, I I~o~rnd:iry Inycrs arc 

for~ncd in which two or morc gases mix with one anot.lrcr by din'~minr~.

400 XIV. Boundary-layer control 

Owing to the definilion of w, we must have Ze, Wc = 0, and for each component i we may 

write 6hc law of maus conservation in the form 

div (er wi) = div (e{ (W -t W{)] = 0. 

(14.24) 

Upon summing ovcr all components, we obtain the continuity equation 

which has the faruilar form of eqn. (3.1). 

tliv (e W) = 0 

111 tho rrbsonco of nxbrnnl ficlds, tho clifiuuivo flow is drivcn, cssenLially, by conccntration 

grndicnta as well as by tl~crmal difiusion which prorlucea n flow of musses in the presence of n 

tenipcrntr~re gratliont. In tho case of a binary mixture, we may write the law of tliffusion in the form 

c, W, - - Dl, (gmd c, -t kT grad In T) , (14.26) 

whcrr I),, clonotes the cocfficient of biuary difiusion, kT is the thermal diffusion ratio, and c, = el/e 

is tho masq conccntration of lhe first gas, assnmod to be the one which emerges from the wall. 

'J'hc c:orflicinnt of binary tlifiusion dcpends only litllc on concentration and is affected by temperntnrc 

in tho uamo way as the kincmntic viscosity. Thc thermal difiuaion ratio, kT, depends essentinily 

on concont.rntion and is frequonlly npproxirnabd hy the rathcr crude relation 

kT = me, (I -c,) 

(14.27) 

duc to Onsagcr, Furry and Jones. Here, thc cocfficient of thermal diffusion, a, is assurncd to be 

a constnnt for wcry specific cornbinntion of gases. 

Inserting cqn. (14.26) into thc law of mass conservalion, eqn. (14.24), written for the first 

component, and taking inlo account cqn. (14.25), we obtain 

Wr may now introdwe the normal honndary-layer simplificatiolls into the right-hand side of this 

rqnnt.ion tl~ns t~rglrrtin~ krms in a/& with respect to those in a/&~. In this manner we obtain 

1 he ro71rrnlmlion rpolzon 

A corrcspontling oqunlion is valid for tl~c scxoncl component,; I~owcver, this second equation 

brcon~cs txivinl wlmn (,hc niotlificd form of cqn. (14.28) is uscd because c, 4- c, = 1. For this 

rrnson. t.11~ S~:COII~ (:q~l:ition is ropl:lc.crl by thc continuity equntior~ (14.25). 

Tho ~non~rnt.rttn cq~~nlions 

wril.tn~~ 

for :I, gas rnixluro :rre identicnl with those for a sirlgle gss antl are 

a7' =0, (1 4.30) 

a!/ 

wl~rro now Q nnd p clrpcntl 011 roncrntr:lt,ion in addition t.o their familiar dcpc~~dcnce on tetnpc. 

rat.urr. . I 

, 

IIC rncrgy cqnat,ion for a gaseons mixt,nre must bk formulated with due rcgzrd being peid 

tm Ihc r~ormal tl~orn~nl oo*~~lnction, to the transfor of h6at by diffusion, and to that by thermal 

~~~I'IIS~OII. Itc~t~riclillfi our ~0114i~lcriltiolls 10 pcrfcrt plscs, w 1111rod11~o ~hc mixhrc ct~tl~alpy 

h = cI ha -{- C, h, . 

(14.31) 

Sincv: tho tloriv:rl.icul is Irngl.l~y, wc rncmly qnot.c thc rrsult. in which the boundary-layer approxirnnt.ion8 

hnvc nlrcntly I~ecn introtl~~cod: 

Ilrrn R sI:intls for 1.l~: ~mivrrwl gns onnsln~~t~. If lll~rr~n~1I (~~~I'IIR~oII is ~~rglw:t.rd, ~.II(! nntl(-rIii~(.(l 

trrn~s nrc: tlclotod. III the tlcrivaLion of this cqualiol~ IIRC 1111s lw(:n ~nnd(! of OIIHI~~(!~'H pri~wiplo 

arrortling lo wl~ich tho corfficirnt of Wlo oonccntmLion grnrlicr~t in UIC brat-flux vrctor in tho 

samr :IS thnt of thc trn~pcrat~ure gradient in tho mass flux. 

and in view of oqn. (14.20). wc olhin tl~c conrlit.io11 t,l~:rl 

(grad el -I- kT grad 111 7') 

I

402 XIV. Dounclery-layer control 

1.2. Exact solntions. In order to solve the coupled pnrtinl difFcrct~tinl cquntious of the parabolic 

typo we Iinvo, nt. prencnt, nt our disposal n variety of riumerical methods [97, 421 as well es 

fast rleot,ronic comptitcro. With tho aid of thcoo, it. ~ RCOI~RR po~~ibh to obtain ahnost nrhitmrily 

close npproxinint,ionn to the cxnct solutions wit.11 n tolernhlc expenditure of Lime. The properties 

of the Iluid can be conccivcd as qunnt,ities thnt vary with position, nnd nrbitmry boundary 

ror~ditionn cnu IIC 1)rencriberl. It is posnihle t,o obtain similnr solutions if the external velocity, the 

blowing velocit,y, n.s well RR tho temprnturc on t11c wall, arc prescribed in n dcfinitr: manner. In 

ouch oases, the sy~lmn of 1mrLiaI diffwc~itittl oql~:ttion~ rednccs to n ~y~t.c~n of ordinnry clilTcrcntinl 

equntionn, nnd tho Inlhr can 110 inhgratccl nun~cricnlly. T11t:ro exist such nunioricnI rcnulb for 

incomprenniblo wedge-llows (inclusive of shgrintioii flow [102,29]), comprrmihle flow over a flat 

plntc nL zero incidcnm, nntl oupr~onic boundnry Inycrs on wcdgeo and cones [IlO]. 'l'lw diagram 

in Fig. 14.21 illuotdr:s, by wny of cxnmple,th Inminar vclocit.y, tempcrnt,urc, nntl conc~iit~rntion 

boundnry layor on n cone with l~clirrm injection. 

A tnct.liocl tlcoigncd to cnlculnte Inminnr, hyporsonir, hinn.ry-mixt,ure hountlnry layers wns 

givnn by ,I. SLei~~bcuer 101 J who npplicd it to t.he exan~ple of cooling by ablation with tho mid of 

pyrdizing tollon. 

All of tbo numcricd ralorilnt,ions montioncd RO far neglect the brnms which stern from thermal 

diffusion, that is t.hc terms which hnvc bccn undcrlinctl in eqn. (14.32). Sucb n simplification is 

aonwtimcs pwmi~siblo RR far nu tho compuhtion of nkin frictiou and hcnt-trnnnfcr rnto in rouccrncd. 

Expc?ritncutn show that LIIC eqnilihrirrm ternpcrnt,c~ro on an ndinbntio w~rll docs not clnnronrro ill 

the prrscnce of thcr~rral diffusion, but ~:ilcuIntions bnncd 011 thin nin~plificd ~(:IIOIIIC nlwny~ predict 

ouch n tlcrrrasc. 

I':xact, (::rI~:~~I:i.t.ion~ on t,~~-suhnt,nnr.r. ho~tndnry layers wbi(:b occ11r ill flows with ovnpornt,ion 

or ~11llli111ntio11 ~)rcsrnt. 11s wit,h wm~idrr:thlr diffi(:~~lt.ic~.'l'I~cclist.ril~~~tio~~ 

of vclority oFt,hccvsjtornt.illg 

nl~l)nLnn~:c (i. o. of t.11~ velocity of blowing) :tnd 01 IonipwnLurc ttt tbc phnnc houlidnry in L11c 

flow clirootioii c:nn no longar hc prrncrihctl ;wbit.rerily. I111lh di~t~ribut~io~is :brine s~)011tn11C011~~y nl 

n r=qult of thn coupled bent, nnd Inwa Lmt~sfcr and ncit,hcr is known n priori. In this clolnain, 

W. sl~lctkton~nrr [00] calculntcd n large number of solutions in wbich tbeevnporation rateas well 

:m t,l~o lodly snlinfirtl energy hnlnnce Imvc I~mn evnl~~ntctl on tile basis of' eqn. (14.53). 

F. 1':iofcld 1211 lmblinl~crl solrit~ions for flow of binnry inixt,weo tht n.risc in the prcne~~cc of 

t.hc nclinl):tt.ic: c~n~~~r~rtion of n lilm of c:trbon tlioxitlc wit.11 n specinl n~st.hemnt,icnI form nssumccl 

for t.he law of ovnpornt.ion. 111 thin work, he dincovcrrtl t.Imt t.he proccm of ntlinhntir. cvn.pornt.ion 

of n plnrw film lc;uls to self-similar solutions,~/. (hp. VIII. 111 snc11 cnnrs, it t.~lrns outl t.l~nt t.11~ 

locnl rtrte of cvnporntio~ niurt follow n l/~z-typc Inw. Thin in tho tlintrihntion of Lbo normnl 

vc+n.it.y ill owt~i~n or blowing on R h t . plii.t(: id m:ro inri(lr~~(:c h t Irndn to wlF-similnr soI11tionn, 

nn il111slrnt.ccl in Vig. 14.12. :I'lw t,~n~~icritt~~~rc 1111d wn~~~~tr:tt.ion nt. 1J1c snrfn.cc of 1.I1o lilm turns 

out, to he uniForm. 

1.3. Approrimate mlutions. Jt in possible tn simplify t h problrm hy nsquming I.hnt thc 

rrntdt,l numhrr, P. rrwl Lbn Srhtniclt, nt~mhrr. S, -- v/l),, nrr oq~d to u11it.y :d Lhnt the viscosity 

i~ n linenr funot.ion of tr:mpor:th~rc. With t,l~cno nnnntnl)t,ions, C. It. k'anldcm 1251 onlculnhd the 

sbrnring ~ tmw nL Lbc wnll wl~cn n light gw is injcctcd: lie co~~ni~lcred C ~ C R of vnryi~~g motcculnr 

nmw ratio wilh rrspr.ot, tn) t.lto II:L.C g.1~. More grncrnl rnncs 01 oxtcrnnl-velocity n d it~jecl~io~lvr1orit.y 

clist.ribut.icmn ran IIC nn:tlyzrd with 1.11~ aid of the integral cquntions [log]. 

Rrfcrrncea 403 

[I] ~Ickerrt, d.: Ihs Rotorsrhiffund sritic physiknlisch~ (Irnncllngcn. Vn~~drr~l~ot.ck 1111tl 1{11y. 

precht. Got tinnrn, l!)25. 

[GI htro~;, .I.R., and Scot.t, P.E.: Some rnnns t,mnsfcr rcs1111~ \vit,I~ exter~inl ~Io\v II~~RRII~C 

gradientan. JASS 27, 025 --ti26 (1 9W). 

171 lktz, A.: 1)ic IVirkungnwaiae von ~mt.crtoill~en I'liigclprofilc~~. Ilcric:l~t.r IIIIC~ I\I)II. Winn. 

Gencllnrhnft f. Lnft.fnhrt, No. O (1922); NACA 'I'M 100 (1!Y!2). 

[a] Uet.z, 8.: 13eeinflurwung dcr Itciburig~scl~icht und ihre prnkt.iclcl~c \'crwcr(,~~~~g. Svhrift.ct~ tlt. 

Akad. f. Luftfnlirtforscl~nng No. 49 (1939). 

[!I] BeLz, A.: J1intor.y of bonndnry lnyrr contd rescnrch in (:c:rmn~~y. 111: I%o~ctl&ry I;~~vr rrlltl 

flow cont.rol ((:. V. Lnclirr~nnn, cdj, 1. 1--20. Imnhn 1001. 

[I01 Ilrnnlow. A. I. .. I111rrown. I). I,., 'l'ol.(:rvi~~, N., ILII~I Vin(:o~~I.i. I?.: I ~x~II.~~III~II~.~I.~ 

IIII,~ t.l~(:ot.t:(i(:~~I 

stritlic~ of nrm sriction for the cont.rol of the laminar bountlnry laycrr. NACA JLcp. 102.5 ( I!)51). 

(1 11 I3rown. 15.: Exnot solr~tionu of the In~ninnr bountlnry lnycr cquntionn for n ~~oronn 111rr1c wit.11 

variahlr fluid ]~rol)erties nnd a. prrssurc grarliont in 6110 rnniu utren~n. Proc. I'irnt US Nut. 

Congr. Appl. Mcch., 843-852 (1951 ). 

[I 21 Ilrown, Mr. 11.. nncl J)or~oughe, 1'. L.: 'J'nblo of cxnct ltiminnr I~ountlnry lnycr uol~~t.iona when 

the well in porouu and flttid properties are vnrinblc. NACA '.L'N 2479 (1!)5I). 

[12n] Cliang, P.K.: Control of flow scparntion. IIernispliere I'ublisbing Corpornt,ion, \V:~nl~ingto~~ 

1)C (1976). 

fhrri&o, 1':. and Eirhclbrct~ncr, E.A. : Theory of flow rm.l.trrchrr~ent. by n tnngc~~t.inl jrt, dinclinrging 

ngninst n ~t.rong adverse prcns~~rc gradient. 111: Jhdtrry 1:rycr 1tnt1 Ilow (:ontrol 

(G.V. I~ncbmann, ed.), 1, 200-231, London, INil. 

Clnrkc, J.Ii., Xlenlres, H.R., and Idbl)y, I'.A.: A provinionnl nnnlynis of turhulcnt. ho~indnry 

layers with injection. JAS 22. 245-200 (1955). 

Curle, N.: The catirnetion of laminar skin friction including clTcctn of tlistrih~~tcd nuct,ion. 

Aero. Quart. 11, 1-21 (1900). 

[10] Culick, F.E.C.: Integrnl n~cthotl for c:tlculnting hcnt and mnnn Lrndrr in Itiminnr ho~~nclnry 

layers. AIAA J. I , 783-703 (llK1). 

[I71 Dn~menhorg, It. E., nr~d Wciberg, J . A. : ISlTcct of t,ype of porons s~~rf:tc:o nntl suct.inr~ ve1onit.y 

d~stribution on the chnracteristics of n 10.5 per cent tl~iok airfoil wit,h area sl~ct,ion. NACA 

TN 3093 (1953). 

[la] von l)oenhoff, A. E., and Jmftin, LIL: PrescnL stntns of 1rscnrc.11 on bormtlnry lnyrr rontrol. 

JAS 16, 729-740 (1049). 

[In] I)ot~ooglw, P.L., and Livingnod, J.N.R.: Exact aol~itionn nf lntrriinw boi~ntlnry Inycr rqmtions 

with constant ~ropertv . - " values for porous wall with varinblr kmnrrnt.urr. NACA llcn. 

I229 (1955). 

[20] I)orrancc, W. H., arid .Dore, F. J.: Tl~e clTcct of maw t.mnnfcr on t.l~c romprcsniblr t,nrhulent, 

boundnry lnyer skin friction nnd hcnt tmnsfcr. JAS 21. 404---410 ( 1954). 

[20n] Rckcrt, JC.12. (:.: 'J'hcrt~~otly~~n~)iiscI~c I

404 XIV. 13oundnry-laycr control References 405 

1251 Ihnldrrs, C.R.: A notc on 1;irninar Inyrr skin friction under the influence of foreign gnn 

. . 

injnction. .JASS 28, I(i6 - 167 (I!WI). 

1261 Ihvrc, A,: (:ot~t,ril~~~t,io~~ h I'6tude expi:ritnent;rle des mouvetnentn I~~drod~narni~uesbdcux 

elitncnsie~ns. 'lXhesis Univcrsit,~ of Paris 1!1:38, I-- 192. 

[27J I~lnt.1, .I.: 'I'hn I~isLory of boundary Iqycr contxol rcsearch in the United Stah of America. 

In: Honntl:iry I:ivcr and flow control (G. V. I,achm:um, cd.), 1, 122-143, London, 1961. 

1281 l?liiecl. (:.: l~r~e:l~nissn BUR dem Sl,ri~~nl~l~g~itlstiL~~t dcr 'I'eeltni~~l~nn 1~0chschnl~ T)anzig. .Jb. 

' ~ch~ff1;arrtcc:h;;. (:csellsr:ll:ift 31, 87 -- I l:f (l!)RO). 

[2!)] Fox, H., and I,ihby, P.A.: Ilcliu~n injection into the boundary layer at an axi~~mtnetric 

stagnation point.. JASS 29, 921 (1962). 

12!ta] (:C~RI.CII, I

Ni\('A 'rM !I74 (1!)41). 

1861 Sinlinr. I

CHAPTER XV 

Non-steady boundary layers t 

a. Gencrnl remarks nn the cnlculntiott of non-wteady boundary layers 

The oxamplos of solut,ions of the boundary-layer equations which havc been 

considarcti until now rcfcrrctl to stcady motion. 't'hcy arc by far the most important 

cases cncountcrctl in pmctic:al applications. Ncvcrtl~elcss, in this chapter we propose 

to consitlnr scvcrnl examples of motions which tlcpcnd on time, i. c. of non-steady 

1)ountIary Iaynrs. 

The most comnlon oxnntplcs of non-stcatly bountlary layers occur whcn the 

motion is slnrted /rom rest or whcn it is periodic. When motion is started from rest 

both the body antl the fluid havc zero velocities up to a certain instant of time. 

The motion begins at that instant and we can consider either that the body is 

tlraggcd through the fluid at rcst or that the hods is at rcst and that the external 

fluid motion varies with time. In this lattcr cam an initially very thin boundary 

layer is formed near the body, and the transition from the vclocity of the body to 

that in the extcrni~l flow takes place across it. Immediatcly after the start of the 

motion t.hc flow in the whole fluid space is irrotstional antl potential with the exception 

of a very thin Inyor ncar the body. Tho thickness of the boundary layer increases 

with time, and it is important to investigate at which instant soparation (reverse 

flow) first, occurs as tho I)ountlary hycr cont,in~~cs to build up. One such example 

was ;rlrcatly consitlorccl in Src:. V 4; it was the exact solut,ion of the Navier-Stokes 

equations for the flow noar a wall which is accclcmted impi~lsively from rest and 

rnovcs in a clircct,ior~ p:~rallcl t.o itself. Also, the start of the flow in a pipe (See. V 6) 

I)clongs to thc sanlc category. 

1'11rthrr cxnn~ltlrs of non-stjcady I~ourtctary layers occur when eithcr the body 

performs a poriotlir motion in a fluid at, rrst, or whcn the body is at rest and the 

fluid rxoc:ut.rs a pcrioclic motion. The motion of a fluid ncar a wall which oscillates 

in its own plnnc: (Srr. V 7) :~ffords an example of this type of problcm. 

1. Ilnundnry-lnycr cquntinns. The funtlan~cnt~al equations for non-steady boundary 

layers have already Occn tlcducetl in Scc. Vlla. In t11e general case when the flow is 

c:omprcssil)lc and iron-stmtly I)ut two-tlimcnsional, ,we must resort to the following 

cq~~a.tions for the vclorit.y n.ncl t.rmpcmturc ficltls (cf. eqns. (12.50a to e)): 

.~ - -- 

1. I nlll illtlrl,l.rcl l.o I'roScssor I

110 XV. Non-~teldy bortndnry layers n. General rcmarka on tho calcolntion of non-st.cady boutlclary layrrs 41 1 

Srorn cqns. (13.80) nntl (13.82). Whnn the: flow is incomprcssiblc, these relations 

simplify t.o: 

i I a ad air 

1 jJ2 ;,/ ('J2(12) I (1 I 3 (rR 53. == 

-- - -- 1 P (;;jZ (I!, 

0 

2. Tlln 111cr11nd or n~~ccmsivr npproximntinnr. 'l%e int,cgr:~t.ion of t.ltc norl-stec:atly 

I~oun11n~r-y 1ny1.r oq~t:~t~ions (15.1) t80 (15.3) (:an l)c (wried outt in IIIOS~~ onscs 11y :I, 

procrss oS s~twwsivt: ~~~)~)roxit~~:~~t~i~~tts. 

1,111: 111~4l1otl 1)cing l):rs(:tl on tJtc r~)Ilo\ving 

p11ysic:~I rc:~soning: In 1,Itc first. inst,anl,, :rll.c:r 1.llc tnoI.ion I I : ~ startotl from rest., the 

bountln.ry layor is vcry thin nncl 1.11~ viscorts term ~(i?~it~/r3?/2) in cqn. (15.2) is very 

In.rgc, n.llerrn.s t.11~ ronvcct,ivc t,crms rct.ain 1,lwir normal valuns. The viscons t.rrni is 

t11rn I)nlnncotl l )j~ (.lie notl-shtly arcclcml,ion r31s/i# 1,ogctltrr wit,l~ the pressure (.win 

in wlrich, :[I, lirsl.. t,ltc. cont.ril)ut.ion of alJ/ill is of major i~nport~ancc. Selnct.ing a syst,eln 

of roortlinnl,rs \vl~ic.l~ is al, rrst, with rcspc>c:b t.o 1,Itc Imly ant1 assuming t.I~at thr ll~titl 

niovrs willt rcy)c~c.I, t,o 1 .11~ I)orly at mst-, \vv WII III:I~~ 1.l1r. assl~mplion th:~t. I,he vrlocit,y 

is c.ontlws~~(I of' 1.1~0 1.rrtns 

ITttclw lsl~rsc c*ontlit.it>ns 1.hr first, n.~)~)rositrt:lt.io~~, it,,, snt.islics t.lte 1incn.r tlifTorcnt.inl 

(v111:t ti011 

whore 

nntl

412 

XV. Non strarly Imnndnry hycrs 

'J'ltc c~sontial .simplification of the theory consists in retaining only the throc underlinetl 

tnrrns in cqn. (16.22), which is thereby linearized and reduces to 

Ry rstimnting ordcls of magnitntlc it can be shown that the preceding approximation 

is a valitl ont? if the r:~tio of the so-callcd "ac" boundary-layer thickness, 

formrtl with the frequency n of the oscillation, is small compared with the steady- 

state 1)oundary-layer thickness 0 which would exist if IJ(x, 1) were equal to TJ(s). 

JIcncc, for the approximation to bc valid we must have 

wltiol~, in ~r;lc!l.icc, restricts the t,hcory to vcry high frcqucncics. It will be recalled that, 

the quantity a, cqn. (15.24), occnrcd in the solution to the problem of an oscillating 

plate which has hen considered in Scc. Va 7. 

Equation (15.23) which is linear and related to the so-called heat-contluctio~i 

oqu:~t.ion (6.17) describes the oscillating component ul of the boundary-layer profile 

and can he solvccl in terms of the given oscillating component U1 of the potential 

flow alone, bcoa~~sc t,lic process of linearization has made it independent of thc mcan 

mot.ion. Thc normal componcnk of the flow can be calculated from the equation of 

cont.inuit.y (15.1 ) ivliich can be split into an average part 

Ilaving solvotl for the oscillation ~ ~ ( y, z l), , n,(x, y, 1) we can rctnrn to eqn. (15.21) 

nncl c:~lculate tlic function F(a, y) which appears in eqn. (15.20). Tho lattcr now 

dcscrihcs tho mean motion d(z, ?I). 

It should he notctl that t h cquation for the mean flow, cqn. (15.20). has a 

form wl~icli is identical with the steatly-state version of the boundary-layer equation. 

I'hc only tliiroronco consist.^ in the a.ppearance of the acl(litiona1 term F(x, y); t,his 

now plays tJic same part, as tht: term If . tlV/dz which originates in the pressure graclicrit,. 

Both tt?rm.s mprcscnt, known f~lnct,ions in the diffemntial equation. The only 

tIifirrnc:c consists in the fact tliat, t~ic mean pfcssure gratiient 17 . dIf/dx is "irnpressed" 

on tho hor~nclary layer and is intfepentlcnt of the trsnsvcrse coordinate 11, 

whcrcas the :~tldit.ional term F(z, y) dcpcnds on it. 

Owing t,o the existence of oscillatory compont:nt&, the average flow is tliffcrcnt, 

from l.hnt. which wonlrl be ol,t,aincd if Llie potential velocity Il(z, 1) were averageti 

a. General remarks on the calculation of non-steady boundary laye 413 

from the outaet. The difference is clearly brought into evidence by tlte appcaranep 

of the function F(x, y) ; it has its origin in the non-linearity of the differcnCial equation. 

It will be stated later in Chaps. XVIII and XIX that the essential charaoteristic 

of a steady turbulent stream consists in the faet that on the mean velocity 

of flow there is superimposed a random, three-dimensional, quasi-periodic oscillat,ion. 

Cbnsequently, problems involving turbulent frce slrcnms cxl~ibit tho same featnrcs 

as those being discussed now; they involve changes in dire~t~ion as well as in the 

magnitude of the free-stream velocity IJ. Tn most cases it is cust,ornsry t,o tlcglcct, 

the free-stream oscillation and to calculate as if the flow wore stcady and :LR if the 

potential velocity were given by 0 (x) instead of lJ (x, t). 'l'his is cquivalcnt to omitting 

the additional term F(x, y) in eqn. (15.20) and necessarily leads to an average velocity 

profile which isdiffercnt from ti (XJ) Tho preceding remarks show clearly t.11:~t~ tltc order 

in which the two operations, averaging and solving the c:cjn:~l.ions, arc pt:rSortr~cd is 

not immaterial and aKccts the final msult. 

4. Expan~ion inlo a series when a steady stream is per~urbed rligldy. Very oftr:n, pr.c~l~lcn~s 

in non-steady bonndary l~iycrs involve nn c:snenl.ially nfencly flow crll \vlric:h 1,lrt.n: is ~~~lwrirt~l,c~srtl 

a small non-stcndy pcrlurbntion. If il in 1~~sumct1 lhal 1110 prL1rr1mLi011 is ~rn:dl t:nn11):irc(l wiL11 

the steady basic flow, it is porisiblc In split the eqnalions into a non-linGw bounclnry-layer equation 

for the steady pcrturbetion. A well-known exatnple is that for wlrich t.bc cxternnl st,rr:un 11n.s tllo 

form 

U(z,t) = d(z) -1 s U,(z,t) + . . . , (15.28) 

whcrc E denoh a very small nntnbcr. Tl~c rnosl itnporhnt ~poc:i;d cilst: ~IICII the estrt.n:d pt:rtarbation 

is purely harn~onic wx.s studied e ~lta~~~li~~ly by M. .I. I~igl~t.l~ill 

of 

linrarizalion can be c111ployecI when the i~n~pcrnlure et Lhe wall is rr~~rosc:r~lntl 1)~. l.1~ c:xprcssio~r 

127). 'I'll(: S:HII(\ t,y~)~ 

lTw (z,t) = pw (z) + E TTw, (z,t) (15.29) 

or when the wall ikelf performu smell, norr-steady. pcrt.t~rl>ing rnot.ions (oscill:~ting 1)oclics). 

In such cases we start with the assnmplion t.l~;rl tl~c sol~~l.ion.q for 1.l~ (Iyn:~n~it. :LR \~~cll as 

for the thermal boundary layer nrc of the following forn~s: 

'J'lie postulated forms from eqns. (15.30) arc introdnced into eqns. (16.1) to (15.3) and Lho losnlting 

terms are ordcrcd with respect to the powers of E. From the rcquirctncnt that, the tlini?rcntial 

expreaaions which mult,iply enell power of s muut vanish singly, we obtain a msc:rtlo nf cliffcn.ntinl 

equations. We list them for t.he cmc when Q = const, wlicn th external llow is of 1.11(. (i)rn~ of 

eqn. (15.28), and when the wall temperature is given by eqn. (15.29): 

Equations for zeroth order (steady bmic How):

414 XV. Non-st.cncly hour~tlnry Inycru 

with tho I)o1111(1nry contlit,itms 

,J - 0 : 16, -. a, -- 0 ; 7'" = T'," (,) , 

1, - m: U" = (7 (z) ; T', = T', . 

ISq~~ntionn of fir& ortlor (purely non-stcndy): 

- 

au, au, au" 

1 

a~ 

, 

I . Iw cy~~:~tir)~~rr of 11igl1c.r ttrtlrrs I~nvc: corrrapo~~~linp nl.r~lrt~l~rrn. 'rho prccoding nyst.c~nn ofrqr~nt~ionn 

IYIII I)(* rolvrd one. :I l'lcr l.11~ ol,lwr, it, IIC~II~ 11ol.w1 LI~rrt trll, oxccpt tl~onc of zc.rcrt 11 ortlnr. nro litwrr. 

Il oq~~:llions (15.1) to (15.3) wrrc t,o psscss rsnvt solutionn of t.he postul.zktl lorn1 (l5.:10) I I Lo ~ 

orilrr P, t.I~rn, g(!ncr:tIIy slwdting, tho solutiolm nrrivctl nt by t,llc prccrtling scl~c~nn worlld tlill'er 

l'ro~n t.lw C*SILI% solut,ion I I ler1118 

~ of or(1cr 141 I I. 

= 

h. llorlntlnry Inyrr for~nnt,ion nltrr impr~lsivc utnrt of nlotion 415 

5. Sinlilnr mid ~emi-similnr solutions. When we stuclictl t,he throry of' st.rntly, 

two-dimensionel boundary layers (see Sec. VIIIb), wc clcscri\)otl as similar that, 

class of solutions for which the depcntlenco on t#hc two vnriablcs 3: :~nd y c-oi~ltl I,(: 

rcdl~cctl to that on n singlc variable 71 hy the npplicat,ion of a st~iL:~l)lo simil:~rity 

tran~format~ion. In s.11 analogous manner, we say that a solution of a non-steady 

two-dimensional problem 1)elongs to the class of similar solutions whnn thc three 

independent variables x, y, 1 can be reduced ton single variable TI. 11. Scli~~lt 1461 and 

Th. Geis 1101 have intlicatxd all such solutions for which a rctluct.ion ton single v~rinhlc 

is possiMc, t.liat; is, si~ch as arc of tho form 

For example, cxtmnal flows of the form IJ (z, 1) = mx/L and the cascs when IJ (x, 1) .- (=tn 

mentioned in See. XVc belong to this class. The similar solutions for ctn cxtfrrnnl 

stream of tho form 1J (2, I) -- x/(n -1 Id), whcrc a and b are consta~~ts, wcrc : i~i:~ly~~l 

by K. T. Yang 1711. 

If a transformation can he found whiol~ reduces Lhc IJlrcc indopcndcnt, vn.rial)lcs 

x, y, 1 to t.uro, we say that the resulting solution is scmi-similar [21]. In particnl:~r, 

when the vnriablcs are rctl~~ced to y and x/t, the solutions arc also called pseutlostmcly 

(r/. 171). A soll~tion of thi~ t,ypc was tliscovorctl by 1. Tani I.561 for the (::tsc 

wltcn the cxtcrnal flow is given by U (2, 1) = (lo - x/('Z' --I), with 11" and rl' tlonoting 

constar1t.s. A wider class of semi-similar solutions W:LS consitlcretl by If. A. Ilnssan 

[In]; scc also rcf. 1211.

416 XV. Non-steady boundary layers 

where TJ(x) tfcrrotcs the potential flow about the body in the steady state. In this 

particular casc we have aUpl = 0, and equation (15.12) of the first approximation 

hccornc.~ simply 

- va2L1, =(, 

(15.37) 

at av1 

with 11,~ 0 hr -~= 0, ;I.IUI M,) = (J(x) for ?J -= m. This cqu;~tion is itIcntic:~l wit,l~ 

t.llat, for one-din~ot~siorrd ltcat contluctior~. It was solved irr Sec. V 4 for the casc 

of a plnt,v st.;irt,ctl in~p~rlsivcly in its own plirnc, while tho fluid was at rest at a large 

tlist.;~ncc: frotn it. It was tl~crr possiblc to introtlucc a new tlimcnsionlcss varin.blc 

(sl:in.iltr~il~y Imn,s/orn~wlion.) : 

!I 

'l=21/;i. (15.38) 

In t.lris m;rnnor wc ol)t,ain the solution in thc form 

IL,(Z, y, 1) = U (x) x Cljt(q) = U (z) erf q . 

(15.39) 

'I'his is l,hc first, ;~pproxirnntion botjh for the two-dimensional and for the axi-sym- 

~nct.rirnl case. I'nrt.lrcr, if the pot,cnt~ial volocit.y is inclopenclcnt of z, i. c. if TJ = 

: (1, :- const (II;rt plate ;I.!, zero incitloltc:c:), oqn. (15.39) constitr~tcs the exact solution 

of cqn. (16.2). since thc: c:onvot:tivc trrms in eqn. (15.13) vanish together with the 

prrssurc torm so that TI, E 0. I Iowcvcr, the solution arrived at in this way does not 

c:onst.it.t~t.c t,he complctc solubion to the prol)lc~n and applies only sufficiently far downst,rcmn 

whcrc thc influonce of the ctlge is negligible and where the flow behaves as if 

th(: plate wcrc infinitely long. Strictly spraking, the complete solution must also 

satisfy the condibion that ~(0, ?J, 1) = 0 for all values of I/ and 1. The complct,e solution 

is givcll in ref. 1541. 

111 the gonrrn,l casc, wl~c:n Llrc external flow U(x, 1) tlcpends on the space coortlinatn, 

it is nwcssary to make a distincti~~~ between the two-dimensional and the 

I. Two-climenuionnl cnue. Wc shall begin by considrring the two-climrnsiot~d 

c.:tsc. 19,r this rmo wc assume a power series in t,imc for the stream function ~tipulating 

th:rt, it, has tl~t, form 

~rrs;d,irtg thrsr cxprc-ssions into cqn. (15.12) we obtain the ~IiKcrer~t~ial equatiol: of 

b. Boundary-layer formation after irnp~tlsivc ntnrt or nrotior~ 417 

with the boundary conditions 5, = 5,' = 0 at 17 = 0, anrl to' = 1 at ?I =- oo. 

Equation (15.42) is identical with eqn. (5.21) and the solution for C,' is intliratctl in 

eqn. (15.39). The function 5,' is shown plotted in Fig. 15.1. 

Combining eqn. (15.13) with (15.40) we obtain the differential equation for the 

second approximat,ion C1 (91) in the form : 

C,"' -1- 2 q el" - 4 el' = 4(5.,'2 - ~ O ~ O - " 1) , 

wit11 the bounclary contliLions C1 = 5,' = 0 at 11 -= 0 ant1 [,' = 0 at 11 -- m. 'I'ho 

solrrt,ion tlcrivctl hy XI. 13l;~sius is: 

Fig. 15.1. 'l'hc fw~ctions t;l nut1 = 

and tib For t.1~ velocity distribution in tho 

11011st,cndy horlr~dnry Inycr, cqns. (15.41) 

;u~tl (15.50). for itnpulsive tno(.ion 

The function (1' is shown plotted (as funct,ion (I,') in Fig. 15.1. 'I'hc initial sl01)cs 

of the two functions, required for the calculation, of sci)nratior~ arc givcn 11.y 

An exact expression for the next term of the expansion of tlrc stream funohu 

it1 t,nrms of time was obtxinctl by S. (:oltlst.cin ant1 1,. 1Losonhr:cd 1141. I

418 XV. Non-slmrly l~orrnclnry Inyrr~ 

I5 x a 111 1) 1 t- . Ci~c~tltrr r!/litctla~ 

I'or tJto rircwlar cylintlcr of ratlitis 11 in n strcntn of vrlocity I/,, we obtth: 

as swtt from rrp. (15.45). 'l'llc tlist.nnco covrrctl uut,il sapration begins is R, = 1, I/,, 

st, t.Ir:~t, 

b. Bo~lnclnry-lnyrr forinntion nhrr inipnlnivc dart of niotio~~ 

Inserting tho vnlucn from cqns. (15.47) intorrp. (I5.46), wr find that I.l~c tin~r cln11s~tl IIIII~I t,lie 

onset of ucpnrntion is 

I'ig. 15.2. I)int,:~nce 8 

t.ravrrsccl by rlliptic cylinelrr 

r~nt.il thc: onset elf 

scpnrntion in t.hc vnuo 

of i~npnlsivc? :~.ctdt.r:~t,ion 

fro111 red, 

4 

y, = 0 for Px < --- 

3 ' 

For k = I cqn. (16.48) t,r~~nnfortn~ inlo cqn. (16.4f;) for 1110 cirex11:ir e:yIit~tlvr. Ibgit~t~i~~g with 

tlh vnlr~o the ti~nn L, for tho onsct of sc[)nrnl.ion clccrcn~as will1 i~~e:rc::tninl: k -7 h/w. IIII~I 1.11~. 

position of tho point of nepnrntion move8 Crom the end of &xis rx f,ow~lrtls LIic cntl of nxia b. 111 

the limit D/n -+ m, i.e. br a plate at right angles to tlw tlirrc~fion of motion, wc 11:tvr I , 

and !I, - 11. Iirnce the onsct of srparation is i~n~nctlint*? for t.Itc c.:~st: of n fl:lt pl:lfn ~wrpc*ntlit.~~inT 

10 t.hr rlirrction of rnot,ioti, n.ntl it tnkes plrmc nt ths ctlgc.

t.11rsis I,rrsc~nl,rtl in 1924. 111 t,his cnsc sop:~ml,ion is s~~ppresscd on that side of t,ltc 

t:ylintlcv wltcrv (.It(: tnnpt~t~t.i:al vrlorit,y has tl~c same direction as the velocity of flow. 

'I'11r process of n(:ccI~r:at,ion for an clli1)tic cylindcr at an angle of incidence has 

\)ern trcnt.t:tl in a paper by 11.J. 1,ugt 128al. In it,, the aut,hor succeetletl in calculating 

t.11~ rortnat,ion of t,ht: st.art,ing vort.iccs at. Reynolds nnml)crs in the range R - Vd/v = 

IT, to 200. Wo wisl~ to rcfw 1.11(? rcntlcr also to n pnpt:r by I). I)umit,rcsc:~~ and M.D. 

Cnzacu lOnl whic:h discusses t,llc same problem but for a flat plate at an angle of 

incitlcncc. Scc also Via. 4.2 for (.he plat.(: at right angles to tthe stream. 

2. Axinlly sy~t~tl~ctricnl prtrl,lcrn. '1'11~ process of bountlnry-l;~yc-r fortn:at.ion 

a.l)o~~t, :an xial ally syn~n~cl,rica.l I)orly accclcr:~tcd imp~~lsively wa.s iiivcstigatetl by 

15. lMt,zt: I!)] it1 ltis (hcttingcn thosis. We consitlcr the 1)onntlary Ia.ycr on n htly 

of rc.volut,ior~ whose s11:~pe is tlcfinccl by r (x), Fig. 11.6, and which is set in motion 

at, t - 0. '1'110 accclcrat,ion is impulsive, anrl the cylintler moves in the tlirection 

ofi1.s :tsis. '1'11~ rc~lcvnnt nqrt:~~l.iorts an: now cqns. (15.2) antl (1 1.27b), antl the sol~ition 

t::111 :,pin IK: rc~)rrsc:~~t,ctl RS :a slltn of n first, :~pproximat,ion, u,,, antl :I. sccontl approxirn:t.t,iotl. 

7, tlt-lirlc:tl by cvlns. (If,. 12) anti (IR.13) rcspc(:t.ivcIy. in view of the (:hanged 

Torn1 c,l t.lto cwnl,itl~~it,y cvlu:at~iotl we it~t.rotlrwc x tlifli~rcnt, sLrc:arn f~~nctio~~, nnn~cly 

:~ntl wr nssnmr it. to Itc or tht: form 

, 

I . hc. variable 17 has t.hc sarne tnca~~ing as in the two-di~ncnsional problem, eqn. (15.38). 

l'lw rlilFrrrnt,i:al cquatiot~ for [, resulting from cqn. (15.12) is identical with equatiolt 

(15.42) for thc two-dimensional problem, as alre:ady mentioned. For the sccor~d approxirnat,iort 

in t,Ile expansion in terms of time we now obtain from eqn. (15.13) tmllc: 

following tIiiTercntin1 equations, defining C,, nnd i,,: 

Thi: cq~~ntion for CIS is itlcntical with that for of the two-dimensional problem, 

and the equation for

422 XV. Non-st,cndy hoixnd~ry Inycrs 

'1'11~ process 01' t IIP form:~t,ion 01' it I~o~~ntlnr~y I:i,yt,r on n rol.nting tlisli was stntliccl 

I)y I

424 XV. Non-atearly boundary layers d. Experimental investigation of the starting procem 125 

[Jpon comparing with cqn. (l5.45), it in seen that for equal values of dU/dz separation occurs 

rarlirr whrn thr nlot.ion is startrd impulsively than whrn the acceleration is uniform. 

ll. Illasius mlrtrlnt~rl t.wo fnrthcr t(.rrns of thr vxpansion, and with their aid the equation 

for 1, is ohtninrd in Ihr following ~notlified form: 

For t.ho c.:isc. of :i cylindrr whir11 is 111:iocxl symtnd.rically with respect Lo the direction of flow the 

last tnrm v:~~~ishrs :I(. Ihc. clownsl rcwn sl.ngnnLion point, and we obtain 

U(T, 1) = 1 ~(r) -- 2 6 1 sin 

R ' 

wl~rrr? h clcnolcn 1.11~ c:ot~st:int :ircrlrration. I-lcnce 

z dw 2b z 

1 ) - - 2 b sin - ; - 

R dz R-R' 

'I'ho pint :I& \vhi(:Ii s

420 XV. Non-nl,mdy 

Fig. 15.58 

Fig. 15.5b 

Pig. 15.5d 

d. Exprimcnt,nl invcstigntion of tho ~tnrting process 

[9b, c]; t,he preceding two pnpers cover t,hc stendy as well na thc nonstcndy case. 

Reference [9c] estnblishes the limits of tJre Rcynolds numlwr mngo in which thr 

"twin" vortices, shown in Figs. 16.6tl cmtl 16.50, cnn cxisC nntl ntlhcrc 1.0 t.l~cs I)otly. 

Separation: Thc process of scpnrntion is much morc difficult, to clescrihe in thc 

case of non-steady lnrninnr boundary layers nntl in the cnsc of moving walls than for 

steady flows along n solid, stationary wnll. In the lnttrr casc, srpnrntion is tlctcr~ninrtl 

by the simple condition that the shenring stress at the wall must vanis11: to = 

,u(au/ay)o = 0. It was shown in a paper by W. Srarz~ and T1.P. l'rlionis [47n], ns 

~Irrdy intimnted in earlirr papcrs by P. JC. Moorc [33] nnd N. Rott [38), thnt in non-' 

steady flows scparntion occurs when the shearing stress at an internal stagnation 

point vanishes. Thus, for sepration 

u = 0 and au/ay = 0 in the interior. 

This condition is known as the Moore-Roll-~S'ears criterion. l%ysicnlIy, this condition 

describes a blow-up of the laminar bonntlary Inycr. Such n ~opnrnt~ing, non-st.c:ntl,y, 

two-dimcnsionnl houn(I~~~r.y lr~ycr cxhihil,~, to 11. ~ ~ L I L CX~II,, 

~ I I 1,111, HILIIIU cIII~,Iw(~~I. 11 

threc-climcnsionnl bountlnry lnycr li)rtr~ccl in the: nnglc 1)c:lwc:cn n flat, plr~tc: 

body mountcd on it. In this casc, shown in Pigs. 11.20 and 11.21, L11c Ilow t'ornrs n 

separation surface; see nlso refs. [47 b, c). 

An extensive review on the unsteady flow around blunt bodics with many cxcellent 

flow pictures has been given by S. Taneda [66n]. 

In conclusion, it may be worth mentioning that thcsc separntion proccsscs 

occur on a much reduced scalc in the casc of slcndcr bodics, such ns c. g. slendcr 

elliptical cylinders, wl~osc longcr axcs arc p:wnllcl to Lhc direction of Ilow, or or 

acrofoils, Consequently, the cxpcrimcntd pressurc distribution around such bodies 

agrees, " in most cnms, vrry closcl~ wit-11 that given by potcntinl theory (sco also 

Fig. 1.11). 

Fig. 15.6. Prcssl~rr tl~st,rilmlion Incnsrrrcd 

nronnd a cirrulnr rylintlcr tl~trirlg 

thr ~tnrting procrw, :hffn.r M. Srhwnbc 

[47] 

P-f 0 

ied 

1 

0 

427 

rirtrl 11 H~~IIJL(.

XV. Non-skdy borlntlnry layers c. I'criodic boundary-lnycr flows 

e. Periodic houndnry-layer flows 

1. Oscillnting cylialder in fluid at rest. In orrlcr to give an cxamplr of a prriotlio 

I,or~ntlary-layer flow we now propose t,o calcr~lat~e t h bo~~ndary layer on a body 

which pcrforr~~s a rcaiproc:rt,ing, harmonic oscillatiorl of smnll amplit~~dc in n fluid 

:I{, rvst. This is :in (:xI,cnsion of t h proltlnn~ of the I)ol~ndary layer on a Il:~b p1nt.o 

prrforming h:irmonic os~ill:~t,iorls in i1,s 1)I:inc wl~idi wi1.s :tlrcxdy tliscrrssod ill Soc!. V 7. 

Tt will be shown in this scotion l,hatt small osrill:~t,ions of a body in a fl~lid at 

rcsL indnce chnmct.crist.ic scconclary llows whosc n:lturc is such that a stcndy 

rnotion is impartfed to the wllolc fluid in spite of the fact that the motion of the 

body is purely EKcct,s of this kind occur, c. g., when dust pattcrrls arc 

c-rtrat,ctl in a Knntlt tube arid arc of somc importance in acoustics. 

Snppose that the p~t,(-ntiaI v~IociI~y tlist,ribution for the cylindrical body wlkh 

wc sl~:l.ll now consitlcr is givon by fJ,,(x). 'I'hc potfcnl.ial flow in the case of periodic: 

osc:ill:~l,ions wil,l~ n c*ircrll:ar frrclucnc.~ 11. is t.l~cn given I,y 

\Vc. shall rtow :wstrrnc :a sys(~cm of roortlinnt,es linltetl with (,he solid body. 'l'hus 

V~IIS. (15.1) and (15.3) may be :ipplictl. 1.110 prcss~~rc ~listriI~ut,ion being given I)y 

cqn. (16.0). 'I'ho 1)ountl:lry conditions arc: 7~ = 0 for 9 -;. 0 and 7~ = IJ for ?j = 00. 

It is possil)lc to :~tt.ctnpt to solvo t.his problem by the method which was used 

irl t,lw cttsr: of :~ccalcrat,ion from rest, i. e. 1)s calc~~latir~g sr~cccssivc? approximations 

lor 1,hc v~~l~~t:il,y-~lisl,ril~~~t~i~~~~ 

function :is cldintxl in cqn. (15.1 I) :ind \vit,I~ t,hc tiid 

of wjtts. (15.12) an(1 (15.13). 

'I'his rnct.l~otl appc:ars to 1)c ntlmissihlc if 

Now 11 i)(I/an: - 1Jm2/rl wliorc tb t1~11ot.c~ a linc:~r tlimcnsion ol t h 1,ocly (e. g. thc: 

tlintnc.t,c:r of 1.11(: o,ylinclor). On tho other hancl aU/at - (I,,, x ?r, where (I,, dcnotc~ 

thr ~nnxitn~~rn vt:lonil,y of the body. 'l'h~rs we have 

* 81 7Id ' 

ao /a; - urp1 

'I'hr III:L~~IIIII~I vrlorit,y (I,, is proport.ior~al to n x s, where s is the amplitude, so that 

with the convention that only the real parts of thc complrx quantities in cjuestior~ 

hctve physiral meaning attarhcd to them. lntrotluring n dimrnsionlrss roordinnt,~ 

tlcfind by 

:d assuming that tjhc firsf, approximation to bha stream fr~nctio~~, y,,,, is of t,l~r fornl 

=v+ - 

s (x, y.t) u0(4 io(n) eln1 , 

and henre 

\\it41 the bountlary roditions to - (,,' - O at 11 -- 0 atd to' = I :it 11 

sol111 ion is 

m. '1'11~ 

Cot = 1 -rxp{-(1 -i)77/J2). 

Itrvrrting to the real notationt we obtain the functio~l 

llo(., y,t) = lJo(x) [em (nt) - c-sp (-- ,1/p'5) cos (nl - - ,,/1/2) 1 (I 5 62) 

n.hic.l~ rcprcsent.~ the first approximation to the ~clocit~~-tlisl~riI,~~~io~~ 

i't~nct~ior~. 'l'lbis 

is t h s:bmc ~ ~olnt~ion as t.hat for the oscillatir~g fl:tt. p1nt.c in oqn. (5.26:~) 

If tho second s~ppr~xiniat~ion icl(x, y, 1) is now cnl(:~~l:tt.cd from rclrl. (15.133), it, 

is scrn (.hat the convective terms on the right.-h:intl sitlc ofthe cclni~t.ion will c~ot~t,ril)~~t.o 

l.vrrns with cos2 7t 1. Thcsc, in turn, can be rctlr~cctl 1.0 tcrmx with cos 2 71 t, sin 2 71 1 

nntl stcndy-sl,at.c, i. c. t,ime-intlcpcndcnt t.orms. 'L'&king into ILCCOIIII~ t11csc cir(:~lrl~ill 

t,l~r 

st.anrc3s we can express the stream frlnctior~ of t,hc src:ol~d :rpproxi~nat~ion 

:. 

420 

fhrn~

XV. Non-utcncly boouclary lnyers 

whrrr tltc bar ovcr tltc symhls tlcnotrs the rcspect,ive conjugate complex quantit.ies. 

'I'hc nortnnl and t,sngcnt.ial componrnt,s of the prriotlic cor~t.ril)ut.ion must vanish 

at. t,lto w:~ll, whercns :~t :L large tlistnncc! from it only the tangential component 

vnnislws. l~utting 11' -- r/l/2 we ot)t.:~in 

ltrgardirtg t>he steady-state cont.riht~tion it is found that only the bounclary 

rontlikions at the wall can bc satisfied, and that at a large distance from it is possible 

t,o n~:tkc the tnngcntial component, finit.(: but not zero. Thus 

3 1 

Slb' = - ;t + -- exp ( - 2 77') 4- 2 sin 71' cxp (- 17') -k 

4 

'I'hr srrontl approximatht is seen to rontnin a strady-st,ab tvrm wliich does not 

vattislt nb a Iargc tlisLntice from thr body, i, c. oukide tho boundary layer. Its magnit.utlr 

is given by 

3 dU, 

u2 (2, 00) = --- (I5 68) 

4n '0 dz . 

,. 

I itc precetling nrgunlcnt has thus Ircl its t,o the remarkable result that a potentrial 

flow whioh is periodic with respect to time induces a steady, secondary ('strenmirtg') 

mot,iori at a Inrgc distance from the wnll as a result of viscous forces. Tt,s magnitude, 

givrn hy rqn. (I5.63), is inrlrpcntlcnt of the viscosit,y. The steady-state c~mponcnt~ 

of t.11~ vc+wit,y is ,such t,lml tl~tiil pnrt.ic1r.a arc seen to flow in the direction of decreaqing 

nniplit.rttlc of that component of the potential velocity which is parallel to t h wall. 

An c~xntnplt: of s~tch a niot,ion, viz. the pntkrrn of streamlines of t.l~c stcatly 

flow al)out a riroular cylinder which oscillntes in a Lluitl at rest, is shown in Fig. 15.7. 

I'igt~rc- 15.8 cont.:~it,s a of the flow pnttcrn al~out a cylinder which performs 

an osci1lnt.or.y mot,ion in a tank filled with watrr. 'I'hc camera with which the photograph 

was t.nltrn nlovctl with t . 1 cylintlcr ~ nnd the surfmx of t,he water was covcrcd 

with fine ~nct.n.llic pnrt,iclrs which rnntlcrctl the m~tiorl visible. Thc particlcs show 

up as witlc I)nntis in the pict,ltre owing to the Ivng exposure time antl to their 

rcciprocat.ing mot.ion. 'l'l~e fluid partic:lcs flow t,ouwds bhe cylinder from above and 

from hrlow, and move away in both tlircrt.ions pnrallcl tm the reciprocat.ing motion 

of t.lic cylintlrr. 'l'l~is is in good agrccn~rnt wit11 t,hc Ihcorct,ical pattmn of streamlines 

show'ti in Fig. 15.7. Siniil:~r phdtogrnpI~s wrrc also I,lll,lisl~cd hy 15. N. Antlradc [I], 

who it\rlrirc.tl st.:~n(lit~g so1111t1 W:LVCS~~OI~~. ;I circulat nylintlcr and rcndcrctl Lhercsulting 

secondary flow visible I)y tlic injection of smokc. 

Fig. 15.7. l'n1,tmn of utmnrt~linru 

of the fitcnily uccontlnry tnolio~l in 

the n~ighl~onrhood of no osrillnting 

rirculnr t~ylind~r 

e. I'eriodic ho~~ntlnry-lnyer flown 43 1 

l'ig. 15.3. Scconclary flow in tho n~.i~lil,owllootl 

of n.11 oac:illnI.ing circ~~ler c:yli~~clcr. l'hc 

cnmorn niovcs wiL11 tl~c cylilidor. 'l'hr ~rict.n.llic 

pnrticltxn which wrvc to rcl~tlrr f,l~o flow viuihlc 

sl~ow 111, n.s wide bands owitla tn t,llc 

long exposure tilllo nncl to thir rccipror.;~ting 

motion, nftcr Scl~licl~l.itlg (441 

Tt is importmt to notticc hare that tllo first npproximat,iot~, 11, itt ccp. (16.62), 

shows t,lti~t, t,ltc tliffcrcnt layers in the fluid oscillab with clilTcrcnt phase shifts 

compnretl with tltc forcing oscillations, and that their amplitudes dccrmsc ont.warcls 

from the wnll. 'I'hc sn.tnc f~at~~~rcs were nxhil)it.cd hy tit(: solutions tlisi:~~rtsivl in 

Cltnp. V. 'rhc first approximntion, u,, as wcll as the so111t.ions in Chap. V wim 01)t,ninctl 

from c1ilh:rcntial eqttntions which (lid not, contain the convcctivc tnrms 

It. mn, t,hcrcforc, I)c stated tltnt y-dcpcntlcnt. phase s11ilt.s :rnd amplitirtles tlrc.aging 

with distance from the wall are caused cxclrtsivcly by the nct,ion of viscosit,y. On 

t,hc other hand, in tho second npproxirnatioll, ul, there appears a tmm which is not, 

periodic antl whiclt rcprcser~t.~ stcntlg stmamiug supcrirnpos~l on the oscilln.t~ory 

motion. Ilcncc, it can also be st&cd t,l~at sccontfary flow has its origin in t,l~e convcctive 

tmms and is due to tlte intcraction bct,ween inertia and viscosit.y. TL slto~llrl 

be borne in mind that simplifications in wlticlt the convcctivc tcrms 'tavc I)ccn on~it.t.c~l 

le;d 1.0 solutions which arc frw from streaming and may, thcrcforc, give a rnislcailir~g 

representation of the flow. Streaming docs, in general, appear only wlicn the? solut.ion 

is carried to at least the second-order approximation. 

'I'hc phenomena under consideration offcr a sirnplc explanat.ion of I

432 XV. Non-ntcndy boundary layers 

a Iargr rlistanrc from the wall tAc vrlocity must, evidently, change sign to satisfy 

tho rontinrriLy rcquircmcnt,. 'I'his intlurcs 'streaming' effects, tl~c sl~ifting of thr 

part irlrs of c111st, ancl causes thm to form littlc heaps at tho nodes. 

lt is clcar frorn the prcc:ctling t1csc:ription tha.t the quantity of dust used to 

protlucc IZ~~ntlt patterns is of grcat importance. A large quantity of dust will become 

n.git,atotl ;~ntl m;ry rc::~c:ll the rrgion of innor flow when vil)mtiorls of the tube arc 

cxc~il,crl. (~onsc:q~~rrltly it may not, be possil)lc to cause the tl~~st to move away from 

(,IIC points of ni;~xim~~m amplitwlc. If, howcvcr, only a small quantity of it is taken, 

t.l~c: infl~tenc:c: of the flow m:Lr the wall mill t~c strongcr a.nd t h points of maximum 

nrn~di1.11c~c: will soon Itoconro I'rc:o of dusl.. I'rolhns c:ont~eclatl with st,cacIy 11101,ion 

whic:h :wconlparl.v os(:illatio~~s havc I~ccn t,rcntrctl in grcatm tlotail in publications on 

:~.c:o~~stic:s, c/. 1081. 

An an:~logor~s invcstign.tior~ of thc flow al~out an axi:dly symmetric ellipsoid 

w11ic:h ost:ill:~t,c:s almut its axis of symmct,ry in a fltlicl at rcst was carried out by 

A. Goslr 1171; c/. also I). Iboy 1-40, 411. 

2. C. C. I~II'S tlienry of horn~onic oscillntio~is. 111 1.111: pw:c:tling soction wc: 

h:~.vo c:orlsitlrrrtl Lypic:;d ox:~~nplcs ol' osc~ill:i.lrions involving fluitis at rest. I'rol~lcms 

in wllic.l~ lhc: osc*ill;~t,iot~ is snl~critnposctl on :L s1,rr:l.m are much more irnport.:'nt in 

:~pplicn.t,ions, IJII~. n.lso 1n11c:h more clif'lic:~~lO t,o analyst. A certain insight into this 

t.ypc: of proc:rss c.:rtl I)(: ol)l,:rinctl with 1.ltc :tit1 of (!. C. 1,in's theory 1281 tlcscril)etl 

ill Soc:. XVa. 

(J (:I:, t) :~= (J) -1- lJl (2:) sin w, t , ( 15.64) 

e. I'criotlic botlnt1:wy -l;iyrr Ilo~ s 433 

A cliagram of this ftinct.ion is seen plotlctl in Fig. 15.10. 'l'l~c c~xprrssiolt (1 5.6fi) sl~ows 

t,hat tlcviat,ions bct.wecn the true mean vclocity p.ofilo 77 ancl tht: quasi-st,c.ntlS vr1oc.il.y 

profile 7~, which would cxist if we were to assumo F(J, 11) = 0, tlrlwnd cssnn(.i:llly 011 

t.11~ an~plit~tdc CI, (z) of t,hc oscillation i~nd 011 i1.s vi~rint.io11 tllIl/tl:r: :~.lot~g t,l~r Ilo\v. 111 

p:~,rt,ioul:rr, oven a I;~rgc amplitntlc of oscillation will ~)rorluc.c: 1111 c:ll:al)gc: ill 0I1r vc.lot-if..< 

profilc if it rcrnains constant along t,ltc flow, i. c. il' I/, - c:o~~st.. Icroln t.hc tli:1gr:1rri 

in l'ig. 16.10 iL cn.n bc tlctlucccl t,l~at tllc I:~rgost rrlnl,ivc rnotliIical.ion of 1,111: vc~loc-il.s 

profilc occurs near the wall, bccausc F(y/d,) has Lhc Inrjit.s(. V:LIII(: F(0) -- 1 thrrr. 

Sincc t,I~c fluid partielas nearest to the wall niovc r~nclrr rrl:~t.ivrlv s~n;dl ;~cc:clr~.;~t ions. 

1,llc ntltlil,ionnl 1)rrsstlrc gr:~tlit:nt will ~rrotluc:c: 1.111: g~~t:r~trst, I , ~ I : I . I I I~ I ~ ~ ~ ; I l11(: ~ 1v:111. 

Fig. 15.10. PloL of the funot.ion 

F(y/6,,) from cqn. (15.67) for n 

single, lisrmonic component in -11.5 

tl~c cstcrnsl strcam 

If tl~crc were a spectrum of harmonics of frrqucncics kn. (k = 1, 2, . . .), i. c. 

for a frcr-stream velocity 

wc would obtain simply 

U (x,t) = u(z) + C Ulk (x) sin (knt) , (15.(iH) 

k 

F (z, y) = C 4 

k 

o, =

434 XV. Non-stcxcly boundary layers 

From what has I~ecn snit1 hefore it is clear t,I~at the position of the point of 

laminar scparat.io~ is aKcct,ctl by the cxt,ernal osciliat.iona nnd t,hnt the point of 

separation must osrillat,~ it,sclf. Finally, C. (1. ],in's mcClrotl lends to t,he valr~able 

conrlt~sion t,li;~t t,lw f~rncl;uncnI,nI oscill;tt.io~~ iirtlrlcrs 11iglrr.r harmonics in thc boundarylayrr 

o~rillnt~ion. 

3. Extrrnnl llnw with etnnll, l~nrlnonic prrturLntins. The c ~ when c t.hc extcrnnl flow 

perfor~nn small, hnrn~onic oucillnt~ions hnn bcrn tronbed in n nutnher of publications. The method 

employcd was lhnt of a scrirs rxpnnsiori in t,ho pcrturbntion pnran~etcr described in Sco. XVn 3. 

We nnswne that the extcrnnl flow is of t,l~e form 

IJ (x, 1) = Tf (XI -I- rr, (x) ~'"l , (15.70) 

and note t.l~nt,, for it, nod, investigntions rest,rict t,l~en~selves to the cnlculnt.ion of t,hc first 

approxin~nlion, that. i ~, of thc fnnctions n,,ol, and l', from eqn. (15.30). M. J. I,ighthill [27] 

for~nnlntrd an npproxin~nt.c n~etllotl for tlrc solution of cqn. (15.32) for arbitrary forms of tho 

fttnc:t.ion o(s) nnd iJ,(s). The particular cnse when both functions cnn be represented in tho 

form of power series ltnn been ronsirlcrcd by 1';. Hori [24], whereas N. Ibott and M. I,. Itoscnxwrig 

[3!)1 o~nn~inocl t.110 CXRIII~IO W I I the ~ t.wo Ftn~ct.ions #(x) nnrl #,(z) nro ui111p1e powers 

or r. 'l'llc c?xrltll~)~o of ~t.;ignat,ion flow st.lldiccl by kt. I$. (:lnllcrt 1131 nlld N. /Lot.t [:!)I nR wc?II 

as tlrc Ilow along n Il:lt, plate nt 7,cro inciclcrlrc dincnssccl by A. Gosh (171 nncl S. (:il~bc:l:rLo [I 1, 121 

oons1it.ut.c n111)-c:nsw of t.11~ 1nt.tc.r. Finally, A. (:tmll 1171 nncl 1'. (:. Hill ant1 A. 11. Strnning 12.71 

pcrforn~cd exporitncnt~nl ~nc:murcmclltR on non-sleady ho~~t~clnry layers. 

If the oxtcrnnl flow is of the form 

U (r,t) - csm (I + E einl) = if (I + E ci1I1) 

(15.71) 

thrn rqns. (15.31) Itwl 1x1 Iho familiar d1(rrrrntia1 rqt~ntions for uiniilnr sohltions, C(1119. (9.8) and 

(!).Rn), nntnc.ly, 

with 

Asnutning in cqns. (1G.32) that 

u, = E ei"' V @, (6, 11) , 

wo arc? Ircl 1.0 t.hc ftlllowing tlilTcrenl.inl cqnntic~nn for t.lw rwxilinry functions @(E, 7) and O (E, 7): 

nt + 1 ln -1- 1 

-- - 

fDq.t'l 

/ (D,," - (€ t 2 m/')@, -1- 1'' @ - ( I - in) 1' E @,,E 4- 

1 2 

2 

+ (1 - m)/"€@t + [ .I- 2m = 0 , (15.76) 

with t,hc bonndnry conditions 

r. Periodic boundary-Inyrr flows 435 

The precwling clilTcrmt.ixl cqnntions arc, normnlly, ROIV~~ in t.hc form of srrics expnn.siorls. first. 

for smnll vnlucs of F and lhcn for Iargc vnlucu of F. Assuming thnt 

for small vnlucs of t, we arc lerl to ordinnry dilTcrcntinl eqnntions for tho fnnctions dik(l,) nntl 

Ok(7). The derivnlivc~ 

loenl Nnssclt nnmtmr. In this mnnnor we: rnn clwivo ll~nt, 

and that 

at q = 0 mrvc to cnlculnb tho slrenring ulrcnn nt t.ho wnll IIR wt!II IIH OIC 

I 

Arrortling Lo 1'. K. Moore 1.711 (am nlno A. ( h l [I71 ~ nntl S. (>il,l)c~lntn 112]), 1 1 r:rsr ~ of t.lw 

flat plntc at zero incidence is reprcacntcd by thc cxpreasion: 

nntl 

SubstiLuting n = 0, wo rewvcr tho uasi steady solution, which signifies thnt nt every inst,ant 

the solution behnvw like the shady Jutio; for tho instantnncous cxternal vcIooity The a penrnnco 

of an imaginary term nt n =!= 0 moans that the boundary layer aull'ers n phase shift wit( respect 

to the cxternal flow, the shift being diflercut for velocity nnd blnpcrntr~rc. Wl~ereas the rnxxirna 

in shearing st,rcsa lend thc tnnxima in the cxternnl Ilow (in the limit n x/IJm -+ CT 1.11~ pllnse 

ar~gle tmda to 459, the mnximn in lnmpcrnturo Ing hchintl t.l~rrn (in the limit, ?l,:r/!~,., -t m~ 

lhc pltmc nnglo tendn to '30"). 111 ntldition, iL turns ont t.l~trt nt lnrgt! VIIIIIOR of n ~/ll~., t . 1 nn~pli- 

~ 

tudc of thc ahenring-strrxw oncillntion incrcnaca withont bound, wl~crens t,l~nL or I.l~c? I~nnt, llnx 

slowly decays tm zero na n %/Urn is mndc Lo incrcnac. 

When thc solution of the system of eqttntions (15.33) is corrictl to second ordor, it is hnd thnt tho functions u,(z.y,t), v,(z,y,l), and l',(z,!/,l) cont,nin n Irnrmonic pnrt of donldt: f'rotp~cncy 

and n ~upplemnntnry, abndy pnrl which in inclopontlt\nl, or 1.itno. 'l'l~c~ Inl.lr.r tnotlific~~ I.ltc\ 1111sic: 

flow and cnn I)o intcrprcbtl na a secondnry flow in cot~~plnhr 11111tlogy wi1.11 1.1111t, t.~t(~t~I(~r(~cI 

ill 

the solutions of tho pwccding seelion. 

For shgnalion flow, wc hnvc Ul(a) = consl, anel it in fo~td 

t.lmt t.hnrl u,, o, and all 

higher-order term vanish, as demonstrated by M. R. Glnuert [13]. Conneqnently, the basic

436 XV. Non.stmdy borlndnry lnycra e. Periodic boundary-lnycr floe.s 437 

flo\\- nt~g~~rntwl I)y t.l~c: t.rr111s tl1 :1.11(1 vI rol~slit~utrs nn exart ROIIIL~OII, one, ~nor~ovrr, ~l~irl~ is 

:~lso c-s:tr,l for lhc; vo~~~p~(~Lv Nlivirr-Stokrs cqllntions ('/. also rvf. [67]). I$y :t ~llit~I110 1XR114fi~rtti:iIiot~ 

of v:iri:~l)lrs, t h 11rtwdi11g r:isc c:i~t bv III:L~C tn yield l,Ito solt1l~ion8 for st~agt~aliot~ 

IIo\r OII MI osc.ill:lti~~~ \wll first. Rivcl; in rrfs. 113. 67, 21. A solution for tho caso of nn ililinite 

ll:it pI;itv with suc~t.iol~ tui

438 XV. Non-atcndy boundnry inyer~ 

fig. 15.11. Velocity dintri5rtt.ion in 

oacillrd,ing pipe flow not ditTerent ins- 

tnnta of one period, after S. Urhitlo 

[G3]. 

If the distance from the wall y = R - r is small compared with the pipe mdius R, 

the ratio R/r can be rcplacctl by unity. Thus, introducing thc tlirncnsionlcss dist,nnce 

- - 

from the wall q = (R -r) 1/i/2 Y = y 1/42 Y , we have 

K212 P(!/) n2 = 1 - 2 cos q cxp (- - 7) -t cxp (- 2 '1) . (I5 80) 

'I'hr vnrintion of tl~is mean is seen plot,tc~l against, in Fig. 15.12. '1'11e maxirnuni 

valur dnrs not roinci(1c with the axis of the pipe (large distance), but occurs near 

tho wall at I/ - 1, )/it12 11 r- 2.28. 'I'ltis vnl~ic a.grccs very well with rnc?asurcrnent 

(E. G. I~icl~arclson's 137). "nnnular nKct:t"). In this connoxion the reatlcr is also 

refcrrctl to M. Z. J

440 XV. Non-st.cncly bortntlrrry lnycrs f. Non-steady, cornprrwsiblo bourttlnry Inyorn J,l l 

whir11 rr1)l:irrs tl~r original, three varinblrs z, IJ, t. Assuming that thc stream function 

is of tl~v form 

edge 

-. -1,-U, I . 

shock wave 

6.~~6 

- . - -- - - - 

(15.93) 

--$ 

Fig. 15.13. I'ortnntiort of n I)orlr~clnry I:ryrr 

Iwhintl n ~iortnal nl~ork wnvr lnoving will1 

7;77; 

boundary layer 

;r vrlocil.y /Is 

S~tl)slihtt.ion of tltc irhovn fortn for tltt: stmatn fi~nc:tion togcther with tho corrcslmt~tling 

form 

T - TmO(i~) (15.95) 

for 1.11~ l~m~wr:~I.~~rc tlistril~~~l~io~i intlo cqrts. (15.1) to (15.5) allows us to (lorive the 

folIowinx, ortlinnry tlifTcrant.i:rl rqrrnt,ions for tlrc functions /(q) and 0(7]). 'I'11r.se arc: 

TII~; soltttio~~s for II/IJ,., --: /'(rl) rrtm (YIII. (l5.!)(;) irt; S C ~ I plnlttcd in Fig. 15.14%. 

'I'ltt. p:tr:inirf,vr Ir,.,//l, fnr t.l~c Iiimily of cwrvos cKar:~ct.crizt:s t,l~c st,rcngtfi of the 

sltot-li \v:L\.r. 'I'IIO I~igl~cs(. ~,ossil,lc vn111c: for ~J,,./u, is (iJ,,/fJ,),,, = 2/(y t I ) ant1 

rtit~t-sl~~~~rcls I,o :tt~ it11inil.c.ly sl,rong sltnr:lc ; with = 1.4, t,his yicltls (fJ,,/lIs)t,,,z = 0-83. 

Nvg:~.ti\.t- V:I,~III,S of I~,.,/IJS (.orrt:s110t1(1 1,o li(:I.il.io~~s, non-st,rady, cont,iriuo~~s cxpansiott 

I':IIIS. wt.11 irt~:~git~t*tI I~oIII-(*III,KLI,~~~I in n singk front.. In the p:~rt~iculnr case when 

I . / - \v(. arc. Ictl t.o thn so-c:allrtl 1t:cylt:igh prol~lom (Stokes's first problem, 

n) 

Urn 1,) (.) 

Fig. 15.14. Vt*Itwil,y JLII~I I~twr~:rrnl,~trt* ~lisl~ril:~~l.it~~~u 

~IYIIII ~Y~IIS, 

Inrninnr 1)orrncllrry liryrr Imhincl n ~ior~r~trl uliodc wrrvc of ~OIIRIILI:~~ vdwily, nf1.1.r 11. hlirc+i 

of 1.11~ WRVC 

The p:~ran~ctcr /Im/(ls cltnrnctcrizcn the ~lrc~rjil.l~ 

(15,!),l) 1111tl (l!;.lOli) i t) 

Sec. Vn 4) which tlenls with tilo impulsive sLart of n flat wi~ll. 1 tf is sccn front Pig. 15. I.In 

that the tlticltncss of a boundary lnycr behind n normal sltoclr cxcortls th:rt, li)r tltc 

so-cnllctl IZnylcigh prolhn. This mcn.ns t11at. II~OII t.11~ I:L~)SC of n wrt.ni~~ l,in~(* 1 .r/ ITs 

after thc p:~ss:~gt: of t.11~ sl~oclr wave, the boundary 1:iycr :LL n givc~i posil,iot~ 11:~s 

grown thiclrcr than on an impulsively stwt3ctl p1at.c aftm t.hc s:tmc pc:riotl of t.irnc: 

has lapsctl from start. The opposite is t.rnc for csp:~nsiot~ w:~vos. 

,, 1 he soltttio~~s for t h lincr~r tliffrrcnt.inI rcln:ition (1 5.97) for 0(?1) r:~n Iw rrprcsrr~( - 

cd in the form of a linear cwmbination of t,wo basic solut.ions, tlrfir~r(l :IS follous 

Thc functions r(q) and s(q) arc solutions of the following ordinary dini.rc~nt,ial 

equation : 

together with the boundary conc1it.ions

442 XV. Norl-st,aady boundary Isycrs 

'I'hc solut.ions for P -= 0.72 have hrr~ plottctl in Figs. 16.14b and a. The nunlericnl 

vnlnr r(0) is x mm.surc of t,l~e rccovory t.empcraturc, TI,, th:rt is, of the temperat.rrre 

a.t.t.11c: s~~rfnnc of nn ntlinl):btic w:~ll. 111 t,l~is CRSC, we havo O'(0) = 0, anti hct~ces(?~) = 0. 

It, follows from cqn. (1 6.W) tJiat t,he adinhatic w;dl temperature is 

\Z'hrn P -r I, wr have r(0) - I, antl thc adiabatic wall t.cmperatVnre Iwcomcs identical 

with the stagn:~tion tcn~prrnturr Ic/. rqn (13 17)] Wlicn the Prandtl number 

of thr gas difrrrs little from unity, it is possible, according to H Mirels 1291, to 

rmploy thr npproxirnnt ion that 

with 

r(0) = P" , 

a-039- - 0'02 for urn :-- 0 (comprcsaion waves) (15.104) 

I -- (U,./Vs) (1.9 

0.13 Urn 

a: 0.50 -- - for < 0 (cxpnsion waves) 

I - j u, 

,. 

I hns, finally, thr t,cmprmt,~lrc distril)ution becomes, 

For the skin-frict.ion rorfficirnt, 

Onrr ngnin. ncw~~eling to 11. Mirrls [29l, whrn t11r Prnntltl numl)cr is near to unity, 

it, is possiblr to rrsort to Lhc following approximations: 

where for compression waves (11411, >0) 

antl for cxpnnsion waves (1I,/1JS < 0) 

The I~o~indary-hycr thickness exccccSs tho so-nnllctl Itnyloigli val~ic wl~cn the 

wave is cornprcssivc; this en11sc.4 1 h sh~~ritt~ ahrrus, the ~kit~.rri(:tio~~ (:ot:l'fi(:i(-t~t, 11ttcI 

tho Nt1ssc11, n~t~~tlwr 1.0 II~:(YXIIII XIIIILIII:~ ~ott~l)~~re!cI wit.l~ I.11oir ICnyIt:iKI~ v~~lttt*~. 'I'l~t, 

opposita is trlio for cxpnsion W:LVCR. In t11c spnci:~l c;rsc when P - - I, t.ttt: Ilt::rt.transfer 

formulae rct111cc to Chc simplc Itcynolds aldogy 

known to the render as oqn. (12.55). 

r. 

1110 precrtling problom which cliscussctl the hounc1:rry layer brl~irrd n shock 

wave of constxmt velocity co~~stit,~tbs an idcnlizccl special c:rsc in thaL iL call bv 

mduccd to a stoatly problcm 1)y the fr1iciI.011~ choi~c of :L coordinate syst,rm in wllic11 

the shock wave is at, rest. More gcnrrxl sol~rt.ions of t h same problem have boon 

trmtctl in the works of R. 13ccltcr 13, 4, 0, 71 ant1 11. Mircls and .J. Ilammnn [301. 

2. Flnt plnte nt zero incicler~cc with vnriable free-strenrn vrlocity nrd mrfncc 

temperature. In our scmmtl osaniplc wc col~xitlcv tho cotnpressil)lc I)or~nrl:~r,y Inycr 

on a flat platc whcn thr frcc-sbrcnm vcloc:ity, 11,(1), as wcll as tho tcmpcmt,urc at 

the surfacr, T,(t), vary in t2hc corlrsc ol't,imc. 'l'hc strc:brn rt~nction y) rrom cqn. ( 16.90). 

and the t~rml)crnt.urc clistribution 

in which the pmss~lrc-grntlicnt tcrrn lixs I~ocn dclotctl. 'rho variablc has I)rcn ele-firwd 

in cqn. (15.9l), :tnd (I,, anti 7', tlvnolo the tlcriv:rtivos of fr~o-strc::~~~~ vclo,'il,.y 

RII~I swface t,rmpcri~t.urc yith rcspcct. to kinrc, rc~spo~t,ive;ly. 

solutions, thc following series cxpn~tsions arc postulal.ctl : 

111 orclcr t,o :rrrivcx :if.

444 XV. Non-st,mdy I~oundnry Inyora 

Urn2 

r] = -- 

22 voo 

tit,finrs a II~\V, tlirnt~nsiot~lrss coordinntc, ant1 the following al)brcviatfior~s have been 

mi ployrd : 

r 7 

I hr lwct:c(ling fortns arc snl~stit~~tntl into the difkrcntial cqrtations for the bountlary 

Inycr ;~ntl it, is li~ttrd that, thc futlctiotis F(q), /O(tj) , . . . satisfy ordinary diITcrentia1 

ctlttnt.ions. Solrrtions for thcm wlron P .-. 0.72 11avc been given in refs. [35, 491. 

'I'lre functiorts F(,,), O,,(T]) ant1 A'(71) arc idrntical with the solutior~s for the steady 

prol~lt-1x1 witlr CI,., in(.rrprc:t,cxl as the insl.ar~tanoor~s vrlocil,y (quasi-steady flow). 

, 

I . hc rcrn:iining t.c:rt~~s tlcst:rilm t.hc? clop:~rt~tros from tho q~~:~si-st~twJy s~l~ttion. 

(:orrc~sl~)~~tlirtgIy, IJIV mlio of hcv~h fir~xrs :it, tltc wall for P = 0.72 (c/. 1501) is 

drscrilwtl Ily 

I 

+,# 

--- . 

-- I", 

, -,,--k... 

I , 

The theory of laminar, non-steady boundary layers has been dcveloprtl cons~tle~ - 

ably in the last years Information on this phnac can 1~ fortntl in thrrc vol~ttnrs of 

confcrcnrc pro~w-tlings. '1'11~ fitst, rtlilrtl Ity 15 A 15icl1rll)rrnttrr, rc%l)orls (111 1 1 1 ~ 

IU'I'AM Symposium "ltcrent Research on Unstcady I3ountlary Layers", Qrwl~rc 1072 

[74]. The second, edited by R R. Kinn~y [76], concerns a symposirtm on "IJnstcndy 

Aerodynamics" held in I975 at the University of Arizona,. l'hc third is tlcvotcd lo nn 

AGARTI mecting lrcltl in 1077 [7BJ. A rcvicw pnpcr by N ltilry may also tnct it comparison 

[37a]. 

[I] Andratlc, E.N.: On tho airculnt.ion musod by tho vibr1lt.ion of :rir in 11 I.IIIII.. I'rov. 1111y. Stw. 

A 134, 447-470 (1931). 

12) Arduini, C.: Strnto limite incomprcnail~ilc Inrninnro ncll'int.orno do1 pnnt,o tli rist.ngt~o tli 1111 

ciliuclro intlofinito oac:illanlo. I,'Aorolcc:nii:l~ .I/, 34 1 34lL (l!)lil). 

[:)I I)ct*kor, I?:.: 1)m AIIW:LC~IUC~ dcr C~OII~.H(:II~(:II~~ in IIII~I 11iut.t:r (:illor I ~ ~ x ~ I I L I I H ~ ~ I I I H ~ v IIIU.. 

~ ~ I I ~ ~ . 

Arch. 25, 155.- 103 (1957). 

[4] Reckor, E. : lnahtioniirc Crcnzscl~icl~tcn l~intor Varrlicl~tr~~~gsst.iinn(!~~ II~I~I I':xl~n~~sio~~~\v(~l 

ZPW 7, 61-73 (1959). 

[5J Bccker, E.: Dic lnminnre inkomprcasible Grcr~zscl~irl~t nn rinrr tlural~ I:rufrntlc \Vrllcu 

deformierten ebenen Wnnd. ZFW 8, 308-310 (1960). 

[fi] Bccker, E.: Instationnre Grenzschicl~ten hintrr Vcrclict~t~~ngsst,iiase~~ unrl Ex~~nnsio~~sa~c~IIe 

Progress in Aero. Sci. I (A. Ferry, D. Kiichc~nnnn, nnd L. I

440 XV. Non-st,rndy ho~~nclnry layers 

/I61 Oiirth, 11.: (:rct~nsc~hioI~tc~~tat~:l~~lng nn Zylindern hei Anfiahrt nus (lor Ituhe. Arch. (I. 

Math. I. 1:IX--- 147 (1!)4H). 

1171 (:osl~. A,: (7ont.ril1ul.io11 A I'6b11clo dc In cor~rl~e linlih lnrninnire inst,nt.ion~~nirc. I~t~l~lirnt.ions 

Scientifiq~trs PI '1'ccl1niq11cs (111 MkxistArc de l'Air No. 381 (1961). 

[IS] (hibhrn, It. .I. : 'Tilo Intninnr I~or~ntl:~ry lnyer on n hob cylinder fixed in n flr~ct,r~nt,ing stmnm. 

J. Appl. Mr.rh. 28, :%:l!) -- 340 (l!Mil). 

[I!)/ Ilnsnnn, II.i\.: 011 11ns1cndy Intniniir ho~n~tlnry lnycrs. .ll'lM I). :100 --304 (I!)(iO): scr nlso 

JASS 27, 474 --476 (I!)fiO). 

[201 Jl:~ynsi, N.: On situilnr sn111tio11s of the IIIISIC:I~~~ q~~:~qi-t\vo.(li~~~c~~si

448 XV. Xon-stcady boundary layers 

[GI] Trimpi, It. I,., nntl Cohen, N.B.: An intcgral solution to the flat plate laminar boundary 

I:lyrr flow cxinting insidc and aftcr cxpan~ion wavcs moving into quicucent fluid part1cu1n.r 

npplicnt,ion t.o the coinplete shock tube flow. NACA TN 3044 (1057). 

1621 . - Ikuji, W.: Not,c on the solution of the unsteady laminar boundary layer equations. JAS 20, 

z:)r,-znn (ma). 

1631 Urhitln, S.: 'l'h pImt.ing viscous flow superposed on the steady laminar motion of intwnl~r~vwil,lo 

1l11itl in n t:ir&~lnr pip. ZAM1' 7, 403---422 (I!)RO). 

[ti41 Wnrlh\vn, Y. I).: Ilonndnry hycr growt.11 on n u~)inning body; accclcrntc:tl n~otion. Phil. 

Mag. 3 (8). 152-- 158 (19.58). 

[(is] Wntmn, 11:. J.: Rountlary Iaycr growth. Proc. Ito~. Soc. A 231, 104-1 I6 (1955). 

i(i61 Wnt.son, .l.: A mlution of the Nnvicr-St,okes-equations, illnst,rat.ing tho recrpo118c of n 

Isminnr 1,onntlnrv lnvrr t,o a aivon chnngc in thc cxtornal strcarn velocit,y. Quart. J. Mcch. 

nljld hhtll 11, '305-326 (1658) 

1671 W:~t.son, .l. : l'lw two tlitnc~~sionxl lanii~~ar flow ncar the stagnation point of a cylinder which 

11m an nrh~trnry trnnsvcrsc motion. Quart. .I. Mcch. Appl. Math. 12, 175-190 (1959). 

1681 . . \Vcntrrvclt, P.J: The theory of steady rotational flow generated by n sound field. J. Arouat. 

Soo. Amcr. 25. GO-- 67 (10%i). 

I691 \Irnrst. W.: (:rrni..uchicht,cn nn eylindriscl~cn Itiirpcm nlit nichkt,ationiircr Qtrerbowcg~~tlg. 

~. ~ 

17 11 Yane. 1

450 

XVI. Origin of tnrhulence I 

pying trnnsil.ion from I:~niinn,r 1.o h~rh~rlcrit flow is ol' fctntlamc?nl~a.l iniport,anc:c 

for I.hc whole snirncc of flnid ~ncchnnics. 'rho iricidcnrc of t,url)ulcnc:c was first, rocngnizcd 

in rclt~tion to flows tl~rorrglr strn.iglit pipes n.ncl elrn.ti~iols. In n flow at vcry 

low ltcynoltls rinrnlwr t.l~rongh n straight pip(: of uniform cross-sccttion ant1 smoot.11 

wn.lls, every llnitl p:irf.icln tnovrs wit,lt :I uniform vrlo~it~.~ long a straight path. 

Viscous forces slow down the p:~rl.iclrs nwr t.11~ w:d1 in relation to tliosn in t,l~c cxtcrnnl 

core. 'l'lic flow is wcllortlcrctl and pnrtic:las tmvcl alo~lg noighboirring lnycrs (1n.niiriar 

llow), Pig. 2.22a. Ilowcvcr, obscrvation shows t,l~at this onlorly pttcrn of Ilow 

(:oases to cxisL nt higher Itcynoltls r~ntnl~crs, Vig. 2.22b, :d t11:i.t sl,rong mixing 

of all l.lic pnrtiolcs occurs. 'l'liis rnising ~)rot:oss can ho mn.tlc visil)lc in a flow t,hrongh 

a pipc, as first shown hy 0. ItcynoI~I~ (711, hy feeding into it R thin thrci~cl of li(\uid 

dye. As long as the Ilow is 1arninn.r the tlrrcatl maint~ins sharply drfincd Im~n~lnrics 

all nlong tho sl,ro:i,~n. As soon a.7 the flow 1)ccomes turl)ulont the tfl~rcacl diffuses into 

bhc stream ancl thc flnict appcnrs uniformly colonrctl at, a short tlistnnce clownstmatn. 

In this ease thcrc is supcrimposctl on thc main motion in thc cliroct,ion of the axis 

of t,hc pipo a snt)sitliary motion at, right nnglcs to it wl~inh clli?c:l,s mixing. 'l'hc pattc.rn 

of strcnmlincs at a fixed point bccomcs snl)jcotccl to continnorts fluctuations ant1 thc 

sul)sidiary motion causcs an exchnngc of momcntnm in a tmnsvcrse direction beeausc 

each particlc su1)stantinlly robins its forward momcnt,tim while mixing is taldng 

placc. As a conscqiicncc, the vclocity tlistrihntion ovcr the cross-section is considcrably 

morc uniform in turbulcnt than in laminar flow. The mc~urcd velocity distribution 

for these two types of flow is shown in Fig. 16.1, where the mass flow is the 

samc for both cases. 111 laminar flow, according to the Hagen-Poiseuille solution 

given in Chap. I, the velocity tlist,ribut.ion ovcr the cross-section is parabolic (see 

also Fig. 1.2), bnt in turbulcnt, Ilow, owing to thc transfer of momcntnm in tho tmnsverse 

direction, it becomes considcrnbly more uniform. On closer investigntion it 

appears that thc most, essential fcature of a turbulcnt flow is the fact that at a given 

point in it, thc vclocity and tlic pressure arc not constant in time but exhibit very 

irregnlar, high-frcqucncy flnctuations, Fig. 16.17. The velocity at a given point 

can only be consitlcrcd constant on the average and ovcr a longer ~eriod of timc 

(q~~asi-steady flow). 

Thc first syst,cmatio investigation into thcsc two f~~ndan~cntnlly tliffcr~rit~ patkerns 

of flow wcre conducted by 0, Rcynolds [71]. 0. Rcyr~olds was also the first to 

investigate in greater tlctail thc circnmstn.r~cos of the transition from laminar to 

tnrbnIrnt flow. l'hc provionsly mcntionctl tlyc oxperimcnt was nscd by him in t,his 

coru~cxion, antl he discoverctl the law of similarity which now bears his name, and 

which states that, trnnsition from laminar to turbulent flow nlways occurs at nearly 

tho samc Reynolds nnniher 171 dlv, wlierc t3 = (;)/A is thc mean flow-velocity 

(Q -1 volrlme m1.c of Ilow, A = cross-sectional area). 'rhc numerical value of tho 

R.c~ynoltls n~inibcr at, which t,rn.nsition occurs (critical ltcynoltls number) was 

Fig. 16.1. Vrlority di~t,rihtll.ion it1 pipc; n) Inminnr: h) turhulrnt 

a. Some ~xperin~rnl.al rrrrtrltn otl transit ion from lnrninar to turl~r~lrnt flow 45 1 

~stablishcd as bring approximately 

Accortlingly, flows for which thc Itcyrioltls nnm1)cr R < R,,,,, arc snppost~l t.o he 

laminar, nncl flows Ihr which R > R , nrn cxl)cctctl t,o IIC tt~rhrrl~~t~t~. 'l'li(: t~i~tii(~i(d 

value of the criCical Itcynolds nurnt)cr tlc~~cnt1s vcry st,rongly on th: vo~~tlit~iotin 

which prevail in Ihc inil.inl pipe lcrigtl~ ns wrll as in t,lic n.l~l)roac-h to il.. 1':vc~n It~~~t~oltls 

tlionght that the cril.ionl ltcynoltls ri~~ttil~cr inc:rn:~w.s as thc tlisI,itrl~:~nc:vs ill t.Iit: 

flow bcforc thc pipe arc tlccreasctl. 'l'liis fact was confirmed csprrimc~ltally I)y 

11. T. 12arnes and 15. G. Colcer [I b], antl latcr by L. Schillcr 1801 who rc-nrhctl 

critical vn.lurs of the Itcynoltls numl)cr of up to 20,000. V. W. 1Slrman [24] succ~:ctlcd 

in mai~it~nir~ing laminar flow up to a critical Itcynoltls nun~bcr of 40,000 by providing 

an irilct which was mnclc cxeept~ionnlly frcc from tliat urh:~nccs. 'I%(: 111)pt:r 

limit to which tho critical lteynoltls nl~mhnr can IK: tlrivcn if cxtrcmc: cnro is !,:II

ITig. 10.2. Variation of flow vclority in a pipc in thc tranuition range at dihrent distances r 

from pilw axis, as 111cas11rrt1 

by .J. I n d ~ k nurn0~r A = ii.d/v = 2550; axial tlistnn~e z/d = 322; E x 4.27 m/scc (P 14.0 ltlsec); vclocitien given 

in nrlsrr. .rhr.rr vclorily plols, oblninnl with tho mid ol n Iwt-wire nncmomatrr, rle~nonstrate the i~~termitlont naluro 

vf llm Ilow in fltxl pcrindr nl Inrninnr nnd lurhu~lrnl flow aurcerrl crch othrr In time 

d 

Fig. 16.3. Intermittct~cy factor 

y for pipe flow in the transition 

range in ternls of the 

axial distance z for different 

Itrylrolds nurnbers R, as rneasr~rpd 

by J. Rotta [75] 

Ilrrc y = i dcnotcs rnnlinllnll~ly tllr- 

IHIIPIIL, IIII~ y = 0 1-onti1iunil5~y larnlrlnr 

Bnw 

rango frorn R 1 2300 t,o 2600 ovcr which transitpion is completctl. At Rcynolds 

rtllrnhtw rlcar l,l~c lowcr limit, the process of tmnsit,ion to f~lly dcvclopcd turbulent 

Hr)li(:J1 1,l1(, (lo,,r pS~,t:tl(lS ~ V ( T very I ~gc tIist,anccs mcasurctl in thousa~ids of tlialnctcrs. 

Mc~asrlrc:tnt~nt.s ol. i,llis kind have been reccntly amplified by J. Meseth [GO]. 

a. Some exprrinicnlnl results on transition from laminar 1.0 L~~rl~ulrnt 

flow 453 

important ones being the prcssure distribution in tllr rxternal flow, tthr rl:lfurc- or 

tlir wall (ronghr~rss) and the nature of thr disturbanrrs in t,hc frro flow (ir~irtisiIj~ 

of turbulcncc). 

Blimt Ilodies: A pnrticularly rcvn:wkal)lo phenomrnorl c:onrlrnt,rtl wit,l~ i,rt~nsil it111 

in thc hour~tlary laycr occurs with blunt bodies, for cxnl~~plc spl~cros or (:irc:~tl:ir 

cyIintl(:rs. It is seen frorn Figs. 1.4 :~nd 1.5 that thc r1r:i.g c:ocflicic~rlt or a sj)l~c:rr: or 

cylindcr tlccrcascs :~l)r~~pi,Iy at Itcyr~oltls rn~mbcrs R :-: IrI)/v of al)orlt :1 x lo5. 

, .I I his almpt drop in thc tlr:~~ corfficicnt, noticctl lid 11y 0. 12ifli.l [231 in rcsl;rt ion 

to sphc:rc?s, is a conscc~~~cnc:c of Lr:lrlsiUor~ it1 the I ~ I I I I ~ 1:iyc:r. I : ~ ~ '1'r:l.tlsil.iotl t.;tttsi*x 

tltc jminl. of scp:lr:lt.ion to move clowt~st.rcar~~ wllicll consitlrr:~l)ly tlt:crc~:~sos i,l~(: 

width of the walrc. 'l'hc truth of this cxpl:in:~tion was tlc1nonsi.r:il,nt1 i)y I,. I'r:r11111.1 

1411 wl~o nlor~ntctl n thin wire hoop sotncwl~:~t .-llrntl of tht: cquator of a si~llorc:. 'l'liis 

causes artificinlly the b0undar.y layer to 1)ccomc turbrllcnt at a lowcr Ilcynol~ls 

numhrr antl protluccs the same drop in drag as occurs w11t:n I,lic Itoyt~oltls IIIIIIIIIW 

is nmtla to incrrasc. 'l'hc stnolrc photogrii~)l~s in l'ig. 2.2.1- nntl 2.26 S I I ~ ) +:~rIj~ 

~ ~ ~ 

thr cxtcnt of t,hc waltc on a sphcrc: it1 thc sub-critic:~l Ilow rcgirnc t,llc \v:~ltc is uitlt. 

arid the drag is large, antl in t-bc supcr-crit,icnl regime it is narrow nntl thc clrag is stnall. 

The lattrr flow rcgimc was here crcatctl witll-the a.itl of L'rantltl's 't,ripping wire'. 

These experiments show coriclusively that the jump in the drag curve of a sphrrc 

is due to n boundary-layer cKect and is caused by tmr~sitio~~. 

Flat plate: Thc procrss of transition on a flat plate at zrro incitloncc is sonrrwhat. 

simplcr to understand than that on a blunt hotly. Thc prorcss of t.r:rnsit.ion in t.11~ 

bountlary layer on a flat plate was first stntlictl by ,J. nl. nrwgrrs 161. 13. (:. van 

der IIcggc Zijncn 1411 antl Iat.er by M. ITanscn antl, it1 grca1.c.r clrt,ail. I1.y 11. I,. 

1)ryclrn 116: 17, 181. According t,o Cl~:lp. VlT, t,llc bo~~ntlnt~y-layer t.l~ivlztlrss on :i flat 

plntc incrcases in proportion to j/z, whcrc s tlcrmtlcs tl~c tlistancc from thc Ic:atling 

edge. Near the lending edge the bountlary Iaycr is always Iamit~art, l~cnorning i,urbulent 

further downstream. On a pl;~tc with a sharp lratling edge ant1 in a. tlormal 

air stream (i. c. of int,crisit,y of turbulcncc T = 0.6 %) t,mnsit.ior~ t.:il~s p l ; at, ~ ~ :I 

distance z from it, as detcrminecl hy 

On a Axt plaLe, in thc same way as in a pip, the c:rit,ic:il Itcyt~oltls n11~11)c.r ran 1)r 

incrcasctl by provitlitig for a dist,urbancc-frcc cxt.orna1 flow (vc.rp low ir~t.c*~lsil,y or 

tllrl)ulcncr), c/. Scc. XI1 tl 2. 

'I'r:~rtsit,ion is casirst to pcrrrivo Oy a s111tly of t.lw vrloc~i1.y tlisl.t~il~i~iiot~ ill (11t- 

1)oun~Iary la,ycr. As SCCII from Fig. 2.23, t~r:insit,io~~ is shown ~)rnt~ti~~(-~~I,l~y 

1j.y :I, sit~ltlt:~~ 

incrrase In the boundary-layer tliick~lcss. 111 a lalninar l)ou~iclarv Iavor tlrc tli~ncq~ion- 

. . 

-. . .. .. 

less I~ountlary-1;~ycr trhickricss, d /i v :,;/Urn , rcrnnins rorlst,:~nt ant1 rtl11;11, :I l)pro,yimatcly, 

to 5. 'I'he dimcnsiortlcss boutidary-layrr t01icknrss is scct~ plot,t.c~tl :~z:~ir~st. 

thc length Rcynolds numbcr R, =-: U, z/v in l'ig. 2.2:) nlrmtly mc!~lt~iot~rtl: :I(, 

R, >. 3.2 x 105 n sutltlcn incrcaso ill 1.llo I)o~~t~tl:~~r~-l:~~~~r 

I.l~i(.l

XVT. Origin of turbulcncc I n. Some rxprrirnrntd reuulte on t,rn~~sitiott iron^ lnmir~nr to turb~~lcnl flow 

Fig. 16.4. Vcloc:it,y profilcs in n 

boundary lnycr or1 n flat plate in 

the trar~sit~ion rcgion, ns nic:buurrd 

by Srhuba~~cr nnd Klcbunoff 

[83] 

(1) laminar, I$la~iua pronle; (2) lerba~lcnl. 

I/,-lh pwcr law, d = 17 lnln 

(= 1.30 in), ext.orna1 vclocil,p ti, = 

27nl/scc (89 ltfscc); t~~rln~lcnce inlensily 

T = 0.03% 

tlistril,~~tion curve. 'l'l~c changes in the velocity profiles in the transit.ion region have 

becn plottctl in Fig. 16.4. They arc based on ~noasurcmcnts pcrformctl by C. R. Schubauer 

and 1'. S. Klcbanoff [$3] in a stream of very low turbulence intensity and 

it is seen that in this case the transition rcgion extends over a range of Reynolds 

numlwrs from about R, = 3 x 10Vto 4 x 10" In this mngc, the boundary-layer 

profile cl~angcs from that of fully tlcvclopccl laminar flow, as calculatecl by Rlnsius, 

to fully tlcvclopctl turl)uloit flow (see Chap. XXI). The process of transition involves 

a large tlccrcnsc in the shape factor lIlz .-.. d,/d,, as seen from Fig. 16.5; here dl clenotcs 

tlie clisplnmmont thickness anti cSz is the momentum thickness. In the case of a 

flat plxta, the shape factor tlecrcxscs from IIlz w 2.6 in the laminar regime to Illz w 1.4 

in t,lio f.~trl~~~lt?rit, rcgirnc. 

'I'his change in the velocit,y di~tribut~ion in tho transition region can be utilized 

for the convcnierit tlctcrminntion of the point of transition, or, rather, of the tmnsition 

region. The principle is explained with the aid of Fig. 16.6. A tmtal-hen11 

t,ul,c or a I'itot tube is moved parallel to the wall at a distance which correspontls 

t,o the maximurn tlifirc~noe bctwccn tho velocities in the laminar and turl~ulcnt 

rrgirncs. On I,cit;g rnovctl clownstrca~n across the t,ransition region, the tubo sl~ows 

a fairly sudtlcn it~c:rcasc in the total or tlynamic pressure. 

'I'mnsitiorl on n flat plate also involves n large change in tile rcsistanrc to flow, 

in this rnsr in t.11~ skin friction. In laminar llow the skin friction is proportiorid 

to I 111. I .5 pnwt~ of vrlocity. cyn. (7.33), wlir~-as in t ~~rl~~lrnt, llow the powrr inrrrnsrs 

to :~l)oul, 1.85, as shown a long time ago by W. I'rordc (2!)1 who prrforrnc~l towing 

~xprrimenta wit11 platcs at very high Itrynolds rininbers In this conncxion the 

I 

reader may also wish to consult Fig. 21.2. 

Morcrcrcnt cxpcrimrnb performed by 11. W. Etnmons [25], and G. 73. Scl~~tbaucr 

and 1;. 8. Itlrbanoff [83] have ~hown that in the case of a flat phlc tfhe process 

of tr:~nsit,ion is also intrr~nitt~ent and ronsists of an irrrgular scqncriro of Inminar 

ant1 t,11rI1111rnt. rrgions. As cxplnin~cl in Fig. 16.7, at. n given point in the boundary 

laycr there occurs sudtlcnly a small t.~~rhnlcnt area ('turbulent spot'), irregular 

Fig. 16.5. CIIRIIR~ in L11c s11nj)c inctor 

Illz = 0116~ of t,l~e hol~ndnry Inycr for n 

flat plnte in the t,rnnnition rcgion nu men- 

28 

L6 

uttrctl by Schuba~~cr nnd Klcbnnofi [R3] 

qltot.otl from [ti51 

14 

1. / 1. 1 I 

Fig. 16.6. Erplnnntion ofthe rnrtl~ocl oiclclrrrnining the 

poiition nf thr point of trnrmit.ion with tho nid oin totd- 

I--l-- 

'F JW rW 600 1 

hend tube or n Pitot tube I- bminar A- trawhbn 4- turbulent -- 

in shape, whicl~ then travels downstream in a wedge-shaped region, as shown. Such 

turbulent spots appear at irregular intervals of time and at different, mntlomly 

distribulad points on tho plntc. In the inlmior of the wodgc-like tlo~nain tho llow 

is prctlo~ninantly ttnrl)ulentt, wht?rcns in the adjoining regions if, nltcrnirt,os co~~t,it~r~oirsly 

het,wecn being laminar and turbulent.. In this conncxior~ sco also rcf. I I:)]. A 1)a.pr.r 

by M.E. McCormick [57n] deals with l.hc problem of thc origin of sr1c:11 tturl,r~lt3~~t 

spots. St turns out tht an artificially created turbulent spot docs not persist w11r11 

the Rcynolds number has a value lower than Ril = GOO; Ibis is consist.ct~l~ wil.11 t,ltc: 

value of the criticnl Itcynolds nu~nl~cr cnlc~~lntctl with 1hc nid of I.hc li~~tsnr #I.nl~ilit,y 

t,heory, cqn. (16.22). Vcrg dctnilcd cxpcrimcnt,ml invrstigntions t,t~rl,ril~trt, sl,ot.s, 

and in particular of the velocity distril~t~t.in~~ in them, havc been cnrricd out by 

J. Wygnanski et al. [lo$].

I+. lli.7. (:rc)wtlt (11' :LII :~rtili(hl ~II~~IIICIII, 

:rt zc\r.o i~~~icltwrr :IS ~t~c~ns~~rrtl 

spot, in a h11i11:~r boutld:lr,y 1:iyor on a flat pht~? 

by (:. H. Scsl~~~l~:r~~rr tuid 1'. S. I

458 XVI. Origin of turbulence I b. Prinriplr~ of the throry of stnhility of l~i~nir~nr flows 459 

to thc approprial.~ rlilTrrrntinl cqr~ntions. 'rliis is tlic mrlhod of small dislur6nnces. 

This scroritl nirtliotl has led to complctr success and will, for this rcason, be described 

with somr tlrtail. 

Wc s1i:ill now considor a two-tlimonsional incornprcssiblc mean flow and an 

cq~~aliy hwo-tlimcnsiond distnrl)anco. 'L'hc rcsulting motion, clcscribcd by eqns. 

(16.2) a.ntl (16.3). sat.isfios tlio two-tlirucnsion:~l form of thc Navicr-Stokes equations 

a,. givcn in rqris. (4.4a, b, c). Wc shall further simplify the problem by stipulating 

that tlrc mean vclocit,y TJ dcpcntls only on y, i. c., IJ = U(y), whcreas the remaining 

two componct~i~ n.rc supposed to be xoro cvcrywhorc, or V W z Of. Wc havc 

cr~c:ount~crctl s~~oli flows cnrlicr, tlcscril~ing thcrn :IS pwdlel &OWR. In thc casc of :L 

chnnncl with parallel walls or a pipe, such a flow is reproduced with great accuracy 

at a sufficient dishncc from tho inlet section. The flow in the bountlary layer can 

also he reg.anlcd ns k good approximation to parallel flow because the dependence 

of tlic velocity U in the main flow on the x-coordinate is very much ~matlcr than 

that on y. As far as thc prcssurc in the main flow is conccrncd, it is obviously nccessnry 

to assume a dcpcntlcncc on x as well as on y, i. e., P(x,y), becausc the prcssurc gradient 

i)P/ax maintains the Ilow. Thus we assume a mean flow with 

Upon the mcan flow wc nssnmc snperimposcd a two-tlirnensional disturbance which 

is a function of time and space. Its velocity componcnts and pressure arc, rcspec- 

tivcly, 

?~'(x,y,t) , vl(x,y,t) , pl(%yJ) . (16.6) 

IIenco tlio rcsnltant motion, according fn eqns. (16.2) and (16.3), is described by 

It is assumctl that t,lic nmLn flow, cqn. (16.4), is a solution of thc Navicr-Stokes 

equations, and it is required that thc resultnnt motion, eqn. (16.6), must also satisfy 

tho Navicr-Stokes equations. The supcrimposerl fluctuating vclocities from eqn. 

(16.5) arc takwi t,o 11e "small" in thc scnsc that all quadratic tcrrns in the fluctuating 

cornponcnts may be ncglccted with respcct to thc lincar terms. The succcctling 

section will cont,ain a morc dct,ailctl tlcscription of the form of the clistarbance. 

Now, tho task of tho stability t,hcory consists in clcbrmining whether the tlisturhancc 

is amplified or whotlicr it clccays for a givcn mean motion; the flow is cynsitlrrctl 

nnst.:~l~lc or stahlo tlcpcnding on wlicther the former or the latter is the case. 

Substituting cqns. (1 6 6) into tlic Navicr-Stokcs equations for a two-dimensional, 

incomprcssiblc, non-steady flow, cqns. (4.4a, b, c), and nrglcrting quadratic terms 

in the tlisturbance velocity components, wc obhin 

'J'hcw arc rramns In nopponr, a4 sl~n~vri hy (:. 1%. Scli~rba~~~r nntl P. S. IClcbsnofT [831, that 

t,l~mc t:or~~pot~c~rl~ nrt: ~I\V:LYR prtwmIr in rcnl IIOWR, particularly in flow^ pmt flat plnte~. I'hcir 

r~~ngnit.r~tlo is rwgligil~lc for tno~L lwrl)oscs, l ~ thcy t mern Lo play n part, not yct fully clucidatecl. 

ill thc proocas of trnn~ition; RCO nl~n foo1.1iotc on p. 468. 

wherc V2 dcnotes thc Laplacinn opcrator a2/i)22 + a2/&/2. 

If it is considcrctl that the mrnn flow it,sclf sntisfics the Navicr-Stokcs qua- 

tions, the above equations can bc simplified to 

Wc havc ol~tainotl throe cqn:~i,ions for IL', 11' and pl. 'l'11c I)o~~ntl~~ry contlitio~~s spwify 

that the turbulent velocity components IL' aid v' vanish on thc wtills (no-sli11 coridition). 

The pressure p' can be easily climinatcd from the two equations, (10.7) ant1 

(1F.8), so that together with thc cqi~nt~ion of continuil.y IJicro arc t\vo cqn:~i,ions for 

u' and v'. It is possilh to criticize the a.ssnrnrtl form of tl~c: rnctLn Ilow, ccp. (I(i..l), or1 

tlic ground that the variation of thc coniporicnt lI of t.hc vclocity with x as wdl :I,S 

the normal component V havc hccn ncglccbtl. Jn this conncxion, howcvcr, .l. I'rctsch 

[44] proved that the rcsult.ing terms in the eqnations arc unimportant for the 

stability of a boundary layer (see also S. J. Cheng 171). 

3. The Orr-Sommerfeld equation. The mean laminar flow in the 2-direction 

with a velocity U (y) is assnmcd to be influenced by a disturbance which is composetl 

of a number of discretc psrtlial fluctuations, cnch of which is said to consist of n wave 

which is propagated in tho x-direction. As it, has already bccn assrirnctl i,li:tt tlic 

perturbation is two-dimensional, it is possible to introduce a stream function yi(z, y,t) 

thus integrating the equation of contin11it.y (10.9). Thc stream function reprcscnting 

a single oscillation of the distnrbance is assumcd to bc of the form 

Any arbitrary two-dimensional disturhancc is assumctl cxpantlcd in a I'ouricr 

scries; cach of its terms represent8 such a partial osc:illation. In cqn. (16.10) a is n real 

quantity and A = 2 x/a is the wavelcngth of t.lrc clis1urb;uicc. The q~iant,ity is 

complex, 

P = P, I- i PI , 

where p, is the circular frequency of the pnrt.inl oscillation, wlicrcas P, (amplification 

factor) determines tho clcgrec of amplificnt.ion or damping. The t1isturl)anc.n~ arc 

tlanipocl if P, < 0 ant1 tslic laminar mcan flow is st,al)lc, wlicmas for PI :. 0 i~~st,:~l~ilii.y 

~cts in. Apart from a a.nd it is convcnicnl to introtlucc tlic4r ratio 

t Tho convenient coniplc?r noLrrt.ion in 11ort1 Iwr. l'hysivnl mr~u~ing is nt.trwl~t.tl only to I.hc rral 

part or the ~tmani unction, tlwn

4GO XVI. Origin of turbulence 1 1). Principles of the theory or stnbility of Inrninnr flows 4G1 

JIerc c, t1cnot.c~ the vclocit,y of propaption of t,hc wave in the z-dircction (phase 

vclooity) wl~crcas c, again dcLertnincs t,ho tlcgrce of damping, or nmplificntion, ticpending 

011 it,s sign. I'hc arnplit,~~tle function, 4, of the flnct,tlation is ass~~~nccl to tlcpcntl 

on y only ~C(:ILIISC the mcm Ilow tlcpcnds on y alone. I'roni cqn. (16.10) it is possible 

to ol)t,ain 1.11r c:ornponcnlss of thr: pcrt,rrrl)nt.ion vcloc:it,y 

11,' - 

t)l/, 

- 

i)y 

: ,#,'(?/) (.""' PI' , (10.12) 

1) -= 

Ry1 -- 

ax 

i u +(?I) &my PI) . 

Irlt.rotl~tc:ir~g I,l~c:sc- valnrs inh rqns. (16.7) and (10.8), we obtain, after the elirninat.io11 

of prrssnrc, l11c following, ordinary, fourth-order, differential cquation for the 

n.tnl)litwlc 4(y) : 

I 

When the mean flow IJ (11) is specified, eqn. (IF. 14) contains four pamrnnt.rrs, n :~n~cl~ a, 

R, c, anti c,. Of t.hcse the rtcynoltls nrlml)or of the moau flow is liltcwiso spccill(:d 

and, f~~rthcr, the ~avcl~ngth i = 2 n/a of the disturb:m(:c is to bc consitlcrctl given. 

In t.his rase the differentid equation (IF.l4), togcthrr with thc Imlntlary c:ontlitiorrs 

(16.15), f\rrnish one oigrnf'uncLion +(?I) and one complex cigcnv:~ll~r: c == c, 1 i ci 

for (wh pir ol' v:~l~tt.s a, R. I Ivrc cr rt:~~rcsw~l~s I,IIc ~)II:LSC v(hwiI,.y 01- 1.11~ l~rw(:ril~t:~l 

t1isturbnncc wltcrcas thc sign of c, tlcLcrnlit~cs wllctllrr bho wave is amptilictl (ci :.O) 

or tla.tnpctl (ci < O )t. For c, < 0 the correspontling flow (IJ, R) is stnth for lllc ~ivcn 

valur of a, wllcrrns c, > 0 tl~notcs il~stabilit~y. Thc limiling case c, - 0 corrrsponds 

1.0 nrnLml (indiKcrmt) dist~rrl~ancrs. 

11nm1)cr is the critical Reytrolds nuinher or limit 

of Ian~inar flow untlcr cot~sitlcrxLion. 

Fig. 16.8. Curves of neutral stabi1it.y for 

two-di~nensionnl borlntlary layer wi(.tr 

two-dimensional disltlrbrmccs 

(a) "non-visrolls" inslal,ilily; in thc rnsc nC vclocity 

~~rnillrs or 1yp :t wilh pin1 *d i~~flrxion 

PI, thr rllrrc or 1w111rnl slnbility is or typc :, 

(1,) "visrow" Instahllity; in I11e raw or vriorit,) 

proIllrr nl type 8 uilhotrt poi111 of lnflrxion, llw 

rurve ol erutml shbilily is or lypc b 

Tl~n nryrnglotm Tor ll~n rrlrvc or rwulral stnbility 

a al. R --+ r, arc ohlxincd rroln llw "rrirlionlrar" 

slal~ilit?. rrlllalio~i (16.16) 

r , 

I he rxprritnrntd rvi(Irncr eonwrnit~g t,ransit,iot~ fro111 l:~n~in:t,r 1-0 I,II~I~~III~II~, 

flow rcfrrrrtl 1.0 ~)roviorlsly I(.nds 11s t,o cs~~rcl, that, ;LI, SIII:LII Ib(~,y~~oI~ls I I I I I I I I ~ for ~ ~ 

wl1ic.11 I:~minar llow is ol~servrtl. all \v:rvrl~ngt.l~s wo~lltl l)rorluc*v ot~lg sl.:rl)lr tlist.llrl):~l~c~s, 

wl~rrrn.s :lip I:~r.gt-r ltc!g~~oltls rtlln~l)rrs, li~r wllic~l~ I.II~IIIII~:III, IIow is 011svr\~1~1, 

ul~st,:~l)lc tlist~lirbarlccs o~tght, t,o corrcs~~ontl to at, I(ywt, sonto \v:Lv~I~II~~.IIs. IIo\r.(:v(~, 

it is nwrssnry t,o rcn~nrk at tallis poitlt t.l~:rt~ t.11~ vrit,ic:n.l I

462 

XVI. Origin of tnrb~rlmcc 1 

numl)er obscrvctl :~t the point of l,ransition. If athnt.ion is fixed on t,hc Ilnw in the 

honntlary hyrr along a wall, then t,llc tllc:orctical critical Itcynoltls nnml)cr indicates 

the point. on t,111. wall at. wllirlm :~niplific:ntion of somc individual tlisturbances begins 

ant1 prorreds rlownstroam of it. The transformatkm of sucli ;~mplificd disturbances 

i11t.o t.nrlnllrnw t,altrs I I somr ~ timc, and tho nnstal~lc tlist,ltrl)nncc: has had a chancc: 

to t.r:tvel somc tlistn.nrc in the tlownstrcsm direction. It must, therefore, bc cxpcct-cd 

that, t.hc o1)scrvctl posit,ion of thc point of transil,ion will be tlownstream of the 

calculat~ctl, thcorctica.l limit of stability, or, in othr words, that the experimental 

critical Reynolds number cxcccds itFl thcorctical value. This remark, cvidcntly, 

applies to Rcynolds n~~mbcrs 1)asctl on tlmc curmnt lcngth as well ns to those bsscd 

on the bourdary-layer thickness. In order to distinguish bctwcen these two values 

it is usual to call the thcorctical critical Reynolds number (limit of stability) the 

pint o/ l:nstabilit?y whcrcas the experimental critical Reynolds number is called 

the point o/ trnn~itiont. 

Thc st,nbiiitg problem, briefly described in t,hc preceding paragraphs, leads to 

cxtremcly difficult mat,llcmntical consitlcmtions. Owing to tllcse, succcss in the 

calculation of thc critical Jtcynolrls nnmbcr eluJccl the workers in this field for 

several cleoatlcs, in spite of the greatest efforts clircctcd towards this goal. Consequrntly, 

in what follows we shall he unable tx, provide a complete presentation of 

tl~e stdility t.hrory nnd will be forccd ta restrict ourselves to giving an account of 

the most important rcsul th? only. 

5. Genernl properties of the Orr-Sommerfeld equntion. Sincc from experimental 

cviclcnce thc limit of stability c, =O is cxpectcd to occur for large 

valucs of tlmc Itcynolcls number, it is n:~tural to simplify thc eqnation by omitting 

the viscous bmms on t.he right-hand side of it, as comparctl with the incrtia tcrms, 

beca~~se of t,lmc smallness of the coefficient 1/R. The result.ing differential equation 

is known as t.110 /rictionlr.~s .~lnhilily ~q7mlion, or Ru?~lri~~A's equntinn: 

(IJ ---c) (4" - a24) -- I/"$ = 0 . (16.16) 

It, is imporhrmt. 1.0 note Ilcw that of t h four bountlary contli(.ions (16.15) of t h 

cornplctn equation it is now possiblc to satisfy only two, bccausc the fricl ionlcss 

stability cqu:~t.ion is of tltc sccon(l ortlcr. 'Umc rcmai~mirmg boundary condition to bc 

sat.isficd is t,l~c vanishing of t.hc normal componrnt,s of vclocity near thc wall of 

a CII~~IIIICI, or, in l)o~~r~(I:~.r.v-l:~y(:r flow, tlwir vanisl~ing al, I,lmc wall nn(l at infinity, 

,. 

I bus, in the I:~l.l.cr case, wc 11avc 

y=O: +=O; y=m: +=O. (16.17) 

,I 

Lllc onmission of the viscous tcrms constitutes a tlra,st,ic simplifirat.ion, hccmse the 

ortlcr of t,lmo cqnn.t,ion is roclurctl from four to Lwo, and t,llis may result in a loss 

of imporl,:mO prol~c!rl.irs of tho gnnom.1 soluI.ion of thc complctc cqnat.ion, as comparctl 

wil,ll its simplilictl v(:rsion. Ilnrc we n1n.y rc~pcn.l.'tlmo rom:~.rks nol.c:cl provio~~sly in 

C11n.p. IV in conncxion with the transition froi? the Navicr-St,okcs equations of 

a viscous IIrritl Lo those for a frictionlcss fluitl. 

- ~- 

t nlrrruly cxplnirtrrl in Src. XVIn, rocnnt, expcritnrntnl rcsulL~ (11. \\I. ICrnrnons 1251, and 

hdl~~l)n~tor RINI J

d(i.4 XVT. Origin of hrbulcnce I 

of wavol~:ngblts; ill t.l~c tlircction of (lcwmsing Itoynoltls numbers, this rangc is 

scpnmtctl fron~ tl~c stal)lo rnngo by tho anrvo of nc:utral stability. 

tn contrnst, with t.11~ l~rccctling C:LSC, IJ~SC~LS inatahilily is associatcd with a curvc 

or ncnl,r:~l st:l.i~ilil,y of sl1n.p b, also sl~own in Fig. 10.8, and with 1)ountlary-laycr 

prolilrs pnsscssi~~g no point of i~tflcsion. At Itey1101~1s nwnl)crs tending to infinity, 

I,hr r:~.t~gc of t~nsl,:~.l)lc w:~.vrlrngl~l~s is rnnl.r:~cl~~~l to a point, anql (1om:tins or ~~nst,:~l~Ic 

osc:ill;~l.iot~s :IIV swt~ 1.0 oxisl, otlly for fi~~il,o Ilnyrioltls t~nrnl)ors. (~cncrally spr:~.Iting, 

t,ho n.tno~tnt, of :~.rnplificnt.ion is m11c11 largcr in t,hc casc of frictionless in~tabilit~y than 

in tho c:~sc: of visc:ous it~sl.:~.l~ilit,y. 

, , 

I llc vsislrncSc: of visco~ls insl.:~l)ilily (XII bc tlisc:ovr~~ctl only in c:onncsion wil,ll 

a discussion of the fitll Orr-Sommcrli4(1 equatior~; it const.itut,cs, tl~croforc, t,hc 

moro tliffirrrlt, nn:~l~t,icnl c:tsc. 'l'hc simplcst case of flow, nan~cly t,hat along a flat 

plal,c: with zero Iwrssurc gmclicnt belongs l,o the kind for which only viscous inst,abilit,y 

tlocs occur; it, W:IS s~~cc:rssl'r~lly taclilctl only comparatively recently. 

TI1 corc. rn I I : Tllc sccontl impor1,:lnt goncr:~l theorem sl,atxs that tsho vrlocit,y 

of 1~011nptio11 of nnutrnl tlis1mrl):anccs (c, = 0) in n I~ountlary I:~ycr is srnnllcr tll:~n 

the m:urtnlrnl vrloci1.y of 1.11~ mcnn flow, i. o. tht, c, < (I,,,. 

'I'his t,l~c:orwn was :rlso first provctl by J,ortl IXnylcigll [70J, albcit itnclcr somc 

~.(~sfli(*t.iv(* :~s~r~mpl~ions; if, was ~I.OVC(I ngnin by 14'. 'Yollmien [I001 for more gcneral 

conclit,ions. It, :lsscrt.s th:tt in t,hc intcrior of the flow there cxist-R a layer wllcrc 

IJ - c = O for nc~ltral tlist.r~rl~anccs. 'I'l~is fact,, too, is of funtlanlental importance 

in t,hc t.llcor,y of st.:~.l)ilit,y. 'I'l~o Ia.ycr for which 11 - c = 0 rorrcspontls, namely, 

to a singttlar point, of t,lto frictionless st.n.l)ilit,y cqrration (16.16). Att this point #" 

I~ccwncs ir~fittit~c! il 11" tlocs not v:~t~isll 1,ltcrn simttltnncor~sly. Tllc (1isl~:tnt:c = 

wltorc: 11 -~. c is c::i.ll(~l t,11(: crilicctl. Inyrr or 1.110 mr:an flow. If [I," 4: 0, thxl 4" tends 

to infinil,.~ :w 

' ' 1 1 

. - .. 

(I,/,. !/ ---y/< 

in 1.11~: ~lc:iglll)o~lt.l~oo(l of 1.11(* c:rit,ic::d 1:tyc.r wllrrc it, is pcrtnissil)lc t,o pnt IJ - c = 

--- 1Jlfr(?y - yJi) :~.[~l)roxit~~:~l.cl.y; (:onsrq~~~~nt,ly IJIC x-ootn~mncnt of lhc vclocil~y can 

br writ.t,rr~ :r.s 

'I'hrts, a.ncortling t,o Ihc frictionloss stability cquation, the component, IL' of the 

vrlooi1,y wl~idt is p:~rallcl to t,llc wall l)cc:otnos infinite if the curvature of the velocity 

profile at, the critical layer tlocs not vatdsli simnltnnconslg. This mn.thmatical 

sing~tI:~.ril,y in t.ho fric:Lionloss sl.a.bilit,y ccl~l:~l~ion poin1.s l,o the f:i.c:t tll:~t t.11~ cff'rcl 

of viscosil.y on (.Ire c(111:~I.ion of motion tnttst noL be ncglcct,ctl in Ihc ncighbor~rhootl 

of t,hc oriticxl I:~ycr. 'J'llc irlnl~~sion of t,hc effcct of viscosity removes this physica.lly 

a11~11rd sing111:1rity of 1.11~ frictionlrss st,:~l)ilif.y ccl~lat,ion. 'l'hc n.na.lysis of the cfial, 

of (,his so-c:rllotl viscous corroct.ion on 1.h~ soluticfn of the st,abilit.y cquation plays 

a fut~tlnmcnl.n.l part in tltc tliscussion of st.nbility. 

'J'l~c two tllnorcms tluc to Lord Raylcigh sl~ow that tho curvature of thc vclocity 

profile n.ni.ct,s st~al)ility in nfi~ntlatncnt.:~l w:~y. Sini~~ltar~cously it has hen dcmonstmtcd 

t11:lt the c~:rlaul:ttion of vrlocit,y profilcs in laminar bountlary layers must proceed 

with vrry high accnraoy for the investigation of stddity to bc possible: it is not 

enongh t,o cvsluato U (y) wit11 sllfficicnt dcgrcc of accnrnry Imt. i(,s srconcI (~rrivativc 

d2Ultly2 must also bc nccurotely hown. 

c. Itcsultn of tl~c theory of stnbility us tltry upply to the bo~r~~tlr~ry Iayr nn XI flut pl;~tc 

at zero ir~citlcncc 

Velocity U 

ffm

466 

XVi. Origin of t.r~rhr~lrncc 1 

whereas concave corners. Figs. IC,.!)r, (I, nIw:~.ys Ivad t,o ins1.nl)ilif.y. This invcst.igat.ion 

matle it plausil)lc t,o s~~pl)osc t11;~t volocit,y profiles wit,lt points of ittflcxion, 14'ig. 

lO.9g, arennstablc. The tr~tt,li of this supposition was later dcmonstratctl by W. Tollmicn 

[loo], a.s aIrra(Iy stat,cd in See. XVI I), Theorem I. 

In ortlvr l,o ol)t.ait~ :I lin~it~ of stal)ilit,y oxprvssctl ill t,crnis of a ltcynoltls number 

for nnst.:rl~lc: volooit,y profilrs (I'ijis. lC,.!)c :rnd tl), tho largest, visro~ts terms appearing 

in t,l~c c.on~pl(~tc st,nhiiit.y cqn:it.ior~ ( 1 (i. 14) wrrc hkrn in1.o acco~~nt., nntl it was crxpoct,ocl 

tht, thry will promolr tlnn~ping. 'I'l~c: i~r(lncncr of viscosit,y on Lho tlist.~~rl):~.nccs 

rxl.rntlctl hrrc only ovrr a vrry small rvgion of (,II(: whole vc,locit,y profilo, being 

loc:ilrti in Lhr i~nmt~~li:rl~c~ ~lc~igl~l)o~trhc,otl ol' Lhc wall, in ortlcr 1.0 sat.isf.y 1.11~ no-slilt 

contlit.ion. 'I'ho c.alr~tl:~t.ions ~)crlormctl I),y 0. l'iel,jrns let1 to t,l~c vcry uncxpcc1,ccl 

result, 1.l1:tt t.11~ introtl~trliot~ of a smnll vnluo of viscosiby inLo tjl~c! cxluat,ions tlitl nol, 

protlurc damping but amplificat.ion for all ltcynnltls numbcrs, antl for all wave- 

Irngth of tho tlisturl)ancrs. Morrovcr, this rosult was ol)tainctl not, only for 11nstahlc 

velocity profilrs (JFip. lC,.!)c:, tl) 1,111. nlso for tho prolilcs of typc: a antl 1) in Fig. 16.9, 

which have l)ocn shown to be Slnl)lc whc:~~ viscosit.y was noglcct.ctl. 

An interim revicw tlcscril)ing lwogrrss achieved betwccn the years 1920 and 1930 

was given by I,. PrantltJ [67a] on t,he occasion of the annual GAMM mecting (German 

Society for Appliccl Mntl~rmat~ics and. Mcollanics) in lZad Elster, 1931. 

2. Cnlc~~lation of the curve of neutral stnbility. A sa.tisfactory explanation of the 

above pnrntlox wn.s supplird by \V. 'I'olltnicn l99l in the year 1929. 11c demonst~rntd 

t,ltat. tlw inflnrncc of viscosit,y on tlist~urbnnccs rn~~st I)e taken into acco~tnt not 

only in the itnmrdiat.c nrighl~ourhood of t,he wall, as supposed by 0. Tiet,jrns, but 

t,hat,;in atldit.ion, it mrtst be ncrountcd for also in t,he ncighbonrhood of t,hc crit,icnl 

layer, where the vcloc:it,y of wavc prol~ngat,ion of t,he tlist,url)anccs becom~s equal 

to the velocity of t,hc mn.in flow ant1 w11c1.c. as shown in Sec. XVI b 5, t,he component 

u' hecomes infinitn: according to the simplified, frictionless theory, the curvature 

of the profile bring tlilTcrrnt. from zero. 'l'hr rxistcncc of viscosity rauses large changes 

in t.his criticnl lny~r, while it is also cvitlrnt. t,l~at in rca1it.y u' remains finitc thcre. 

IIowcver, the inflltrncr of viscosit,y lwwncs cvitlent. only if the curvat,urr of the 

velocity profile is not Icft out, of account. Tl~cse considerations demon~t~ratcd that 

it. was necessary Lo st.~ttIy t.he 1~eh:~viour of sn1aI1 dist.urbances with respect to curved 

velocil~,y profilrs ((121J/t1?/2 f O), and wit.11 visco~it~y takcn into account bot,l~ in t>he 

r~eiplthor~rl~ootl of t.hr wdl and in the cril.ical layer. This programme was carried 

out. by W. 'l'olltnien in t,hr palwr q~tot,ctl rarlicr, and ns n res~~lt,, htr was able to find 

a limit of st.:ll)ilit.y ((:rit.ical Iteynoltls n~~ml)cr) for the cxamplc of the flow in the 

bonntlary Itcyc-r on a f1n.t pIn,te at. zero incidence which agreed well with experiments. 

Generally spcnlring, for net~t~ral oscillat.ions we find 1,ltat. 

and, consequent.ly, $1 and $2 represent the slowly varying solul.iot~s, wllc:rens $3 

and $4 become the fast varying solut,ions. The pair of solutions +I, #2 satisfies both 

the frictionless disturbance equation (Raylcigh's equation) anti the V~SC~IIR. Orr- 

Sommerfcld equation, eqns. (16.16) and (16.14), as y + oo. By contrast, thr pair of 

solut,ions $3, q5.4 satisfies only tho viscous dislurbar~cc cquntion. For Ihis rc-ason $ ,, 

95:! are rcfcrrctl to ns the fri~t~ionlcss solutions, wl~crcns #3, #4 nrc r:cllctl ~.II(- viscvris 

so111 Lions. 

with the boundary condition that $ = 9' = 0 at y = O.The non-viscoussolution 

does not satisfy the no-slip condition at the wall (y = 0) because $1' + 0 there. 

Furthermore, at the critical layer given by U - c = 0 we discover that 41' + ca, as 

explninect earlier. It follows that the contribution frorn frict.ion becomes particularly 

large at those two locat.ions, and t,hat t,he rcquired particular solution &(y), as well 

as the general solution #(y), vary with at a fast rate there. As a conaeqncncc, it. 

becomes vcry tedious t,o calculat,e the characteristic function $(y) and the eigenvalue 

c = c, + i ci, whether analytically or numerically, for a given pair of values of n a.nd 

R. When numerical methods are used, the special diflicult~icsste~n from the fact that 

the highest derivative in t,he Orr-Sommcrfeld equation, $"", is multiplied by t.he 

very small factor 1/R. Mat,hematically spcaking, t.hc lnrge diFTerence bct,wccn the 

course of thc funct,ion $(y) at the wall and at the critical lnycr as dcpicted by the 

frictionlew (Rayleigh) equation and the equation containing friction (Orr-Sommerfdd) 

stems from !.he fact that the order of the di~crentinl equation is rcd~~octl frorn 

four to two when the viscous terms arc deleted in it. 

An attempt to calculat,e numerically the characteristic functions 4(y) of the 

Orr-Sommcrfeld equation (16.14) for a large set of prescribed pairs of vnlucs of the 

reciprocal wavelength, ar, and Itcynolds number, R, puts enormous detnands on the 

capacity of a computer. This explains why 0. Tietjcns [9R] and W. ITcisenbcrg 1421, 

who ntkncltcd this problem in t,ltc twcntics, failed to achicvc success. 111 t.hc c.nd of 

t,hc t.wcnties, Tollmien rcvcrtcd t,o this problcrn antl forind no other way but I,o f:dl 

back on a very tedious nnalytic procedure. Nevertheless, these time-consuming ana- 

lytic methods proved eminently successfult. Det,ails of these calcdlations can be 

found in the original papers of W. Tollmien (99, 100, 1011 and D. Grohne [38]. There 

is no need to summnrize this work herc, because t,he calculations have been rendered

468 

As :I. srcl~trl 

XVI. Origin of tnrbulrnce I 

1.0 rcf. 1.471, t,llc cfi:c.f. of' a slighf, st,rc:amwise change in 1.he 1)asic 

flow \vns sf~~cli(vl a. nun1l)rr of 1.imcs 12. 4n., 31, 4fi:1., H4a., 1061. As alrrn.11~ point.ctl 

ottf. I)y .I. I'~.c~f.sc:l~ [6!)1. Illis c*lT(~t. is stn:~.ll. 

.. -. . 

'1'11~ VXIIWI~(I ~ O III I 01' I I I O~~.S~IIIIIII(~~~~~I(I 

~ 

w111:tli1111 

i~l~rc,d~rc.cvl 11s Ill(. I:wh ol p:~rnllrlisni in Illo hsir IIow r:tn I~rr fot~~id 

:IIII/ ,\. II. Kit)l;.l~ (8.4i~J. 'I1111~r0 nrc six :t~l~lilion:~l kr~ns. 

~ I : I I I ~ it) ~ % IIN: ,r-~li~wl im 01th~ :~n~plit~~dc oft IIC (lisl~~rl):~nre, t,wo ~ C~IIIS 

~IYW twlnl~o~lrnt 01 I I I ~ 

(lli.14) \!hid) f~o11lai11s I,llr :~~lclilic~n:~l ~ITIILS 

in :i IxLpcr I I \\'.S. ~ Saric 

'l'\vo trr111s arc: i~~t.~.o(ltcr,rtl hy thr 

arc :~~lclod t~y 111:: t,rnns. 

velocily ill llw Imsir Ilow, onr Inorr trrnl is d ~ t.o ~ llw c rl):lngc ill t,l~c 

uxvc.lrl~gtl~ of Ill(: elistt~rI)nn(~r in tl~c ~:-dirrt4io11 nnd. finnlly, the sixth terln rorrc~sponds to 

I~~pl~~~r-nr~l~~r 

I~WIIS ill I)or~ntl;~rg-kiycr I Iirory (srr (!I~:ip. jX). 'l'lw presence of s~~rtiori or blowi~~g 

~ I V W risc lo I'11rl11rr IWIIIS. An invrstig:ttio~l of tl~r 11111nvrir;il sol~~t~ions of LIIC so nodi if id Orr- 

Son~~~~rrl'rlcl IY~II:I~ ion lor v:eriyus vc*loril,v ~~rolilrs of I IIC I":~llrnrr-Siinl~ scvks, 12. 31, IO(i], lniletl 

10 ~ I I ~ I I I ( I$111 C ~ :l(l(liIion:~l lwms in IIIOS~ I~:ISVS. For I.llis rc:~son il is ~Iil'liv~~ll to 111:1lie :i ro~np:~risnn 

IWI\IIWI s1n.11 so111Iio11s its \wII :IS II~~\\(Y~II 111(v11 :~nd 01r SO~II~~IIIIS 01 Ilw "19in1l)lilir~l" Orr- 

Son~~~~rr(i.l~l (~~e~:llion. Ilo~wvrr, in IIIIISI (YISVS 1111. ~I:IIIKV ill IIIV limit (~I'sl:~l~ilily (IIID lo li~rk 

nf ~~:w:~llrlis~ll III~IIS 0111 lo I~r~~nall. Il'~~nw~.ir;ll ~ J I I I ~ P I ~ I);~vr 

R IIWII givrn 1)y F. C. T. S11en ct.nl. 

~85ll~. 

The result,s of st.a.1)ilit.y r:slcul:tt~ions porforrnrtl in :~c:c:ortI:rncv wi(.l~ (.In. nrc~f.llotl 

tlesc:rilwtl in 1.11~ prccrtling scct,ion arc sllown in Pigs. 16. IO :l.ntl I(i.1 l as wc.ll :IS 'l':~.l)le, 

16.1. The st~nl.rl)oinfs along l.llc? cr~rvcs II~c~nlsc~lvc~s 1~*prrsr111. ~IPIII.I.:I.I II~SI~III.II:IIII.I~S; t,I~i: 

rcgioti rrnl)rac~d lty the ~ r v cor~~es~)on(ls 

c 

to nnst~:~.l)lc ~lisl.~~t~l):~t~c~t~s, 

:III(I fI1:11, onfsi~lc 

it, contains sf.:~l~lc: point,s. ?'hc two l)ranc:I~(~s of the (:II~v~ of nctlt.ra.l st.:ll~ilif.,y f.c.ntl 

to\vartls zero a.t, very I;~rgc Itc~y~wlels lu~tnbc~s. 'l'hl: sn~:~llvsl. I

470 XVI. Origin of h~rlwlcnce T 

I 

R,,,, '5.0 

Fig. IB.11 

Fig. 16.10. Chrvrs of nc~ltrnl skd)ility for t h tiint.rtrbnncc freqllency /?, and thc wnve velocity c, 

n*i n fnnclion of llnynoltls ~nrtnbcr for t.hc hnundnry layer on n flnt plnb nt zero incidence (Blmius 

prolilr). 'J'11ror.y ncxwding tn W. 'l'olltnicn (991: nrrlncricnl cnlc~llntions by R. Jordinson [47]; see 

nlm 'l'nl~lr 16.1 

Fig. 16.1 I. Cnrveu of nerrtrnl nt.nhilit,y for t.lw tlist.~lrhnncc wnvelrngtll n 61 nn n f~rnction of Rayndds 

ni1nt1wr for 1.11~ 1~ound:1ry Inyor on n 1hl. plntc at zero inciticncc (Biwills prolilc). 'l'llcory nccording 

to W. 'l'olllnicm [W]; nnn~rric:nI mIccllnt,ionn by 1%. Jordin~on 1471; see also Tnblc 16.1. The 

nrnplitm~lc? tlist.ribrition for dist.nrlmnc:cn 1 nnrl I I in given in Fig. 16.20 

r 7 

I Itis is t11c ~wint. (11- inst:~l)ility (iw t .1~ 1~or111tl:i.ry I:~.~cr on n Hat, plxtr.. 11, is wtnnrkal~lc 

t.hn.1. only n c~orrly~:l.r:~l~ivc:ly n:Irrow mngo or'wnvclcngtl~s n.ntl Sreq~tencirs is "dnngcrow" 

li~r t.110 I:~.rninnr I)o~rncl:wy layer. 011 t,hc one Ilnntl, t.lwrc is n lower limit. for 

the Itcytol(ls numl)cr, on t,ho otllcr, thcre is an uppw limit for t h cl~nrnct.erist~ic 

mn.gnilwles of t,llc: tlist~urhnccs. Once the Iat,tc:r arc cxcceclcd no inst,nbiLit.y is cnusctl. 

The nuinrricnl v:~Iucs :we: 

A tlctxilcd c:ompn.rison bot,wccn the precctling thcorctid results and experiment 

will be given in the next section. Hcre we shall only remark tlmt the position where 

the boundary layer bccon~cs first unstd~le according to theory (point of instabilit,y) 

must. always be oxpect~~l tm lit: ~~pst~rcnm of the experimentally observed point of 

tmnsition I)ocn.~~sc nctunl tmh~~lencc is created along the path from the point of 

itlst~hiiit~y to the poink of ltran~il,ion owing t,o thr nn~plifirntion of thr rrnslable 

disturbanrcs. 'rhis condition is satisfied in 1.11~ cnsc untlcr consitlcmtion. \\'(a 11:~ve 

rr:l 

= 960 (point. of tr:wsil.io~~) , 

. I , he clist,nncc bctrwccn tilo point of ins1,:~l)ilit:y nntl lhn point. of 1m11sil.io11 tlclw~~tl.r 

on t h d~grec! of n.mpli/irnlio~, and IJIC kid of tlist8url)nr~ccs 11rcscnt in th(: cxternnl 

stream in tens it.^ of turbulence), lmt. t11c R ~IIRI n~ccl~nnism of :1.11111lilicnt,io11 ~:III IN: 

ol)t,ninctl from t,he sLdy of the ~nngnit~tlcs of tho ~~nr:trnctr!rs in the inlr:rior of l.l~r: 

ctrrvc of ncutml std)ility, P, > 0. (~nlc~~lnt,ions or this Itin(l wc:rc: first, ~)e:rli~rrrlc:tl 

hy 11. Scl1Iicl11,ing 1761 l'or 1,110 fhl, 1)111l,c; I,ltcy IIILV~: I)c(:II rq~v~~.I,(;cl l),y S. I*'. SIICII 

185 1. 

In order to g~in n clenrcr insight intn the mcc:Itnnics or Lltc oscillnLit~g rnot,ion, 

JI. Scl~licl~tirlg [77] dctcrmincd the cigcnfi~nct,ions $(?I) for scvcrnl ncutml tlistturhances. 

This enabled him to draw the pn.t,tcrn of strc:rmlincs of the tlis1,url)ctl motiotl 

for neut1ral oscillnt.ions. An exnmplc of such n pnt.t.crn can he foun(1 in Fig. I(i.14. 

7 7 

I he tlin.grntn in Fig. 10.12 illr~sI.t.:~l.cs tho nn~j)lilic:nl.ic~n of IIIINI.ILI)I(~ 

in the bounclary layer on n flat, plah. 'l'h tlingrnm, based on n recent, cnlculnt.ion 

performed by 11. G.Onibmwski ct nl. [GR], extcnds ovcr n wide rnngc of R.cynoltls 

nurnbrrs. It turns out t.hnt the rnaximrttn nrnplificntion rntc does not pl:~co it.sc4l' at 

very higl~ Reynolds n~~mbcr (R -+ m) 11nt is locntntl in the motlorntc rango of R =- 10" 

to 10". ?'Itis is due to the fnot, t.llnt. t,lw rrtrvc of nc~lt,rn.l sl.nl~i1it.y fcrr n !In( pln.tr is of 

Fig. 16.12. Curves of cot~st~nnt, temporal 

amplification for the bonndnry lnyer on a 

fiat plnte nt zero incidcncc ovcr n wide 

"I 0.25 - 

(list.l~t.lt:il~(:(~~

111 1:1tvr I.imos. ,I .'l'. Sh1n14 19, !)(I] ;~ntl I). Gt.oI1nc 1341 ~n:~tle nn at.l,c~npt, to tlot.ert~~inr 

the (:ot~rst, 01' I,II(\ :~,t~~~IiIi(;a,t,ior~ 

01' 11ns1~n~l)It~ ~Iist~urh~~ccs taking into a~rcnntit, t,he 

cII'ec.1, of thr ?fo?f-livrnr t,c!rms in 1.11~: c:clnn.l,ions. In t.his conncxion it is irnport.n~tlt fro 

~x~aliz(~ III:I.I 1,110 an~p~iIic:~l~ion or l h ttnsI.i~l~lc 

~ t~isi~~~~~mnrvs 

(%IIS(\S lh: 111?:k11 fIt)\v h(:O11trc~vt, 

(111ik (~t111si(lt~r;1,111~1. 'l'l~is. in I ~1.11. ~YIIISOS a rhangc in lht: t~rn~~srrrnf cnvrgy from 

I IIP III:I.~II n101 ion Lo the osril lal.i~~g inol,ion. sincr it, is propo~~I,ionnl t,o rI71/11?/. The tn:~,in 

csll'~.ct of t.l~is is l.l~:it, :it n Int.t:r stngc 1.11~ nt~st.n.l)lc tlist,r~rl)n.nccs no longvr amplify in 

IU.~~IIW~~~)II 

I,o cs~) (Pi r) I~rlt, tcntl 1.0 a li1)it.c vnlnc: which is intlepcntlent of the initd 

vnI11c. 

'I'llr tlist,:itrcc bet,wcctl t,hc point of t.r;~nsit,ion ant1 t,llc point of nc~~ll.rnI sl,:il)ilit.y 

tlrl>(~n& consitlernbly on tl~e 1urh11,l~~m inl~mity in atltlition lo its dcprntlcncc on 

nrnplifirat.ion (scc also Set:. XVI (1). 

Fig. I(i.lB. Curven of con- 

~tant. .~pdn1 nniplification 

for 1110 I~ot~nrlnry lnynr on n 

flat pIaI,e at, zero incidence 

in t,lw lower mnge of Reynolds 

n~lmbrrs xs cnlr~~lnt,ed 

I)y It. .lordinson 1471 

d. Compnrison of the tl~rory of sI:lbilily with rxl~rrirnrnt, 473 

More t,Iin,n n (l~n(lt~ was l,o cIi~1)sc l~vforc a11 r:xprrin~ct~I.:~l vvriIi(-:ltimt or the 

nl~ovc t.l~cory of s1,al)ilit.y co~~ltl be ol)l.:~int~tl. 'I'ltis W;IS I~rilli:~t~l.ly :~rl~ic:vc,tl I)y 

(1. 13. Srh~~l~:lrtrr :it~tI 11. I

474 

XVI. Origin of l,t~rl~~tlct~rr 1 

1%. (:. vnn dcr llrRge Zijnen 1411 and M. Tlanscn. 'i'hcsc mcasurernents led to the 

rc.sult. t.11nt. t,hn crit,ic:d Jbynoltls numhcr was cont,ainccl in the rangc 

Soon nfi.cr, 1 I. 1,. I)rytlcn I 16. 171 and his c:ollaltornt~ors nndertook I& vcry t,horougl~ 

s.nd r,aroli~l inves(.ig~t.iorl of this l,ypc of flow. I)tlring the course of tht:sc invcst,igat,ions 

rxt,cnsivc? t1at.a. OII t,ho volorit.y clistrihl~l.ior~ were cnrcfullg plotted wit,l~ the aid of 

Imt-wire a.ncn~omrt.crs in t.rrrns of spncc roortlinntcs n.nd timc. Ilowc~nr, t,llc: srlcct.ivc 

an~plilication of (lisI.urImnc(~s ~~r(dicI.r(l Ity the theory could not yet Itr det.cct,cd. 

At nl~o~rt the same time, cxpcri~ncr~t~s cn.rrict1 out n.t Goct~tingcn on a Ilnb pinto 

ill a w:rtcr chnnncl yicltletl a qunlit,at.ivc confirmation of tho t,hrory of stahilit,y. 

'i'l~e plrot~ogr:rphs in Fig. 16.15 clcpict n t,urbrilent region which originated from n 

tlist.nrl)nncc of long wavelcr~gth. 'J.'ho similnrit,y between these pl~otogmphs mtl the 

!.l~corct,ic;~.l pxl,tcr~~ of nt.ream1inrs of a neutral clistt~rltnnce shown in I'ig. 16.14 is 

irrcfttta,I~lc. 

111 tiiswsing (,r:~nsit.ion it is necessary to introduce one very important pnrarnrt,rr 

which rncn.surcs t h "tlcgrcc of dislmrl~nncc" in the external stream. Ik im- 

~tort:t,nc-r wn.s lirsl, rccogniscd whrn mrnsttrctnc~lts of t,ho dmg of sphrres were per- 

(i)rnwI it1 tIi(lr.rcnt, win11 h~nnrl~. 111 this (:onnexion it/ was discov~rcd that the critical 

Ilrynoltls nrrml)or of a sphcrc, that is that valuc bf the Reynolds number which 

rorrcspontls to the abrnpt, t1t:crrasc in tllc dm.g cocflicierrt sl~own in Pig. 1.5, depends 

vtq rna.rl~r:tIly "11 t.11~ sl,rcngt.h of tho tlisturltnnccs in t,l~c free stream. This can bc 

~~~cnsttrrtl clt~nt~t,il,at,ivoly with tho aid of 1.110 time-averngc of [.he oscillxl,it~g, tur- 

Intlrnt, vrlocilirs :\s thry occur, for exn.nlplr, I)~hin(l a S~~IY>II (see nlw Scc. XVlll f). 

d. Compnrimn of tho theory of stnhility with cxlwri~ncnt, 

where rJ, denotes the mean velocity of thc flow. Tn gcnernl, nt n c:rrta;hir~ cIist,at~t.c. 

from the screrms or Itoncyeombe, tho torbeloncc in a wid tu1111el boco~~~es iwtropic, 

i. e. one for which tho mean oscillations in thc three components arc cqttal: 

- - 

,'I2 = "'2 = p. 

111 this cnsc it is suficicnt to rwtrict oncsclf to thc o~cillnt~iorl u' irl tJ~c dircat,iorl 

of flow, and to put 

T = ~/Z/U,. 

Fig. I(i.15. Flow along a flnt plnta; tt~rbu- 

Ience originating from n disturbmcc of 

long wavelength after I,. Prnndtl [A81 

The photngraplw wcrc Inken with thc aid ors alnw 

molion-plcbro eamcrh, which lravrllcd on s lrollcy 

nlong will1 the flow; con~cqurnlly, lhc camara In 

trsincd on the nnnlc group or vorllcrs nil tbc 

time. Tho flow la made vinihlc by uprinkling 

alumlnirtm dust on Lhe wafcr rn1rCar.e 

478

476 

XVI. Ori~in of turhulrnce 1 (1. Comparison of the throry of stal~ility nith rspcrinlr~~t. 477 

. . . 

' 

Fig. 16.17. Oscillogra~n of the a'-rotn- (37 2.23 --.-----..A"ponent 

of fluctualions caused by 

Lr181\ 

ranclorn ("11:1t,t1ral") dislr~rl)nnccs in $ 152 2+8 *- , -- 

the Intninar bor~ndary hyrr on n flat 

plate in a stream of air. Mcasvrementa 

on t.mnsition from htniriar to L~trbrt- 

S 

% 

8 I . 2.74 

D I . 

- 

. . . . . . . . 

1)uritig t h cxpcritncnt-s wit,Il urti/iri~rl (~;.v/~II./J,III(.c.G :I t II~II IIIVI,:II st,~ril~ t.\ I ISII(IIII~ 

ovvr :L wi(lt,li ol'nl)out, 30 ctn, 045 mm I,ltivlc :t~ttl 2.5 nlm tlwl) W:IS ~~l:lcnl :I(. :I tlisl;111t.(. 

of 0.16- nlm fronl l,ltc wall nntl wn.9 cxcit.c.tl I)y :I tn:~gnc.lic: lic.l(l in(l~~c-vtl wiI.11 IIIV :~i(l 

of an alt.crnat,it~g crtrrrnt.. 111 this wnjr it, \\.:IS possil)lc: 1.0 in(l~tw L\vo-(litt~(~t~siot~:~l

478 

XVI. Origin of I.rtrb~tlmcr 1 

tlishrl,anc:rs of ~)rc:soril)o~l frc:cl~~nnry, as stipolat.cd by the tflieory. This gave rise 

t,o nmldilircl, tla~npcd and nr~lt,ml oscill:rtions sim~lt~aneonsly. They were again 

rnrnsurcrl wilh 1.11~ aid of a Iwt-wiro nnomornctcr. Itcsolt,~ of s~rnh mcasuremrnt.s 

:rrc! sl~ow~~ ~)loI,tc.tl ill l'i~. Iti.18. 'l'l~c. c~spcrimrnt.:d poi~~t.s, wl~ic:li are joincvi by a 

I)rolic.~i lint,, rrl)rrsrnl~ rnc~:~s~lrcd nc:11t.r:11 oscill;ll,ions. 'I'hc t,llcorc.t,ic.:iI ctlrvc: of nr11tr;ll 

I)nt~c.c-s :II. v:~rying tlisl.:lnc~c l'rotn I,II(: w:dl. I'ig. 16. I!) shows osc.illogr:~tns of ~IIC sin~tsoitl:il 

niot.ioti for tho c:onipient, 11'. Ih:h oscillogmm rnnt,nins t.wo sitnnltanc~ons 

cnrvrs, OIIC of \vl~ivh w:~s :d\v:~jw tdcn :\,I, tJ~r wnic dist:~.~~cc from thc \wIl, thc ot,lwr 

having Imrn t,:~ken at various distances. The variation of the amplitude of the U' 

oscillnt.ion over t.11~ I)or~ndnr~ I:~ycr witlt,l~ is shown in Fig. 16.20. Tho din.grn.m 

rcpre~cnt~s thc results ol)t.aincd by Schnba~~cr and Slrrarnstnd and refers to t,lic ncut,ral 

tlisti~rl)ancc~s mnrltccl I and 11 in I'ig. 16.1 1 . 'l'l~crc is good agrrcrnent with t.11~ tlicory 

(111~ t,o 11. Schlic:l~ting 1.771. 

Vvry rnrrfnl rsprrimcwt.~ of t,llis kin(\ IIRVO Iwnn pcrfornied more rct:erit,ly by 

,J. A. 1104s (-1.11.1. 1741 who ltntl at !.l~oi~.disposnl n, wind tunnel ol'tJ~r wry low ~JI~~II~CIICC 

int.cr~sit.y T -- 0.0003. '1'11ry report cqrlxlly good agrecmcnt betwccn tlieory and exprimcnt. 

\\lc. II:LVC :~lr(ml.y rronnrkctl c.n.rlic*r t.It:~.t. t .1~ oxpc:rirnrnt,:~l vorificat,ion of the? 

st.:l.t,ilil.y I.l~cwry W:IS first. m:rtlv ~mssil,lc: wl~cm :I st.rr:i~n of vory low l.~~rl)r~lt:ncc int.cnsit,.y 

(:ottlrl I,n protl~rcrtl. 'l'llc oltlvr os~wrime~ll,s wl1ic41 wcrr 1)rrfcorrnc.tl at a t~lrl~nlrnrn 

int.cnsity of T ..-. 0.01 c.ot~lirti~ctl the- c~s~wc*t,:~t,ion thn.t, t.11~ ol~srrvrtl pint of transition 

lirs tlownst.rc~:~n~ of t,Ii~: pint. ol' inskll)ilit..y prc:(lict.cxI l)y tlwory. Ilowover, the clist,nnc-o 

I,c~t,\vrc.r~ thr ~~oints of illst.:~l)ilit,y :III(~ t.r:~t~si(ion (lopn~~cls t.11 5% n~nrlzctl degree 

on t.rlrl,~tlrnc-c. int~c~~lsil,y. It. is t.o be c.xl)cv:t,n~l lht. illis clist,anrc s11011ltl tlrrrc:nsc as the: 

int~rt~sit~y of t~~trI)~tlrnc.c is it~cr~ws~~il IKYYIIISC 

in ~IIV prrscnrc of high t,urb~tlrnre a 

sn1:1.11 nrnottnt. of :iml,lilic~:~t~ior~ srtflic~cv to l)rotlttc.c- I~~trl,~tlc~ncc from t.ho unst.nl)lc tlisr 

. 

t1114,:mc.c~. I IIC grnp11 (III~ t.0 1'. 9. (:l:rnvilln 1361 :LII(~ S~IOWI~ in Fig. 10.21 ill~~stmt.cs 

this point. in r~~htion to t,l~c IIOIIII(~:L~~ 

l:~.vcr on :I fht, plate. T11r clifTcrrncc l~ct~wec~~ 

t.I~c: Itc-y~loltls n~lrnl)c:rs fortnntl tvit.l~ the tnotnrnt~r~rn tl~icltness at the points of transit,ion 

:incI inst.:~l)ilit~y, 11a111c~ly 

1:;s. I(i.18. Curvcs of ~mut.r:~l slaldity 

for 11~11lrnl frrq~trncic~ 

t,nt~c:cs 011 n flnt, pl:rtr :rt mro 

i~~ri~lrr~rc. Blrwwrv~~~(*r~ta d11r lo S~III- 

Im~wr ILII~I S~~:LIIIX~II.(~ (821. 

clue to 'I'olln~ic~\ [W] 

'I%(:ory 

cl. Co111pnris011 of tllc t.11rory of nt1r1dit.y wit 11 c:xpcri~~~rt~l. 

of ili~lt~r- 

I. l!l ~I~:LSII~PIIIPII~~ 

ill 

l,l~r h111i11:ir I)w~ttlnr.y l~ryi-r 111dor111r11 11.y 

Sch~~hn~ter nncl Skrn~l~ut,ntl [82] 

1 . 2 0 . \':lri:bt.ion of alnplit~ltln of tho 

Sim~lllmrow recorrlirt~ or vvlorily will1 Llir nirl of ?~'-llual~~:rLio~~ for t\vo ~lcntr:rl dist.urlmnces in 

Iwo Id-wirr ~llrlllOlllr1rrJ placed nl n didasm of 

30 c n ~ bohind ll~r rlril!. Ttw lowcr ctlrvo enrrragonds n. lanlir~nr Im~ntl:~ry lnyrr on 11:lt pl:rlc nt. zrro 

lo n ltol wirr plnrrrl nl :t clirl.anec wf 1.4 lnrn frmn i~ic:itlcnr.e. htc.:ts~~rr~t~n~t.s hr l,o Sdlul~;utcr 

lllc wnll: 1111. IIpl!rr rltrvr carrw)mncls Lo n Ibol. wirr :rnd Skrn~nd.:rrl (82.1 '1'11rory ~IIC 1.0 Srhlic:l~plnrtvl 

nluryitt~ dir1ntw.u froltl ll~r wnlf nr it~di~~l~vl. 

'~IIv qlrlp vnr pl:wwl nl n ~lirlancr or It0 rm III.IIIIIII t.it~g [771 

1 1 l 1 I a I I Yrntlwnr?, 70aee'. 'I'lw rrrvrr l11lw11vd 1 ttttd 11 wrn:s1wt~d 10 111~ LWO 

nct~lrnl ~1i~llIrl~;~nc~~ I nnd I1 it) Iqg. lli.1 I 

vrlucity Urn = 13 lnlsec 

011 nsi:ill:rl,io~~n

XVI. Origin of turbulonce I d. C'ornpariflon of the theory of fltability wilh rxprritnrt~t 481 

Vig. 16.21. Mrasnrrn~c~~l~s on t,r:~nsit.ion 

ol a 11:~l, plalc, nftrr 1'. S. l:rn~~villr 

(:Hi]. Dillivw~vr. I)cl.wc:ct~ t.hc Itcynoltls 

numhers at, t,lw poinb of trmsit.ion 

and insl,:~hilit.y in lrrn~s of twl~rrlcncc 

inlnnsity. As t.rtrItrtlcnce intnnsity 

incrrnscs, thr point of transition 

moves closer to t,hc point 01 instalti- 

1it.y 

All vspritncvlt,:rI poitits 1.r:lc.c a sitlglr: cltrvc. 'I'ho point of t,mnsit,ion does not coincide 

wil.lt t,hr poittl, 01' inst.:ll)ilit.y r~nl,il vrry high tmrbulcncr int,cnsilics of a h ~ T t =0.02 

t,o 0.03 I~avc, I)rnlt rr:~rlicvl ; c/. I 1. 

Other vclncity profiles: Wr now l)roc:cctl t.o tlescribe briefly i~ivcst~igat,iol~s into 

t.11~ st,nl)ilit,y of'o(.l~rr vdoc:iI,y 1)rofilw; a more tlct,ailctl account is given in Chap. XVII. 

A I):L~WI. IIJ S. Ilollirtgtlnlr [4:,l nont.ains a contribut,ion t,o t,l~c stdy of the 

st.n.l)ilit.y of vrlot:it,y prctfilrs in t,hc w:rlre of a solid body. 'l'hc stlability of laminar 

,ic.t.s w:~s statlic.tl by N. (h1t.1~ (IOJ. Mrntion may, finally, bc made of lhr work of 

I\. Rl id~alltr :~t~cl II. Sclln,elc (581, l'. 'l'nl,surni [!)Oaj, I,. N. IIowartl [4G], and C. MT. 

(.~lrnsh:~.\v ant1 I). 15lliot. [ I I]. 'I'll(; last rt~fcrcnce est,al)lishetl a limit of stability of 

RCrit - - 6.5 for :t 1)l:~nc. jet,, t.hc It~ynoI~Is nun~I)cr hing formcd with the jct wirlt~lr 

at, 11:dI' hvight,. 

and J.T. Stuart [DO] : see also R.O. J)il'ritna ct a,I. 1141 antl .I.'I1. Stlrn~+ I!)I 1. i\ srtnjttrary 

of t,llis prol)lrm area was givrn by A. R4icl1allrr. M. llzrtl:~ 14(il)] is :~lso oI'it~lt~~.wt. 

A clear idca of the tlctails of thc mccl~anism of an~plificaliot~ can I)(; fi)rmcd 

by studying the smoke pictures of the zone of t,rnnsit.iott ir~ thc bonntlary Inyw or1 

an airfoil takcn by IT. ncrgh [B] and rcprotlucctl in Fig. 16.22. TIN: :~rtific:i:~I clist.ur- 

I)anccs wcrc protluccd with tlic aid of a loutlspcaltcr; t.l~c:y arc sccn to itttluvc: in 

the boundary layer a succession of amplified, regtrlar waves, their amplitude incrc:wing 

in t,l~e downstmam direction. See also [I]. 

Three-dimensionnl flows. The cxprrimcntnl cvitlcnrc ntltltlc:ctl so 1i1.r shctn.s bltnt. 

transition is st,artcd an a result of t,hc amplification of t,wo-tlit~lc:llsiot~:~~I tlixt.~rrl):tttc!t:s. 

'l'he growth of such tlist,urbances was investigated in great tlrt,ail I)g G. 13. S(:l~ultanrr 

and IT. I

482 XVT. Origin of tmbulcncc I c. Effort of oscillnhg frrc? st,rcn~n on t.rnnnit,ion 483 

process of transition from ltbmin~r to turbulent flow is the conseq~~cnce of an inshbility 

in t.1~ laminar flow, enilnciatcd by 0. Reynolds, is hereby completely vindicated. 

It. certsinly represents a posaiblr: and observahlr: mechanism of transition. The question 

as to whet.11~~ it pints a complet,e picture of the process and whethcr it constit~ltes 

the onhy nloahanism oneountcrcd in naturo is still at prtxent an open one. The latter 

cjt~:st.iotis now oconpy t.hc at.tcnt.ion of many research workcr*. 

e. Effect of oscillnting free stream on transition 

Aftrr it hntl bccn dincoverccl with 1.11~ aid of the oxprin~cnt drmrihed earlier thnt. the intensit.y 

of turbulenc:c of tho cxternnl stream, thnt. is thnt the preaence of an irreguhr tittle-dependent 

flnrtnntion in t,he free strcnm, cxert.ed a strong inflnence on transition, it was natural to nndertnke 

st,ndicu on t,ho chct of regular fluctnnt.ion in the free stream on tratlsi6ion. The effect of a 

nnperin~ponrd flnct,unt.ion of smnll nmplit,udc (E < 1) in an cxternnl stream U(z, 1) of the form 

on tho ~trtwt,urc of n lntninnr bonndnry lnyer w~ diucnwed in SBCB. XV n 3 nnd XV e 3. 

Sincc the Il*ynolils rlnrnbrr nt. trnnnition clrrrmnr*r ronnidorsl~l~ RR tho in tan nit^ of turbnlcnrc 

inrrrnru, it iu plnllniblc Lo suppoue Lhnt n ainlilnr erect nhould occur a8 the amplitude d u = 

~111 of the periodic cxternnl stream is 111;de to increw. The effect of an oscillation euperimposed 

on the external strcntn on thc transition of a hminnr boundary layer was clarified experimentally 

by J.H. Obren~ski nntl A.A. Fejer [63a] arr well aa by J.A. Miller and A.A. Fejer [60a]. Them 

inveetigntionu conwntrated attention, in the first place, on the boundary layer on a flat plate 

(Bl~ius profile). In this cnse, the velocity clintribution in the externnl stream is 

Hcrc {J, is the tin~c-avcrngc of thc frco-strcnm velocity which is independent of z, dU is the 

nn~plitudc of the temporal Rnctuntion in the external stream, and n denotea ita circular frequency. 

'rho mcnsurcn~enb rcported in 163~1 werc performed in an incompressible stream with 

and with frequencies of n = 4 Lo 62 sec-1. 

Them very carcful experin~entnl invent.iptions yielded the following emntial reaults: 

(a) The c-riticnl Rrynolds nnmbcr of tho start of transition, R,,fr = Urn zw/v dcpends only on the 

arnplitudc AU/Um of the external fluctuation. 

(h) Thc dimcnsionlr.s trnnuition length, thnt in the distance between the start of trnnuition and 

itn ronil,lrtion, R,,f - RZsc, depcntls only on the frequency of the externnl oscillationt. 

(c) '1.11~ record showing the varintion of velocity with time demonutratm that the line of transition 

is chnrart~rized by n regular and intermittent transition. The meaaurementa led to the conclusion 

thnt trnnnition rnn hc t~rscribrcl by the following "non-steady" hynoldu numbw: 

Sinrc thr rhnrnrlrrirtic length of tho cxtrrnal, oscillnt.ing stzrnm iu C - llm/n, it, iu po~sible to 

cxprrw the ~~non.~tt~ntly" I~tyold~ nnmbcr in thc form 

uz (A u/rrm) 

RNC = 

2nvn r' 

llrre All/l?, in the rlitnenuionlr~~ nn~plitude of tile impredsed oscillntion and ns/lJ: is itR dimensionlr~u 

frrqrrt-~~t*y. 'I'hr n~mnnrrn~rnb showd tlint the llcynolds number Rz.t, = 11, ~tr/v nt 

- -- 

t Start of t.rnnnition nt R,,(, .- (1, .rl,/v -- lower curve in Fig. 16.10. Complct,ion of t,ranuition 

nt. R,,I, - 11, .rl./l* - nppcr ctrrvc it1 Fig. I ti. 16. Over the t1iut.nncc from zt, to xr it is observed 

thnt Ihr intr:r~~~ittr:r~cy fndor incrcnucu fro111 p = 0 to y = 1; this is interprctcd by t,hc statenwnt 

thnt in t.liin zone we ohnerve "trannitional t~~rbulcnce". 

the point of t8rnnuit~ion warr always con~itlcrnbly reduced con~pnrctl wit,l~ that for nhtionnry llow 

when t,hc %on-st.eatlyV Reynolds number am Inrge, i. c. whcn RNS > 27000. In t.l~c.sc: rxpcriincnt.s. 

t.rn.nnit.ion on tho flnt p1nt.c in ut.nlionnry Ilow s(.nrl.ctl nt. R,,,, -- 1.8 x 10". At.t.or,li~lg to 

Fig. I(i.l(i. thin vnluo of Rz.1, cor~.cnp~ntI~. np~)rt~~in~nLoly. t.o n tnrl~rtlet~co itrlr11nit.y of T 0.230/,, 

in the cxtnrnnl nt.rennt. 

So far, n nnt,isfnctory theory of ut,nbilit,y for houndnry lnyors in t,hc prcocnco of nn oxt.crrinl 

o~cillnting utrcntu doen not cxiut {I I a]. 011servnt.ion of intern~ittnnt t,nrl~ulcncc i r ~ t.11(! I~rrwt%tlrr of 

n frro-st,rcntn oncillntion ~IIOWR thtl its freqrroncy fl, is of thc antnc ortlor of n~np~it.~~clo nn thnt, of 

nntnrnl. neutrnl disturbnncer of the Tollrnie~l-Scl~licl~ti~~g typo fro111 ntnhilit.y t,l~eor.y; sm also 

I'ig. Ifl.18. The freqnenry, ,I, of 1.l~ oscillnt~ing flow invcst.igntml Irwc \vnn ~ninllrr 11y 11 I':u:tor of 

ahont 100 t.hnn thnt in t.hc nnl.urnl, nc~rtrnl tlint.nrbnnccs. 

A review of Ilte prorcns of trnnsition in thr prcnrnrc of frrc-strrnnl o~rillilt~io~ln uils rrrrntly 

prrhlinhrd hy ILJ. l,ochrkc, M.V. Morkovin, nntl A.A. Fcjcr 14HhJ 

(G) Conlcscencc. of 1.11rLulcnt spot.s ink) :I fr11l.y t1cvc~lopc:tl t.~~rl)~tlt*~rt. I)OIIII~~:LI.~ 1:13't~. 

In most CRRCR, t,hc t,rnnsit.ion frotn t,r~t.l)ulc:nt spvt,s (.o I.ully tlevclopotl (,III~IIII~W~I~ is 

associated with the furrnntion of a separat.ion bubblr, ns nlrwtly ~nont.ionctl in c:otlncxion 

wit,ll Fig. 10.10. At, tho present, time, only stngcs (1 ), (2) ant1 (3) nrr :~mrnnt)le t.o :L 

t,hcoretical analysis. The complete clnrification of the ~~~nnining stngrbn will rrq11i1.e 

rn~lch ntltlit,i~)nnl t.l~corc~t~icid resenrcl~ work. 

(1) Stnbtr now 

(2) I~'11~1nI~lr '~f~ll~~~l~~~~-H~~l~li~~l~li~~u 

WIWR 01--x I?,,,, 

Q 099 9 9 

lominor I- Ironsilion 

A,,,, 

R c 

JdTig. 16.23. Itlcnlizcd sltclrh of t.rnnsil.ion zone ill lhr I)O~III~~ILT~ 1:1~t:r 011 11 Ililt l~llltP 111 Z~.I.O i11t.i. 

clcncc after P. M. While 11073

484 

XVT. Origin of t~~rbulence I 

74, 34 1 -:!ti0 ( I !)05). 

121 Ihrry, M.I).J., nnd Jtosa, M.A.S.: Tlic flat. plnto boundary Inyer. Pnrt 2: The eKcct of 

i~~vrcnsing I.l~ic:kncns on stnbilit.y. .lFM 43, 813--818 (1070). 

[3] Ilrn~~y, I). -1.: A non-litwnr theory for oscillnt,ionn in n l~nrnllel flow. J IPM 10, 209 -2:lli (l!)61). 

141 Ilrtrhov. It.. nnd Cri~nin:~lo, W.O.: Stnhility of pnr:dlcl flown. Acetlc~nir I'rcss, l!)(i7. 

, 8 , . 

I4n] Ilout,liirr, M.: Stdilit.6 li116nire tlcs 6co~~lenirntfi prmqnc pnrnlli4es. I. .Jonrnnl clr MLrnnique 

11, 5!I9 W21 (1!)72). 11. I,n couchc lin~ito tlc I3lnni11n. Jor~rrlnl de M6cnniquc 12, 75-!I5 

(l!)73). 

I51 I5ergI1. 11.: A niet~liod for viwnlizing periodic bormdnry lnyer phcnonlcna. IUTAM Symposiuni 

I{onn~lary-layer rescnrch (fl. (;iirl.lor, otl.), Ik:rlin, l!)5X, 373-- 178. 

(61 Jhrgcrs, J.M.: The niot.ion of n fluid in t,he ho~tntlary lnyer nlong n plnnc? ~111oot~l1 snrfnec. 

I'roc. I'irst, Intern. Congress for Appl. Mec.11. 113, I)elft, 1924. 

171 Chng, S.J.: 011 the ut.nhilit.y of Isniinnr hor~ndnry lnyer flow. Quart. Appl. Mat,l~. 11, 

346.- 350 (1!)5:!). 

[XI :l~rnrr, I)., il:irry, M.l)..J., nnd Ross, M A.S.: Non-linur stability tl~cory of the flat plate 

~o~~ncl:iry Inyrr. ARC Cl' No. 1296, (1'374). 

[!)I (hrt.tc. M.: Ift,i~tlcs sur le frot,terncnt clrn liouiden. Ann. Chim. PIiyn. 21, 433--510 (1890). 

(10) (:uric. N.: Ilydroclynnti~ic ~t.nbility in r~nlitnitcd fieldn of visror~s Ilow. l'roc. Itoy. Sac. 

1,ontlnn I\ 2.78. 4W-501 (1!)57). 

11 I] (.'ler~sl~nw, (I.\V.. nnd l$lliott, I).: A ~ui~nericnl t.rcnt,nlent of the Orr-Sonlnrerfelcl eqn~tion 

in t11c c~i~sc of n ln~ninnr jet. Qrrnrt,. J . Mrrh. Appl. Mnth. 13, 300-333 (I!)liO). 

Illn] l):~vin. S.: '1'11~ nt.nl)iIil,y of prriodic flo~v~ AIIIIII:~ I

486 

XVI. Origin of k~rbulenre I 

1481 Jordin~o~~, 13.: Spr~1,r111n of rigcnvnlnrs of tho Orr-Somtnorfrld equation for lllnsilla flow. 

l'l~ys. I'l11i11s 1.1, 2535 -2537 (l!)71). 

I48nl I < ~ ~ I I ~ I

488 

XVI. Origin of t~rrh~rlence I 

(HRnJ Slirn. S. I?.: Stnbilit,y ol I:~niintw flo\vu. High Sprd Aorodynnn~iru and ,let, Prop~~lsion 4, 

71!) H5:l. I'rinrrlon nntl Oxford, l!)(i4. 

IX5l,] Slirn, IP.C.'l'., Chrn, T. S., nnd Ilrtnng, I,. hf.: The elTert8 of nlninflow r:~dinl vrlority on the 

s(:~l,ilily of tlrvrloping lnniinnr pip flow. .l. Al,pI. Mccl~., Trans. ASME Srr. IC 43, 200-212 

( I !)7li). 

InIil Sotn~nt~r(i~Id. ,\.: Kill I3rilr11g znr l~yclrt~tIy~~:~r~~i~cI~t~~~ 

I':rkl5rung drr t.t~rI>~~lrnten Pliinsigl~t*i~sl)t*\%t*g~~~~gv~i. 

:\{ti tlvl 4. ('o~~pr. lnttm~;~l. th:i M:L~.. III. 111; --124. 1ton11~. I9OK 

IN7 1 Sq~tirv. I I. It. : 011 t IN* st~ttl)iIily 01' I l~rt.t\-tli~~~r~isit~~~nl tlist.ril)~~t~it)n ol' \~isoow l111iti l)t-I.\vrrn 

I~:~rnllc:l \v:~lls. I'rot.. I

490 XVII. Origin ol t,urbulencc 11 a. I3hct of pressure gradient on tmnsitioti it1 bountlnry layer along ~moot.l~ wnlln 491 

a. Effect of prennl~re grndierit on trnnsitian in boundary layer along ~n~ooth walls 

r 7 

J lw bountl:~.ry hyor on a llnt pl:lt,c a.t zero inridcncc whose stal)iliI,y was 

investigated in Chap. XVI has the pcc~rlixr characteristic that its vclocity profiles 

at tli&rcnt tlist,anccs from tho lending ctlgo are similar to each other (cf. Chap. VII). 

In t,his case sirnilnrit.y results from the ahsencr of a pressllrc gm.tlicrit in the external 

flow. On thc other h:md, in the case of a cylintlric:~,l htly of arbitmry shapc when the 

pressurc gmdient along thc wall changes from point to point, t,llc rcsult.ing vcloc:it7y 

profilrs arc not, grncrally speaking, similar to each other. In tho rangtrs where t,ho 

pressure docrc.ascs tlownstmam, the v~locit~y profilns have no point3s of inllcxiol~ 

and are of t,llc typo shown in I'ig. 16.9~ wllcrcas in regions wl~crc t,llc: prossure inorcnscs 

downstmam thoy arc of the type sli0~11 in Fig. 16.9~ ancl do posscss points of 

inflexion. In t,hc caso of a flat plate all velocity profilcs have the same limit of 

stability, namely R,,,, = ( fJ, d,/v),,,, - 520; in contrast with that,, in the case 

of an arbitrary body sl~npc, the intliviclual velocity profiles have marlrrtlly tlifiront 

limits of stabilit,~, I~ighcr than for a flat plate with favourable prcssure gratlicnt.s, 

and lower with adverse prcssuro gradients. Consequrntly, in ortlcr to dotormino 

the position of the point of instability for a body of a given, prescribed shape, it is 

necessary to perform the following calculations: 

1. Dctcrminat,ion of the pressure tlist,ribution along the contour of Lhe body 

for frictionloss flow. 2. I)etcrminat,ion of the laminar boundary layer for tha.t pressure 

distribution. 3. Dctcrmination of thc limits of stability for these indivitlual velociby 

profiles. The problem of determining the prcssure distribution bclongs to potential 

throry which supplies convenient met.hotJs of computation as, for example, tlescril)etl 

by T. Theodorsen and J.1S. Garriclc 1242) and F. Riegels [193]. Convenient nlethods 

for the calculat~ion of laminar boundary layers were given in Chap. X. The third step, 

t,he st.al)ilit.y calculntion, will now be discussed in detail. 

It is known from the theory of laminar boi~ndary layers, Chap. VIT, that, 

generally speaking, the curvature of the wall has littlc influence on the development 

of tho houndary layer on a cylindrical body; this is true as long as the radius of 

curvahre of the wall is mnoh larger tjhan the boundary-lager thickness, which 

amounts to saying that the effect of the centrifugal force may be neglected when 

analyzing the formation of a boundary layer on such bodies. Hence the boundary 

layer is seen to develop in the samc way as on a flat wall, but under t1he influence 

of that, pressnrc gratlicnt which is tlctermined hy the potential flow pnst tfhe body. 

The same applies t,o the tl~t~orminat~ion of the limit of stability of a boundary layer 

with a pressure gradient which is different from zero. 

Tn contrast with Ihc case of a flat plate, whrro the external flow is nnilbrin 

at [J, 1 const,, wr now h:~ve Lo con0cncl with an ex1,ernal strram whose vrlority, 

I/,(T), is :L f~lnrtion of the lrngth roordinatc The velocity Urn (z) is related to the 

prrssuro gmtlirr~b tlp/tlr through tho Rrrnoulli orpation 

to work witsli a nicnn llow whose ~olocit~y Il (y) rl~pcn(ls only 011 hllr 1 r:it1svrrsc 

coortlir~~t~c y. 'l'ho inffl~oncc of Lhc I)~I:RSIICC gra(/irnt 011 ~l,al~ilit,y ~na~~if'rsl.~ iI,svlf 

t.hro11g11 tltr form or tdic vclocit,y 1)rdiIc givv11 by fI(?y). \\'c IIILVI: nlrwtly wicl in 

See. XVLb thnb the limit of st,al.)ilit,y of a vcloeil.y prolilo drpt:ntls shongly ou il,s 

shape, profiles with a pint of hllcxion possessing colisitlrrably 1owc.r li~nitw of 

stability than thosc without ono (poit~t-of-inflexion criterion). Now, since the pressure 

grndjent cont,rols t,hc curvature of the velocity profile in accordance with ~(111. (7.15) 

the shng tlcpentlencc of the limit of stability on the shapc of tho vrloni1.y profile 

n.mount.s to a largo inflt~cnce of tJlc: ~)~CSRIIPO gmclicnt on sl,al)ilif~y. If, is, I,ltc~rc.li~ro, 

true t,o say that accelerated flows (tlp/tlx < 0, clUrn/tl:c: > 0, f;~vour:lblc pressure 

gradient) are considerd~ly morc stable thn clccclerat,ctl flows (tlp/clz>O, clIJ,,,/dz

492 XVII. Origin 01 turbulcncc I[ a. Effect of pressltro gradient on tranxition in honndnry layer dong ~mooth wnll~ 493 

st.iplnt,e. for the SR~C of simplicity, a one-parameter family of laminar velocity 

profilns. An cxnmpie of sunh a one-parameter fnrr~ily of volooit,y profiles, which, 

nlorrovc.r, (:o~~sl,ih~ll~c exwt, solrltk)n~ of Lhc bor~rdnry-1n.yor eqr~alh~s, is rrprcsrrlt,ncl 

by I lnrl rcr's wctlp. flows. Their free-sl.re:~m velocity is givrn by 

nr~d 1.l1c ass.ioc:inl,ctl ve1oc:il.y profilcs can bc fourd plotkcd in Pig. 9.1. IIcre m denotm 

tho slt:~,pc farlor of the profiles and tho wctlge angle is = 2 m./(m..I- 1). When 

ni, (1 (ttcol.cn.siiig prrssurc), t.11C1.e is no point, of inflcxion. AS cerly as 1941, J . J'rctxh 

[I 78, IT!)] carried out ~JIC sta1)ilit.y cnlc~~lal,ion for a scrics of profiles of this one-paramc:t.rr 

family. l,at.er, in 1969, thcsc cal(:rllations were considerably cxtentlwl by El. G. 

O~nbrcwslzi ([G:!] of (~ha.p. XVI); Ilc cvaIuat,ed not only the critical Reynolds number 

In~t. a.lso 1.h~ an~plificat~ion ra1.c of t.hc ~~r~stsblc: tlistnrl)anc:c-s. 'l'hc ealc~~In.f~ions rcvcd a 

s1,rongc.r d~y)cwtlrnc:o of t,hc cril.ic::~I Iky noltls I~IIII\)IT on 1.11~ sl~n.~)(: l'n.(:I.or 1)) l.hnn tlitl 

varlivr wo~~lz. '1'11~ (li:~y,r:~n~ ol' liiy,. 17.2 dwcri1)cs OIIC rcs~~ll.

404 XVII. Oriein of tnrbnlencc TI 

Fig. 17.5. Sl~nrlograpl~ pict,r~re of reverse t.ransit.ior~ front t~rtrl)r~lnnt, to l:rn~it\ar flow in n \~orr~tdary 

layer it1 s~~pcrsonio flow rot~r~d a corner at. M = 3, nfkr J. Skrnl)erg [215j 

wig. 17.1;. Srltrtitnt.ic rrprcncnl,nt,ion of 

the flow in t.hc Imr~ndnry hyrr in RII~Waonir: 

flow nrot~ld n vornrr, nf1.w .I. Sternberg, 

c/. Fig. 17.5 

t l'hr vnlr~c R , - A45 givcw hcrc for A - 0 tliffcru so~ne\vhnt from the value 520 givrn previor~uly 

in JGg. 16.1 I. 'l'ltiu is duc 14) the clilli:rc~~cc bctwccn tl~c exnct I%lnniun vrlocity profile 

11srd ~wvvio~~sIy nntl nll ~b~~l~roxiltl:ttr ollr rlltl~lo,vcd for 1.h ~)rol~ar:bt.io~r of rig. 17.2. 

graph of wn.ve-like nt.rcnk I~IICR in a writer vl~n~it~el ~d~l.~illt*d with 1,h? nid of lrhc t.cll~~ritt~tt 111t?t11od 

by IT. X. Worttnnnn 1257, 2581; di~trlrbnitco crcnkd nrlificinlly hy n.11 oscill:~t.ing sllrip (3 x A00 

>: 0-03 III~I). The strip is lovntcd nt n st.iiIiou wltnre R1 =- 750; the at,rcnk li11c8 itre c:rnnlrtl xt 

R, == 950 (Irft. border of figure). The rolling up ofntrenk linrs tlow~~ntrrn~~t 

instnldit,y of the prrt.t~rlmt.ion wnres. 'l'hc figrlrra t1c11ote 1listnnc.r~ ill rm 

in n rotwicqrlrtlre of t,l~e 

Tho phot.ogra,pli of Fig. 17.7, tdtcn by F. X. Wortm~nttn [2M. 2561 it1 a wn.t,c:r 

cl~anncl conveys n clcar iniprrssion of unsf.cLl~lc oscillnlic)its in n. 1ntni11n.r I)ollntla.ry 

Inyw. The picture was obtained by the tolluri~~rn ~nt:t.hotl /256]. l'hc a.rt.ificnl dist.nr0nnces 

were generated wit,h t,ho aid of an oscillnt.ing strip placctl near t.hc nall, 

in n mnnnrr siri1ila.r t.o l,hat crnploycd by Schulmr~cr. ~lld ~krr?tnst,atl ant1 tIcseril)~(1 

in Chap. XVI. 'J'hc ~)rrssnrc rise a.long thr wirll is so srnnll t.l~nt. tho I'ol~lhn~tsc~~ prntnct.rr 

from rqn. (17.3) llxs t.11c vnlnr A - --8. At. f.hr shlion whrrr the. tlist.t~~.l):~nrc 

is gctivtnl~tl I.II(. 1oc:trl Il.c!ynol~ls IIIIIIII)I'I. 1111s 1.l1r \ ~ ~ I IRbl I \ - 750, IIII~I l.11(* (litllwsiol~lcss 

wa.vrlt:~~gLlt of the tlist,url)nnco is nl (TI -= 2 nOl/l -- 0.48. This [wir~t. is loc.:~.t.ctl 

far in thr ~lnst,nl)lr ficltl of Fig. 17.3. '1'11~ instmit.n.ncv)~ts s~lnpsl~o; of I.Itv sl.rcdc linrs 

in Pig, 17.7 sl~n\vs 111~ fii1:11 11I1nsc ol' Iht% t~\vo-tli~~~ct~sio~~;~l 

~I(:~~~~IoI~~II~*III~ 

of' 1111. (listrrr1tnnc:c 

nI)out, 20 wnvc4rngt.11~ tlownst~rc~nrii of (.l~c osril1:rf ing strip. 'I'his tlisl.nr~~:~nrc: 

nmplifirs in cornplt~t,~ n.grrrmrnt of l.l~rory wil,li c!xl~rritnri~l.. 'I'l~c tl~~et,~~n.licin, wl~icl~ 

is stmill two-tlirnrnsior~nl near t.hc Irf't. rtlgo of I.IIc pict,ttrr I~reonws tlisl.or1.cd in i1.s 

rnitltllc by 1.Iic: o~~cotning longitutlir~n.l vort.icos. At thr right, ctlgc! ol' I,III% pit:I.ttt.r if. is 

nlrrntly ~)ossil)lr t,o tlisccrn "t,~~rl~~tlent~ rows". 'l'l~is ronfit~tns our 1.c.lnn.14ts (YIII(:W.II~II~ 

t,l~rcc-tlirr~rnsiotinl tlist,urbanccs givrn nl, t.hr cwl of t,his cli:tpt.cr.

490 XVII. Origin of turbrllence I1 b. Drt.crminntion of the ponition of thc point of inshbility for prrscril)rtl body shnl~c 407 

On sevcral oc:cnoions wc have sttrcsscti thc fact that n pressure increme along 

a 1)oundary lnycr sttrongly favours tmnsition to t,url)ulent flow in it. Conversely, 

a st,rong pressure tlecrease, such ns may bc crcat8cct bchi~id sharp cdgcs in supersonic 

flow, cnn car~sc a turbulent boundary laycr to become laminar. Interesting observations 

of this kind were made by J. Sternberg 12151 who employed a cylinder 

providctl wit,li e conical forc-body. Figure 17.5 shows a shadogmpll of the flow 

along the conics1 fore-body at a Mach number M = 3. The boundary laycr turns 

turbulentr at thc tripping wire provided for the purpose. Further downstream, 

hohintl t.11~ corner formed at the jnnct.ion of thc two bodies, the t~~rhulent bountlary 

1:ry-r hrns lamirrar again, Fig. 17.0. This phrnorncnon is explainctl by the circunnstnncc 

t11:lt t11c lnrgc f:tvo~~rnhlc pressure gradient at the shonlder impresses a very 

strong acrolrr:tl,ion 011 the flow and this, in turn, ~xt~inguishcs the t.nrl)ulrncc, in 

a w:iy rcniinisc:rnt of the eKccl of a strong contmction placed ahead of the test 

scct,iori OF a wind tunnt:l. Q~lalit~ative indications on this process can bc found in a 

~a1)or I)y W. 1'. ,Jones nntl E. 15. Immtlcr 1 low. According to thcsc nuthors, rclnminnrixa.Iion 

(nxt,inc:tion of t,rlrbrtlcncc) occurs in incomprossil~lc st.rcnrns w11c:n t,l~c 

tli~nc~nsionlws acc:rlcr:~t,ion paramctcr sat,ixfics the incq~ralil.~ 

Introtltrring I'ol~lhnusen's shape fartor A from cqn. (10.21), anti using eqn. (17.3), 

wc ran tra~~slntc the prrcrding contlit.ion to rcari 

'I'hc: t.ransit.ion fro~n n t,ctrl)~tlrnt. to n laminar flow pat,t,crn in a tnbo of cirrular 

cross-scot,ion was invcsbigat.cil in dctail cxpcrimrntxlly by M. Sibnlltin ns cnrly as 

1962. In partirr~lar, this investigation cxtcntlcd to a study of the attenuation of 

longit,ntlinnl t.r~rl~rrlont flnctuat.ions and discovered that t.llis is st,rongcr ncnr the wall 

t.l~nn in thc ccnt.c:r of the pipe. 

. I . he prrwtli~~g rrs111t.s will cnal~lo us 1.0 ralrulat~c in thc following secfiot~ the 

posit ion of l.I~i, pint of in~t~ability for the casc of two-tlirncnsional flow past a body 

of arbit mry .;hapc. 

11. I)c.lc.rniinnlio~~ of the position of the point of instnbilily for prescribed hotly shape 

., 

I 11c: tlct.crmina.l.inr~ of tl~c position of tJtc pdint ol tmnsit,ion for prescribed 

I)otlj~ sl~:~pcs (in t,wo-tli~r~c:r~sior~al flow) becomes very emy if IISC is made of the rcsults 

c:o~~l.:ririrrl it\ Figs. 17.3 nntl 17.4.'I'lrc essc:nt.i:d atlvnnt:r.gc of t,hc tnrt.Iiotl t.o \)ctlcscri\)ctl 

II(*I.~ consisl.~ in I.lt(: f:~:l, 1~1r:iL no ft~rtl~cr l:iI~orio~is ~nlc~tl:it.ions arc rcquirotl, t,he Lctlious 

j)arI, of the work I~nvil~g 1)ocn con~plctctl once nntl for all whcn compnt,ing the diagran~sii~ 

Fig 17.3. 

We 1)cgin wit,ll t,he evalrlnLion of t8hc laminar 0onntla.ry 1:hyc.r from thr pot,cwtinl 

velovit,y tlist,rilrnl.ion ll,,(x)/U,, whicl~ is regnrdrtl :i.s known, I)y t.11~ IISV 01' 1'0111- 

II:IIISCII'S :I~)~)I~~X~III:I~,

498 XVII. Origin of turbnlmco II 

t,hr RXILIII~~IC of MI cllil)l.ic: rylirttlcr whosr major axis, o, is re1;~tctl to its minor axis, 

11, Oy 1.11~ r:~t.io rr/O :- 4. l'hc flow will he assumed parallcl to the major axis. 

The pot.rnt,ial vcln~it~y-clist,ril)uLion fur~ction for such a cylinder was already give11 

in l'ig. lo.!), and the rosults of the calculations pertaining to the boundary laycr 

are shown in Pigs. 10.10 and 10.10b. From the variation of the shape fact.or wit,]l 2, 

Fig. 10.1 lb, and wi1.h the aid of Fig. 17.4. it is now possible t,o plot the variation 

of t,l~e locnl critical Itcynoltls num her, R, = (U, d,/v),,,,, as shown by tile crlrvc 

marked limit o/ slnhilil?/ in l'ig. 17.8. I'ron~ thc calculat~ion of the larninnr bounrlary 

layer wc ca.11 also t,:t.ltc I,hr vari:~t.ion of tho tlimensiot~lcss displ:~ccmcnt, thicktlcss 

((r,/l) (I/l/,l/v), as SIIOWII in Vig. 10.IOa. For n givon Imly Reynolds ~,ttn~l~or 

71, llv, it is now possil~lc to cvnlu:~t.c the locnl Itcynoltls number, IJ, d,/~, basc,tl 

on t.11~ clispl:tc~c:ttton1. 1I1it:l~nt:ss;sit~rr 

whcrc! tthc valuc of U,,(z)/fJ, is known from the polcntinl vclocity funct.iot~. 'J.II(: 

curves of C, r?,/v in trrrns of t,hc arc Icrlgth, zll', have also been draw11 in Fig. 17.8 

for variotts values of tho Iboynoltls nrtn~bcr TJ, l/v. The points of intersection of 

thesr r~~rves with 1.110 liwiil 01 s(rrhilil!/ givr thc position of the point of instal~ilit.~ 

for thv ~~c~s~~wt.ivo v;~ltlc. of t.I~o Itryl~oltls nrrrnl~c:rt. 'l'ho point,s of instability for n 

falllily oft-lli11t.ic c:ylitttlrrs ofsl~:ntl~w~cw r;tt.ios n/b -- 1, 2, 4. 8 arc shown in Fig. 17.9. 

It is rcvn:~rk;~l~lo t.lt;~t, t.hc sltif't. of (.It(: ~)oint of irtstabilit.y \vil,l~ an increasing ltcynoltls 

nun111c.r is vcry slt~:~ll for t,ho rasc of :L rirc~ll:tr c:ylintlrr. 'l'his shift becomes more 

prot~o~t~~~~~:tl as tJ~r slcwtlvrt~rss r:~t,io is irtcrrnsctl. 

'I'hr lwsif ion of 1,I1r ~wiitL of inst;~.l,ilil.,y [or MI arrofoil ran be cosily c:alculat.c:tl 

in :I sinti1:rr m:lnrlcLr. 111 l.llis c:onnc?tion it, is pnrt.ict~la.rly importan(, to tletermino 

(III. tlrprt~tl(~~~c.c on 1.I1t. at~glo of' inritlcwcc: in a(ltlit,ion to that on t,ltc: Iteynoltls 

~ltrlnl~c.r. 'l'hr rcw11l.s of s~tc:lt c:~.lrnl:lt,ions for thc C:LSR of a syrn~nct~rical Zhukovskii 

1- 'l'l~r I,IIVWS I[,,, hI/v 1'1,r v;~rio~tn va111rs of (Ic,> l/v C:LII 11t: tlr~iw~t 

l l ~ t w III :I (liwclio~~ ~:IIXW 11) 1111- :\xis l>r ~)r~li~uil,vs, if IL Iw::~rith~nic 

'I'hiq is :I vvry mnvtv~itv~l si~~~l~lilit~:~lit~~~ 10 IIW wl~tw 

fro111 rnch oll~rr 11y t,rn~~slnt,it~g 

w:dc is I I W ~ for I~Iw IILI.IIT. 

:I grnphi~d ~~~t~lhotl is vt~~pl(~~ixl. 

h. 1)ctcrmination of thc posibion of lllr point of ina1al)ilil.y for prrsrril~rtl 1)otly ~hnpc 499 

1Fig 17 9. I'onitiott of l)oi~lta 

of instnbility for clli1)tic 5- 

c.ylinilers of ~lrntlrvww --. I - 

rnlio n/b - I , 2, 4, 8, co (IhL 

plate) plotted against tho 

body llrynolds nun~hcr R , 

, _,> , 

6 fl5 4s 5 4s

500 XVTT. Origin of turbulence 11 

of minimttm prrssttrr irlstal)ilil.,v and, ronsequently, transition sets in almost at 

onrp rvm at low lteynolds numbers. 

I'igurc 17.1 1 SIIOWS, furt.l~cr, the pmit,ion of (,he point of instnbilit,~, as clet.ermined 

cxpcrimrnl:tlly for a. NAVA nrrofil, which possrssrd :in almost, idrnt,ical pressurr 

disl~rib~tl.ion with l,l~:il, ol' tl~c Zlt~tliovsltii :ic:rofoil ~ttttlrr ronsitlt.r:~f~ion. 11, is sccn 

t,l~:~t, l,Iir, lwit~t, oC tr:tnsiI.io~~ 1it.s Iwl~it~tI l l t ~ l)oit~t, ol' it~sl~:~l)ilil~y 1)11t, it1 fr0111, of l,ltt* 

poittL of l:~.l~lill:~r ~r~):il.ilI.ioll (i)r :III \':1111t'~ of Ib~ytlol(ls 11111nbcr ant1 Lift rocffioict~l, 

:IS rsprcl,rtl fron~ tl~rorc~lic~~l t.onsitlcr:it~iot~s. Sorontlly, the shift, of thc point of 

l,ratisibion wiI.11 a v;iryitlg Ibcynoltls n~tnll~c:r :rntl lift, cosfficirnt, follows l,h:~t, or 1.11~ 

poinl, of insl:iI~ilil~y. I ~ ~ ~ s I Iof I I . sysl,ctn:il,iv ~ ~-:~I~ml;il~ions on t,hc position of ~,II(? lwi111, 

of tra~~sili~n I'or :~(*roli)ils of varying t,I~iclit~

Tlw tlisl.nr~cc Iwt,wccn the point, of iwt.n.l~ilit,y and t.11~ point. of t,mrlsit,iolt can 

bo rcprescnlccl in t,llc form of (.he tli1li:rrrlc:c: Irct,wce~t the Ttnynoltls numl)orn forrnctl 

wit11 the aid of the rnonwnt.~ltn t.lli~lzncss i~t t.llcsc tewo points, as wn.s a.lre.zdy clotlo 

in Fig. 10.21, that is, as (llr?,/~~)~, -- (116,/v),. Fig. 17.14 sllows a 111ot. of thin quantity 

in tmlns of tho lncan I'ohll~:ll~so~~ 1)~r:1n1rtcr J? :~ntl is hn.srcl on t,hc vnl~lcs fonntl 

by 1'. S. Gm.nvillc 1751. llorc we 11:~vc: 

i,rr~ninnr nrrdtd~: 'i'hr st.:llrilil,y c~:~lc.lll:~liolls s~~nlln:rrizrtl ill I?igs. 17.9 nrltl 

17. 10 clvrnnnst.rat~c. very c:onvinrit~gl,v t,llnt. I It(, ~)~~cssu~.c~gt.:~tlit.nt. llaxnclvc~isi\~c: i~r(lrtrl~c.c* 

OII sl:~l~ilit,,v ant1 tmnxition ill VOIII~II~IV :1.xrf~c.tn(-111, wit,l~ 

'l'h (lnsig,~ 

~II(*:ISII~I-~~I~*II~~~.

504 XVII. Origin of k~rbulencc 11 

positmion of t,l~c point of tmnsit,ion is shown in addition for aerofoil R 2626. It is 

seen that transition occurs sliortly after the pressnre minimum in complete agreement 

with t,Iw t,heorctical results in Fig. 17.10. Figure 17.16 shows, furt.her, plot,^ of drag 

cocfficicnt.s in terms of thc lift coefficient for three aorofoils of equal tl~ickness but 

varying caml)er. It shonld be noted that hy increasing tho camber it is possible to 

canse a sllift in the region of vcry small drag in the direction of higher values of 

lift,, Intt rvrri so, t,lic rngion of rctlucctl drag still extcntls over a definite witlth only. 

Needless to say, in the case of laminar aerofoils t.he int,crart,ion between t.hc estr~.nal 

stream antl t.hr bountlary layer is very import.ant; mct.liotls for the c:i~Ic~tlation of 

sncli effects have been tleveloped by R. Eppler [BO]. At this point, it, is nrccssary t,(~ 

remark t.11ut cert,ain rircumstances c:ause consitlcrablc difficulties in t.he pract,ical 

application of laminar arrofoils. Principally thcse are dnc to t.he great. drmn~~ds 011 

t,he smoot~l~ncss of the surfaces in order t,o exclnde prwnat,ere transit.iot1 owing to 

roughness. In this conncxion we wish t,o draw the reader's at.tent.io11 t,o a paper by 

I,. Speidel [212] on lnniinar aerofoils placed in a Iiarn~onically dist,urbed free streatn. 

Fig. 17.15. I'rr*nw~rn dist,rih~tI,iot~ lor l:~.tninnr 

arrofoils at zero incitlmcc (c, 5 0). i\erofoilsOOI2, 

65, -012, 66, -012 from rrl. [I]; 

wrofoil It 2525, nftrr IIort.~ch (Dl] 

'I' = posilinlt or point or trxnsilion for R -- :1.5 x 10' 

Fig. 13 Ili Corfficirnts of profile drng. c,,,, 

plotted ngninst lift coefficient, c,,. for three 

Inniinnr nrrofoils with vnrying rwnhrr, 

R - 9 x 10" from ref. [7]. The rrgion of 

smnll drag mows townrds higher lift roef- 

ficients, c,,. as rnmhcr increases 

b. Dcterrninntion of the position of the point of instnbility for prrsrrihrtl horly nhnlw 505 

'L'hc discussion in this section may bc suminarizctl as follows: 

1. The tJicory of std~ilit,y sliows that, tlic prcssure gmdioll; cxcrln an ovcrwl~c~ltnir~g 

influence on the stability of the Imninar bountlary hycr; a tlrcrc-as(: in prcssurc 

in the downstream directlion has a stal~ilizing cKcct,, wltcrcns increasing prrssurc: 

leads tjo instnbility. 

2. Jn consequence, tlic position of the point of maximum vclocil,y of t.lic pof.rnLi:~l 

velocity distrit~ut~ion function (= point of minimurn pressure) inllucr~c:cs tlccisivcly 

the position of the point of inshI)ilit,y antl of t,hc point of t,ransitio~l. It ran I)c 

assumctl, as a rough guiding rulc, 1Ji:~t at nwlium 1tc:ynoltls nrirnl)crs (R =--: {Of; 

to lo7) the point of inslability coincitlcs with the poinL of minimurn pressure 

and that the point of transition follows shortly afterwards. 

3. As Urc angle of incitlcnce of an acrof'oil is incrcasctl aL a constanl ltrynoltls 

number, the points of instability and transition move forwards on the suction 

side and rearwards on the pressure side. 

4. As t,he R.cynoltls number is increasctl at const.nnt incitlcnce the points of inst,al,ilil.y 

and t.ransit,ion move forwards. 

6. At very high Reynolds numbers antl with a flat prrssure minimum, t,l~e polnt 

of ins1.nl)ilit.y may, nntler ccrt,ain circum~t~ancrs, sliglit,ly precede the poitit of 

niinin~nm prrssurc. 

6. Even at low Iteynolcls nurnbcrs (R = 10Vt.o 10" t11c points of inst~:~t~ilil.y :I.IK~ 

hnsition precede the point of laminar separation; nndcr cerhin circunistnnres 

the hminar boundary layer may become soparabed and may re-at,t.ach as a 

Flexible wall: Anothcr effective rnethod of stabilizing n larninnr bo~mdnry lnycr is to rnnke 

the wetted wnll flexible. In connexion with the obsorvetl antonishing swimming performance of 

porpoises [go], it hns been suggested that these nnimnls have n very small skin-friction coefficient 

bernuae the boundary lnyer on them remains laminnr even nt very Inrge Rcynolrle numbers 

owing to the flcxibility 01 thcir skin. Jri ordrr t,o put t.hk hypot,hesis to the tcst,, M. 0. Krnnter 

[110] performed ~ncnuuremenLq of drng on olwt.ic cirrulnr rylin(lcrs plnced in a stmnn~ pnrallel to 

their axes. Indeed, reductions of the order of 50% in drng, compared with rigid cylindrrs. have 

been observed in the range of Reynolds numbers R = 3 x 10" to 2 x 10'. 

Furthermore,T.B. Benjnmin (41 and M.T. 1,nndnhl [I201 instit,uted comprehett~ive thcoretical 

analyses on the stnbility of boundnry layers on flexible plntes with the aid ol the method 

rxplninetl in See. XVIc. Thcse revealed t,hnt, in nddition to tho Tolltnien-Sclrlirhtitig wnves which 

occur in n form ~notlilied by tho flexibility of t.1~ wall, there appcnr tnodifietl c1nst.i~ wavcs in tho 

wnll itmlf. Such elnstic waves are creatod owing Lo the prwence of tho flow outttide the wnll. 

F~rt~l~ermorc, there appear waves of the Kelvin-Helml~olCz type, rnther like those observed on 

free shear layers. The first effect - the n~otlification of the Tollmien-Scblichting wnves by the 

flexibi1it.y of the wall - may, taken by iteelf, explain the drnatic displacement of t.hc point, of 

neutd utnbility in the upstream direction. However, tho three effectn which depend on t,lic int,ernnl 

friction in the wall counteract each other t,o a certain extent. For this rcnuon, we would expect 

only a small overall effect. Thus, M.O. Kramer's experiment.al results appear to be confirmed by 

the ~1.nhilit:y throry only qrlnlitntively hut not qunntitntivoly. 'l'hc suppo~ition that M.O. Krnrnor's 

rwt~lltt wuld lwrhnp 1)o oxplniti~d hy t,ltn inll~~onw 01 wnll llcixihiIil,y on thr~ 111Il.y rlovc~lr~lj~~cl 

t.urbulcnt boundary layer induccd U. Zimrnormnnn [25!)) to rlnclcrtakc o thoorctionl invcntigntion 

into thin problem. He came to the conch~sion that the flexibility of the wall could lead to a roduction 

of the shearing stress on the wall of the order of 10 per cent,, nt lcnut in the presrnce of a fluid 

of high density such as water. In the nbsenc~ 01 n co~nplete theory of turbulc~~ce, it. is impossible 

to view these rwulta nu more than est,imaks. 'l'he pnpcr. [259], contains references to additional 

contributions which concern themselveu with the effect of wall flexibility on the stability and 

turbulence of boundary-lnyer flown.

c. Efict of ~urtintl on trnt~nition in n hn~rntlnry lnyer 

It, has alrrntly Iwrn poitrI.ctl 0111, in CII:I~I. SIV t.hnt Ihc application of suction 

I.o a 1:rminnr ho11rtt1:~ry hyrr is an rKcc1,ivc mmns of rrtlucing drag. The clrect of 

surtio~~ is t.o st.nOilizr I hc h~ntlary laycr in n way sirnilnr to the cffcct of the prcssrlrc 

gratlirnt tliscussctl in tl~o prccntling srctior~, antl the rrduction in drag is nchicvctl 

1)y 1)rt:vc11ling tmnsilion fro111 I:~tnit):~r t,o IIII.I)IIICII~. flow. A marc (Ichilc(1 :~n:~lysis 

rrvcnls t,l\nt. t.lrc it~fluc~~~cc of s~~vlion is (111~ to t.wo cficts. First, s~~ction ret111ccs thc 

l)or~~l~l:~ry-l:~.yrr t~l1ic4tnrss n.ntl a t,l~it~nc.r I)ol~ntl:~ry hyrr is loss prom t,o I)ocome l.t~rbrllrnl.. 

St~:ontlly, s~~t:l.ion crr:~l.rs :I. I:~n~it):~r. vc~loc~il.y 1)rofilc which ~~ossc:ssc~ higl~rr 

limit. of stal1ilil,y (c:rit.ic::d I

608 XVII. Origin of turhulcnce TI c. Effect of suction on transition in n hounclnry lnycr 509 

It nligllt be rc~narlzc*cl 11c:rc LhaL n more nccurnte rnlc~~lation would prcsu~nnldy 

lr:~11 t,o n higiwr value: or tho volumc: coc.ffic:icnt,. This is tluc to the Fact that tho 

:~syrnpt,ot.ic: vclorit,y profile, on whosc oxistcncc thc nt)ove c:~.lculat,iorl was based, 

clcvelops only nt n, cc:rt,:~irl tlist,:mce from the laatling edgc. 'l'lle vclocitty profiles 

Imt,wrc:n th:tt, point and the leading ctlge :Lrr of different shapes, changing gmtlunlly 

from t,hr I~~:I,S~IIS form wit.11 110 s11ct~ion at short distances behi~~d t,he leacling edge 

t.o the :hove: asy~nptmtio I'orm. 'l'he profile shapes ill this initi:tl, stmt,irlg Icngth for 

the lamirlnr boundary Iaycr with suction have been plotted in detail in Fig. 14.8. All 

thcse vclocit,y profiles hnve lower limik of stnt~ilit~y than the asymptotic one, ant1 

it, follows that t.hc qu:tnt,it,y of fluid to be removed over the initial lengtll must; be 

larger tlmn tho valuc given in cqn. (17.11), if laminar Row is to be msintainetl. 

111 order to analyze this matbr in greater detail it is necessary to rcpcat the stability cal- 

culation for thc scrica of velocity profiles in the starting length taking suction into account. These 

profiles constitutc n one-parameter family of curves na shown in Fig. 14.8, the parameter being 

given hy 

and changing from 6 = 0 at tho leading edgc to 6 = w for the nsymptot,ic profilc. In practice, 

however, it may be assunlcd thnt t,hc starting length ends with 6 = 4. The resulting critical 

Reynolds nun~hers have bccn compubd by A. Ulrich [243] and are given in Table 17.1 ; the corresponding 

curves of noutral stnbilit,y haw been plottcd in Fig. 17.17. The nmplification of unstnble 

dist,urbancrs for the asymptotic profile has been calculated by J. Prcbch [180]. The highest 

drgree of amplificat.ion obtained in this calculation was about 10 times sn~nller than that for 

Fig. 17.18. Determinat.ion of 

critical value of voluinr coef- 

ficient for maintmlance of 1%- 

minar flow through suction for 

houndnry layer on flat, plate 

Table 17.1. Dependence of critical Reynolds number of velocity profilra witb suction on 

dimensionless suction volume factor [, after Ulrich [243) 

-- 

= I

Q " 

tho flat plnto (I3lnniuu flow) in Fig. 10.13. With tho rc!..ulLR of thiu cnlrulntion it is now cvlrry to clot.er- 

mino the volume coelficicnt of suction which is sufficient to ensure stnl)ility ovrr 1110 st,nrting 

length. It can be obtained from Fig. 17.18.in which the limit of stnl)ilit.y from Tnhk 17.1 and t,lle 

variation of the dimensionle,ss displacement thickneaq 

for :r pcsr:ribc~l vnluo ~rf c~ - (--v,,)/Um hvo I~oon plol.lntl ~~gninnt. t.ho cli~~~c:r~niol~lr*n~ Icy~gt,Il 

coordinate. lierc (-8,) ijl/v is known in terms of { from tho calculation of the ho~lnrlnry layer, 

Table 14.1. It is sccn from Fig. 17.19 that tho limit of stability is not crossccl at nny point over 

the whole Icngth only if the volume coefficicnt is kept at a val~~e larger than 1/8,500. Hence, the 

critical valuc of the volume coefficient beconler, 

We are now in a position to answer the question which wm left open in Chap. XIV, nan~ely, 

t,hat concerning the acLunl dccrcn.qc in the drag on a flat plntc at zero incidrnco wllonc: t)o~tndirry 

Iaycr is kept larninar by suction. Figure 14.9 cont,ninccl a plot of the cocfhirr~t of slcin 

friction under these conclitions cxprrssetl in terms of tlie llcynolcls nntnber with the: voll11110 

coefficient eQ nppearing as a parnmcter. If the curvc wl~ich corresponds to cQCrcl from eqn. (17.12) 

is now plotted in the diagram, it is possible to dcducc thc variation of t,ho cooffic:icllt of skin 

friction for a llnt plate untlnr condition^ of oplimum ~~~ction, nn RIIOWII in Pig. 17.1!). 'J'hc clintancc 

between the curvc marked 'ol)timum suction' ancl that markcxl 'Lurbulcnb' rorrcspontls 1.0 1.11~ 

saving ili drag en'ccted by thr application of suction. 

Fig. 17.10. Ckxfficicnt of skin frict.ion of :i 

flat platc at zero incitlcncc. Oplirnum auction 

dcnoles smsllcst volurnc coeffiricnt cqrrrc - 

Fig. 17.20. Itc4:ttivc snving in t1ra.g 011 flat 

pl:rt.c nt. xrro i~lc.iclcnrc! with sc~clio~~ III:I~I~hining 

ln~ni~~ar llow :~t, O ~ ~ ~ I I L W I.snr/ion I I from 

= 1.2 X 10 whii4i just sufliceu to main- I'ig. 17.19 

tain laminar flow '4r~ -s '1 rurb - '1 iataiwtr milh sudim

510 XVl I. Origin of Idmlrnro I I 

'I'l~c rlTc.rl, of srrc.t,iol~ on lhr litnil, of st4:~.l)ilil.y t.ogelhrr with t11n.t of :I. prcsstlrc 

grntlicwt, ran It(: rrprcsrntc:(l gr:lpltit.:lll~r 11y plotking t,ho cri1,icn.l Rcynoltls nnml)cr 

:rgaitwf, 1,ltc s11:spc I':ic:I.or Illz --: B,/dz of (,h: I)ountln.ry layer profile, as was clor~t: in 

Fig. 17.21. '1'11~ c,ril.ic::~.l Iky~~oltls nnrnbcrs for a fht plnte wit,l~ zrro prossllrc. gr:l,tlict~l, 

I 

cl. ICITrrt of hotly forces OI\ trnrlaitiot~ 

Couette flow: l'hc st.nhilit.y of Ianlinnr 110~ IK~.WCCII I,WO con~rnl.~.ic, ~d,i~.l.ing 

cylinders (Couet,to flow) is govcrned t.o a 1wgc ext.ent I)y t,Iw ccwt,rifng:rl forecs. 'l'l~c: 

velocit,y dist~ribut.ions which occur in ttl~is cnscl wrrc. givvn in Fig. 5.4 on tlw Imsis of 

exnet, solut,ions of tl~r Na.vier-St,oltrs oquat,iot~s; t.l~cby rovc:rctl various v:ll~tc~s of Ihc 

ratio of radii n = rl/rz, nntl c,oncornctl two 1)n.sir cnsrs: (I) innrr c.ylintlcr rotn(.c:s, 

outer cylinder at rest,; (11) outer cylintlcr rot,nt,rs, inncr c:yliritler at. rrst.. In (hsv I

512 XVII. Origin of turbulence I1 d. Effect of body forcea on transitiou 513 

of (.his ar~.a~~grincnf,. I

514 

, 

XVII. Origin of turbulence 11 

e. Effeotu due: to hcnt trnnofer nrld compreusihilityt 

1. ln~rotlurtor~ rctmnrk. 'I'll(: t.ltrorc:t,icnl nntl ~xprritn(:nt,nl rns~llt,s coticcrtri~~g 

tr:l.tlsil.ion closcril)ctl itt tho prcv-(:(ling sc:c:t,ions are valitl only for flows at modcmtc 

sl~:ctls (itic~otlrI)rc~ssil~ln Ilow). 'l'lln rll'cc:l. of t,hn compressibilit,y of thc? firlit1 on tmnsit.iori 

h:~s rcrntll,ly Iwcn rxl~n~tsl,ivcly invnstignt,ed urltlcr the ~t~itiid~s from nerort:~.t~lic*nl 

c:t~ginrc:ring. In t.he c::l.so of'cornprrssil)lc Ilows, apart, from tho Rlnc:h nrt~nl)rr, 

it, is t~crrcss:~ry t,o hltr it1t.o :~cc:onnt, ono ntltliteional, itr~port~nnt pnr~~nct~rr which is 

ro~~~~c,rt.rtl \vil.l~ t,l~o ~:rt,c, of Ilrnt, t.rnt~sl'c:rrctl Ixtwccn t,llc: ll~~itl ant1 the wall. \Yhcn 

I,II(% I111itl is i~~(~o~~~~~t.(*ssil~l(:, 

lt(~~t, WII IN: vxd~;t,t~gwI l~~:t,\v~:rn Ih: w:i~ll :I.II(I I,IIc ll~ti(l 

i( 1 11(, (,(~t~~~)t~~~:~t,~~~~: 

of t,l~(: \v:~lI is I~igltcr or lowrr t,h:~~ th:~t, of t,he fIui(1 hvitlg 

IKISI. it.. 111 1.11~ c:n.sc: of' :L (:otn~w(~ssil)lc Il~~itl, 1.l1c h(~iI, cvoIvc(1 in 1.l1r I)ont~(l:~.ry l:iy(\r 

I~ro(l~~c~t*s :III :i.tltlit,ion:bl, itnl)orl,:~trI, inflnonce, ns nlrontly sltown in (:II:L~. X I1 1. 111 

rit l1c.r (.:IS(: :I t~Itcrtnnl l~ont~,li~ry I:~yrr (I~:voIoI)s in a(l(lil,iotr 1,o tho vc:Io(:it,y I)oun(l:try 

1a.y~-r ant1 plnys it,^ pnrt in the dctcrmin:~t,ion of tho inst.nl)ilit.y of n smnll dist,urbnnoe. 

r 1 

Lhr t~l~rorct.i~aI ar~tl cxpcrirnc~ntd rotisitlcrnt~iot~s which wr are abor~t t.o discuss will 

show t.11nt. for t.he s~~I)sonic flow of n ~:LR, lwnt. trnnsfer from t,lle 1mtntln.ry Inyer to 

t,hr wnll exrrt-q n st.n.l~ilizing influcnc-c, wltilr Irrat transf'rr from t,he wnll to t .1~ pas 

11ns the opposit.c cfT(~:t,. I3ot.h of t.hcsc nrr reversed for the fiow of a liquid. For srlprrsonic 

flow. n new t,ype of unstable rlist,urbnnco is possil)lr ~ hich responds to t,he tmnsfrr 

of 11ca.t. in nn entirely tliffcrcnt mnt1nc.r. 

2. Tlw effcct of lmnt frnnsfor in iucompressible flow. Sornr of the main frat,urrs 

of t,l~c rlkct of f.ltc t,r:~tlsfi~r of 11c:n.t from t,lx w:dl t.o t,lw fluid on t.he st.aldit,y of n 

Inn1inn.r I~ot~tttln.ry 11iytv. rrtn ltr ~v:tdily r(w)gnizrd rven in the rrrsc: whrn tho flow is 

incotn))rtwil,Ic. Wc shrill. t,llcrcli)rr, cxl~lai~~ it. first. in t,llis simplifird form. The first 

exprrirnrntnl invrst~ig:it.iotts on t.he inflltener of 11rn.t t,ransfer on t.ransit.ion were perli)rrnrd 

some t.imc ago by W. ],inkc [ 13 I 1. \\'. I,inltc measured t,hr c1ra.g of n vcrt,icnl 

hrakd ~)la.tc plncrtl in a horizontal strrntn in a rnngc of length Rcyllokki nlllllbrr 

R = 105 to 106, nntl ol)srrvrd t.li:rt. Ilc~nling rar~sc:rl it. to inorrase by n Inrgc ntnor~nt,. 

11e conol~ttlcxl ftwm t,llis incrca.sc, qnilr rotwrt.ly. t,llnt, t.hr l~cat~ing of the phtc cnrlficd 

t,he t,rnnsit.ion Itrynoltls numl~cr t,o dvrrrase. 

Now. if t.lw wall is Ilot.trr than Ihc. llllitl ill thr frcn strmni, we Irnve TI,, > T, ant1 

tIw t.t-mpcr:it~~t.c* gt.n.tlirnt. n.t Iltc. wnll is nc*)mti\.c: (t17'/tly)w < 0. Since for a gas the 

vis(.osily ittc.r(~asc-s wit11 tc~t)i~)t~~~:~t~~t~(~ :iwotding to ccln. (l3.3), we mnst. have (d/l/dy)fi; 

< 0. Since t h vclocit,y grndient is positive nt t,Ilc wnll it follows from ~clll. (17.15) 

tlmt. 

A nnmcrirnl cnlcul:tt.ion l)y '1'. Crl)rc:i nntl A.M.O. Smith [221 fi)r air cwnfirmrtl 

the (l(:crcnsc in t,Ile crit.icn1 IZ~~ynt~I~ls IIIIIII~N:~ for I,ltc ons~t, of i~~sl~al)ilil,y 

flnt plntr, nntl n sinlilnr tlrcrrnsc in t,ltc: I.ru.nsilion I (;On(:, tllc tlinit~nsion:~l 

irlc:rrnsos in invcrsc. proport.iot~ 1.0 dl. 'I'll(* rcsrtlls for :I. coolctl \vr~lI sl~ow tilt! 

cxpcctcd tlc-sl.nbilizing cll'cct for liquids. In t.hc t.ltc:o~.y of' A. It. Wrizz:l.n, t . 1 only ~ inflr~encc 

of hmt, transfer, other thn on t,ho rncnn vrlooil,y j)r.olile, is t~11rougll t.Ilr krnperaturn 

rlcpenclence of the viscosity. A more oornplrt.e (11c:or.y by It. I,. 1,owell and 

wntor boundary laycr on :L flnt platc, .aflcr 

A. It. Wwmnn, 'l'. Okilr~~r~rn atlrl A. M. 0. 

Smil.l~ [2RO)

516 XVII. Origin of turbulcnco XI 

E. ltrsliot,lto 11461 inclutletl the t,cnipcrat.nrc and dcn~it,~ Ilnct.unt,ions, but. led to 

almost iclcntical riumcrical rcsult,~. A st,ahilit,y experiment. by A. St,razisar, 

.I. M. I'ralll and 15. Itcshot.lro 12271 vorilirtl t,llr prrdic:l,rd shifl. ofl,llc rninimutn crit.ir.al 

ltcynoltls numbcr witli a small amount, ol' Iw~ting. 

I'rce eo~~vection: Transition of a frrc-cot~vcctior~ houndnry layer on a vertical 

Ilc~l.t.c~I II:rt. 11In.t~ wns fitxl, rc.ht.ctl t.o t,llcs a~nl)IiIicnt~ion of sm:l.ll tlistnt~l~nnoc~s by 15. 1%. (:. 

IC(:k(.rt and 15. S(icltrigcn 157, 591. 'J'lic rratlrr is referred to articles by 13. Cebhardt 

[79, 80, 811 for n comprchcnsive review of this field in which mrich progress has 

been rnatlc in explaining ot~sorvrtl t.rnt~siOion phenomena by rncnns of accurate 

n~ttnrt~ical calculnt,ions Oascd on thc n~c(~llod of sriiall dist,arbances. 

Wllcrc;~s for the vert.ical hent.cd plst,r the insf.abilit,y originat,es from progressing 

wavcs of t,hc 'I'olln~icrl-Scl~IiaIit,it~g t,ypr, on the irtolincd heatled plate standing unstable 

vorticw wit,li axes along the dircct,ion of flow havc been observed; t,hesc are 

of thr Taylor-(:oort,Irr t,ype, see 1147, 228, 811. 

. . 

'l'llo st.nl)iliiy of a ~ I W c.onvrc:t.ivc st,rcmn on a hval,rd vcrticnl plr~f,c- WILS invc-sti. 

g:~t.cvl I)y 1'. It. Nnchtnlic~irn ( l(i7 1 who rnil~loyc:cl (,hr mc~t.hoc1 of wrril.ll clist.~trl~n.ncw. 

I he vc,loc:it.y :~ttcl t,ompc:rat.nrc tlist.ribul.ions were those of Figs. 12.23 ant1 12.24, 

~cspcct.ivc:ly. Velocity prolilvs wil,h a st,rong point of inflcxion, such as tlltose in Fig. 

12.24, arc intrinsically characterized by a Inw limit. of stability. The inclusion of 

t,rmporal t,rrnpc~rat,urc fluctuat.ions on tol) of the velocit.y fluct~at~ions produces an 

adtlit,ional st,tvng tlcstal)ilizing cfict of t,hr nlain flows, because this mechanism t,ransliw 

cwrrgy from thc main motion 1.0 t . 1 tlisl.nrbancc. ~ 

'l'hc calculat.ion leads to two 

c:ouplcd clilTc-rcnt.inl equat.ions which now rvpln.c:c: t,hr Orr-Sonirncrfdtl equatkm (1 6.14). 

Onr of t.llrm t d c w to vc.loc*il,y and t.ho otl~cr to t,cmprrat.urc. These two equations 

contai n t.hc I't.nntl t.l numlwr ant1 l,hc (:r:i~shof number in addition to the Reynolds 

nnmbrr. In this connexinn t,ltc ~mcler shol~hl consult t,he papers by E. Edrert ct al. 

)S!)I, A. Snc.\vc:zyli 122!)1 antl '1'. I 1 wit,ll phnnc vc.loc:il,y c, -- c:" --- (I,, m, 

and n = 0. Ncn1,ral supersonic tlist,urb:mccs wc: pos~il)lc~ ill wrt.:litt fIows, I)ut 110 

general conditions for their existence havc bccn given. Figurc 17.24 shows 1,11c: clirncnsionlrss 

phnsr vclocitics cJUm and co/U, of t.ho nc~~t.rnl sul)sonir and sonic: tlist.~~rl~:incv 

as fullct.ions of Mm for n fnrnily of ntli:rb:~t.ic flat.-pln.t,c I~onntlnt~y I;ryws. . 111c~ . 111(~:111 

bountlary laycr profilcs which were usrtl in t.hc cnlculat.ion of c,*, ant1 will h: usc:tl 

throughout this Section, are aceurat.c numerical soluLions of I,hc coml~rcssiblc: In.tninn.r 

boundary layer equations for air with both t,he viscosit.y cocflicient, nnd I'rnntltl 

numl)rr fr~nct~ions of t.emperature, and with a frcc-st.rcatn stngnation t,cmpcrnl,~lro of 

31 11< up to M, = 5.1 where Tm = 50 K. At higher Mach numbers, Tmremajns at 50 I co > 0 in Fig. 17.24, all of the boundk~y layers of t,his family 

sat.isfy the conditions of the extended theorem and are unst.al~lc tto frictioulcss ({is- 

turbnnces. The movement of the generalized inflexion poinL 1.0 larger y/S wit.ll ill- 

creasing M, is similar to t.he movement of t,hc inflexion point \vil.ll increasing nt1~r.t.s~. 

pressure gradient. in incompressible flow. l'igure 17.24 also givrs t,hc tlimct~sior~lt.ss 

displaccmcnt thicltncss dl vl/,/x v, ns a fnnoth of M, for Lltr fnmily of ntli:~I,;,.t,ic: 

boundary layers. I,. 1,ees and C.C. Lin were gblc to prove th:~.t l.llc wnvo n~rrnl)c~~. O~'I.IIV 

neut,rrtl subsonic disturbance is unique as in inco~nprcssil)lr flow, provitlccl t,llnt. t,Ilc 

mean flow relative to the phase velocity is everywheresubsonic, i. e. h2 

t.he boundary layer, where M = ( IJ - c,)/a is the locnl rrlntivc~ M:tch IIIIIII~)~~. 

Although t-heir proof t,ltat cqn. (17.17) is a s~ifficionl, contlil.ion for 1,l1c: inst.:ll)lit,y h:stl 

t,he same restrict,ion, it appcnrs from rxlensive nnmrricnl c:rk~~ht.ions l.l~:rl, rtltt. (1 7.17) 

is a true snff cient condition even when M2 > 1. On the contrary, L.M. Mack [ 1521 show-

518 XVI I. Origin of turhr~lcncc 11 c. Effects duc to heat transfer and cornprcsnihilil.y 510 

rtl 11y n~lrncrirnl rnlrulnt,ions thnt with n region in tho boundary layer whcrc lk > 1 

tI1rr.c- nrc- :In inlinitc nrlmbcr of nculml wnvc numbers, or modes, with the sarnc plrasc 

vrlocity c,. 'l'hr mult,iplc modes arc n result. of the change in the govcrning tlilPcrc:nt.inl 

rquation for, say, thc: ~mwurc oscillntictn from rlliptic whon MZ < 1 to hyperbolic 

\vhrn M" I. 'l'l~e first motlo is t.lx snmn as in incomprrssiblc flow, nntl was first 

rotnl~~trtl I'IW c~otnpr(~ssil~lc flow 1)y Ir. IATS tu10 1':. Itc:sllot.ko 11421. Thc ntltlit.ionn1, 

or I~ighrr. rnotlw hnvo no incornprc~ssil~lc ronnt,crpxrts. c, = c,, MZ, first, reaches 

11nit.y at M, 2.2, nntl the uppcr I~ountlary of t,hc rcgion of supersonic relative flow 

is nt ?I/(> -- 0.16, 0.43, 0.50, Lor f& =- 3, 5, 10, rcsprctivcly. 

.. 

lhc multiple 11tmt.rn1 di~t~url~nnrcs wit,lr phnsc vdocit,y c8 are not tho only ones 

possiblo wllcw ME, > 1. Thcrc nrr also rnult.iplc ncut,rnl dist,urbances wit.h U, < c, 

< Il, -t a,. Tllrse dist,nrbnnccs (lo not. tlcpond on the boundary layer having a 

gc.nrrnlizcd inllrxion point,. I~urt.llcrmore, there nre always adjacent amplified tlist~rlrlmnrrs 

of t IIC .w~rre t?/pr idh, plrass 11e1ocitie.s c, < I/,. Co~tacquentl?/, the co~rr.prossil)le 

Ooi~?tdar?/ /fl?/rr is isnslable lo frictioltkss dislicrba~~ces rrgardloss of any other f~aturen o/ 

thc idoci/?/ rr~rtl hrg~cr(r/rtrr profilrs (1,s lonq n.p th,rre in a rqlion idtrre M2 > 1. 

A limiting fn.rt,or in t,he amplifim.t,im of first,-motln tlisturbnnccs is t.hnt c, must 

lir I)c:t.wcvw ro n.11tl r,. Any lhing t,llnb incrcnsrs the diffcrrnoo cr - co nlso incrc-asps thc 

nmpIilic:nt~ion f:~.c:Ior /?,. As shown Iiy Iqig. 17.24, t,llis tlifkroncc can be cl11it.c stnnll. 

Tho co~lst.rn.int inlposrtl I)y co, which tlnlilrc c, is nnrclnkcl to the boundary-layer 

profile, call only 1)o rrnlovctl 11y consirlcring n moro general form of disturbancc than 

has I)crn usrtl up to t.llis point,. With 

As n rcsult of c, - co increasing wit,ll {I t.ho ~naxirrlr~tn nmpliBcnt,ion fnet,or of 

t

620 KVII. Origin of tr~rbrllonce li 

lly M, -- 3.5, thc maxirnntn atnplificat,ion factmr of both t,llrec- and bwo-dimensionnl 

dixt~~~rl~nnccs occrtrs at. R = oo. It, is in this second rrgion, whcrc the inst,abilit.y 

nssrtmcs nn c:sscnt.ially frict,ionlrss ni~t.uiv, 1.lln.t. nn unst,al)lc bantl of frcqttrncics associakcd 

with thc sccot~tl mode first appears for R < 2.26 x 100. In the third region, 

M, .:- 5, t,l~c nmplilic~rt,ior~ fart.ors tlro.casc st.rdily in pt.oporl.ion to the incrcase in 

(TI sllowrl in I'ig. 17.24. 

For thr low snpctl flow of a gas, wc havc alrcatly disrusscd the tlcst,abilizing 

cffcct, of ,z hcatrtl wnll and tltc stabilizing effect of a cooled wall. Lccs 1123, I241 cal- 

cl~lat.etl similar rlrrcts for cornprcssihlc air 1)ountlary layers, antl, in addition, prrdictctl 

the possit)ilit,y of co~nplct~cly st.ahilizing snpcrsonic bountlary layers by cooling. Alt,llouglr 

t,his prrtliction anti s~lbscclnentcalcnlations of tllc cooling require(1for complete 

st,aI~ili~n.t.iot~ by M. Tiloom [I I I alltl E. 12. van I)ricst [32, 331 were bascd on the asymptotic 

t,l~eory of t.wo-ditncnsiond dist,~~rbances and took no account of the higher modes, 

tJ~c more rrccnt, oomput.cr calcnIat,ions have verified t,l~a.t, suffioient cooling will indeed 

~omplrt,~ly stal~ilizc, or nrn.rly so, bot,h two- and t,lircc-dimcnsiond first.-tnotlc tlist,ttrl):cnrrs 

ovrr n witlc: hlac:l~ nnml)cr rnllgr. I'iq 17.27 shows, from t,llc: ft.ic1.ionlrs.q 

tl~rory, t.llc rat.io of (/Ir),,,, to its vallro for thc adiabnt,ic wall, (/?t),nnz,ad as n ftlnetion 

of 7',,./7',a, the rnt,io of t,l~c wall tempemt,tlrc to the adiabat,ir, wall t,empcrat,urc. The 

st,nbilix:it.ion of t,l~rrc-dirnc~nsional first.-motlo distnrl~ancrs is clearly seen, with t1hc 

st.at~ilizat~ion tlccrcasing with increasing Mach number for the same tcrnperat,ure ratio. 

On t,hc contmry, sccontl-tnodc dist,urbnncrs, far from bcing st.abilizctl by coolil~g, 

arc tlc-st,abilizrtl, The rrnson for t,l~is tlin'crent, behnviour is, once again, t,llat the 

gcnrralizcd inllcxion point,, which is stmngly inflncnectl by cooling, has no importance 

fi~r t.hc ~lnst~al)lo Iligl~rr motlcs. 'J'llc irnportnnt, q~tanl,it~y, t.1~ cxtcnt of the supersonic 

~~c-lal.ivc-flow rrgion, is lil,t.lr inllucncrtl I)y cooling. 

Simil:rr rrsult,s t,o (,l~osc showlt in I'ig. 17.27 a.rc ol)t.n.inctl from the viscous t.hcory, 

exc.cpt, (.hat for M, > 3 Irss cooling is roquirrd for stabilim1,ion a(, any linit.c ky- noitls nnml1c.r than is given by the frict.ionltwi t,l~cory. In this corlncxion sec also 

pa~wr 11y I

on the bnsis of t.hc tl~corct,ical rcs~rlt~s just, prcscnt.ctl. An import,ant point to keep in 

mind is t.hat. altJ~ongh a l~oundary Inycr has definite inst,abilit,y properties, its transition 

Jtcynoltls nrrm1)cr dnpcnds not only on these propertics hut also on the t.ypc nnd 

int.c?nsity of the tlist,url~nnccs prcscnt in thc How. 'Tho only facility in which it is convcnicnt, 

to stwly atliabntic: I)ountlnry Inyrrs is {,he supersonic: wind tunncl which has 

its own spc:cin.l tlist,itrl)nncc rnvironmcnt. I3t:low M, := 3, t,ransitiotr rncasnrcmonts 

difli:r witicly for tlillrcnt, tunnels. Il'or M, > 3, hth .J. 1,nufcr 11351 and 1C.R. van 

Driest nntl ,J. C. 1his1)n 1341 have sl~own t,llat t,urbulcnce from the supply scct,ion docs 

not all'cct t.rn~tisit,iorl in t,ho tmt scc!l.ion. Infitcad, the primary tlist.urbnnce sourcc 

rc~sl)onsil~l~: liw t.t~:i.~~sil~io~~ 

i~ IN: nt:ot~slk rn(li~i.I,io~~ rrotn I.lt(: l,url~r~l~:tlI, I)o~t~~tl:~.ry Inycrs 

on t h tunnc~l wn.lls. In atltlit,iot~ t,o t,lle rll'c~t:l, on trnnsilion tncasurctnerlt.~ of tlill'ercnccs 

in t,llc dist,urbaric:c cnvironmrnt-, t.hcrc is also the problem of defining nntl tncasuring 

the transit.ion Itcynoltls numhcr in n consist,cnt mnnncr. An instruct.ivc comparison of 

five dilrcrrnt mothods of measuring trn.nsit8ion has been given by .J. 1,. Potter and J. 1). 

Wl~it,fielcl [172]. 'Ute method of small dist.urbanccs can properly bc applictl only to 

the calcuint.ion of n start.-of-t.rnnsition ltcynoltls number. 

r 7 

Lhc nnmrrwus wind-tunnrl t.ransit.ion dn,t,a for Mm > 3 accunl~~lat~ctl hy S. It. 

I'ate :~nd C. .J. Schnclc:r (I831 fiw fiat plnt,es nntl by S. R. I'atc 1181 1 for cones, formed 

bllc I,n.sis of thcir corrclat.ions bascd solely on parameters of the ac~ust~ic radiat,ion. 

. 'Ihsc data, togct.hcr wit.11 measrlrcnlnnt,~ nt & < 3 by J. T,nufcr and J. R. Martc 

11381 on (*ones, I,y I). Colrs 1241 on n fiat, plat.(!, a single observation at Mm = 1.6 I)y 

,T. M. Kcntlnll of laminar Ilow on n flat. plnt,c n,t, R = 4.3 x 105 in t11c same tunnel used 

by I). (:olcs, and tnonsnrc?nwnts on cones in t,mnsonic tunnels by N. S. I)oughcrt,y and 

1'. W. St.cirllc [4!)], suggest, t,hc fi)llowing pnt,t,crn for the Mach numl)t:r dependence of 

tlrc ttrnnsit.ion Reynolds number in a good wind t.unncl: An init.ial incrc,ase for Mm > 1 

with a peak, ~~crlla~)s rnt.llcr broad, 1)c:twccn Mm = 1.5 and 2.0, follow~d by a 

decline, anti t,ht?n, st.nrl.ing son~e\vh~rc brtwt:tv~ Moo = 3 an(\ 5, a monotonic increase 

wl~icll cont,inucs t.o at. least, Mm = 16 arcording to n~easl~roncnts in a helium tunnel 

[157j. It is of particular int,crcst f.llaC tl~csc t,llrec Mach number regions corrcspontl 

roughly to the thrrc rrgions tlisaussctl prrviorrsly in conncxion with Fig. 17.27. A 

more tlirnct cont~cxion wit.11 st.ni~i1it.y thnory was rnntle by .I,. M. Mack 11541 by rncans 

of a simplified cn.lculn.t~iotl of t,lrc start of tmnsition on a flat plate based solely on a 

crit,ionl amplit.utlo A, of the most, amplifirtl single-frequency disturbitnce, as given by 

ecp. (I 7.18 a). 'Slllc rcsult.~ of t,llis celculat~ion dong with some experimental Nut-plate 

dnta 124, 451 nrc: shown in l'ig. 17.29. With A(l the value of A at the nnufrnl-std)ilit,y 

point,, Llrc uppt~ curvc rcsult~ from assuming tht. A. is intlcpmdcnt of Mach number, 

and the lower curvc rcsult~ from ,wsulning A. a M& for M, > 1.3. It is the lower 

curvc which corresponds t,o t.m.nsit,ion in a wintl tunnel, where J . Laufrr 11361 has 

clc.tmrninrtl that from M, -- 1.6 t,o 5.0 t.lw frcc-strrnm rms tlis1,rtrl)nncr nrnplitudc 

vnrivs rssrn1,inlly as M,:. 'l'llc pwrnl sitniln.t.it,y of t,l~is curvc h the mensurcmcnt~s 

fully suppork the viaw t.ha.1, tmmsition in s~~~)ersonic bountlary layers rosulLs from thc 

alnl~lilirn.tion of pnrticular flow tlistlurl)ancrs in acyordance with the rnetOlod of small 

tIist.ttrl,nncc:s. 

111 a11 rxpcrirr~twt,n~l invr-sl,i#nt.ion of Lhr c~ni.t:t, otl t,rnnsit,ion of a flow paran~ctm 

sltn)i ns M:~c:lt nun~\)cr, it. is ncwssary 1.0 Itcop t,hc unit Itcynolds number U,/v, const.nnt,, 

n.s wit,lr I.IIv t~~c~:~srlrc-~llrtit.s gi\+rn in Vig. 17.2!). 'I'hc clcprndence of thc transition 

lbc~.noltls n111n1,t.r on rtnil. I

524 XVII. Origin of turb~~lrnce 11 f. Shbility of a boundary layer in the presence of thrco-dimensional disturbancra 525 

'l'hc rcsult~s of an investigation by 13. R. van lhicst and J. C. Boison [34] on a 

conr at Me = 1 .!I, 2.7 and 3.7 arc shown in Fig. 17.31. Thc increase in the transition 

Itrynoltls nrlml)or 1vit.h cooling is clcarly seen, as is the reduction of thc st,:~bilizat~ion 

ofloot. wiL11 increasing Mar11 number. 'l'lic Int.t,cr trend continues to llighcr Mach numhrs, 

n.s mn,s shown by 1.h~ small stal)ilizat~ion cffcct found on flat platcs at M, = 6.0 

ant1 6.5, respccl,ivcly, I)y A. M. Gary (211 and 1). V. Maildalon ( 1561. These cffccts of 

w~olilrg arr in c:oml)lc*t,c. acrortl with t,hr I~cllnvioirr of first-mode tlist,nrl~ances 

shown 

ill It'ig. 17.27. Ilowt:vcr, in lwo cxpc~ritncnki by N.S. I)i:r.conis, tJ.lt. .Jack and It..l. 

\\'islli(~\vski 147, 101 J, at. M, = 3.12, cooling Iwyontl the rcgion of st,al~ilizat,ion shown 

in Fig. 17.31 rrsnlt.crl in a tlecrrosc rat-ller 1.han an increase in the t,ransit,ion Reynolds 

nn1111~t~. This pl~enon~enon has bccn rallctl "l.ransit,ion reversal" because it is contmry 

1.0 1.11~ cxprct,cd t.rcn11. It 1la.s :dso INWI OI~SCI~VC(I in sI~o(:Ic tunnrls by I$. E. 

Itic:ll:~rtls nnd ,J. 1,. Stoilcry 1195, 1961 011 a flat, at. M, = 8.2, and by K. F. 

Fig. 17.31? Expcrinrcntnl tmtisilion dn1.a ohtnirrcd 

on a iOo-cone at. zero angle of incidcncc showing 

~tnbilizing cffcct of wall cooling nt three 

Mnrh nunrhers in a sr~personic wind trinnel, nftm 

15. I

626 XVII. Origin oI t.11r011lonco I1 

~oncent~ric cylintlcrs of wl1ic11 tho inner cylinder is in motion and the outer cylinder 

is at rest nhrds nn cxnmplo of an ~inst~nhlo stratification caused by centrifugal forces. 

r 1 

.Lhe fluid pnrt,ic:lcs ncnr thc innor wall cxpericnce a higher centrifugal force ant1 show 

n tendency t,o I)rring proprllctl outwnrtls. '1'11~ st,abilit,y of this type of Row was first 

invrst.igat.cd I)y I,ortl Itzylcigh I 191 I who nssun~cd t,I~:rt the /hid wa.s non-viscou..~. 

Ila fount1 t,l~nt tho flow I)t:cornrs ~tnst~;~blo wl~rn tJlc j,eripl~cral vcl~cit~y, IL, tlcorcxsrx 

with th: ratlius, r, more strongly than 1 IT, t,l~at is, whcn 

const 

?L (r) = ---- -- \vit,l~ n > 1 (unstnl)lc) . 

r 

The case of n V~.~OIIS fhid was first, invest.ignt,cd in detail by G. I. Tnylor [I401 

who used tht? frnmc\vork of n lincnr throry for this purpose. When n certain ftcynoltls 

nun11)cr has 11rrn cxcrrtlrtl, t,l~rrc nltpcwr in t.hc flow vortices, now known as Taylor 

vort.icrs, whosr axrs nrc Iocntcd along the c.ircurnfrrencc and wl~ich rot,n.t.e in altrrnnl,rly 

opposilr clircct.ions. I'igrr~.t- 17.32 cont,nins n .whemnt,ic represcntnt,iol~ of t,l~is 

~not,ion whit:l~ is clrnract,rrizrd 1,y il~r I':act t,l~nt. tho nnnnlus Let.\vccn t,he two cylintlcrs 

is cornplct.cly fillctl by t,hcsc ring-lilir vort,ices. 'l'l~c contlit,ions for tl~c flow t.o 1)ccornc 

nnsta.l)le can ho expressed with t.hc nit1 of& cliarnctcrist.ic numl~er known ns t.he Taylor 

w?m~.bcr, T,, of t.11~ form 

wllrrr tl t1rnoCc.s t.11~ witlt,ll of t.11~ ga11, Ili tho innrr m.tli~~s, nntl I/, the pcriphcrnl 

vv1oc.il.y of l.11~ in~~rr c.ylitlt1t.r. (:. 1. 'I'iaylor's st,nl)ilit,y criterion is ill c.xrcllent ngrro-

528 XVII. Origin of turbulence I1 

IIICIII, \vitJ~ tncn~~~rctr~r~~l~~. 

This (::~n l)c infcrrotl very clearly from the pictures of snch 

'l':~.ylor vorl,icc,s ol)t:ainotl I J I?. ~ Schultz-(lrnnow and 11. IIein [204], several of whicll 

hnvo I1t:t:n rq~rotlnrcti in I'ig. 17.333. In their cxperimcntal arrangement in which the 

pip 11:t.tI I.lw tlimc~~sion or d .= 4 mm, and the inner radius was R, -- 21 mm, the 

vcwl.it~r.s :i.pp~:~~:d ;LI> :L ~wril)hcrnl vclority, I/(, which corrcspontls to n lboynoltls 

n~~tnlwr R (Ii d/v : .: !)4.5, Fig. 17.33:~. It, is t~ol~cworl,l~y t,h;~I, t,l~t: flnw rctn:iint:tI 

lihtni~t:ir :I(, L11t' IIIII(:~ I~igl~t:r ~bt:ynohIs I I I I I I I I ) ~ of ~ ~ R r--- 322 (T, --= 141) a11t1 R -:= 8(i8 

(T, =- M7), IGgs. 17.33 IJ, c. 'l'urbulcnt flow did not bccomc tlevelopctl until a Rcynoltls 

rli~mlm R -= :l!)fiO (T, = 1715) had bcct~ mncl~ctl, Pig. 17.33tl. It, should I)c sl,rcsscd 

rn~pl~n~.ir:~lly LhnL tshc first nppcnmncc of ncutral vortices n.t thc limit of stnl~ility 

in nocortlnncc wit,h cqn. (17.20) and thc pcrsistcnce of arnpliIict1 vorticos at higher 

Tnylor numl)rrs tlors not in any way imply that thc flow has bccome turbulent. On 

t.lw contrary, cvcn if the limit of st,ahility is exceeded by a large margin, the flow 

rcmains wt,Il ortlrrcd nntl Inminnr. Turbulent flow does not bccome developed until 

'I'nylor, ant1 t.l~rrcfort:, ltryr~oltls numbers vastly cxccctling the limit of st.al)ilit,y 

:~rr nl hit~ctl. 

.I. '1'. Sh~nrl, 12181 s~tc:c:twlcd in con~puting thc flow pattern of the unstal)lc 

Intnin:~r Ilow in l.11~ prcscnrc of Taylor vorticcs nntl with thc non-lincnr terms in 

(.llr r(luat.iot~ of' mo(,ion rctninctl. Ilc disrovnretl the sxist,once of equilibrium bctwccn 

Kg. 17.34. I'low hot\rsc:cn t.wo conoentxic rylintlrr.q: tor+io cocflicicnt for inner cylinder in t,rrms 

of t.hr 'I'nglor nnml)cr, T,,. 

f. Stability of a bo~inclary layer in Lho proscnm of tl~rcc-di~ncnniotlrbl tlisl,t~rl~:inws 52!) 

the transfcr of energy from the base flow to the sccondnry flow ant1 t.11~ viscous 

energy dissipation in t>hc secondary flow. The t,ransfcr of cncrgy fron~ the Onsc 

flow t,o thc secondary flow causes a Inrgc incrcnsc in thn torqrtc rcq~~irrd to roht,c 

t.l~(! inner cylinder. 7'hc diagram in Fig. 17.34 cotitair~s n compnrison I)ct,wrcn Ihc 

t.l~rorrt.icn.lly dcrivctl :inti thr cxpcrirntwt.ally mcns~trrcl v:~l~~rs of 1 h t.orcj~tc, roc*f'L 

ricnl, C,,. '1'11t- I:~l.tvr is clcfinwl as 

Xi 

C, = ------- - . - . . (17.21) 

R,~ 1' ' 

-4- n ~ ( 2 

wit,l~ 1~ as tthc l~cight, of I,hc cylintlcr. '1'11~ 1inc:tr l,l~(:or~l wil,l~ SLII:LII rrhtivt: g;111s, 

(//I(,, yields 

In :tcldit.ion to the? txtrvc wllic:h (:orrt~spot~l~ 10 IJiis lint.:t.r t.lt(!ory, IIII(I wl1i(41 I(-:I.(IH 

lo :L Ior(11tc t;ot:l'fit~it:t~l, (,'M . - 047/T,, Sor d/11', O.OW, lht* (li~~pxtn (:o~~t,;~it~:+ 

t,Iit: 

cllrvc provitlc:d 11y .J. '1'. SI,~~nrt,'s ~IOII-lirtwr t1~:or.y as wcll ILS on(: givt:tt l),y ;L I.Iwory 

lor turl)ulcnt flow; thc Intber leads to tho formul;~ tlial. Cnr - T,,-".2. 111 all, wt: may 

tlisccrn t,hrcc rcgirncs of Ilow, cnch circ:umscril)c:tl I)y 1.h~ 'l':~ylor r~~~rnltc:r in tho 

li)llowing way: 

T, < 41.3: laminar Coucttc flow, 

41.3 < T, < 400 : laminar llow with 'I'nylor vorl,ic:os, 

T, > 400 : lurl~tllor~t Ilow. 

Agrccmcnt bctwccn theory and cx~wrimrnt is cxecllrnt in thc first I.wo rangost. 

An extension of Taylor's thoory can bc found in a study hy Ii. Iiirchgarssner [IOG]. 

A detailed experimental investigation of Couettc Row, particularly in transition, 

was carried out in 1965 by 1) Colcs [291 

Ekct of an axinl velocity: The preceding stability calculations have been 

extended by 13. Ludwieg [I 32, 1331 to includc the case when the two ryl~ntlct s arc 

also axially displaced with respect to each other. Let u(r) denote the tangential 

velocity, and let w(r) denotc the axial velocity. If we now introduce the dimensionless 

velocity gradients 

- r dzl r dw 

u=-- and GI=--, 

u dr u dr 

wc can writc the stability criterion for n non-viscous fl~tid in the form 

t l'lic cxperirner~tnl msulkq displnycd in Fig. 17.34 dernonstrntc furtl~or that an increase in the 

Taylor number, that is, that an increase in tho lteynolrls number at a constant value of d/.R,, 

cansen n trflnnition from cellular to tnrhulcnt flow. Whcn thc flnw is tc~rI)nlcnt (1, > 400), 

wo have CM - Td-0.2. and I~cncc, nt constnnt d/Rt al~o CM w ((I, dlv)-o.Z - R (1 2. 'l'lm sarno 

~(:RIIIL WIW discov(:red IIY {I. It~it:J~nrtlI, ((201 in Cl~np. XI X) WIICII 110 ~t.ndi(:~I I h I:ILH(: or I~III::~~ 

Couette flow between flat parallol walls. It is remarkable that ll~c same dependence of the 

torqm coefficient on Iteynolds number exists for a disk rotating in a llnid at rcst, eqn. (21 30).

630 

XVII. Origin ot t.~~rhr~lrncc J l 

This ineqnnlity contnins Rayleigh's criterion from eqn. (17.19) as a special case 

ant1 rcs~~lt.s wherl 7o = 0 is ns~nmcd here; we then find that 1 + 5 > 0. The stability 

calcnlntion which led to eqn. (17.23) took into account disturbances which were 

not ~icccssnrily axially symmetric; the 1n.th.x turned out, to be the "moat dangerous" 

oms ant1 detcrminctl (he litnit of st.:~l)ilil.y implied by thc ineqrlnlity (17.23). l'ignro 

17.36 shows an example of an unstable flow which contains vortices in the shape 

of spirals. If. Ludwirg's tllcory has bccn compared withexpcrirnent.al resulk [134] 

in Fig. 17.36. Every bnse flow invcstigated experimentally is represented by a point 

in t,Ilc I;, 271 plane. The opcn ant1 full circles characterize stable and unstable flow, 

respectivciy, it being riotcd that vortkcs were observed for the latter. It is seen 

t.hn.t, IT. I,utlwicg's st.:~l)ilil.y crit,crion from rqn. (17.23) is fully confirmed hy cx- 

Fig. 17.37. R.nngen of Inwinnr nnd l,rrrl)~tlet~t, flow in n~~tii~luu I~C~IYC~II two concctltric. cyli11tlc.r~; 

innrr cylinricr rotntrs. outm cylitldrr nt. rn~I, in prmRltro of nxi:d flow: plot. in t.crr~lu of 'l'r~~lor IIII~II. 

I,t.rT,, 1111rl IIIIIIII#('). R.,; tlll~ll~11~l~llll'l1(H 1))' #I. I

632 

XVII. Origin of t~~rbr~lenco 11 f. Stability of n boundary layor in the prcacnco of thrcc-din~rnsionnl tlisturbnncc.~ 533 

d/N1 < 0.2. 'l'hc. out,c,r spl~crc was at rest, whereas the inner sphere rottat.etl. 'l'he 

charact.t-r of the Ilow in ~uch a spltcri~al annulus is also tlct.crn~incd by tho Taylor 

nurnlwr from rqn. (1 7.20) and t,hc Iteynoltls number formed with the annulus width, 

d, and I.hc poriphcral velocity, (II, that is by 

In t.11~ range of validit,y of linrar theory, that is bcforc t,hc appearanoc of Taylor 

vortices of the kind shown in Wig. 17.32, the torque noting on t,he inner sphere is 

with rlenoting t.11~ torqnc, and Rr the inner radius. 

Whereas in t.hc ~~rcretling msc with rot,at>ing conccnt,ric cylinders thc entire flow 

fidtl is either laminar or t.~~rl)~tlcnl,, dcprntling on t.hc valucs of t,he Taylor and Rcync~ltls 

nr~mI)ers, t,l~c: case of l.l~c- sphere is more complcx, bccause different flow regimes 

cat1 occur sirrl~~ll,at~ro~~sly sidc by sidc. AS t.hc Reynolds number is increasctl, Taylor 

vorticrs. and hence also t

XVII. Origin of t.urh~~lencc TI 

7 ! Alr:~s~~rwnet~t of tltr pni~~l. of I.r:~t~sition oil slighbly ronrnvr wnlln. nftrr H. \V. 

I . 

I,iq)i!~n.~tn 1 127. 12Hl; (n) cril ir:d 1Z.ryol1l~ IIIIIIII>~T 

. ~ 

q~~ar~t,it.y r1wf21r vcmm f:f 

b, - I~WICII~.II~ lI~I(.k~~(.ss: It - rndills OF rltrvn1.11rr OF wnll 

IJm 421, 

1' 

62 

vrrsus z: (b) t.ho oharactcriut~ic 

f. Stability of n hountlnry layer in t.ho ~roncnre of t,hrcc-tlitnct~uic>t~nl clist~~rl~n~~cm 535 

A very t,horor~gl~ cxpcritnc.nt~a1 invrstigntion or t.rnt~sitiiru nloug n ~.OIIV:IV(~. c.111.vrt1 

wnll was rrccnt,ly mt.ric!tl out. I)y 11. Ilippc~s 1 161 who cwployrtl tnotlrls tll.nggc.tl n.lo11~ 

a wnt.cr r1in11nc.l. 'I'l~esc, c~xltwit~~c~n(s l,l~t.o\v ligI~L on t,hc origin of lonKit 1ttli11:11 \.o~,t i(cj 

liltc those in I'ig l7.X 1). 111 t.llis ront~rsiorl stsr the: p:rpcLrs I)y I". X. 1Vol.t ttinlln I %%(iJ 

and 11. (:oert.lcr ntid 11. Ilnsslcr IH3J. 

,. .Il~e consitlrrnt.iot~s co~hit~cd in t,hc prcscr~t sec:t,ion togc?t.l~rr wit.11 those ill 

Chap. XVI ant1 Sees. XVll a, 11 lent1 to the follomit~g pivtctro of' t.mnsit,ion irt tllc 

bour~tlary layer of n solid body (c. g. nri n.r:rofoil); tm.n.silhi on flat, anel convex 

walls is governctl I)y tho itlst,i~l)ility of l,ra.vclling, t,\ro-(li~ncttsiot~n.l 'I'olllnion- 

Scl~lirl~t.ir~g waves wl~crc~s that on c:ono:ivc udls is povc:rnctl by the st.:il.ionnry 

Taylor-Gocrt.lcr vortices. 

Fig. 17.40. I%ot.ogrnph illtt~trntitlg 

t.mt~nit,ion in t.ho bountlnry layer on 

n disk rotnting in n fluid at rest after 

N. Crrgory, J. 'r. St.~tnrL ntd \V. S. 

Wrilltrr (771. 1)irrct.ion of rotntion iu 

ro~~r~tnr-clocka~iri; upaccl n = 3200rpm 

radius of disk = 15 om 

Stnlionnry vor1irr.r nrc rf!rn forming in an nnnlllnr 

rcgion or innrr radius lli = 8-7 cm End 

ouLer rn

536 

XVlJ. Origin of t,urhr~lcnce 11 

for which tlic tlrt,nils of the laminar laycr arc known from See. V b. A pliotograph 

iI11rst~rat~ing the prnt~css of transition on a rot-,ntirtg tlislr ant1 t8nkcn by N. Gregory, 

.J. '1'. Sttiart. rind W. S. Walker [77] is rcprotlucctl in Vig. 17.40. 7'11~ photograph shows 

t.hnt. in an nrtnnlar rcgion ttherc nppyr st,at.ioriary vort,iccs which assumc thc shape 

of log:trit,litnic spirals. 'l'ht: inricr ratllns of this region marks t,l~c? onset, of inst.nOilit,y 

:tntl f.rrlt~sit~ion oc:c:urs :el. I h oltt,t:r mtlills. 'l'lic: intwr r:~(Iius (:orr~spon(ls 1.0 :L ItcynnItIs 

nt~titl~rr of R, -1 ltt2 (L)/v = I.!) x 105at~tl ;et tlic outer radius we havc R, -- RO2 (01, = 

=. 2.8 x 10". ,J. 1'. Stt~art complcn~ontctl t,he cxpcrimcntal work with an analytic 

stmtly of thc stahilit,y of such a motion. In it,, IIC assumctl t.11~ existence of threetlimrnsiorial, 

pcriotlic dist~lrbanccs whose forms incl~ldcd 0.s special cases the progrrssing 

'I'olltnirn-Sclilicfitirlg waves as well as thc ~tat~ionary, t,l~rec-dirnensiol~d 

'I'aglor-Gorrtlcr vor1irc.s 'I'hc rcsults of his calculations sliowcd qrtalitntivc agreement 

with Ihc cxpcritnc~ntal results of Fig. 17 40. 

Anot,lier case of t,l~is kind occurs on a yawed flat, plate in supersonic flow when 

the :~ssorin.t,cti lantinnr lw~~ntlnry Inycr t)rcoinrs unst.at~le. As ~liown cxpcrirncntdly 

\,y .J. ,I. (:inoux 1841, t,hc \ IOIIIKI~I.~ hyrr tirvvtops ~ongit.w~ina~ vort.ic:rs which product: 

trnnsit ion. 

g. The i~~fluer~ce of rougl~rless on transition 

I. Introtl~~ctory remark. The prolhrn which we arc about to examine in this 

sec.t.ion, namcly the questhi of how t.hc process of transit,ion tlepentls on tllie roughness 

of t,hc solitl walls, is one of consitlcra1,lo practical importance; so-far, however, it 

has not. I)cen possiblc to annlgzc it t.ltcwrct,icnlly, Tllc prolA:m untlcr cot~sidcmtion 

has g:i.inctl in irn~mrt.anc:e in the rcccnt ~~ast,, ~~artic~~larly sincc thc adventl of laminar 

arrofds in acronaubic;d applications. 7'hc vcry cxtcnsive cxpcrimcntal material 

nolh:ot,cd np 1.0 d:'.t,c incl~drs information on thc efrcct of cylindrical (two-dimensional 

roughness elcmc~it~s), point-like (t,hrec-dirncnsional, single roughness clcn~cnts) and 

clistril~ut~ccl rorlghncss clcmcnt~s. Many of the investigations include addit,ionnl data 

on t,ltc: inflt~crtcc of pressure gratlicnts, t,~~rbulcncc intensity or Mach number. 

Generally spearking, the prescncc of rougllncss favours transition in tl~c. sense 

t,liat utltlcr oblicrwisc itlcnt,ical cordit.ions transition occurs at a lower lteynolds 

nt~nibcr on a rough wall Llian on n smooth wall. That t,liis should bc so follows clearly 

from t,lic I,llcory of stability: ti~~_e~i?tencc of roughncss elements gives rise to arldit,ional 

tlistur\)anccs in tho laminar stream which have to bc atltlcd to those gcncratG4 

by t~~trI~~~Ien(~~ 

ant1 alrcady present in the boundary layer. If the disturbances creatctl 

by roughncss arc bigger tiinn those due t.o turbu~cncc, we ijiiistckji+ !,h

638 XVII. Origin of t~~~rbnlc~~rr II 

As the Ilright k is incrc:rsctl, the position of Lhe point of tjrarisit1ion ztr moves closer 

to tllc ronghncss clcn~cnt, which nlcnns tht tlic curves in Fig. 17.41 arc trnversctl 

froni left to right. The expcritncnl,nl points begin to tlcvint,e from this curve upwnrds 

as soon ns the point of tmnsit.ion has rcncl~ctl tho rougllness element, i. e. when x,, -. x,. 

Thy t.hnn lie nlorig tho fnrnily of stmight lines which contain r,/k as a parameter 

and is given by 

Pig. 17.41. 'I'l~r rri1,ic:nl Ikynolcln nulnbrr for lntninnr Im~~~clnry Iayrr nn o fitnc:Lion of t,l~r rnlio 

of t~riglil, k nfn~npl~nr~n elcincwL 14, tlltr dioplnrcntcnl, I I~icknracr of I lw I)onnclnry l~~ycr nl. Ihe poniI.ion 

of f.lir rongl~nms rlrtncnt,, dl,. for single, t\vo-~Ii~~~r.~~siotid ro~~gliness rle~nc~~it,s in inron~prr~~~iI)I~ 

fl0\\. 

Tlw ~w:~~t~rrtn~~nIs 

wv ~:~li~C:~vl~~rlly inlrr~~~~lnl~~d 1,s rqn (17.28) 

t 

.- 

[I* 

it is nlso RIIOWII in Fig. 17.41. According t,n Jnp:tncse ~nonsuremcnls [237), the 

l~ypcrbole-like branch of the curves in Fig. 17.41 possesses 11nivers:~l vnlitlity, 

both for flows with cliKcrent, weak pressure gmtlicnt,~, nnd with different inlensitics 

of turbulence. Increased turhnlence eauscs merely an enrlicr deviation of the cnrve to 

the left,, in the direction of the tnrb~rlc~~cc-tlepet~(let~t crit,ionl IEeynoltls nuinbcr of 

, 

IIRVC Itvl I

540 XVII. Origin of turhulonce 1 I g. The influence of roughness on transition 54 1 

'I'ltc rorrrspontling curvo is rcprcscntcd in Fig. 17.42. 

According t,o TI. 1,. 1)rydcn [30, 401, it is possible to take into account the 

vnrintion in tjllc t,nrl~ulcncc intensity by plotting the ratio of the critical Reynolds 

nl~rnlwr for n rough wall t,o that for n smooth wall, namely (R,t,),o,,h/(R,tr),mOolh, 

:I.S n. fnnction of 1;/Olk, TGg. 17.43. When j~lottcd in this sgstcm of coorclinatcs, the 

r(w11l,s of tn(:rls~~rcrncnts wit,ll diffcrcnt itltcnsitics of tmrI)ulcncc fall on a single 

cnrvc which mcn.ns that, the ratio (R,t,)ro,,,/(R,l,),monlh is a function of tho single paranlct.cr 

k/R1,. Tht: t,llrcc questions posctl at thc cnd of the last section can now be 

easily nnswcrctl wit,lt t,lre nit1 of t,l~c t.hrrc graphs of Pigs. 17.41, 17.42 and 17.43. 

Very tlcl.n.ilrd cxperimcnt,~ concerning the influence of n two-dimensional, discrete 

roughness elcmcnt (wirc) on transition wcre performed reccnt,ly by P.S. IClebanoff 

ant1 I

\vi(,l~ a (-ylit~(lw envcrv(1 wiI.11 sit.11rrs, 

l.11~ S:IIII(* is svrtt 1.0 be t.r~~c :I,IK)II~, 1110 l):~r:~l)olic vcIo(~ity profilrs it1 a, pipe. 'I%r sn,t~~c 

conclrrsion w:ls rc~nel~otl by (:. R.1. C'otw)s nntl .I. 12. ScIInra ( I HI, by C. I,. I'c*l

544 XVII. Origin of t~~rbulcnco 11 

The rnrn.surctl jmints represent obscrved neutral disturbance vortices at the 

Im~mlary bet~wccti clamping and amplification. The agreement between theory and 

expcrinwrit is vcry good. 'l'hr thcory confirms the supposition that small velocity 

cwtnlwnrnt.~ in 1.t~ t,n,ngrnl,inl tlircctions cause Ilagen-l'oisenille flow to becomc unstal)le. 

,I. Ilot,l.:~. whosc~ work was disrus\qcd in (lcl.:~il in Scc. XVIa, performed rnoasurcn1t~nt.s 

011 (,IIo it~I.c\rttiit~t,ct~~y f:tctor of lwgc tlisLurbances propagdcd downstream 

in t.hr inlrt, sc~c:f,iott of :L pipe. Similar cxpcrinicnt,~ were performod by E. It. Lindp 

~ 11.701 n who rnaclr I,hc tlisturl)ance visihlc by the use of polarized light and a 

I)~-~.c~f~.ii~gctit,, wcalc solution of bcntonitc. IC. R. Lindgrcn was ablc to show that 

cvrrl strong initial disturbances decay in the inlet length when the Reynolds 

number of the flow (based on the pipe diameter) is small. At Reynolds numbers 

from nlm~t R = 2600 npwards the process of transition begins. It is characterized 

by an amplification of the initial disturbances and by the appearance of self-sustaining 

t~~rbulrnt flashes which emanate from fluid layers near the wall along the tubc. 

Thc preceding peculi:~ritics of laminar flows through pipes forcc us to re-considcr 

thc relation between the theory of small clisturbances and txansition and, in particular, 

to pose tJic question as to whether transition can alwmys be said to be duc 

to an nmplificat,ion of sn~nll disturbances. No conclusive answer to this question can 

at pr(:scnt be given without further work on the behaviour of small, three-dimensional 

disturbances. In this connexion it should also be remcmbercd that the limit 

of stability for plane Poiseuillc flow which lies at R, = 5314 as stated on p. 480, 

c:onsidcrably exceeds the critical Reynolds number for transition observed in 

cl~nnncls. This is inconsistent with the theory which asserts that t h limit of stability 

must always occur at a lower Reynolds number than transithn itself. However, 

at the prcsent, stage of knowledge, and in the face of tho present interest in the 

subject,, juclgerncnt must be reserved until further results become available. 

The stability of a laminar boundary layer on a body of revolution was also 

invrstipt~ctl I)y ,I. I't~t~cll [17C,J; in t,his coi~ncxion consult a paprr by 1'. S. Granville 

1821. 111 raws whew t,lir rat,io of' bonntlatylaycr t.hiclrncss to curvature is vcry small 

c:o~nl):irwl n.il.11 unit,y, the rcsult,ing st.n.l)ilit,y cquntion for t.hr axially symmetrical 

raw I)rron~cs itlvnt,ical wit.li t,l~at. for thc I)l:~nc case. llcnc:r, all rrsults obt.nincd for 

the Iatt.cr rnn bc extrntlctl to apply to the former withoi~t, rcsrrvation. 

111 Ahbolt. J.11.. van 1)ornlwK A.15., and Stivrrs, L.S.: Sltrnrnary of airfoil daL7. NACA 

L 3 

Il.cp. ~ 24 (1954). 

[2] Althn~~n. I>.: Sl,ut,tgnrt.cr Profilk:~t.alog. Inst. Acrodynamil~ of SI,tlttprt Univ. (19721. 

1:)) ARC Ilk1 24!)!): 'l'riinsi t.ion and thg rnrasnrct11c.nt8 on the 1lonll.ot1 I'xrtl sample of Inminnr 

flow wing c.on.rt,ruc:t.ion. Pnrt I: by .J.H. I'rcston nnd N. Gregory; J'art 11: by K.W. 

I< imlwr; l':~rt, I I I : .Joint, I>iscnssion. 

[:!:\I I~r:tdcy. .l. A,: (!:~lc~~l:lt.inrl of Lhc Inrninnr I,ountl:+ry Iayor nnfl prctlict.ion of (.rannition 

on n shrnrctl wing. AM! Ibhl 3787 (1976): ItAI': 'I'R-7:)150 (1!374). 

[4) I ~jnt~~in, 'l'. IS.: lI:lTc(:ts of IL flexible bc~tntrlary on l~ydrotl~nntnic ntnbilily. JFM 9, 513- 

.Kt2 1 l!Mi I \ 

[7] Bradow, A. I,., and Visconti, F.: lnvest,igation of bonnd;rry Iaycr Rrynoltls nrrtnl)cr for 

transition on an NACA 05(215)- 114 airfoil in tho 1m)glcy two-dir11c11nior1111 low-t.t~rhu- 

Icncc prrsnllre tunnel. NACA TN 1704 (1948). 

[8) Ikinith. P.F.: Ro~tt~dary layer transit,io" at ~nch 3.12 wil.11 rind \vil.l~o~~f nittRlr ~ OII~III~U.SS 

elcrncnt. NACA TN 3267 (1954). 

[9] I3ussmann, K., and Miinz,H.: i>io Stsbilitiit der Iarninarrl~ Itc~ib~t~~gxnc~hicht ],lit 1\1J4:111 - 

gnng. .Ib. dl.. I,rtftfahrtforsol~r~ng I, 36 -- 39 (1942). 

[lOl I~llaslrlanrl, I

646 XVII. Origin of turbulence TI 

Frrnkiel, 1'. N., lanrlnl~l, Al. 'r., rind I,~~mlny, I,.: St.rnct,nrc of t,url)ulrnrr an0 c11-;1g r~duc.tion. 

ILJ'l'Ahl S~IIIII.. Wnsl~ingtm, I). C.. 7 - 12 .JIIIIC 107fi. 'l'l~r I%ysirs of I~'l~~i(ls 20, 

No. 10, I'orl, 11. p. S I ~ - 2!)2, S l!)77; :ilw I$. A. '~'II~IIII in l'rnv. III~WII. ('o~~grrss ~ I I

M8 

XVIT. Origin of t.nrhltlrncc II 

. . 

lag, Mninz, I00 - 1!)5, l!)W 

[!)I] Iliggins, IL.\Y., and Pnppnn, C.C.: An c.uprri~~~c~t;~l invcsLigalion of thr rfTrct. of snrftccc 

l~c~tine on lmrtntlnrv " Invrr . tr:rnsit.ion on n f1:ct plntc in snprrsonic flow. M;\(:rl 'l'N 2351 

(l!)51):' 

[!I21 Ilols1,cin. (1.: hlnssnngen zur I,:~~ninarl~:~lt~tng tlor lZc:il~rt~~gsscl~irl~t,. I.ilict~t.hel-llrric.l~t S 10 

17- 27 (1!)40). 

(931 tlrtnng, I,. hl., nnd ('hon, T.S.: Stnl~ility of tlrveloping pipe flow snbjrctrtl t,o no11-nxisynltnrtricnl 

rlistnrl)ancrn. .I lchl 8.1, 183 - I!):! (1!174), scr also 1'11yn. Iflnicls 17, 245- -247 (1!)74). 

1!)4] Van Ingrn, J. I,.: A snggrst.cc1 swni-rntpirict~l ~nrthorl for the ralculntion of t,hc bonndery 

Inver transit.iot~ region. 'l'echn. Univ. Uo1). of :\eronnutics. I)elft,. Report V.T. H. 74 (1956). 

19.51 Jnrk, .I. Jt., and IXaronis, N.S.: \':rriation of bonntlary-lngrr transition with heat tmnsfrr 

on Lwo bodies of revolntion at a Mach nntnhcr of 3.12. NACA TN 3562 (IDRR). 

[90] .Iacol~s, 13. N., ru~l Sl~crn~rcn, A\.: Airfoil nc:ction rl~~~r:rrLrriuLirs as rtnbctr.tl I'g vnri~ctionn 

of the ILeynoltlnnun~bcr. NACA TJt 5Mi (1!)37). 

[97] Jcffrcys, 11.: 'l'hc insLabilit.y of a Iagcr of flnid hentcd below. Phil. Mag. 2, 83:i-844 (1926); 

see also I'roc. Jtoy. Scc. A 118, 1!)5-208 (l!Y28). 

1081 Jones, 13. hl. : I?light cxpcritncnL~ on tl~c bonndnry Inyer. \\Iriglrt Brctl~ers Lect.ure. JAS 

.5, 81- 102 (19:IIl); also Aircrnft 1':ng. 10, 135--141 (1938). 

1991 Jones, 13. M., nnd Head, M. R.: Tho reduction of drag by distributed suction. Proc. Third 

Anglo-An~erican Aero. Conference, llrighton 109-2330 (1951). 

[I001 Jack, .J. It., and I)iaconis, N. S.: VarinLiott of bo~tntlery-layer t.ransition wit.h heat transfer 

on two hodics of rcvnl~~bion at a Marh nnrnlrcr of 3.12. NACA TN 3562 (1955). 

[I011 Jack, ,I. It., \Yisnic\vski, Jb. .I., and I)iaconis, N. S. : Effcctn of extrcn~c surface cooling on 

I)ortndnry Ingcr t.ransitiott. NACA TN 40!)4 (1!)57). 

[102] Jillir, l).Mr., ant1 Ilopkins, E.J.: 15fli~t.s of hfaclt-nnn~ber. Icading-edge blunt,ncss and 

swrep on honnctnr,y-layc:r transiLiotl nn rr. flat. plnto. NASA 'I'N 1)-1071 (1M1). 

[103] Jonrs. W. P., llnd I,nu~~tlcr. 13. I

550 

X\'ll. Origin of t,11rh111rnrc TI Itefcrences 55 1 

(I:18\ I,:~nirr. ,I., nntl Mnrlc, .I. I:.: lksrrltn nwl n criticnl c\iscrrssion of trnnsition-1loy11olcls- 

IIIIIII~)(:I. 1nvil~111rr1nents on insul:~trd rows nntl llnt plates in st~personir n.incl tunnrln. Jet, 

Propnlsion I.:tI)., I'nsntlcmn, (hlif.. Rep. 20--!I0 (I!J55). 

[I391 I,nufcr, J .. n.nd Vrrlmloviclr, 'I'.: Stithili1.y r~nrl 1.rnnnition of a s~~prrsonic: Inn~innr I)oundnry 

Inycv on :i flirt. plntc!. .lIVkl 9, 257- 2!)!) (I!)liO). 

11401 I,&R, L.: 1'110 nt.:tl,iliLy of t,l~c I:uninnr 1)01111dnry Inyrr in n con~prwsil)lc flow. NI\C:II TN 

I:%(iO (1947) and NI\(!I\ I

552 

XVII. Origin of tnrhnlonco TI 

Tin41 L-..., ltr*shot,ko. --.. ~. IC.: St.abilit,v I,hcorv na a mido " to the evaluation ot t.rnnsition data. AJAA J. 7, 

1086-- lO!)l (l!)69). 

ll94nJ Itesl~otko, I

554 XVII. Origin of t,rlrbuloncc TI 

[23HhJ Tnni, 1.. 1111~1 S:~tn, 11.: lloitndnry Inyrr t,ra~inition hy roupliness elemenk. J. Phys. Soc. 

Jnpn 11, 1284.- 12!)1 (IM(i): src also IXr Congrco Internntionnl de Mi.canique AppliquCe, 

AcLcn. 11'. RG93 (1057). 

12391 Taylor, O. I.: Internnl waves and tnrhulencc in n flnirl of vnrinhle drnsity. Rnpp. Proc. 

Verb. Cons. Jnternat. your 1'l':xploration dc la Mcr. LXXVJ Copenlingen, 35-42 (1931). 

12401 Tnylor, G.I.: Ell'ects of vnrialion in tlennity on the stabilit,y of superposed streams of fluid. 

I'roc. Roy. Soc. A 132, 49!)-523 (1931). 

12411 Taglor, (:. I.: Stability of a vi~rons liqnid e.olit,nined het.wcen two rofnting cylintlers. Pliil. 

Trans. A 223, 28IIb343 (1923); scc also l'roc. Roy. Soc. A 1.51, 4!)4-512 (1035) and 157, 

540---A64 nntl 565-578 (I 936). 

[242] l'l~roclorsen. 'r.. and Gnrrirk, J.: (2cncrnl pot,cnt,inl thory of ztrhit,rnry wing section. NACt\ 

TI< 4V2 ( l!X33). 

\ , 

12431 Ulrirl~. A,: 'I'hcorrtinrlic Utit~cr~~lcllilligrrl iihcr die \Vitlrrwtnntl~crs~inr11in cl~lrrh 1,nniinnr- 

Iin.lt~tng lnit, ~\bsnilg~~ng. Nrllriftm (10. Altncl. (1. Li~ft,fnlirt,forrrcl~~i~~g 8 13, No. 2 (1!)44). 

12441 \\'cntlf, L".: 'l'r~rl~nlcntc St,riin~r~l~g z\visrheu zwci rot.iornllclon konxinlc~i %yli~~d~rr~. 1)i48. 

(:iitt.ingcn 1!)34. I ng.-

6M XVI I I. Pundemcnt~nls of turbolent flow b. Mean motion nnd Il~~rt~~ntiot~s 557 

Fig. 18. l a. C:&nlcrn vcloril.y 12.1 5 ctn/scc 

- -- 

Kg. IX. I 11. C:~n~or:t vr:lority 25 c~n/scc 

Fig. 18.1 rl. Cntncr:t vclocit.y 27.6 cnl/san 

Tn following this path it Itas at lonst provctl possil~lc t.o ~st,nl)lisl~ rmt.ait~ tllcoretical 

principlcs which a.llow 11s t,o int,rocl~~cc n 1nr:lsuro of ortlcr into the exprritnct~t~al 

mnterinl. Morcovcr in ninny cnscs it provctl possible to prctliot, t(11cw tntwn v:~lllc.s 

untlcr t,llc nssurnpt,ion of ccrtnin plat~siblo I~gpotllcscs nntl so to ol)t.nin gootl nglwment 

wit,ll experiment. The following clixpt,c~s will givc an nncount of such a srlniempirical 

t,licory of tnrbrrlent flowt. 

b. Mean motion n d fluctuntiot~s 

Upon closc invcst,igntkn it appears t.llnt, t.Ile most, striking f~::it,urc: of t~~rl)~rl(*nt, 

motion consist,s in the fact tlmt the velocity :lntl prossllrc :~t n fixctl poit~t. in s1,:ic.c 

do not remain const.nnt with tirnc but, pcrforni vcyy irrrgulnr fl~rc:t.~t:~t~ions of high 

frequency (scc Fig. 16.17). Thc lumps of fl~~itl whic:h pc:rfnrnl srrc:lt fl~tc:tu:~tions ill t.l~c 

tlirect,ion of flow and nt right a.nglcs to it, do not, co~isist. of single 11101~~ct11t:s :IS ~ISSIIIII(-~I 

in the: Itincl,ic theory of gases; they nrc tnnc:roscol)ic Iluitl I):dIs of varyi~~g sn1:111 S~XI'. 

It nmy be notctl, by way of cxnmplc, t.l~nt~ nlt.l1011g11 t,I~t: vcIocit8y Il~~(.t,~~:~t,iori ill 

ch:annel flow does not exceed scvcral per cent.., it, ncvcrt,hclcss has n drrisivo inllucnc:c? 

on thc whole course of the motmion. The flr~ct,l~:~l.ions rrntlcr cor~sitlrt.:~t.iot~ nlily IIC 

visr~nlizetl by realizing t,llnt, ccrtnin bigger port.ions, of t,hc fluid h:~vc: t,hcir own int.rinsic 

nmtion which is supcritnposctl on t,hc 1n:1i11 flow. S1tc.11 /hid 1~111s or 11i111ps :~rc! 

c:learly visihle in t,he pl~ol~ogmphs, Pigs. 18.1 11, c, (1. 'I'llc: sixc of sr1c.11 lluitl I):~lls, 

which c.ont,inunlly ngglornemt,o i d disint,(yy:it,(:, (It+ml~invs 1.11~ nrule o/ I~o.I~IPIIcP; 

thcir size is tlct,crtn~nccl by the cxt,crn:ll c.ontlit,ions :~ssoc*i:~.tt:tl wiI.11 1.11~ flow, t11:lt is, 

for example, by the rncsl~ of n screcn or Ilot~cgcolnl~ t,l~rortgh whirll tJ11: st,rc:~tn l~ntl 

p:~ssctl. Scvernl qunnt.itnt.ivc nienslrrcniont.~ of t.lic mngniI,~~tlcs :~ssoc-i:~t.c~l wi1.h s~tcl~ 

fl~rc:t,rt:it.ions will be givon in Scc. XVl l l (I. 

-- - -- - - . -- 

t Scvcral workers, in pnrt.icnlnr J. hl. Jhrgrrs, 'I'll. ~on Ji:irn~;in :ttlql (:. I. 'l'nylnr cl11i1c c.:u.ly 

dcvclopcd a tllcory wllich cxceetls t.llrsn li~nik nncl wllioh is I)nsrtl on sl.nl.ist,ic:nl ~w~rt.pl.s. 

JIowcvrr, (his I.llcory has not so far I)cctl nblc 1.0 solvc 1.11r ~IIII~~:IIII~II~:I~ 

1~t11)1r111 t~lwlir~~~rtl 

cnrlicr. We do noL propose l,n consi~lcr I.llis sl:ttidit~:tl tlwory of l~~~rl~~~lrt~rr- 

in t 111% rwt1:1i11111\r 

of I h 1)ook IWII rrl'kr I,ltc rtwlor lo 1110 ~~o~~~~~rr-l~rt~~ivo 

tvvirw~ Ivy ti. I

558 

XVITI. Fundamentals of tr~rbulelit flow 

Tt, has already bcrn pointed out, in Chap. XVI that in describing a turbulent 

flow in mathcrnatiml terms it is convenient to separate it into a mean motion and 

into a flududion, or eddying motion,. Denoting the time-average of the wcomponent 

of vrlo~ity by 12 and its velocity of fluctuation by 7~', we can write clown the following 

rrlations for the vrlority components and pressure 

as indirntcd in eqn. (16.2). Whcn the tmrlwlent. stream is comprrssil~lc (Chap. XXTII), 

it is necrssnry to inclutlc ll~~ctr~ations in tho tlonsity, Q, and in the temperature, 

T and to put - 

Q z 3 1- Q' ; 7' 2' 4- yl' . (18. 10, f) 

'rhc time-averages are formed at a fixed point in space and are given, e. g., by 

In this connrxion it is understood that the mean values arc taken over a sufficiently 

long int.crval of timc, tl, for thcm to be complet,cly inclepentlcnt of time. Thus, by 

clcfiniLion, the t.imc-avcmgcs of all q~~antit~ics describing the fluctuations are eqnnl 

to zero : 

- - - - - - 

u' =O. , v' =O; w' =O; p' =O; Q' = 0; T' = 0 . (18.3) 

The fcc?n.t~~re which is of f~~ndamcnt,al import,ance for tho course of turb~~lent motion 

consists in the circumstnncc that the flnctmations u', v', 1u' influence the mean motion 

7Z, fi, 1% in s11c11 a way that the latter exhibits an apparent increase in the rcsistancc 

to tloform:~tion. 111 other words thc presence of fluctuations manifests itself in an 

apparent, incrcasc in the viscosity of thc funtlamentnl flow. This illcreased rcpp.renl 

visco.sit?/ of tlic mean st.mnm forms thc central conccpt of all t,l~eorctical considerations 

of tnrl~~tlont mot,ion. We shall begin, therefore, by endcnvouring to obtain 

R c:loscr ir~siglil~ into these relations. 

11, is uscful Lo list here soveral rnlcs of operating on mein time-averages, as they 

will be rc:qnirctl for rcli:rcncc. If / and g arc t,wo dependent variables wliosc rileall 

v:tIurs arc to Ijc li)rrnctl :11i(1 if s t11:nnk:s :III~ 0110 of the indcpcnd~:nt vn.ri:~ldcs s, y, 

z, 1 lhrn tl~c following rules :~pply: 

c. Additional, "appnrent" turbulent stresses 

Before tlctlucing the ml:~t,ion lmtwccn the mean motion :mtl tltc: np[)arc:nI, st,rosscs 

caused by the fluctuations we shall give a physical explanation which will illustrate: 

their occurrence. The argument will be based on the momcnt~~rn theoron. 

Let us now consider an elementary arca dA in a turblllcnt stream wliose vclocity 

components are u, rt, w. The normal to tho area is imaginctf parallel to the 2-rrxis 

arid thc directions y and z aro in the plan0 of dA. The mass of fluid passing tJ1ro11g11 

this area in timc clt is given by CIA . ~u - dl and h011c:c the flux of ~nomc:nt.urn in tlic 

z-direction is cIJ, dA . Q u2 dl; correspondingly the fluxcs in thc ?/ and z-tlirockions 

arc (1.1, = tlA - p n 77 . tll and (1.1~ = tlA . p id io - dl, rc:sl)cc:l.ivc:ly. Ilt:mcn~l~cri~~~ 

I h l , 

thc tlcnsit.y is co~~stmit we can caIc111:~bo the following tinlc-:~vcr:~gcs for tlic lluxcs of 

momcnturn per unit time: 

By qn. (18.1) we find that, e. g., 

applying the rules in eqns. (18.3) and (18.4) wc find that 

awl that, similarly, 

u2 = us A- ~ ', 2 

Iience, the exprrssions for the momentum flnxcs per unit timc become 

7'11esc q~~antitics, dcnoting I.ho ratc of chango of momcntum, Imvc the clirncnsioii of 

forces on t,he elementary area cfA, and npon dividing by it we obtnin forces pcr unit 

area, i. e. stresses. Since the flux of momentum pcr unit timc through an arca is 

always oqnivalcnt, to an cqunl and opposih forec excrtcd on tlra arcs hy the surrountlings, 

we conclndc that the arca undcr considcmtion, which is nor~nal to - the 

s-axis, is act,cd upon by the strcsscs - e (iiz + p) in the 2-direction, - p (ii 6 -1- u' v') 

in 1.11~ y-~lirt~t:t,ion am1 - p(Q tZ -1 IL' IIJ') in tho z-dire~:tion. Tho li~,sl, of Ih: I,l~rtx: is 

a nornlal s1,ress nnd the I:bt,ter two are sitraring strrssw. It is t.1111~ scrn titat thc 

s~~prrposit~iori of flnct,n:blhns on the mc:m motion givcs rise 10 three atltlitional 

strrsscs 

- - - 

ozr = - Q u'2 ; tyzr = - Q U' V' ; t,,' = - Q 11' w' (18.5) 

acting on the rlemrntary snrface. T11c.y arc tcrmctl "app:~rc~~t" or It?c~no1& stres~es 

of t~~rbulrnt flow and nlust be adtlccl to thc strcsscs caused by tho steady flow as 

explained earlier in conncxion with laminar flow. Corrcsponcling rxprcssions apply 

in the case of rlcmcntary areas normal to the two remaining axcs y ant1 z. 'l'l~ry

560 

XVIII. P~tndan~cnlal~ of turbulent flow 

lorm topctllrr a c*omplrte ~trtss tensor oi l~~rbulenl /low. Eql~ationst (1 8.6) were first 

tlctlncrtl 1)y 0. ltcy~oltls 1431 from the equations of motion of fluid dynamics (see 

also the next srct ion). 

It, is easy 1.0 vis~~alizc - that f,he time-averages of thc mixed products of velocity 

Il~~ct.u:sl.ions, SII(~I as C. g. TI,' 11' - do, in fact,, diffcr from zero. The stmss component 

a,,' - T,,' -- - - Q 11' 11' c:~n he int,erprct,cd n.s the t,ransport of z-morncntfnm through 

a surfacc 11or1nn.l t,o tho ?/-axis. Consitlcring. for example, a mean flow given by 

IZ == I:(?/), Ij --- 221 --- 0 with t14/dy > 0, tPig. 18.2, we can see that the mean product 

16' 11' is tlilTcront. I'rom zero: 'l'llc pnrl,iclcs wl~ich travrl upwards in view of the trrrbulcnt, 

Fig. 18.2. Transport of momentum due 

to torbrdent velocity flrrctuation 

- 

ottly tlilli:ro~l, from 7,cro l~t, also nrg:l.l,ivc:. 'l'llc shearing stmss a,,' = - p 11,' 11' is 

~)osit,ivct in tllis c.:~.sc nntl hns tJ~e snmc sign as the rrlcvnt~t Inrninnr sllcaril~g stress 

T, -- 16 tl~i/tl?y. 'l'llis fact, is nlso csprcssctl by sl,a.l,ing that thcrc exists a co~wlnlion 

I)c*t,wcotl 1.l1o longit.n(linal and t~mt~svcrsc Iluctuation of vc4ocit,y at a givcn point. 

cl. nrrivntion of rlw ~trrss trrisor of nppnrctlt tt~rbrrlcnt friction 

from thc Nnvic-r-Stokcs cqrrntior~s 

ll;~\,it~g ill~~sl.r:~(.c(l t.11~ origin of t,I~r n,~l~lit,i(~~~:~l I'nrccs ca~tsc(I l>y l~~trl~nlc~~l~ 

Ilt~c-(.~~:~l.ion wil.lt l,11(: :lit1 of :I, pltYsic:~l :trg~trnc:nt wo sh:~,ll ttow pt.o(:(~~~I 1.0 tl(*rivc 

tit(- snlllc rxl)rrssion in :I. tnoro forn1:~l ~ 1.y :~11(1 dirrd.1y from t.llr Navicr-SI,oIzcs 

cvl~~:~tio~~s. 'i'lw ol,jwl. of 111~ sncxwxlirtg :I~~IIIIICQ~, is 1.0 (Icrivc thn cqun.l.ions of 

~noliou \vhic41 tnr~sl, Ilr sn.l.is~ircl 1)y 1.11~ tin~c-avcrt~grs of t.llc vclocit,y com~~onc~~t.s 

ii, i;, 171 :111tl of tlw pr(.ss~~rr p. '1'11~ Nnvi(~r-Sl,ok(~s cclu:~I.ions (3.32) for incompressible 

llow WII I)(% r~x\vril,k~~ in tltc, rcu-111 

wltcrc V2 de~~ot~cs 1,apl:wc's oprat,or. Wc IIOW in troducc t.hc 11y pol.l~cscs t.rg:~tdil~g 

the decomposit.ion of velocity componcnt,~ and prcssnrc int,o tlwir tirne-avcr:~grs 

and fl~~ct,~lat.ion tmrns from cqn. (18.1) antl form tirnc-averagcs in tPhc rcwtll,iltg 

cqunt,ionst t,erln by tcrrn, t:lking inLo aec.onnt thc rult,s from cqn. (18.4). Sincsc 

a?/az = 0 et,c. t.hc equation of continuity bccomcs 

From cqns. (18.7) nntl (18.6~1) we obtain also thaL 

It is seen that the time-averaged vclocity components and tl~c fluctu:lt.ing coniponents 

each satisfy the incompressible eqnat.ion of cont,in~~it~y. 

Tntroducing tJle assumpl,ions from cqn. (18.1) ido the rcl~t:~t,ion.s of tnot.iolt 

(18.0a, b, c) we obtain expressions similar to those givcn in 1.lln ~)rccctling scv:t.ion. 

Upon forming averages antl considering the rules in cqn. (1 8.4) it is not,icctl t,l~nt. the 

quadmt.ic tcrn~s in thc menti values rernn.in unalt.crctl 1)ccausc l,l~ry :~rc :~Irc:t~ly 

const,ant in t.imc. l'hc tcrnls which arc lincar in the Lurb~~lcnt con~poncnts sud1 

as c. g. ad/at and a2u'/ax2 vanish in view of cqn. (18.3). 'rho sntnc is true of t.1~: 

nlixetl t,crrns such as c. g. G . IL', bllt t.hc cl~~adr:~t.ic tcrtns in t h llirc:l,n:~l.itt:! c:o~tl- 

-- 

ponents remain in the eqnat,ions. Upon averaging they assume t,lrc form 1.'" 71,' v' ctc. 

llrncc, if the averaging process is carried out on cqns. (18.0), ant1 if silti~)lific~:~(io~ls 

arising from the continuit,y equat.ion (18.7) arc int.rod~~rc*tl, the follo\~ir~g syslctn 

of eqnatlions rcsults 

r 7 

.I 11c cl~~a(lrnl,ic terms in tnrlmlcnt vcloc:;l,y cotnpo~tr~~t.s It:~vo l)ecl~l I,r:~nsri~~wtl lo 

th: rigl~t,-I~:~~n(l si(1o for :I, r ~~~son whid~ will soott I)(WIII(: :I~I)IIIWI~,. ICeps. (1S.S) 

togctl~rr with t,l~e ccl~~:~I.ion nf cont,in~~if~y, (Y~II. (18.7), (I(~l.~wtlit~c 1.110 11r01~1(-111 t11111(.r 

consitlcralion. 'I'hc Icftr-Irnnd sitlcs of ccpls. (18.8) arc l'~~rtn:rlly itlcnt.icd will1 (111: 

steady-state Nnvier-Stokes equations (3.92), if 1,llc vc1ocil.y con~ponrnl.~ I(,, v, 111 arc

562 XVlI1. R~nclnmont.nIu of tur1)ulant flow 

replacctl by kheir time-averages, and the samc is true of the pressure mtl friction 

f.crms on tho rigtit-hand siclc. In atltlition the oq~lations contain terms which tlcpcnd 

on Llie t~~rl~nlorit fl~~cti~ation of the slrcarn. 

Comparing eqns. (18.8) with cqns. (3.11) it is secn that tlie additional t,erms 

on t,lic right-liand siclc of cqns. (18.8) mtn be int~rpmtcd as componcnts of n, strcss 

tcnsor. I3y eqn. (3.lOa) the rcsc~lf,:rnt snfi.~e: force per unit. :m:a dw to 1,hc n.tltlitional 

tcrnts is s(:m Im I)(: 

Carrying tho analogy with cqns. (3.11) stmill filrtlior we can rewritc eqns. (18.8) in 

tho form 

antl iipon comp:tririg cqns. (18.9) with (18.8) we can see tliat the components of 

the strcsq knsor tl~tc to the tr~rbillcnl vclocit,y componcnts of the flow arc: 

r 1 

Illis stresq tmsor is identical with tho one obtainad in eqn. (18.5) with the nid of 

thc: mornenturn c?q~rat.ion. 

l'rom tho preceding argument it can 1)c conclutfctf hhat, the components of 

tho mom velocity of tur1)ulcnt flow satisfy thc samc eqnn.I,ions, i. c. eqns. (18.9), 

as those satisfied by laminar flow, except that tho laminar stresses must be incrensctl 

by atlclitional sl.rcr3.qos wliicli are givcn by tlie strms tensor in eqn. (18.10). Thcse 

atltlitional stresses arc known M upparent, or virtual S~~CRSPIF of turbulent /low or 

Ile?/tsol*lm stresse.~. 'l'hoy arc due to t~~rbiilont flucti~atiori and arc givcn by the tiincmean 

vnlr~cs of the q~~adratic terms in tho t~~rbnlent components. Since thcsc stmsscs 

are atl(ic:cl to tl~c orclinary viscoi~s terms in Iyminar flow antl liavc a similar influence 

on the conrse of tjlie flow, it is often said tli& thcy arc caused by edh/ viscosity. Thc 

t.ol.al strrssc~s arc Ihc: snms of the viscoi~s ~(.rcsscs fron~ cqn. (3.25a,I)) and of tlrcso 

:bpp:i,rcnt, strcsscs, so thtt, e. g., 

Crnemlly speaking, the apparent, strrsscs far outwrigh thr visro~~s romponc~ifs and, 

ronwclurntly, tlir Iattrr may 1)c ornittrtl in many nc411al r:isrs wil!~ a gootl clc.pc.c 

of apl)roxitnal ion. 

Boundary-layer eqnations: At this stage it may be iiscful 1)riefly to o~~t,linc thr 

form ctf the boundary-layer equations for turbulent. flow. In the rase of lwo-di~~aertsionczl 

flows (15 = 0) cqns. (18.7) ant1 (18.8a, b, c), ~norlifiotl by the Iwutitla~y-1ayc.r' 

approximations as outlined in Chap. VI 1, lead to 

(two-dimensional, tr~rbulcnt boundary layer) 

Due t,o the boundnry-layer simplifications, the term 

which is generated by the normal st,resscs, can be neglcctcd. A compnrisoii with the 

equations for the laminar boundary layer, eqns. (7.10) and (7.1 I), leads to the follow- 

ing rules : 

(&) The velocity components and thc pressure, u, v, nnd p, are to be rrplaced by 

their time-averages Z, 8, j5. 

(b) The inertia terms and the pressure term remain unchanged, whereas the viscous 

term v ??2u/??y2 must be replaced by 

This is equivalent to stating that the laminar viscous force per unit volume a~~/ay 

must be replaced by 

where tl = palllay is thc laminar shearing stress from Newton's law, and .ct = 

- Q TG'? is the apparent turbulent stress from Reynolds's hypothcsis. 

Do~~ndnry conditions: The boi~ritlary contlitions to bc satisfied I)y thc mrriri 

vrloritey ron~poncnts in cqns. (18.9) arc the Anme as in ordinary Ianiinar flow, 

namely they all vanish at solid walls (no-slip cotltlition). Moreovrr, all turbulent 

components must vanish at thc walls and they are very small in tlirir immrdiatc 

nrighl~oi~rhootl. It follows, tl~rrcforc, thnt dl componcnts of the trrisor of :1pp:1rcnl, 

xtrcssrs vanish at Ulc solid walls and thc only strcsscs which not nriw t,licrn arc the 

viscous stresses of laminar flow as they, generally speaking, do not vanish thrre. 

F~lrthrrinore it is secn that in thc irnmcdiatc ncigh1)ourhood of n wnll the nppnrent 

stressts are small compared with thc viscous stresses, and it follows that in cvrry 

tnrb~ilent flow there exists a very thin laycr next to the wall which, in rssence, 

behnves like one in laminar motion. It is known as the lnrminrcr svh-loycr ant1 its

564 XVIII. I~~~ndnrncntals of tr~rbulent flow 

C. Swnc mcn~urernenta on ll~~ctunting t~~rbulcr~t vclocitiea 

In cxpcrirnc~nt.al work on tnrbulcnt, flow it is nsual (,o mcasurc only the mean 

val~~cs of prcssorc and vnloc:it,.y bccausc thy arc t h only quantities which can be 

~nci~sured convenicrtt~ly. l'lw mcasurcmcnt of the turbulent, fiuctuatir~g components -- 

u', v', . . . t,hcrnsclvos, or of their mean valucs such as 7~'~, u' v', . . . is rather difficult 

arid rcqrlircs cl:rl)o~.ato equipment. Itcliablc ~nc:isurement,s of the fluct~ration-velocity 

roml~mcnts IIRVO O(:cn o\~t,:~incd with the aid of hot-wire ancmometcrs. The measurement, 

of tho ntcan values is quite sufficient for most practical applications, but only 

tt~rotrgh the actual measurement of the fluct~tating components is it possible to gain 

n tlcepcr untlersta.ntling of tlic mechanism of turbulent flow. We now propose to give 

a short account of sonic c~perimcnt~al work on the measurement of the fluctuatingvc1oc:it.y 

con~poncnt,~ in odcr t,o present, a rnore vivid physical picture of t,he plienorncni~ 

and i11 or(lcr to give some jnstification lo the preceding rnat~l~cmatical argument,. 

I I. Iteic:lia,rtlt 141 J carried ont, such measurements in a wind tunnel with a 

rect~ariglllar test scctior~ 1 In wide and 24.4 em high. The variation of the mean vclocit.y 

over t,hc height of the tunnel, ii(?y), is seen plotled in Fig. 18.3; measurements were 

matlc in the ccntml section of tl~c tunnel. It is seen to be a typical turbulent velocity 

profile with a steep increase ncar the wnll and a fpirly uniform velocity near the 

ccnt.rc-line. The maxirnurn vclocit,y was U = 10 cmlsec. The same diagram contains 

also plots of t h root-rt~can-scl~larc values of the longitmtlirial and tr:tnsvcrse components 

,1/2 ancl 115 rc~pcctivcly. The transvcrse flucLnatio~t does not vary grcally 

ovcr the Iwigllt, of t.he cllanrwl and its avcragc? value is about 4 per cent. of f7, but 

I.llc longit.~tdinal t.url~ulcnt component cxl~it~il~s a pronorrncctl stmp maximum of 

0.13 17 vcry close to the wall. It is clearly Reen frotn the clingram th:tt I)ot,l~ t.~~rl)ctlcnt, 

components decrease to zero at tho well, as st,atctl earlier. Figure 18.4 shows a plot. 

of tile mean value of the product - 7=, which - is cq~l.zl to the t,rtrl)~~lcrtL sl~twing 

stress except for a factor Q. The value of - 76' v' falls to zero in t.l~o cont,rr 0 1 t,l~c: 

t,est section for reasons of symmetry, wltcreas it.s maximulrt ocrnrs ncar t.11~ wall 

sho\villg that tnrbnlcnt friction has its largest value there. 'I'hc brokrn line t/o sl~ows 

the variation of shearing stress which was obtiiinctl from the ~nr:~surt:tl prcbssllrc 

tlistlribution and independently of the measurement of vclocit,y. 'She tewo wrvrs 

nrnrly coincide over the major port,ion of the Itrigl~t, of the t.cst, srct.ion, atitl this ntay 

I)c inl.arprrt.atl as a ~ ood cl~eck on the rncnsurcmc~~ls; it. :dso sl~ows t11:~t n.lnrost. :tll'of 

th(? sl~caring stress IS due to t.urbulcnce. 

- l'hc two nttrvcs under consitlcr:~t,iot~ clivc.rgt: 

rlpar t,Ile w;Lll, t,)lc cllrvc of -- TI,' I,' tlw:rcasi~~g t,o zero. Irc~c~:~~~sc- I,III.I~IIIVIIL (III(.~,II:I~.~~IIs 

&c out near the wall. The tlifirencc bctwecn the two curvcs gives I:~n~it~:lr fri(.t ion. 

Finally Fig. 18.4 contains values of the correlation coe//icie7rl, VJ, I)ct,wccn t,ltc 10npit~11clinal 

nntl transvcrse fl~tcl~r:~l.ions at t,l~c same point,; il, is tldit~cvl l).~ 

Fig. 18.4. Blrnsurrt~~r~~t of Il~rc-l,~~n.l,it~p rotllponcnts 

in n clmnnrl, ~~ft,c:r 

llrir4mrtll 141 I 

TI," pro,lll,.L iF?, 

lltv shcxring slrvw T/,,. :111tl I IIV wr- 

r,411tI,,n v ~n4l\,~l~~lll 

v*

The corrrlnt,ion co~ffiricnt~ y~ ranges ovcr values up to y~ = -0.45. 

More rxlr~~sive rncasurcment~s on t,hr t,urhlrnt, fluct,~lat,ions have also been 

pcrformctl it1 t,hc I~ountlnry hycr of a flat. plate n.1, zrro incidence. Figure 18.5 reprntlnocs 

some of t,llr rrsults ob1,ninccl I)y 1'. S. IZlcba.noff 1251 irl a boundary layer 

on a fht. plak n.ssociat~cel wit11 a st,rmm of t,he very low t,nrl)~~le~lce intrnsit,~ of 

0.02 0/, (c/. Scrs. XVIcl ant1 XVI I1 f). at, a ltoynoltls nnrnl)cr R, = ( I, z/v = 4.2 x loe. 

,. 

I ho prnfileof 1.11~ t,en~poral me:m of the vrlocity, d, cxllibits ashape whic:h is very much 

lilte t,llnt in :I c:h:cnncl, I'ig. 18.3. 'Sllc vari:~t.ion of tho lon~itmtlinnl fluct.untion v

568 XVIII. F~~ndamentnls of turbulent flow 

,I 

1 his is cxplnint\d 1)y the reqnir~mcnt~ of cont,inuity accortling to which, as we know, 

t,l~e r:~,l,c. ol' Ilow through any cross-scction ronains constant in time. The integral of 

1.11~ c:orrrl:~tion function It, that is, thc quai~t~ity 

IS I.llc sc:c.ol~tl vrloc.ilty, 11~' in eqm (14.1:1). is n~wsurcd at. tht: same 1ora.t.ion 

l)111 :II. a (~iIl'(w111, ~IIS~.:III~< of I,imn (71~' at, inst,ant tl and 11.; at instant t2 = tl -1- 0, we 

obtain the so-called autocorrelalion funcliott. The provision of spacc-t,imc corrcl t .1011s, ' 

that is, of observations of two velocity components, oath measured at a tlilrrrrnt 

location in space and at a different instant in tirnc, allows us to gain a gootl tlr:tl 

of insight. As an example, we reproduce in Fig. 18.8 snch space-time corr~lnt~ions 

Dist.anco from wall: 

y/d = 0.24 

T5orrrtdary-layer t.hickncss: 

8 - 16.8 nini 

Fig. 18.8. Space-time correlations of velocity flnctuatio~~s in Lhr turlnrlcnt Iwu~ltlary lnynr on 

e flat plate, as measured by A. J. Favre, J. J . Caviglio nnd 11. .J. lhnns [tfi] 

ol)t.nincd by A. J. Favre and his coworkers (161 in tl~c Lnrl~ulcnt bountl:~ry Inyrr on 

a fl:l.t plata. 'l'l~e 1nrnpor:tl tlispl;~cctncl~~l,, t,,,, of t.hn m;tsimcr~~~ oI'(-:~(.ll (:Itrvv is inll)osc(I 

by 1.h~ passage of turbulent ctlrlics; tho odtlics move wit,ll nn a~~~~rosit~~;~l~c: 

vc*lotil.y 

which is equal t,o 0.8 U,. 'l'11e tlccrcasc in the maxima is thc rcsult of ;L procrss wl~ich 

can be visualized as follows: With the 1)assago of time, tho t,url)ulc~~l, ntltlirs losr t,l~cir 

intlividnality t,hrougll mixing with the surroutlding I,urbnlc~~t, flt~icl. Concurrc:nt,ly, 

ncw etltlics c:ont,inttously spring into being. 

An albrnativc tlcscript,ion of the st,nicturc of turbulcricc is obt,:~inctl wltc~~ a 

fr~q~~e,zcy analysis of the motion is provided instcad of a correlation fu~irt~ion. 'l'l~is 

leads 11s t,o the concrpt of the speclr.w~rr of a t.rrrl)~~lcnl, st,rc:~ni. f,ct I, clcnot.r thc -- Srcquency 

a.7~~tl F(n) (171, t.11~ fr:wtional con1,rnt of tlw root-t~~r:~~~-s(~~~ar(: 

v:~lur, IL'~, of 

t.11~ longil.utlinal Iluct,uat,ion wl~icl~ bclongs t.o 1 .l~ frcqucnvy ir~l~c~rvnl fro111 11, - t.o 11. I ,111. 

'I'IIO fnnct,ion F(n), which rcprcsent,~ - t,l~c tlcllsi1,y of 1.11~ tlislxil)r~l.io~~ ol' IL'VII ?I, is 

Icnown as t,l~c .spechd rlistrih~~tio,a of 71,'" l3y tlcfi~~itiorr, wr: must, I~avct

670 XV111. I'nntlntnrt~t~nlrr of tt~rhrrlrnt, flow 

A. N. Kolmogorov, C. 1'. von Wcizsncclzer 1641 and W. Ilciscnbcrg. As the frcqrtency 

bcconirs cvcn I:irgrr, F(ir) clecrensee under ~IIC acttion of kincmnt,ic viscosit,y at, R 

fa,ot,er mtc st.ill. Arcortling to W. IIciwnl~crg's t,hcory [Inn], nt vcry high vnlt~cs of 

I'requcncy wc s11o111d nbscrve that F(a) - w7. 'Chc t,wo t.heoret.ica.l hws arc representrtl 

in Fig. 18.9 hv thc t.wo stmight lines lnhrllrtl (I) niid (2), rnspcct.ively. 

IPig. 18.9. Ikx-pmcy spectrttrn of 

the longitudinal fluct~~ation in the 

turbnlrnt horlndary layer on n flat 

platc ~neas~~red by 1'. S. Klcbanoff I251 

Cnrvo (I): F - n-513 

Curve (2) : F - 

Theory rlnc to W. Hciscnberg [I!)a] 

J . h1nrCel1n.l [36n] pc:rfortnrtl tlct.n~ilccl rncnsurctncnt-7 on tile frcy~trnry spect,runl 

in flows wit.h liolnogmrons turbulence. In particular, 11c invcstigat,ctl the effect, of n 

strong t,wo-tlimrtisiond contrnction of thc st.rcam. 

prrsswc Il~~ct~rintions it1 t,hcsc znncs move, nt, nn inst.n.ntmumr~s convect,ivc velocity of 

40 t.o A00/, of n1rn.n stmnni nntl ill t.hr tlircct,ion of t01c mrnn stwnm. Tllc wnvo form 

of s~trl~ fIuct~~l:lt,ions cl~angcs slowly ~it.11 titnc. See also the papers by W. I

572 

XVIIl. F~~ndnnwntds of turl)ulcnt flow 

For this rrnnon, cqunt,ion (18.17) enjoys very wide np licnhility. A dilnensional argument which 

w n fir~l, ~ ndvanccil I)y A. N. I

574 XVIII. 1'undnnient.alu of tnrl~nlrnt. flow 

Jhtnilctl invcstigntions wl1ic:h were rnrrictl out by (1. I. Taylor [5:1] and 11. L. 

Dryden [II] Iwt to the conclusion tht, t,hc drag in n stream cannot be adequately 

tlcscril,c!tl . .- Ijy ,sxcifying tl~c mapnit.utlc of t.llc: lln,ciu:~tio!~ of thqy+city . coppor~cnts . 

alone, I~ccnusc it is nlsn :~.ll'~~(:l.t:!l l~y t.11~ structure of thc turb.ulcnt. stream. On the 

bnsis of a t.llcory of trtrbulcn& 11cvcloy)ntl Ijy himself, G. 1. Taylor prr>J)%ecl thet 

t.hc vri1,irn.l ltt~yt~oltls 

tlr~rn\,c:r or :I. sl1l1c:rc: tlc:l~c:ntls on the p:~m.tnctcr 

whcrc: I, is t . 1 scnlc ~ of tnrl~ulcncc, th~t, is, t,llc intcgrnl of the corrclat.ion function 

dclined in cqn. (18.14), and L) is tho rlinmetm of the sphere. II. U. Meier et a.1. [36b] 

investigntctl the influence of the scale of turbulence, L, on the turbulent Iwnntlnry 

1n.yc.r nt low turh~~lcnce intensity. They obbained nlnximunl values of t,he wall shenr 

strcss wl~rn t,Iw scale is of t8hc ordcr of tho bonndnry-layer thiclrness. 

Fig. 18.10. Relation hct.ween the crit,icnl 

Reynoldn nun~bcr of a ~pl~ere nnrl t11c 

intensity of t~~rbulcncc of the tnnncl, nflrr 

H. L. Dryden and A. hl. ICucthe [8, 101 

[I] Iht.cl~clor, G.K.: The t11eory of hoinogenco~~u tnrbr~lcncc. Cnmhridgc, 195:). reprint 1970. 

[In] Rlnkn. W.K.: 'l'urhnlcnt houndnry Inycr wnII prcnsrlro Ilnc~lrrrrtir~r~u ot~ RIIIOO(.~I ILII(I ro11gl1 

wnll. .JlW 44, 637- 61iO (1!)70). 

(21 13ratlshnw, I'.: An int,rodnction to tr~rbulcnce nnd it8 tncnurcnlent.. I'crgnmon l'rm, 1971. 

[3J RowJcn, K. 17.. Prcnkicl, Y.N., and 'J'nni, I. (rd.): Ho~~ntlnry Inycm and t11r1)nlcnro. I'roc. 

JUG(:/IU'l'AM Syrnp. Kyoto I!)($, l'hyn. J%~ids Soppl. (1967). 

[:hJ 111111, M.K.: Wall ~~ress~tre fi~~ct~rntion~ msocintnrcl with ~nhrronic t.11rbn1cnL 1)onnrlnrv flow. 

?JPM 28, 719-754 (1!)67). 

141 Bnrgcrs, J. M.: A n~nthcmntionl n~odcll illr~sLrnt.ing thc theory of 1.urh11lrnrc. Arlvnt~ccs in 

Aj1111. Mcclt. Val. 1 (1%. von Mirrcs :ind '1'h. vot~ I

576 

XV1 I I. 1~untlnment.nls of turbulent flow Iteferences 677 

. . 

(l!l5l). 

[:%I 1 LnuTcr. .I.: New trendu in expcrirnrt~tnl turl)t~lcnce rmenrch. Annual Review of Fluid Mech. 

7, 307- 326 (1975). 

[:I21 Imrfcr, .I.: l'lw str~~c:t.urr of t.~~rli~~lcnrc in fully tlcvclopcd pipe flow. NACA Rep. 1174 (1954). 

(33) I,nurrnrr, J.C.: Intensity, uc:nlo, and spr~Lrn of t~~rli~tlen(~e iu mixing region of frcc ut11>sonir 

jrt. NACA R.cp. 12!)2 (1056). 

[34J I.rslir, I).(:.: Dcveltqmnnk in t.1~ theory of tnrbnlcncr. Clnrendon I'r&qs, Oxford, 1973. 

[:%5] Iin. C.C. : St.nt.ist,icn.l t.l~corira of turbulrnrc. High Speed Aerodyt~nmicu and .Jet I'rop~~lsion 

Vnl. IT, Srr. C, 10G -253 (IN!)). I'rit~rrton. 

[:Hi] Iin. C.C., nntl Ibc4, W. If.: I'~~rl)ulrnt flow, f.l~eoret.irnl nsperts. Hondl~. I'liynik (S. Fliiggr, 

ccl.) Vol. 1'111/2, Springer-Vcrlng, Ilrrli~~/(~iit.ti~~gcn/~Iri~IcII~erg, 1963. 

[3G;1] hlnri.cbnl, .I.: IC1.11tln cxp6ri111enlnle tlc In cli.for~nnt.ion plane cl'une turhulenc~c homogimc. 

J. Mi.ranint~c II, 2(i3 -2!l4 (1!)72). 

I:%lil~] hloi~v. Il. 11.. nml I

CHAPTER XIX 

Theoretical assumptions for the calculation of turbulent ilowe 

a. Fundamental equations 

It is not very likrly that scicncc will cver achieve a complete understanding 

of the merhanism of tllrbulence because of its extremely complicated nature. The 

main variablcs which are of practical intcrest are tho mean velocities, but so fnr no 

rational theory which would enable us to determine them by calculation-has-been 

formulated. For this reason many attempts have been made-to-create-a mathematical 

basis for-the investigation of turbulent motion with the aid of semi-empiricalhypotheses. 

The empirical assumptions advanced in the past have been developed 

into more-or-less complete theories, but none of them succeeded in fully analyzing 

even a single cnse of turbulent flow. It is necessary to supplement the original 

hypothesis with additional hypothcscs which vary from case to case, and the form 

of certain functions, or at least certain numerical values, must be derived experimentally. 

The aim which underlies such empirical theories of turbulence is to deduce the 

still missing fundamental physical idens from results of expcrimenta;l measurements. 

The-turbulent mixing motion - - is rcsponsib!e - not only for Fn exchange_@ momentom, 

hut, it rilG GnhRnces thG transrer of heat an? mass in fields of .flow ~ ~ c & @ d 

with non-uniform distributions of temprrature or concentration. The methods for 

the mlculation of turbulent flow, temperature, and concentration fields developed 

so far are based on empirical hypothcscs which endcavour to establish a relationship 

between thr Reynolds stresses produced by the mixing motion and the mean values 

of the velocity components togethcr with suitable hypot,heses concerning heat and 

mass transfcr. Thc morncntum cquations for the mean motion, eqn. (18.8), as well 

as the differential eqr~ation for temperature (not quoted in Chap. XVIII) cannot 

acquire n form whirl1 is snitablc for being integrated unlcss assumptions of this kind 

have brcn inlrotl~iccd bcforcharid. 

.T. IZo~lssinrsc~ 17, 81 wns tltr first, to work on the problrm stnterl in the preceding 

srrlhn. 111 analogy wilh tho cocffisirnt, of viscosity in Stoltos's law for laminar flow 

i)u 

Tl=/L --, , ay , 

Iw introdricctl a miring cnr//icienl, A,, for ihc Reynolds stress in t-urbulent flow by 

pr~tbitlg 

- , , dS 

t, -=-pn v =A, - 

(19.1) 

dy . 

b. Pmndtl'a mixing-length t,llcory 5711 

The turbulent mixing cocfficicnt, A,, corresponds to thc viscosity, p, in la- 

minar flow and is, thcroforc, often cnll(x1 "appnrcnt" or "virtc~nl" (nlso "eddy") 

viscosity. 

The assumption in equation (19.1) has the great clisntlvantagc that the cdtly 

riscosity, Ax, is not a property of the fluid like yr, but clcpends itself on tho mean vclo- 

city Q. This can be rccognizetl if it is rrotcd that viscous forces ill turbulent flow arc 

approximately proportional to the squarc of the mean vclocity rather than to its 

first power RR in laminar flow. According to cq~~ation (l9.l), this would imply that A,. 

is apprnximntcly proportional to thc first powcr of the mran ~clocit~y. 

Often, use is made of the apparent (virtual or ctldy) kinrmatic viscosit,y E , - A,/@ 

which is analogous to the kinematic viscosity v = p/p. If this is done, thc cquations 

for t,lw shcaring stress arc rewritten 

and 

It is now possible to introduce into thc Navicr-Stokes equations for thc mran flow, 

eqns. (18.9), the boundary-layer simplifications. In the casc of thc velocity 1)ountlary 

layer these will be similar to the considerations discussrd in Scc. VIIa in connrxion 

with laminar boundary layers. In the casc of two-di~~~cnsior~:~I, ineomprcssil)lr, turbulent 

Bow, with due regard being given to cquation (19.1), wc obtain thc following 

system of tliflcrcntinl equations: 

which ~honld bc romparetl with ecjns. (18.12) ntitl (18.13). The preceding sct of 

equations corresponds to equations (7.10) and (7.1 1) for Iarninnr flow, and tlrc bound- 

ary conditions for the velocity components are identical with tllosc in the laminar 

case, rqn. (7.12). 

h. Fmndtl's ~nixing-length theory 

IIC IISC~ 

'1'11~ Ilypot,hrsos in eqns. (19.1) antl (19.2) C ~IIIIO~ for 1.11~ c:~Iwla(.io~~ 

of actual cxrcrnples if 11othin2. is known about the dcprntlcncc of A, on vrlocifsy. 

In order to tlcvclop thc prccctling rncthotl (irritiatccl by I~oussincsq) it is II~. v : ( :- ssary 

to find empirical relations bctwcen the cocfficicnt.s antl the mean vclocil,y. Jn tliscussing 

these, we shall confine ourselves in the prcscnt scchion to thc velocity ficld 

in incomprcssiblc Row because the latter is then intlcpendcnt of the tempnmt~~rc 

field. The calculation of comprcssiblc-flow Grltls and OF temperature fmltls, ~ 1~1, in 

particuhr, of the rates of host transfer in trlrl)~~lcnt motion, will bo taltcn trp in 

detail in Chap. XXIII.

580 

XIX. 'I'lrmrclic~l ~~suntptions for the calrulation or t,nrhrtlrnt, Itown 

In 1925 L. I'rantltl 121 j made an import,ant aclvanoo in tfhis direction. In tlcveloping 

his I~ypot~hcsis we shall rcfcr ta tttc: simplest cast! of parallel flow in which the 

velocity varies only from st,rcarnlinc to st,rcarnline. 'I'hc principa.1 clircct,ion of flow 

is ihss~~n~rtl pnrnllrl to the x-axis ant1 wo havc 

- 

u=E(?/) ; 5=0; 5=0. 

r 7 

I It(: pmc:otlir~g t,ypc: of flow is rcdixc-tl itt :I rrc:tSangctl:~r c11nnnc.l for \vliic:lt 1,11(: rcsrllta 

of rt~t:astrrcrnc:nt, on t,~rrl~olc!ot, vnloc:il,y c:orn~~onc:rtl.s wcrrc? givtw in I'igs. 18.3 :tntl 18.4. 

In t.11~ prt:sott. c:wc only tlrt: slw:cring stres~ 

rrtn:iins cliffrrrnt, from zero. 

- 0. Simil:irly :L Ilttnp of fluitl wl~iol~ trrrivcs a$ ?I, frorn the I:~mitia at -I- 1 

~)osscs~s :a vrlo~i(,~ \vltic:11 oxccccls that, around it,, the tlifircnce bcing 

( 'I'lir trrm inirlurr lt-nglh l~nn also hcrn usrd. 

Ilerc a' < 0. The valocit,y tliKerenccs ca11scc1 1)y the 1,r:~nsvrrst: motion cnn I)(. 

mg:irclctl as t h Lrlrlmlcnt velocity componnnt.s at ?I,. 1Ionc:c wc: can c:i.lcul:~t,c t.ltt5 

time-avrrage of t,lic nl)solitttc value of this Ill~ctmntion, :~ntl we obt.:~.itt 

Equation (19.5) lratls to t,ltr following p11ysic::d ir~l.c:rl)rrl.:~.l~ion of t.lm mixing Ic:rtgt.l~ I. 

'I'hc: mixing Icwgt 11 is I.l~:t,t, tlist~lnc:c: it1 t.l~c: t,r:it~svc:rsc: tlil.tv:f.iott whit41 ntrls(. Iw t:ovt.rc~~l 

by :III a&)n~w:ath of Il~tid p:~rt,i(:lrs t,ravc:lling with it,s origin:t.I III~YIII vc~lc~c:il~.v 

in ortlcr t.o nl:ilrc f.hc tlilk~rcncr bet,wct:n it.s velocity :and Lt~c vc1ocait:y ill t.11~ tw\r 

lamina oqu:rl 1.0 t,hc mc:m txansvcrsc: Il~~ct.u:at.ion itt t8t~rl~ulc.ttt. flow. 'l'ltc: cl11c4ort 

as to wl~cthr tllc lump of fluid complekly retains tltc vc1oc:ity of it.s original Intl~in:~ 

as it moves in a t,ransvcrse tlirection, or whctl~cr it pnrt,ly assumrs tha vc:loc.ity 

of t.1~ crossed lan~itta xintl cont,innc.s itcyot~tl it, ill :r (Ir:~.~~~~c~~c: 

clit~c:ct.iott, is 1tvr~ 

lc:l't, cnt~ircl~y o p ~ 1'r:~ncItJ's . concrpt, of :L tnixittg It-ngl~l~ is :III:L~~~OIIS, IIIP t,o :I vvrhi~r 

point,, wit.11 l,hc: mean free IKLLII in Lhr Itinctio t,ltcory of gases, tho nt:~in tlill;*rc~ttw 

being that. 1.11~ I:at.t.er conccrns it.sc:lf with t,ltc microsco[)ir rnotiot~ of 1nc~1~:c:ult~s. 

whereas the present concept c1cal.s wit,h t.lte ntacroscor)ic: rnot,iort ol I:trgc% :~gglonwrat,ions 

of fluitl p:crt,iclcst. 

It may bc imaginctl tht the transvcrsc vcloc:it,y Iluntuatior~ origin,

682 

XIX. Thcoroticnl wsltmptionu for the calculation of tnrbrtlcnt Rowa 

representation that the lumps which arrivc at layer y1 with n positivc value of V' 

(upwards from below in Fig. 19.1) givc rise "mostly" to a negntivc c' so that their 

product 7~' v' is negative. The lumps with e negative value of v' (downwards from 

nbove in Fig. 19.1) are "mostly" associntetl wit,h a positivc u' and the product IL' v' 

is ngsin negative. The qualifying word "mostly" in t,he above cor~t~cxt. expresses 

tho fact that the npponr:trtc:c of p:wl.ic:los for wlticli %' I I : ~ the opposite sig~t t80 tho 

nbovo is not completely - exclt~dcct I)ut is, ncvertl~elcss, much less freqncnt. 'L'hus, 

t,ho t,ornl)ornl avcmgc 14' v' is tlircrcnt from zero, nntl nc~tivc. lIcncc, we assume, 

- 

,ivr :- cm. m. (19.611) 

with 0 < c < 1 (c f 0). Nothing is known about thc numerical factor c but, in 

esscnce, it Rppcars to be idcnticnl with tltc correlation factor defined in eqn. (18.12). 

Thc experimental resultts plotted in Fig. 18.4 give some idea ns to its behaviour. 

Combining cqns. (19.5) ant1 (19.6) we now obtain 

It, sltoulcl be iiot,ed that the const.a.tlt in the above ~~11atio11 is diffcrcnt from that, 

in eqn. (l9.6), RS the former also contnins t,lw factor r, from cqn. (1 9.6a). 'l'hc constant, 

can now bc inclr~tlctl with t,hc still unknown mixing Icngth, and wc lnny write 

Conscquent.ly, the shearing sLrcss from cqn. (19.1) can bc writkn as 

l'aking it~t~o areo~tnt t.hnt the sign of 7, tn~~st, change with that of tlii,/tl!/, it is feud 

t,h:1tf iL is tnoro correct t,c) writc 

This is I'rrrnrlll'a ~~tdsi~~q-lr.trylh. hly~lhc.~i.~. It, will be shown lnt,cr Lhnt it. is very useful 

in the calc:ulnt,ior~ of Lurl)ulont Ilnws. 

Con~paring cqn. (19.7) wit11 the Boussincsq hypothesis in cqr~. (l9.l), we find the 

follo\ririg oxl,rcssiot~s for the virl,rtnl viscosit.~ 

and for thr v irt,unl kinrmnt,ic viscosit,y from cqn. (1!).2) 

It is known from cxprrimcnt.nl cvitlrnrr t,l~nt, t,urbrllent drag is roctglrly prol)ort.iorlnl 

to thc squsrr of velocity and thc same result is obtained from eqn. (19.7) if the mix- 

c. Fnrtlirr nam~tnptione for the t,nrhnlent eltenring streas 583 

ing length is assumed to be indepentlcnt of the rnngnitt~do of velocity. The mixing 

length, ~rnlike viscosity in Stokes's Inw, is still not n propcrty of the fluid, htt it is, 

at least, a purely local function. 

In numerons cases it, is possible lo cstnblish a simple r~lntion l)ct.~ccr~ t,l~c 

mixing length, I, ant1 n cl~nrnctcristic length of the rcspcct.ivc flow. For exnmplc, in 

flows n.lor~g smoof,lt w:~lls b ni~~sl. v:lttisl~ :ll, lthc w:~.ll itself, Ilt.c:ltls~ 1~r:~tt~vcrst: tno(.in~ts 

arc inhibited by its prcscncc. In flows along rough walls t,hc mixing Icrigtflt ticar t,l~r 

w:d1 must t.crd to n vnlrtc of l,hc same ortlrr of ~nagnit~tttlc ns the solitl protr~~siot~s. 

Pr:~.ntI~.I's cq~tnt,ior~ (I 9.7) II:IS I~CCII SIICC~SS~IIII~ n.pplic:tI to tI~c RI.II(I~ 01. iur1t7I1c;tt 

moliou nloq ~r~nlls ((pip, clrxnncl, pli~tt:, 1)ountlnry Inycr), nntl to t,hc problctn of socallcd 

/rw lurhtle?~t /6w. 'L'hc Iatt,rr trrtn r.~fc!rs to flow wit,l~ont, solid wn11s, s11c1i ns t(.I~c 

mixing of s jet with t h snrrountling still air. Exnmplcs of such npplic:at,ions will he 

given in Chaps. XX, XXI, and XXIV. It. A. M. Ualbrnith ct nl. [13a] provitled 

good cxpcrimc.nt.nl support for the ut,ilit,y of the niixing-lrngt.lr concc.pt,. 

c. Further flssumplions for tho trrrb~~ht slleflring elresa 

l'r:tt~clt~l's cqtmt,ioti (19.7) for sIt(!:trit~g stress itt t~~trl~rtl~~nL llo~ is st,ill itt~s:~tisliit:l,nry 

it1 t,lt:~l. the :~.pp:~rt:nt~, kittt:ndi(: viscosit.y r, t-tin. (l!).7l~), v:~ttisl~(;s :bl, pink 

whcrc cl?Z/th/ is cqilnl t,o zcro, i. o. at, poi111.s of n~nxitntlt~~ or ~nit~itntt~~l vcloci1,y. 'L'his 

is certainly not thc cnsc bccn~rsc t~irbrtlt~rt, mixing t1oc.s not. vnnielt :~t pni~~k ol' 

maxin~utn vc4ocit.y (rcnt,rc of clrnnnrl). 'l'lir I:~t.t.ttr viaw is c:onlirmc-tl l)y I~r~ic:lr:~.rtlt,'s 

t~rc:~~srtrc-~nc~~t.s on l.ttrb~rlt:~lt~ Il~t(:Lu:bt.io~~s, Vig. 18.3, ivlticl~ sliow l,l~:ll. in tlic (:c:l~trc o(. 

t h (:11a.t1n(4 t,l~c: longitrttlit~nl :inti t.ransvrrsc Ill~c:t.r~nt.iot~s bol.l~ c1illi.r fro111 xrro. 

In orclcr t,o countcr thcsc difficidtics Id. 1'rnndt.l [23) rst,nl)lishcd n t:onsitlt.mbly 

sirnplcr cquxtion for tltc nppnrcnt kincmntic vi~cosit~y. It, is vnlitl only in the case 

of frec tpurbulcnt flow nnd was tlcrivctl from cxt,cnsivc cxpcrirnct~t,nl tlnt,:~ on frec 

turbulent flow duo to TI. Itcichnrtlt [24]. 111 mt.ting 111) this nrw hypnt,l~csis I,. Pmntltl 

nss~tmetl that the tlitncnsions of t.11~ lumps of lluitl wllich move in :I t,rnnsvcrsc tlircc- 

Lion during turl,rtlcr~t mixing arc of tllc s:mc ortlcr of m;~gnit~tttlc :IS the witlLI1 of tJtc 

mixing zone. It will I)c rccitllcd thnt, tltc prcviotts I~ypot,ltc;sis itnl)lit:tl that. 1,ltcy wcrc 

small compnrcd with t,lic t.ransvcrsc clin~cnsior~s of tJte region of flow. Tl~c virttlnl 

kinematic viscosity, s, is now formcd by multiplying the tnaxirnum tlifTcrcncc in thc 

time-mcnn flow velocity with n IcngtJr wl~ic:l~ is nssrrrnctl to be proportion:d to the 

widt,l~, I), of the mixing zone. Thus, 

t On comparing this cqtmtion with cqn. (19.5~). it is sccn tht. nccorcling 1.0 Llrr prrsrnt, I~yl~otI~c~iu 

thc transverse fiuctuntion v' is proporlional to 17,,,,, -1T,,(,, and I.lmt t3ho mixing longth I' 

is proportional to thc width h. 1\11 nltcrnntive h)pof.lrcxis ~rltiolr rc!lntc:n to thc npp:rrc.nt kincniatic 

vinronity F, nnd is wry similar to tl~id. in cqn. (1!).9) \\.a8 fortrrulittcd by 11. Itrirlti~rdl. [24].

wtl~:~.c t.I~t* otigini1IIj1 (:otlst.nn~. en is now ~nl~It.ipIicd Ily thc il~t.c~.rnit,t,rl~c:y fn.cI.~~ y. Jn 

t,urn. :III~~ in n.t:rortlnncr with ~ I I V ~ncvwl~lvrnc.nt,s of 1'. lZlr\)n.11olT (S~W \pig. 18.6), t h ~ 

i~~l.(*t.~~~il,t~(~t~(*y 

fa(d or is :I,II~)I~~~~III:LI~(~(I l)y I~Iw rcl;~l,io~~ 

SCV :~lso rcl'. I!) 11 1. 

A r(w111,. sitnil:ir t,o l,l~c on(: conhinwl in ~111. (I 9.71, 11:~s IKCII ol)l,:rinc(l 11y 

(:. 1. 'I':~.ylo~ (:{21 OII t.11~ l):wis of his vorl,ic:il,y t.~xns~)orl, I,hcory. In I'r:rl~tltl's t.llcory, 

(.llc* :~ss~ltnl)hn is nl:ltlc i.11:1.1. t.11~ rnc:tn vc,locity 17 rcwlnins ronsl.anl. tl11r.ing 1.111: t.r:~nsvcXt.sn 

tnol.io~t of :i Ir~mp of' Iluid; 'l':~yIor's thcory sl~l)sl.il,ul.cs for this the 1lyl)ot.hcsis 

1l1:1t LIIC roi:~tion, t,l1;1,1, is t,l~nt (lC/(l?y rctn:iit~s consl.nnl,. T11is yicl~ls I,IIC w111:ition 

wl~i(,l~ (lill'(w I'ront (~111. (l!).7) tncr(~1y l).y I,lw I':i(:l,or. 112. 'I'his n1c:tIls l,l~:it, I,IIP mixing 

I~~ngt11 ol' (:. 1. l';~y\or's vorl,i~~it,~~-t.r:it~sS~:r tl~;or.y is lnrgcr l ~y a f:tct,or 115 t11:in I,ll:~t, 

ill I,: I'~~:~~~tlt.l's I I I ~ I ~ I ~ ~ I ~ ~ . I ~ I I ~ - I.~I(YI~Y. 

~ . I ' ; I ~ I'1'1111s I S ~ ~ ~ ~ lu - : 112 1.. 011 IIIC I~nsis of his 

ot~silr:~io~~s, . I. 1'1lor o n t l I I : I I Iilsion of trnirrnl~rc tlilli.t.c:ncc:s 

:IIIO vorli(.il.y in 1.11(, nlisil~:: xonc 1)c~hintl :r c-ylit~tlricnl rot1 oi~mr in t.onformity \vit.l~ 

I 

tl. Von Klrrn~hn's fiirni1:wity 11ypot.11nsis 585 

identical laws. This is in essential ngrccmcnt with cxpcrirnenta, ntd t,hc cxplnnat,ion 

turns on thc fact tht here the axcs of tllc vortices nrrango tl~otnsclvcs princip:~lly 

at right nnglcs to the main stream and to tho clirocl,ion of thc vc1oc:il.y gr:ulitrt~t.. 

By contrast, in a flow field in the proximity of a solid wall there prctlorninat,e vort.iccs 

whoso axes arc pnrallcl to tho flow tlircct,ion. Vor tllis rnason, tl~c tcmpcml~tlrc Iioltl 

I)ccon~cs similar to the vclociLy licltl tlircct,ly. 

d. Von Kirmiu's siniilarity hypothesis 

It wo~llcl bc vcr.y convcnicnt to possess :L rulo w11ic:ll :~llowctl us to clctcrnlinc t,l~c: 

dependence of mixing lengt,l~ on space coordinates. Th. von Krirm&n 1171 tnadc 

an attcmpt t,o txt,al)lisl~ s11c11 a rdc assl~lnir~g that trrrlwlcnt fl~~c:tnat~ions :Lro sinli1:~r 

at all point of the field of llow (simihri/?y wk), i. e. t,I~at they (IiIPcr from pint to 

point only by time nnd lcngth scale factors. A velocity which ;s charnct.erist.tc of the 

turbulent,, fluct~lating motion nnn bc formod wit,ll I,llc nit1 of' thc t ~~~~l~nl~~nl~ 

sIlr:l~~illg 

: 

st.rcss by tldining it., wil,ll I,llr n.itl of cvlll. (I!). I ), r1.s li~llows 

'She cl~m~t~it~y v, is cnllcd the /tiction veloci!?/ and is a rnc:asltrc of 1.111: inI.(:n~it.~ of 

turbulent, eddying and of the correlation which exists 0et.wcen the Ilr~ct.lmtincr com- 

? 

ponents in the z and y directions. For the si~nilarit~y rule under considcrnt.~on we 

imagine a two-dimensional mean flow in l,hr .r tlirccl~ion, RIICII t.llntl ?j - N (y) 11.1ld 

5 = 0 (parallel flow), and an auxiliary nlot,ion wl~icll is also two-tlirncnsion;II. 111 t,lris 

case it is possible to show t,hnt the rnle that 

const,itutes a necessary contlit,ion to secure co~npatibilit~y I)ctwcrn the simil:r.ril.y 

hypothesis and the vorticit.y trnnsport equat,ion (4.10). 

Introducing an empirical dimensionless constant x, von IChrtnh matlr tllc as- 

sumption that t,he mixing length sat,isfics t,he equntion: 

In accordance with the above hypothesis, thc mixing Icngt.11, I, is intlcpnntlrnt, of thc: 

magnitude of velocity, being a f~inction of the vclority clis(.ribut.ion only. 'l'l~o mixing 

Icngth bccorncs n purely locnl fnnction as :ilra:l.tly rc:cl~~irc:tl c:irlic:r, :i11

586 XIX. Throretical aaotin~ptiona for the calculation of turhdcnt flow8 

A. 1lrl.z [4] gave n very Iuctd derivation of eqn. (19.18). In latcr times von 

J

I lorv t,hr c:onst~;~,~~(, of in t.c:gr;rt.iotl, C, ~nnst I)c tlet~crtninctl from the condit,ion at the 

wnll :~ntl sc~vcs t.o fit, t.lw t.url)tilrnt vrlocity clistrib~~t~ion to Llrat in tl~c laminar sub- 

Iaycr. Ilo\vc.vcr, c:vc:n wil,I~out tlvt.nrrnining C it is possible to tlcdr~cc from eqn. (19.27) 

:L I;l\v an:~logotts t,o t.I~:ll in cqt~. (19.21). 111 spite of t,hc f;lc:t tkrt cqn. (19.27) is v:llid 

only in the ~~cighI)o~~rhootl or tlto w;~ll, I)(Y:RIIRG of the ass~~~nptiot~ thati t -- const., 

we shall at,t,rrnpt t.o ~tsc it, for thc whole rrgion, i. I?. up to 11 = h. Since at y =: h we 

Irnvc u -7 u,,,,,. wc ol)t,:~in 

7',,,ar - 7 11 I h 

u,,,, = "*O In li -t C , 

- -- In - ; (y = disb:~nrc from wall) 

"*o K Y 

, 

I 

, 

his 1111iv(:rs;11 vcl~(~i(.~~-(l~fc(;l, 

I:LW tlrw 1.0 I'rnndt,l is shown plotlet1 ;ls c:urvc (I) 

in ITig. ,1!).2. In t.11~ prc?c:c:tling argument we sr~cccetlccl in deriving a urrivcrsal vclocityclist.ril,~tt.io~~ 

I:lw from I'r:~n(It.I's law of friction in cornplcbc analogy wit,h that in 

o(111. (1!).21), whic:h was olll.:linctl from von IChrrn~in's si~nilarit~y rrrlc. The only clifferent:(: 

is in t.l~c: form of t.llc: functions of y/h wl~ich'appear on the right-had side of 

cqns. (1!).21) : L I I ~ (1!).28) rrspc.cl,ivrly. on rcflcxidn this will not appear incomprchc?nsil,lc, 

if wc: f.altc intm nccount the tlifkmncc in t*hc assumption concerning the 

sl~r:~ring st.rcss. Vorl 1C:'Lrrn;in nssnmctl a linear slrcaring-stress distrib~ttion, bltc: 

mixing Icngt,l~ bring 1 - u'lu". On t h other hand, Prantltl assumed a constant. 

sl~cnriitg st,ross and 1 - ?I. l'igurc 19.2 cor~tair~s a comparison hetwccn t.hcsr: two 

I:rws. A I~rt.l~c:r cornprison with cxpcrirncnt is clcfcrrcd to Chap. XX. 

It* may be worth 11ot.ing in pnssing Il~nt, it. is possil,l(: lo ol)l.:li~l 1.I1v sin~l,l,: I.(~SIII(. 

l.l~:it. 1 .- x ?I front tl~c vclocit,y-tlrfi:c:t. I:LW (I!).27), t.ogot.lwr wil.11 vot~ I tlisc:~~ssrd in gr(~~t.vr (I(~l,i~il in 1 . l SII(YYT- ~ 

(ling clrnpt.cr, give a value of x -- 0.4. 11'110 scc:on(l co~rst,:~nt,. 0, tl(,l)c*~ltls 011 t,11(, 

n:ttarc of t,hc wd1 snrf:icc; rdov:l~~t, n~~rncrirnl v:rlnc,s will IN! givrn in (%:I;). XS.

5l)O XIX. Tl~rorrt,icnl nmumptions for t,hc mlculntion of td~i~lrnt Ilows 

'I'll(: nnivc?rsnl vc~loc:it~y-elisl.ril~~~t,iot~ 

I:Lw, nqn. (I9.:1:,), wl~ic:l~ has now I)ecn dcrivctl 

for th: (:tist? of :L fht, wnll (rcc:t.:ingul:w c:l~:~.nnol) rct.airw its fu~nlnmcntnl irn1)orLancc 

for flows t,l~rough circular pipes, :w will be wcn in t,l~c next chnpt,c:r. We may now 

st.ntc, in anticipation, that it loatls to pot1 agrccmcnt wit11 cxperimcnt. 

Tn nonclutling this c:hnptrr it may 1)e worth stmssing once again tht thc two 

nnivcrs;il vclocit,y-clis(.ril)~~ttio~i Inws in cqns. (19.21) ant1 (10.27) were obtained for 

t,~irl)nlcnt~ flow, and took i11t.o nc:nount,, ap:lrt from t,hc small sub-layer mar the 

wnll, only tmrhnlont shmring strosscs, ant1 it sho111tI 1)e realized trIrnt3 snch an assumpt,ion 

is s:ttlisfi(:(l x,l:~rg!r l~cyn?ltls nurnl)crs~only, (h~scq~~ent~ly tile vclonit~y-tlist,ri- 

1)ul.ion law, p:~rt.ic:~~ln,rly t.l1:11, in cqn. (I!).33), must I n rcjinrtlctl as nn asymptlotfie 

law a.pplic:xi)lc to very 1:~rge ltcynoltls numbers. For smallgr-Rcyr~oltls- ritlnll~c~~ 

w11c-1~ .I?n~it!:~r. fi.ict,i(?l~ I:xo~(,s,. somc ~II~~IICII~C outside th~very~-tl~it~ ,q~b-l:~ycr, 

- .- - . . -. t:xp . 

riihcnt Icads to a power law of the form ,.. 

. .. 

. 

wlwrr the exponent 21, is approximately rqunl to :, but varies somewhat wit.11 the 

Itrynolris nnml)cr. 'l'his point will also be t,akcn up agnin in the succeeding cl~aptcr. 

'rhc c:~e of so-cxllod Co~wtt,c flow I)ct.wcen two parallel flat plates which nre 

tlisplnccxl rcl:it,ivc t,o circl~ other (Wig. 1 .I) const.it,ut,rs a very simple cxaniplc of a 

flow it1 whicl~ the sl~c::~ring stress rcrnnins c:or~st~ant,. 'l'he sl~c~wing st.rrss T rrrnains 

Fig. 19.3. Vdority profilcs 

in prallel Coucttm flow 

betwren two parallel plates 

moving in opposite dirrctions, 

after H. Rcichardt 

[25, 261 

At R - I200 Lhc flow Is Inmimr; 

nt R .- 2000 nnd 34.000 ll~c flow 

Is L~~rbulcnt 

rigorously constant in trrrbnlcnt ns wcll ns in laminar flow, xncl is cq~~al l,o t,11:1t, :it, 

thc wall, to. 11. Ltcicl~a.rtlt. 126, 261 carrictl out an oxtcnsive invcst.igat,ion of this cSasc; 

somc of his rcsulh can bc inferred from Fig. 19.3 which sl~ows several vclorit,y pwfilcs 

observed in Couctt,c flow. 'l'l~c flow rcmzins laminar ns long ns the I~C~IIOIIIS 

number R < 1600 sncl thc velocity distribution is t,lwn linear to a good tlcgrrc of 

approximation. When the Reynolds r~umbcr R excccds tl~c value 1500 the flow is 

turbulcnt. Tllc turt)~~lcnt velocity profilcs arc very flat near the centre ant1 bccornc 

very steep near the walls. A profilc of this kind is to be cxpcctccl in tnrl)ulont flow 

if it, is rcmcnibcrcd thxL t.11~ shearing stress nonsists of n I:~.tninnr c:ont.rit)~~t.ion , 

, anfl :i t,~~rbulcnt~ conI.rihl,ion 

clue to turbulent mixinp. Ilcncc 

wl~crc A, donotes the mixing cocfficicnt tlcfinctl in rqn. (19.1). 111 t,llis matinor the 

velocity gradient turns out to be proportional to I/(p -t A). Since A varies from 

zero at the wnll to its maximum in the centre of thc cl~anncl, the velocity profilc 

must bccomc stmp at tltc wall and flat at t,Im centre, as confirmctl 1p.y thc plots in 

Fig. 19.3. The turbulcnt mixing cocfficicnt increases with an increasing Reynolds 

number and the curvature of the vclocitpy profile bccomcs, correspondingly, more 

pronouncrtl; compare the paper I)y A. A. Szcri [nlnl. 

f. Further dcvelopmcnt of theoreticnl hypotheses 

The cnlculation of Lurbulcnt flows on 1.111: bn4k of t h difl'crcnt s~n~i-c~npirirnI l~yl)ot.l~(:scs 

discusset1 previously, and mrricd out in rict,nil in thc succoctiing cl~npkrs, is not sntisfact.ory 

in so far as it is still itnpossiblc to analyze t1ifli:rcnt kinds of turbulont llow \vitl;'iho &I of t.hc swnc 

hypothesis concerning trrrbulcnt friction. Ipor cxatnple, Prnntltl'a hyp~tl~c~is 011 1.11~ mixing length, 

cqn. (19,7), fnih cotnplctely in-tl!? casc of RO-cnlld isotropic. turbulence ris it, c?xint,s bcl~i~~cl n 

scrccn of $tic ~cs11, bccnuw in tliia cost the: vciociLy giatlicnL of the biuic flow ik ct111nI 1.0 zc:r~ 

cvery.wJ~crc, Tho liyPo~llcscs lor bhc cilct;lniio~~ or clovclopccl LurlmIcnt flow, (~~HoIIRR('II in Sees. 

XIXb and c, have been considerably cxtentlcd by I,. I'ranclt,l 1221 in an attempt to rlcrivo n universally 

valid system of equations (turbulnnt flow near wall, frco turbulcnt flow, isotropic tmrhu- 

Icncc). 

Energy eqemtioa: L. Prnntlll bnsctl his 

- 

ncw dcvelopmcnt - on t.lw consitlcrntion of t.hc kinotic 

encrgy of turbulcnt fluctuation, R = o(r'2 + 11'2 + z), nnrl cnlc~rlatctl thc rltange of t.1~ 

energy of the suhsirliary motion with tinrc, UR/J)l, for n particle which nlovcs with Lhc basic 

stream. This is con~poncd of t,l~rec Lcrnw: of the decrcnsc. in cncrgy t111c to internnl fric.t.ion in the 

motion of tho lumps of fluid, of thc tmnsfcr of cncrgy from tho hnsic motion Lo tl~o sut)sidiilry 

n~ot,ion - this term heing proportionnl to (dlJ/dy)z - and, linnlly, of thc trnnsfcr of kinetic 

energy from the more turbulent to the lem tmbulcnt zones. The encrgy balnnce I)ct,wren t,llc.~e 

tl~rcc terms leads t,o a differential equation for the cncrgy of the t,urbulcnt'sr~h~itlinry motion 

which must be added to the systen~ of differential equations for the Incan niotion; it has the forni 

cliasipntion production diK~~nion

502 XIX. Throrrl.icnl nsaun~ptions for the cnlrulntion of turbulent, flows 

flnre j -- 0 for t.wo-tlimrn~ionnl Incnn flown, j =; 1 for nxially symmetric nlenn flows (y-radial 

clist,nncc frorn nxis). 1,. l'rnndtl referred to the preceding as to the firot fundn~nental equation. 

A sroond equntion rclittcs the turbulent shearing stress with the velocity gradient of the mean 

flow nnd is nnalogoua to thc old mixing eqnnt.ion (19.2), but also contnins the energy of the 

turbulent nr~lr~itlinry notion, lhnt is 

, 

I . hr two rqunlionn - (I!).%i) ant1 (19.37) - ront.nin the three frce constnnta c, k, k, ahich in~~nt 

I)r rlnrivrcl 1)). :L rrfercnrc t,o cx~)erin~rnt~nl rrsnltn. 'rho length scnle I, is n locnl funct,ion which 

rc!l~rrsrntn. rss(wt.i:dly. lhr mixing length OF eqn. (19.7). The defi~~itiotl of this q~tntltit,y can, hornevrr, 

also I I 1):1sc:tl ~ nu :ui intcgml oft 111: eorrrl:it~ion funot.ion of t.lie velocity ro~npo~~cnls niens~~rorl 

:II I\UI lwinls (sw J.(:. l

594 

XIX. 'rheoreticnl nssornptions for the cnlculat,ion of t~~rhulcttt flows 

[I] 13ntcl1rlor, (!.I

Turbulent flow through pipee 

n. Experirnentnl results for smootli pipes 

\\'111*n :i Ilttitl is :11Io\vrtl to cnlcr :L circ111:w pip front a largc ronl,:~inrr, tltc 

vc~loc~i(,y tlis(,ril,r~( io11 in t.11~ rross-sections of tho idrt lan~~th vnrios with thc tlisCnncc 

from the: itliti:il cross-swI.ion. In sections (:IOSC to that at cntrancc the velocity 

tlisl.ril)~lt.ion is nearly uniform. Il'11rt1wr t1ownstrt:am tlhc vclocity tlistribr~t,ion 

cltnttgrs, owing 1.0 t,Itt: ir~ll~tr~tcc: of frict,ion, rrnI.il :I f~llly tlcvclopctl velocity profile 

is :~l,l.:~inc:tl :rt, :L given cross-scc:l.ion :LII(~ rcmains constm~t~ downstream of it,. Tllc 

vn.ri:ll~ion of t11t: vr1orit.y profile in the inlet length of a pipe in lam.innr flow was 

tlcwrilrctl in Scc:. XI11 (l'ig. 11.8). 11,s length is approximatdy 1, = 0.03 rl . R 

so I Il:tI, for R := 5,000 l,o 10,000 it, rangcs front 160 t,o 300 pipe-din~net~crs. 'rhc inletf 

Irng1.h in /?o.b~drt~l flow is consitlrmltly sllort.cr t,h:rn in lan~inar flow. According 

1.0 1.11~ IIIC:I.SI~~~III~-II~.S p(vforn1~t1 by I[. I

698 XX. 'I'~~rl~ulrnt flow f,lrro~~gl~ pipen 

Fig. 20.1. Vrictionxl rc:sistsancc in a ntl~ootl~ pipe 

R- y 

v I I I 5 . 1 o r 1c1-1i1i11c Cor ln~~~i~trr flow: curvr (21 rrwn cqn. (20.5). afkr DI%qi.il~s (51 Tor 

I'ig. 20.2. Vnlocil,y diu- 

I.ril~t~t.ion ill utnootli 

pip for varying Rey 

nolds number, after 

Nikr~rsdao 1451 

whnrc tllc oxponcnt n. varies slightly with t.hc Itcynolcls number. Y'hc plots in IGg. 20.3 

show that thc assumption of a simplc: l/n-th-power law agrees wcll with rxpwimcnt, 

.zs the gmplls of (u/(J)" againsl y/R, fall on straight lir~cs, wlicn n sr~i(,:~.l~It: cl~oicc: 

for n has l~orn madc. Thc valuc of t.hc cxponcnt 7s is n = 6 at. tl~c lowost, ltcynoltls 

number R = 4 x 10R; it increases to n = 7 at R = 100 x 10%nnd 1.0 71. = 10 at, 

thc liighcst hyrloltls numhr, R -- 3240 X ICYR, nttn.inotl in this invcstig:~t.iorr. 

found that 

Fig. 20.3. Velocity distribution in arn0ot.h pipes. Vcrificat.ion of tho ruurumption in eqn. (20.6)

'I'nblr 20.1. Itnt.in of IIIC:LII to nmxin~u~n velocity in pipe flow in trrrn~ of the exponent n of 

t h vrlocit.y tlisl.rib~~t.ion, according to eqn. (20.6) 

b. I~clntinn Lctwcrn Inw of friction n d vrlocity ilistributinn 

'1'11~ cvlu:~tio~~ liw t,lw vt~loc:it,y tlist,riltul,ion (20.6) is rclat,cd t,o 13lasir1s's law of 

l'ric.t.ion in ccln. (20.5) rintl t,l~is rclntion, first tliscovrrcxl 1)y 1,. I'mntltJ 151 1, is of 

I'~~ntl:amc~~t.nl i~nltorl.n.nc:c in t.11~ theory of t,~~rbulent flow; it allows 11s tlo draw con- 

(-Insions from pilw cxporimcnt.~ which art: valid for the flat plate [321; use of them 

will I)r marlo in Chnp. XXT. 

011 suhstit~i~ing the v:dnt: of 1 from (yn. (20.5) intm eqn. (20.4) we obtain the 

follo\ving t>pr(wio~~ for t .1~ shearing str~ss nt the wall: 

v,/v = 9, which wcrc nlr~atly r~sctl in cqns. (19.31) :1n(1 (1!).:

602 

XX. T~rrbulcnt flow thro~tgh pipes 

For fut,ure rcfcrcnco we now propose to write down an cxprcssion for the friction 

velocit,~ 11, from cqn. (20.10). We obtain 

v, = 0.150 u ' 

This cnn also be written in tlimr,nsionless form as 

where c; denotes the local skin-friction coefficient. This relation, which is equivalerlt 

to the one in eqn. (20.5), is known as 13hsius's Inw of skin fcicthn in pipc flow. This 

relation will be used later. 

c. Universal velocity-distribution lawe for very large Reynolds nutnbera 

r 7 

1110 fact that the exponent in the law of pipc resistance as well as in the cxpression 

for velocity dist.riht.ion clccrcascs with increasing Iteynolds nurnbcrs suggcstR 

that both must tend asymptotically to some exprrw&ms which are valid for very 

high Ibynolcls nirmbcrs and which must contain the logarithm of thc independent 

variable, as it is the limit of n polynomial for very small values of the exponent. 

A tlct~ailcrl cxaniination of expcriment,al results for vcry large Reynolds numbers 

shows that such lt~garit~limic laws do, in fact, exist. I'hysically such asymptotic 

laws nre chamctcrizcd by the fact t.h:rt I:~minar friction becomes completely ncgligihlc 

cornpmxxl with turbulent friction. 'rhe great advantage of such logarithmic 

laws, as comparrd with the Iln-th-power laws, consists in their being a~ymptot~ic 

expressions for very large hynolds numbers; they may, therefore, be extrapolated 

to arbitrarily large values beyond the range covered by experiment. On the other 

hand, when the I/u-th-power laws arc used the value of the exponent n cllanges, 

as the range of Reynolds numbers is ext,cntled. 

S11c.h an asyn~ptot,ic log.zrit,hmic law has already been givrri in eqn. (19.33) for the 

rase of flow along a flat plat,c. It was dcdncecl frorn Prandtl's ecluntion (19.7) for 

t.ilrl)~~lcnt shearing strcss nntlcr t.hc assi~n~pt,ion that the mixing length is proportional 

t,o 1.11~ tlist.:anc:c frorn t.hc wnll, 1 = x!/, ~ n was d valid for small w:dl (li~tanccs y. 

.. l his rquat,ion h:ts 1-hc: form: 

4 == A, In 3 4- Dl (20.13) t 

.~ . 

t 11. lkic:har~ll. [fir,] intlirntcd a refined cxpremiort for thp vclocity distribution. It covcrs the 

wholc rangc of distances, froni the wnll of the pipe at ye= 0 to the centre-line at y = R, i. c., 

it is also true for thc la~ninnr sub-layer, to which eqti. (20.13) does not apply. It is also valid 

in the ncighbortrl~ood of bhe ccntrc-line, wherc ~ricasurcd velocity-distxibution curves show systctnatic: 

clrviations fron~ cqn. (20.13). In particrtlur, the transition region shown aa curve (2) in 

Fig. 20.4 iu wcll rrproduccd by thc forrnltln. Thin ~~nivcrsnl velocity-distriblltion hw wan dcducccl 

with thc aid of bl~rorcticnl rabin~ntinns arid vrry r:rrrf~d riicns~trc*~ncnlrt of tl~c turb~llrnt mixing 

coefficient '4 rlrlinrcl 11y cqtt (I!).I). (hmparc also a pnpcr by \V. Sznbicwski (741. 

r. IJtiivcrsaI vc1nt:iLy-di~trib~ltioti laws for very large R.ryttnlds ni~t~~l~cru fi09 

with A, = 11% and Dl = - (11%) . In as free constants. Wcshallapl)ly this equation 

without ch:tngc to pipe flow. Comparing it with the rnt:nstrrcmcnt.s pcrforniccl I)y 

,J. Niknratlsc!, as shown by curve (3) in Irig. 20.4, it is seen thnt cxc:c:llont ngrecnlor~t 

is obtained not only for point,s near t,he wall but for thc wholc rango 111) 1.0 the axis 

of the pipc. The nilrricrical values of the constants :arc fortntl to IN: 

A, - 2.5; I), = 5.5 . 

'I'his gives I.llc following V:LIII(:S of x atitl fI: 

llt*tit~: l,Itr i~niv(wal v~~lo~:il.,v-tlisl.rilit~l~it~~~ 

I:LW for vcry l:~r~t% lLt*y~~t)I,ls 

has the Ibrmt 

4 = 2.5 In 3 -k 5-5 

4 -- 5-75 log Y) 1- 6.5 . I (slllool.ll) 

I I ~ I I I ~ ~ B ~ 

I5.y a reasoning siinilnr 1.0 I.l~c ow givw in 1.h~ ~nwwling wction it, is 1,ossiblt: t,o 

arrive at a corrcspontling universal asyrnl)t.ol.ic rcsisl.:rnc.c: forlnitln front t11c- :~l,ovo 

~rnivcrsal velocit.y-clist~ribiil~i~~ii crl~int.ion. 

1Scl1lat.ion (20.14), hcing one for l.~trI)ulont Ilow, is v:alitl only in regions where 

1.11~ I:rrninn.r shc:aring strcss c:~n be iw,nlwI.c:tl in cwinl):lrisori witlt the I.~rrl)itlc.nt 

stress. In I.hc irnmctliaf~o ~inigl~I~ot~rl~ootl of I,llo w:all, whc:rc Iflie I,~~rl)ulc:t~l. slic:~ring 

sl,rc:ss tlw:rcnscs 1.0 zcro : ~ r d I:amirt:ar sl.rcwc:s prctloinin:atc, tlt:vi:~tiolts froni t.his 

law must, be expected. 11. Rcichartlt. [54, 551 rxtcntletl l.l~is kind of ~neasnrcmc~nt to 

incli~tlo vcry small tlist.nnccs from t,llo w:~ll in n Ilow in a channcl. Curve (2) in 

Fig 20.4 rcprcscnts the transition froni t.hc Irrlninar sub-layer (c/. Scc. XVlllc) 

1.0 l,ltr I~trl~~rlotit. bonntl:~ry Inycr. 'l'hc c:~~rvc! tlotto1.cvl 11.y (I) in 1.11~ almvt: tli:~grnin 

c:orrosporitls to laminar llow for whit:ll T~, ---- 11 11/!/. \Vitl~ T,, = pv*2 we ol)l.:tin 

Irron~ (.his it can bc sccn t.1i:~t 

fric.l,ioit may be complctdy ncgl~:c:tctl c.ornp:aretl with 1:rniinar friction. In Lhc: r:Ln,ne 

5 < 1 17*/11 < 70 botfh conf.ril~ut.ions :in: of tho snrnc ortlcr of magnitutlc, wllc-rws 

for y v*/v > 70 l,l~c 1amin:rr con1 riI)ut,ion is twgligil~lc rotnparwl wil,l~ I 1tr1)11Itwl, 

frit:l,ioii. '1'1111s : 

for wrlurs y 11,111 < 5 (.lie cont.ribut,ion fro111 t~trht~It:~~t, 

'* > 70 : p~ir~ly t~irln~lcnt. friction . I

604 XX. 'hrl)r~lcnt now t,hro~tgll pips 

We now propose t,o compare the cxpcrimcnt.n.l rrsnlts on vrlocity-rlistril)utior~ 

mcasr~rerncnts in pipc flow with t.hc altcrnntivc ~~niversal equation, which was 

clcdl~rc~l in Ch:~p. XIX in the form (I/ --u)/v* .- /(l//R). It will be rccalletl that 

it followcd hot11 from von I

606 

XX. 'l't~rhl~lc~rit flow thro~~glt pipes 

wlirrc /(?/)/I{) - > I for y/Jt + 0. lnt,rocI~tring v* = iT"/b :~ntl con~l)initig cqn. (20.17) 

with rqn. (20.IC,) we? ol~t.niri the following tliffcrcntinl equation for I.ho vcilocity 

dist.ril)~~t~ion 

whrncc:, l>v intc~r:tt.ion 

Wig. 'LO.(;. V:iriiJion of nlixinp hgld~ ovcr 

pip tlinnickr for rough pipea 

Curw (I) rrom rqn. (20.18) 

IJerc lJic lowor litnib of in(.cypt.ioli :1t. yo, wl~orc t.11(: vc:locit.y is ccp~l Lo zero, is ol' 

tlic orcler of l,lic tllic:knc~s of tlic 1:tminar sr~l~-lnycr and, tlicrcforr, proport.ional 10 

v/n, 3.s seen from cqn. (20.15a). 'l'li~~s yO/II = F, (n* R/v). Tilo ~naxirnum ve1ocit.y 11 

in t,hn mnt.rc: of tlw pip c:~n I)c? (Ic~II~M~ from cqrL (20.21 ) and I~coo~ncs 

r 7 

1 hus wc? hxvc :xgnin 1xwi h1 to t h univ(vx~I \~cl~(-ity-Oi~I.l'il)tt(.iO~l litw. (~111. ( 19.2 l ) 

am1 qn. (l!).28). 'l'lic: cssrnt,i:~l g(:t~cr:tIiz:~t,io~~ wliic~lt I I : ~ now I)c:oti :tt4ticvc1l cot~sist.s 

in t,lir fact, t,liat thr ~~nivc:rsnl law in cqn. (20.22) is vnlitl for rougli :IS wdl :IS liwsnioot.lt 

ljipcs, tt.lic f~~n(:t.ion k7(?j/H) 1x:ing t.11~ s:\ni~: in I)ot.Ii oases. I':(III:I~ ion (20.22) :~ssc:r(.s I ltp~. 

c.11rvc-s ol' vc.1rwil.y tlislrilnll,io~l ~jlot~l.c:tl ovvr t.11~ 1)ipc rntlitts c~tn~l~r:lc~l, illlo :1, si~tgltr 

wrvc for ull v:~.Itws ()I' l~~~ynol~ls nt~nil~~ m(I for all (1c)grcc:s of rot~gl~~t~s, il' ((~---~I,)/IJ* 

is plol~t8~:~l ill t4(~rnis or !I/ It, lTig. 20.7. It, ni:~jf IK: nol~xl ~JI:LI, 1 . 1 ~ :I I)ovv I~WIII ol' I It(! 

vclocil,y-tlist~iiI)i~t~ioti law was lirsL tlrtluwtl I)y 'l'. 15. St,nnt,ol~ 1721. Ati c:xl)lic.it csxpression 

for F(?//R) cor~ltl Ipc ol)t.ninc:cl I)y ovnltlnl.ing tlic inl.cgr:~l in cc1n. (20.21): 

it is, Iiowev~~r, sitnplcr 1.0 ni:~kr 

IISO of 1.110 :tlro:uly Itnowti Ihrni or tlio vc:loc:il.y-~lisl.~~il~~tlion 

Inw for smooth piphs ns give-11 iii 1:(1u. (20.14). Ilonc:c,, ill :t wa.y sirni1:tr 1.0 c.clris. 

(20.9) nntl (2O.I0), wo Iinvc 

r. -- I1 

. 

R 

= 5,75 log - . (20.23) 

"* Y 

Pig. 20.7. IJtlivcm;rl vcloci1.y-11isl.ril~trlion Inw 3 

for smooth nnd rough pipe8 

6 

5 .. 

4 - 

C~~rvc (I) from rqn. (20.23). I'randtl; 

rl~rv~ (2) from cqn. (20.24). vnn K4rtnh: 

1 -- 

eurvc (3) from egn. (20.25). 1)arcy -. 

o* 

I

'I'ltr ~tnivvrs:rl voloci(,y-tlisl.ril,rtt.iot~ Inw c;ul I)c tlctlr~cccl also from von K:i.rmdn's 

sirnil:~ril.y I;L\V, cqn. (1!).21), wltonno we ol)t.:tin 

\rit,ll y rlcnoiing l.lro tlixt.nnc:c frorn t.lw w;tll. 'l'his rq~~;ti,ioll, sl~ow~r RS CII~VV (2) in 

Il'ig. 20.7, also n.grc:cs well with tllc c:xpc~rirnrr~t:rl v:d~~c:s, if x -: 0.30 is c:hoscn. 

1Gg11ro 20.7 coni,:~ins ILII :t~ltlit,innal mrvo (3) which is l)nsc(l nn f1, lhrcy's I

GI0 

XX. Turbulent flow throngh pip 

and from t h nniversal ~clocit~y-dist,ril)~~t.ion law, cqn. (20.14), we have 

which coml~inctl wit.11 aqn. (20.20) gives 

We can introduce the Rcynoltls number from 

so that we obtain front cqns. (20.28) and (20.29) 

According to this result the univcrsd law of friction for a smooth pipe should give 

a straight line if 1/dj is plotted against log (R da). This feature agrees extremely 

well with experiment, as seen from Fig. 20.9, whcrc the results of measurements of 

Fig. 20.9. Univernnl I:rw of friction for n n~nooth pip 

Cervc (I) rrom eqn. (20.3n). I'mwlll: cttrvc (2) rrota rqn. (20.5). Illnslsl 

I 

e Nikuradse 

Saph a n d W 

0 ~urself 

e Ombeck 

lohob and Erk 

m Stantonandhmd 

Schillerad~ 

d. Univcrd rcsint,nnce Inw for smooth pipes nt vory lnrgc Rcynolrln nilnil)crs 61 1 

many authors have been plotbcl. The nnmerical coefficients for tho avcragctl cnrvc 

passing through the experimental results differ only vcry little from the preceding, 

derived values. The straight line (1) passing thrortgl~ the cnpcrimcntal points in 

Fig. (20.9) can be represented by the eqrietion 

This is Prandll's w.niversn1 lnw of friclioiz for smodh pipcs. It hns been vcrifiecl by 

,I. Nikuradse's 1451 cxpcrimcnts up to a Reynolcls number of 3.4 x 10~ntl tho 

ngrcemcnt is seen to be cxccllcnt. From its derivation it is clear that it may bv 

extrapolated to arbitrarily large Reynolds numbers, and it may be stated that 

measuremcnfa with higher Rcynoltls nnrnl)crs arc, thcrcforc, not rcq~~irctl. V:dws 

computed from cqn. (20.30) are given in Tahlc 20.2. The ~iniversal law of friction 

is represcntcd by curve (3) in Fig. 20.1. 

Teblc 20.2. Coefficient nf resint.nncc for smooth pipcn in tcrmn of the 1L:ynolds nwnhnr; nee nlnn 

The universal equat,ion agrees well with Rln.sir~s's rqtl:~l,ior~ (20.5) up to R .= los, 

hnt, a,t higher V I ~ I I ~ l3lasi11s'~ 

S cqt~ntion dcvi~~~t,cs progrwsivrl.y tnoro from ~,III! r(:sttIl.s 

of n~e;~sr~remr~~L, wlicrras cqn. (20.00) maint,ains good ngrccrncrit. 

The flow of gnscs throngh srnool.li pipos at vcry higli volocitins was invcs1.i~l.ctl 

hy W. lh~~sscl (lU]. 'l'hc vl~ri~~tion in 1)r(*sstirc httg ;L pi~w I'I~ 1liIhw14, ~I:I.SS llow 

mtcs is rcprcsentnxl in Fig. 20.10. Thc num1)ers shown agninst tho curvcs ititlic::t\a 

the fraction of maximnrn mass flow through a nozzle of cqnal dinmctm and with 

cq11a1 stagnation pressure. The curvcs which fall off to the right refer to subsonic 

flow, whereas the increasing curvcs apply to supersonic flow. 'Che lathr curves 

include jumps to higher pressures and subsonic flow cFTectcd by a shock. The cocfficicnts 

of resistance are not markedly diflercnt from t.l~ose in incompressible flow,

XX. T~lrbrrlcnt flow tl~ro~rgh pipcs 

Fig. 20.15. Srconclnry flows in piprs of triancross-section 

(urllcn~atic) 

g~tlnr R I TCC~II~II~R~ 

~ 

Fig. 20.16. (hrvcu of constant vrlocit,y for n. 

rcrtnng~~lor opcn rliannol, nfkr Nikt~ratlsc (431 

* 

tli;igrams of sccontln.ry flows in t,riangul:ir and roct.angular pipcs are shown inXFig. 20.15. 

It is swn that, the sccondnry flow in (.he rectangular oross-section which proccetls 

from t.ho wall inwnrtls in Clic neiglibourhood of the cnds of the larger sidcs and 

of t.lic mitltlle of the shorbr sitlcs creates zoncs of low velocity. They appear vcry 

clcnrly in (.he pic:t,urc of curvcs of constant velocity in Pig. 20.13. Such secondary 

flows romc int,o play also in opcn channels, as cviclcnccd by the pattcrn of curves 

of const,ant, vrlooity in Via. 20.16. Y'hc maximum velocihy docs not occur near thc 

free surfiicc I,nt, at. about, one fifth of 1.11~ tlcpth down, and the flow in the frcc surface 

is not, at n.ll two-clirnensionnl a.s might have been expected. When the cross-section 

of the cllannol cont.ains a narrow region, transition does not occur simulbanoously 

ovrr t.licr wholo or the Ilow. I'or cxnmplc, in the rrgior~ within an acute angle of a 

tfrinngulnr cross-section, the flow remains laminar to very large Reynolds numk)crs, 

whcrc.~ in thc bulk il 1i:t.d tnrrlctl turhrlcnt long ago. Such a state of affairs is seen 

illustrntc~l wil,li tlic nit1 of Pig, 20.17 which represents the results of mcasuremenLq 

Fig. 20.17. Bollndnry betwrrn laminar 

nl tl t.llrlmlrnt flow in nn n(:utr, t.rinw 

gular 1 cl~annel, dct~rniincd visually by tho 

we of smoke injection, after E. 1L G. 

Eckcrt nnd l'. E. Irvinc [13] 

R, - h,drrulir. rndius = dh/2 

pcrfornlctl by E. R. G. J3cltcrt ant1 T. E. Irvinc [13]. At a Itcynoltls nilnil)cr of 

R == 1000, the flow remains Inrninar ovcr 40 per cant. of [.he hcight of 1,hc triangle:, 

t.lic region of lamirmr flow dccrcnsing :is the Itcynoltls ri~trrtl)cr is iric:rc::lsc:tl. 

13. Meycr 1381 invcst.ignt,ctl the prcssilrc antl vclociljy tlist,riI~irtion iri a Ilow 

through a stmight channrl wit11 :i cross-sc:ct.ion whose sllapo varictl but whosc crossscc:t.ional 

nrca rcmainctl const.nnt,. IIc 11sc:tl a chnnr~cl in wllic:li ;I cir~i~li~r (:rmsscvtion 

was gmc1u:illy trxnsf'ornictl int,o n roc:t.n.nglc: wilJi ills sitlos in t.lrc r:il,io 1 : 2. 

'I'rn.rlsition was clli:ctctl in hot11 tlirt:c:t.ions over two rlilTc:rcrit~ I~:ngths, :rntl it, w:is 

tlisc:ovort:tl I,hnl. I,hc prcssnrc: loss in l11t: ~~ortioii with Im.nsit,ion from c:irc:lc: to rc:c:l.:lriglc: 

co~isitlcr:hly cxc:cctlctl t.Imt in the oplmsito tlirc:c:t.iori. 

Most, pipcs llsctl ill cngir~coring sl.ruc:tirrt:s c:nrlnot, IN: rc:g:lrtlctl :w hoing I~~tlr:ir~lic:n.lly 

smoot,h, ;it, Ic:lslf att highor Itc:ynoltls rll~nil)crs. 'I'll(: rc:sisL;~nc:c: 1.0 Ilow olli:rc:tl 

I)y rongh walls is larger than tl1:~t irnplictl I)y thc prc:c:t:tling oclllnlions for smoot.ll 

pipes. Conscq~icnt~ly, the laws of fric:tion in rough pips ;rrc of grc:;lt pr:ic:tic::rl importmwc, 

awl cxp(:rimcnlnI work on thcrn I)c:gan vory wrly. 'l'llc tl(sirt: Lo c:xl~lorc: 

the laws of frictionof rough pipcs in a systc:matic way is frr~st~r;it~(d l&.t.llc: funtl;~rnc:nt.~~.l 

. - . - . . cliffi~~~t,.~~.that the numbcr .of parn.nict,cr~ tlcscribirig roilgllncss is cxtmortliri;lrily 

large owing to t11c grcnt tlivcrsity of gcornc:t~ic forms. If wo consiclcr, for cxnrnplc, 

a w:dI with . .~. itlcntjcal prot.r~~sior~s ~~~~~~nt:i~~~t~Ilccor~clr~siv~i.bI~:it 

its tlr:tg clt:l)t:ritls 

on .. lhc . ,, tlcnsitty -. -. - of ... :-.-. tIist.riOnt+t~ .. of siic:li rougl~nrsscs, i. c. on their nllrnl)t:r IFr unil, :Lra:L 

as we n.s on thr shapc antlhnight nr~tl, fin:llly, also on t.hc way in whtt:h they :m: 

tiist,ribtit,&l over i!s surface, It took, ~hcroforo, ;a i o r t.irnc ~ ~ to fortnnlat,~ oIn:w ant] 

- - - -. . . 

sirnik ~ W which S tIcscril~c the flow of flnitls tl~ro~gll rough pipes. I,. Tlopf [25] 

rn:do :i comy)rol~cnsivc rrvicw of 1.110 nllnicw)ils cnrlicr cxocrirnont,:d rtwults :~ntl 

found two l.ypcs of ro~lghncss in rthtiori to 1.l11: rcsist~in(:(: fnrmuh for rough pipes 

and open c:ltnnncls. 'l'hc first, ltilltl of rongllrloss c::ll~sc:s :,. rcsisl,:irlc:~~ whic:ll is proj)orlSion:d 

to tljc.glunrc .of the vclocit.y,; !,I~is~~l~cans t,h;~t, t,ll(: c:ocfficiorrt of rc:sist,:l.nc-c: 

. . . 

Is itdnpt:ntlcrlt~ of 1,hc ltcynoltls ni~rnhcr ant1 corrosl)orltls to rcIat,ively coarse n.ntl 

tightly sp:icccl roughness clcrncnb SIIC~ as for cxninplc c:o:srsc s:lnrl gr:tins gluctl on tJl(: 

surf:lcc, cerncnl, or ro11g11 cast iron. In st~nlt cxscs lllc: nalirrc of Lllc ro~lglltlt:ss ~:;III 

bc cxprcssccl with tlic aid of n sitlglc rougllncw p:~r:imc:l.cr k/l

Icror~~ ()I(, phj.sic*:~l ~)oint. of vicw it ml~st I)c conclutlnd that 1.11~ mtio of t,hc height 

(IS prot,t.~~sions t,o t,l~r I,o\~~~cl:~r,y-l:~yc~r t,l~ic*lz~~rss sholtltl I)(: the tlct.rrmining factor. In 

l);~rli(~~~l:~r, 

I,IIv ~~II(-IIOIII~~I~O~I is cxl)rcl.c(l 1.0 rlrl)n11(1 on 1,111: I,l~idrrl~w or t,l~r In.minar 

s\~lr-l:lyc.r d,, so I.II:II. k/ii, ln~~st, I)c rrg;~rclvtl :IS an inlport,atlt tlirnonsionlcss ntltnbcr 

\,:Ili(.l~ is (*l~:~r:~(*l,~~risli(; of l,hf: ki11(1 of' ro11g~111~:ss. If, is d~ar 12hn,k roltgh~lcss \trill ca11se 

IIO i~l(-rt*:~sr in ~I&I:III(~ ill (::ISVS \vh(w: I.II(\ I)I.(~~IIS~OIIS arc: SO s111:~ll (or 1,111~ 1)011rl(l:~ry 

I:~yrr is so II~ick) t.llnl, I,llc-y ;~rv all c-orlt;rit~c.~l wilhit~ 1.l1c I;lrr~ic~:rr scrl)l;~jw, i. c. if 

k < (St, and t.11~ wnll may I)c cwwidcrrtl I~ytlr;ir~lic:~lly smoot,lt. This is simi1a.r to the 

:I,I)s~~IIT or t,l~c ~ I I ~ I I I ~ of ~ I ( ro~~gltn(w 

* ~ on rrsist:tncc in lln~g~~~~-l'oisc~~ill~: flow. I:I,sII~~:~)I(-II~,s 011 ro~tgl~ pip(-s 11:l.v~ IWCII 

w1.1.ivg1 o111, 1)). .I. Nilr\~r:t~lsc [.4(;J-\-, ~vllo IIS(YI (~irv111:lr 11ips (-ov(~rc~l 011 I,IIc it~si(Ic $IS 

t igltt Iy :IS ~)ossil)lv \vit.l~ s:~t~tl of :I t1cfinit.r gr:\i~~ sizt, glc~cvl 011 ($0 l llc w:~ll. 13y c-hoosing 

1)il)cs of v:~rying (Ii:~n~r~t.rrs n~ttl Ity (hnging IIIC sim: ofgrnit~, IIC wn.s :10lc 1.0 vary the 

rc-l:l( it.(, ro~~gl~ncw f.,/ll liom nbol~t 1l.500 1.0 1/15, 'l'ho rrgl~l:~rit.irs of I)rhaviour tlisc~)v(.r~~l 

(111ri11g 1.l1r ( WII~SC of l.I~rsv mr:~s~~rrtn(mts can .bc: corrrl:~t.ctl \vit.l~ l.l~osc hr 

stno01 11 pil)~*s ~ Ia, I sitt11)lc tn:~tlnw-. 

\\'v sII:III Iwgi11 I)), (l(w.ril)i~~g Nik111x11sr's III(~:LSII~(-I~I(~II~S :III(I w: SII+III t,It(:n SIIO\V 

t.11:11. 111r rrl:~I~io~r I)c~l~wrc~~~ t.l~c ~wisl,:~nc~r hrlnrtl;~, nntl I.llc: vrlori1,y tlisl.riI)~ction, \vl~ic.h 

\\r I;)IIII(I rilrlirr ill t11r (.:IS(- 01' sn1ool11 oil~vs. van I)c esl.rntlr~tl t.o t,l~r rnsc of ror~gh 

l)il~~,s ill :I II:II 

IICII W I ~ . 

t 111.whnl. Follows wc, sl~nll 11sr t.llr sylnhol ks Lo rlcnotr LIIC grain size ill Nikuratlnc's narltl rorlgll- 

wsu, wnrrvit~g I.hc ny1111)ol k for :III$ ol,llrr ltillrl of rollghncsn. 

: rl'l~c nu~~~t*rir~:ll ~:~IIII.s of t,? I-*/v q11otw1 11rrr will Iw hivc(1 1:tI.w ~~OIII 1.11~ velocity dist.riht~inl1 

law. Tl~cy :IM: v:lIirl OIII~ for rrw~l~~~rssrs ol~k~incd with mtld. 

Fig. 20.18. Rosiatance formuln for rough pipes 

The size of t,hc roughness is so smdl that, all prottrusio~~s :ire conf.ni~~c:tl \vit,hit~ the 

laminar snl)-layer. 

Protrusions cxtcntl pnrl.ly out.sitln Ihc I:rminnr sub-l:~ycr ;1n(1 1.11~ atltlit,io~~:~l ~.rsisl:~nty~, 

as cornp:irc:d with a srnoot.l~ pipe, is ~n:iitrl~y cl11o t>o lhc limn tlr:ig c:spr.ric.~~c.c-tl Ipy 1.11~ 

pl'oLrusions in the bo~~rltlary layer. 

Vclocity dis~ribution: '['he vc.Ioci(.y gr:uliotlt no:cr :I. rollfill \v:1.11 is Ivss slwp I.Il:rr~ 

f.l~al. nrar a sn~oot.l~ onc, ns can I)c scrn from Vig. 20.1!), in whicl1 l.I~r vcloc~i1.y ~-:~l.io 

71/11 has Gccn plol~ctl against the distance ratio y/ll for a smoolh a~ltl hr scvcwl

618 XX. Turb111cnt flow through pipen 

rough pipes, all Imving heen measurotl within the range of validity of t h square 

resistance law. Expressing thc velocity distribution function again by a power for- 

mula of t h type of cqn. (20.6) wc obtain cxponcnts of # to &. The variation of mixing 

length over the cross-section calculat,ctl from these curves has already been plotted 

in Fig. 20.6 from which it is seen that it is exactly the same for rough and for smooth 

pipes. It can be represented by the rmpiricd cquation (20.18). In parlicular, in the 

neighhourhood of the wall we have 1 = x ?/ = 0.4 y. 

r 

Rg. 20.19. Velocity diatri- 

bution in rough pipen, aftm 

Nikuradse [40] 

It. follows, thrcfore, that, the logarithmic law for velocity distribution, eqn. 

(10.2!)), remains valitl for rough pip, cxcept that tho con~t~anl of intcgmtion, yo, 

must 11c given R clilTcrcnt numcricnl vnl~ic. Furthormorc, it is natural to malto it 

proport,ional to the roughness height k,, i. e. to put y, = y k,, so tl~at cqn. (19.29) 

now becomes 

I 

tho c.onst,nnt, y still clrprtltling on t,lw nat,urc of thr pnrt.icular roughness. Comparing 

this cclui~tiou with J. Nikr~mtlsc's measurcmenk, we fild that they can, in fact, be 

rrprrsc~ltstl by an equation of the form : 

where the constant 2.5 = l/x = 1/04, whereas I3 assumca different vnlurs for tllc 

tl~rrr rauges of roughness discussed previously. In tlm rango of the complcl.cly rougl~ 

regime, we have I3 = 8.5, so that in this region 

u - = 5.75 log $ + 8-5 (completely rough) . 

v * 

F 

(20.32a) 

The corrcspontling slmigllt lirto is scon to agree well with bhc rcsnlt~ of measuremcliL, 

Fig. 20.20. Gcncrt~lly spcalring M is a function of the rougl~ncss Itcyrmltls 

nurnlm v, ks/v. Thc valuo which corrcapords Lo I~ytlraulicnlly smoolli flow Idlows 

at ol~cc from dqrl.s. (20.32) ;~ritl (20.14), and is 

v* k, 

B = 5.5 -1- 2.5 In -- (I~ytlmnlically smooth) . (20.33) 

U- u R R 

- -- - = 2-5 Ill -- = 5.75 log -- , (20 23) 

v* Y Y

620 XX. Tnrbnlcnt flow tlvough pipes f. Rough pipes and cquivalcnt sand rough~icsa 02 1 

Fig. 20.21. Ilonglincss futlction Jl in hrn~s of ?I, ks/v, for Nilturarlsc's sand ronghncna 

t'urvc (1): I~ydrr~tllrfilly stnootk, eqn. (20.3:l): rurve (2): II - 8.5; completely rongl, 

onc:c morc. It li:~ hccri fount1 to :~pply to smooth pipcs in conncxion wit,h Fig. 

20.7. In ortlcr to soc morc c:lcarly the conncxion bctwccn the velocit*y distributions 

for smooth and roiigh pipes, it is uscful to re-plol tlic rcsults for rough pipcs in the 

form of a rrlation I~cLwccn thc tlirnc~~sionlcss vcloc:ity IL/V* = 4 and tho Rcynolds 

1iuni1)cr y v*/v = 91, as was tloric in cqn. (20.13) ant1 Fig. 20.4 in rolatiori to smooth 

pipcs. Writing cqn. (20.32%) lor the ror~gh pipe in thc form 

= 5.75 log v: + Dl (complctcly rough) , 

v* (20.33a) 

and comparing it with cqns. (20.33a) and (20.32a), we olltain 

ks "* 

D, = 8.5 - 5.75 log - (cornplctoly rough) . (20.33 b) 

,. 

l his vclocily tlisldnition is sect1 plothtl in Fig. 20.22, .alter N. Scholz [W]; it 

rc:prcs(:rit,s tlic voloci(.y rlishibulion for smooth pipcs as wcll as that for rough pipes, 

in acaortlnncc with cqn. (20.33a). 'L'hc diagram consists of a family of parallel straight 

litics wiIih V+ kS/v playing thc part of 8. parameter. 'L'hc value of v, ks/v = 5 corres- 

ponds 1.0 hytlrartlir::~.lly smooth walls, thc range bctwcen o, ks/v = 5 to 70 corresponds 

t,o I.r:~nsit.ion frorti t.lic Ilyt1r:rulically smooth to thc cotnplcLcly rough regime, and 

for a, k,/v > 70 1.11~ flow is c:omplcl.ely rongll, as mdntioncd previously. Tn particular, 

t.llc: tliagr:rrn shows cl~::~.rly l,llat the laminar sub-laycr which reaches as far as ?/ v*/v = 5 

in hy~11.:111li(::~Ily stno0t11 pip, has no itnpt~rt.an(:c for c:omplctciy ronglt walls. 

Ildnti~m lwlwc.c.t~ rc.sistnncr: fortnula and velocity distributin~~: 'I'liis type of rcla- 

I ion (.xist.; for rottgli pipcs nlso :r11(1 ran lw tlrdnc:cd in the same mnnnrr as was clone 

Fig. 20.22. Universal velocity 

r o e or turuent o w 

tl~rongli pipcs which is valid 

for smooth as wcll as for rongh 

walls, after N. Scholz [65] 

fO 

+-f 3 

-+ 

(1) amonlh, lnminnr rublnyer, + - q 

(2) amonlh, twbulcnt. cqn. (20.14) 

(3) ro~~gll, turbalcnl, eqn. (20.33a) 

m 

Ld 

with D, from eqn. (20.33b) 0 

. . .- 

li j i i b 

in See. XXd for the case of smooth pipes. The relation is simplest for tile complelely 

rough regime. We begin by calculating the mean velocity from cqn. (20.23) in the 

same way as in eqn. (20.26): 

13. L- I1 - 3.75 v* . (20.34) 

Substituting U = v, (2.5 In R/ks 1- 8.5) from eqn. (20.32a), we have 

i. e. 

C/v, =. 2.5 1n (R/ks) 4- 4.75 or 118 = ( ~,/6)~ = [2.5 In (R/ks) -1- 4,761-2 , 

1 = [2 log (Rlk,) + 1.G8]-2 , 

which is the quadratic resistance formula for complclcly rough flow. It was first 

derived by Th. von KLLrmiLn (Chap. XIX [17J)from thc simi1arit.y law. A comparison 

with J. Nikuradse's experimental results (Fig. 20.23) shows that closer agrrcrncrit, 

can be obtained, if the constant 1.68 is replaced by 1.74. ITence the resislnnce formula 

for the completely rough regime becomes 

The experimental results lie very closc to a straight line in a lht, of 1/1/1 ;tpinst 

log (Rlk,) and it is worth noting t.lrnt rqn. (20.35) may bc ;q)plictl to pipes wiI.11 nori- 

e--- -- 

t An equation which corrclatos tho whole 1r:tnsiLion rcgion from liytlrnnlirally stnootli to 

completely rough flow was cstabliuhcd by Colchrook and Whik [GI: 

For ks - + 0 t.his equation transiorms into cqn. (20.30), valid for Iiyclranlit::~lly nn~oot.l~ pipos. 

For R + m, it transforms into cqn. (20.35) for Lllc con~plctdy rough rcgimc. In t,lic: tr:msition 

region eqn. (20.35a) plok 1 against R in a way wliicl~ rrscniblns the curvc labcllcd 

"commercially rough" in Figs. 20.18 and 20.25.

circ111n.r cross-sectional nrcas if R is rrplnccd by the hydraulic radius R, = 2 A/C 

(A -- area; C -= wettcd perimeter). 

If, is also c:wy to tlcrivc t,he relat,ion between the resistance law and the velocity 

tlis(ril)ut.ion it1 t.hc Irnn..~ilion rrgion,. From cqn. (20.32) we have 

On (.he other h:~.ntl, from cqn. (20.34) we ohtain 

and the preceding equation gives 

Fig. 20.23. Reaiatance forrnul:~ of snntl- 

roughened pip in completrly rouglr 

rcgitnc 

Cllrvc: (I) from cqn. (20.35) 

'I'hr last cquat,ion dc:t,rrminrs tlrr. valr~c of the resist.ancc conl'licient 1 if the cunst,ant 

I3 is Irnown from the vc4ocil.y tlisLribntion. 011 t,hc ot.l~cr hnntl, cqn. (20.36) can be 

~tsrtl 10 drtcrminc t,hr ronst;~,nt I{ ns :L function of a, k,/v either from t,hc velocit,y 

tlist.ril)t~i.ion or from 1,ho resist,nncc formuln. 'l'hc plot in Fig. 20.21 agrees well with 

t.1tr rrsnlLs from cii,l~rr of t,l~rso ~nct~ltotls and provcg (.hat the calcnlat,ion of t,hc 

vc~lorif~y tlistril)uI.ion front I,III! rt:sisfs:~nce I'orrnr~ln is prrtnissildc for rough pipes too. 

'l'hr 1imit.s bctwc:rn the thrcr rogimcs, namrly t.11oso of l~)~tlrn~~lirn.lly smooth flow, 

the t.rattsil.ionnl rrgitnr, :mtl t.hc con~plvt,cly ronph regime, whiclt 11n.v~ I~rrn givcn 

c::~rIicr, can t~ow I)o t.nlton tlircclly from Fig. 20.21. Wc have 

and plott.ed in Fig. 20.4. The limit of t,hn I~~clmt~lically smooth rcgin~c 11, k,s/~l - : 5 

givcs t,he thickness of the laminar sub-layer ant1 coincitlcs with 1.h~ limit of Lhc r:iny,c: 

in which tho Ilngen-Poisc~tillc, purcly Intnin:r.r, vrlot:it~-tlist~ril~~~f.iot~ I:lw rc.l.:~it~s il,s 

validity. 'l'hc! litnit, of v, ks/v - 70 for 1.h trxnsil.ion:rl rogirnc: ;dso c:oinc%lrs \r ilh 1.11~: 

point whrrc 1.I1o mcnsurcd vclority tlistriImt,iott goes over ta.l~gct~I,i:~.IIy ittt.o 1.h~ 

log:lrithinic formula (20.14) in fully turbulent frict,ion. 

S. Goldstein [19J sncrectletl in detlucing the litnit. of v, ks/v == 5 for l,lto hytl~irtlically 

smool,l~ regime from the criterion t,l~nt at 1.hnd point a von I

dimensions 

Fig. 20.26. 1tc.sist.ancc of commercially rorlgh pip after 1,. P. Moody [40] 

Fig. 20.24. IZc~nlts 

of rnea-mrernents on 

regrllar roughness 

pattcrnr after H. 

Schliehting [G3] 

k - nctllsl hcight ol 

protrnsinn; k, - cqui- 

\rirnl sand rouglanesrs 

k, - oquivnlcnt rnnd rni~ghners, to he dctrrminnl in psrtim~lnr cwcs frnnl tlic suxillnry graph in Fin. 20.26. Tlm 

hrokrn llnc Inclirnln tho houndnry of tlm complctcly rongh rcpimr wlwrc I.lte qaadrslie law or friction n~pllcs 

g. Other types of tor~ghnms 626 

in Fig. 20.18, wlrere rcsult,~ of t.hc mc~s~lrc!rnc:~~f,s c~r-rictl o ~~t, by 1%. I ~ I I W :I,III~ 

P. Cnlavics [3] on a "eomrncrcially smooth" stccl pipe wilh n flow of hol, w:t(,rr ;Ire 

seen plotted together with Niltumtlsc'a vnlrrcs for pipes ro~~glronotl witch sn,ntl. 

The clifliculty in applying tho nbovc cnlculnt,ioris Lo ~)r:~ct.ic:n.l c:~st~ lies itr (.II~. 

fact t,hnt, the value of roughncss to 1)c ,zscril)ed t,o a given pipe is not knowl~. Vcry 

extensive cxpcrimcntd results on tlic rc:sisLnncc of cornrric:rt:i:~lly rough pilci ;IIY: 

ro~it~ninctl in n pnpor by 1,. P. Mootly [N]. Fig. 20.25 SIIOWS I.IUI.(~ l.11t- gt.:1.1111 01. 2 

ngnind R for tlilrcro~rl vrht:s of k,/d is it1 csswt:c itlo~~lit::ll will1 ,I. Nilt~~l.:~(Is(~'s ~li:~. 

gram in Fig. 20.18. 'rhc intlivitll~al valucs of ccluivnlcnl, rcl:rtivc s:~t~tl rot~glt~~t>ss 

ks/d can be obtn.inccl from thc nuxilin.ry grnplr in rig. 20.20 wllorc: piprs nv(: SWII 1.0 

have bccn nrrangctl in thc ortlcr of v:~l~~cs on Nilu~r:ufsc's c:q~~iv:tlcnt s:~.~~tl-ro~lgllltt~ss 

scale. This follows from tJic fact tht the vnlt~cs of 1 ill l,c!r~ns or ks/d ngrt:t- wi(.l~ 

Nikuratlse's velurs from Pig. 20.18 in the complctcly rough reginlo. 'l'lio h.:ursil,io~~ 

from hydmulicnlly srnoot.li conditions at small ltcynoltls rrrtml)ors to cotnplrk 

rouglincss at large Itcynoltls nurnl)crs occurs n~llcli more gr:ullrnlly ill sr~cl~ cw~rr~ncrei:~l 

J'ig. 20.26. Auxiliary diagram for the 

evaluation of equivalent relative sand 

ronghnesa for cotnnrercial pipes, afkr 

1,. P. Moody [40] 

a) rivclcd steel 

b) rrinrorcrd concrctr 

c) wwc1 

cl) cnal iron 

a) ~alvanizcd ntccl 

I) blt~~mcn-rontod slcel 

g) olrllctllrnl nnrl fnrgeil slwl 

11) clrnwn plpcs 

It is somctirnos irnpossiblc to fit cornrncrc.ially rough s~~rfnccs snbisfacLorily into 

the scale of sand roughness. A peculiar type of roughncss giving very Inrgc values 

of thc rcsist.sncc cocfficicnt was tliscovered in thc water duct in t,hc vnlloy of t,I~c 

15clwr 168, 821. This pipe hntl a tliamctcr of 500 nun and nftcr n long ~)criotl or tts:ip 

it was noticcti t,hst thc mass flow tfecrcasctl by more t,hari 50 per ccrit. IJpo11 c.xntnination 

it was found that thc walls of the duct, wcrc covcrccl with a riblike tlcposit, 

only 0.5 mm high, the ribs being at right anglcs t.o thc flow tlircct.ion. Tlrtls thc 

geometricsl roughness had the small value of k/R = 1/500, but the effect,ive sand

626 

XX. 'rnrb~rlent flow through pip@ h. Flow in cttrvctl pipes nntl rliffuscrs 627 

roughness showctl vnlurs of ks/R = 1/40 to 1/20, as calculated from the resistance 

coefficient wl~icli was, in turn, dctcrmincd with the aid of the mcasnred values of 

niasn Ilow. It appcnrs, therefore, that rib-like corrugations lead to much higher rcsistances 

thnn sand roughness of thc name absolute dimension. Extcnsivc experiments 

on tho increase in the rcsistnnce found in commercial ducts, for example in mine 

shafts, can be found tlcscribed in a paper by E. IIuebner [26]. 

Purt.licr tlct.nils concerning t,hc rrsisl.:~ncc offcrcxi to flow by rough wnlls, perticularly 

t.hosc due to single protrusions, will be givcn in Chnp. XXt in conncxion 

with t,lic (lis~nssion on t.hc r~sistnncc of flat plntrs. 

h. Flow in curved pipes ntd diuu~cr~ 

Carved pipca: 'l'lir prcc.rcling considcrntions conrcrning pipc flow n.rc valid only 

for sl.r:ligI~t~ pipes. JII curvcd pipes thcrc rxisb a, scronthry flow hcrn~wn t,hc pnrtielcs 

ncnr t.hc flow axis wllicll 11:tvc a higl~er vclocil,y arc act.ccl upon by n lnrgcr ccnlrifugal 

force t.hnn t.ltc slower pnrticlcs near the walls. This Icntls to 611e cmcrgcnce of a 

sccontlnry flow which is tlirectctl ontwxrds in t.hc rcntrr and inwards (i. e. towards 

t.ht ccr~t~re of rr~rvnt~nrc) nrnr t h wall, Fig. 20.27. 

r 3 

J he inflr~cnrc of rt~rvnt~trr is strongnr in 1nmin:~r than in t,r~rl)~~lrnt, flow. C. AT. 

\Vllitc [RO] untl ]\.I. Atllnr (21 rnrrictl ont. cxprrimonl~n otl 1n.mirmr Ilow. 'l'l~c! I~~rl~t~lcnl, 

case was invcsLigntcd c!xprdrnont.nlly by 11. Nippcrt [47] and 11. ltichtcr [5G]. 

Theoretical cnlculations for the laminar case were carried out, by W.R. Dean [lo] 

nntl RI. Atllrr 121. '1'11e rl~a.r~ctrcistic tlimrnsionlcss vnrinl)lc, whir11 tlct.crmincs t.1~ 

inllrwrlcc of r~~rv:~t~~rc it1 /he lmni~mr cose, is the Deem number 

Fig. 20.27. Flow in n c.nrvrtl pipe, aflrr 

Prnntltl [52] 

may be u sd for values of the parameter D exceeding about 102." TThc rcsults of 

mcnsuremenB urc approximated with a higher dcgrcc of precision by the following 

empirical equntion, first given by L. Prnndtl 1531 : 

This cqunI,ion givcs good n.gmcmcnt wit,h cxporirncnLnl rcsulta in 1.h~ r:l,nfio 

C). M. Whil,c [RI] has Pouncl thal thc rrsisl,nnc:r cocfkir~~t for lr~rhulwrl lloro ill a. 

wrvrd pipc can hc rcpresentctl by the cquation 

whosc form indicates rlcnrly that thc ])ran numbrr can no longer srrve as n ~llitnl~lr 

independent variable. In more recent times, JI. G' Cuming [R] carried out an irivcsti- 

g,ztion into tho phcnomrnon of scrondarg flow in rurvccl piprs. 

in wlricl~ JZ -T= 2 D. If the nt~rncri~~d cocffiricl~t, o~~l.~itIc tlhe I )~~C~I~IICSVS is rrltl:tc~t1 

by 0.101, the equation gives good agreement with experirnentnl res~~lt~s in the range 

of K > 30. 

In the ti~rbulcnt case Ito 1271 hns sl~own thcoreticnlly that the ratio of 1.11~ ~.osintance 

coefficients, I/,Io, may bc exprcs3ed in terms of t.hc ditncnsionlcs-q vnriahle 

R (R/r)z. The experiment.al results of Jto [28J can hc represent,ed with sufncirnt i~cc~lracy 

by the equations mentioned in thc footnote. 

In flow through a bend or elbow t.herc is not only some loss of enorgy within t,he 

bend itself, bnt n part of the Ions protl~~ccd by the bent1 hkcs place in the stmight. 

pipe following it. 15xtensive measurements of the loss cocfficicnts for smooth pipe 

bends and n correlation of results wore given by 11. Ito 1291. Thcorcticd rrs~~lt.s nrc 

reported by W. M. Collins et ai. [Sb]. 

In flow through a radially rotating st,raight pipe, n. secondary flow sirnilnr t,o 

that found in a curvcd pip is sct, up by the action of n (hriolis forc.n; it. fiirrs risv 1.0 

n Inrgc incrcc~sc in resisl,r~ncc. I3xt,c11sivc: ~ncnsurrrncn~a :1.11cl 1.11corct.ic:~l c:t~lcl~l;~l.ions 

on this subject were carried out by 11. Tto and li. Nanbu 1301. 

t H. Ito [27] givcs: 

and 

These differ somewhat from, hut are in general nprecmcnt with, C. M. IVl~itc'~ cqnnt,iot~ RI)OVC.

628 XX. T~lrbnlmt flow through pipes i. Noa-shady flow throngh a pip 

15xttcnsivc mcasurcmcnt~s and tlicorcLica1 calculations on frictional losses in 

turbulcr~t flow hnve also bcen carried out. hy R. W. Dctra [ll] who includcd curved 

pips of nonrirculer cross-scclion in his investigations. It is found t,haS the resistranee 

offered by an cllipt,ic pipe is grcater whcn the major axis of the ellipse lies in the 

plnnc of CII~VA~JI~C than when it is pcrpcndiculnr to it. 

- - 

11:. l

630 XX. 'I'r~rl~ulrmt. flow t.l~rouah pipc:s 

j. Drag reduction by the nddition of polymers 

Tn turb~~lcnt flow, t.11~ prrssnre drop in n pipe can be considerably reduced relative 

to eqn. (20.00) by t,hr ntltlition of sn~nll qunntitics of polymer particles. In laminar 

flow, similnr nrltlilivcs leave the pressure drop prncticnlly uncl~angrd. The extent of 

drag rcd~tetion tlcpcnds on the tnolcculnr weight of t,he polymcr and on its concentration. 

7~hc gmph of Fig. 20.29 dcucribcs 1.11~ rcsist,ancc cocflicicnt,, 1, as a function of 

tho licynoltls nurnbcr R = S'd/vp - where vp is the kinematic viscosity of the solut,ion 

- for dill'ercnt valucs of the conccntrntion c of the solution. 'rhe measurerncnts 

were perfortnrtl by R.M1. I1nt,ct~son and P. 11. AbcrnntAy [50nJ. As the concent,ratio~~ 

incrcnscs, the rrt1ucc.d rrsislmce cocfficirnt tends t,o a curve of maximum reductiot~, 

curvc (3), indicated by 1'. S. Virlc 1771. 'The diagram of Fig. 20.29 tlisplnys poitits 

obtnincd at two measuring stations along the pipe. The diflkrence in t,he values 

of thc resistnt~co coefficient nt hhosc two sections is cxplsinetl by the fact that the 

polymer molecules ?re tmn npnrt in the turhulcnt st,rcarn resulting in an effective 

tlecre~se in concentration in the downstrenm direct,ion. In spite of intensive 

rcsearch into this phenomenon (c/. the pnpcr I)y M.T. IAnndahl [36]), no satisfactory 

cxplanntion for its occurrence has yct been advanced. Nevertheless, experiment,^ 

demonst.rate unmiutnlrahly that thc retl~~ction in tlrng is linked to changes in thc 

stxncture of tthc tnrlmlrncc. The proccss is bcst illustratetl with the aid of t.he experimcnt,n.lly 

dct,erminctl ve1ocit.y-tlisttributsion Inws. 

nn - kinrn~ntir visrvelty of t,hr rr~lutlon 

0 - ~wnsl~ri~~~ alntlcrn at 214 cl frrrnl lolet 

A - n~rnsrlrin~ st.nl.lon at 1641 d from inlrt 

(1) lnn~irrnr flow with 1 = 64/R 

(2) twt~elcnt now. Nnwtonlan nllla, elin. (20.6) 

(3) nryn~ptots for rnaxin~nnl rlrn~rccl~~rlionnlter 

1'. S. Virk [77),rqn. (20.46) 

Fig. 20 2!). Ilroistn~~creorfficir~~t, 1,of ~111ootl1 pipes in t~~rbulctrt flow of polyn~ersolutiona ns nfunction 

of II.~ynoIds nr~wbrr as ~nrns~~rrd by R. W. Pntcrson nntl F. H. Ahernathy r60]. Solntionn of 

1mIyc*l I~ylcnr oxide of givrn roncentrntions r ; 1 pp111 dcsignatcq I g of polyn~rr per 106 g of watt-r 

Arc:r)rrling l,o I'.S. Virlc [78, 793, it is necessary to distinguish three velocitydist.ril)~~l,ion 

zones: 

r 

$ 

(1) 'I'hc Intninnr snl~lnyer (0 < 17 < 10); this corresponds t.o curve (I) in Fig. 20.4. 

(2) 'IYic fnlly tlovrlopc.tl t.nrhnlent zone. TTcrc the distribr~t~ion follows the law nccortling 

t.o rqn. (20.13) with A 1 = 2.5 rrgnrdlnss of th- physicnl propertics of the 

solut,ion. 'l'hc c:onsl~:~.nl. 14 vnrics st,rongly wi1,h tlw conc~nt~r~tion. 

(3) 'I'lic zonr tlcsrril)rrl as elastic^" 'I'his zone plnrc,s itsrlf I)ctnrrn 1 hr In n~innt. 

sublayer and the fully tlrvcloprd turl)ulcnt region 1771. llerc fl~c vrlocity is 

rrpresrntrtl by the log:withtnic Inw 

. . 

Illis law is vnlid For nll int,ent,s t~nd prposrs ILN fn,r :LS 1,111: c:c~tit.~~v of the* pil1c Sot, 

sul'ficirnl.ly high concc~n tmtions. In iinalogy wit.l~ ccln. (20.30), we c:nn in tcg~.;~t.c it, 

to drrive t.11~ rcsistn.nce formuln 

( I] ;rkeret., J.: (:rcnzscl~ichtrn in gcrnclcn unrl gekriin~~nI.rti I)ill'~tsorrn. 111'1'~1bl-Sy1nposi11111 

t(reiburg/Br. 1!)57 (H. Gvrtler, etl.). Ilcrlin, 1958, 22-37, 

[In] Ackcwt, .I.: AspcoLq of inlarnnl Ilow. lfll~~iii nrrel~~ini(.~ of inf.c.rnnl flow ((:. Sovrirn. c.rl.). 

lhcvit*r I'ublinhing Conri~nny, AIIIsI~~~~~IIIIII/I,~II~IoII/Nc~~ 

l'ork, l!)l;7, I - - 24. 

[2] Atller, M.: Stroniung in gckriim~nten Jlohrcn. ZAMM Id, 257--275 (1!)34). 

131 Bauer, B., and (Mnvics. F.: I':xprrin~cntdlc unci tlioorctiscl~o Untrrsncl~r~ngon iilwr din 

Rol~rreibung von Heiza~nsserleitur~~~~. Mitt. d. J"rrnl~rizk,afL~wrkc:~l tl. IC'I'II Ziirich I!):l(i; 

see nlso: F. Galnvics: Scl~\wizer Archiv 5, 12, 337 (1!)3!)). 

[4] Becker, E.: Beitrag zur ~erechnung von Srk~~~~tl:irnIrii~~~r~r~ge~~. 

ZAhlM Jfi. spccinl issue, 

3--8 (1956); seo nlso: Mitt,. Max-l'lanck- Inst,. riir SI.rii~~~r~ng~forsrl~~t~~g 13 (1!)5li). 

[4nj L

632 XX. Turhulcnt flow through pipcs 

1141 Eckcrt., 15.12. G., nnd Irvine, T.F.: Incon~prcssible friction factor, transition nnd hydrodynamic 

cntrnncc-length studies of ducts with trinngular rind rectnngulnr cross sections. 

Pnpcr prcsenbd nt Fifth Midwcstcrn Conf. on Flnirl Mecl~. 1957. 

[IS] Pox, H.. W., and Itline, S.J.: I%w regimes in curvctl subsonicdiffuscrs. J. Basic Rng., Trans. 

ASMIC $4, Scries I), 303--312 (1962). 

[15n] I'renkicl, F.N.. Lnndal~l, M.T.. and I,umlcy, -1.L. (ed.): Stn~cturc of turlmlcnce nnd drag 

rotiuctiou. IU'I'AM Symposi~~m, Washington 1). C., 7 -- 12 Jur~c 1!)7fi, 'l'he Physics of I~l~ricts, 

20, No. 10, I'nrt 11, S 1-S 209 (II177); scc nlso 13.11. l'on~ in I'roc. Intern. Congr. Rheology, 

North-Holland, A~nstcrtlarn, 1949, Scc. 11, IB5. 

[I61 I~ribch, W.: IGinfluss tler Wandrauhigltcit nuf die tnrhulc~~te Gcscl~wir~tligkcibvcrtcil~~ng 

in Rinncn. ZAMM 6, 109-216 (1928). 

[I71 k'romm. I

634 XX. Turbnlont flow tltroogh p ip 

Seifertlt. R.. anct Kriiger, W.: Uherrmclmd hohr rtoibangaziffer einer Fernwnaserleitung. 

Z. VDl $2, IH!) (19.50). 

Spnlrling, 1). B.: A single fortnuln for the "law of the wall". J. Appl. Mech. 28, 456-458 

(JMl). 

[08h] Sobey, J.S.: Invi~rid ncrondnry tnotionn in n tnbe of slowly varying ellipti~it,~. JFM 73, 

G2l -039 (1976). 

[an] Sprmgcr, 11.: Mr~nungrn nn J>iffuooren. VDT-Bcr. 3, 10-110 (1955); see also ZAMP 7, 

372--374 (J950). 

[70] Sprengcr, ti.: 1':xprrimentdle Untomnchnngen nn gernden und gekriimmtcn Diffusoren. 

Mitt. Jn~t. Aerodyn. EII'H Ziiricli No. 27 (1!)59). 

I711 Stanton, T. R.: l'lie mccllnnicd viacoaity of flnida. Proc, Roy. Soe. London A 85, 360 (1911). 

1721 Stnntnn, T. I

636 

XXI. Td~nlcnt houndnry layers at zoro promure gredicnt a. The smooth flat platc ' 

R = I/, 1/v (Urn - frcc-strca~n vclocity; 1 - length of plate) are so large t1hat they 

cannot bc strbjcctcd to mcasurcment in a laboratory. Moreover, even at motlerate 

Itcynol(1s numhcrs it is murh morc difficult to carry out mcasuremcnts in the 

bounclnry Iaycr on a plate than in that inside a pipe. It is, therefore, very atlvantageons 

that it is possiblc to calculate the skin friction on a plate from the extensive data 

availnblc for piprs hy thc nse of a method dne to 1,. Prandtl [40] and Th. von 

IGirmi~n [30J. This ralcnlation of t,hc skin-friction drag on a plate can he rarrictl out. 

both for smooth nnd for rough walls. A good summary of this work was given by 

P. R. Hama [23]. 

a. The smooth flnt plate 

The approximate method to bc applied to this problcm is based on the momentum 

integral equation of I)onntlary-laycr theory as givcn in eqn. (8.32) of Chap. VIIf, 

the vclocily prolile ovcr the boundary-laycr thickncss bcing approximatccl by a 

suitable empirical equation. Thc morncntum equation thcn provitlcs a relation 

hrtwccn t,hc chr~mderislic prumelers of thc boundary laycr, i. e. bct,wcen displacement 

tdlicltness, morncntum thickncss and shcaring stress at the wall. 

Tn thc following argument we shall assume at first that the bountlary layer is 

turbulent already at thc leading edge (x=0) antl we shall choose a system of coortlinatcs 

as shown in Fig. 21.1, h tlcnoting the width of the plate. The boundarylayer 

thickness O(x) increams with x antl on translating the tlatn for n pip into 

t.hose for a j)lat.e we not ic:c that the maximum velocity, U, of the former corrrsponds 

to thr free-stream vclrx:ity, U,, of thc lnttcr, the rntlius, R, of thc pipc corresponding 

to the boundary-laycr thickness, b. 

At this stage wc introduce with I,. I'ranrltl the fundamental assumption that the 

vr4ocit.y dis~ribution in tlic bonnclary lnycr on a plate is idcnticnl with tht inside 

a c.ircular pipc. This assumption cannot, ccrtainly, be exact, because the velocity 

rlist,ribnt.ion in a pipc is formed unclcr the influence of a pressure gradient, wllcrcas 

on a platr t,hc prcssurc gradient is zero. However, small differences in thc velocity 

clistribntion arc unimportant, bcca~~sc the drag is calculated from the intrcgral of 

morncntum. I~urtllrrrnorc, thc cxperimcntal results obtained by M. IIansen [23a] 

ant1 .J. IW nnrgrrs [6] prove that this assumption is well satisfied at least in the 

Y 

Fig. 21.1. Tnrbulent, boundary lnycr 

on n flnt plab nt zero iricidcnce 

mrrgc of rnotlrratcly large Rrynoltls nurnbcrs (IJ, l/v < 10" They hot11 found 

that thr vrlority profilc in thc bountlary laycr on a plate can be described fairly 

well hy a powrr formnh of thc form of eqn. (20.6), as found for a pipe We shall 

revert once more to this problem (p. 643), when we 'shall discuss some systematic 

deviations between the velocity profiles in pipes and on plates at larger Rcynolds 

numbers. 

The skin-friction drag D(x) of a flat plato of lenght x on onesidcsatisfics the follo- 

wing relation as seen from eqns. (10.1) and (10.2) in Chap. X: 

IEerc to(%) dcnotes Lhe shcaring stress at a distnrlcc x from tl~c Icading rtlgo, :~lld 

the secor~d integral is evaluated at x over the boundary-layer thickness. Introd 

ducing the momentum thickness d2, dcfincd by A, 1Jm2 = / 7l(r!, - TI) dy ill 

eqn. (8.31), we can rewrite eqn. (21 .l) as follows: 

From eqns. (21.1) and (21.2) we obtain the local shcaring strcss : 

Equation (21.3) is identical with thc monlent,lrn~-int,cgral equation of ho~~~~dary- 

layer theory, eqn. (8.32), in thc case of uniform potential flow U(x) .= IJ, -- COIIS~,. 

We shall now perform thc calculation of the drag on a flat plate on the assump- 

tion of a f-th-power law for tho velocity profile which is trrro for modcrntc Royriolds 

numbers, and we shall then confinc onrselves to quoting thc results for thc hgarith- 

mic law which is valid for arbitrarily largc Itcynolds numbers, Fig. 20.4, bccarrsc 

the complete calculation for this case is fairly tedious. 

1. Resistance formula deduced from the 4 -th-power velocity distribution Inw. 111 

accordance with the preceding argument and with eqn. (20.6) it is seen that the 

+-th-power law of velocity distribution in a pipc leads to the following volonity 

distribution in the boundary layer on a flat plate 

whcrc d = B(x) tlcnotcs tho I~oun~lnry-layer ~II~CICIICSR wl~ioh is a function of rlist,nnt-r-, 

x, and is to bc clclcrminctl in thc course of thc calculation. 'l'hc :msumption in 

cqn. (21.4) implies that the velocity profiles along a Rat pletc arc similar, i. e. that 

all velocity profiles plot as one curve of n/U, versus y/d 

Tllc equation for shcaring stress at the wall is also taken ovrr tlirwtly from 

the circular pipe, cqn. (20.12a) : 

.From eqns. (8.30) and (8.31), togcthcr wilh cqn. (21.4) we o:~n ca1culat.c t,l~c tlispl:tcemcnt 

thickness, dl, and thc momcntnn~ tliickncss, (1,: 

637

638 XXT. Tnrbnlent, honndnry Inyeru at zero prcmnre gmdirnt. 

l'ront rqns. (21 3 ) oncl (21.0) we have 

6 - . a 

7 

,-g; 6,=;i,jd. 

wllicl~ is the- tlill'vrcnl in1 rqltation for (Y(x). lntrgrntion from thc initial vnll~c: fi = 0 

at 1 =- 0 gives 

S (x) = 0.37 z (y)-" (21.8) 

The l)o~~nclary-layer t.hickncw is seen to iricreasc with thc power 21-f the 

dist,ancc, wl~creas in I:iminar flow wc had S - ~ ' 1 ~ The . total skin-friction drag on 

a flat, plate of length 1 and width b welf,ctl on one side is, by cqn. (21.2), given by 

'l'he drag on a plate in turl~ulcnt flow is sccn to hc proportional to iJWR15 and l4I5 

compnrctl with 11,"2 and 1'12, rcspcctivcly, for laminar flow, eqn. (7.33). Introducing 

tlimensionlrss coefficicnta for the local and thc total skin friction by putting 

we obtain from eqns. (21.3) and (21.2) that 

1ianc.0, fromeqn. (21.9). wecan writccff =0-0876 (U,~/v)-J1~andc, = 0.072 (U,Z/V)-'~~. 

Tllc last equation is in very good agreement with experimental resulh for plates 

wlloso hountlnry layers are turbtllent from thc Icading edge onwards, if the 

numerical constant 0.072 is changed to 0.074. Thus 

'1'111: rrsist,ancc formula (21.11) is seen plott.cd as curve (2) in Fig. 21.2. The range 

of vali~lit~y of t.his formt~la is restricted to U, 6/v < 105 in accordance with the 

limitation on Blasius's pipe resistance formula. JJy eqn. (21.8) this corresponds to 

U, l/v < 6 x IOfi. Since for R, < 5 x 10"he boundary layer on a plate is fully 

. . 

t In. tho genornl cam of a power law u/U = (y/d)lln we have: 

Fig. 21.2. histance formula for amooth flnt platm at zero incidcnccr; cornpnri~on brt.wrcn 

theory and rncnsnremcnt 

Tl~eorellcnl carvcn: curvc (1) mom eqn. (7.30. Inn1111nr. Illnrrlnrr; cwvc (2) Tm~tt eqn. (21.11). l u r l ~ ~ ~ I'r~1ldt.1; l ~ i , 

curve (3) from eqn. (21.18), lurbnlonl. Prandll-Scl~lirl~llnp; curvo (3n) from rqn. (21.IOn), la~si~~nr-lr,-tt~rh~~lrnl 

tranaibion; curve (4) rrom efin.('L1.10~).111rb11im(. Selmlt~-Ctr~~now 

laminar, it is possiblc to specify tl~e following mngc of valitlitty for cqn. (21.1 1 ): 

6 x lo5 < R, < lo7, using round numbers. Int.roducing tl~c ~iccessary corrections for 

the numerical coefficients we obtain the following expression for the local corfficiant 

of skin friction 

Equation (21.11), as already ment,ioned, is valid on the assuml)t.ion t.hat the 1)ourlciary 

layer is t,urbulcnt from tthe lending edge onwards. In rcalit.y, t.he boundary layer 

will be laminar t.o Iwgin with, and will cl~at~go to :I t,~~rbulent, onc furtlwr tlowns(,reanl. 

The position of tho point of transition will clepentl on tho intlensit,y of t,ltrl)~rlcnce 

in the extternal flow and will bc tlcfinecl by t.he value of the oritirnl ltoyrlolds 1111tnI)er 

which ranges over (TI, x/v),,,, = R,,,, = 3 x 10Qo 3 x 10" (sro Sco. XV1 a). 

The existmce of t,hc I:minnr scc:t.ion cntlscs tho tlrtq t,n tlcc!rcnsc> II.II(~. l'ollowit~g 

I,. Prantlt,l, the dccrcasc can be cstimatcd if it is assumed tallat 1)c:hintl lhc point, 

of tmnsition tthc turhulont boundary layer hel~avcs as if it* wcrc tnrbulc~tt, from 

the leading edge. Thus, from the drag of a wl~olly turl~ulcnt boundary l;~pc?r it is 

necessary to subtract tlic turbulent c1ra.g of the lengt,h 111) to thc poiht of t,ransitiott 

at xcr,, and to add thc laminar drag for the same IcngtJ~. 'Sltus, the dccrcnsc I)rc:onw,s 

AD = - (~12) Urn2 6 xcrl, (eft -el,), whcre c!, and cf, donote tl~c cocfficicnt of 

turbulent and laminar skin friction, rcspect~vely, for the total drag at the: scct.ion 

where tran~it~ion occurs, i. e. at R,,,,. Hence the correction for cf is

610 

XXI. Ttrrbulent bor~ndnry layers st zero premure gradient 

1'11t~ting Acj = - AIR,, we find that the value of the constant A is determined by 

the position of the point of transition R,,,,, namely 

Conscqaent,ly, the coefficicnt of total skin fricLion including the eflcct of tlic laminar 

initial hgth becomes 

Taking cjt from cqn. (21 .I 1) and cj, = 1.328 R,-'I2 from the Blasius formula, 

eqn. (7.34), we obtain the following values for A : 

, 1 3 x los 1 a x lo6 1 lo8 / 3 x loa 

- -- - -- 

A / 1050 1700 1 3300 / 8700 

2. nesistance formula deduced from the logarithmic velocity-distribution law. 

The Itcynolds numbers which occur in practical applications in connexion with 

flat plate problems considerably exceed the range of validity of eqn. (21.13)t, 

and it, is, thereforc, necessary to find a resistance formula which would be valid 

for mucli higher Reynolds numbers. In principle such a formula can be derived 

in tlie same way as before, except that the universal logarithmic velocity-profile 

equation slioultl be used instead of the f-th-power formula, in analogy with 

eqtis. (20.13) and (20.14) for pipe flow. Since the universal logarithmic formula, 

as shown rarlicr, may be cxtrepolatecl to arbitrarily large Reynolds numbers in the 

case of pipe flow, wc may expect to obtain a resistance formula for the plate which 

would also lcntl itmlf to extrapolation to arbitrarily large Reynolds numbers. In 

any case, it is again implied that pipe flow and boundary-layer flow on a flat plate 

exhibit identical velocity profiles (see also p. 643). 

The derivation is not so simplc for the logarithmic law as it was for the f-thpower 

forml~ln. This is mainly doe to thc fact that the application of the logarithmic 

law t,o thc flnt platc does not lcad to similar profilcs any longer. We shall, thereforc, 

refrain from reproducing here the details of the calculation, referring the reader to 

T,. Prandtl's original peper [40]. 

l'hc lognrithmic: formula for pipc flow was derived in eqn. (20.14) in tho form 

t In largo and fnst aeroplanes tlie Rcynohls numbers of the wing are of the order of RI = 

- 8 x lo7; a large, modern fnst ~tmmer reaches about Rl = 5 x JW; see also Tablc 21.3, 

p. GFI. 

a. The stnooth flnt plntc 

denoting the characteristic vclocity formed with the wall shcaring strcss to. In bhc 

case of pipc flow considcretl in Chap. XX, the constants wcrc intlicat,ctl to II:LVC the 

numcrical valucs A, ---- 5.75 and Dl - 5.5. Ilowcvcr, cxt.cnsivc ox~~c~rinic~n~.nl irivestigations 

(sec Fig. 21 3) have clcmonstratcd that thc vclocity profilcs in the two 

cases under consideration, in a pipc ant1 on e flat platc, arc sorncwll:~t tliffcrcnt 

and it becomes necessary to modify tthc numcrical valucs to 

'lllc calculat,ion leads to a fairly cumhrrsomc sct of cqnations for tho local :wtl t20t,al 

cocfficicnts of skin friction in tcrms of tltc Icngtli 1k:ynoltls nuni1)cr R, - : I/, ll~. 

In the process, a formula for the dimensionless boundary-layer tliickriess 11, d/v = r],, 

is also obtained. The numcrical rcsults arc shown in 'hl)le 21.1 and the grnph of 

c, versua R, has bccn plott,ctl in Fig. 21.2 as curvc (3). 

Since the exact formulae from which the resistance law rcprcscntd by Table 

21.1 has bccn evaluated is exceedingly inconvcnicnt, IT. ScIilic11t.ing fittcd thc 

relation between c, and R, from Tablc 21.1 into an empirical cqnation of t.hc hrnv 

In ordcr to make an allowance for tlic laminar initial Icngth, it is mquirctl to makc 

the same deduction as before, cqn. (21.13). Thus 

where the value of the constant A dcpcntls on tllc position of thc point of transition 

as specified in the Table on p. 601. This is the Prnndtl-Schlichting skin-/ridion /ormula 

for a smooth flat plate nl zero incidence. It is valid in the whole range of Reynolds 

numbers up to R, = 10hnd it agrecs with cqn. (21.13) up to R, = lo7. It is seen 

plotted ,as curve (3a) in Fig. 21.2 wltere A = 1700 was choscn, corresponding to 

transition at R, = 6 x 105 13lasius's curvc for laminar flow corrcsponcling to 

cj = 1,328 R,-'12 is also shown for comparison, curve (I). 

A very similar theoretical calculation for the skin friction of a flat platc was 

tlcvised by 1'11. von lCbrmi~n [20]. iC. 15. Scllocnllorr [50] ~nntlo use of' von IZrir- 

mbn's schcmc and derived from it the cxpressior~ 

1 

- - - = 4.13 log (Rl c,) . 

1/CI 

ltesults of numerous experimental mcasuremcnts arc sccn plottd togcthcr with 

these theoretical curves in Fig. 21.2. The measurements performed by (1. Wiescls- 

-- 

t The reaulta for the coefficic~~t of local skin friction, c;. in 'Cable 21.1 can nlso be fitted into 

an empirical equation ns follows 

c f ' = (2 log R, -0.65)-~"

642 XXI. Ttlrbulent boundnry lnycn nt zero prcwtlrn gradient 

Tnble 21.1. Itcniutnnce forrndn for flat plnte computcd front the lognritlimic velocity profile 

in eqns (21.14) and (21.15); sec curve (3) in Fig. 21.2 

R,. 10-8 

0-107 

0.225 

0.355 

0548 

0,864 

1.20 

2.07 

343 

6.43 

9.70 

18.7 

34.3 

51.8 

102 

229 

125 

768 

1576 

l~ergcr [67] on cloth-covered glazctl platcs lio somewhat above t,he turl~ulent. curve (2), 

which would indicate that, there was no substantial laminar length in his expcrimenb 

mil that the roughness was small. The measurcme~it~s duc to P. Gebers [10], which 

mngc from R, = 10"m 3 x lo7, fall on tlic transition curve (3a), cqn. (21.1Ga), at 

the lower end of the mngc. At thc higher Reynolds numbers his results lie on curve (3) 

from eqn. (21.16). 'l'he measurements reported by K. E. Schoenherr [50] also show 

good agreement with tlicory. 'l'hc highest Reynolds numbers have been whieved 

hy (2. I

644 XXI. Turbulent boundary laycrs at zero pressure gradient a. The smooth flat plate 645 

K. Wiegl~artlt [M] ntlvanccd an cxplanntion for the difference between tlic velocity profile 

in a pipo and tlrnt on n phk, pointing or11 tllnt t,he influence of t1tr1)ulrnce at the outer edge 

of t h boundary lilyor tliNcm in the two cnRca. In t,ho cam of n platc a low degree of turbr~lcnce 

in the cxtmtd ~t~re~i~ii gives risc to vclorily fl~~ct~~ntions which arc practically mro at the oubr 

edge of Ll~o honnrl~~ry Iiycr, wllcrcas in tlrc crntro of the pipo thcy would hnve an apprcciablc 

mngnit.r~do bccarrsc of t,lw inllitcncc of 1l10 other side. To the srnallcr intensity of turbulence 

on a plnlo there corrcsporirls a slacpcr incrcnsc in velocity and l~criro a thinner total houndary 

Inycr. lle wns idso able t.o ~llow t,hat thr vclocil.y profile on a platc bccon~ca vcry clono Lo tlrnt 

in pip flow if tlrr cxtcrnal llow in niatlc Iiiglily I.tlrbtrlcnt. 

J. Nilz~~rndsc [38] alxo condnctcd a vcry comprcl~enxivc series of cxpcrimonh on flat plnlcs. 

Ire found that in the range of l:qe Rnynolcls twtnbcrs of R, - 1.7 x 10' to 18 x 106 the volocity 

profilcs arc similar, if ?i/U is plot,lcd against y/dl, whcrc 6, clcnotes the displaccmcnt thickness. 

'J'lic univcrnnl vclor:ity-clist.ribrltiol~ law w/ll = /(y/~!~) turns out to be indepcndcnt of 

lhc Iteynolcls number. 'l'l~r loca! nncl total cocfficicnb of skm friction have heen calculated from 

tho nicns~~red ~rlooit~y proliles wit11 thc aid of tho ~noment,um tl~corem. 

Tho following intcrpolnlion forn~ulac wcrc ol)t,ninctl for tho velocity distribution, dis 

lhic4~nc~, and roefficicnt~ of &in friction, rcspcct,ively: 

plarcrncnl thickness, IIIOI~CII~~I~III 

.- ~ ~ = 

urn 

0.1315 

0737 ( ) , 

'.?.!L = 0.01738 ~zo'86L , 

v 

In conncxion with t,lie calculation of skin friction on a platc, the papor by V. M. Falkner [15] 

may also be consulted. In a paper hy D. Colerr [Ea] the velocity profiles are reprcselntrd by n 

lincnr con~hinat,ion of two universal functions, one of which is c:allcd the law of the wake, the 

other being the law of the wall ns already mentioned. 

Mc,muretnent,s pcrforrnrtl hy 11. hlotzfcld [3GI concerncd tlre~nnclves with the turbulent 

boundary lmyer on a wavy wall. II. Schlichting [46] gave some eutimatcs concerning trrrbulcnt 

boundiiry layers with suction and blowing. When homogeneous (that is, continuously and 

unifornily dist.ribnted) suct.ion is applied, thc asytnptotio boundary-layer thicltncss remains 

constant in lho name manner an for a laminar boundary laycr. However, in the turl~nlcnt case 

thc borrntl:~ry laycr is ~nr~clr inorc scnsitivc to clrnnges in the snction HOW-rate than in the 

laminar. Vcry cxknsivc tncasurcmcnb pcrforn~ccl in tr~rbtllcnt boundary layers on porous flat 

walls by A. l'avre, R. D~~mna and E. Vorollet [lo] show that the npplication of suction exerts 

a st.rong inll~~cncc on t.he 1.11rb11lcnt motion. 

4. Errect 01 finite dimensions; boundary Inycrs in corners. Wl~cn a flat platc of finite span 

is pllrccd in a ulrrn~n which llow~ in t h tlircction of ik Icngl,l~, il is I;)ontI t.l~nl, nonr tJre docdgc 

1110 honnclnry layer is no longcr two-din~rnsionnl, ns it is along tho centre-linc of t,he platc. 

~xperinicntri pcrfi~rtnctl by .I. W. 1Tlder [13] dcnionslrated that near the edges there arise 

sccolldary flows wl~icli arc similar to'lhosc ol)scrvcd in pips of iron-circulnr cross-~cct,ion (cf. 

See. XXc). 'l'l~ix causes n large incrc;mr, in the locd skin-friction cocfficicnt along the edges. 

I\ccording tn 1':ldcr'~ mcasuron~cnL~, and rcmnrltnbly cnongh, thin additionnl dmg, always 

avcmgcd over t,lic span, hrns out Lo ho intlcpcndcnt o[ tlrc lcngl,l~ Ilcynolds number, Rl, or 

the width of thc plate. Irowcvcr, t.hc rcgion sit,uatd vcry close to tlic lcacling edge of thc plate 

forms an cxcrption, tlic lorn1 skin-friction coefficient varying irregularly in the flow direction 

and gt right anglcs to it. Still weording to ISltler's mcasurcmants, the incrcasc in drag is given 

hv 

The second term in this equntion accounta for the rapidly deoaying erect of the lending edge 

(on this detail the reader nmy also refer to A. A. Townsond [04j). 

A similar effect arim when two platea ali ned with the flow are made to form a concave 

corner. The interaction between the two bounfary layers for the cam of a rectan lar corner 

waa studicd by K. Ceraten [20] who indicates the existence of an additional drag oKnsgnitude 

where, according to K. Geratcn, the interaction contribrrbion is 

and 

8.76 

Ac = - -- in laminar flow, 

I Rl 

Ac = - -- @;: 

in turbulent Row . 

f 

The supplementary drag hna turned out to be negative, which mcnna that the drag of two platcs 

which are wettcd only on the inner side of the corner and which arc joined at riglrb angles, is 

smaller than the drag of a flat plate of equal total area. 

E. Eichelhrenner [12] examined the case of a corner of arbitrary angle. 

5. Boundary layers with suction and blowi~~g. Mensurenrent: In this section wc nhnll intaroduce 

brief remarks concerning turbulent boundary layers on a flat plate with suction and blowing 

which may serve as an cxtcnoion of tho conaidcrntiona of Chap. XIV on Inminnr hounclnry layora 

with suction. Thc first tllcorcticcil study of this Inpic wan ~nntlo 

(21.21 a) 

nfl erirly IUI 11M2 11y II. S~~l~lir~l~l.i~~g 

146, 471. In modern t.imcs experimental as well as tl~eoretical studics have been perfortncd by 

J.C. Rotta [44]. 

Some of Itotta'a expcrimentol results are shown grapliically in Fig. 21.4. This is a dingriim 

showing the variation of the momentum thickness dl(%) along a porous flat plate with I~onrogcncoua 

suction and blowing at various values of the auction velocity, v,,,, at the wall. The external 

velocity was Urn = 20 to 30 m/sec and the normal wall velocity ranged from o,, -= -0.10 n~lsec 

(auction) to 0.13 m sec (blowing). The volume coefficient varied from ca = v,,,/fJ, = --0.005 to 

+0.005 and was, t I IIIR, vcry smnllt. Thcse rncmurements confirmod thc well-kno\vn fwt. that tlio 

rate of hountlary-layer thickness growth in the downstream direction increases MI the blo~ring 

Fig. 21.4. Turhulent boundary 

lnyer on a flat plntc with mi- 

form suction or injection: nio- 

mentum thickness 62, according 

to eqn. (7.38), along the plate; 

mea~urements by J. C. Rotta 

[441 

t Suction and blowing startd at a short distanrc from the lending edge rathcr th-11 nt t,hc leadi~lg 

edge itself.

646 

XXI. Turbulent houndary layers at xcro presRure gradient, 

ratc incrcwcn. For CQ = - 0.005 the boundary-lnyer thickneu~ reaches n constant vnll~e downutrcrrnl 

n ~ cw~ntitutca d an twyn~ptc?tic honnclwy layer in the sense of Sec. XIVh. 

The stndy of tnrhnlent boundary layers with suction hns marly applications. Among them 

we may ~ncnt~inn tht tlie int,roduction of a foreign gaa inb tlie boundnry lnyer t,hrough a porous 

wall or ttlirough doLa ronsLit.i~Lrs a very effcct,ive Incano of film or tranalliration coolin Thin 

rcduccs the rate of 11mt trnnnrer from the hot, sbrramirig gas t,o the solid hody, as is done kr gaa- 

Lnrhine blarleu. Si~nil:rrl,y, thin is n Incans of rcrtncing the ratc of heat flow frorn t.he boundary 

laycr rendcrccl very hot by kitict,ic heating on a body flying at a hypersonic velocity to it8s wall. 

Hlowing can also produce a considerable reduction in drag. A very good review of such rpplicat,ionn 

wan pnhlinhcd hy L.O.P. .Jcromin [28]. 

Thcory: In order to cnlrnlntc the nsynlptot,ir: tnrln~lent boundary layer on a flat, plate with 

Iiomogcncous ~nct~ion, we ohservo fro111 cqn. (18.13) t.hat the normnl vclocil,y v = fa, iu constant 

over the whole thic:kness of the Inyer. Hcnce, we can inkgrata the equation of motion in the z- 

dircrtion wit.11 rcnpcct. to tho nortnal pliroct.ion, and thns ohfnin 

Int,rodncing t.hc frict.ion velocity 17, = (r,,/e)I12 and taking into account, t.he fact that at large 

tlistanrca from t,hc wall (i.e. ontrritle the lan~inar sublayer) - it is po~sible to ncglcct the viscons 

hear v(?w/a!/) with reapcct t~ the tnrhule~lt stress -dvl, we derive from eqn. (21.22) that 

With l'randtl's mixing-length aaaumpt.ion 

from eqn. (19.6~). and putting 1 = x y, we deduce from cqn. (21.23) tliat 

Hem x - 0.4 dcnoteu von KkrrnBn's connt,nnt. The preceding equation immediately proves that 

the velocity disLribution can be given the following dimensionha form: 

Pig. 21.5. Tnrhulent honndnry 

lnyer on a flat plate with uni- 

form suction or injection: theo- 

retical velocity distribution 

according to cqn. (21.26) after 

J. C. Rotta [44] 

Pig. 21.6. Tnrhulcnt boundnry laynr on a flat 

platc with nniform snction or injevtion: VCIOcity 

diutribntion in the honnclary lnyer according 

to eqn. (21.26) for different v~rlnes of the 

snclion pnrnnictw v&, nncr J. C. Rotla 

1441 

0 rxporl~~~rnt 

- rnlcwlnllon 

Here q = y v,/v is the dimensionleas distance frorn thc wall from oqn. (18.32). Tho integration 

of eqn. (21.23) given 

Equation (21.26) can he regarded aa a goncrnlisntion of tho universal velocity tlistribntion 

. law . for impernleable turbulent boundary layers, eqn. (10.33), to the cmc of pervious walls with 

e~ther surtion or blowing. In order to inclnde in our considerations the existence of a laminar 

sublayer, it is pertinent to int,roduce E. R. van Driest's [ 101 damping term, eqn. (18.1 I ). 'rhe 

result of such a calculat~ion is shown in 1Pig. 21.5. A comparison with the experinicnts of .J.C. Ilottn 

is given in Fig. 21.6. The agreement is satisfactory if a suitable value is cl~osen lor Lhc nr1jnstd)lc 

constant C. 

Experimental invcstignLions on turbulent boundnry layers with injection of tho sanlc or 

another gun through porous wella into a compressible stroam at Mach numhcrs up to M = 3.0 

have been performed by L.C. Squire [58]. Calculations show that the nsaurnption of Prantlt,lls 

mixing length here too leads to satisfactory result9. 

b. The rotating dink 

1. The "free" disk. The flow in the ncighbourhootl of a rot.ating disk is of grnat, 

practical importance, p.articularly in connexion with rotnry machines. It, bccorncs 

turbulent at larger Reynolds numhcrs, R = U R/v > 3 x lo5, in tho sarnc way as tlrc 

flow about a plate. Here R denotes the radius anti U = n) R is 1hc t,ip vclociLy of 

the disk. The character of this kind of flow was described in Scc. V I I, w11ic:li 

contained the complete solution for the laminar cnse when the disk rotdes in an 

infinitely extended body of fluid ("free" disk). Owing to friction, tho fluid in the 

immediate neighbourhood of the disk is carried by it and then forced outwiwds by 

the centrifugal accelcration. Thus the velocity in tho boundary laycr has radial 

anti a tangential component, and the mass of fluid which i,s driven o~it~wnrds hy cxwtrifugal 

forces is replaccd by an axial flow. Making n simple estimation of thr, balarrcc 

of viscous and centrifugal forces in laminar flow it was possible to show that the

048 XXI. Turbulent boundary layers at zero preaeure gradient 

boundary-layer thickness 6 is proportional to 1/72;, and hence, independent of 

the radius, and that the torque, M, which is proportional to p R3 U/S, must be 

U2 n3(U R/v)-11" The exact solution for 

the laminar casc showed, further, that t,he dikensionless torque coefficient, dcfined 

as 

given by an expression of the form M - Q 

for a disk wetted on both sidcs, is given by cqn. (5.56), and is equal to 

C, = 3.87 R-"' (laminar) , (21.28) 

where R = R2rn)/v is thc Roynolds numhcr, Fig. 5.14. 

It is now proposcd to make the same estimation for the turbulent casc basing it 

on the same resistance formula for turbulent flow as was used in the case of the flat 

plate, i. e., in the simplest case, on the +-th-power law for the velocity distribution. 

A fluid particle which rotates in tho boundary layer at a distance r from the axis is 

acted on by a centrifngal forco per unit volume of magnitude e r w2. The centrifugal 

force on a volume of area dr x ds and height S becomes e r w2 dr x ds. The shearing 

stress to forms an anglc 0 with the tangential direction and ita radial component 

must balance the centrifugal force. IIcnce we have to sin 0 dr xds =erwV~Jr xds or 

On the other hand, the tangential component of shearing stress ran be expressed 

with the aid of eqn. (21.5) which was used in the case of a flat plate, replacing U, 

by the Lmgential velocity r o. Thus 

to cos 0 - e (or)"' (v/S)ll' . 

Equating T, in thcsc two cxprcssions, we find that 

8 - ral' (v/Up6 . 

It is sccn that in the turhnlcnt casc thc bonntlary-lnyer thickness increases outwards 

in proportion to r3/bnd docs not rcmain constant as in the laminar case. Further, 

the torque becomes M - to R3 N e R W~(V/CO)~/~ RRI6 B3 so that 

Th. von ICiirrnhn [30] investigated the tnrbulcnt boundary layer on a rotating disk 

with the aid of an approximate method based on the momentum equation and 

similar to the one applied in the preceding section ijb the study of the flat plate. The 

variation of the tangential velocity component through the boundary layer was as- 

surnctl t40 obcy the 4-th-power law. The viscous torque for a disk wetted on both 

sidcs'wa.. shown to be equal to 

b. Tl~c rot~ting disk G49 

and tthc torquc coefficient dcfined in cqn. (21.27) bccomcs 

C, = 0.146 R- turbulent) . (21.00) 

This equation has been plotted in Fig. 6.14 as curvc (2). It shows very good agrccmcnt 

with the experimental rcsults cluc to W. Schmidt and G. ICctnpft for 

R > 0 x 10" Tho numerical factor in the cquntion for ((he 1)ound:~ry-Iayrr t hi&n~ss 

which was left unclctcrrninctl bccomcs 

3 = 0.520 r (v/r2 o)' 15, (2 I .3 I ) 

and the volume of flow in the axial direction is given by 

as comparcd with cqn. (5.57) for laminar flow. 

An approximate calculation based on the logarithmic vclocity-dist~ribr~l,iot~ Inw 

u/v* = A, In(?/ v,/v) -1- Dl was performed by S. Coldsteiri [21], who found i.hn following 

formula for the torque : 

1 

= 1.97 log (R 1/c) + 0.03 (turl)ulcnt) 

7"M 

It is n~tewort~hy thnt this equation has tho same form as the ~~niversnl pipe-rraistancc 

forrn~rln, cqn. (20.00). Tho nnmcricnl fac1,or~ have bccn 'atlj~rstctl lo obLl~i11 I,II(- IwsI~ 

possible agreement with experimental rcsults. This equation is sccn plotted as cnrvc 

(3) in Pig. 5.14. On this topic see also P. S. Granville [22]. 

2. The disk in a housing. The dislc in turbines or rotwy compressors 111os1Iy 

revolve in very tight housings in which the width of thc gap, a, is small compared 

with the radius, R, oC the disk, Pig. 21.7. Consequcntly, it was found necessary to 

investigate the case of a disk rotating in a housing. 

Laminar flow. The relations become particularly simple when thc flow is laminar, 

R < lo5, and when the gap is very small. If the gap, s, is smaller than the boundarylayer 

thickness the variation of the tangential velocity across the gap becomes 

linear in thc mmncr of Co~~ctto-flow. lFcncc, tho shcaring strcss at a distancc r from 

the axis is equal to T = r(up/s and thc torquc of the viscous Corccs on onc sitlc of 

a disk is given by 

n 

Consequently for both sides we have 

J 

2M 

=n w R4,+ , 

and the torque coefficient from eqn. (21.27) becomes 

t Soc refe. [10] and [31] in Chap. V.

650 XXI. T~lrhulent boundary lnyers at zero preaaure ~rndicnt 

Bwnday layers 

Fig. 21.7. Rxplnriatiori of sym- 

bola for the problom of n clink 

R - RL 

7- 

Fig. 21.8. Viacottn drag of dink rotating in o houning 

Curvr (I), rrom cqn. (21.341, Imnllnnr: rurvo (2), rrorn cqn. (21.35). 

laminar; crlrvc (3). rrom eqn. (21.36), tarbulcnt. Theory with no Lous- 

ing (Tree disk) sce Pig. 5.14 

C,, = 2n - (laminar) . 

9 R 

t 3 

J his equation is secn pk)thd as curve (1) in Fig. 21.8 for a value of .v/R = 0.02. It 

shows very good agreement with thc cxperimentnl values due to 0. Zumbusrh 

(sw rrf. [54j). 

C!. Schmir~lcn [49] invrstigat.rtl I,l~o influcncc of 1.11~ witlt,ll n of t,lm lateral spacing 

of R clislc in a cylindrical housing, Pig. 21.7, on the assumption of vcr.y small Rcynolcls 

numl~ers (creeping n~otion). Thc Navicr-Stoltcs equations can bc simplified because 

of t,hc vcry low ltcynoltls nurnbcrs (scc Soc. IVtl) and the solution for the moment 

cocffirionC appears in tho form C,, = Ii/R, in analogy with cqn. (21.34). The const.ant 

I\' tlrl)cnds ou t,he two tlimcnsionlcss ratios n/R ant1 a/R.' In t.ho case of vcry smdl 

vnllws of o/R (< 0.1) tho valurs of C,, arc! n~arltctlly 1:trgcr I.l~n.n thoso in cqn. (21.34), 

wl~c~rrns liw hrgo vnlncs of cr/II' cqn. (21.34) retains ib valiclit,y (K = 2 n Ills). 

. . 

'I'lw flow pal,tnrn in t.11~ case of larger gaps diffors considerably from the above 

sin~plc scl~rnic. 'l'llis latrl.cr case was invrst.iptct1 t.~~rorcticaily and cxpcrimcntally by 

1'. Srll~~lt~z-(:r~lllo~v [54]. If 1.h~ gap is a mnlt,ipl# of t,he boundary-layer thickness, 

thcn an ntlditional boundary layer will bc formed on thc liousing, Fig. 21.7. The 

fluid in thr bount1:ary Ia.ycr on tho rotding disk is centrifuged outwards, and this is 

co~nprnsatctl by a llow inwartls in t.hc I~onnclary layer on thc housing at rrat. Tllcre 

is IIO apprrcinldn radial component. in tllo int,t:rrnctlial.c Iayer of fluid which rot,ntcs 

wi1.11 :~l~ont Irn If t,llo angular velocity of 1.l1c clislc. P. Sc1111lt.z-Grunow invcstigatcrl this 

b. The rotating disk 66 I 

flow both for thc laminar antl for the turbulcnt case. Tl~c cxprrssion for t,ho tmqw is 

of the same form as for thc free disk in eqn. (5.56). only thc numerical factor has 

a tlifforent valnc. 'Che frictional momcnt of a disk in laminar flow and wrl,tctl on both 

sitlrs I~cromcs 2 M = 1.334 ,A I 3 x 10' tJw flow arountl a clislc rot :tLing 

in a housing becomcs turbulent as usual. This crwc was also solvctl by 1" S~r1111ltz- 

Gronow who usrtl an approxitnalc mclhod bnsctl on t.hc sc~hcmc of Fig. 21.7. 7'11~ 

tangential vclociLy was assnmctl to obey the 4-th-powcr law and it was sl~own that 

t,hc eorc rcvolvrs with nl~out, half t h angnlnr vc1oc:it.y in t,his rasc t,oo. The monirnt 

cocficicnt wc~s shown bo bc aqua1 to 

C, = 0.0622 (R)-"' (turbulent) . (21.36) 

This equation has bccn plottccl in Fig. 21.8 ns curvc (3). Comprotl wilfh Inwwre!ment 

it laacls to vnluea which arc too small hy ahout 17 par cont., nrd this mwt I)o 

atLribut,rtl to the arntlc n.q.sl~tnl)t.ionu mntlo in tl~c cr~loulnl,ion. 

It is particularly noteworthy that, apart from thc casc of vcry small gaps, 

eqn. (21.34), the momcnt of viscous forecs is complctrly inclcprnttcnt of I(hc witltlr of 

the gap, as secn from cqns. (21.35) nntl (21.36). Comparing tho frictionsl rnon~cul on 

a "free" disk and on one rotating in a housing, cqns. (21.35) and (21.36) as against 

eqns (21.28) arid (21.30), it is seen that tho n~omcnt on a frcc tlisk is grcatm than 

that on a disk in a housing, Fig. 21.8. This fact can bc explained by the existence of 

the core which moves at half the angular velocity. This dccrrascs t.hc transverse 

gradient of the tangential velocity to approximately onc half of what it woultl hr on 

a free disk and, consequently, the drag is also smaller than on a "free" disk. 

The flow process depict,ctl in Fig. 21.7 in which thc boundary layer on thc rotating 

disk flows outwards and that on the casing flows inwards was lat.er invcstigntcd 

e~periment~ally by J. Dailey and R. Nece [Bb]; their measurcrncnts covcred the wiclc 

range of gap widths s/R = 0.01 to 0.20, and a range of lteynolds numbcrs R = 

R2w/v = 103 to 107 and included bot.11 laminar and turbulent, flows. Thc rcsults 

shown in Fig. 21.8 concerning the torque have bccn largely confirnicd. 

coolcr casing at, rest. is irnport,anb in the design of gas turbines. Tl~c tcmpcrnt,urc ficld 

which develops in t,he gal) bctwcen t.he disk ant1 the casing is strongly infinrnrcd by 

the complcx flow pat.t,nrn which prcvails in it; in t,urn, this has a large i~~llurncc on 

the flux of heat from tlisk to ho~lsing. The simpler case of a rot.nt,ing "frcc" disk wxs 

I invest,igated some t.imc ago by K. Millsaps nntl I

652 XXI. Turt)~~lcnt houndnry layers at zero prcssure grndicnt c. The rough plate 653 

gap with a single bountlary lnyer and that of a wide gap with two scparate boundary 

layers, one on tho inner and one on thc outer wall. In most cases good agrecmcnt 

between theory and measorcment of heat flow was obtained. 

c. The rough plate 

1. The resistnnce formuln for n uniformly rough plntc. In mOst practical applications 

conncctcd with thc flat platc (c. g. ships, lifting surfaccs of an aircraft;, 

turbinc bladcs) tho wall cannot bc considcrctl hytlraulically smooth. Conscql~cnt~ly, 

the flow pa& a rough platc is of as much pract.ical interest as that throl~gh a rough 

pipc. 

The rclative roughness k/R of the pipc is now rcplnccd by the quantity k/6, 

where (9 denotcs tho boundary-layer thickncss. Tho csscnt,ial difference bet,ween 

tho flow through a rough pipc and that ovcr a rough plate consists in the fact that 

the rclativc roughness k/6 dccrcascs along the platc when k remains constant because 

8 increases downstrcam, wl~crcas in a pipc k/R remains constant. This circumstance 

causes the front of thc plate to behave differently from its rearward portion 

as far as the influcncc of roughncss on drag is conccr~lcd. Assuming, for the sake 

of simplicity, that thc lmundary laycr is turbulcrlt from the leading ctlgc onwards, 

wc find complcLcly rough flow ovcr the forward portion, followcd by the hnsition 

rcgimc and, cvcntually, thc platc may become hydraulically smooth if it is sufficicntJy 

long. The limits bctwcen these three regions arc tletcrniincd by the climcnsionless 

roughncss paramctcr v, k,/v aa givcn in cqn. (20.37) for sand roughness. 

'l'hc result of t h calc~~lation for pipcs can bc transposed to thc casc of rough 

platcs in cxaciJy the samc way as for smooth platcs in complcte analogy with the 

dctailcd clcsoription givcn in Scc. XXTa. Such calculations were carried out by 

L. Prandtl and 11. Schlichting [41] wit,h the usc gf Nikuredsc's results on pipes 

roughcncd with sand (Scc. XXf). Thc calculatio~~ was bascd on the logarithmic 

velocity-distribution law for rough pipcs in tho form of cqn. (20.32), whcncc u/v, = 

1.= 2.5 In (ylk,) + B. The dcpendcnce of the roughncss function R on the roughness 

paralncbcr v, ks/v is given by the plot in Rg. 20.21. The calculation, which is 

esscntially thc samc in Scc. XXIa, n~ust be carricd out separately for the transition 

and complctcly rough rcgimcs rcspcctivcly. For the dcteils of this method reference 

sho~lld hc made to the original paper. 

The rcsult can bo rcprescntcd in two graphs, Figs. 21.9. and 21.10, in which 

the coefficient of total skin-friction drag, c,, and the local coefficient, c,', have 

been plotted against the Reynolds number R = U, l/v with the rclativc roughness 

Ilk, as a, paramctcr. In the case of the local coefficient, U, x/v and xlk, are used. 

In addition thc diagrams contain curvcs of U, k,/v = const, which can be comp~ltcd 

at once from the previous oncs. The two families of curves have the following 

significance: if the ve1ocit.y on a given plate is changed, Ilk, remains const,aut, arlci 

the cocfficicnt of skin friction varies along a cur,vc ilk, = const. If, on the, other 

hand, the length of the plate is changcd, (I, ks/b remains A constitnt, and the &,ocfficicr~t, 

of skin friction varics along a curve lJ, k,/;= const. 13ot11 grapl~sfliave 

beon computctl on thc nss~lrnption that the turbulent bou~ldsr~ layer t~cgiri; rigl~t 

at the lcading edge. 'lh broken cwvc shown in the tli:~&,ims corresponds to t,he 

limit, of complete rotrghness and it may be notctl that a given rclnt,ivc rorlgl~rlcss 

]pig. 21.9. Ilcsist,ancc formula of sand-roughcnrd plnto; cocfficict~t of lokd skin frict,ion 

Fig.21.10. Resistance formula of sand-rougllcned plate; cocificicnt of local skin friction 

!&.z

654 XXT. l'url~ulmt I~o~~ntlnry lnyrr~ at zero pressure gradient 

causes the coefficient of skin friction to increase only if the Reynolds number ex- 

ceeds a certain value, in complete similnrit,y with pipe flow (see See. XXId). 

In the romplot,cly rough rcginic it is possible to make use of the following 

interpolation forrnnlnc for t h oorffirirnts of skin friction in terms of relative rough- 

ness : 

2.87 1- 1.58 lng -" 

(21.37) 

ks 

which are vnlid for 102 < Ilk, c~ lofi. 

In order to use these clingrams for roughr~esa other than the sand roughness 

assumetl hero, it. is necessary to tlctcrmine t hr equivalrnt send roughness as explained 

in Src. XXg. 

In the cslrulat.ion of t.ltc tlrag on ships it, is important to consider plates with 

vcry small roughness (painted mctd plntcs) as well as smooth plates covered with 

single protuheranccs, srrch as rivet, heads, wrltlctl seams, joints, etc. F. Schultz- 

Grunow [62] carried out a large numl)cr of rncns~~rements on such surfaces in the 

open cl~annd of the Tt~stitut~e in Gont,i.ingcn mcntioncd in See. XXg Atlditionnl 

comprchcnsivc data on ronghncsscs occuring in shiphilding can also IIC fountl in 

several papers hy G. I

656 XXI. Turbulent boundnry layers at zero prcssure gradient 

Fig. 21.12. ltcnistance coefficient of 

circnlor cavitics of varying depth in a 

fiat wnll, ns tncnsurcrl hy \Yicghnrdt [66] 

Figurc 21.12 presents thc increase in drag caused by circ~~lar cavities shown in the sketch 

(diameter d and dcpl.11 h). Since the definition of q adopted previously loses ita sense in this case, 

the drag wns mnde dimensionless with refercnco to the stagnation pressure outside the boundary 

layer, Aco = ADlf q n d'. Tho increaso in drag is smaller for smaller values of the ratio of the 

depth of t.hc cavity, A, to the hondary-layer thickncm, 6. It is noteworthy that all curvcq have 

a common rnnxinn~m at hid r~ - 0.5. l%rlher, small locnl nmxima occur at -hid 0.1 and 

1.0. The rninin~a bctwccn them occur at - h/d = 0.2, 0.8, and 1.35. Depending on the depth 

of the cavity it. may nometi~nes hnppcn Lhat rcgular vortex patterns arc formed in it, leading 

to tl~c diRcrent, val~rcs of drag. As swn from tho symmetry of tho curves about h/d = 0. ~l~allow 

cavitiw of up to - d/h = 0.1 givc the same incrcaso in drag as cormpnding small protubcrancoa. 

ltor~gh~~css in the form of rifling or ridges on a plate cut normal to the flow direction have 

been the subject of modern studies by A. E. Perry et n1. [39a]. 

1 

Fig. 21.13. Curves of conatant velocity in tho flow field1 bchind a row of spheres (full lines), as 

n~easurcd by 11. Schlichting 1451, and accompanying it the secondary flow (broken lines) in 

the bounr1:iry Inyrr bchilld sphere (I), ,zs calrulnted by F. Schultz-Grunow [65a]. In the neigh- 

bourhood of the wall, tho vcloeity behind the nphorcs is larger than that in the gaps. The spheres 

produce a "ncgntivc wake cff&t7' which irr rxplnincd by the existence of secondary flow 

1)ismrtrr of n1rIirrt.q d - 4 mm 

The flow patlcrn which exisb bchind an obstnclc loccd in the boundary lnycr nmr n wnll 

diRcm msrkcdy from that bchind an ohshclo placc(f in tho irco ntxcnm. 'Shin cirnntnst,ancc 

cmcrgca clcarly frorn nn oxpcritnc~~t pcrformctl by 11. Scl~liol~ting [45] nncl ill~~nt.r:itnxl in 

Iqig. 21.13. Thc cxpcrin~ent consiuLCd in tho ~ncnsurcmcnt of 1.11~ vclocity liclrl I)chind a row of 

sphcrc~ plnccd on a smooth Ilnt s~~rincc. 'l'hc pnlhrn of curvcs of conntrrnt vcloc-ity clorwly nhown 

a kind of negative wake e//ect. Thc smallcst vclocitira Imve bcon mcn~urcd in Lhc frcc grrp in 

w11ic.h no spherrs are prcscnl ovcr Lhe whole 1cngt.h of thc plate; on thc olhcr hrrr~tl, l.11~ lnrgcst 

vc1oc:itine havo bccn n~cnsrrrcci behind tho rown of upl~orcs whcrc prcciscly t.ho nmnl1c:r vc:loc:il.ios 

would bc cxpectcct to exist. W. Jacobs 1201 corriccl 0111 rr tuorc dc:t;iilccl invc~st.ig:iLion or thi~ 

peculiar cllect. According to $1 remark made by P. SchulLz.Ur~~now [RR], Ihc rcnson for n~~ch 

behrrviour seems to be conncctr,d wit.1~ the cxintmw of wcondary (low of a kind which is similar 

to thnt on a lift..gcnerating body. 'rhc strcamlinc~ of this secondary llow havc I)C(YI shown 

skct,nhcd in Fig. 2];13. 'l'ho cxistcnce oi Lhia cllect was oonlirmccl by I). 11. Willi:in~n nnc1 

A. 1". Brown [68] who pcrformcd mcnar~rcmcnb on an ncroloil provitlctl with rows of rivctn. 

Thcre i~ in existenno a very cxtennivc literature concerning thc ror~ghncsa of acrofoiln 

[9, 24, 251. 

3. Trnn~ilion from n smooll~ lo a rough ~~rrlnee. W. .Jncnbs [27) i~~vrntigslotl 1.hi Ilow p:rtlearn 

near a wall which co~eisted of o smooth wction followd by a rough one, or vice versa. 

The problem is of some interest in meteorology and oeeurs whcn a wind passcs from ncn Lo 11111d. 

or from land to sea, flowing past nurf~rocs whose roughnesscs dimcr connitlorably from cnch 

obher. It is noticccl that the vclocity profilo which corrwpontls lo thc clownst.rc:rn~ nrc4.ion of 1.11~ 

wall forms only at a cerhin distance bchinel the bonndary bctwcc~~ 1110 two sccliol~u. 'Ik, a vnri:il.ion 

of nhcxring st,rcss ~alcnlntcd frorn tho mcnn11rc:c1 velocity prtdilc with I h :lie1 of l'rr~11cIl.1'~ hypo- 

Lhcnis, i. c., r - plz (ei~l/t~?y)~, is ncxw ldol,t~!cl in b'ij?,n. 21.14 11111~ 21.1fi. 'th? ~ I ~ # L ~ ~ ~ HII

658 XXI. Torhulcnt boundary layera at zero preuaure gr~liorit 

Fig. 21.14. Varint.ion of slicnrit~g stmm in tho bonndary layor on pnaaing from a amoolh to a 

rough portion of wall, na ttl~a~ured by W. Jacobs 1271 

Fig. 21 I5 Varinlir~n or dimring ~trr911 in ~IIP I~ontidnry lnyer on pauning from a rough to a 

rmoolh porlion of \$:ill, ti~rnsi~rrtl 11y \I1. Jnrol~n (271 

InJ!le case of lur.bu~?nt.~bo~m+ry hyrs roughness has no effect, and the wall 

is hy&aulic$llfjmooth if all_ potuberanc& iife co_n_tai+ -within the laminar sub- 

GeTAs mcntionctl befor&, the thicltnrss of the lnttcr in only a small fr,zvtion of 

the boundary-laycr t.hickness. In conncxion with pip flow it waq founcl that, the 

condition for a wall to be l~ydraulically smooth is given I)g rqn (20 37) which statrd 

that the dimensionless roughness lteynolds numbert 

- 

-- v* -k - < 5 (Iiytlmnlicnlly smooth) , 

wl~crc v, = \/to/P denotes thc friction velocity. This result can be consit1t:rccl valid 

also for the flat plntc nt zero incidence. 1Iowcvcr, from tlto prnctical point. of viow 

it sccrns more convenient to spccify a value of rclativc rouglincss kll. Itcfcrring to t.lie 

diagram in Fig. 21.9, which represents thc rcsistancc formula for a plate, we can 

obtain the admissible value of k/l from the point nt which a givcn curve Ilk =-. (:onst. 

deviates from the curve for a smooth wall. It is sccn that the ntlnlissiblc vnluc of k/l 

decre,wes as the Reynolds number U, l/v is increased. Ronncletl-on' vnl~rcs from 

Fig. 21.9 are Listed in l'al)lc 21.2. They can be sitmmarizccl by tlic following simple 

forrnula : 

.--- . = lw, 

(21.42) 

wl~ose approximate validity can also be deduced dircct,ly from Vig. 21.0. 

Y 

Table 21.2. Admiwihle height of protr~bcrances in tcrlns of the Rcynoltln nu~nl~c~ 

This fornir~la givcs only onc v:~luc of k,,,, for t,lir: whole Icngt.1) of t.hn pl:tto. 

Sincc, howcvcr, tho bouncl:wy-layer tJ~iclzncns in smnllcr ncSnr (,IIc: It*:ltling rtlgc:, 

the atlmissibb vdue of k is srnaller ~q)st,rcani thn ft~rl~llcr tlownst.rcnm. A fi)rtnrlla 

which takes this circu~nst,:cr~nc into acaonr~t is obt;~inctl wllrn o,2/(1,2 = t,,/~) 11,,2 - 

= 4 cf' is introtluced, cf' clonot.ing the local cocfliciont of skin friction, as given in 

'I?ahlc 21.1. Thus wc obtain 

Umrknam 7 

660 XX I. Tl~rhlent bo~~ndary layers at zero pressure gradient 

For mall Reynolds numbers R, < loR eqns. (21.42) and (21.43) give practically 

the snmc results, whereas at larger Reynolds numbers eqn. (21.43) gives somewhat 

greater values. We are, thus, jr~stified in retaining the simpler equation (21.42) 

because there is noclanger of finding values of k,,,, which are too high. Equation (21.42) 

stdes that the admissible height of roughnes.9 elements is independel~t of the length of 

the plate; it is tlt+rminctl sololy by tt~r velocity and by tho ItincmaLic viscosity in 

:~ccortlancc: with thc contlitior~ 

(21.44) 

It follows that t,he absolutc values of admissible roughness for a modcl and its 

original arc equal if tile velocity and kinematic viscosity arc the samc in both cmes. 

Vor long botlios this may lead to extremely small admissible roughnesses as compared, 

with t,hcir linear dimensions, see Table 21.3. 

Fig. 21.10. Atln~issible roug11ne.w 

k,dm for rough platcs at zero inci- 

dence, and aircrnft wings from 

cqn. (21.44) 

For ~xnct'icnl applicntions iL is still more convenierlt to relate the admissible 

valuo of ronghncss directly to the Icngth of the plntc, 1, or more generally, to the 

length, I, of t h I)otly undcr consiclcration, (c. g. length of ship's hull, wing chord, 

blntlo chortl in t~rrl)inrs or rol.ary cornprcssors), I)ccnr~se this lcatls to a more graphic:

662 XXT. Tiirltnlcnt bo~tntlnry lnyrrrr nt zero prcssnrc ~rndicnt 

mrnsure for tlw rcqttirrtl snrfarc smoothnc~s. To achieve this, eqtlation (21.44) may 

be rewrilttcn ns 

wlwrc R, - Il.,, I/v. 'l'hn tli:~grntn i ~ t 

l'ig. 21 .I0 rnny Itc: uscd to fn~ilit~nta calculnt,ions 

with the nit1 of' cqn. (21.46). 'L'hct tlingrnm cont.;rins a plot of admissible sizes of 

prot,ul)crn.t~cen ngnirlst Rcynoltls ntln~l)(?r, wit.11 the cl~nractcristic length as a pararnctcr. 

'l'hc mngrs of Itcpnoltls n11rn1)crs cnrot~nLcrctl in variolts engineering applications 

(ship, airsllip, nirc:r:l.ft,, conlprcssor I)ln(lcs, .sI,cam t,~crbinc blndcs) I~nvc been 

shown at, t,llc bottorn or thc diagram for convenience of reference. In addition, Table 

21.3 gives n sumrnnry of scvcm.l examples which hnve been computed with the aid 

of Fig. 21.16. rri thc cnsc of .d~ip' hdl~ atlmissiblc roughnesses are of the order of 

scvcrnl Iiuntlrcclths of one ntillimcbrc (scvcrnl t,cntlls of one t.housantlt.l~ to scvcrnl 

t,ho~tsnndf.l~s of an inch); surl~ vnltrcs c~tinot 1)c attained in practice nnd it is always 

necessary to allow for a consitlrra1)lc increase in dmg due to roughtlcss. The same 

is txuc of nir~hips. As far ns rcircm/t surlncea are conccrnctl, it is seen that atlmissible 

rongllncss tlimcnsions lie bcl.wcen 0.01 and 0.1 mm (0.0004 and 0.004 in). With 

vcry carcful pmpnmt.iori of t,Ilc surface it is possible to meet these demands. In the 

r:rsc of nrodel nircrnlt and compressor blnrlcs wliicl~ rcqnire the same order of smooth- 

IIPSR, i. 0. 0.01 to 0.1 mm (0.0004 to 0404 in), hydraulicnlly smooth surfaces mn bc 

ol)t.nined wit.llot~t nntfnc clifficully. 1'11c Itcynoltls numbers encountered in steam 

twbimx arc coml)arntivcl.~ large ~CCRIISO the prcss~~rcs arc cornpnrntivcly lligl~t 

in spitc of Lllc smn.11 linmr tlimcnsions, nntl ndmissiblc rougllncss values are, conseqt~cnt~ly, 

vcry smn.ll. 'I.'hc rcqrlimtl vnlucs of Iwtwccn 0.0002 to 0.002 mm (1O4to 

10 in) c.n.1, I~artlly Ire al.tfninctl on newly manufacturetl blnrlcs. They are certainly 

cxccctlcd :rfl.er n prriotl of opcrntion due t.o corrosion and scaling. It may now be 

rc~mnrltctl tht, 1.hc prcretlirlg consitlcrntions apply to t1igl~tjly spaced protul)erances 

wltic:h corrrsporid l,o sand rotlgl~ncsn. In t.hc rnsc of widely speccd obst.aclcs and in 

1.l~ msc of w:lll wnvincss the atlmissil)lc vnlr~cs arc sorncwl~at larger. 

r 3 

1I1c in(l~irn~:c of rougllnrss 011 the Iossrs in :r st,cnm turbine stage? tlepcntls to a 

gwnt, rxtant, on t,lw prrssnrc tlrop :wross it.. i . e. on t.Im tlcpree of reaction of the stage. 

, 

I . his point crnolpcs clcnrly from Fig. 21.17 whicli rcprcscnts the rfsdts of measnren~cnts 

pcrfolmrtl by I,. Spriclrl\68l on turbine cnscntlcs with varying sand rougllness. 

, 

1 . IIC tlia.grnm cont.nit~s n plol of t,llc loss cocf'ficicnt

664 XXI. Turbulent boundary layers at zero pressure gradient 

We shall now calculate the valuc of k,,,, for a wing of length 1 = 2 m (about 6.5 ft) in 

air (v = 14 x m2/scc) at a velocity U, =83m/sec =3OOkm/hr (about 185 mph). 

We have R, = U, l/v w lo7. Consider a point on the wing at x = 0.1 1, i. e. at 

R, = U, x/v m lo6. The boundary layer can remain laminar as far as this point 

owing to the existence of a negative pressure gradient. The shearing stress at thc wall 

-- 

for a laminar boundary layer is given by eqn. (7.32) and is to/~ =0.332 UW2 1/1'/~~X = 

= 0.332 x 6000 x lo-? m2/scc2 = 2.20 m2/sec2. IIcnce v, = itole = 1.52 mlsec. 

Inserting into cqn. (21.40) we havc 

v 15 

kc,,, = 15 - = ~ 6 X 3 0.14 x lo-' m = 0.14 mm (about 0.0056 in) . 

u 8 

This shows that the critical size of a protuhcrancc which causes transition is about 

ten times largcr t,linn thc valuc of about 0.02 rnm (0.0008 in) in the turbulent boundary 

layer, as calculatctl in Tablc 21.3, for the case in hand (small aeroplane). 

Thc laminar bo~~ndary laycr "can stand" much largcr roughness than the turbulent 

boundary layer. I(. Schcrbarth [481 carrictl out experiments on the bchnviour 

of laminar bountlnry layers on walls provided with single obstacles (rivet heads). 

It was mccrtained that behind the obstaclc t,herc forms a wedge-like turbulent 

distjnrbcd rcgion whose angle of sprcad is about 14O to 18'. 

r 1 

I hc very rxlmsivc mmsuremenl~s carried out bv E. G. Feindt 1171 h:~vn I d to 

k J - -.. 

a refincnicnt~ of the criterion for t,l~c critlical height given in eqn. (21.46) as mentioned 

in See. XVIIg. 

Kg. 21.18. hag on circular cylinders 

at varying ronghness, aftm Fage and 

Warsap [I41 

Thc inflnrnct of rouglmcss on form tlrag can be surnrnarized a$ follows: bodirs 

with sharp rtlgrs, such as c. g. a flat plate at right angles to the st,rearn,'nre quite 

insensitive to surface roughness, because the poitjt of transition is determined by 

thc edges. 011 the othcr hand, thc drag of bluff bodies, such as circular cylinders, is 

very sensitive to roughness. Thc valne of the critical Rcyrloltls number for which 

thc drag shows a sudtlcn drop (Pig. 1.4) tlrpcnds to a rnarltetl degree on thc roughness 

of Ihc surfacc. According to mcasurrmcnts, [I, 141 as shown in Fig. 21.18, the critical 

Reynolds nurnher decreases with increasiug relative roughncss k/R (d = 2 R = dia- 

meter of cylinder). Tllc boundary laycr appcars to I)c tlisturbctl by rougl~ncss Lo 

such a tlcgrec that transition occurs at considerably lower Iteynoltls nurn1)rrs t.lr:~n 

is the case with smooth cylinclers. Ronghncss has, Olwrcforc, thc samo clTc:ct as 

Prancltl's tripping wire (Fig. 2.25), namely, it does reclncc tlrag in a ccrl.nin rnngc 

of Reynolds numbers. Jn any case the drag in thc supcrcritical r:tnge of Itcynoltls 

llu~nbcrs is always Iargcr for the rough than for t.11~ smool~l~ cylintlrrs; scc I ~tm IC,OJ. 

[I] Ackerct, J.: Schweiz. Bnuzeitung 108, 25 (1936). 

[la] Alltonin, It.A., nntl Wood, D. H.: Cnlculnt,inn of a tnrl~r~lent I~onntl~wy Inycr clo\vnslrt~:~n~ 

of a an1a11 step cliange in surface roughncss. Asro. Quart,. 26, 202--210 (1!)75). 

[Z] Rammert, K., sad Fiedler, K.: Dcr Rcibungaverlnat von rn,~heri TII~~~IICIIS~~I~ 

13rcnnatoff-Wiirmo-ICraft 

18, 430-436 (1966). 

[2n] Banner, M. L., and Melvillc, W. K. : On the separation of air flow over water nnvcs. ,I TM 

77, 825-842 (1976). 

131 B~mnlert, K., and Ficdler, K.: Hinterkantcn- und lteibun~~vcrluut in '~urbineti~c.l~:r~~fcIg11,lern. 

Forschg. 1ng.- Wca. 32, 133- 141 (1066). 

[4] Blenk, H., and Trienes, H.: Str~m~~ngsteehr~isclre 13eitriige zuni Wintlsch~rtz. (;r~~ntllagcn 

der Landteehnik. VDI-Verlag, No. 8, 1956. 

[5] I%rndrrhaw, P., and Grrgory, N.: The dotorminnbion of locnl tnrl~nlcnl, ~ltin fri(.I.ion fro~n 

observations in the viscous sub-laycr. AltC JtM 3202 (I!Nil). 

[O] Burgers, J.M.: The motion of a fluid in tl~c bountlnry Inyrr along a plnnc? ~nionl.l~ RII~~IWC. 

Proc. First Intern. Congrese Appl. Mech. 121, Delft (1824). 

[en] Caly, R.: Der Wiirmeiibergang an ciner in1 geschlosaencn Gehause rotierenden Sclleibc. 

Thc~is Anchon 1966. 

[7] Chapmann, D.R., and ICester, It. H.: Mcasnrenicnl~ of tnrbnlcnl skin friction in cylintlcrs 

in axial flow at subsonic and supersonic velocities. JAS 20, 441-448 (1083). 

[8] Colea, D.: The problem of the turbulent boundary laycr. ZAMY 5, 181-202 (1054). 

[Ba] Coles, D.: The law of the wake in the turbulent bonndary layer. JFM 1, 191 -226 (1986). 

[8b] Daily, J., and Nece, R.: Chamber dimension effecta on induccd flow npd friction resintnnce 

of enclosed rotating disks. J. Raaic Eng., Trans. ASMIP. Series D, 82, 217--242 (1960). 

[9] Doetsch, H.: Einige Versuche iiber den JCinfluss von Obcrf~ric~lcnstarrlngcn auf die Profileigensclraften, 

insbesondere auf den Profilwidcrstand irn Schncllflug. Jb. dt. Luftfnlirtforschung 

1, 88-97 (1939). 

[lo] Van Driest, E.R.: On turbulent flow near a wall. JAS 23, 1007-1011 (19N). 

1111 Dutton, R.A.: The accuracy of tneasuretnent of turbulent akin friction by means of surface 

P~tot tubcs and the distribution of skin friction on a flat plate. AltC IZM 3058 (1957). 

- 1121 - Eichclbrcnner, E.: La touche-limite tnrbulente B 1'inti.ricnr d'un dihdrc. Itech. A6ro. I'aria 

NO. 83. 3-8 (1961). 

1131 Elder, J.W.: The flow paat a flat plate of linitc width. JFM 9, 133-183 (1960). 

[I41 Fage, A,, and Waraap, J.H.: The eKects of turbulcncc and surfacc rougl~neas on the drag 

of circular cylinders. ARC RM 1283 (3930). 

[IR) Fnlkner, V.M.: The resi~tance of a smooth flat phtc \\it11 turbulent bonndary hycr. Aircraft 

Engineering 15 (1843). 

[I61 Favre, A., Dumaa, R., and Verollot, E.: Couche limite sur paroi plane porcuac avcr a~piration. 

Publications Scientifiques et Techniques du Ministdre de ]'Air, No. 377 (1961). 

[I71 Feindt, E. G.: Untersuchungen iibor die Abhangigkcit dcs Urnschlages laminar-turbulent 

von der Oberflachenrauhigkeit und der Drnckvertcilung. Disa. Braunschweig 1956. Jb. 

Schiffbautechn. Ges. 50, 180-203 (1957). 

[17a] Fomter, V.T.: Performance loss of modern stream-turbine plant due to surfaro r~ugl~nras. 

The Inst. of Mech. Eng., Preprint, London, 1967. 

[18] Gadd, G.E.: A note on the theory of the Stanton tube. ARC RM 3147 (1960). 

[I91 Gebers. F. : Ein Beitrag zur experimentallen Errnittlung des Wauscrwiderstandru gegrn bewogte 

Kiirper. Schiffbau 9, 436-452 and 475-485 (1008); also: Daa Bl~nlicl~keitclgesetz fur 

den Flaclienwider~tand in Wauaer gcradlinig fortbcwegter polierter I'lntten. Srhiffbau 22, 

687 - 030 (1920/21), continuation8.

666 XXI. Turbulent boundary layers at zero prensure gradiont References (if17 

[20] Gersten, I Schiichtinn. 11.: Exncrimcntelle Untcrsuchungon znm Ita~~higkoitrrprohlcnl. Ing.-Arch. 7, 

1-34 (19%). NAC~ TM 823 (1937). 

r461 . . Schlichting, k.: Die Grerlzsch~cht an der ebenen Plntte mit Ahsaugung und Auuhlwen. 

~rnftfal~rt&rncl~~~n~ 19, 293-301 (1942). 

r471 Schlichting. H.: J)ic Gren7achicht mit Absnngon nntl A~~shlnscn. J,~lftfill~rt,ft~rurl~~~~~g 1.9, . 

L- 2 

179- 181 j1n42). 

1481 Scherbarth, IC.: Grenzschiclltnlessungrn hinter einer punktformigcn Sttirung in Inminnrcr 

Strdmung. Jb. dt. Jn~ftfahrtforschung I, 51-53 (1042). 

[49] Schtnieden, C.: vber don Widentand einer in ciner Flunsigkeit roticrcndon Schrihc. ZAhlAI 

8, 400-470 (1028). 

[50] Schoenherr, K. I:.: Resintnnce of flat aurfacecl moving througll a fluid. Trans. Soc. Nav. Arrh. 

and Mnr. Eng. 40, 279 (1932). 

[51] Schofield, W. H: Mcaaurements in arlvernr prwure gradient turhnlcnt honndnry lnycrs wilh 

a ntep rhange in surface roughness. JFM 10, 573-693 (10713). 

[52] Schultz-Grunow, F.: Der hydraulischo Itcibungawiderstantl von Platton tnit tniinnig rnuhrr 

Oberfliiche. insbeaondere von Schiffaobcrflachen. Jb. Schiff bnutcchn. Gcs. 39, 171; -198 

(1938). 

rr;m Schnltz-Grunow. P.: Neucs Widorstnndngcsctz fiir glnt.t,c l'laltnn. I,uftfnl~rLfornt:l~~~r,a 17, 

L J - 

239 (1940); also NACA TM 980 (1941). 

[64] Schultz-Grunow, F.: Der lteibnngswitlrrnt~r~~d rotiercncler Srl~rihcn in (:ol~~instw ZAMM 

--, 1.5. 191-204 - 11935): sce also: H. Fultinger: ZAMM 17, 356-358 (1937) and I

CHAPTER XXII 

The incompreesible turbulent boundary layer with 

preseure gradient J- 

In tho present chaptcr we sliall discnss the bchaviour of a turbulcnt boundary 

layer in the prwrnce of a positive or nrgativc prcssr~rc gradient along thc wall, 

thus providing an extension of thc sobjcrt matter of the preceding chapter in which 

the boundary layer on a flat plate with no pressure grhdicnt was considered. The 

present case is pzrticnlarly important for thc calculation of the drag of an aeroplane 

wing or a tutbinc blade as well as for thc untlcrstanding of the processes 

which takc plarc in a tliffuscr. Apart from skin friction we arc intcrcstctl in knowing 

whether the boundary layer will scparal.c under given rircumstanccs and if SO, 

wc shall wish to detcrmine tl~c point of separation. The existmcc of a ncgstive 

and, in particular, of a positive prcssnrc gratlicnt exerts a strong influcncc on the 

formation of the laycr just as was thc case with laminar layers. At the present timc 

these very complicated phcnomcna arc far from being understood complctcly but 

there are in cxistcncc several scmi-empirical mctbods of calculation which lead to 

comparatively satisfactory results. 

In the year 1962, J.C. Rotta [86] prepared a comprehensive and careful review 

of this vast ficld of knowledge. In order to develop methods of calculating incom- 

pressible, turbulent boundary layers with pressure gradients it is necessary to derive 

from experiment relations which go beyond thosc employed for pipes and flat plates 

at zero incidence. For this reason we shall begin by giving a short account of some 

experimental results. 

a. Some cx~mrirncntal results 

ILrly systc~natic cxpcrimcnts on two-dimensional flow^ with pressure drop and 

prcssuro rise in convcrgcnt, and divcrgcnt clianncls with flat walls have been carried 

out by F. Doench [28], J. Nikuradse [71], II. Hochschild [45], R. Kroener 1571 and 

J. Polzin [76]. Measurements on circular diffusers, and particularly on the efficiency 

of the process of energy transformation, are described in papers by F. A. L. Winternitz 

and W. J.Ramsay [123]. These experiments demonstrate that the shape of the velocity 

profile dcpcnds very strongly on the pressure gradient. Figure 22.1 shows the 

velocity profiles which were mcasurcd by J. Nikuradse during his g~~erirnent.3 with 

t Tho new veruion of tliiu chnplcr wo.8 propnrcd by Profemor E. Truckenbrodt whose nssistance 

I I~ercl~y grnbhlly ac:knowletlgo. 

a. Some experimontnl rusulta 

Fig. 22.1. Vclocit,y diutri1~11Lion in coni~ergcnl 

and. divergent cl~nnnols with flat wall^, as 

n~cmrirccl by J. Nikurnduo [71] 

- ImIr Included nnglr; It - wicllll of ctlnnrlcl 

-1.0 -0.6 -0.2 0 0.2 0.6 LO 

L6 

Fig. 22.2. Velocity distribulion in a divergo11 

chnnnel of ldf includotl angle n = 6" and 

a = Go, as measured by .J. Nilruradse [71]. 

The lnck of Qmmetry in the velocity distribution 

signifies incipient separation 

Fig. 22.3. Volocity distribution in n diaergent 

cl\aiincl of hnll inclrtdect analc n =: X", 

rnctwr~red by .I. Nikr~ratlnc [71]. Itcvcrsc flow 

is coniplebly dcvclopcd. Tlicr flow oncillntcs 

nt hgt?r iiikrvah het.wm\ pnthrll~ (a) twd 

(b)

670 XXII. Tho incomprcaniblo t~urbnlont bound~ry laycr 

slightly convergent or divergent channels. Tl~c half included angle of the channels 

ranged over the valrles a = -ao, -4", -2", 0°, lo, 2", 3", 4". The bo~~ndary-layer 

thickness in a convergent channel is much smaller than that at zero pressure gradient, 

whercas in a divergent channel it becomes very thick and extends as far as the centxeline 

of thc chn.nne1. For semi-angles up to 4' in a divergent channel the velocity 

profile is fully symmetrical over the width of the channel and shows no features 

associated with sepamtion. On increasing the semi-angle beyond 4" the shape of the 

velocity profile untlcrgoes a fnndament.al change. The velocity profiles for channels 

with .5O, Go and 8" of divcrgence, respectivcly, shown in Figs. 22.2 and 22.3,cease t,o 

bc symmctricnl. With a 5" nnglc of divcrgcncc, Wig. 22.2, no barlz-flow can yet bc 

disccrncd, but separation is about to brpin on one of t,he channel walls. In addition 

the flow bcnomcs unstable so t,l~nt, depending on fort-uitous disturbances, the stream 

adheres alternately to the one or the other wall of t,he channel. Such an instability 

is characteristic of incipient, separation. J. Nikuradse observed the first occurrence 

of separation at an nnglo bet.ween a = 4.8" and 6-1". At an angle of a = Go, Fig. 22.2, 

the lack of symmetry in the ve1ocit.y profile is even more pronounced, and the reversal 

of t.he flow intlicnt~cs the start, of separakion. At n = 8" the witlt,h of the region of 

Pressure distribulion 

Try- 

Fig. 22.4. llol~~lrlnry lnyrrotl wing nrrofnil. nn rncrmltrcd by Stilrprr [IOR]; mensuremmtrr in flight; 

lift rooffic:iollt r,, =- 0.4; IZoynolrln nt11111)cr R - 4 x loo; chord 1 = 1800 nlm. 't'l~e boundary 1a.s-er 

in turhulrnt all nlonp tho prrn.wrc sirfr owing lo xtlvotnr, prmnttro grnrlintlt; on tlle aertio~s ridr it. iu 

lnn~innr uprtrcntn of prcfxwrr tnit~i~~i~~tn nnd trlrl~rtlc~~t downut,rcnl~l from it 

reversed flow is considerably larger than for n = 6", and frequent oscillation of the 

stream from one side to the other is observcd, the phenomenon being absent at a = 

6" and Go. However, the duration of one particular flow configuration is sufficiently 

long for a full sct of readings to be obtained. As tho nnglc of divcrgence is incrcnscd, 

the region of reverse flow becomes wider, and the beats are more frequent. 

The diagram in Fig. 22.4 shows an example of a turbulent boundary laycr formed 

on an ~erofoil and measured by J. Stueper [lo61 in free flight. In the case represented 

here, the boundary layer on the pressure side is turbulent from the leading edge 

onwards, because here the pressure rises over the whole width of the wing. On tho 

suction side, t,hc point of t,ransition plnccs itsclf a short distancc behind thr pressure 

minimum in agreement with the description given in Scc. XVII b. The fact that the 

boundary layer has become turbulent is inferred from the sudden iorrrasc in its 

thickness. 

Very thorough expcrimcntal in~est~igations into t,ho bchnviour of l.url)ulcnt, 

boundary layers with pressure gradients have been later perfnrmctl by G. 1%. 

Schubauer and P.S. Klebanoff [97], by J. Laufer [68], and by F.H. Clauser [21]. 

The first two of the abovc papers contain, in particular, rrmlts of mcnsurcments on 

tmrbnlent, fluctuations and on thc correlation cocfficicntw which wcrc clolincd in 

Chap. XVTII. Thc last paper contains cxtcnsive rcsu1t.s of mca~urcmrnt~s on shnaring 

sLrcssns. 'l'hc c:nlcul:~l,ions closcril)otl in tho following oonl.ic,~t~ c::ru c:vitlonI.ly nlq~lg o11l.y 

to [flows which adhere comptctcly to the walls, tirat is, to cnscs which are sitnililr to 

the one shown in Figs. 22.1 and 22.4. 

b. The cnlculntion of two-dimc~~siorlnl turbulent lto~~nclnry lnyers 

1. General remarks. To this day, all methods for the calculation of turbulent 

boundary layers rely on semi-empirical procedures, because the apparent nornml 

and tangential stmss componcnta crent.cd by the turbulent fluclmations as well as 

the thus released energy losses cannot be ~alculat~cd by purely theorct.ical means. 

Furthcrmorc, it. is still necessary to int,rotlucc hcrc empirical relations of the t,ypc of 

Prandtl's famous mixing-length formula invented in 1925, because the statistical 

t,hrory of t,urbulcnce has yet t.o produce a replacement. for it. [t is nst,onishing thatf 

1'rnndt,l1s hypot,hesis, half a cent,ury after it8 discovery, still plays a vcry important 

role in the lit,erat,urc on the calculat~ion of tu~.bulcnt boundary Iaycrs. Mosl contcmporary 

met,hods are approximate; thy make use of t,he momentum and energy 

equat.ions of t,he velocit,y layer (as distinct from t.he t,hcrmal layer which will not be 

discussrcl in this scction) and of certain relations t.hat follow from them. Thc corrcslmding 

rclitl,ions for Inmi11n.r I)ouneln.ry In.ycrs wrro tic-tivctl in Clr:~ps. S :me1 XI. 

The procedures for the calculatio~~ of turbulent boundary laycrs available today 

can bc tlividcd into two classes: methods based on in,legral fornts of t,hc principal 

equations and methods based on diffarcnlid equations. The former can be traced t,o 

work t.hat was done by Th. von IGirrn.in in 1!)21 1.: +!:is procedure, thc partial tlifferer~t,ial 

equations are reduced to a system of ordinary dillercnbia~ cquat,ions in that. an 

ana1yt.i~ int,cgrat,ion in the t,ransversc dircot,ion is first performed, cf. C11n.p~. VllI 

and XITI. It1 the ot,l~er class of cnscs, thr pnrt.inl tliITrrential ~clnat~ions arc int.cgratctl 

dircct.ly I)y the n.pplicnt,ion of nnrnoriral rnrt,hotls, suc:h as the mrtltod of fi11it.c tlifircnces 

outlinctl in Scc. IXi, or by finitc clcmcnt.s. It, is c:vitlrnl, thn,t t.hc nniount, of work

672 

XXll. 'l'llc incon~prcssiblc turbulent bol~ndary lnycr 1). 7'110 cnlci~lnLion of two-rlimcnsional lrtrbulont boundary lnynrs 673 

involved when differential equation methods are used is substantially larger than in 

tho case of integral methods. The former require the use of a very large digital 

computer equipped with a large memory, whereas the latter can be done on a small 

calcrllator or, even, with the aid of a slide rule. 

In the following paragraphs we shall confine ourselves to the des~ript~ion of 

methods which rcsult merely in the calculation of time-averaged values of such 

variables of the turbulent flow as t,he velocity, the local shearing stress and the region 

of separation, because we subscribe to the view that only such mean values are of real 

interost to the engineer. Thus we rcfrain from calculating all those quantities that 

result from fluctuations, for example the correlation coefficients, the intensity of 

turbulence and its scale. Readers interested in these aspects are referred to more 

specialized publications, e. g. [lo, 813. 

Rcsearch into turbulent boundary layers was considerably advanced by the 

Stanford Univcrsity Conference organized by S. J. Kline in 1968. The results achieved 

at the time have been published in two large volumes edited by S. J. Kline, M.V. 

Morkovin, G. Sovran, D. J. Cockrell, D. E. Coles and E.A. Hirst [64]. In the appended 

[79] "morphology" prepared by W. C. Reynolds, the reader will find a de~cription 

of 20 integral and 8 differential methods and characterized according to their respective 

physical basis (status = of 1967). They differ, principally, in the empirical closure 

functions which are introduced in ordcr to malre the system of equations solvable. In 

addition, the conference had at its disposal 33 sets of experimental data which served 

as testing material for the computational algorithms. About ten years later, W.C. 

Reynolds [81] provided once again a summary review of the very large number of 

computational schemes; this appeared in his contribution to the Annual Reviews of 

Fluid Mechanics of 1976 (cf. the same author's 1974 contribution in Chemical Engineering 

[80]). In 1974 there appeared the book by F.M. White [I191 which describes 

20 integral and 11 differential procedures. It is difficult, and we shall not attempt, to 

select a "best method" from among the very large number proposed so far. 

A summary of many of these methods, principally integral ones, was prepared 

earlier by A. Walz [116] and J.C. Rotta [86, 871. A review of differedial methods is 

conlaincd in P. Bmdshaw's contributions [9, 12, 13, 141. Further, the book by T. 

Cebrci and A.M.O. Srnit,l~ [20] and two earlier papers by the same authors [18, 191, 

contain good reviews of many calculational procedures. The two earlier reviews by 

L. S. C. I$Y . 

2. Truckenbrodt's integral method. Before we proceed with the description of the 

details of E. Truckenbrodt's [I141 method, we find it helpful for its understanding to 

preface it with a few historical remarks. As already mentioned earlier, all computa- 

tional algorithms for turbulent boundnry layers rcly on ecrtnin empirical relations. 

As time progressed, and, in particular, since thc middlc of t,hc tl~irtics, tho empirical 

basis, and hencc also the semi-empirical and theoretical computational proccclurcs, 

underwent a process of continuous improvement. 

The first method for tho calculntion of t,urbulcnt boundnry Inycrs wit.11 prcssure 

gratlicnts was formulated by E. Gruschwitz [40] in 1931. The cxpcrimcntd data on 

which this mrthod wns basod wcro Intcr irnprovcd by A. 1

074 XXIJ. Tho incomprc~~ihlo t,t~rbnlcnt boundnry lnycr 

uniqwly, as evidcnced by the graph of Fig 22.5 This fact is expressed by a relation 

Iflz = /(If32), if a dight, residual rlcpendence on the Reynolds number is neglected. 

Guided hy the prenrding ohservaf ion, IS.'rruckenbrodt [I 141 introducetlt the nrodt/i~d 

shape factor 

The reference valuc (IIz3)m = (l/Naz)m hns been chosrn as t.he loner limit, of integmtion 

because it. reprcsent.~ an avcrage value for flows wit'hout a pressure gradient,. 111 

the casc of t~urbulcnt boundary layers wn choose = 1.3. The nunierical evaluation 

of the relation in eqn. (22.4) can tie undert,alren on the basis of a relation 

intlicntcd 1)y 11. Fernholz [33]. The result is seen plotketl in Fig. 22.6. 

In the case of flow with zero pressure gradient, we find that I/ = If, = 1 by 

definition (mean value in the casc of tnrbulrnt flow). Flows wit11 atlversr pressure 

gradicnta (pressure rising in the downstream direction) are characterized by If < 

H < 1, wlierc.ns for accelerated flows (pressure decreasing) we find that 1 < H < 

Hop where Ifs denotes tlie shape factor for the velocity profile with incipient separ- 

ation, and Ifo denotes the shape factor of a two-dimen~ional stagnation-flow profile. 

According to I 

(Hlz)s > 4.0 or 11s w 0.723, wl~creas A. \\'nlz [I 161 proposcs the values 1.50 < (1132)s < 

1.57, or 0.736 < If,. < WXil. According 1.0 A. I\. 'I'ownseritl (I Ion] (cf. St.vatfortl (1041) 

a vanishing xhcar st.rcss occurs for (I/12)s - 2.274 or IIs = 0.784 in tlie case of 

profiles crcat,ed by an ext,crnal flow with U(x) - sP with p = -0.234. The various 

shapc fact,ors for incipient separation haw been indicat,ed. in Fig. 22.6. The values 

of the modified factor Ils fluctuate n~uch less than those of (Ill& and (H23)S. 

Refrrcnce 11141 indicates t.l~at acparalion can occur for 

The rangc Ifs - 0.723 < II < 0.761 = If; I 

describes vrlority prolilcs thnt nre prow to separiatc. 

: l'hr ri~rnirric*nl 

nv:~il:~blc nL llw I~nre. 

roncltnntn in rqn. (22.51,) hnve heen ndjusted to represent the experiments 

b. Thc cnlculntion of two-dimcnsionnl tr~rbulcnt honndnry Injqyv 675 

Fig.22.5. l'hr ratio of houndnry 

lnyrr thirknrclncn Ifaz = &/rh plottcd 

against If12 = fill&, aftcr J.C. ltottrr 

1831 and I

070 XXII. Tlic incompreaaibto lnrl)~rlent bounclary layer b. The calculation of two-dimenaionnl turbulent boundary lnyera 077 

As ttlie hnsic rqiia~f,ions for nromen,tum. thickness &(x) and for e?asrg?/ thickn,esa 

03(2:), we obtaint : 

and 

rmpcct.ivcly. Ilere CT is the skin-frict,iorr cocfficicnt and c~ is thc.dissipntion cocfficicnt. 

r 7 

llrr prrretling two coeifiricrib rela.t,ctl to thc shearing stress depcnd st,rtingly on the- 

ltcynolrls nurnbcr, R, arcording to eqn. (22.2), and on the shape factor 11 in conform- 

ity with cqn. (22.4). 'l'hrr following power-laws for their description lrnve withst.oot1 

the fcst of time: 

I I 

'l?lic cxprcssions contain the factors a([{) and P(N) which are kque functions of the 

shape factor and a specified power of the local Reynolds number~ Rz or R3. Thc dingrams 

in Figs. 22.7% and b represrnt the quantities n. and h as well as a' = n/nm and 

p' = fi/j?rn together with am and /?, (denoting values for zero-gradient flow) as 

f~lnrtions of II. The rrsprctive form~rlac are quotcd in the captions. Jt is seen that 

/?' varirs slowly with 11, whcrcas n' assumes the vnlue af(ll = Ils) = 0 at scpnration 

antl t>llcn inrrcascs fast with incrcasing If. 

Equations (22.8a, b) are now substituted int,o eqns. (22.7n, b) and this lends 11s 

to the modified forms of thc momcntum and energy equations for &(x) and 63(x), 

Pig. 22.7. Shearing stresses in turbulent bonndary layer/corresponding to eqn. (22.8) after [33], 

[I 161; a) exponent8 n and b in terms of If; b) fnctors a and P in terms of If 

t The above eqnntiona ncglect t.hc effcct of the normal components 3 and of the tonsor 

of lteynolds stressrs from oqn. (18.10). Among othcrs, references [85, 87) contain indications 

on how to modify bliooe equnt.iona if this uimplification is not acceptable. 

In order to complete the evaluation it is still necessary to know the ahape-laclor 

/un,dion H(x). It is shown on p. 487 of [I 131 that eqns. (22.7n, b) lead to tho working 

forms 

for the cnlculation of H32(x) and of HZ3(x). The shape factors can now be cnlculated 

either by the use of a coupled pair of cquations, namely (22.7a) antl (22.98) or (22.7 b) 

and (22 9b) To distinguish bctwcen these two possibilities we speak of the morne~tl~~rn 

method in the first casc and of thc energy method in the second case. In most. proccdr~res 

use is made of the momentum method, whereas E. Truckcnbrodt [I 11, 1141 favours 

the energy method. Thc lnttcr choice has bccn madc for two rcnsons: (a) The Icft-lland 

side of eqn. (22.7b), unlike that of cqn. (22.7a), does not dcpcrld explicitly on tho 

shape factor. Thus instead of eqn. (22.7 b) we can also write 

(b) TIlc dissipntion fnc(,ol. c~ on thc right siclc or c(,n. (22.70) munt bc (::llc\~l:~(rcl rroin 

eqn. (22.8b) by performing a quadrature extended ovcr the boundary-layer tlricltness 

0 ( y < &x), whercas the skin-friction coefficient c~ on the right-hand side of cqn. 

(22.7~) tlcpcnds only on the local shcaring strcss at thc wnll, cqn. (22.411). 'l'llis 

signifies that the dissipation work depends much less on the shape factor than thc 

shearing stress at the wall. This is confirmed by the graphs of n'(I1) and P'(I1) in 

Fig. 22.7b. Thus, in the energy method, the coupling between the equations determining 

the boundary-layer thickness (energy equation) and the equntion which 

determines thc shape fnctor turns out to bc much wcnlccr than in thc n~omcntum 

method. 

Reference [114] s11owsI1ow the basiccquntions for t,lre boundary-laycr t.llicltncsses, 

eqns. (22.7a, h), can be transformed into equations that dctermine the local ltcynolds 

numbers defined in eqn. (22.2). Similarly, this reference shows how the basic equations 

for the shape factors, eqns. (22.!h, b), can be transformed into equations for 

the modified ~hape factor defined in eqn. (22.4). In this mnnner, we obtain 

Table 22.1 summarizes tl~c expression for the contractions m, di and y. The quantities 

R, m, @ and p are provided with subscript 2 for the momentum mcbhod, and 

wibh subscript 3 for the energy method. 

4. Quadrature for the calculation of plane turbulet~t boundary layers. Under crrtain 

simplifying assumptions it is possible still f~nrther to simplify the system of equations 

(22 lla, b) In this manner it is possi1)le to derive explicit cxprcssions for R(n) and 

N(r) by quadrature for an arbitrary velocity variation, TJ(x), in the outer flow, that

678 XX 11. Tho inconiprcnnihle t~~rl~alont boundnry lnyer 

Table 22.1. Su~nn~ary of thr qnanlitir~ whirl1 occur in the equations for the cnlrulation of the 

din~ensionless ~non~entum tlrirkness, Rz, of tlw din~rnsionlcss energy thickness, R3, and of the shape 

factor; RCC eqns. (22.1 In, b) 

BIonwntum method 

(wbscript 2) 

Energy method 

(subscript 3) 

is for zcro, atlvrrsc, or favouraMc prcssnrt. grat1icnt.s. The int,rgrat.ion is here cxtentlccl 

only over c~rt~nin powers of TJ(z). We now proceed to derive such approximate, cxplirit 

form~ilae and to show how a, suitable choice of approximate procedures can 

lead to a step-by-stcp improvcrnent. 

Analogy with lan~innr borrndary layer (momentum method): In a manner analogous 

wi1.h I

FRO XXII. Tho incornprcs~iblo turbulent boundary layer 11. Tho calculaLion of two-climcnsional turbulent boundary lnycrs (iX I 

tribution U(x). Since the valuc of the shape factor, N(x) = 1, has already been 

assigned, the only quantity that we need to calculate is the local Reynolds number 

determined by eqn. (22.11a). Since m = const, we can contract the two terms on 

the Icft-hand side and solve the problcm by performing two integrations, one each 

for Rz(x) and R3(x). In ~ont~racted form these are 

The relations and numerical values to be used for the exponents i, e and n, as 

well M for the modified lrinematie viscosity v' are listed in Table 22.2, separately 

for the momentum and for the energy method. The constant of integration is 

DifTercntiat.ing eqn. (22.17) wit11 rc~pcctr to x and taking into account thc contmctions 

dcfincd in Tablrs 22.1 antl 22.2 wc can tlcmonstrat~ eonsist~ency with eqn. (22.11a). 

In the case of the moment,um m&hod, eqn. (22.17) becomes identical with eqn. (22.16) 

if wc put i = n, R = Rz antl IZ = Ez. 

Comparing the nurnerical data of Table 22.2, we find far-reaching agreement. 

In spite of considerable difl'erences in the assumptions for the shape factor and for 

the shearing stress at the wall we discover that thc two explicit cquat.ions for the 

calculation of the momentum thickness are equivalent. The following specific numeri- 

cal values can be recommcndctl: 

The cnergy mcthod is discussed below. 

Analogy with self-similar eolutions (ener~y method): Self-similar solutions in 

boundary-layer thcory are generally described as equilibrium /lows when they occur 

in turbulent motion. They are charnctcrized by the fact that the velocity profiles 

u/U nt varying positions x become similar for certain velocity distributions U(x) of 

the ontcr flow. 'J'his means that, the shape factor H(x) remains constant with z, that 

is that dlfldx = 0. Figure 22.7 implics that all quantities which depend on x in 

gcnrml n~ust bcrome constant for such equilibrium boundary layers. 

We now suI)st,itutc in cqn. (22.10) the expression for cn from cqn. (22.813) and 

note that the integration with respect to x can be performed in closed form with b = 

const and p' = const. The Reynolds nnmbcr formcd with the energy thickness is 

thus given by 

. . 

I hc numerical vn.lucs for 6, r, n, and v' are to Ip sclcctcd in accordance with the 

relations in Table 22.2 and l'ig. 22.7s. 

I 

In the spceial rase of separation-prone flows for which Hs < I1 < Ilk, we find, 

for example, that. 1 4- b$ = 1.004 and I + b, = 1.152. Tliesc two values differ by 

about. 5%. Such a disrrcpancy can t)c tlisregardcd in view of the i~ncert~ainties in- 

Iirrent in s~wh appro xi mat,^ mctzhods. In othcr words, this signifies that it is possiblc 

to perform calculations using numerical values based on t,hc flat-plat,c ani~logy. 'rhc 

quantity P' = PIPQ) that appears in eqn. (22.19) also depends only wcaltly on the shape 

factor; by way of approximation, we let it be P' = I. 'rhns, the calculabion of ~(IIc 

cncrgy t,hickness with the aid of eqn. (22.10) can be bascci on tdlc followitig numerical 

valucs : 

R += R3: h = 0.152; e == 2.3 ; n = 3.3; r' = 78 v; P' =- 1 . (22.20) 

With thrse assumption, eqn. (22.19) transforms into cqn. (22.17) bearing in mint1 that 

i = b, R = R3, and /C = "3, as cxpcctcd. 

Refcrence [114] diows t,hat eqn. (22.1 la) suffices by it,sclf to solvc the prnbleln 

when the energy mct,llocI is used. 13y contrast, when t.hc nionlent.urn mctliotl is usctl, 

t.he coupling between cqns. (22.1 la) and (22.11b) cannot hc disregardcdt. 'Hie Iatkr 

Icads us t,o the trivial result that Rz = IIz3R3 in view of thc dcfnit.ion Ifzn -- 02/fi3. 

At. t.hr icvcl of approximation - - 

considcrcd so far, thc momcntum mctliotl t.urns out 

t,o be itlcntical witJr t.11~ cncrgy mct.liot1. Ncvcrthclcsrr, tho two proc:rtl~~~.c,s tlilli,~, 

csscntially from one another in that tlic momentum mcthod employs tho t,wo basic 

cquat,ions (22.11~~) and (22.1 lb), whcrras t,hc cncrgy mct,hotl gct,s by with cqn. (22.1 la) 

nlonc. As far as tlic devclop~nent of furthcr approximat.ions in the for111 of simple 

inkgrals is concerned, we have cxhaust,ctl t.hc pot,ent.inl inlierent in thc rnomrnt.um 

mct,horl. In the energy mcthod, cqn. (22.1 1b) is used to tlcrivc n forninla for t,hc shape 

fnclor by closed-form int.rgrnl.ion, as wc arc about, t.o how. 

Integration method due to E. Truckenbrodt: 13. Trucltcnbrodt [I 11, 1141 worlrcd 

out an npproximnt,~ mc:l,hotl for 1.11~ cxplicil inttcyrnl.iol~ of t,llc cqi~n.Iio~is of t.111.1~111v1it~ 

boundary layers which serves to obtain the boundary-layer t,l~iclrness (cnergy tliickness) 

as well as the (modified) shape fact,or. The first vcrsion of the method [Ill] has 

proved to be practicable for calculat,ions in engineering applications. It was, therefore, 

tliougl~t useful to modify it in the light of more rccent discoveries. Employing 

the classification introduced above, we tlescribc this as an energy method. Tlrc method 

can be used for two-dimensional as wellasforaxiallysymmct~ricflows,c/. Sco. XXlX cl. 

Tlic method is based on cqn. (22.11a) which is used to calculate the Iteynolds 

number formed with the energy thickness. Hence we put R = R3, m = 2 and (IS = 

@a = (21~) /? Rab in accordance with Table 22.1. If we assume 1,hat 6 = const and 

p(r) is known, we can integrate eqn. (22.11a) with rcspcct to x and obtain 

1: 

1 B4x) 

{R3(4I1+* = -;; [C1(2)1C where E3(x) = E3(x1) + JP' Un dx. (22.21) 

The numerical values in cqn. (22.20) are valid up to P'(.T) = P(:c)/Pw. Ilowcvcr, an 

inspection of Fig. 22.7b shows that /I' docs not clcviatc much from the valuc 1.0, 

and we may calculate with /I' = 1.0 by way of approximation. In this case, eqn. 

(22.21) transforms into cqn. (22.19). Thus, if no great dernands of accuracy are made 

on the values of the Reynolds number, we obtain 

-- 

z 

2% 

\ lllltbl 

t By way of amplificstion, we mcnt,ion that the result, derived in Tal~le 6 of [I141 in also valid 

when a -= const nnd b = ronut,.

082 

XXII. Thc incomprm~il~lr L~lrbtrlolt bn~rndnry lnyrr 

wherc bhr following numerical vnlncs (cf. Table 22.2 - energy method) have been 

employed : 

b = 0.152; v' .= 80 v; with flq(rI) = v' {[U(r1)J2 R3(xl))lCb. 

(22.22 b) 

This explicit formula cont,nins only t.hr extmnal free-strcam vclocity U(x) which 

mny IIC known from pot,cnt,ial t.hcory or from mcasarement. The positmion .2: = xl 

constit.ntes t,he st.nrt,ing point for t,he calculation. 

Apart from the velocity U(xl), the constant of integrntion E3(rl) rontains also 

the encrgy thickness 63(.cl). If the station xl coincides with the point of transitmion 

the cnergy thickness should be ralc~llatcd over the laminar boundary layer in t.hc 

rangc 0 2 x xl Here x = 0 rlenotcx thc start of t,he boundary layer; for example, 

the leading edge of a plate or the stagnation point of a blunt body. It waa shown in 

11141 that eqn. (22.22a) is also valid for laminar boundary layers whenb = 1, vi = 

v/Q& = 0.91 7v and lC3(xl = 0) = 0 should be specificd. In this case, with a laminar 

starting length, the constant of integration becomes 

112 

(point of transition). (22.23) 

If the boundary lnyer drcntly is t,urbulent at z = XI, it is necessary to substitute 

into eqn. (22.22) for E3(z1) t,he local vnlue R3(x1) = n3(x1) U(xl)/v. 

In many practirnl npplicnt.ions it. is not enough to know the \)cl~aviour of t,hc 

boundary-layer thickness, lwre t,hc encrgy t.hickness &(x). This is the case wit,h 

ueparation-prone or separat,ctl boundary hyers. If, for example, it is necessary t,o 

makc n statcmcnt about t.he possibiliby of separation, it is necessary to know the 

velocity parameters along the wdl. All mct,hods discussed in Sec. XXIIbl provide 

procedures for t,he calouletion of sonir xhnpe fnctor in addition to tht of a boundarylayer 

t~liiclrncsx, such as the niomcnt,~~m thickness &(x) discussed t.here. The shapc 

factors nre defined differently in diff'rrent methods and different differential equations 

are specified for thcir ralrnlation. A revicw and intercomparison was given by J.C. 

Roth [85]. 

l'hc diPFrrcwt,inl rq11r21~ion (22.!h, 1)) for t.hc shnpr. factors 1132(x) and Hz3(~) were 

obtained by tlic coupling of thc ~~ior~~rnturn-integral and energy-integrnl equnt.ions 

(22.78, b). Thc pereding dilTrrent,inl equntions det,ermine the shape factor in a unique 

way proviclrcl that onc-para~net~cr velocity profiles Hlz.= f(H32) or Hlz = f(Hm) 

nrc post,ulnt,ctl and npproxiniak cxprrsxions for t,he shear-stress coefficients CT and 

co are s~hst.itllt.c11 from rqn. (22.Rn, b). The dct,crmining equntion (22.11b) for thc 

shapr factor cnn br writ.tm in t,crms of thc naodijied shape lactor If = f(llzs) proposed 

by 15. l'r~~rlrcnl~rotlt.. l'ogcl.hcr with rqn. (22.11a), this relnt,ion forms a system of 

simultn.rous tliffrrrnhl equations for t,l~e Reync~lds number formed wit,h t,lte energy 

t,hickncss, Ra(:v). and for thc shapt factor 11(a)! According to Table 22.1 we must 

put TN -.- 2 =- const 1tr1 Ihr rnrlgy ntcthotl clisrussed herc 7'hc forms of the funrtions 

cl,3(RR, 11) nnd lli3(R3. If) arr lo Irr taltrll Irom thc same talrlc Rcference[ll4]summarizci 

cqns (22 1 I n) nntl (22 1 1 0) as follows : 

with 

and 

Z 

G1(x) = G(q) + j y' un-' dx, 

where n is listed in Tablc 22.2 (energy method). The correction function y'(x) = 

yf(R3, H) can be calculated with the aid of a(ll), b(H), u(H), /l(II), If12(x), 1fS2(H), 

as well as R3(2) and N(x). 

The correction y' differs by a larger or smaller amount from the value 1.0 in the 

case of a turbulent boundary layer, nnd cannot bc determined with an adrqunte 

degree of reliability. 13y way of approximation, we assume y'(x) = const = 1.0 and 

introducc a new quantity c = const = 4.0 in order further to simplify the analytic 

solution. The qunntily c has been. so dct~ermincd as to nchicve optimum ngrccmcnl 

between available measurements [64] and theoretical results; see also [114]. The 

modified shape factor is obtained from the equation 

which is the result of somc algchraic txansformations not reproduced Itcrc. llerc thc 

influence /unctions of the external velocity distribution are defined os 

The initial valuw, i.0. the constants of intcgration are 

We take the numerical constants as 

The integral expression (22.25) for the calculation of the shape factor contains 

only the external velocily distribution U(x), as was the case with the corresponding 

integral expression (22.22) for the calculation of the Rcynolds number. The detm- 

mination of the inllucncc function N(r) rcquircs thc pcrformnnco of n doublc inLc- 

gration with respect to x. The position x = xl once again represents the starting 

point of the calculation. 

The constants of integration G(s1) and N(xl) contain the shape factor lI(xl) 

in addition to the'velocity U(xl) and thc Reynolds number R3(xl). If the position 

xl coincides with the point. of transition, it, is necessary to require that the encrgy 

thickness of the laminar boundnry layer must be equal to that of the turbulent. 

boundary lnyer in accordance with eqn. (22.23). On the other hand, the shapc factor 

may change it.s value at the point of trnnsition. The numcricnl values of bl~c shapc 

fnctor lic in t,hc mngc 1.0 2 11 2 Ils = 0.723. 

2,

684 XXII. 'Yhc incon~prcssible Lnrbulont boundary layer 

The t,l~cory of tJic origin of turbulence presented in Chap. XVIl leads to the 

conelusion, which agrccs with mcasurcmmt,s, that tmnsition from laminar to turbulent~ 

flow in the boundary Inycr occurs at n place which tics n small distance downstream 

from the vclocity rnaximuru of t,lic external stcream. For t,his rcnson, and by 

way of approximntion, it is permissi1)lc t,o base thc calculz~tion at the point of transition 

on the value that, corresponds t ,~ an cxtcrnal flow with a zero pressure gradient. 

According to the definition of II in qn. (22.4) the latter is equal for laminar and 

turbulent houndary Iaycrs, namely 

If thc boundary laycr is already tdrbulcnt at x = q , it is necessary to employ tho 

corresponding local valucs G(zl) and N(xl). 

5. Applicntion of the mcthod. The approximate mcthod described in the preceding 

paragraphs can bc applicd wit,h ease bccausc only simple integrations are required. 

Such dct.ailcd caloulat.ions have been performed for all experimental data (33 sets) 

collected in 1543; in particular, using cqn. (22.22), thc calculations yielded the variat.ion 

R3(x) of the Reynolds number formed with the energy thickness as well as the 

corresponding variation Il(z) of the modified shape factor aftcr eqn. (22.2B)t. In this 

manner, thc pract,ical calculations inclutled vcry diverse extcrnal flow rcgimcs and 

so covercd a wide range of applications. Figure 22.8 illustrat~es the comparison bctween 

theory and mcasurement for an aerofoil in an adverse pressure gradient. Similar 

comparisons for ot,hcr meas~iremcnt.~ are shown in Figs. 22.9a, b:. The latter diagram 

contains a comparison of calculated and measured valucs of the Iteynolds number 

and of t,l~e shape factor for the measuring stlation located furthest downstream. 

Deviations from the straight linc const.itut.e a measure of the quality of the approximate 

method. The comparison for logR3 contained in Fig. 22.9a is satisfactory, 

particularly if account is taken of the fact that excessive demands on the accuracy 

of calculat.cd valucs of the Reynolds number arc of no great practical significance. 

According to tlicory, the six sets of rneasurcments illustrated in Fig. 22.9b for 

which I2 < Ils cxhibit incipient scpamtion. Measurements have confirmed this, arid 

Ref. [J 141 cont,ains a more dctailcd discussion of this circumstance. The sets of mea- 

surements designated ldent 1500 antl Idcnt 2600 show particularly large dis- 

crepnncics hctwxn Lhcory antl measurement. Thc casc Itlent 1500 rcprcscnts a re- 

attached boundary laycr bnhind a ledge. It is understandable that thc preceding 

method is not quit,e sat,isfactory in this casc as far as the calculation of the Reynolds 

number antl of t,hc shnpc factor is conccrncd. Case Idcnt. 2600 rcfcrs to n so-callcd 

equilibrium boundary hyer formed uncicr an external stream with U(z) - x-0.255. 

Townscnd [IIOa] investigated a similar boundary layer, na.mely one with TJ(x) - 

x-0.234. IIc obtained t.he value B = 0.748 for the shape factor which differs consider- 

ably from t#hc measured value Ei = 0.823. Thc appyoximate method yields If = 0.731. 

At Lhc prcscnt time it is not possible to explain the reason for these discrepancies. 

To conclutlc, we wish to draw the reader's at$ention to the fact that the simple 

t We have introduced corrcct.iot18 for t.lircc-tli~~~er~niotlal eflccts ill order t,o account for a possible 

bouvcrgcncc or divnrgenco of nt.rcntnIincn. Tim correction wns based on t,he methot1 of J.C. 

: 

Ihttn 1861. Cf. mmnrk on p. 676 

'rhc dingrnnis inchtlc t01c crms of axially ~ytnmct~ric flowd discus8ed in See. XXII d 1. 

1). 'I'll(: calc:ulntion ol two-climrnsionnl lurbulcrit bounclnry lnyrrs 

Fig. 22.8. Turbulent boundary layer on a wing aorofoil in adverso premuro grndient [R4]; cnsc Idcut 

2100: a = (z - xl)/(x - sN) where zl = initial lneasuring station (start of measorement), XN = 

final measuring station (end of measurement). Memured points by G. B. Schubaucr and 1'. 8. 

Klebnnoff. Theory - full line - after eqns. (22.22) and (22.26). a) Reynolds n~rn~bcr; I)) uhnpe 

fnctor~ Ill* nnd II 

Fig. 22.9. Turbulent bouudary layer data taken from 33 sets of mensureruents with diKercnt velo- 

city distributions in the free stream; plotted points refer to end station at ZN. Measurements (sub- 

script Illms) nftcr [R4]. 'rheory (nub~cript I'h) ns in eqns. (22.22) nnd (22.25). n) hynoltls ru~mber 

RB; b) shnpe factor 11 

assumption regarding the coefficient CD from cqn. (22.8b) for dissipated work is only 

conditionally valid because it describes merely ita variation with the local Rcynolds 

number and shape factor. A more accurate calculation would have to include the 

effect of the upstream portion of the boundary layer on CD (cf. here the investigation 

in [86]). 

In cases when the external velocity can be assumcd to be proportha1 to a 

power of x, say U(x) - xp with p = const, the application of our method becomes 

very simple. Let us assume that the turbulent boundary layer starts at x = 0 with- 

out a laminar inlet portion so that the constants of intcgmtion in cqns. (22.22a) a d

(22.26) vi11k11. 'S11r reql~ircrl int,cgrnls can bc writkrn in closed form, and we obt,ain 

- 

with 1) = 0.162, -- 2(1 4 1)) /?, = 04127, c = 4.0, r = 1 + (3 -+- 28) p and n = 

I -1 2 (1 -1- b) p. I'or n givcn vnlr~c of p t h shnpc factor is Il(z) == const,. 'l'hia means 

t.hnt, for 11(:1.) - .7:p IVD arc tlcnling will1 n sdf-sirnilnr solrgtion (cquilil~rir~r~~ I,ountlnry 

Inyrr). 'I'ho c:nsc p 5 0 rq)rewnI..s n Il:rt plnt.o at zrro incitlcnce wit,11 lJ(:r) = U, = 

co11st.. 

08 

0 7 

O' O.? dL 0:6 i8 1.; 

lheory 

0 02 OL 06 08 10 

X 

Fig. 22.10. Tnrhnlrnt honntlnry - laver . on a bodv 

of rrvolution with initially strong prcssnrc tine 

end tmnsition to constant prrssnre [R4]; cnac 

Sdrnt 4000: ii - (x - XI)/(% - XN), where xl 

= initial measuring station (start of meaaure- 

rnrnt). ZN = find measuring station (end of 

n~cnsurrmcnt ). Measured pointi by Mosrs (case 5) 

Theory (fnll line). 

a) ILynolcls nnrnher R1; b) shape factor Ill2 6. Rc~narks on the behnviour of turLulent boundnry layers in the presence of n pressure 

grndirnt. 'Slw applicntinn of the method described in Chap. XXIlb4 to turbulent boundnry layers 

lrndn 11s to the cnlc~~lation of the vnrint.ion along the llow of the Reynolds number Ra(z) fornlcti 

wit11 tlm energy t,l~ivkneru, fin(%), and of that of the modified shape factor H(z). Adrlitional quantitics 

pcrtnining to the bonndnry In.yer cnn be obtnincd hy adding .the re la ti or^^ depicted in Figs. 

22.6 and 22.7. 

, 

I . hr Ilr,ynoltla stressos (lo not change rn~trh dong strna~nli~~rs in rclnl.ivnly short t ~~rb~~le~t 

hn~nthry litycrn in 1.11~ ~~rcscn(.c of atFOng p'W8111.C gradicnLq. It. O. Doissler 1261 dc~nonstmte~l 

that, t,hc n~snn~pt.ion ofa constant shnaring stress can lead to good agreement betarell ealcr~lat.ion 

and ~ncnsnrcmrnt ; Ito also snccccded it1 calculating heat-t.rn118fcr coeflicients for tr~rbul~~lt bollnd. 

nry 1:tyrr.s 1261 by tho nsc of the same met.hod. 

I 

Boundnry layer tl~ickness: When tho vnlucs of II(k) n.rc know11, t,he diagram in Fig. 22.0 

yields the reln.tion 111z(a.) = 1ftz[1f(z)l and Ilzn[ll(~)J 111 turn, employing the deAnit.ions given 

in eqna. (22.31), c), we ran cnlcnlate the disj~laoen~ent thickness and the motnent11111 thicklless 

I). 'I'hr rnlc~tl:rf ion of t.\vo-dimrnnion;II I.nrbnlrnt. hom~tlnry Inyrrs 687 

rcspcctively. For cqvilibriuni bowdwy laynr~ for which Ille(z) = eonst nnd Ilm(r) - ro~~st. 

we obhin 

hl(z) - &(x) - &(x) - z(I-~D)~"+~), 

as seen from eqn. (22.281~). 

Total drag: 'The form drng of n hody in a &ream consisb of skin frirtio~~ ant1 I~~CSSIIIC tlr:~~. 

Tho skin friction is the integral of shearing stresses taken over tho surface of thc body. 1Svon in 

cnnna \viIll~w~t ~rpn~~:~.l~in~~ it in nc(:cunary to n01l 1.110 prn.s.viirc flr(~q tm skin frif4ion. 'I'h(- ~ri~il~ of 

the 1)rcssurc tlrng lies in the f~wt that thc boundtiry Isycr cxcrh a displncctncnL artit~n OII 1l1o 

external utream. 'l'l~c stren~nline of the potential flow are displaced from the contonr of the hody 

hy nn amotrnt, eqnnl to t,l~c tlisplncement thickncrrs. This motlifirs sorncwhnt thn prmsrlrr rlistrilmtion 

on tlte bwly ~nrfnoc. In contrnst wit.11 poLc:ntial llow (cl'Alcn~bwt's ~~TILIIOX). Iht: rcs~~It.i~(~t 

of this prcsnnrc disl.~~ibnLion ~notlilirtl 11y friclion no Irmgrr vnnisl~r~ ~IIIL prvd~~rw n 1)rrswrc drng 

which IIIII~~. be added t,o skin friction. The two togeLhcr givo lorw drog. '1'111- c-nlrnlntion ol- for111 

drag which is determined by t,lle momcntnm tl~irkncss at t.hc t,rniling edge will ho tlisc~~sscd in 

det.ail in Chap. XXV. 

Non-sepnrntina boundnry layers: Thc ~~rcnuurc drag remains smnll only if scpamtion mn be 

avoitled. 'Shis cnn be nchicvctl by llw I)ropc!r clc~ign of t.he nhnpc of tho 1)otly. 'l'hc srlf-~imilnr 

Inminnr flowu dincnss~d in (;thnpn. Vl 1 1 and IX tillbrcl cxnlnplcfl of flows whic:h (10 nol. Irnd to 

separation in the prtxence of an adverse prcssnrc gradient. When 1.h~ external flow follows the 

power law U(z) - zp, separation occun in lnminar flow for values of ps < --0.09. Thr corrcsponding 

value in tr~rt~ulcnt flow is obtnined from cqn. (22.28b) by suhstitdng in it. N = Ns < 0.723. 

Thi~ qivw p~ < - 0.27, wherem A.A. Townsend [IlOa] indicntm the valrre ps < - 0.234. This 

nignifien tllat a tnrhulcnt honndnry lnyer cnn sustain a considerably lnrgcr adverso prcsaurc 

grdicnt. wilht~t, srparnling t,l~nn tlocn n Inmin~w h~ndnry lnycr. Self-nimilnr nnl~~l~ion givf* 11 hint, 

on how 1.0 nrr~rngc? 1.110 prr-vxurn dinlribation ill ortlt?r In) n~~skrin 1hc 111r4(wt. ponnilh IIIIVI'~H(! JlrI~HHIIIo 

gradient without separation. A pressure dislribution that stark wiLh a large and continues with 

n decren.hg adverse pressure gradient generates a thinner boundary laycr and makes it possible 

to ~~~stnin n Inrgo t,otnl I)ressnro incrrn~c t.llnn n, uniform gratliont, wonltl. This fact wnn rt~nfirmrd 

cxporirnontnlly by G. U. Sc11ul)auor and W. C. Sl~nngor~borg [!I&!] rrnd by B. S. Ft.reLl;,rtl (10.iJ. I\ 

critical review of different methods of calculating the position ot the point ol' separation is contnined 

in [17]. 

Re-attaching boundary layers: More recent contrilmtions concerning t.lm partic.ulnrly intcresting 

me when a separated shear layer re-athchos itnclf to the wall and clcvelops furt,l~er as a 

boundary layer in the downstream direction arc rontainctl in the papor 11y 1'. Rr1rtls11:iw ~rncl 1'. Y. 

F. Wong 1141 na well na P. Wauschkuhn and V. Vnmntn ltnm 11 171. The tliscussion rclatrs to a 

boundary layer which has separated at a backward-facing stcp. The esscntinl dill'crcncc bct\vecn 

such a houndary layer and a "normal" boundary layer, for cxamplc on n flnt pla.te or nn aerofoil, 

consist8 in the fact that its turbulcnce structnre h~ hcromc st.rong-ly disturbed by the prior 

separation. Such a pcrtnrhation in st.ructure n~altcs it very climcult to formulate a proccrJurc for 

calculation. P. Wanschltuhn nnd \'. Vnssnta Ram [I 171 report measurements of wall nhcar stress, 

mean-velocity disttihntion and Reynolds ~t,ress in the rc-attached laycr and describe romparisons 

with uevcral evalunt,ion procedures. 

7. Turbulent boundary layem with suction and injection. The possibility of ir~flncncing the 

Bow in a boundary lnyer by blowing or snction in of some practicnl iniportnnco. parlicnlnrly with 

a view to increasing the maximum lift of aerofoilu. The promd~~ro for cnlculnting laminar boundary 

layers with suction was given in Soc. XIV h; tho corresponding mcthotl for a b~rhr~lrnt. ho~~~~tlary 

laycr wns discusbed in Scc. XXIa. 

A procedure for the calculation of a tnrhulcnt houndary laycr with I~omogenrous 

suction 

and blowing on a flat plate at zero incidence wna first fortnulatcd by H. Schlichting [!)0]. Expcri- 

mental invcstigatione and a comparison between them and theory were discvbed in See. XXIa. 

The preceding procedures were extended by W. Pechau [75] and lt. Eppler [32] to inclnde the 

ewe of an arbitrary velocity distribution -vo(z) of suction velocity. The rtxulk obtained by 

these methods are discussed in [92, 941. They contain further calculations performed with the aid 

of this procedure; they illustrate the effect 01' the magnitude and position of the suction zone on 

the minimum suction flow required to eliminate sepnrstion on ncrofoils. It turns out that the 

optimum arrangement is to concenLrate the auction zone in a narrow region on tho suction side 

of the mrofoil and to place it at n short distance behind the noso. This is undcrstnndahlc, beca~~so

688 XXJI. The incon~prcssible t,urbulent boundary lnycr 

the Inrgcst local advcrse prcssurc gradicnta occur in that region when the angles of incidence arc 

large. The required mir~imuni suction mta, ns described by the suction cocfficicnb co. ,{, are 

of the order of 0.002 to 0,004. A. R.wpet [78] performed flight mensure~nonts on wings provided 

with suction at the nosc. 

AnoI.hcr clli?cl.ivc? mcl~hotl lo incrrrr.qe wrrrin~lo~r li/l 

*! )arl,icnlarly in wings with a large flapdcflcct.ion 

:~nglc, consists in Ll~c injccliou of :t I.l~in jet of atr of largc vclority closc to the nosc of 

the flap, I'ig. 22.11. 'J'his dcvicc inll)arl~ a considamhlc arnonnt of enrrgy to the turbulcnt bounrl. 

nry lrtyor and causes it lo adhere lo 1.hc wing. 'Ulc pin in lift nchicvcd by this method can be 

esbin~at~tl by comparing t,hc pressnrc distril)utiorm of the flap wing with nnd witl~ot~t ~eparixtion, 

rwpecthly. According to J. Willinrns [122], t~l~ceffcotivencssof (.he jet can be judged with reference 

to t.lm tlin~rnsionlr~a ~nomcnlnn~ cocfficicnt 

wl~cre 11, dcnolcs t.hc vclociby of tShc jet and n rcprcnolta ib witltl~. F. 'l'l~o~itns [log, 1101 

~~wfortnod rxlcnsivc nic:murrn~cnt.~ on t,hc rlli:ct.ivc~mn of injcction Tor tlrc incrcnne in the lift 

of Ilnp wings. 110 W.ZS RIRO able to formul:rh n procedure which allows us to calculnte t,hc value 

of thc motncntnni c~ocflicicnt rcquircd to avoid separation h injcction through a slit into a 

turbulent boundary Iaycr. In addition, I?. Thomas [I001 perLrmed detailed measuremente in 

Lhc turl~nlcnt 1)ound:rry Inycr I~chind an injection slit. Sirnilnrly, iuvratigations were performed 

by P. Cnrrik nnd 15. A. Eichclbrenner[lC,J on t.11~ q~wkion of the rct,urn of a ~epsrakd boundary 

Iaycr in a large ildvcrsc presstire gradient throngh t,hc application of a tangential jet. 

H. Srhlirht.ing rgl] gnvc n ~hort snmn~nry of irivc~t.igal~ion~ into tho pro1)lcnl of increasing 

the n~aximum lift of wings by snit,nl~ly controlling the boundary layer. 

If a dilfcrctit gw is injccLcd into :r turbulent I)OIIII~I~~,V I~rycr, wc nrc ngail~ fnccd witell a 

Oinnr!y lnyer, .w wns the cwc with laminar flow (Scc. XIVc), in which the concc~~tratiotl varies 

throughout tlw flow field. Various physical hypotheses have bcen proyoscd in order to bc in s 

position to annlyssc the procrss of injection into a turbnlcnt boundary layer. 11. L. Turcotte [115] 

amumcs tlint the proccss of mixing is c~scn1i:rlly complete in the laminar sublayer and derives 

in this manner an approxin~ate formula for t,he shcnring atreas at the wall for the case of an inc:on~prenail~lc 

Iluirl. The formnln wm extended to include compressible boundary Iaycn; ih form in: 

111 this cquntion, the sobacript tu rcfcrs to the wall, thc subscript 0 relaks Lo the ewe without 

injection and thc subscript 1 dracribcs tllc frce ~trenn~. Tl~c validity of the preceding equation 

haa been confirmd by n~enn~~rclncnta lwrformcd by several authors on plaks and cones at Mach 

numbers ranging from 0 to 4.3. 

Extensive mcamrementa on the c(fect of the injection of an other gas on the shearing strew 

at the wall in boundarv layera formed on conea in wmpresaible flow have been reported upon 

by C. C. I'appna and A: I?. Okuno [731. 

M. W. Jtnb&n aud C.C. Pappna [UU] propo~cd a mixing-length theory for tho calculation 

of the effects of the injection of a foreign gas into a turbulent boundary layer. This waa applied 

to the calculation of thc rate of hwit transferred from the wall, and the corresponding results 

for the injcction of hclium and hydrogen are shown in Fig. 22.12; they have been plotted along 

with experimental results for comparison. The latter show an even larger decrease in heat transfer 

mka than predicted by the theory. By contrast, the dcovery factor seems to be affected but 

little by the injection of a lighter gtw, in a t>wbuIcnt as w&II aa in a laminar boundary layer. 

Expcrin~enb in which a heavy gnn (freon) was blown into a tarbuicnt boundary layer of 

air yiclded npproximnlcly idrnticn.1 velocity profiles ns those in which air was discharged, even 

though t.lie deosit,.y ratio of the gasos between t,he wall and outer edge of the boundary layer wna 

ns hig11 as 4. &x(:ept for t,l~c cnec of an adverse prcwure gmdicnt or of very vigorous blowing, 

the phcnomenn rnn Im described quite well with the aid of Prandtl's mixing-lengt,ll theory. 

b. Tho cxlcnlntion of two-dimensional turbulent bonndnry ltrycrs 

1) without Injection 

gain in lit1 

"due lo injeclion 

separated llow 

dl with injeclion 

Fig. 22.1 1. Flnt wing wit,h injection thrortgl~ n ulit at thc nose of Lhc 11111) for 1.ho IIII~~OHV ol'i~~~witsing 

maximum lift; a) separated flow, without injection; b) adhering flow with injeclion; c) prcsstm 

dist,ribution: d) vclocit,y distribution in tho boundary layer( 

Fig. 22.12. Heat-transfer rates for a binary bouiidary layer on a flat plnte at zero incidence with 

the injection of hydrogen or helium int,o air in a turbulent bonndnry Iaycr, after M. W. ltubcsin 

and C. C. Pappna [UU]. Comparison between theory and mennnren1cnt for tho St.anlon number 

S = q/el ti1 cpl (T, - T w) 

689

690 XXII. Thc iticompre~niblc ti~rbi~lorit boundnry lnycr 

8. Bonnclnry lnyers on cnmhered wnlls. Two-dimensional boundary layers on 

curved walls have been investigated by 1%. Wilcken [121] (see also A. Retz [4]). If 

the wall is concave the faster p~rticles are pressed against it by centdugd forccs and 

slower particles are deflected away from it. Thus the process of turbulent mixing 

which takes place between faster and slower fluid particlcs is accentuatecl and the 

intensity of t~~rhltlence is increased. The rcvcrsc is t,rue of mnvc.z walls in the ncig11bourhoocl 

of wliicll the faster particles are forcctl riway from the wall, the slowcr 

particles being pressed towards it, and turbulent mixing is impeded. ConsequentJy 

with equal pressure gradients, the thicltnws of a turbulent boundary laycr on a 

concave wall is greater than, and that on a convex wall is smaller than, the ihiekness 

on a flat plate. 11. Schmiclbaucr [!%I cxtcnded Grusehwitz's method to include the 

case of convex walls. Further results were provided by G.L. Mellor [lola, 101 b] and 

R. N. Meronry and P. ~radshaw [G5a] and 13. R. Ramaprian and 13. G. Shivaprasad 

[77a]. 

c. Turbulent boundary layers on nerofoils: maximuna lift 

A very comprcl~ensivc survey of thc prol~lcm of high-lift of ae,rofoils has reccntly 

been given by A.M. 0. Smith [101]. In the following, we proposc to dealwith t,hc theoretical 

t~qpocts of calc~lat~ing the maximum lift of aerofoils. 

It is wall known that t,hc maximum lift of an aerofoil is a.ssot:iat.sd with {.he 

separation of the boundary layer on its suction side. 'J'hus the theoretical preclict,ion 

of the n~aximurn lift must deal wit.11 the prcssurc di~tribut~ion of an aerofoil section 

with partly separated Row and with the int.eraction between this pressure disbribution 

and t.he bo~~ntlnry laycr. This prohlcm 11n.s bccn at,t.nclted by K. Jacob [47]; sce also 

the summary articlc by G. K. Korbacher [55]. Figure 22.13 refers t.o a. prof le at the 

rather large angle of incidence of ac = 10.7", and presents some theoret,ical and experimental 

results for the pressure distribution. The pressure distributions (a) and (b) 

for the two Reynolds numbers, R = 0.4 x 105 and 4.2 x 105, differ considerably: 

for t,he low Reynolds number the flow on t.he suction side of the profile is nearly fully 

separated; at t,he higher Reynolds number, the flow is only part,ly separated, S being 

the point of separation. 130th pressure dist,ributions are eharact.erized by a rat,her 

long stretch of nearly constant pressure on the suction side of t,he aerofoil. In t.he 

separated acre in terms of the potential flow theory, these pressure distribut,ions are 

calculated by nssun~ing that there exists a region of "dead air" on the suct,ion side 

wit,h approximat.cly constant pressure at its boundaries. With a surface singularity 

n~ethod such a region can be simulated hy an out,flow region produced by a cerhin 

distribution of sources on t.he aft part of the suction side of the profile. R~alizing this, 

the main prohlcm now is to determine how the Reynolds number influences separahion. 

'l'his is achieved wit>h t,hc aid of boundary-layer theory in the following way: 

in t,he potential-flow cnlcr~lnt~ion the 1ocal.ion of the point of sepnrdion is treated as a 

free paraniet,cr. TIIS determinat.ion of this parameter is achieved by combining the 

cnlc~ilnt.ion of the pressure disl,ribution of thc potential flow with separation wit.11 the 

calc~~lation of t.he laminar or turbulent boundar layer generated by this pressure 

distribution. An "atlecluatc flow" demands t,hat t I le point of separat,ion of the boundary 

laycr must coincicle wit,h the point of' separation of the potential flow with a 

tlmd-n.ir region; the rcquirctl rcsult is achieved by iteration. In this way the point of 

scpnmt.ion cart be located. The calenln.l.ion bring.9 to hear t.hc influence of the Reynoltls 

numlwr, h~cause the lomlinn of the point of sepnrat,ion of n tnrbulent bound- 

S - sepnrntlon; 

T = trnnsltlon 

c. Turl~~ilcnt boitndnry lnynr~ on ncrolniln: mnxinintn lilt 691 

Fig. 22.13. Prwure distribution on an 

wrofoil in aepnrated flow, after K. Jacob 

[47], nt two different Reynolds numbers 

R = Vllv 

Fig. 22.14. Lift coefficient eL against angle 

of incidence a for an nerofoil with a slat. 

Thsory by I

002 XXI1. Tho inro~nprc~sil~lo tt~rb~rlent boundary lnycr 

ary layer dcpends on the Reynolds number. Figure 22.13 shows that for the pressure 

distribution of the profile Go 801 thcre cxists rather good agreement between experi- 

ment and the trheory undcr consideration. 

The theory was extended to multi-element aerofoil systems with separation [48]. 

Additional results, especially on the lift, are presented in Fig. 22.14. The diagram 

dcmonst,ratcs that, the curve of the lift coefficient versus the angle of incidence Cr,(n), 

and especially the maximum lift coefficicnt C[,max, for an aerofoil NACA 64-210 

with a slat is considerably improved by tho slat. The agreement betwecn theory and 

experiment is quite satisfactory here, too. Finally, Fig. 22.15 shows the dependence 

of the maximum lift coefficient, CL,,,,, of the profile NACA 2412 on the Reynolds 

number, R. The increase in thc maximum lift coefficicnt with increasing Reynolds 

number, which is ohcrved in experiments, is well confirmed by the theory. 

Calculations of maximum lift of wings in laminar flow have bcrn performed by 

G. 11. Goradia rt, al. [37, 381. 

11. Three-dimensional boundary layers 

General remarks: The phys~cal nature of a three-dimensional boundary layer is 

charact,erized by the fact that the direction of the velocity in the interior of the boundary 

layer deviates considerably from that in the outer flow. This is brought about 

by a pressure gradient that acts at an angle to the main flow. As a result, there occur 

vigorous scconclary motions, cf. Fig. 11.1 in Chap. XI. A good example of such a 

flow pntlrrn is rontnir~ccl in thr mmsurrmrnts pcrformcd by ILC. Snrhdcva and J. TI. 

Preston [a!)] in thc boundary laycr on a ship's hull. 

There exists a summary account describing the calculation of three-dimensional, 

incompressible bounciary layers prepared by J.C. Cooke and M. G. Hall [23]; it deals 

predominantly with laminar boundary layers. A comprehensive monograph on turbulent 

three-dimensional boundary layers was published by J.P. Nash and V.C. 

Patel [70]. The analytic calculation of a general case, for example that of the boundary 

layers on swept or delta wings, is still very difficult, even though numerous 

proposals of such met.hods exist. &re we may mention, for example,-the work of 

N.A. Cumpsty and M. R,. Head [24], J. C. Cooke [22], P. Bradshaw [7], L. F. East 

[29], 1t. Miclwl ct nl. [FG], and A. Elscnanr and R. van den Berg 1.311 anrl F.M. White 

et, :~l. [L 18:1]. 'l'hc prcsrnt st,al;ns of rcscnrch in this ficld was rcvicwcd by Fannelocp 

at n symj)osinn~ hcld in Trondhcin~ in 1076 [30a]. In what, follows, we shnll describe 

sevcrnl simplcr examples of t,l~rec-dimensional turbulent boundary layers. The state 

of the thcorg is, howcvcr, still unsat,isfactmyt. 

1. Boundary layers on bodies of revolution. C.B. Millikan [67J'was the first to 

enlcnlat~ a turbulent boundary laycr on a body of revolution, the method having been 

based on thc momentum integral equat,ion. The relevant momentum equation was 

given in eqn (I I 39). Using our prcsrnt notatio?, we can write it as 

IJrre R(T) tlrnolcs thc radius of the local cross-section of thc hody of revolution. 

e, 

At the aft portion of n body of revoh~tion the two derivatives, rlN/tl:t: :lnd 

dR/dx, become negative. It followsfrorn the precedingequation that thr ~norncnt~urn 

tdliclzncss dz(x) increascs and becomcs very Iwgc there. This may orcntc circumstmccs 

which nullify thc main assumption of boundary-layer thcory, nxrnrly t,l~at 

6, < IE. As a consequence, the calculat,ion near the butk ol t,he body of revolution 

may become crronco~s anrl thc posit,ion of the rrgion of scpnration cannot IJC tlctrrmincd 

reliably. According to F.M. White [II!)], cquation (22.31) romnins us:~l~lc 

when the local Reynolds number satisfies the condition that 

U(l) R(2:) > 1000. 

v 

1'. S. Granville [39] formulated a multiparamcter procedure for tho cnlc~~lntion 

of t.urbulcnt boundary layers on rotationnlly symmctric bodies placccl in an asidly 

tlir~ct~ctl st,rcam. 'l'hc nlcthod hinges on trlic cnlculat~io~? ol' momcntmm t~l~irkncss and 

of a shapc factor and can bc used for the aft portion of the body where t.hr I)onntlnry 

layer thickness is of the same order of magniturlc as thc local radius of t.11~ I~otly. 

In a manncr similar to that used for two-dimensional I~onntlnry Iayrrs, 'I'I~II(:~ZCIIbrodt 

[Ill, 1141 was able to show that t,he use of the encrgy intcgral rquathl leads 

to an explicit integral formula for .the calculat,ion of the mcrgy thickness. If a: tlcnot,cs 

the current, arc length measured along n meridian, and ]((a) the radius of a, scc.t,ion 

normal to tho axis of symmetry, then tho cxt.cnsion of cqn. (22.22n) Tor 1.11~. Itc.y~~oltls 

number formed with the energy thickness can now be written 

r 

The numerical constants b and v' should be taken from eqn. (22.22b) and the constant 

of integration is 

In the more recent formnlation [114], the equation for the modificd shape /aclor in thc 

axially symmetric case contains the function describing thc variation of the body 

radius.This is in contrast with the earlier formulation [l 111 according to which the 

modified shape factor was the same for bodics of rcvolut,ion and t,wo-tli~t~c~t~sior~al 

bodies. The generalized form of eqn. (22.25) is now 

whore the influence functions for the radius and cxternal velocity distril)ut.io~~s nrc 

~(1) = ~ ( x ~ ) 

Z 

Z 

+ I Ill+b U2(l+b) dz; N(x) = N(q) 1 c / IIl-'h lJ2("h)I'GC-' tlz. 

2, 21 

The constants of integration are 

G(x1) = v' [l€(rl) {IE(x1)}'+"U(x1)}'+26 {R3(rl)}1+b]; 

N(xi) = [U(xi) G(.zi)/I~(xi)lC. 

'I'hr nr~mcrical ronstants follow from rqn. (22.2Gb).

J'ig. 22.17. 1,ovnl lift. cwefhienb. c,, nt. vnriot~~ rntlinl sections on n rot,llt.ing propeller nccordirlg 

tn IIII('ARIIRIIICIIL~ 1wrfor111cc1 by 11. lli~ntnolskntl~p [44] 

Tho diagrams in Fig. 22.10 show a comparison bctwccn tlmwy and mrnsurcrnrnt 

in a flow past an axially symmetric body; the diagrams plot the Itrynoltls number 

formed with thc cncrgy tl~icloicss and the modificrl shape factor. 

Tn order to take into account correct,ions due to thrcc-di~~rc~~sio~ralit~y r:111sr(1 by 

the possible convergence or divergence of streamlines, J.C. Itotta [8F] proposcs to 

base the calculntion on an clTccLive radius R(z). Numcrioal valuos for R(n) nrr snmrnnrizctl 

in [86] for all ~ncasr~rcmolts catalog~~rcl 

mrrils 1)y Mi.\\'. \Villmart,h ct xl. 1122~1 nntl A.M.O. Stnit,lr [IOh]. 

in 1541; comI)nrc Iirrt: t,lic III~~:~,sIII.(~- 

2. Bn~tt~clnry lnycrs on rntnting hndiea. 'J'l~t: calt:ul:~t.io~~ ol' I:intin:~t. 1~)111itl:i1.v 

layers oti rotating bodies placcd in an axiril strcn~n was clisoussotl in Scr. S 1 c. 

The mctl~od of calculation which maltcs usc of ntomrntt~n~ int.rgl;ll t~~n;ltions, 

formulated for the meridional ant1 t:irc~~mfcrent.ial tlirrc:tions rcsprc~tivc4y, hs been cxtc~~cletl by R. Trucltenbrotlt 11121 to inclutlo t.hc ~urbulrnt, c.:~sr. IT(: wn.s, 

moreover, fortunato to succcctl in giving convcnicnL intcgr:ils li~r the rn.lt~~tI;tt ion ol' 

the parameters of thc boundary layer. JCxperimcntnl and furtltcr t11cwrct.ic::l.l i~~vrs(.igntions 

into the boundary layer on rotnting strcamlinc botlics wcrc c:nrrirtl ot~t I)y 

0. Parr 1741. Jn this casc, tho bountlary laycr grows rajjitlly wit.11 t,lro rot.;~l.iotr 

paramctcr 2 = (11 RIU,,,; hcrc (1) tlcnotcs tho nng111w velocity, It t,l~t: I:~,t.grst, r;uIius 

of thc body, nntl (1, is tho axial rofrrcnc:~ vcloc:il,y. 'I'lrc t,~~rln~lt!r~l. I~ott~i(l:~ry 1:iyt:r 

on a rotating body of revolution placctl ill an axial st.re:ltn can IN: (::dt:~tl:~.lt:~l witl~ 

the aid of thc system of equations (11.45) to (11.48), in whicl~ thc shr:wit~g strcss 

must bc ass~tmcd to vary with the rotat.ion pnr:mct,rr. 'I'hc tlin.gr11.m ill I'ig. 22.16 

compares Lhc onlculatctl nntl n~cas~trt:d vnlt~cs ol tho rnomcntr~nr I.l~ic:lit~c:ssrs A,, :111t1 

a , as rcportcd by 0. l'nrr [74] for :I cylintlrical botly provitlctl wit.11 n. sp11vric.nI Ilosr. 

'L'hc ngrccrncnt is good. 'L'lrc rcgion of tmnsition l'ron~ laminar to Lnrl)ulcnt flow 

moves forward as thc rotation paramctcr incrcascs; its position coincitlos with the 

point at n~hiclr thc momentum tlrickncsscs incrcnsc al)r~~j)(Jy. Scc also Scc. X1112. 

A mctliocl for thc calculntion ol tll~rrr-tli~~~c~tisionnl bo11ntl;~ry I:~yc:rs or1 st.;~t.io- 

. ~ 

rtnry botlics as \vdl as on rotating orics, suc:lr as pro[)cllors or 1)l:rtlcs of rol,;lr.y cornpressors 

and turbinw, was inclicatcd by A. Magcr [Fl]; comparat,ivc mcasurcmenb 

are contained in ref. [621. H. Himmclskamp [44] carried out mcasurcmcnts 

in thc boundary laycr on a rotating airscrcw ant1 tlctcrminctl iooal lilL cocffioic~rb 

of t,hc blade from mcasurcmcnts of prcssurc tlisLribuLions. Somc of his rosulbs arc? 

sccn reproduced in Fig. 22.17; they arc given in the form of plots of lthc local lift 

cocfficicnt., c,, at various radial seclions, in term^ of thc nnglc of irlciclcnce, or. 

Corresponding mcasurc~nents on n stntionary bladc placcd in a wind tnnnnl are 

also shown for comparison. Figure 22.17 shows that ~narlrr?tlly itrt:rcasctI lift coefliri~~~f,~ 

arc obt.aincd near the hub, and the dcct can LC trnoctl Lo soparation 1)cing clc1:~yccl 

to larger angles of incidence. For cxamplc, the scc:t,ion closcst to the hub has I, 

maximum lift coefficient of 3.2 compnrcd with 1.4 on thc stat,ionary blatlc. The 

tlisplacemcnt of s~pirat~ion towards larger anglcs of incidcncc is cxplai~rctl by the 

appearance of an additional acceleration which acts in thc flow direction ant1 which 

is crcatcd by Coriolis forces; it has tl~c samo cffcct as n favonrablc prnssurtr gmtliont. 

111 addition, but to a lesser extent, the ccntrifugd forces acting in the bound;rry 

layer carried with the blade exert a bcncficial influcncc with rcspcot Lo soparntion. 

Ii'luict p,zr~icles in the hountlarg layer are actctl upon by a centrifugal forcc which

is prol)nt~t,iot~nl to the rnclius. Consctl~lcl~tly, loss fluid is transprtctl to each blntfc 

from t h contm tAan awry from it and outwarclu, ant1 tl~c bor1ntlnry layer is thinner 

t.llnn woultl be I(II~ casc in t~wo-t~itiicnsiotl:rI flow about the same slrape. A. Betz 161 

gave some t(t~eorct.icnl argumcnta on l.his point. F. Gutscl~e [42J made tile flow on 

a propellor I~lntlc visible: I)y ~~ainting Chc former with a tlyc. Ccl~t~riftlgal forces also 

rxcrl :r hr~c itl~lut:ncc on the J)rocc:c%s of 1.mnsiI.ion. I[. M~lcsnlal~n [(j8] sllowctl ill 

his t.hc.sis t,l1:itp, otl~cr thit~gs being cqnal, trarlsition occurs or1 a rotatirlg propeller 

I)la.tlc at. :t. considcrat)ly lowcr Ltcynolds number than on one wl~icll is stationsr.y. 

Fig. 22.18. ConvcrgrnL :tntl clivcrgent 

hounclnry Inycrs; ~yrrtmn of 

coordinnten; 

a) divcrgc~~t~, 

a -1 z > 0 ; 

11) oonvcrgcnl., a I- :r -: 

0 La- ' "n*, 

3. (:rcnvcrgc~lt ~ IICI clivcrgeni bo~rrdnry layers. 'rhc methods for tllc calculat,ion 

of t,r~r\)t~lcnt 1)ortntlary layrrs wlrith were tlcscribed in Scc. XXlTb have been ext,cntlntl 

11y A. l

(308 XXIT. The incomprc~~iblc turbulent bo~rndnr~ layer 

[16] Carribre, I'., and Eichelbrenner, E.A.: Theory of flow reattachment by a tnngentinl jet 

diachargin ngninst a strong adverse prmmre gradient. Boundary layer ~ n flow d control 

(G. V. l,acknnnn, ed.), Vol. 1, 209-231, 1961. 

[17] Cebeci, T., Mosinskis, G. J., and Smith, A.M.O.; Cnlculation of separntion point8 in incompreasiblo 

turbulent boundary Inyers. J. Aircr. 9, 618-624 (1972). 

[I81 Cebeci, T. and Smith, A.M.O.: A finite-difference solution of the incon~prewible tnrbulent 

boundnry layer equntions by an eddy viscosity concept. AFOSR-IIW, Stanford Conference 

on Con~pntntion of Turbulent Roundnry I,ayers, Vol. I. 346-355 (1968). 

[In] Cebeci, T., and Smith, A.M.O.: A finite-difference method for celculnting comprrssible 

lnminnr and turbnlent bonndnry layers. .I. I3nsic Eng., Trans. ASME, Series D, 92,523-535 

11 ,-- 9701 , 

[201 Cebrri, T.. and Smith, A.M.O.: Annlyni~ of tnrbnlent bonndnry layers. Amdemir Press, 

New York, 1974. 

[21] Clauser, P.M.: Turbulent bouudnry layers in adverse prcasnre gradirnk. JAS 21, 91 - 108 

1 , lc)fid\ * ., ., . 

[21a] clnuser, F.N.: The tnrhulnnt honndarp layer. Adv. Appl. Mech. 4. 1-51 (106.5). 

[221 Cooke, J.C.: Boundary layers over infinite yawed wings. Aero. Quart. 11,333-347 (1960). 

[231 Cooke, J.C., and Hnll, M.G.: Ronndnry lnyer in thrce tlimcnsions. Progmss in Acronauticel 

Sciencca 2, 222-282 (1062). 

[241 Cumpsty, N. A., and Head, M. R.: The calcr~lation of t.hree-dimensional turbulent boundary 

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Part 11: Attachment-line flow on an inlinito swept wing. Aero. Quart. 18, 150-164 (1967). 

Part 111: Comparison of attachment-line calcnlntions with experiment. Aero. Quart. 20, 

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1471 Jacob, K.: Berechnnng der ahgelosten inkompressibleti Stronmng nnl Trngfliigclprolilc urtd 

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~ t.nrb~~lonl.o It~*il~~ll~~u~r.l~ii.l 

Diw. Gottingen 1942; 1ng.-Arch. 13, 293-320 (1943 

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flow. JI'M 37, 625- -242 (1969). 

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[741 I'nrr, 0. : Untcr~url~ungcn dcr drcitlirncnsionalrn Gmnzscl~icl~t an rot,icrentle~~ Drel~kiirperl~ 

bri nxi~ler Anst,riimung. l)iss. 13raunsch\vcig 1902; 1ng.-Arch. 32, J93--413 (1903). 

17.51 I'ecl~au, W.: Nin Niil~crun~sverfnhrcn zrlr 1Zerechn11n~ dcr ehenen ru~d rotntionnsvrn~~~etriscllcn 

t,urbrrlentcn Grenzs~hicht mit beliel~igcr ~bs,&pn~ oder Ausblns~lng. .fb. \VGL 

1958, 82- V2 (1959). 

(761 I'olzin, .J.: Striin~~~ngs~~ntm~cl~~~~~gcr~ 

an cinem ebenen Diffuser. 111g.-Arch. 11, 001 -- 385 

(l!)40). 

1771 i'rete&, ,I.: Zur thcorctischcn I3crenhr11111g dcs I'rofilwitlcntnt~tles. Jb. (It.. I~~ftfnhrt,forschune 

I. 01-01 (1938). 

[77a] ~nrnnprinr~, B.lt., and Sl~ivaprasad, 1%. G.: Menn llow mcasnrcrnenb in t~~rhulcnt houndnry 

I:t.yers : h ~ n~itll~ g curved s~~rfnccs. A[AA J. 15, 189--1!)0 (1!)77). 

[78] Rmpct,, A,, Cornish, d.,J., and Grynnt, (?.I).: Delay of the stdl by suction t.hrough distributed 

perforations. Aero. Eng. Rev. 11, 0, 52-00 (1952). 

1791 Reynolds, W.C.: A morphology of thc prediction n~cthodo (of turbulc~~t I)oundary layers). 

Article in [A41 Vol. I, pp. 1 - 15 (1969). 

[80] Ilcynolds, \V.C.: Rccent advnuce*r in t,llc computation of t.orIndent Row. Advances in 

Clicmirnl Icnginrering 8, 1!)3--240 (1974), cd. by T.B. Brcw ct al., Acatlen~ic Press. 

I ] y n o l l \ : omputtion of 11r011lnt OV. Ann. v I l i c l . 9, I 204 (1!170). 

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(1!)5I) nnrl Max-l'ln~~clz-IIIS~.. fiir St,rii~i~i~ngsfors(:I~~~ng (:iiLt,ingcn Itrp. No. l (1950). 

[83] JEotCa, J.: Scl~~~bspnnn~~~~gsverteil~~~~g 

und Encrgiediwipation bei turhr~ler~bn Grenzscl~icl~tcn. 

1ng.-Arch. 20, 1%-207 (19.52). 

[84] Rot,t~, J.: Nilheru~~asvrrfnl~rc~~ zor Uerechnuna turbulcnter Grenzschiclrten unter Benutzung 

(leu ICncrgi~safxes. Mr~~-l'litnek-ln~t. fiir ~t.rij~r~~~~~~sforsel~~~ng 

(:iittingell Itep, No. 8 

(1953). 

[05] itolti. .J.: 'J'urbulcnt boundary laycrs in incotnprrssible flow. Progress in Aero. Sci. 2,l-219 

(1!)f12). ,- ,. cd. IIV ., A. k'crri. I). Kiichcmann and L.H. G. Strrno. , Pernnmon u Press. Oxford. l!J02. 

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Dissil)nt~io~~sgcsetzc~~. 111g.-Arch. 38, 212 -222 (IVO!)). 

ltott.~, J.: 'l'urbulentc Striimungcn. Stuttgart, 1972. 

ltubcsin, M. \V., nnd Pnppn8, C.C.: Annlysis of Ll~c turbulent boundary-lnyer cl~arnctcristics 

on a lht late a.it,l~ tlintribut.ccl light,-gns injcction. NACA TN 4149 (11158). 

Ituhin, S. I:.: IIICOIII~~CSS~~I~C flow dong n corner. JPM 26, 97--- 110 (I!)C,Ii). 

SnchJcv:~, It. C., and I'rrst.on, J. If.: Investignt,ion of turbulent boundnry layers on a ship 

moclcl. S~:l~ill'sbrrh~~ik 23, 1 --45 (1!176). 

Srhlirl~ting, 1-1.: Die C.mnzschicl~t an clcr ehcnen Platto init Absaugung uud A~rshlxsen. 

1,11ftfaI1rt,f~)rscl1111g 1.9, 2!)3-301 (1042). 

Srhlicht.ing, It.: 15inigc neucrc ICrgeh~~issc iiber Gron7.~cl1iol1tbeeinfl11ss1111g. Proc. First Int. 

Congr. /\c?ro. Sci. hlndritl; Atlv. in Acro. Sci. It, 503-58fi, Pcrgnmon Press, I,ondon, 1959. 

Srl~lirhting, 11.. nnrl Pcrl~nu, W.: A~~ft,ricbserI~ijl~r~~~g von 'Vragfliigeln tll~rcli kontinuierlich 

vcrLriItc /\hs~tugung. ZIPW 7, 113.- 1 I!) (1!)5!)). 

Srl~lirht.il~g, H.: 'I'l~ree-tli~ncnxionnl hountlnry lnycr/flow. Intern. Assoc:. Ily~lmulir Iteswrcl~, 

IXth (hngr., I~~~l~rovnik, l2(?2--- l2!)0, (I!)(il). 

Sri~Iichting. II.: Acrotlynntnisclrcr ~~roI~Ic~nc drs IIiic~I~ntn.r~ft.rieI)~~x. I,cctore at '1'11ird ~nt. 

Congr. Arro. Sci. (ICAS) Stocltlioln~, Sweden, 1902; ZI'W 13. I--14ql9G5). 

Sd~lirhling. 11.: 15inige llcucrc ICrgcbnissc nus der Acrodynnmilt rlrs Trngfliigcls (Tenth 

I'mndtl hlrnwrinl I,rrt,~~rc? I!l(i(i). .Jh. W(:l,lt 1966, 11 --32 (1907) 

ary layer. N,lCA Ihp. IW) (1951). 

[!Is] SCIIIII);LIIC~, (:. 13.. a d Spnngcnherg. W.O.: Icorcctl !nixing in I)oun(l;u.y I:~yrrs. .ll'Rl 8, 

[104n] Smith, A.M.O.: St.rat.fortl's turlmlcnt srp;trntion rritmion for axially ayn1111(4ri(. IIow. 

%AMP 28, !I28 - 938 (1977). 

(l!)54). 

[I071 Sznblenski, W.: Turhulcnte Striimungcn in tlivergcntcn Irr~~ss~~l~i~~l~lsl.rii 

gen mit Druckanstieg. 111g.-Arch. 23, 2!)5--301; (1!)55). 

[10!)] 'rl~o~~lnrt, F.: Untersucl~ungcn iihcr dic ISrhiihung flea Aultriebcs von 'l'rnglliigoln ~uit~t.i:Is 

C~rcnzscl~icl~tI~cai~~fl~~ss~~~~~ 

thrcl~ Ausl~l:rnc~~. I)iss. I3rn111mol:i I ! ; I 0, 4 li5 

( I!)Kq. 

[I 101 Thomas, F.: Untersuchungcn iiber die Grcnzsrlricht an einer \V:~nrl sl~ro1n:~l1\r.iirt.4 v011 ri~~cm 

Ar~sblnsspnlt. Abhnntll. k~iss. Gcs. Urnunsrh\vrig 15, I - 17 (l!)Iif). 

[LlOn] Townsrntl, A. A,: The tlevclop~ncnt of t~url)~llont I)ountlnry I:iycrs \vit.ll nrgligil~lc wnll 

strcs~. ,J IW 8, l4:$--. I55 (l!)GO). 

[I I I] Truc~kenl)rotlt, 142.: Ein Qr~:ttlrntnrvc~rf~tl~w~~ zur Ilcrrcl~nung dcr litn~innrcn und l.~~rl~ult:~~tcu 

I~eihrtngnscl~irl~t. hri cbcncr und rotntionsnym~~~c.Lrisc!li~~r Sl.riinrung. Ing.-Arrli. 20, 41 1 - Z'L8 

(1!)52). 

[I 121 l'ruc*ltenhrodt, 15.: Ein Q~~actmt~~rvcrfnl~rcn zur IJrrccl~n~~ng tlrr I~c:iI)~~ngmc~l~irI~t, nn :~xi;xl 

an~estriitnte~~ rotierenclrn I)rehkiir~crn. hg.-Arch. 22, 21 -35 (l!)54). 

[I131 'I'ruckcnl)rodt. E.: St.riir~~r~~~gsl~~c:(.~~iIz. Springer. I%orli~~/lltitl~:lI)crg/N(*\~~ York, I!t(iR. 

[I 141 Trurkcubrodt, E.: Neucre ICrkcnntnissc iibcr die Bcrcc:l~nung von Slrii~~~rr~~gngrc~~~xsc.l~ir:l~l~c 

rnittels cinfacl~cr Q~~ntlrnt~urformel~~. Part I : Ing.-Arrh. 43, !) 25 (I 973); l':~rt, I I : lng.. 

Arch. 43, 130 - 144 (1!)74). 

[I151 Turcottc, I). I,.: A sublayer tllcory for fluid injcction int.o 1.lw incon~p~rssil)~~ t,~~rl)ul~wt, 

I)ountl:wy It~yrr. .JASS 27, 075078 (I9(;0). 

(1 161 Wdz, A.: SLriin~ungs- und 'Vr~~~l~cr:tt,~~rgr~~~~zsrl~irI~tt~~~. 

I~IIII, 1

Turbulent houtldary layers in compres~ible flow t 

n. Grncrnl rcmnrks 

It has been domonstfrnt.ccl in Src. XTTTa tht tho presenco ot'ltigl~ velocities in 

tho I~onntlary In,ycr gives riso to sii(:h 1qc tcmpcratnre tlifi'crcnccs tht it becomes 

ncccssary to f:&c into account thc elT'cct of temporatwe on the properties of tlic fluid 

in addition tm that of tlic clinngcs in its volume. Beyond this, it is found that the 

transfer of hcat plays an esscnt.i:rl part in the bchaviour of a compressil~le bountlary 

hycr; its presence leads to the appcxm.nce of a strong interaction bct,wccn the velocity 

ficlrl arid the t~cnipcmt~~rc field:. 

I. Turbulent hcat transfer. When :I liquid or a gas of non-uniform trmptraturc 

is caused to move tnrl)ulcntly, it is found that the tmrbulent mixing motion creates 

in it tsemporaturo fluctuations in :ctltlit,ion to thc more familiar vclocit,y Il~rct~uations. 

In analogy wit,li rqn. (18.1 ) for v~locity flt~ctnations, we may r~presrnt the fliictuating 

tcnipcmture 

T=T+T' (23.1) 

in tlie form of the slim of a temporal average, 7', antl a pure fluc:tuat.ion, I". These 

fluctuations givc rise to a supplementary heat flux which is analogous to the flux 

of momentum evolved by the velocity fluctuations. In order to show this more clearly, 

we assume, as we did earlier in Sec. XVllIb, that throngh a surface element, dA, 

whose normal points in the x-direction, there Bows a mass of fluid dA eu dl during 

timc dl. The cnthalpy of this mass per unit volume is e cp T, and tho convective 

flux in the x-dircction has a value dQx = dA e u cp T. If we now introduce the expression 

for u from eqn. (18.1) and that for T from eqn. (23.1) and form the tempornl 

average of the heat flux, we shall obtain 

It is secn that tho presence of vclocity and temperature flr~ct~uations generates the 

- 

supplementary heat flux rlA e c, u' I" in the x-direction. Corresponding expressions 

are obtained for the supplementary fliixes of hdat in the directions y and z. We 

-. 

t I km indebted to Dr. J.C. Rottn for the text of this chapter which is new. 

: A comprehensive aummory of the theory of turbulent boundary layers in compressible flows 

is given in the book of S.S. Kutateladze and A. I. Leont'ev [SO]. 

a. Grnrrnl rcrnnrks 703 

concludc, therefore, that the three components of thc additional heat flux (q~iant~ity 

of heat per unit arca and timc) arc: 

q; = e c , , m ; qyl =ec,,v(rllr; qi =ecp?OI[I1). (28.2) 

It has bccn assumed llcrc that there exist.s a statistical corrclntion bctwcrn tltc 

velocity and tcmpernturc fluctualions. The existcncc of such a correl~itiori in the 

presence ofagratlicnt dT/tlyof t,he mean tcnipcratrtrc can bc den~onst~ratctl in - the satno 

way as that usetl earlier to tlcrnonslrate the cxistcncc of the corrcl:~t.ion IL' v'. 'l'llc 

:~rg~~rnrnt atlvancctl in t.he last paragrnpli of Scc. XVIII 11 rct.airts its forc:o if is 

subst,itutctl for 12 antl 7" for I*'. In sue11 circurnst:~nccs 1.hcrc will :~riso a corrclnt.ion 

v"l". 1t follows fi~rt.ltcr from this :~rgntncnt. that, t,lie sirnnlI.nncons esist.cncr of t,lic? 

gmdicnts ch;/tly and dT/tly must impose a strong corrclntion betwxn TL' i~n~l 'I". 

%'his conclusion has I,ccn confirmed by mcasurcments with hot-wire ancmomctcrs 

in notnprcssil,lo [47] nnd inc!ornprcusil~l~ hot~n(lary Iay~rs forrnctl on n I~c~:tt.cvl wr~ll 

141, 421. According to rneasurcmcnts pcrforrnctl by A. I,. IZistlcr 1471. the c:orrclalion 

coefficient. - -. 

u' T' 

-. .- -- - . - . - 

i* f? 

2. The frtndnnirntnl cql~aliona for comprrnsibic flow. 'I'crnpc?r:~t~lirc fluc.bttnl.ions 

togclltcr with I.ho prcssurc Ilnetnnt.ions mcnl,io~lctl c~~rlicr in SCC. I\: VI I I 1) 11rotI11t'(? 

tlcnsity flnctuat,ions. lror this reason it is assutnctl that the tlcnsity 

e=@+e' 

(23.3) 

is also equ:ll to the sum of a tirnc-avcmgc, $. and a tlcr~sil~y flt~ctunt,ioli, p'. The 

fluctuations in tcmpcmturc, pressure, and density arc rcl:tt,otl throiigll thc ccl~rnl,iort 

of state of the gas, eqn. (12.20). When the gns is trcatd as pcrfcct., and wlicn t,hr 

fluctuations are small, we may put 

to a first approximation. In addition to thc turbulent transfcr of heat,, thc presence 

of density fluctuations constitutes tho second important ncw pltcnomerlot~ which 

occurs in compressible, tr~rbulent st.rcanls. Evidently their prcscncc may not bc 

ncglcclctl when cxprcssions for tho tensor of nppnrent strcsscs (SW.. X.Vlllc) is 

derived. Formally, when eqn. (23.3) is tnkcn into accorlrlt, cqns. (18.5) n~t~sl I I rc- ~ 

placed by tlie following additional terms due to turbulctlce 

-- 

Here e'u', e' v', and e' play the part of tho components of a turbulent flux of

70-t XX I1 I. 'Ihrbulcnt bonrwlnry lnycrs in comprcwiblc flow 

mass in the thx directions: z, y, z. On averaging, the equation of continuity for a 

c~omprcssible strewn, cqn. (3.30), leads to 

1bcgn.rcling tho clensity flnctuat.ions, it is possible to say at first that p'/@ is hanlly 

likely 10 exccctl u'/S. Sinco, now, u'/d < 1, it appears possible to neglect the last 

tcrtn in caoll of cqns. (23.6) with respect to the first. Further simplifications rrsult 

when aI.l~(:nt,iot~ is confined to bountlary layers in which d & 12. .I. C. Rottn [SO] 

clc:~nor~str:~l.ccI l.l~:rt, in such cnscs if, is possiblo altogct,llor to eliminate thc tlcnsity 

Illlct~~ations from t,hc cc(nnt,io'ns for boundary layers if, as is customary, tho nornial 

st,rosses tllnmsclvrs arc ncglcctctl. First we notice that 6 < ~i in the eqlmt,ion 

for t',, in (23.5), so that only two terms need be retained. Purthcrrnore, since 

r?p/(7r < ae7/a!/, thc contitluit,y equation (23.G), written for a t)onntlnry layer 

which is two-tlimcnsior~d on t,hc average, aeqr~ircs the form 

'l'l~c bountlary-layer equation i.9 clrrivecl from eqn. (12.50b) in that cqn. (12.50n), 

mult,iplictl wit,l~ u, is adtlcd, with eqns. (18.1) nntl (23.3) s111~sLiLutccI ; thc ro~ult is 

thcn averaged in nccortlanco with cqn. (18.4). When the above-mcntionecl t,ertns arc 

neglnct.ctl, tho following, final form for the boundary-lnycr equation is ol)t,ninetl: 

It is nol.cd that in tyns. (23.Ga) and (23.7) the tlensity fluctnat.iot~ appears only in 

the form of fl adtlctl to @ 6. It is, therefore, convenicr~t to re-introduce the original 

- - 

expression fo; t.ho mass IIIIX p v -;. 17 -1- p' v' in the y-direction, ancl to tlcfine the 

tfurl~l~lnnt,, appnrcnt. strc:ss as 

In any cam, the exact value of tho mean velocity component at right angles to the 

wall, 17, remains untlet,crnmincd, boing of little interest anyway. The energy equat.ion 

(12.19) can ljc: treatctl in like manner. Introducing the turbulent heat flux 

we obtain tht: following set, of equations which describe the processes in compressible, 

t~rrbolcnt, bour!tlary layers: 

Ilere, (.he t,erm ~rcpresents the mean valuo of the (lissip:~t ion, ancl for it,, I hc following 

npproximat,iori may bc employed: 

, @ = ( p 

ail 

-- -1- rt ) 

The set must. t)c nugmcntctl by t.hc a.pproxirnato form of the rqn:rt,ion of st,nl,c for 

mean values : 

- - - 

P"PR?'. (ZI.!)) 

Tho prccccling syst,crn of cquntior~s for co~npressil)lo, l~rrrl)l~lcnt l)o~lntl;~ry I:~.yc!m rcplnccs 

crpnt.ions (1 2.6Oa) to (1 2.ROtl) liw corrcspontlitlg I:~n~in:~r flow. 'l'l~c I)OIII~~~:II~ 

contlitions rcmn.in 11nc11:rngctl (cf. Chap. XII). 

In order 1.0 explore the tlchils of t.rnl~nlcnt motion in comprt?ssiblc mctlin, it, is 

necessary to untlert,akc tncasuremcnts with hot, wires. 'l'his is matle cliffic~rlt. 0~1 thr 

ncctl to unro~lplc 1.11~ c:fTcrt of Lcmpc:rat,urc ant1 vcloc:it,y Ill~ctnn.l~ions \vit.l~in ;I si~~glr 

signal. 'L'hc problems which :wise in this way form f.hc subject of tho pl11)lic.n.t.ions 

[49, 651 by I,. S. G. Kovns~nny and M. V. Morkovin, rcspc:cl,ivcly. 1,cavirrg n1)art the 

appe~rnncc of density :rnd temperature flr1ct,r1:~t,ions, it is found thatf 1.11~ flow rrmnir~s, 

in its gcnar:d oublinc, tho same :IS in :LII inc:oml)rc~ssil)le Il~~itl. Ilowc~vc:r, :IS I,IIv hl:rc.h 

nnml)or is incm:~sctl, the volooity Iluc:tu:~Liot~s loso ill iill,t:tlsit.y, :I.$ ~lc!~r~o~~sl,r:tl,~:(I I)y 

Lhn oxpc:rim~~n(,n.I rrs11ll.s tlrtc! 1.0 A. I,. IZistlrr 1471 : LII~~ SIIOWII ill Icig. 23. I . '1'11t- t*lli~4~ 

of tlcnsitv fluctnat,ions which go bcyoncl those inc:lwlrcl in ccltls. (23.8:~) to (23.8~) 

Fig. 23.1. Distribution of turbulent vclocity 

flnct,nntionn in tire houndnry Inycr 

on n flat pink placed st, zero incidence in 

a sulvxsonic strcnrn. Monsorc~nc~~b clnr tx, 

A. I,. Itiatlcr [47J nntl F. S. Kle11nnoR [48] 

In ortler to render the system of cqnntio~~s (23.8;1.) to (23.8tl) more :~mctl:~l)lr 

to practical calcnlnt.ions, it is possible, as was (lone in Ch:rp. X [X, to iritrothco i1ll.o 

it empirical ass~irnptiorls for momentturn and heat. t,mnsport. lCcluat,iorl (19. I) for thr 

app~rent shearing stress t, = t',, is usnnlly taken over ul~changcd. As far ns the Lnrbulent 

1ica.t flux is conccrnetl, it is custon~ary to givc it a f rm rc~ninisccr~t of Fourier's 

law of ther~nal conduction, cqtl. (12.2), according to which we have 

aT 

--k.- ?I 

Q I - 

(Iamiriar) ,

706 XXTII. T~~rbulont bo~~ndary loyors in comprcaaiblo Row h. Rnlntion bctweon vclority and tcmfnmturc dintxihotion 707 

and to postulate that 

aT 

q, = - c, A - (turbulcnt) . 

(23.10) 

ay 

It is ~ N C that the exchange mechanisms for momentum and heat, arc similar; neverthcless, 

they arc not identical. The cxchange coefficients A, and A, for momentum 

and heat, respcctivrly, havc, therefore, dillkrent values in general. Taking into account 

cqns. (19.1) and (23.10), together with eqn. (23.8d), we can transform the system 

of equations (23.8a) to (23.8d) to the form: 

3. Relation between the cxchangc cocfficienta for momentum and heat. We 

have stressed in the past that the occurrence of a fluctuating motion in a turbulent 

flow causes momentum to be exchanged vigorously between the layers of different 

velocities. It also causw an increase in tho tran~fcr of heat and mnss when tcmperaturo 

or concentration gradients arc present. For this reason, there exists an intimate 

connexion between heat and momentum transfer in general. In particular, we must 

expect the existence of a relation between the heat flux and the shearing stress at 

the wall itself. The existence of such an analogy between heat and momentum transfer 

was first discovered by 0. Reynolds [76], and for this reason we now speak of Reynolds's 

analogy (cf. Sec. XIIe 3). This analogy enables us to make statements concerning 

the transfer of heat from the known laws of drag in s turbulent boundary 

layer. The exchange coefficients for momentum and heat - A, and A, - both have 

the dimension of a viscosity, p(kg/m see or lb/m see in absolute systems), so that in 

addition to the molecular Prandtl number P = p cJk, it is convenient to introduce a 

comcsponding, dimensionless, turbulent Pradl number 

Thus by definition, 

The totd rate of heat tratisferrcd assumes the fofm 

The turbulent Prandt-l number can be dctcrmincd with the sit1 of simultaneous 

determinations of velocity and trmparatmc profiles; unfort~~nately, the level of 

Fig. 23.2. Ratio of the turbulcnt trnnafcr 

coefficiente A,/A. over the length of a 

radiua in turbulent pipe flow, after B. 

Ludwiog [65] 

Rcynolda numhcr R - 3.2 x 10' tn 3.7 x 10' 

confidence with which thc results of such measurements can be accepted is low owing 

to the difficulties of measuring locd temperatures in flows in gencral, and to the 

uncertainties in the values of the gradients dzi/dy and dP/tiy. It turns out that P, 

varies with the distance from the wall. In an investig.ztion performed by I-I. Ludwicg 

[56] it was found, as shown in Pig. 23.2, that thc ratio A,/A, = l/Pl vi~rics from nbo~~b 

unity at the wall (r/R = 1) to about 1.5 in the ccntrc of a pipc (r/R = 0) and is indcpondent 

of the Mach number. Similar rcsulb were rcportcd by 1). S. Johnson [42] who 

made measurements in a boundary laycr on a heated wall. According to tllesc, thc 

ratio AJA, increases from about unity at the wall to approximately 2 at the edge 

of the boundary Inyer. A. F'xgc and V. M. I~alltncr (cf. ref. [R7]) and IT. Rcichartlt [72] 

mcasrtrccl a vnluc: 0f2, thc I'ortncr in the waltc bchincl a circulrw c:ylitlclor,rrntl t h Ird.Icr ill 

a free jet, both in a11 inc~rn~rcssible st.ream. According to the preceding measurements 

the ratio Aq/A, is smaller in a bountlary Iaycr thn in a frcc strcern owing to thc 

inflncncc of tlic wrdl on 1.110 ho~~ntlnry I~~ycr. I I, R~OIIIR, Itliornroro, plnr~aiblo to nswnto 

that thc mtio Aq/Ar Itas a valuc of unity at the wall (according to Imlwicg the value 

is 1.08 giving P, w 0.9) atid increases to a valuc of 2 (P, =.. 0.5) away from the wall. 

In practice, frequently, n constant value of Aq/A,=l (Pt = 1) or of 1.3 (Rcichnrdt, 

kiving P, = 0.769) is nssumcd. It, mnst, howcver bc poirltcd out that thc nlanncr 

111 whic11 tho turbulcnt 1'rantlt.l number varies across a boundary laycr has not Lcen 

dotcrmincrl beyond tloul,t,, ant1 that 1,l:crc oxist, cxpcri~ncnt~al rcs~~lts which arc in 

conllict wit11 the prccetling oncs, as rcl)ortntl in the ~nmmarics by J. I

708 XXIII. Turbulent bonndnry Iaycrn in co~npmibl~ flow 

wumed that the same mechanism causea the exchange of momentum as well a8 

of heat. Since the velocit.y and temperature profilm are identical, we can then writo, 

that 

k Tw- Tm 

q (2) = - 

/a urn TI, (4 

The ~mcccling equation can be easily re-arrangcd to the form 

N, = t R, c,' (Reynolds, P = P, = I) (23.16) 

describcd earlier as the Reynolds analogy. It is seen that the relation of direct 

proportionality between the Nussolt number and the coefficient of skin friction 

which was derived in Chapter XTI for the case of laminar flow paat a flat plate at 

zero iricitlcncc, (el. cquation (12.156 b) remains valid in tho turbulcnt wso. Equatior~ 

(23.16) retains its validity in the prescnce of compressibility, just as was the case 

with laminar flow, on condition that the Nussclt number is now formed with the 

temperature difference Tw - T,. t 

As already mentioned bcfore, the principal difficulty in studying turbulent 

houndary layew and turbulent heat transfer problems stems from the fact that 

the eddy or exchange coefficients A. and A, are not properties of the fluid, unlike the 

viscosity p or the thermal conductivity k, but that they depend on the distance from the 

wall inside the boundary layer. At a sufficiently large distance from the wall they 

assume values wllich are many times larger than the molecular coefficients y and 

k, so much so, in fact, that in most cases the Iattcr can be neglected with respect 

to the former. By contrast, in the immediate ncigtibourhood of the wall, i. e. in the 

laminar sub-layer, the eddy cocfficients vanish because in it turbulent fluctuations 

and hence turbulent mixing arc no longer possible. Nevertheless, the rate of heat 

transfer between the stream and the wall depends precisely on the phenomena in 

the laminar sub-layer and so on the molecular coefficients p and k. It is fortunate 

that eqn. (20.16) remains valid throughout, regardless of the existence of a laminar 

sub-layer, because whcn P = 1, as shown in Section XIIg, the velocity and tempc- 

rature distribution in the laminar sub-layer remain identical. The assumption that 

P, = 1 in turhlent boundary layers leads, as a rule, to useful results; by contrast, 

the Prandtl number in the laminar sub-layer can differ appreciably from unity, as 

is the case, for example, with liquids (Table 12.1). When this is the case, cqn. (23.16) 

loses its validity. Extensions of the Reynolds analogy to cases whcn P # 1 have been 

formulated by many authors, among them L. Prandtl 1701, G. I. Taylor [96] and 

'l'h. von 1CArnitin [44j, and R. G. Dcisslcr [20, 21, 22, 231. 

I,. Prandtl nssumctl that P, = l and diviclctl the boundary layer into two 

zoncs: tlir laminar sub-layer in whirh the eddy coefficients vanish, and the turbu- 

t Frequently, insCmd of th Nr~rrselt nnrnbcr nse is mfde of the so-called Stanton number 

If this i~ prcfrrmd, t h Itcynolds analogy from eqn. (23.16) becornea ' '?, 

s = ; c,'. 

Tho rrnminir~g nql~ntionn can bo owily I.rnnnforrnd to rcplnco N by S. 

lent, cxtcrnal boundary layer, in which the molecular coefficients p onel k con be 

neglected. Under these assnmptions, cqns. (19.1) and (2:1.14), wriLtcn for 1,111: laminar 

~ub-layer will Icad to the form 

k dT 

- (1 -- .. - - -- 

T 11 du ' 

whercas in tho turbulcrlt layer thy will lend to 

d 7' 

fl 

- 

- -- CP a*,, . 

IlcIncnlbcring t,hat at the wall r = 0, as sum in^ t,hat the tcml)cmt,urc :L& the wall 

is r,or~sLnnt al1ct cq~~:rl 1.0 It,, nnd clcnoting the volocity nnd t,cn~[~crnt,~~rc, rcspoc~ivaly, 

at tlIc 0111cr cdgc of t h laminar sub-layer by u, nncl 7',, rrntl in the: free sl.rc!a~n 1jy 

[I,, 'I1,, 1'raI1dtl i~~tr~d~~~ccl thc assnmption tht tho rrltio q/t remains const:tntj 

across the width of the I)oundnry hycrt. lntcgration over bhc Inminor sublaver 

will t.hen load to 

Sirnilarlv, integration over the turl~~lc~lt zonc will lcncl to 

I4;quating the two right-hand sidcs we obtain 

Ilcncc. the local cocfficicnt of heat txnnsfcr bcconlcs 

On introducirlg eqn. (23.17) we have 1 

C T 

a -? 

L i u U (P - 1) (J, 

wC express Lllis rcslllt in terms of tho Nassclt numl)cr : L I ~ 

ore: led in this ~ :ty 

to tllc extension of the Reynolds analogy which was clorivcd indcpcnclcrltly by 

1,. l'randtl and G. I. Taylor:

710 XXIII. l'urhnlrnt. honntlnry lnycrn in rornprrmihlo flow 

In order to apply the preccding equation to particular cases it is still necessary 

to malzc a suitable ass~~mption about the ratio of thc mcan velocity at tho outer 

edge of the laminar suh-laycr to that in the free stream t. In the particular casc when 

P = 1, thc I'randtl-Taylor equation (23.18) reduces to Reynoltls's cquntion (23.16). 

In clcriving tho Prar~tlt~l-Taylor equation (23.18) it was supposed that tllc bountlary 

laycr coulcl bc sharply divitlcd into a turbulcnt laycr and a laminar sob-layer. 

In actual fact onc merges intp thc othcr in a continuous way antl it is possible to 

discern the cxistcnce of an inkrmcdiatc, or I~uCfer layer in which thc mngnit~~rlcs 

of tlic molccl~lar and tml)ulcnt cxchangc arc comparable. 'S?Il. vo11 I

712 XXI 11. 'I'~~rl~~~lemt bo~~ntlnry lnycrs in comprwsiblc flow b. Rclat,ion bctween velocity and ternpcrnture distrilmlion 7 1 3 

Tnhlc 23. I. l'lw conntnntn n nrd 11 for bhc cnlrulntion or the coclficior~t nf hent transfer from cqn. 

(23.20) nnd of tho rcrovrry htor l'ron~ rrp. (2:1.27), nfler H. Ibichnrdt [73] nrtd J. C. Rottn [El]. 

'l'lic tcmpcmturc tlistribution in turbulent boundary layers on flat platcs in the 

prescncc of an arbitrarily vn.rying, turbulent T'mndtl number, P,, was studied by 

R. R. von Ilricst 1283 :rnd J. C. Itotta 1811. In thc I;~tt.cr rcfcrcncc it is showti t,l~at 

only thc vsl~rcs whidi Ll~c turhulcnt I'r:dtl numbcr, P,, assumes c1o.s~ to t,he wall 

detcrrnine the rate of hcat transfer antl the tempcraturc distribution; conscqucntly, 

tlic details of the variation of P, away from the wall are less important. The variation 

of P, with tlist,:incc: from tfl~o w:Jl is hrougl~t to Iw:w only through tho int.crrnctliary 

or the cp~ntily A wlion in t.hc rcrnnintlcr thc val~~c of P, at the wall is substit~ited. 

A suitable value for this scems to be Pt = 0.9. J. R. Taylor [98] performed such calculations 

for boundary layers with variable pressure and temperature along the wall. 

2. The transfer of heat from rough mrfaces. tt tins bccn clcmonstratctl in 

Secs. XXf and XXIc that rough surfaces develop considerably larger values of 

skin friction in turhulcnt flow than do smooth ones. The samc is true of thc coefficient 

of heat tmnsfcr. Normally, however, the percentage increase in the rate of 

hcat tmn.sfcr is smallcr t,l~arl that in skin friction. This is understandable, because 

a part of the turhulcnt sharing stresscs can bc transmitted to the wall through 

pressure forcw cxert~tl on protubcranccs; but thcrc exists no analogue for this 

mcchanism in Iicat, transfer. Expcrimenhl invcstig,ztions on the transfer of heat to 

a rough ttrbc wcrc carried out, among others, by W. Nunncr [66] and V. F. Dipprey 

and R. IT. Sabcrsky 1261. 'rh~ latter autliors made mcasurcmenta at different values 

of trhc I'rancltl nunibor. l'l~coretical conaidcmtions clue to 1). F. Ilipprcy and R. TI. 

Sal)crsky [26] :13 wwrl 113 to P. It. Owen antl W. It. Thomson [67] arc bnscd on thc 

hypothesis that the elfcct of roughness on the rncchanism of exchange is confined 

to the regions locatd in thc proximity of thc wall. Starting with this hypothesis, 

it is possible to derive an equation which has the same structure as cqn. (23.20), 

and differs onlg in that the term (P - P,) must be rcplaccd by a quantity, P, which 

is a function of the Prandtl numlm, P, and of the roughncss. In the particular case 

when P, = I, we othin 

I c ' R P 

N z - --- -- .--;:r _-%._ _ . .. (r)' ipprcy, Sabersky, Owen, 'I?~omsorr ; P, = I). (23.22) 

I -4- 4 c,' /I (a, k/v; P) 

Pig. 2:1.4. Tho roughnc~n funct,ion (p 4- 8.5) P-0.44 rts n ft~nr.t,ion of lJ+ k./v for sand FOII~IIIICRR 

nt vsrioun Prnnclll nu~~~bcrci, from 1110 lncn~r~ro~~~o~~l~ 

by I). IT lI)ipprc*y 111rt1 It. II. S1\11t!wI 70. '1'110 grnpl~ ol' fho 

function p togctlicr with tlic expcrimcntd rcsr~lt,~ ovcr Lhc wholc rarlgc of roughncss 

Reynolds numbcr v, ks/v is shown in Pig. 23.4. Owcn ant1 'l'homson corrclntcd 

experimental results from various sourccs, including thosc from refs. [25] and [66], 

and coricludcd that 

(23.24) 

Proccdurcs for thc calculatio~~ of heat-transfer ratcs in turbulcnt flows with nonisothermal 

surfaces havc been worked out by D. U. Spdding [88], and J. Kcstin antl 

coworkers [36, 46, 461. Extcnsivc mcasuremcnts under such conditions awe performed 

by W. C. Itcynolcls, W. M. Kays, and S. J. Klinc [77]. 

3. Temperature distribution in comprceaiblc flow. In ortlw Lo ut~tlc~rat~ancl the 

laws which govern the tcmpcraturc distribution in compressible flows, tho rcadcr 

may wish first to rofcr to tho rolovant consirlcrntions for lnminnr I)o1111tlary lnyors 

wliicll wcrc advanccd in See. XlItb. Wlicn tlic pressurc remains (:oIIHI.:LI~I~ I I I ~ 

P = P, = 1, the tcmpcraturc distribution satisfies cqns. (13.12), and cqn. (13.13) 

in tho general case with hcat transfer, both owing to the evolution of frictional I~cat,. 

When P + P, $r 1, it is possible to cvaluatc thc recovery t~mpcrat~~irc on an (atliabatic) 

wall by tlic usc of cqn. (13.19), i. c. by 

The rccovory factor, r, is somowhat larger in turbulcrlt flow than it was in inrnir~ar

714 X X I I I. 'hrhulcwt. I)oc~ntl:~.ry I:rycrs in conq)rcrrsiblc flow 

flow, rxpcrirnr~~ts showing tltat on the avcragc its value placcs itaelf between 0.875 

ant1 0 88 (see lj'i'ig 17 31). 'I'hc diagram in l'ig. 23.5 rcproduccs 1,. M. Mack's [56] 

comparison of valucs of the rrcovcry fartor, r, measurcd on concs at cliffcrcnt Mach 

numhrrs and at cliffcrcnt Itrynoltls numbrrs In order to rstimate the cffcct of 

Pr:tntlt,l nurulwr, many rtutllors quote thc formula 

whic:l~ yioltls r .-I 0.896 at. P :-: 0.72. It is rtlso possildc t.o obtain this csthatc thcorcti- 

oally, in a mallncr analogous to that used I,r thc crtlculntion of tltc cocfficiont of 

11eaL transfer. For this I)urpost? it is ncccssary to start with the cncrgy equation 

(23.1 1 (:) ant1 10 irlclutle Lhc cffoots of Cllc molecular and of tho turbulcnt transfer 

mechanisms in aocortlancr! whicll the hypot.hcsis contraincd in eqn. (23.14). Proceeding 

in this way, .J. C. Rotta [81] obtaincd the cquation: 

Tltc quantity h is a function of the raLio PIP, and accounts for, like the quantity a 

in cqn. (23.20), tho procwscs taking place in the laminat' sublayer. It is given by the 

intrgral 

Wind lunncl 1 Mm 1 Typo of pone 

I 

k~,ercirca 

(:AI.CIT 5 x G In. 

Aatm I x 3 lt No. 1 

JI'L I8 x 20 in. 

I ' 18 x 20 in. 

J I I?( x 20 in. 

J I 12 x 1% in. 

I , 12 x 12 in. 

2.18 10' wood 

6.0 20" ccrnrnlc 

2.0 PO' l~olluw; slrcl 

4.60 5" nl~rcglnss 

I.o:l 13' lucitx 

4-50 13" llrcitc 

1.W 13" lf~cilc 

2.54 10" lllcitr 

c. Influcncc of Mach nunher; laws of fricbion 715 

Numerical values have bcen incluticd in Table 23.1. The factor 13 dcpcnds on P, and 

somewhat on dc,'/2 . According to Rotta, we may take 

When the turbulrnt I'mndtl number varies over the thicknrss of thc bouutlary 

layer, it is necessary to insert into eqn. (23.24) the value assumed by it at the wall. 

Whcn the Pmndtl n~lml)er, P, as well as the turbulcnt Prantltl number, P,, differ 

from nnity, it is worth noting that, normally, eqn. (13.21) givcn in Chap. XTII for 

laminar bountlary layers constitutes a usable approximation for the tcmpcmt~lrr 

distribution in a comprcssiblc turbulent boundary layer. 13. SchulCz-Jnncl~ 1!15] 

dcvelopcd n procedure for the calculation of temperature distributions in turbuicnt 

comprcssible boundary layers. 

c. Infl~~enee of Mnch nun~ber; lnws of friction 

To date, the calculation of turbulcnt boundary laycrs in inconl pressiblc flow 

has not developed to a point where it could be classccl as bcing morc than a semiempirical 

theory. It is, therefore, not surprising that the same remark applies to the 

cnlc~tlation of comprcssil)lc tur1)ulcnt boundary layers. In tho rnsc of incomprcssil)lr 

turbulcnt bounclwy laycm a starting point is proviclctl by tllc hypothcscs whicll wc-rc 

tliscusscd in the prcccding chnptcrs, narnrly by l'rnntltl's mixing-lcrlgt.11 Ilyl~otlw~is, 

by von Kilrmh's similarity rule or by Prandtl's universal velocity-distribution law. 

The authors of numerous contemporary papers have cndcavourcd to create a semiempirical 

theory of comprcssiblc turbulcnt boundary layors by transposing thcsc 

hypotheses and by adapting them to thc compressible case. This ncccssitatcd thc 

introduction of additional ad hoc hypotheses. In the absence of detailed investigations 

into the mechanics of compressible turbulent flows, thc transposition of thc semiempirical 

theories of turbulent flows from the incompressible to tho cornprcssible 

case involves a good deal of arbitrariness. 

Prom the practical point of vicw, thc tlimcdtics incrcnsc bccnusc, on tl~c one 

hand, there arc two additional pamrnetcrs- thc Mach numbcr, A&,, of thc: froc 

stream and the temperature, T,, of tllc solid surface- which influence the flow, and, 

on trhe other hand, tile available experimental results are not cntircly frcc of contmdictions. 

Tllrcc mcthotls should bc singlcd out from among tl~c numcrous propos:ds 

for handling the problem, bccause they havc bcen employcd parlit:ularly freq~lentIy: 

(1) Introduction of a reference temperature for the density and viscosity of the gas. 

(2) Application of PrantlLl's mixing length hypothesis or of von Kitrmbn's similarity 

hypothesis. 

(3) Transformation of the coordinates. 

Over and above, the litcrature of thc subjcct contains expositions of mc4lotls 

which cannot be classified under any onc of the thrcc proceding hcdding~. In an 

impressive comparison, D.R. Cllnpmnn and R.11. Kcstcr [I I] brought to thc forc 

the large diffcrcnces which result when cliffercnt methods arc usctl to calculat~c skin 

friction (cf. [30]). An extensive comparison betwcen twcnty cliffcrcnt ~omput~ational 

schemos and cxisting, expcrimental rcsults was carried out by D. R. Spnlding and 

S. W. Chi [89].

716 XXIII. 'I'urbr~lcnt bor~ntlnry lnycra in con~prcmiblo flow 

1. The flat plate at zero incidence. The guiding idea of the methods of class (1) 

is tho hypothesis that the laws of incompressible flow remain valid in tohe compressible 

case on condition thnt the values of density, Q, and viscosity, p, are bken at a suit- 

ably choscn rcfercnce temperature, T*. Th. von Khrmicn [43] was the first one to 

utilizc this possildity and sclccted the tomperaturc at the wall as his refcrcnce 

tempcratnrc. Starting with tho law of friction for a flat plate at zero inridence in 

incomprcssiblc flow embotlicd in eqn. (21.17), von lCdrmAn obtained the following 

equation for t.he skin-friction coefficient in the compressible mse: 

whcre M, = tJ,/c, dcnotes the Mach number of the free stream. The preceding 

equation is valid only for an adiabatic wall; in it, the viscosity function was assumed 

in the form p/p, = l/T/To. Various attempts have been made to improvc the method 

of thc reference temperature by choosing a value T* which lies between the highest 

end the lowest values of temperature, T, encountered within the boundary layer. 

E. It. G. Rrkcrt [29, 301 proposed to place the referencc temperature at 

T* = To -t. 0.5 (TI, - TI) -1 0.22 (T, - TI) , (23.30) 

whore TI denotes the tempcmturc at t,hc cclgc of the boundary layer, T, is tho sorfacc 

te~npcrat~urr at thc wall and 'f', rcprcsents the recovery (i. e. adiabatic wall) 

tcmpcmLrrre. lilckcrt's formnla inc:l~~tlcs I.hc c:wc with hcat transfer. Thc int~roduction 

of a mfcrcncc Lcmpcmturc const,iltntcs the simplcst way of accounting for thc influcncc 

of Mach numbcr and heat transfcr on skin friction ant1 lcatls to results which arc often 

adcquate in criginecring applirntions. For this reason, M. 11. Bertram 121 carried out 

a Iago programme of calculations of skin-friction coefficients covering a wide mnge 

of Mach num bcrs and trmpornt,~lrc rat,ios. 

Thc itloa of applying Prancltl's mixing-lcngth hypothesis wa.9 taken up by E. It. 

van Thiest [27]. Ilc ~Lipnlatcd that 1 == x y, as given in oqn. (10.22). The cffect of 

coniprcssibilit~y is brought to bc:w by allowing the dcnsity to vary thus causing the 

boundary-layer thicknrss to change too. Ilc obtained explicit formulae for turbulent 

ski11 frict,ion on ;I ll:~t plale, with and wiLhout, hcat tr:msfcr, which acconnt for the 

influcncc of thc 1L.ynoltls ant1 Mach numI)crs simultancoosly. For tl~c case of an 

adia\)atir, wall thc formula for thc coefficient of total skin friction has the form: 

whcrc 

and M, = U&, denotes thc frce-stream Mach nwnbcr. The symbol u) denotes the 

exponent in the viscosity law p/p, = (ll/T,)" from eqn. (1 3.4). This equation differs 

from (23.29) by the factor (sin-' 1)/1 on the left-hand side and by the appearance 

of t,l~e exponent cr, of thc viscosity law. For M, -+ 0 eqn. (23.31) transforms into 

Fig. 23.0. Cocfficicnt of lob1 akin friction for nn ndinbntio flirt plnh nt mro inc:iclc.nro for In~nitubr 

ant1 turbulent bounclnry leycr. 'J'hcoroticnl curvca for tr~rlwlc?nt flow rrotn cqn. (2:).31). :tftn.r 

E. R. van Driest [27]; y = 1.4, cu = 0.76, P = 1 

- Theory dl10 to Wllson 

[I021 k,r an adlnbxlic wnll nnd 

zero prewlre grnrliant: Lllc rntio 

T,JT, vxrirn helwaen 1.8 Tor 

M -- 2 and 21-0 for M - I0 

Tlwory dlic lo vnll Drirat 

(271, wit.11 ltcat trnndcr, wro 

pressure gradient 

Menserotnml.s: 

(1) xclinhntic wnll, zero pressure 

grndinnt 

(2) wi1.h I~anl 

trnsnfer, zero prrs- 

sure grnclicnt 

(3) with i~cnt lrsnsfcr, T,,IT, 

= 8.0, favourablc prCSsUrC gra- 

dient 

- theory, If%son, without heat trmh- _- _ _ 

--- the04 ran Dies[ with heat Imn* I I 

Fig. 23.7. Skin friction coeficient of a Rat plab at zero incidence M n function of Lhc Mach 

numbcr for a turbulent boundary layer; compariaoti betweon theory and measurement; R, w lo7, 

from 1381

718 XXIII. Turl~ulcnt boundnry layem in comprossiblo flow 

von 1Z:irm;in's i~icomprtssiblc resistance formula, eqrl. (21.17). Fig. 23.6 gives a plot 

of eqn. (23.31) ant1 a comparison with experimental results. The measure of agreement 

bctwccn thcory and experiment is not satisfactory in all cases, but ill this comexion 

it, must bc pointd out that mcasurcmcnts at high Mach numbcrs are somewhat 

uncertain. R.E. Wilson 11021 cnrricd out similar ealculationq, but based them on 

von JZ6rm6n's similarity hypot~hcsis, cqn. (19.39). Limiting himself to the case of an 

:ulial~at,ic: w:dl, IIC tlcrivccl a resultf which is quite similar t,o cqn. (23.31). I'urtllcr 

cxpcrimontnl msulk arc contninctl in Pig. 23.7 which shows a plot of the ratio of 

tltc skin-friction c:ocfficients in compressible and incomprcssiblc flow in terms of t.11~ 

M:wh n~lmbcr, c:ovcrirlg a rangc which includes very high Mach numbers. The graph 

cont.ains two t,heoreticnl curves; the first one due to R. E. Wilson [102] presupposes 

an adiabatic wall, and the second one, +rived by E. R. van Driest [27], includes 

tl~c cffcct of hcat transfer. The mcasurementa were performed 11y several workers 

[7, 14, 38, 63, 871 and show good agreement with thcory. Atlditionnl information 

concerning the inllt~cticc of hcat transfer on skin friction is contained in I'ig. 23.8 

wllich was also based on van Driest's calculations 1271. The diagram shows that the 

skin friction on an adiabatic wall is sornewhnt smaller than is the case when hcat 

flows from the fluid to the wall. 

Pig. 23.8. Skill fridiot~ codfit~irnt for a 

ht pl:rls at zcro il~citlcr~cc in turbulcl~t 

flow with lieat transfer as a function of 

Itcynolds numhcr for different valrrcs of 

t.lic tetnporakrrc ratio !7',/T,, after 1':. It. 

van Driest [27] 

Coordinate trnnsformntion: 'l'hc coordinate transformn.t.ion dcscrihcd in See. XIIId 

and valitl for 1amin:w flow can also be cnrricd through formally whcn applictl to the 

cliffcrrnlial cquaf.ions for comprcssil~lc tarl~ulcnt bountlary hycrs. The lteynoltls 

stress t',, is trnnsformcd t,o 

and with this substit.ntion, the momcnt~~m equation (23.8h) acquires the form: 

ati a - ac 

6 -1. "j .-'C. = s1 ---' (1 -1- a8a 1 

a~ ag a S) 4- v, --I- + - 

% eo ag . 

(23.33) 

(23.34) 

The symbols uscd here arc identical with those dcfincd for eqns. (13.24) to (13.41). 

Wit.11 the mnthematicnl possil)ilit,y of t.ransforrning thc equations for coniprcssil~le 

flow ir11.o a form it1cntic:d with that for iricomprcssihln flow, ninny nut,liors (r. g. 

B. A. Magcr [57j, D. Colos [15], L. Crocco [16], I). A. Spencc [$I, 921) t:ouploti a 

physical Iiypothcsis, accortling to which the vclocitry prolilcs in the t,r~~rislormc:tl pl:cnc 

rct:~in t.hc samc form as tlir~l, valitl for iriconiprcssil~lc Ilow. Conscclucnt.ly, tl~c law 

of friot.ion as wcll as othor relations rcrn:~in viditl wl~cn the Imtisforn~otl 

arc sub~tit~~t,cd into them. This conclusion, whic:li is ccrtainly valid For Iamir~:w flows, 

tlocs riot ncccss:~rily carry over to I.url~ult:nl. Ilows bct::~~lso t,hc l.rnt~sli)rlti:~t.io~~ < 

coortlinatcs cannot be applictl to the eqnations which tloscribc the IlrrctunLing motion. 

This lcatls to contratlictiorls with rcspcot to all thcorics which arc 1)ascd onI3o11ssillesq's 

assumption cmbotlicd in cqn. (19.1). Thcsc inclutle thcorics whicl~ utilize Pr:~litll.l'.s 

mixing-length hypothesis or von IGrm:i.n's similarity hypothcsis. If wc accept t h 

physically plausible assumption that thc eddy kinematic viscosity E, dcfinrtl in cqn. 

(19.2) is intlcpcntlcnt of clcnsity, we arc f:~ccd with blic fncl 1,liaL a t~.nrlsli~r~li:tl,io~i LO 

t,hc incomprcssil~lc: form ccasrs to be possil~lc. Ilowcvdr, :L t.mtisforrn:ktion to 

~:L~:uII(.~~('~s 

can still be cnrric:tl out. In thi~ case (,lit: ricw rtltly Itinc*tnr~l.ic: vis(:o~iI.~. I.,, is II-IIII~.~I 

to the original quantity, 8, through the equation 

Now, it is known that thc dcnsity mtio e/pl varies consitlcrably with the distance, 

y, from the wall whcn the Mach number is large. Conscqucntly, one of two conclusions 

forccs itself upon us. If wc, assume that the velocity profilcs remain unchangctl oom- 

1)arcd with the incomprcssiblc case, we find that, the clisl,rii)~~l,ion of F has cli:~ngctl. 

If, I~owcvcr, wc admit, that c rcmnins unalt.crccl, we cntl up wit.11 rnoclifictl velocity 

profilcs. The statements concerning tlic cffcct of Mach numl~or on thc vclot:ity profiles 

in tlic original coordinates which can be m:ulc on the I~asis of the two prect:tling 

schemes turn out to be exactly opposite. This observation throws a gootl tlcnl of 

light on the whole complcx of problems which arisc whcn tho laws ol~lainctl in ihc 

incompressible case are tmnslatcd to apply to t.lie comprcssiblo case. 

Further dctailer The effect of Mach number on thc velocity profile is hrought to 

bear through the increase in temperature in the direction of the wall. Since it is 

possible to soppose that the pressure, p, is indepcndcnt of y, it is found that thc 

density distribution in the boundary layer is described by 

As tho Mach number incrcascs ont.sitlc an atlial~n.t,it: wall, it is seen that the density 

nlust dccrease very strongly at small values of y ant1 Chis must cause the l~o~lntla.ry- 

Iaycr tl~ickness to incrcasc considerably. On tlic othcr hand, an incrcasc in thc Mach 

~lnrnher effects an increase in viscosity and a decreasc in the skin-friction coefficient. 

, 

I 

, 

his, in hum, causes I,hc laminar suh-layer lo incrcnsc strongly. An cxa~nl)lc of the

720 XXIII. Turbulent boundary layers in con~prcasible flow 

Fig. 23.0. Meanurcrncnts on 

vclociLy tIistril~uLio~~ in 

turi~ulcnt boundary layer 

on flnL plate at zero incidcncontsupcrsonic 

velocity, 

ahr It. M. O'I)onn~ll [20] 

M, = 2.4; d, - nlomcnlun~ 

lluieknrss rrorn cqn. (13.76); 

TW - "'a 

vclociljy profile in n aom~~rcssiblc Ifnrl~~rlcnt Iwundary laycr is given in Fig. 23.0 which 

contains a plot of u/U, in terms of y/02 for M, = 2.4 as mcasurctl by It. M. O'llonncll 

[26]. Ilcrc, d, rcpmscnls thc momcnt~rm thickness dcfincd in eqn. (13.75). 

In t4hc atloptctl systcnl of coortlinatt?~, thc points for tliffcrcr~t, Rcynoltls numbcrs 

arrange tli~rnsclvcs well on a single curve. 'l'llc t,l~corctia~l curvc shown on thc graph 

tlcviates from the corrrspontling curves for incompressible flow much Icss than was 

tl~c cnsc with laminar flow, Fig. 13.10. As cxpccted, thc 11ountlar.y-layer tl~it:ltncss 

incrcn.scs with Mach nnmbcr ; this is I~rongl~t, into evidcncc in Fig. 23.10 which 

clisplnys velocity profiles up to M, = 0.0. It is worth noting in this conncxion t,l~at 

turbulent boundary layer on n. flat plate 

in'supcrsonic flow at various Mnch num- 

bh, r29 memured'hy I!. W. Matting, D. 

It. Clqman, J. R. Nyholm, and A. C.. 

T1iornn.q [58] 

c. lnfl~lence of Mach number; lawe of friction 72 1 

t.hc momentnm thickness from eqn. (13.75) bccomes smaller comparccl with the 

boundary-layer thickness, 8, as the Mach numl>cr is incrcascrl, bccausc the clcnsity 

tlccreascs in t,hc direction of tho wall. 

The tliagrarn in Fig. 23.1 1 contains a logarithmic plot of the vclocitty ratio 'M/IJ, 

against 71 - y v*/vW of thc type er~countercd in Fig. 20.4, in wliiclr thc valucs of 

clcnsity, Q, ant1 kinematic viscosity, v, havc bccn taken at thc wall tcmpcratnre. It is 

noted that the characteristic shapc familiar from incornprcssiblc flow persists at! 

lligl~cr Rlnch numbers, but qnantitativc tlepxrturcs makc their appcamncc. 'I'his 

follows from I,l~c ex~~crimcntal rcs~~ltn plot,l.ctl in lhc figtlro ant1 due tm It. I

wl~erc c, denotes the vclocity of sound at the wall, S is the Shnton nnmher, and 

c,' t,l~e local skin-friction coefficient. Calculations performed by J. C. Rotta [78] 

under certain simplifying assumptions yielded results which werc qualitativcly correct; 

howcvcr, the elTcct of p, on the 1amina.r sub-layer turns out to be larger in experin1ertt.s 

than that which can be reflected in the calculations. The measurcrnents undertaken 

by 8. U. Mcicr L60, 01, 621 give an indication of the corresponding t,cmperat.ure distributions. 

The eval~lntion of these result* showed that the t(urbu1ent Prandtl nrttnbcr 

increases across the sublaycr and reaches a value exceeding unity; this means that* 

thc factor A, for hcnt tmnsfcr to the wall dccrenacs fmtcr than t,hc corrrspo~~tling 

eddy coefficient A, for momentum transfer. According to H.U. Meier and J.C. Itottn 

[63], it i~ possihlc to descrihe this state of affairs theoretically by txansposing Pmndt,l's 

mixing-lengt,h hypothesis (Chap. XIX) to the transport of heat. Thus, eqn. (23.14) is 

transformed into 

The mixing length 1, for heat transfer diffcrs ~cs to mngnit,r~cIe from that for morncntum 

transfer, 1 in eqn. (19.7). In analogy with I3.R. van Drinst's equat,ion (20.15b), 

it is assumed that in t.hc neighhourhood of the wall we may put 

The dimensionless const.artta xq and A1 have tliffcrcnt, vdues than x and A in eqn. 

(20,1511). The turhulcnt I'rancltl numhcr, cbs drfinctl in eqn. (23.12), becomes 

The variation of Pt acrosu the boundary layer was computed by I1.U. Meier [64]. 

Figure 23.12 allows us to conclude t,hat measured total-temperature distributions 

are reproduced quite well by calculat~ions ba,sed on J.C. Rott,a's [78] law of the wall 

for compressible boundary layers. The diagmms represent the ratio To/Tm of total 

Fig. 23.12. Totnl tompcrnturo To in tho 

tnrbulent boundary layer on a flat wall 

and in tho prcsenco of a weak hent flux at 

sr~pcraonic vclocit,y, nflcr H. U. Mcier et 

,nl. [G21 

, blnrlt nunther Mm ;- 2.0 

, Rry~tolda numtwr Rlrn~ = 0.8 x 10' cn-' 

ntcantlrrnwtts 1,). 11. U. >Icier l6Oj 

--- Illwry ns 111 eqn. (23.378) wlllt (XI+'= 0-9; 

A/AI -; 1.3 

I)ln~enalo~~ler~ heat tranafer roefflrlent 

hc'lw1cp TWQW rJ 

1 

1.ocnI nk111-Trivtlnn rorfflrlnnt rj - r,,/ p , IJ' 

c. lnfluonco of Mac11 nntnl)or; lnwo frict,ion 723 

temperntmres as functions of the Mach-numbcr ratio M/Mm. llere 

When the rate of heat transfer is small (q, m 0), the temperature increases from t.he 

wall outwards and reaches a maximum which is followed by a decrease to a minimum 

and an ultimate increase. 

When the wall is rough thc influence of thc Mach number on slrir~ friot,ion is 

even greater. According to H.W. Liepmann and F.E. Goddard [37. 621, the ratio 

~,~,,,~,,,/c~,,~ for the complotdy rough regime I~ccomrs proportional to the, (Iw~si!.~ 

ratio e,/em, and hence 

"l compr = 1 

el cnc 1 -I- CL Ma 

2 " 

where r denotes the recovery factor. 

2. Variable preosurc. In practical applications, it is frcqucntly ncccss:wy to 

perform calculations for turbulcnt boundary layers in compressible flows with varying 

pressure. The need is particularly acute in the design of co~lvcrgcnt-divergent nozzlcs 

for supersonic wind tunnels, because the displacement cmect of the boundary layer 

in them must be known fairly accurately. As was the case with incomprcssiblc flow, 

the known approximate procedures are based on the integral momentl~m equation; 

in some cnscs, tho cncrgy integrnl cqnation has also baan ctnployetl. 'l'ho Lwo il~lqr~d 

equations in question have been already given as eqns. (13.80) and (13.87) for adiabatic 

walls. As far as turbulent boundary layers are concerncd, these are writtcn: 

momentum-integral cquation - 

dd, d dU 

--+2-(2+N -Mz)=.Y 

dz U dz 12 61 u' ' 

energy-integral equation (kinetic energy) - 

they are valid for P = 1, and are not restricted to adiabatic walls. Here, O3 denotes 

the energy thickness, eqn. (13.76), 6, represents an cntl~alpy thickness, eqn. (13.77), 

and HI, = 6,/6,. 

A number of authors, including G. W. Englcrt [31], IC. Rcsltotlto and M. Tucker 

[76], N.B. Cohen [12] and D.A. Spence [92], applied the Illingworth-Stewartson 

transformation with respcct to thc momcntunt-intcgml cquation (23.39) and thus 

reduced it to its incompressible form. A. Walz [loo] rcduced the two cqnnLior~s 

(23.39) nnd (23.40) to a rclativcly convcnicnt form from tho point of vicw of n~rrncricnl 

computation and oncornpassed the required universal f~~nctions in a srt of tnblcs 

of numerical values. 

J.C. Rotta [84] described a similar procedure for two-dimensional and nxi- 

symmetric Rows as well as for the calculation of a body of revolution in subsonic 

and supersonic flow [105]. The agreement between calculations and mensurrmcnt is 

satisfactory up to a Mach number of M, = 2. The deviations which occur at M, = 

2.4 and 2.8 are cxplnined, partially, by the fact that the curvature of the strcnmlincs

724 XXlll. 'rur1)ulcnt boundnry Inyera in coinpressible flo~ 

in conj~rnction with t.1~ vnrintions of tlenait,y exert^ nn unexpcetetlly large inflrlcnec 

on the tlevclol~mcnt. of t.ltc boundary Iayc-r - an effect not aceor~ntrd for ill t,lle 

calcnlat~io~~. The rcnsona for this effect, of streamline cnrvatrrre were invest.igat,etl 

by J.C. Rotta 1821; n cont.rihulion t,o t,llis prr~l~lcrn was also rnatlc by 1'. Rrarlsltaw (4). 

Methotls of finite tlilTerr.nrrs hnvc also becn atlnpt.ct1 to deal wit.11 l,url)ule~~t bountlal.y 

layers in comprrssilde st~rc:ltns. 7'. Ccbcci and A.M.O. Smit.11 [I)] dcveloprtl n mtttllotl 

Imsrtl on ~nising t>lwory (srv Scc. XIXr) wllosc wlitlil.y lins 1tcc.n rxtcntl(,tl to inc:l~ltl(* 

t.hroe-tlir~~cnsio~~al 1)ountlnry Inycrs I 101. 'l'he ~net.l~od due t,o 1'. 13rntlsltnx\. (sc.e 

Sec. XIXf) that mnlres use of t.he equat.ion for kinetic energy has n.lso becn extended 

to a.pply t,o cornprcssihlc flows (61. P. I~mtls11,zw [5] rcnchctl t,hc conclusion t.hat t,hc 

volumet,ric tlilxtation exerts a deep influence on the st,ructure of the turbulence in the 

boundary laycr. Agreement between rnensnrement, and calculnt.ion could be considerably 

improved by the introduction of an additional term in eqn. (19.42). A method 

of int,egrat.ion for three-dimensional cornprcssible boundary layers was (leveloped by 

PT). Smit,l~ (941; a ~)roposnl in this rnaI.(.cr was tnntlc by J. Cor~st,cix [gal: colnparcalso 

I). Arnnl ct, al'. [Inl n.nd J. Consteix c,t al. [Db]. 

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1671 Owrn, P.R., and Thomaon, W.R.: Hcnt trnn~fcr across rongh surfnccs. JPM 1.5, 321 --Xi4 

- - 

(1943). 

[68] I'nppns, C.C.: Mcasnrcnwnt of beat trnnufcr in the tnrbulorit Iiornltlnry layer on n flat. plate 

in supcrsonic (low and comparison wilh skin friction rcsulta. NACA TN :12'2!! (l!)h4). 

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[70] Prandtl, L.: Eine Beziehung zwischen Wiirtncaltstnusrh nnd Striin~ungswiclerstntlcl (lor 

Fliissigkciten. Phya. Z. 11, 1072- 1078 (1910); see also Coll. Works II, 585--5!M. 

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328 (1!)40); NACA TM 1047 (1943). 

172) - - Jloic:hnrtlt,, 11.: 11111)nla- rind Wiir~t~o~ir~nI.nr~sc.II bci froior 'I'1n~11111c~nz. ZAhlM 2.i. 21% 272 

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[74] fZeichardt, H.: Die Grundlngen des turbulctrten Wiirmeiibcrgn~lges. Arch. Wiirtnotc!chn. 2, 

129- 142 (1061). 

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Munchester Lit. Phil. Soc. 14, 7-12 (1874). 

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13-A--5R .- - -- W .. I , IORRI. - -. ..-,. 

[78] Itotta, J.C.: Ober den Einfluaa der Mnchschen Zahl und dm Wiirmeiibergangs auf das 

Wandgesctz turbulcnter Stramun en. ZFW 7, 264-274 (1959). 

1701 Rotta, J.C.: Turbulent boundary!ayera with heat trnnsfer in comprcssible flow. AGARTI 

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[80] Itotta, J.C.: llemerkung zum Einflnes dcr 1)ichtoschwank~ngcn in turbulcnlotl Urcns- 

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[81] Rotta, J.C.: ~entp~rnturvrrteilun~en in dei turbulenten Grenzschicht an der ebencn l'lntte. 

lnt. J. Hcat Maaa Transfer 7, 216-228 (1964). 

[82] Rotta, J.C.: Effect of streamwise wnll curvaturc on compr~lsiblc turbulent. bonndnry layers. 

IUTAM Symp. Kyoto, Japan, 1966. Phys. Fluids 10, S 174-S 180 (1967). 

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[84] Itott.~, J.C.: POICI'RAN IV - Hechrnprogmmm fiir ~rcnzachichtcn hoi kotnlirrsd~lon 

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. 1861 Schubauer. G.U.. and Tchen, C.M.: Turbdent flow. High Spcocl Aorodyt~trtnics and Jot 

> 

Propulsion V, 75-196, Princeton (1959). 

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[9l] Spcnce, D.A.: Some applicatione of Crocco'a integral for tho turbulent boundary Iayor. 

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I1RlRL 

3 , 

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36, 50-56 (1050). 

Walz, A.: Uher Fortsc11rit.k in Niihcrwigsthnorir r~nd I'mxis tler Rercchnung kornprcssibler 

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Winter, K. G., Roth. J.C., and Slnith, I

730 XXIV. Prec turhi~lent. flowu; jcta and wnkes 

in tjhe tlownst.ream direction. Concurrently the jet spreads out and its vclocity de- 

creases, but the total momentum remains constant*. A comprehensive account of the 

problems of free jets was given by S.I. Pai [26]. See also the book by G.N. Abrnmo- 

vich [ll. 

A wake is formed behind a solid body which is bcing dragged through fluid 

at rest, Fig. 24.1 c, or hehind a solid body which has hcen placed in a stream of fluid. 

The velocities in a wnkc arc smaller than those in thc main stream and the losses 

in the vclocity in thc wakc amount to a loss of momcnta~m which is due to thc drag 

on the hody. Thc sprcad of thc wakc increases as tllc distancc from thc body is 

increased and the cliffcrenccs between the velocity in the wake and that outside 

I)ecomc smaller. 

Qnalitativcly such flows resemble similar flows in the laminar region (Chaps. IX 

and XI), but thcrc arc large quantitative differences which are due to the very much 

larger turbulcnt friction. Free turbulent flows are much more amenable to mathc- 

matical analysis than turbulcnt, flows along walls because turbulent friction is much 

larger than lnrninar friction in the wholc region under consitlcration. Consequently, 

laminar friction may bc wholly neglected in problcms involving free turbulent flows, 

which is not tho c.wc in flows along solid walls. It, will be rccalled that in thc lattcr 

case, by cont.mst,, laminar frict.ion must always bc taken into account in thc imrnc- 

diatc ncigl~bourhood of thc wall (i. c. in thc laminar sub-lnycr), and that causrs great 

mathematical tlifficultics. 

Furtlirrmorc, it will ho noted that prol)l~ms in frec turbulent flow arc of n 

houndmy-hycr nature, mcaning that tho region of space in which a solntion is being 

sought docs not cxtcntl far in a transverse direction, as comparctl with the main 

dirc~t~ion of flow, and that tho transverse grndjcnts arc large. Conscqucntly it is 

permissible to study such prol)lcms with the aid of the boundary-layer equations. 

In tlic two-tlimrnsiond iwomprrssit)lc flow tlicsc are 

Ilcrc T t1c:notcs t.hc I~url~~~lcnt shmring sl.rcss. l'hc pressure term has bccn droppcd 

in the cqrmtion of motion because in all problems to be considered it is permissible 

to assume, at Icast to n firsL approximation, that the prcssnrc remains constant. In 

the case of wakcs this assumption is satisfictl only from a certain distance from the 

tmtly onwartls. 

In ordm to be in a po.sit.ion t,o intrgratc the systcrn of equations (2.1.1) and 

(24.2), it is necessary to exprem the tnrbulcnt shearing stress in terms of the paramctcm 

of tho main ~Iow. ~t present suct~ an rIimin&tiort can only 110 aotticvcd witti 

the aid of sorni-cnipiricd ~~snmptions. 'LY~csc liavc already bcen ciiscuwcd in Chap. 

XlX. In this conncxion it is possible to make use of I'rantltl's mixir~g lcngtll tl~cory, 

eqn. (19.7): 

or of ite extension 

b. Estimation of tho incre-c in width ntid or tho dccre,wo in vrlocity 

where the mixing lcngths 1 and lI are b be rcgartl~rt na purely local functionst. They 

must be suitably dcalt with in each particular case. Further, it is possible to use 

Prandtl's hypothesis in eqn. (19.10), namcly 

t,=o& aa- ~ I L 

,ag - e xl b (urn,, - %,in) 

731 

(24.5) 

where h tlenotcs thc width of the mixing zone and x, is an empirical constant. Morc- 

over 

is the virtual kinematic viscosity, nssumcd constant ovcr thc wholc width and, IICIICC, 

independent of y. In adclition it is possible to use von IChrmh's Ilypothcsis, cqn. 

(19.19) and that due to G. I. Taylor, cqn. (19.15~~). 

Whcn cithcr of thc nssnmpthns (24.3), (24.4) or (24.5) is uscd it is fonncl that 

the rcsolts differ from cach otlicr only compnmtivcly littlo. ?'he bcsl rncnsuro of 

ngrccmcnt with cxpcrimcntal rcsnlta is furnished by thc awnml)tion in ccln. (24.5) 

and, in addition, the resulting cqnations arc morc convenicrit to solvc:. I'or I.l~c:sc: 

reasons we shall express a prcfercncc for this hypothcsis. Ncvcrtl~clcss, sonlo cxar~lplcs 

will be st~~diccl with the aid of thc l~y~otlicscs in cqns. (24.3) and (24.4) in ortlcr to 

cxhibit tho diffcrcnccs in tltc rcsulta whcn clifhrcnt, l~ypoblicscs arc IIRCCI. Morcovt~r. 

the mixing Icngth formula, cqn. (24.3), has rcndcrctl such valuablc service in the 

theory of pipe flow that it is useful to tcst ib applicability to thc typo of glow under 

consideration. It will be recalled that, among others, the universal logarit~limic 

velocity dist,ribution law has bcen dcduccd from it. 

b. Estimation of the increase in width anal of the clecrcnoc in vclneity 

Bcrorc proceeding to intcgratc cqns. (24.1) ant1 (24.2) fnr scvcrd parth11:rr 

cascs wc first propose to make estimatiotis of onlcra of magr~it~utlc. In this way wo 

shall bc able to form an idea of the typc of law wl~ich govcrns thc increase in the 

width of thc mixing zone and of the decrcasc in the 'hcight' of the ~clocit~y ~~roAlo 

with increasing distancc x. The followit~g accourlt will bc based on one first givrt~ 

by I,. Prantltl [27]. 

When dealing with problems of turbulcrlt jck and wakes it is usu;dly assumed 

that the mixing lcngth 1 is proportional to the width of jct, 0, because in this way 

wr are led to 11sdu1 rcsults. Hcnce we put 

t This extension was not diucu~acd in Chnp. XTX bccnuuc it is usrd only very rnraly.

732 XXIV. Frcc trtrbt~lent flows; jets and wakeu b. Estimation of tho incresso in width and of the dccrcnso in volocit.y 733 

III addition, the following rule has withstood thc test of time: The rate of increase 

of t,he width. b, of the mixing zone with time is proportional to the transverse ve- 

locity 1)' : 

Here D/Dt denotes, as usual, the s~bstant~ive derivative, so that DID1 = u a/ax -1- 

+ v a/az/. According to a previous estimate, eqn. (19.6), we havc v' - 1 au/ay, and 

t,hus 

Further, the mcan value of au/ay taken over half the width of the jet may he nssumetl 

to bo approximately proportional to u,,,/b. Consequently, 

1 Z' = const x -- u,,,,, = const x B u,,, . 

(24.7) 

Dl b 

Jet boundary: With t.he use of t h preceding relations wc shall now estimate- 

tho rntc at which the width of thc mixing zone w11ic:h nccompnnics n frro jcl, l~orlntlnry 

incrcascs with t.hc tlist,xncc, z. For the jet bountlar.y we have 

On comparing eclns. (24.8) and (24.7) we obtain 

which mcnns thnt tho width of the mixing zone associatctl with a free jet. boundary 

is proportional t,o t,hc cli~t~ancc from t,hc point wllcrc the two jcts meet. Tl~c coristnnt, 

of inl,cgraLion which mustf, stricLly spcalting, appear in Ll~c nbovc equation can br 

rnatle t,o vanish by a snitablc choice of t.hc origin of the coordinate syste$. 

Two-tlimcnnionnl nntl cireulnr jet: 15qr1ation (24.8) rc:nr;rins vn.litl in t.hc cnsc 01' 

a two-tlimcnsio~~d ancl of a cironlnr jrt,, rc,,,,, 

lirir. 'l'lins in sn(:Ir t:ascs wo :dso Ilnvo 

clr~~of.itrg now I.hc vclocii,y at, t .11~ ccn1.1.r- 

.;I 

In thc case of a two-dimensional jet we have J' = const x p uZ,,,,, 11, wlicrc J' 

denotes momentum per unit length, and hence u,,, = const x h-t/21/.~'/p. In view 

of eqn. (24.9) we have, further, 

u mnZ= const x -- 114 (two-dimensionaI jet) . (24 10) 

iG 

In the case of a circular jet the momentum is 

J = ronst x p u2,,,, h2 

and hence 

[n view of eqn. (24.9) we now have 

Two-dimensional and circular wake: Instend of cqn. (24.8) wc now have 

wlierc u, = U, - 14. On equating tho two cxprcssions, wc obtain 

or 

db 1 

U,--- -U 

dz b l=Bul 

db 

-- - p - (two-tlimcnsiond nntl circular wr~ltc) . (24.12) 

ci z U, 

'J'lte cnlcnlation of momcritum in problems involving wsltcs tli~crs from that 

for the case of jets, because now t,here is a direct relationship 1)ctwcc.n momcntrtm 

and the tlrag on the body. As nlrcntly mentionctl, eqn. (9.26), the momcntrim irilrgrnl 

is 

D=J=eJu(U,-u)dA, 

provitlctl t,l~at 1.11~ control surface has I~ccn plac:ctl so fnr bchintl the body t.h:~L t

734 

XXlV. Prrc b11r1111lcnt flows: jot8 RIIC~ wnkcs 

Inserting eqn. (24.12) for t.lio rate of increase in width, wc obtain 

or 

h N (p x cell d)1/2 (two-tlimonsiond wake) . (24.15) 

Inserting this vr~lur into cqn. (24.14) we fintl that the rate at which the 'depression' 

in the vclocity curve tlccrcascs downstream is rcprcscntcd by 

(two-dimensional wake) . 

Tn other words, t,lie width of a two-dimrnsional wake increases as ii and thevelocity 

tlccreases as 1 /fi . 

Circular wakc : Dcnot.ing tlir frontal arca of thc body by A we can write 

its drag as D = 4 c, A e M,Z and the momentum, eqn. (24.13), becomes 

J - e U, u, h2. ICqunting 11 and J, wc ohtain 

Inserting this v:rluc into rqn. (24.12), wr fi~d Illat thc increase in width is given by 

or 

d 1) 

6' ,iz - P CII A 

h N (p c, A x)1ln (circular wake) . (24.18) 

Tnscrting eqn. (24.18) into (24.17) we fir111 for Lhe clccreasc in the ilcpression in the 

vcloc.ity profile tho rxprossion 

c,# d 

(circular wake) . 

- ( ) 

'I'nblc 24.1. I'owrr Itiws for tho inrrcnso in witl1.h nntl for tho docrcnm in tho ccntm-lino vclocity 

in terms of distance z for problclns of free turbulent flow 

Fro jet boundary 

Two-dimenuionel jet 

Circnler jot 

'I'wo-dintonsionnl wake 

Circulnr wake 

width 

b 

laminar 

-- 

:ent-rc-line velocity 

or Ul 

Z0 ' 

1 

Z-IIJ i 

turbulent 

:entrc-line velocity 

'J,",,: or 'JI 

, .Lhus, . for a circular wakc we find that, Lho wirlLIi of Llic w:dtc incrc~nsos in ~rroport.ion 

to x'I3 arid that the velocity decreascs in proportion to x-~IR. 

The power-laws for thc width and for the vo1oi:ity in ftllo centre 1i:~vo 111:ori 

summarizctl in 'l'able 24.1. Tho corresponding laminar eases which wtm partly 

considered in Chaps. IX and XI have been added for completeness. 

c. Examples 

Tlie prcccding c:stiniatcw givc in Ll~crnsclvcs n vory good i(lo:~ of t,llo OHSI:II~.~:L~ 

features cncountcred in problems involving free turhulcnt flows. We shall, howover, 

now go one stcp f~~rthcr arid shall exaniinc scvcral pRrticl~lar c:ases in muc:li grcatm 

detail deducing the complete velocity tlistributiori function from the ccl~~:~tions 

of motion. In order to achieve this result it is necessary to draw on ono of tho hypotheses 

in eqns. (24.3) to (24.5). The examplcs which hnvc bccn sclcatd Iicro for 

consicler:~tion all have tlic common fc:hire that tho velocily profiles wliicli owur 

in thcm are aim.ilnr to each othr. 't'liis means that thc velocity profilcs at tlifi'~:rcnt~ 

distances x can IN made congn~cnt by n, suitsblc choicc of a vclocity and :r width 

scalc fnctor. 

1. The urnoothing out of a velocity discontineity. As our first cxarnlrlr wc- s1i:~ll 

consider tlic problem of the smoothing out of a velocity tliscontinuil.y wltit:h was 

first treated by 1,. Prandtl [27]. At time 1 = 0 thcre are two strcams moving at 

two different velocities, U1 and U2 respectively, their boundary bcing at y == 0 

(Fig. 24.2). As already mentioned, tlic bonndary ncross wliic4i the vcloc4t.y v:trit-s 

discontinuously is unstable and the process of turbulent mixing sinoothcu out. the 

transition so that it becomes continuous. The width of the zone ovcr which this 

continuous transition from velocity U1 to velocity 1J2 takes place incrcnscs with 

incrca9ing time. We are hcre concerned with a problem in non-steady parnllcl llow 

for which 

u = u(?y,l) ; v = 0. (24.20) 

Thc convectivc terms in eqn. (24.1) vanish idcntically. Making use of I'randtl's 

mixing theory, eqn. (24.3), we can transform eqn. (24.1) In give 

Fig. 24.2. The amoothing out of n velocity 

discontinuity, after Prandtl [27]; a) Initial 

pattern (t =O), b) Pattern at later instant a)

700 XXIV. Prrc tr~rb~tlent flowa; jcla and wakes 

The width of the mixing zone, b, increases with time and b = b(t); the mixing 

length is nssumcd to be proportional to b in the same way as before so that 1 = /I b. 

Assuming that the v~locit~y profiltts are similar, we may put 

wit01 11 = y/h antl 1) - lp. 'I'ltc cxponrnt p in the oxprcssion for the witl1,h can be 

detwnminctl from tho contlit,ion that in eqn. (24.21) the accrleration ant1 frictional 

terms n~ust br. proportional tlo cqual powers of t,imc, t. Thus awlat is proportional 

to 1-I, wllercas thc right-hand sidc is proportionnl t,o 12P-" = 1-P, so that p = 1. 

In this manner we obtain the following ass~lmptions for the problcm in hand: 

The ve1ocit.y u is bcst assumed to be of the form 

with lJ, = & (U, -1- U,) ant1 A = 4 (I/, - 11,). In ortlcr to maltc sure that at the 

edges of the mixing zone, i. e. at y = & 6, the velocity becomes equal to U, and U, 

respect ivdy, wo must put f = jl 1 at 77 = f 1. Tnserting thc vnlue from rqn. (24.22) 

into cqn. (24.21) wc obtain (.he followi~~g tlill'crcr~t,ial rqu:rtion for / (I/): 

l'hc equation has one solution /' =- 0, i. o. / = const, which rcprescnts tho trivial 

case of a constrant velocity. Tf, howcvcr, 1' tliffers from zero, we may tlivitle through, 

whcnrc we f ntl 

q 4- D2.B /Ir =o. 

B 

with c, = - IX/6 /?2 A. 'I'hc above solution satisfies the condition f (0) = 0 so that 

tlm rot~stant~ c,, and C, can br tlctcrmincd from thc condition f(q) = 1 and /I(?) 

nt, y == b, i. e nt 11 -- I. Ilrncr, 

= 0 

Introtlucing t,llcsc values illto cqn. (24.22) we qbtlain the solution in its final form 

with 

1 1 

, t) = - 2 (ul + 0,) + (24.23) 

(ul - UJ [ ; (F) L (6)3] 

6 -- $ 8' (U, - U2) t . (24.24) 

The velocity distribution from eqn. (24.23) is sccn plotted in Fig. 24.2. It has thc 

remnrlrablc property t,hat tho velocity in 1.11~ mixing rcgion docs not go over into 

the two free-stream velocities asymptotically. Transition occurs at a finiGc tlistanrc 

y = 11 with a tliscont,in~tity in @u/ay2. This is a general propertmy of all solutions 

obtained on the basis of Prxndtl's hypothesis (24.3) for tho shearing stmss in Curbulont 

flow. It, const.itut.es what may be called an esthetical tlcficiency olt,l~is hypot,llcsis. 

Thc itnprovctl hypothcscs (24.4) or (24.5) are frcc of this blcmish. 

Tho quantit,y /I = llh is the only empiriral constant which appcnrs in t,he so- 

I~rtion; it can bc tlctcrminctl solcly from cxpcrimcrlt.nl tl:cta. 

2. Free jet bon~ldnry. Thc condit,ions at a frcc jcl boundary arc rloscly rt.lnt,rrl 

to thosc in t,he prccetling examplc. With rcfcrencc to Fig. 24.1 a we shall consitlrr 

the more gencral case when at x = 0 therc is a meeting of two stfreams whosc const,:l~~t 

velocit,ics are IJl and U,, respectively, it, being assumed that U1 > IJz 1)ownstrcam 

of t.he point of cncountcr thc streams will form a mixing zonc whoso \vitll.l~ 

h increases proportionately to x, Fig. 24.la. The first solution to the problcm 

under consideration was given by W. Tollmicn [B2], who madc use of I'rantltl's mising 

length hypothesis for turbulent shear, eqn. (24.3). We shall review hcre the mathcmatieally 

simpler solution due to H. Goertler [I81 who bascd it on Prantlt,l's hypothcsi~ 

in cqn. (24.5). Since tho virt11a.1 Itincmnt~ic visrosit,y E is inclcpontl~~nt~ of' ?/, 

C~IIS. (24.1) nntl (24.5) givc 

au a~ aZu 

1L--+V--=&-. 

(24.25) 

az ay 'ay= 

Putting b = c x we obtain thc following expression for tho virtual kinematic vis- 

cosity, cqn. (24.5a), which is applicable t,o our casc: 

In view of the similarity of the velocity profiles and v arc funot.ions of y/x. Pl~t~ting 

[ = a y/x we can integrate the equation of continuity by the adoptio~~ of a sLrrnm 

function p = x U F([) where U = h(U1 + U?). Then u = U a Ff([) and eqn. 

(24.25) leads to the following differential equat~on for F([): 

F"' -+ 2 a2 F F" = 0 , (24.27) 

where a = &(x, c 1)-112 and A = ( Ill - U,)/(U, + U,). The boundary conditions 

are = & 00 : F'(t) = 1 f 1. The differential equation (24.27) is identical with 

Blasius' equation for the flat late at zcro incidence, cqn. (7.28), but the presrnt 

boundary conditions are different. II. Gocrtler solvcd oqn. (24.27) by nssl~n~ing a 

powcr-series expansion of the form 

with Fo = [. Substituting (24.28) into (24.27) and arranging in ascending powers 

of A, we obtain a system of differential equations which is solved by recursion. 

The first of the differential equations is of the form

738 XXIV. Frcc tnrbnlcnt flows; jctn nnd wnkes 

with the boundary conditions F'l(5) = f 1 at 5 = f oo. The solution of (24.29) 

is given by the error function 

The contriht,ions of the s~lccccding twms of the series in eqn. (24.28) are not significant. 

JJence the solution becomes 

with 

u = IJ, + U, UI - y, 

Figure 24.3 compares the theorctirnl solution with 11. Reichardt's [29] m~asurements 

for the case when [Iz = 0 nntf agrcemcnt is seen to be very good. The quantity a 

is the only empirical constant loft free to be adjustad from experiment. According 

to the mensurements performed by H. Reichardt the width h,,l of the mixing zone, 

mensurcd between stations where (UIU,)~ = 0.1 (corresponding to 5 = - 0.345) 

and (u/~J,)~ = 0.9 (corresponding to 5 = 0.975) haa the value h,,, = 0.098 x, which 

yicltls a -- 13.5. 'rhr virt,unl Itincmatic viscosity 1)ecomes c = 0-014 h, , x lJ1. 

Fig. 24.3. Velocity tlistribntion 

in the mixing zonc of a jot; 

n = 13.5 

Blunt body: The process of turbulent mixing th~t occurs in the wake behind a 

blunt body was explored in detail by M. Tanner [49]. The results are displayed in 

Fig. 24.4. At each edge behind a blunt two-dimensional body or around the sharp 

circular edge behind cylindrical bodies t.here form mixing zones of the kind sketched 

in the figure. Tho velocity distribution across sttch a zone is of the same shape as 

t.liat in Pig. 24.3; it can be described by eyn. (24.30). The similarity parameter a from 

eqn. (24.30n) strongly depends on the angle 4 of the two-dimensional wedge or axially 

symmetric cone. This dependence is represented graphically by Fig. 24.4. The 

pa~nmcter a tlecreascs considerably as the wedge angle 4 is increased. For 4 = 180" 

(plate at right angles to the flow direction) the value of a is only one half of that for 

$ = 0 (frce jet). This signifies that in the walte the angle of spread of the mixing 

Fig. 24.4. Turbulent mixing zone in the wake 

close hchinrl n ~q~int wedgc-liko body RA invwtigntctl 

by M. Tsnner [40]. Tho aindnrity pnrnnmbr 

a front cqn. (24.308) reprwcnted as a function 

of tlie wedgo angle 6 

zone behind a flat plate at right angles to the flow is about double of that in n hae 

stream. However, this is true only for the case when a flnt splitter plate is placed 

in the wnlre to prevent the forn~ation of n von JGrmAn vortex street. 

W. Szablewski [46, 47, 481 extended these calculations, acr well as those given 

in Scc. XXIVcI , t.o cnscs wllcn therc is n Inrge cIifi~rr~~cr in the tIcnuit.itw of I.II~! 

two strwms, IIUL :t s~nall tlil~crcncc in t11t:ir vclocitit~s. 11, tt~rus out, tIt:~t, tht* \ vitlI,lt~ 

of the mixing zoncs are afTcctcd only very slight.ly by this tlihrcncc! in tlrnsit..y. 

Ncvt~rtlicless, as tlic cliffcrcnco in t.llc drnsit,ics is incrrnsrtl, the zonc of rnisin~ I~rt.olnt~+ 

tlispI:t,ctxl in the ~lircction of t11c loss dt!tisc jct. 'l'lto p~wtding W S I I I ~ t::tut :I~SO IN: 

appliccl whcn t.hc two jets differ in tlioir cllomical cotiat?l~t.r:lt,iot~s. 1'. 13. Goocl(~t~ni, 

G. 1'. Wootl and I[. J. JJrcvoorL [I71 cnrrictl out an cxpcritncnt.:d invast.ip(.ion into 

the contlitions at the frcc I~otrntlary of n supersonic jot,. l'hr rcsults SIIOWC(I that 

the mixing zortc is soincwhnt nnrrowcr ant1 the 1cvt:l of turl~t~lrncr is somrw11:~t~ 

smaller than in inconiprrssil~lr flow. 

3. Two-dimensional wake behind a eingle body. Two-dimensional walres wrre 

first investigated by H. Schljchting [35] in his thesis presented to Goettingen Utd- 

versity. The investigation was based on Prandtl's mixing length hypothcsis, cqn. 

(24.3). A solution for the samc problem which was based on I'mnclLI's liypotI~(~~i.~ 

in eqn. (24.5) was later given by H. Reichardt [29J and II. Goertlcr [18]. Wc sh;d 

now give a, short account of both solutions in order to illustrate thc fact that tplla 

two results do not differ much one from the other. 

In the case of n wake, the volocity profiles bccomc similar only at! lnrg~ t1istnnt:r.s 

downstream from the Gody, there bcing no similarity at smallcr distnnrcs. Wt: 

shall restrict ourselves to tho consideration of large distances x so that, thc vrloritty 

difference 

u1 = Urn-u (24 31) 

is small compared with tlie frce stream vclocity I/,. At large dist,nncrs 1.11~ 

stn.l,ic 

pressure in the wake is equal to the static pressure in tho frce stream. Conseq~~enbly, 

the application of the momentum theorem to a control surface which oncloscs the 

body, assumed to be a cylinder of hcight h, gives

Neglecting u12, we obtain 

XXIV. Prce twbulcnt flows; jets nnd.wnkos 

+ m 

n=he U,/u,dy 

y- -m 

Substit,uting D = 4 c, d h e rJ:, whcrc d denotcs 

we obtain 

+m 

t,hc thiclrncss of the cylinder, 

As deduced in,Scc. XXIVb, thc width and the velocity difference vary in a manner 

to give 1) N x1I2 and u, N x-'12. 

Shearing stress hypothesis from eqn. (24.3): Since the term v8ul8y in eqn. 

(24.1) is small, we obtain 

- a14! = 2 la 2 au aau 

--I. 

(24.33) 

ax ay ag 

It is assumed that thc mixing length 1 is constant over the width h and proportional 

to it, i. c. t,lmt 1 = /I b(x). In vicw of the similarity of the velocity profiles the ratio 

7 = ?//h is inlrotlurcd as tho intlcpcntlcnt variable. In agrccrnct~t with tho power 

laws for the width and for the dopth of depression in the vclority profilc wc makc 

the assumptions : 

h = B (en d x)'I2 (24.34) 

Inserting into eqn. (24.33), wc arc led to the following different.ia1 equation for 

the frtnction /(v) : 

-- 

1 

(/ -1 7 /') = 

21J2 

- --- /' /" 

2 U 

wit,l~ the honndary conditions u1 = 0 and aul/ay - 0 at y = h, i. e. f = /' = 0 

at 11 -= 1. I~~tcgrathg oncc, we obt.ain 

whrrr thr constrant of intqption Itas been mn?tle ccpl to zero in vicw of the boun- 

dary rontli tion. I

742 

XXIV. Frcc t,url)rllcnt flown; jctn atid wnkcn 

bllz = 0.441 h, we have 0.44 1 ]/lii /I == : ant1 thus 

1 0 = -- = 0.18. 

b 

The precctling so111th1 const,it~~tcs an approximni.ion for large tli~t~ances X; 

rncnsnrcmont~ ~llnw t,llaf. it, is valid for z/c, d > GO. 111 the casc of srnnllor distanccs 

it is possible to calculate additionid farms for tho velocity, t,hc terms bcing proportional 

to %--' and x-~/~, resprctivcly. 

Shearing stress hypo thesin from eqn. (24.5): From eqns. (24.1) and (24.5) we 

now obtain 

i)ll 

ax 

a2UI 

ayZ 

(24.38) 

The virtrtal lzirwmatic visrosity is here E,= k1 ul,,, h and, hcnec, constrant nnct 

equal to E,, say. Consequently, the tlifferential~equation for ul is identical with that 

for a laminar wake, eqn. (9.30), except that thc laminar kinematic viscosity v must 

be replaced by F,. Thus we can simply copy the solution which was found in Chap. 

IX. Denoting r] = y i- , wo obhin from cqns. (9.31) and (9.34) t.hat 

so that finally 

The valllc of half tJlc wit1t.h at ldf tho depth is I),/, -.. 1.075 I/F,,//I., c,, rl (.r c,, rl)'ly. 

Comparing witall tho prccctline; mcnsurctl vah~e of bIl, it is Ii)ul~(l t.lt:~t, fhc vml)irit::tl 

qunntit,y 6, has t.hc vnlne 

Eo . .. = 0.0222 . 

ar D 

The preceding solution shows that the vc1ocit.y distribnt.ion in t,llc wnko c:w be rcprcscnt.ct1 

by Gausrt's function. The allarnativc sohrt.ion from cqn. (24.3!)) is scott plol.t.cd 

in Ipig. 24.5 ns curve (2). 'rhc tlifkrcncc bctwocn this rtolnI.iol~ it~tfl 1.11:1t in rqn. 

(24.37) is vory sln:dl. 

\V. Tollmien [53] solvcci the same problcm on Chc lxtsis of voll Ii:irm;in's 

tlypothcsis from eqn. (lD.l!)). Tn tho nrigllbourhootl of t,llc point,s of inflrsion in tho 

velocity profile, wl~crc Pir/ay2 = 0, it, Itas provctl nc:t!c:ss:try to in:lkc: :~~l~lil,iol~i~l 

assumptions. Extcnsivc cxpcrimcnts, which wrrc carried olll, by A. A. 'I'ownscwl 1.541 

in tho wako of a cylinder antl which wcrc concerned with t~~rlnllcnt fI~~~t~~:~t,ion 

at Reynolds numbers near 8000, showed that at a distancc equal to al,out 160 1.0 180 

tliamcters thc trlrl~cllent microstructrlrc is not ynt ftrlly tlevclopetl. I~urt.hcrn~on?, 

osoillngrams taken in t,hc strcanl dcmonsLmtc that the flow is f111ly t ~~rl~~~lnnt~ o111.y 

aro~~nd the ccnt.re, nnd ~I~~rtuntcs br.twcen laminar and tliri)ulent, in the ~lrighhlrhood, 

of the outer boundaries of the wake. Mcnsc~remenk on circ~tlar cylinders at, very 

large Reynolds numbers wcrc dcscribcd in Chap. I1 ; cf. 11. I'fcil 126b 1. 

Circular 11111lkecn have Ixcn invcst.ignf.ed by Miss I,. M. Swain [41] who Oiwt~I 

the calrr~lation on the hypothesis in eqn. (24.3). She obtained thr same rxprrssion 

for vclocity as in thc two-din~ensional casc, cqn. (24.37), but thc powcr laws for tllc 

width antl for thc ccnt,rc-line vclocity wcrc found to be tlilTercnt, namely b - XI/:' 

and ulm,, N X - ~ / ~ as , already shown in Tablc 24.1. 

Until recently, it has bccn ncccptctl tht t.hc valociby disLribuLion in :I W:I~C 

becomes indcpentlent of the shape of the body far cno11g11 bchind it,, antl is thrrcfore 

of a universal form. This belief was put. t,o ~ IIC t.est in a scrirs of ~~p(~ritnrr11,rt 

performed by If. Iteichardt ant1 It. J3rmshaus [31] and re1:itctl to wakcs bchintl bodics 

of revolnt,ion. Tt turned out that in cach individual casc the vclocity profiles rcmnin 

similar at varying distanccs behind the body. Ne~crt~hcless, the profiles behind bluR 

bodies (plates, cones with a ratio diamcter/height = 1) tcnd to be fullcr than t,hot;o 

behind lender ones (for examplc a cone with a ratio tlian~cter/hcight = 114 to 116). 

IJiffcrenccs of t.his kind have not bccn observed in two-dinrcnsional wakcs.

744 

XXIV. lhc L~trl)ulcnt Ilow~; jot. nnd wnkr~ 

4. Thc wake behind a row of bars. The wake behind a row, or cascade, of bodies, 

such as that, behind a row which is composed of a very large ntimber of cylindrical 

bars whose pitc11 is eqnal to 1, Fig. 24.7, is closely rclated to the wake behind a 

single botly. Thc prcsant, casc was investigated both theoretically and experimentally 

by R. (:ran Olsson [19]. At a certain distance from the row, the width of the wake 

cast by a singlc clcmcnt of the row is equal to the pitch, i. e. b = 1. Tho velocity 

diffcrencc ul -- ITr,, - it is hero also small comparctl with IJ,, and cqn. (24.1) can be 

simplified to 

1 ar 

.. [JW ?'I =.= . - ... 

ax ay ' 

(24.40) 

'rhc c:alculat.ion for thc caw in llancl bccomes very simple whcn the more general 

mixing longth i~ypot~l~csis from cqn. (24.4) is used. 'rho first step consists in the 

clct,crniinat,ion of t.hc cxponent in the power funct,ion for the decrease of u, with x. 

011 putting u, - XI' /(?I), wc have au,/az -- zp-l. 'l'hc right-hand side of eqn. (24.40) 

becomrs proport,iona,l to atla?/ N (an/$/) . (azu/ayz) - x21', because the mixing 

length, hcing proportiod to thc witlt.h, is constmt. Thus p - 1 = 2 p and it 

follows t,hnt p = - 1, or, that t,he velocity difference ul decreases in proportion 

to a-I. 

In the case of fnlly tlcvelopcd Row the vclocit8y tlistribution must he expected 

t,o he n pariotlie fnnct,ion in y, whosc period is equal t,o 1. Thus we assume 

1 - "1 

x 

I'ig. 24.7. I'low pattern bcl~inti a row of 

ham. Explanatory ketch 

The point y = 0 has here been made to coincide with the centre of one depression 

in the velocity di~t~ribution, and A is a free constant whose value is still to be determined. 

We now form the expression for the shearing stress t from eqn. (24.4) wit.h 1 = const 

and assume that l1 = 112 n, which scems permissible. The result is a very simple 

expression of the form 

Inserting t,llis cxprcssion into eqn. (24.40), we obtain A = (r1/1)2/8 n%nd hence 

the final solution 

.4ccording to the measurements performed by R. Gran Olsson, this cquat,ion is valid 

for x/l > 4. 

Behind a row of circnlnr bars for which 1/d = 8 the magnitude of t,hc mixing 

lengt,h is given by 

1 

7 = 0.103 . 

R. Gran Olsson also studied the case with t from cqn. (24.3) which implics 1, = 0; 

wit,h this nssu~npt~ion the calculation bccomes much moro cu~nbcrsomc-. 11. (:oc~t.lr~- 

1181 solved the same problem with thc aid of assumption (24.5) for t antl found 

that t,he solution was itlcnt,ical wit,h cqn. (24.41)t. A sccontl approsimnt.ion for 

smaller distances from tho cascndc was tlctlnccd by G. Cortlcs [7]. 

Cascades with a very narrow spacing bctjwccn the bars arc often used in wind 

tunnels to obtain a locally uniform velocity disLributsion. But, oftcn several jets 

close in on each other, and this process prcvents the velocity from becoming uniform. 

J. G. von nohl [5] made a more detailed stndy of such phcnorncna nntl pcrfornlctl 

experiments on several rows of parallel, polygonal bars varying the solidity m, i. e. 

the ratio of that portion of the cross-section which is filled by bars t,o thc total 

channel cross-section over the values m = 0.308, 0.462 and 0.618. Wllcn tho value 

of In is small t,hc singlc jets remain parallel; the closing-in of jets occurs 11.t ahont, 

m = 0.37 to 0.46. 

5. The two-Jimci~eio~~nl jet. Tl~c tj~~rl)ulcnt two-tli~nc.~~sio~~~il 

lated by W. Tollmien [52] who used Pmndtl's mixing length hypothcsis, cqn. (24.3). 

In t,he present section we shall, however, give a short account of the simplcr solution 

based on Prandtl's second hypothesis, eqn. (24.5), which was given by 11. Rrichardt 

[29] and 11. Goertler [18]. Measurements of the velocity distribution wcre performed 

by E. Foerthmann [Ill and H. Reichardt [29]. 

The rate of increase in the width of the jet, b - r, antl that of thc drcrcasc 

in the centre-line velocity, U - z-'I< have already been given in 'J'ablr 24.1. Eqrrations 

(24.1) and (24.5) lead to the differential equation 

jot, WIW first. (YIIVII- 

which must be combined with the equation of continnity. The virtrtal kincmalir 

viscosity is given by 

&, =xlbU, 

where U denotes the centre-line velocity. Denoting the centre-line velocity antl Llle 

width of the jet at a fixed characteristic distance 3 from the orifice by U, antl b,, re- 

-- - - -- - 

A 

t With tr=K A(u,,, --u,~,,), we have u - -- -- - cos (2" i) or, on mmpring wil~ 

8xeK x 

eqn. (24.41), K - ~(1/1)~ = 0.103* = 0 0333. Thtm the virtttal kinematic vi~cosit.y herornrs 

&I= 0.0333 A(%mnz-~~m(n) .

746 X XIV. I+cc turhttlrnt flown; jeta and wnkcn c. Examples 747 

spectively, we may write 

Consequently, 

Fnrther, we put 

' 8 

s,= s, (:)' with E, = x1 0, U, 

7 =a-Y, 

where a denotes a free constant. The equation of continuity is integrated by the 

use of a stream function tp, which i~ assumed to be of the form 

Thus 

y) = a-I Us 6"' z''~ F(q) . 

On substituting into eqn. (24.42) we obtain the following differential equation for 

F(v): 

1. F' + 1 

-. FF" + -EL a2F"' =O, 

2 2 us 

with the boundary conditions F = 0 and F' = 1 at TI = 0, and F' --; 0 at v = oo. 

Since s, contain^ the free constant xl, we may put 

This substitution simplifies the preceding differential equation which can now be 

integrated twice, whence we obtain 

FB+F'=l. (24.44) 

This is exactly tlic same equation as that for the two-dimensional laminar jet, 

eqn. (9.42). lksolution is F = t.anh v so that thevelocity is# = Us (~1.9)-lla(l -tanli2v). 

'L'lie chamc~cristic velocity can be exprcsscd in terms of the constant momentum 

-I m 

per unit Icngth: .I -- p / UZ dy. Hence .I = ) p Us% s/a With J/p = R (kinematic 

-03 

momrnt,um), we obtain thc final form of the solution: 

,rllr Vn~IIo 

t,llr siligle cmpiricn.1 constant o was determined experimentally by 

11. Rcicliardt [29] who found that a -1 7.67. Fig. 24.8 contains n compnrisou 1)ctwccn 

the theoretical curve from eqn. (24.46) with the nicasurcmt:nts due to E. I'oertli- 

mann, curve (2). The theoretical curve obtained by W. l'olltnicn [52] on the I)xsis of 

Fig. 24.8. Vclocity tliatril)~~tion in a two-tlilncnnionnl, turbulent jot. hlctw~~rtmct~b 

Foerthmann [ll] 

Theory: rarrv (I) BIIC If* Tnlln~irn [Be]: curvr (2) lrom cqs. (24 45) 

cllto Lo 

Prandtl's mixing-length hypothesis, curve (I), hna also been shown for cornparinon 

The first theoretical curve shows a slightly superior agreement with nieasurerncnt 

as it is fuller near its maximum. 

1.125 I 

From the given nnmericnl value of a we obtain s,= - - 

4n 112 J . or 

E,= 0.037 bl12 U , 

whcrc hllr again denotes half the width at half depth. 

A generalization of this problem consisting in a study of turbulent mixing under- 

gone between a two-dimensional jet with a co-directional external stream was ex- 

plored by S. Yamaguchi [GO]. See also S. Mohnmmadian [24n] nnd TI. Pfcil ct nl. [26a]. 

6. The circ~tlar jet. Experimental rcsnlb on circular jcts wcrc give11 11y W. Zitil~n 

[61] and 1'. Ruder1 [33] as well as by IT. Reicliardt 1291 ant1 W. Wucst IFi!)] So& 

results of measurements on circular jets are also contained in t h scrirs of rrl~orts 

published by the Aerodynamic Institute in Gocttingen [GZ]. 

The first thcorctical treatment of a circular jct was givcn by W. 'l'ollrnit~n [52] 

who based his study on Prandtl's mixing-length tlicory. In t.his cxsc, as ~cll as 

in the preceding one, the assumption for shcaring stress given in eqn. (24.5) lcntls 

to a considernbly simpler calculation. According to Table 24.1 (.lie witl(.li of t.11~

748 XXIV. Free turbulent flowa; jete and wakes 

jet is proportional to x and t h centre-line velocity IJ - x-I. Thus t,he virtual 

kinematic visrosit,y t~rcomcs 

which means that it ronxins constant over t,he whole of the jet, as it was in the 

two-dimensional wake. Consequently, the dirercntial cquation for thc velocity 

distribution bccornes formally identical with that for the laminar jet, it bcing only 

necessary to rcplacc the kinematic viscosity, v, of laminar flow by the virtaal ltinemat,ic 

viscosity, F ~ of , turbulent flow. It is, thercforc, possible to carry ovcr t,he 

solution for the Iarnitmr, circular jct, ccps. (1 1.15) to (1 1.17). Introclucing, once more, 

the constant, kinematic momentum, K, as a measure of the strength of tllc jett, 

we obtain 

I 

3 K 1 

U =- - 

Xn cox 1+ 1 ,2 ( T ) '' 

The empirical constant is now equal to fl/co. Accortiing to the mcasurement 

pcrformctl by IT. Reiclmrdt the width of the jcl is given -- by h,/, =- 04848 X. With 

7 = 1.286 at u = ) u, we hnvc hllz -- 5.27 x c ,/1/~, and hence 

whrrc, as bcforc, I),,, tlcnotcs half t.11~ width at half dcpth 

'I'hc diagram in Fig. 24.0 contains a comparison ,between measured velocit,y 

tlist,ribut.ion point,s and the tlleorcti~d results from eqns. (24.46) shown as curve (2). 

Cnrvo (1) proviclcs a furthcr cornprison wit.11 t,hc thcory due to W. Tollmicn [52]. 

The mixing 1cngt.h tllcory lends hcrc also to a vclocity distrihut.ion curve wlticll is 

sonicwlmt t.oo pointcti near thc mnximum, whereas eqns. (24.46) givc exccllcnt 

agreement ovcr the wf~olc widt.11. 'I'hc pnttcrn of stream-lines is &own plotted in 

Fig. 24.10. IL is seen tllat the jet draws in at its,haoundary fluid from the surrounding 

mass at, rcst, so tl~at thc mass of fluitl carrictl by the jet incrcascs in a downstream 

c. Rxarnplcs 

. . 

Fig. 24.9. Velocity distxibut,ion in n circolnr, turbulent jot,. Menuuromentn duc t,o ltoicl~nrrlt [2D] 

'l'hrory: rurvc (I) dur lo Tullmlcn[6Zl:curve (2) from eqns. (24.48) 

Fig. 24.10. Pntttlrn of streamlines 

in a circulnr, turbulent free jet 

tiirection. The mass of fluid carried at a distmcc x from the orifice can bc ralculat.ed 

from eqn. (11.18). Inserting the above valnc for F,, we obtain 

Calculat,ions on the velocity and tcmpcraturc distributions in two-tlimc~~sional 

and circular jets havc also becn carried out by 1,. IIowartll [21], both on the basis 

of I,. Prandtl's and of G. I. Taylor's assumption conrcrning turbnlcnt, mixing. 'l'llr 

mechanism which governs thc mising of a jct issuing from a circular nozzlc wit,lr 

the fluitl in a large pipe was studied cxpcrirnentally by K. Irikt,orin [%I. 'l'hc 

experiments covered a range of values of the velocity ratio in thc pipe to that in 

the jet of from 0 to 4. Compared with thc mixing of a free jct wit11 the surro~lnd- 

ing fluid it is noticed that the pressure increases in t,hc direction of flow in :I m:rllnrr

wliic:h rrscm1)lrs t,Ile pl~cnomrna now a sucldcn incrcnsc in cross-scct.ionn1 nrcn 

nntl somctitncs tlrscrild as (hrnott's loss. A thcorct,icnl cnlculxtlion bnscd on 

I'mntf(,l's tni sing length hy potlicsis sllowcltl that t,hc vclorit.y dist.ribution i~chnvcs in 

tho s:lrnr way ns in n t:ircwla.r \wkc (witlt.11 - r1/", centre-line vrlocity -~-~1:'). 

\\'l~cn $1, jet, of fini1.r wiclt.11 rmrrgrs it1t.o n ~~nifor~n st-ream, the uniform vclocity 

tlist.ril)nt,ion I)rcwi~rs l.r:~nsfortnrt1 I1c.n.r t,lic n1out.h of tho nozzlc into the prccctling 

prolilc. 'l'lic caso in hnntl wn.s sl,tijlietl by A. M. J

762 XXIV. Frcc td)ulcnt flows; jcta and wnltcs 

The first attempt to describe the circumstances of a wall jet by theory was under- 

taken by M. B. Glauert [16]. Thc former was considerably improved by E. A. Eiehel- 

brrnner et al. [13]. The semi-empirical theory succeeded for the first time in predicting 

the separation of a wall jet. Subsequently, J. S. Gartshore and B. (2. Newman [I41 

established an integral-momentum method which was based on very extensive mea- 

surements. Thesc includcd wall jcts witll injection. The calculation made it possible 

to determine the numerical value of the momentum coefficient that is necessary to 

avoid separation of the wall jet. Further expcrimental results can bc found in the 

papers by P. Bmdshaw and M T. Gee [4] as well as of V. IZrulta and S. Rskinnzi [23]. 

The account by P. Thomas [51] describes expcrirncnts concerning thc mixing of a 

turbulent,, two-dimensional jet boundetl by a wall on one side with nn external flow on 

the other. 

Two-dimensional jete on highly convex, curved walls exhibit the wcll-lrnown 

C:nn.nrln cffrct, that is the adherence of the jet over wide tlistances along the wall in 

the flow directiofi. Expcrimental and t,heoretkal investigations into the pattern 

ereatcd by a plane jet flowing nlong the contour of a circular cylinder have been 

cn.rrird out, by -1. Gersten [Is]. 17. A. Dvorak [lo] deals with the calcnlat.ion of turbu- 

Irnt I)o~~ncla~ry hyors on highly convex, curved wall^, pixying special attontion t,o wall 

jcte flowing along cnrvctl walls. Wall jets are employed in practice for boundary layer 

cont,rol and in film cooling; compn.rc a,lso H.G. Ncwrnan [25a.], A. hTcl,rnl [24c, 24tlI 

anti I). W. Young [ROa]. 

Thrre-rlirnrwionnl unll jrls with a finite ratio of the two sides have been recently 

stutlirtl expcrimcntnlly Ijy 1'. M. Sforzr~ and G. Ilt.rl)st [42], hy 1%. G. Nrwmnn ct, nl. 

[25J, by N. V.C. Swnrny nnd U 11. Gowtl:~ 1431, as wcll as by N.V.C. Swarny and 1'. 

Bandyopatfhyay [44] 'rliese measurcmcnts revealcd a very fast ratc of spreading of 

the jet in the spanwise direction and the existence of a very different fictitious origin 

for the growth of the width of the jet In the parallel as opposed to the normal wall 

direction. 

e. Dill'usion of tempernlure in free turl~ulent flow 

r 7 I hc process of turbulent mixing causcs a tlransfcr of the proprrt,ics of the fluid 

in a tlirot:lion at right angles to tho main stream. On the onc hand the mixing motion 

enusrs ~nowc~tlicm to flow awny from tho tnriin sl,rc:tm, on the otllcr I~:rntl, p:rrt.iolcs 

srtspcntlt:d in the Iluitl (1lo:cting particles of dust, chcmical ndtlitivcs) arc directed 

into the stream, and in atltlit,ion there is a transfer of heat, that is a diKusion of 

a tempcrature field. The intensity of the transfer of a given property in turbulent 

motion is asually itcscribctl I)y a suital~le coefficient. Denoting the coefficient for 

momcntum tmnsfcr by A, ant1 that for heat I)y A,, we can define them (sccScc.XX11Ia) 

k1.y writing 

Ilere 11, antl r, T tlertofc morncntum a,rltl hcat per unit mass, respectively, and t and 

q clcnotfe the flux of momentum and heat (= quantity of heat per unit area ant1 

tinic) rcspeclively. In t,his conncxion 14 and T denote temporal means. Since the 

mechanisms for the transfer of momeritnm antl heat are not iclent,ical the values 

of A, and A, arc, generally speaking, different. However, according to Prantltl's 

mixing-length theory the mechanisms of the transfer of ~nomcntnn~ :~,ntl 11r:~t it1 

free turbulent flows are itlent~ical which means that A, and A,, are nss~ltnr~l c:cl11:11 to 

each other. The messuretncnt pcrformctl by A. 1i'nge and V. M. I~:rlkncr 1.50) in the 

wake behind a row of heat,ctl bars have shown that t,hc tcmprmtrrrc prolilo is witlcr 

than the velocity profilc antl that, by way of approxirnntion, wc mn.y assrtritc 

A, = 2 A,. This rcsull agrccs with Cr. I. T:~ylor's tlicory which was t1isc:ussctl in 

Scc. XIXc, and according to which brlrl)rtlcnt mixing ~not~ion c:~,tlscs :LII c.xc:ltnngc: 

of vorticity rather than momentum. The problem of t,he tliIFusiorl of tctnprratnrc 

in free turbulcnt flows was also consitlcrctl I)y It. 1tcicha.rtlt 1301, who tn:~.tl(> I)ot.l~ 

tlicorctical antl cxpcrirnont,al contribrtl,ions. 'J'he thcorctic.:d work is closoly rol:~lctl 

to t.ltat tlcscribed in thc prccecling scction. First, empirical rclat~ions have bco~l tlrtlucctl 

for the temperature profile from expcrimcntal rcsults in the same way as was tlonc 

previously for the velocity (nlomentum) distsribution, hypothcscs on turbulcnt flow 

having been avoided. On the basis of an argument which we sl~all omit. I~cre, 

Rcichardt succeeded in Jcriving a rcn~arlrnble relation bctwccti l,hc t~ctnpc~ral.urc 

and the velocity distribution. This is given by 

T = (ufn:)Ar'Aq . (24.60) 

Tmnz 

Hers, the subscript mas refers to the n~aximnni valucs, and the sc:llt:s for 11. :111(1 7' 

must I)c so rcrrattgctl ns Lo rcntlnr Lhc poinl,s li)r WII~CII TC r - 0 n~~tl 7' . - 0 t.oi~t(.i

754 XXJV. Frce turhnlent flows; jets and wakes Rcfcrenccs 755 

Exp. I1 

1.36 

0 2.34 

0 3.65 

A 1.19 

A 2-01 

Fig. 24.12. The mixing of coaxial turbulent jets 

of different velocities and temperatures in a pipe, 

after S. R. Ahmcd [la]. Variations of the velocity 

along the nxis of the pipe n) for vnrious velocity 

rntios U* = urro/~so at a constant value of the 

temperature rntio O*; b) for vnrioua values of t.lie 

temperature ratio Q* = Olro/Oso at a constant 

value of the velocity. F* = frro/fso denotes the 

area ratio of the inner jet Lo the whole jet 

two cases: 1, two-dimensional [low above a lincar source of heat placed on a horizont,nl 

floor and 2. &xi-symmetrical flow above a point-source. In both cases the width of 

the velocity and temperatmure profilc increases in clircct proportion to the height 

abovc the floor, x. In the two-dimensional cnsc thc vcloait-y rcmains constant at dl 

heighbs, whereas the temperalure dccreeses as x-1. In thc axially symn~etrical casc 

the velocity is proportional to 2-113, the temperature being proportional to x-514. 

The two-dimensional case was treated theoretically on the basis of Prantltl's mixing- 

length theory (tmnsport of rnomcntum) as well as on thc basis of G. I. Taylor's 

vorticity tmnsport tlteory. The nxially ~ymmctricnl casc could bc invcst~igntcd only 

with the aid of Prandtl's thory bccausc G. I. Taylor's tlicory brcnlts clown in thig 

case. Measurements performed for thc axi-syn~met~rical enso conf rm t.11~ thcorrl~ical 

cslculations. The diffusion of temperature behind a point-source and behind n Linear 

source placed in the boundary layer on a flat plat,c were investigated experi~ucntslly 

by I

756 

XXIV. Prrc torbulent flows; jeta and wakes 

[14] Gartnhore, ,I.%, end Newman, B.G.: The turbulent wall jet in an arbitrary pressure gradient. 

Aero. Q,uart. 20, 26-66 (1969). 

1151 . - Gcrsten, I

CHAPTER XXV 

Determin~ation of profile drag 

a. General remarks 

The tdal tlrag on a Ijotly placctl in a stream of fluid consists of ski~r./;iction 

(equal to thc intcgml of all sllrnril~g strcsscs takcn over the surface o ~~~lle botly) 

ant1 of Jorm or prewurc, drag (integral of normal forces). Tho sum of th6 two is called 

told or pro/ils tlmg. The skin friction can bc c:LIcIII~~~ with some accuracy by the 

rise of the rncthotls of the prccccling chaptrrs. The form drag, dhich docs not exist 

in frictionless subsonic flow, is due to the fact that the presence of the boundary layer 

modifics the pressurc distribl~t~ion on t,he body as compared with ideal flow, but its 

comput~ation is very difficult. Consequently, reliable data on total drag must, in 

general, bc obtained by measurement. In more modern times methods of estimating 

the amount of profile drag have, nevertheless, been established. We shall discuss them 

bricfly in See. d of the present chapter. 

1.1, many cascs the tlcbrminatio~~ of total drag by weighing lacks in accuracy 

bccarisc, when mensurc~rlrnts arc performed, for example, in a wind tunr~el, the drag 

on the suspension wires is too largc compared with the force to be measured. In some 

cases even, such as in frec flight cxperimcnts, its direct determination becomes impossible. 

In such cases the mctliotl of tlctermining profile dmg from the vclocit,y 

tlis~rih~t,ion in the wdzc (I'itot travrrsc method), which has already bccn clescribcd 

in Chap. IX, 1)ccotnrs vcr.y ~uscful. Morcover, it is often the only practicable way of 

pcrforrning this kind of mcasr~rcment. In priuciple it can ody be used in two-tlimensional 

and axially symn~ctricn.l cnscs, but we shall restrict ourselves to the consitleration 

of tho two-dimcnsiontd casc. 

Thc formula in cqn. (9.2'7) which was tlcdacec~ in Chap. IX and whidi serves to 

dctcrn~inc the magriitutle of drag from thc v~locit~y distribution in the wake is valid 

only for com~)arat,ivcly large ctisbnces from the body. According to it the total drag 

on a botlyt is givcn 11y tlic cxprcssion: 

+m 

1) ==~Q/U(U,--u)dy. (25.1) 

y= -OD 

Tlcrc h tlrriot,rs the Irngt,ll of the cylindrical body it, the direction of thc axis of the 

cylinder, I/, is the frcc-stream velocity, and u(y) dcnotcs thc velocity distribution 

t In Cllnp. TX tho totnl drag or1 a ldy ~.la 

111 this chnptcr Llic sy~llbol U is used for iL. 

tlrnokd hy 2 I) (for tho two ~idtxi of the plate); 

b. The expcrimentol method due to nets 759 

in the wake. The integral must be taken at such a large tlistancc from the body that 

the static pressure at the measuring section becomes equal to that in tlic untlisturbctl 

&ream. In practical cases, whet,hcr in a wind tunnel or in frec flight incas~~remcnt,s, 

it is necessary to come much closer to the body. Consequently it becorncs nrcessnry 

to take irho account the c~nt~ribution from trhe pressure brtn and eqn. (26.1) nlctst 

be modified. 'l'his correction term has an appreciable vhluc whcn mcasummct~t,s arc 

performed close to the body (e. g. at distances lcss tlh onc clmd in tho case of 

aerofoils) and it is, therefore, important to have a comp~rativcly accuratc exprcswion 

for it.. The correction term was first calculatccl by A. I3ctz 141 and later by 1%. Rf. 

tJorrcs [2G]. At prcsct~t~ most mcasuremcnb arc bcing cvalu:~tccl wilh t,l~o :&I of t.116 

formula clue to Jones because of its compamt,ive simplicit,y. Ncvcrihrlcss, we propose 

t.o cliscuss Bet,z's formula as well becausc it* clerivatkm cxhibits scvcm.l very 

iritIcrcstting features. 

b. The cxperirnentnl method due to Betz 

Wit.11 rcfcmnet: to I'ig. 25. I we s~lcct a control surface around tho 1)otly as sliow~l. 

In thc rnbry cross-section 1 in front of the botly the flow is loaslcss, its total pressure 

being g,. The total prcwurc in cross-scct.iori I1 I)chintl tho hotly is !j2 ,: (I,.,. '1'110 

remaining cross-srctions of the control surf;lcc arc imaginctl placed far cnoirgl~ from 

the body for the flow in t81iem to be untlistorbecl. In order to satisfy the condit.ion 

of continuity, tl~e velocity u2 in cross-sect.ion It niust in some places cxccccl t,he 

11ndist,ur1)~tI velocity 11,. Applying tlic moinentwn tlrcorcrn to the cont~rol surf:~c:c 

gives t.11~ following expression for the drag on a cylinder of length h: 

In order to atlapt this rq~li~tion to the cvnluation of cxpcri~ncn~~nl rcsn1t.s it, is nct~eswary 

1.0 t,mnsfornt the above int,cgmls so that they necd hrdy be cvn1unt.c.d owr Illat. 

sect,ion of the velocit,y curve which includcs the dcpression of plarlc: I1 in t,hc profile. 

The total pressures satisfy the conditions: 

I 

;it. inlinit.y: 

900 = ~ r+ n e um.z 

The first integml already hns the tlcsirrcl form, 11cm11sc the total prcssrlrtx is rt111aI t,o 

g,, rvrrywlierc outsidr thc tleprcssion In order to transform the scco~~tl inirgml in

760 

XXV. DotcrminnLion of profilo drag 

the same way wc intro(1uce a hypothetical flow u,'(y) in cross-section II which is 

idCllt,i~al with IL, cvcrywhcrc outside the tlcprcssion but which differs from u, in the 

of tllc drl)rc~siot~ in LhaL lhc totd ~rcssllro for 1 ~ is ~ cqllal ' to gm. ' r h ~ ~ 

9,=p2+ 

1 

Zeu,'a. (25.5) 

since t,hc actrlal flow ul, u, satisfirs the equations of contir~nity, the mass flow of the 

hypot,llctical flow rl, 11,' is too largo across scction I[. 'Phis is cquivaleat to the 

exist,ellee of a source which is locatccl, essantially, at the body ant1 whose strcngth is 

Fig. 25.1. 1)ctornlinntion 01 profilc 

drng by tho method due to Bctz [4] 

A source which cxists in a frictionless parallel stream of vclocity U, suffers a thrust 

cqual to 

R =--p u,Q. (25.7) 

We now apply the momentum theorem from cqn. (25.4) to the hypothctiml flow, 

i. e. we assume a velocity ul in section I, and a velocity u,' in section XI. Since 

g,' = g, and since the resultant force is equal to R from eqn. (25.7), we obtain 

Subtracting this value from eqn. (25.4) we have 

~+~~,Q=b(~(g,--g,)dy+-;e/(u;'-~~?dy). (25.8) 

111 view of eqn. (25.6) we have now 

Each of thc above integrals necd only bo evaluated over the wake since outside it 

?I,' = r,. Sincc u,', - uZ2 = (u,' - 7t4 (u.~' f %), the above can be transformed to 

In order to determine tho drag, D, it is ncccssary to measure the total prcssurc, g,, 

and tho static pressurc, p,, over the cross-scction I1 bchintl t,lrc body. Thus wo also 

obtain g, as it is equal to g, out,sicic tho clcpression. Thc l~ypothct~ical vclori1,y IL,' 

is (lcfincd in cqn. (26.5) from which it can bc calc:ul:~tctl. 

In cascs whcn the static prcssurc ovcr the measuring station cquals that ill tllc 

untlistmrhcd strram, i. c. whcn p, = p,, 

transforms back into cqn. (26.1). 

wc also have u2' - IJ.,, :~n(l rqn. (25.9) 

Ilcfining a tlirncnsionlcss cocfficicnt of drag by writing 

where qm = 11; denotes the dynamic prcssurc of t,hc oncoming stmnm nr~d 1, x 1 

is the reference area, we can rcwrite eqn. (25.9) to read: 

c. The experinicntnl nicthncl doe to Jnues 

Some time later, I3. M. Jones [26] indicatcd a similar mcthod for the dctcrn~ination 

of profile drag. The final formula duc to Jones is somcwhat simpler t,l~an that 

due to A. Betz. 

The cross-section I1 (Fig. 25.2) in which measurements are performed is locatctl 

behind, the body at a short distance from it; tho static pressure p, at the measuring 

station is still markedly different from the static pressure in the undisturbed strcam. 

Cross-section I is placed so far behlnd the body that p, = pm. Applying cqn. (26.1) 

to cross-section I, we obtain 

Fig. 26.2. Dotcrminstion of profile drag 

by the method due to B. M. Jones [2F]

762 XXV. Determination of profile drag c. The experimental mcthod due to Jones 7 63 

In order t,o confinc the determination of u, to the use of results obtained from mea- 

surementa in cross-section 11, we first apply tlhe equation of continuity along 8 

streamtube 

euldy, =euzdy. (25.12) 

Hence 

~ = b ~ / u ~ ( ~ ~ - u ~ ) d y . (25.13) 

Secondly, according to B. M. Jones 1261, we make the assumption that the flow 

proceeds from section I1 to section I without losses, i. e. that the total pressure 

remains constant along every strcnm-line betwcw~ the stations I and 11: 

lntroilucing the total pressures 

we see from eqn. (25.13) that 

-- 

= 261dg~ - p2 ( ~ / 9 - ~ i&=-%) 

z ~ dy , 

(25.15) 

where t,he integml extends over cross-section 11. In this case, as in the previous one, 

the integrand differs from zero only across the disturbed portion of the velocity 

profile. Introducing a dimensionless coefficient., in thesame way as in cqn. (25.9a), 

and taking into account that g, - pm = qm, we have 

Jones's prccrtling cq~mtion also transforms into the simple equation (25.1) in cases 

when the static pressure at the me,asuring station is equal to the undisturbed static 

pressure, pz = pm. 

A. 1). Young [75] inclicatacl a transformation of Jones's formula which sirnplifie9 

the eval~at~ion of thc intcgral in cqn. (25.16). The resulting equation contains 

an ntltfitivr corrertion term apart from thc ir~bcgral of the total prrssi~ro loss taken 

ovrr bhr tlrprcssion in t,hc vclocity profile. The correction term depends on the ahnpo 

of tho vrlocity profilc in the measuring station, but it can be computed once and 

for all. A rritiral nppmisal of this method is contained in a note by G. I. Taylor [67]. 

The prcrc(ling two experimental mct,hotls have been used very frequently for 

t.hc tlcfenninxt.ion of profilc drag 110th in flight and in wind tunnel measurements, 

[(i, 12, 16, 20, 38, 39, 61, 62, 69, 701, and have lrd $0 very satisfactory results. 11. 

I)oc.tsc+ [6] dernonst.rnt~cd that both the 13etz and the Jones formulae can bo uscd 

whcn the clist.nncc bctwccn the mcn.suring statmion belhd the acroloil and the aerofoil 

it.sclf is as short as 5 pcr cont. chord. In this casc thc correction term in Betz's 

formnla amou~it.s t,o al~out, 30 per ccnt. of thc first term, Both mcthotls are partvicularly 

suit.ahle whcn t h inflwncc: of strrfacc ro~~glincsscs on profile dmg is being determined 

as well as t,o t,hc il~t.rrminnt.ion of the wry stnnll drag of laminar nerofoils. 

A. D. Yorang 1711 extended t.he applicnbility of Jones's mothod to comprcrrsihlc flows, 

Retracing the steps in that derivation, we apply the continuity cquat,ion for co~iiprwsihlc flow. 

el u, ~ 

and deduce the following formula for drag: 

Y= I el dyt . (25.17) 

Fiem, again, il i8 ncmsnry to cxprcae u, in brms of the qrrnntit,ic:s tnmrcnre~l in plnno 11. In tho 

realm of wmprmible flow it is necrmunry 10 rcplnco Jonra'e nnacmplion thnt g, = g, hy tho 

mnutnption thnt the entropy remains constant along a atrcamlinc from plnnc II to plane 1. 

This lends to the isentropic rclation 

If, now, the stagnation pmure measured by the Pitot tube in compro~siblc flow is clonotctl 

by g, we have 

and it can be vcrificd that e n (2.5.19) also lends to tho ~~onmpbion g, -- g,. 1.11~ vr1orif.y u., 

cnn bc doternlined front t~ic hirnou~li equation for co~nprcssiblc ,low, nnmnly 

I 9-1 1 

In order to solvc the problem in principle, it, is only ncrc.usary to express tlrc vclocit,y 11, i11 terms 

of the measurcd prrasurea g, and p, in plane 11. A mcaaurcmcnt of the totnl and stnlic prcssurcs 

in plane I1 is again sufficient for the determination of the drag of the body. However, the com- 

plic~ted relation between velocitiea and prewures in the compressible Bernoulli eqantion Icn& 

to a very cumbemomc equation. For this rowon, A. D. Young cxpnntied tho vclocitics ?I., a d u, 

into series of the form 

In this manner, the terms in eqn. (25.15) derived by Jonm for the incomprc~siblc cnsc. can now 

be separated, and tlic remaining terms can be nrrangrd in a jmww scrim in tcrnls of 1 110 hlnc.11 

number. Thus 

where ca, ( denotrrr t,lm drag coeflicicnt for tho incotnprcssiblc cnue, na given by eqn. (25.16), 

and the cocfficicnts A], A,, . . . rcpreaent certain integrals which can be calcc~lntetl from tllc 

measured dnta in plane 11. Rcstxicting one~clf to low Mnch numbers, and I~cnce to two krn~s 

in tlrc oxpansion (2.5.23), one obtains

764 

XXV. I)ot.crtninntiotl of profile drag 

Tilo ndditionnl term which d~ponrh on the Mneh number provides n negntivc contribution 

to tilo drag cooffirirnt,. It is poasiblc t.o cvalnxtc lhis additional tom once and for all if a suitable 

nsst~rnption i4 nindc for tho shape of the drpression in the velocity profile in the wake; this was 

also done by A. I). Young. 

'1. Cnlc~~lotio~~ of profile drag 

Mcthocls which can 11c usctl [or the calculation of profile drag antl which arc 

I~a,scd on t,l~c same principl(:s as tl~o abovc cxpcrirncnt:d mcthotls, havc I)con tlcvisctl 

by J . Pretscli [40] and IT. U. Squire arid A. 1). Young [MI. Thcse are tied in with 

t,hc calculat,ion of boundary layers, as described in Chap. XXII. However, in ordcr 

t.o bc in a position to calculate pressure tlrag it is necessary in each case to make 

use of certain additional, empirical relations. See also H. Goertler [19]. 

We tlow propose to give a short des~ript~ion of 11. B. Squire's and A. D. Young's 

mct,llod of calcnlation taking into account some more rcccnt reqults. We shall begin 

by transfornling eqn. (25.1), which relates the tlrag on a body with the velocity 

profile in thc wake behind the body. Introducing the momentum thickness &, from 

cqn. (8.31) nil thc drag coefficient from eqn. (25.9a), we can rewrite it as 

tlcnotcs tllr momcntnm thickncss of thc wake at a largo distance from the body. 

On the other hand, the calculation described in Chap. XXII permits 11s to evaluate 

the momc~tt~nm thickncss at thc trailing cdge, for which the symbol a,, will be used. 

The cssencc of Squire's method consist8 in relating these two quantities, dz, and 

dzl, in such a way as to permit the calculation of drag from eqn. (25.25) when the 

momcntum thickncss at the trailing edge of the body is known from a boundarylayer 

calcnlatior~. 

'l'hr momrr~tum integral cquation of boundary-layer theory, eqn. (22.6), is 

valid also for the wake behind a body with the only difference that the shearing 

strrss s,, must be equated to zero. Thus we have 

whew 11 -- 01/b2 antl U' = dU/dxt. The symbol,^ denotes now the distance from 

thc trailing edge of the body measurcd along the centre-line of the wake. The last 

rqnatiori c.:m also be written in the form 1 

t 'I'hr shapr f;wlor ,Y,/h, will now Iw drnobrd by If, for uilnplicit.y, rather LIian hy [I,,, ns hrfore. 

Integrating over z from the trailing edge of the body (sub~cript 1) to a stntion 

sufficiently far downstream, so as to have U = U, and p = pm, we oblain 

At a large distance behind the body we have H = 1, and consequently 

m 

n-n. 

B-I 

Here HI = 61/6, denotes the value of the shape factor H = 611/821 at the trailing 

edge which is known from the calculation of the boundary layer. This equation 

gives the required relation between BZm ar~d 821, provided that TJ1/U, and the valuc 

of the integral on the right-hand side are known. First we find that 

In ortlcr to bc: in a position to cviiluntc the intrgml, it, is ncccssriry lo know Lhc rc- 

lation between the static pressure in the wake, which determines the valuc of U, 

and the velocity distribution in the wake which, in turn, determines thc valuc of 

the shape factor If. Tho mngriitutlo of In (U,/U) docrcascs monolonically along 

the wake, starting with the valuc In (U,/U1) at the trailing cdge until it roaches 

zero at a large distance. Simultaneously I1 decreases from the value If1 at the trai- 

ling edge, until it reaches unity at a large distance. H. I3. Squire established an 

empirical relation between In (U,/U) and H. According to experiment: 

so that 

On substituting into eqn. (25.27), we obtain 

or, with the roundcd-off' value of HI = 1.4: 

On substituting this valuc into eqn. (25.25) wc obt,ain an cxprcssion for thc coof- 

ficicnt of total drag in the forn~

766 

XXV. Determination or profile drag 

The coefficient of profile drag can be evaluated from the above oqnation, if the momentum 

tltickness at the trailing edge is known from the boundary-lnyer calculation 

ant1 if, in addition, the ideal, potential vclocity at thc trailing edge, U1, is known. 

The latter can be found, for example, from a reading of the static pressure at the 

trailing edgc. According to a method proposed by H. R. Helmbold 1221 the determination 

of IJ,/IJ, can also proceed as follows: We begin by evaluating the momentum 

thickncss at the trailing edge, 8,,/1, from eqn. (22.17) using the value 7~ = 4. 

This valuc is thcn substiti~tcd into eqn. (25.28), and in thc resulting formula IJ1/U, 

is raised to the power -1 0.2. 'I'hus this factor can bc approximntcd by the vnluc 

of unity, because UJU, itsclf (Ides not dilTer much from unity, ant1 the value of 

the coefficient of profile drag for o m side (R = U,Z/v) can be found from eqn. (25.28) 

to he 

i I 

with 

The subscript 1 rcfers to tho point of transition nnd thc vnluc of the constant C 

can bc cletcrmincd from the condition that thc laminar and turbulent momentum 

thicknesses must be cqi~nl to each other at the point of transitmion, bzt = dztwb = 

aZfarn. The value of dzlatn can be found from cqn. (10.37). For uniform ~ot.entia1 

flow 1vit.h IJ = U, eqn. (25.20) transforms to the corresponding exprcssion for the 

flnt plate at zero incidence, eqn. (21.11), if, in addition, we put C = 0 for fully 

tleveloprtl t~~trk~ulettt flow. 

TC. 'l'rucl~cnbrotlt [G8] tmnsformcd cqn. (25.20) replacing tho potential vclocity 

distribut,ion by t,hc coordinates of the acrofoil scction thus, evirlent.ly, eCcct.ing a 

considerable simplificnt.iort. 

11. B. Sq~~irc n.nd A. I). Young [64] cvnlr~atcd a number of cxamplcs by tile use 

of a tliffcrcnt mctltotl. We shall now tlcscribe somc of thern, rcfcrrit~g to Fig. 25.3, 

wltirh csontains a waumk of thcsc rcsrrlt~. Tltc thickness of tlhe acrofoils was vnricti 

from dl1 - 0 (flnt platc) to rill -- 0.25 and tho Reynohls nurnl~crs R -- IT, 1/v 

mnpcd from 10Q,o 108. It is found that the profilc drag is very sensitive to the 

position of the point of transition from laminar t,o turbnlent flow. This lntter parametm 

wns vnricd from x,/E = 0 to 0.4. The increase in profile drag with thicltncss 

is, rssenl~inlly, tluc t,o an incrcnsc in form drng. Fig. 25.4 shows the rclation bctwccn 

form atttl profile tlrng. Analogolls calculations were performctl by J. Pretsch [40] 

in rclation to von I

768 XXV. Determination of profile drag c. Losses in thc flow through cilclcad~s 769 

Fig. 25.5. Increase in the coefficient of pro- 

file clrag plotted in terms of relative thick- 

ncas, as calculnted by Scholz [58] 

Totnl or proflla drag cl)tOt - CU form -1- CI 

metrical cases, applied to rough walls (equivalent sand roughness) as well. From 

a very large number of calculated examples on aerofoils (two-amensional case) 

and bodies of revolution, it proved possible to deduce relations to describe the influence 

of thickness on profile drag. 'Shese are shown plotted in Fig. 25.5. The 

difference Ac, = c, - c,, denotes the i~crcase in the coefficient of skin friction, 

related to the wetted surface, as against, its value for rr. flat plate at zero incitlcnce, 

c,. The curve for the two-dimcnsiorlal case agrees fairly well with the results 

shown plotted in Pig. 25.3 for the case of a fully turbulent boundary layer (z,/l = 0). 

In this conncxion the paper by P. S. Granville [lS] may also be consulted. 

These calculations give an indication about the effect of friction on lift. The 

displacement of the external streamlines caused by the bountlary layer modifies 

the pressure distribution on an acrofoil and causes the experimental value to become 

lower than that givcn by potential theory. This loss of lift was calculated by I

770 XXV. Determination of profile drug e. Lomen in the flow through cascndca 771 

Pig. 25.7. Prcssrlrc clist,rihulion nnd position of point of scpnrntion or n tnrhnlcnt, houndery 

l:rycr 011 tho I)lntlo of n 1.urlho mscntlo for two tlilTercnt nnglca of inflow, afler F.W. Iticgcla [44] 

The work ill rnf. [4G] shows how t,o employ the method outlined in Sec. XXVtl 

it\ ortler t,o c.nlt:nl:~tc thr. losscs of n two-clirncrmsionnl cascade at varying angles of 

illllow. N. Srllolz allti I,. Spcitlel 1601 syst,cmatixecl such calculntio~ls ancl comparctl 

tht!n~ wit.l~ rxlwrin~cntal results. 

'I'ltn vrloc.it,y tlisl.ril)t~t,ion irnn~rrlintcly Idtirlc\f the exit plane of the cascnrle 

shows stmng tlc~~,rrssions wllicl~ st.cnl from t,hc bountlary layers of the iutlivitl~lal 

l~l:~.rins. 'I'url)ulrnlf ntixing rausrs t.I~csc velocity tliffernnce~ to sn~oofh out further 

clownbl.rrnm, thl~s giving rise to an ntltlitionnl loss of energy. 'l'l~c amount of los.9 

tlltc lo n,il:ing r:in IIC t:v:~.I~t:~,,c.tl wit.11 tltc nit1 of t.hn ~nomtmtum tl~rorrm. When 

tlt+~rmi~~i~~g the t.ot:l.l loss it1 the flow tl~ro~lgl~ cascncl~s, it is necessary t,o take Chis 

mixing loss into account in atltlit,ion to the loss of rnergy in the 11ountl:ar.y layers 

of the individual blades. Thus a calculation of losses in n casc:dc cot&trs of the 

following three partial calculations: 1. 1)eternmination of the ideal, potrnti:tl prcssure 

distribution around tho contour of the blacles. 2. Calculations of t,lw (1:rminar or 

turl)rrlont) hountlary layer at a blntlc. 3. DcLcrrninat.ion of the losses clrle to mixing 

in t.hc wake bchintl the cnscadc. 

The tot,al amount, of losscs nssociated with a cascade is best spccifictl by intlicatjing 

thc tliffcrcncc Ag in t h total prcss~~rcs between the nnclisturbctl flow in front 

of IJtc rascnclo rind Lhc "srnoot,llctl out" nclunl flow far bol~ind it. 'Jll~us 

where p2' and wz' denote the pressure and vclocit,~~ in the real (i. e. alfcctctl I1.y 

losses) flow far bchintl the cascatlr, respcctivcly. 'rhcsc sl~ould bo clist.ingrtisl~otl from 

thc values p, and iuZ, respcctivcly, wltich refer to ideal (losslcss) flow. It is convenient 

to render thn Lotd loss Ag tlimensionlcss with reference to thc dynnmio Ilond 

formed with the axial velocity component w,, = wl sin P, = toz sin P,, as it tlct.crmines 

the mass of fluid which passes through the cnscadc. For reasons of rontillttit,y 

iB vnluc must be tho same in front of ns bclti~~tl tltc cascade. We tlto~ il~l.rotlttc:c 

the following coefficient: 

t, = - -9 - 

few,," ' 

Some results of the systematic invest,igntions on cnscntles, rarrircl o~tl :it t he 

Braunschweig Engineering University 1601, alao [49], are shown in Fig 25.8 These 

represent a comparison between measured and calculated values of the loss coefficient. 

All blades were derived from the aerofoil NACA 8410. The variable parameters 

Fig. 2.5.8. LORS co~flicicnt tt from eqn. 

(25.33) in terma of the deflexion coefficient, 

dn = '~tl$f/W,, for turbine cascndes 8.. 

with difircnt solidity ratios t/1, after 

[49J. Men.911rc1ncnta and calculntio~~a by 

N. Scllolz nnd L. Speiclrl [GO] 

Itlarlr prolllc. NArA RllO 

Rcyeolds na~olrer R = w,llr - h x 10' 

.- .- -. . -~ - 

t In t,hc dmign of stcnru turbines it is 11sun1 to employ n aelocit?y coc//icient, 111, wl~icll 

is clrlincd 

as the ratio of thc rod exit vclociLy to its veluc in ideal flow, so tllet y, -- tu',/tc~,. Co:~srq~~rn(ly, 

thc two cocflieicnh snlisfy the rcl:~tion Lt = (I -i/~2)/sinZ 

P2.

XXV. 1)obrlninalion of proRlo drag 

included the solidity ratio 111 (= 0.6, 0.76, 1.0 and 1.25); the blade angle was Ps = 

30" (turbine cascade). The loss coefficient defined in eqn. (25.33) is seen plotted in 

terms of the dcflcxion cocfficient or deflcxion ratio 

ad = Awd/?l~,, , 

where AN?, tlcnolcs 1.h~ tm.r~svcrsc compor~cnt of vclocit,y (i.'e. vclocit,y in circomferential 

direction) created by the cascade. If we first center our attention on the 

middle range of the polars (adhering boundary layers), we notice a steep increase in 

the loss coefficient which occurs as thc solidity ratio decreases. The reason for it lies 

in the fact that the number of bladcs per unit of length of the circumference is larger 

when the pitch is small than when thc pitch is larger. To a first approximation the loss 

coefficient is proportional to the number of blades. At the right and left edge of the 

polar we observc a sudden and large increase in the loss coefficient. This is due to flow 

separation on the pressure side (left end of curve) or on the suction side (right end of 

polar) of the bladc. In the latter case, an increase in the flow angle causes the admissible 

load on the blade to be exceeded. It is remarkable that the polar curves displace 

themselves in the direction of largcr angles of deflexion as the solidity ratio decreases. 

The ~ncasurcmcnts a.nd the calculat.ions were carried out for a Reynolds number 

R -. to, llv - 5 x 10% .TIC calculations wcrc performed on tho assumption that thc 

bountlnry hycr was turhulcnt. all along the hlatlcs. In the oxpcrin~ontal nrmngcrncnt 

the boonrlary layers wcrc made turbulent by the provision of tripping wires ncar 

the leading edges. Thc calculated and measured values of the loss coefficient show 

very good agrccmcnt with cnch other. Furthcr examples and comparisons between 

theory and experiment are givcn in [47, 631. 

Wake: A very dctailcd experimental investigation of the flow in a turbulent 

wake bchind a cascade of blades is described in a paper by R. Raj and B. Laksh- 

minarayana [42]. Measurements included determinations of the velocity distribution, 

intensity of turbulencc, and of the apparent Reynolds stresses in the wake at different 

distances from the cascade. It has transpired that the wakes are not symmetric up 

to a distance (314) 1 bchind the blades in cascades which turn the flow. The decrease 

in velocity downstream from the cascade exit section is considerably slower than at 

a flat plate, behind a circular cylinder or downstream from a single aerofoil at zero 

incidcnce. 

Jet flnp: The angment,at,ion of the turning angle A,'? = Dl - p2 of colnpressor 

cascades by a jet flap has been investigated by U. Stark,[G4a]. 

2. 111flue11ce of Reynolds number: Thc chnngcs in the aerodynamic coefficients of 

n cascadc protlucctl by a chnngc in l,hc itcynol(1s nnmber arc import,n~~l whcn it 

becomes ncccssary to apply the results of tcsts on models to thc design of a fullscale 

turho~nachine. This effcct is excrtetl principally on the loss coefficicr~t, and 

the can be found discussed in a sizeable umber of publications 15, 41, 651. 

From tho physical point of view, the cffect of Rey A" olds number on the loss coefficient 

of a two-tlimcnsional cascndc is analogous to that of the skin friction of a single aerofoil: 

hccausc in eit.her case the cffect originates in the boundary layer. The losses 

sufTcred by the cascatlo stem mainly from the boundary layer if the pressure distribut,ion 

dong a I)latlo in n cascadc is such that no imporhr~t scparntions occur. 

'l'hry nro t,hcn ak:rl.otl hy tllc Ttcynoltls number in aho~lt t,ho same way as t,he skin- 

I 

friction coefficient of a flat plate at zero incidence arid are proportional to R-'12 for 

laminar flow, becoming proportional to R-lI5 in turbulent flow. In both cases, the 

Reynolds number is formcd with the blade length, I. Thc dcpcndcnccof t.hc loss cocffi- 

cient on Reynolds number in the absence of separation can be determined by calcu- 

lation with the aid of a method proposed by K. Gersten [15]. A rcsult of this kind is 

seen displayed in Fig. 25.9. The diagram describes thc variation in the loss coefficient, 

A g 

(12 = - 

;be4 ' (25.84) 

of a cascade consisting of thick, strongly cambered bladcs, over a considcrablo range 

of Reynolds numbers, that is from Rz = wzllv = 4 x lo4 to 4 x 105. I-Icre Ag dcnotes 

the loss in stagnation pressure and iuz is the exit velocity. In order to providc a 

comparison with mensurcmenta, thc diagram contains a thcorctical crrrvc which tnkcs 

into account separation losses computed with the aid of Ref. [GO]. As far as the position 

of the point of transition is conccmed, the calculation was based on the expcrimentally 

verified circumstance that the boundary layer on the pressure side of a 

blade remained laminar as far as the tmiling edge, whcreaa that on the suction side 

undcrwcnt transition at the point of minimum pressure. Thc dingmm in Pig. 26.9 

demonstmtcs t.hnh thcrc cxist.~ cxccllcnt ngrccrncnt I~cLwccn oalcnlntiotl nnql rtltrclrrilrament. 

The magnitude of the losses is strongly influenced by t.he position of the point of 

transition. As thc Rcynolds numbcr is incrcnscd, tho point of transition movcs for- 

ward and this lengthcns the turbulent portion of the boundary layer and causes the 

losses to increase. The forward movemcnt of the point of transition is enhanced by 

increased roughness 1131 or by an increased turbulencc intensity [a], as one would 

expect to find in a turbomachine. At very low Rcynolds numbers the boundary laycr 

can separate before transition has occurred in it thus causing a large incrcasc in thc 

Fig. 25.0. Loss coefficient of a turbine cascade, eqll. (25.34). in brr~rs of the Rryl~oltls tl~it~~bcr Rz, 

after I

XXV. Dctcrminntion of profile drag 

a) rrensurc distribution for vnrioun Rry- 

noltk numbero nt Mz -- 0.3 

b) Loss cocfficirnt (12 from cqn. (25.34) 

nrr n function of tho Reynolds numbcr Rz 

Fig. 25.10. Aerodynamic coefficients of a turbine cnscnde its a function of the Itrynoltls number as 

menm~red by H. Schlichting nnd A. Dns [52, 531 

n) I'rcauure dist.ribut,ion for vnriorls Rey- 

tlolds nulnberu nt MI = 0.7 

I 

b) 'IJous coerficieut. Ira from C~II. (25.34) 

as a function or the Mach number Mz tor 

various vnlura of the ltcynoldu number 

vig. 25.1 1. Aplclrlynnnlir. rotffirirntn of n t,t~rl,inr wnrnclr its n funrtion of Mnvb nr~mhrr nn measu- 

T& by 11. Schlicl~ting and A. I h [62,53] 

c. Los~es in the flow tlwnlgh cnscndos 775 

loss cocfficient under certain circumstances. This large increase in the loss coefficient 

at low Reynolds numbers is illustrated in Fig. 25.10b which refers to a turbine cw- 

cade. At larger Reynolds numbws, Rz = 5 x 105, the transition is spontaneous nnd 

the losses are small. At moderate Rcynolds numbers, Rz = 1 x 105, thcrc is Inn~innr 

separation followed by turbulent re-attachement. Thus under the boundary layer 

there forms a so-called separation bubble and the loss coefficient increases consider- 

ably. At very low Reynolds numbers, Rz = 0.5 x 105, the laminar layer separates 

and stays separated to t,he end of the blade. The losses increase by a large amount 

once more. 

The details of the separation of the boundary layer are once again mirrored in 

the pressure distributions plotted in Fig. 25.10a for three valucs of the Itcynolds 

number. The extent of the scpnration bubble depends strongly on thc Reynolds 

number and on the intensity of turbulence of the oncoming stream. See [8,20,28,37, 

43, 57, 601, and the pnpcr by R. Kiock [30]. C/. W.B. Robcrts 1431. 

In conjunction with our discussion of the cffect of thc Rcynolds number, it is 

necessary to stress that under certain circumstances the surface roughness can have 

a large influence on the losses. In addition to enhancing transition, roughness can 

also directly increase the losses. This occurs whcn thc protubcrnnccs cxccrd n crrlnin 

admissible value; see [3, 561. 

3. Effect of Mach numher : The preccding results concerning the Loss coefficient of 

cascades refer to incompressible flows (M < 0.3). The effect of compressibility can 

be said to set in at M > 0.4. An example of this effect is shown in Fig. 25.11b. The 

plot represents the loss coefficient for a cascade producing a small angle of turn in a 

subsonic flow. The Mach number Mz is the independent variable and t,he thrce curves 

refer to three different Reynolds numbers. The pressure distribution for M = 0.7, 

Fig. 25.118, shows that at Rz = 4 x 105 the loss coefficient increases sharply as the 

Mach number is increased. The sharp increase occurs as a rcsult of shock formation 

in region8 where the local value of the velocity of sound, cp, crit, has bcen exceeded in 

the flow. For the two lower Reynolds numbers, Rz = 1.0 x 105 and Rz = 2-0 x 105, 

the pressure distribution points to a separated flow. The results displayed in Figs. 

25.10 and 25.11 demonstrate that the Mach number exerts a deep influence on the 

flow through cnacades in the range of Reynolds numbers from R = 104 to 105, in 

addition to the large effect of the Reynolds number itself. The preccding measure- 

ments were performed in the high-specd cascade wind tunnel in Brunswick [54] in 

which the Reynolds number and the Mach number can be varied independently. 

The diagram of Fig. 25.12 illust.mt,es the effect of the Mach number on the loss 

coefficient of a cnscade that produces a large angle of turn in the flow. 'rhc cnscadc 

was designed for incompressible flow. The loss coefficient remains nearly constant at 

the value Ct2 M 0.03 up to M2 = 0.7; it increases sharply as the Mach number is 

further increased. The reason for this behaviour is clear from Fig. 25.13 in which it 

is possible to discern the existence of shock waves on the suction side of the blade. 

These cause separation of the boundary layer. 

The effect of the Mach number and of the turbulence intensity on the loss coef- 

ficient of cascades has been studied in two theses presentcd to the Engineering 

University at, Rraunschweig by J. Bahr [2] and 1%. ITcbbeI 1211, respectively. Rcfe- 

rence [50] may also be consulted on this point.

776 

XXV. 1)ctormination of profile drag 

Fig. 25.12. Loss coefficient of a turbine 

cascade, t tz from eqn. (25.34) in brnis 

of the Mach nnmber Mp aftm 0. 

Lawaczeck [34] 

blnclc nnule: Ps = 56'; snllclily rnt,lo: 111 -0.81 

n~iulr nl inlet: /1, = 00'; Iley~~oidn nrtmbcr: 

A, - 0 x 10' 

Fig. 25.13. Transonic Row through n 

tmrbinn cancndc. Phot,ogrnph obtain- 

nd wiI.11 the aid of Schlieren tnetlrod 

by 0. Lawaczeck and H.J. Neinernann 

[32]. Exposure 20 x 10-9 sec. The 

strong shock waves on the suction side 

of the nerofoil cause separation and 

hence large losses, see also Fig. 25.12 

In modern times, the development of steam turbines of increased powcr density 

has caused the outer bladc sections of the low-pressure stages to operate in the transonic 

vclocitpy rregimc. This made it neccssnry to undertalte systematic investigations 

into l,hc behnvio~tr of t,ransonic turbine blades. Here tho Mach nrimber of the appronching 

stream is lower than unity (MI < l), ,whereas that at exit exceeds it 

(Mz , I); cf. 1.11 1. Rcfcrcncrs 133, 341 contain an~accor~nt of t,ransonic flow across 

cn.scadc with n Inrgc angle of turn. 

11. IInas and IT. Maghon [20a] give a comprehensive account of the practical 

appli~at~ions of these research results on flow through cascades as they relate to 

modern developments in steam and gas tmbines. 

References 

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778 XXV. Detcrn~inntion of profilc drag Jteforenccs 779 

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len grillm d'nuhea. Technique et, Science AQronautique 5, 227-232 (1966). 

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Lomnx, IJ., and Steger. L.: Relaxation methods in fluid mechanics. 

Wieghardt, K.: Experiments in granular flow. 

Christiansen, W.H., Rumell, D.A., and Hertzberg, A. : Flow I~mrs. 

Widnall. S1i.E.: The structure and dynanlica of vortex filementa. 

Tien, C.L.: Fluid mechanics of heat pipes. 

Koh, R.C.Y., and Brooks, N.H.: Fluid mechanics of waste-water disposal in the ocean. 

Goldsmith, H.L., and Skalak, R.: Hemodynamica. 

Ladyzhenskaya, 0.A.: Mnthcmaticnl nnalyais of Navier-Stokes equntiona for incontprcsaible 

liqnida. 

Mnxworthv. T., nnd Ilrowand. F.K.: Experinlent8 in rotaLing and stratified flowa: Oceanographic 

application. I 

Lnufer, J.: New trends in experimental turbulence rwearch. 

Itnichlen. F.: The cflect of wavw on rubble-mound structorea. 

Csn~acly. G.T. : Hydrodynan~ics of large lakes. 

Vol.8 (1.976) 

ROIIRR, 11.: Hydranlirs' Int*?st golden age. 

Bird, R.B.: Useful non-Newtonian models. 

Petorlin, A.: OpLical effects in flow. 

Davis, St.11.: The stt~bility of time-pcriodic flows. 

Ccrmak, J.E.: Aerodynnntics of buildings. 

Fischer, H.B.: Mixing and dispersion in esttmrieq. 

Hill, J.C.: IIomogcneons turbulent mixing with cl~ctnicnl reaction. 

Pearson, J.H.A.: Instability in non-Newtoninn flow. 

Reyt~olda, W.C.: Compntntion of turbulent flows. 

Cotnte-Rallot, G.: IIot-wire anc~nomctry. 

Woocling, R.A., and Morrl-St-ytonx, H.J.: Multiphnsc flnid flow tllrongh prons tnrdin. 

Inmnn, DL, Nordstrom, CI1.E.. and Flick, 1t.E.: Currcmtn in ai~htnnrine rnnyons: An nir-sea- 

Innd inhwwtion. 

Rcshotko, E.: Boundary-layer stability and tmnsition. 

Libby, PA, and Williams, F.A.: Turbnlent flow^ involving chentirsl reactions. 

Ruannov, V.V.: A blunt body in a snpersonic sham. 

Vol. 9 (1977) 

Jones, R.T.: Recollections fmm an earlier period in Americnn aeronautics. 

Pipkin, A.C., and Ttmncr, R.1.: Steady non-vincnmetric flown of vincoclnntie Iiquitln. 

Bradshaw, P.: Cotnprerrsible turbulent shear layers. 

Daviduon. J.Jr., Hnrri~on, I)., and (?nede~ 11e Ct~rv~dho, ,J.It.l~.: 011 tho Iiq~~idlikn 

Ilt~iclizcd bccln. 

Tani, I.: History of bonndary-layer theory. 

Williarn~, 111. J.C.: Jncotnpreasible boundary-layer neparation. 

Pleaset, M.S., and Prosperetti, A,: Wobble dynamics and cavitation. 

Holt, M.: Underwater explosions. 

Zel'dovich, Y.B.: Hydrodynamics of the universe. 

Pedley, T.J.: Pulnionary fluid dynamica. 

Canny, M.J.: Flow and transport in plan&. 

Spieltnan, L1.A.: Particle capture from low-speed laminar flows. 

Saville, D.A.: Elcctrokinotic effocta with small particlcs. 

Brenncn, Ch., and Winet, 11.: Fluid mechanics of propulsion by rilin and flagella. 

Hiitter, U.: Optimum wind-energy conversion systems. 

Shen, Shan-fu: Finite-element nlethods in fluid mechanics. 

t~~hvior of 

Ffowcs M'illintns, J. E.: Aeroacoustic~. 

Belotaerkovskii, S. M.: Study of the unsteady aerodyna~nica of lifting surfncw wine, the r.otnprrt.w. 

Vol. 10 (1978) 

Binnie, A.M.: Some notfa on the st,ndy of fluid ~nechanics in Canibritlge. 

Tnck, 1E.O.: I-lytlrorlynantic problenls of ship in rcst.riclotl 1vnt01.e. 

Bird, C.A.: Monte Ca.rlo simulation of gnsflows. 

Berninn, N.S.: Urng reduction hy polyniors. 

Ryzhov, 0. S. : Viscons transonic flows. 

Griffihh, W.C.: Dust explosions. 

Leith, C.E.: Objective methods for weather prediction. 

Callancler, R. A,: River meandering. 

Dickinson, R.E.: Rosshy waves - Long-period oscillation^ of ocenns nnd atmospl~rrca. 

Jenkins, J.T.: Flows of nomatic liquid crystals. 

I,eibovich, S.: The structure of vortex breakdown. 

Lawn. E. M., and Livruey, J. 1,. : Flow tltrongh Rcrrons.


Sherman, I?. S., In~bcrger, J , and Corcon, G.M.: Turbulence and mixing in ntably stratified waters. 

Patkrson, G.S., Jr.: ProspecLq for compultional fluid ~necshanics. 

Taub, A. 11.: Lielntivistic flnirl tnechanirs. 

Rcethof, G.: 'I'~rrl~r~lr~~rc-gc~~rrntrd noine in pipe flow. 

Ashton, G. D.: ltivor ice. 

Mei, Chiang C.: Nwncrical mcthods in water-wave diKraction and radiation. 

IZcllrr, 1-1. R.: Numerical mrthods in bortntlery-layer theory. 

B~~sse, F. H: Mngr~etol~ytlrodyt~alnics of the e.zrt,h'u dynamo. 

A 2. Advances in Applied Mecl~anics, Academic Press, New York 

(only ront.ribrttions to fluid mechanics listed) 

Vol. 1 (1948), od. by R. von Misea and Th. von Krirmhn, 

Dryden, H.L.: ltcront advances in t.hc mcchanics of bonndary layer flow. p. 2-40. 

Unrgcrs, .J.M. : A mathcmat~ic tnodcl illr~rrtrnting tllc theory of tnrhulcnco. p. 171 -1!)0. 

von Mises, R., and Schiffer, M.: On Horgn~nn'o integration method in two-dimensional compressible 

fluid flow. p. 240-285. 

Vol. 2 (1062), ed. by R. von Mises and Th. von KhrrnQn 

von Khrmbn, TI)., and Jh, C.C.: On the st,nt.i~t,ical thcory of iuotropic tnrbnlence. p. 2-19. 

Kuorti, G.: The Intninar boundary lnycr in romprc.ssiblc flow. p. 23-92. 

l~olubarinova-Korhina, Y.Y.: Theory of filtration of liquids in porous media. p. 154-225. 

Vol. 3 (1953), ccl. by R. von Mises and Th. von KhrnAn 

Carricr, G.F.: I3onndnry lnycr prohlcmu in applicd mcchanics. p. 1-19. 

Znltlnatnni, 0.: Tho one-dimcnsionnl iticntropic fluid flow. p. 21-59. 

Frcnkicl, F.N.: T~trbulcnt diffnaion: Mcan concontrat,ion distribution in a flow field of homogeneons 

turht~lcncc. p. 62- 107. 

I,udloff, H.P.: On acrodynamics of blastn. p. 100-144. 

Gudcrlcy, G.: On t,l~c prescncc of shock^ in mixcd suhaonic-sopcrsonic flow patterns. p. 145- 184. 

Rosenheacl, I,.: Vortcx systcms in wakes. p. 185-195. 

Vol. 4 (19,5G), cd. by 13.1,. Dryden and Th. von IZhrn~hn 

Clauscr, F.H.: Tho turbulent bonndary layer. p. 2-51. 

Moore, F.K.: Thrco-dintensional boundary layer theory. p. 160-228. 

1'01. 5 (195R), ed. by H.L. Dryden and Th. von KSrmhn 

Fabri, J., and Sicstrunck, R.: Supersonic air ejectors. p. 1-34. 

Van Do Vooren, A.I.: Unsteady airfoil theory. p. 36-89. 

Fricman, X.A., and Kulsrud, 1t.M.: Problems in hydromagncticu. p. 195-231. 

Wcgnnrr, 1'. I'., ntttl Mnrk, L.M.: Condonsnl.ion in snpcr~onic and 11ypcrs011ic wind tnnt~ols. 

p. 307 -447. 

Vol. 6 (IBGO), ctl. hy F1. I,. Drydcn and Th. von Iprcnnir flow over slender Me aeiated with poser-bu shocks. p. 2-3. 

Rnymoncl. H., and ltobcrt,, 1j.H.: Somo clctnentnry problems in tllngr~eto-llydro(Iy~~;l~tlil.n. 

p. 210-319. 

VoZ. 8 (IBG4), cd, by H.L. Drydon and TI). von Khrmhn, 

Sears, W.R., and Resler, E.L.: Magneto-arrodynamics flow past hodics. p. 1-68. 

Markovitz, H., and Coleman, B.: Incon~pressible second order fluida. p. 69- 101. 

Ribner, 1I.S.: The generation of sound by turbulent jets. p. 104- 182. 

Run~yantsev. V.V.: Stability of motion of solid bodies with liquid filled cavities by I,yap~~nov's 

method. p. 184-232. 

Moineev, N.N.: Jntrodrrction to thc theory of oscillations of liquid-rontnining botlira. 11. L?:!:! -28!) 

Val. 9 (196G). etl. by O.G. Clwrny et al. 

Drnzin, P.G., and Howard, L.N.: Hydrodynsn~ic stability of pnrallel flow of invinoitl Hnid. 

p. 1-89. 

Moiseev, N.N., and Petrev, A.A.: The calc~~lation of free oscillationu of a liqrtid in a ~notionlonn 

container. p. 91 - 154. 

Vol. 10 (1067), ed. by G.G. Cherny et al. 

Lick, W.: Wave propagation in real gases. p. 1-72. 

Paria, G.: Magneto-elasticity and magneto-thrrmorlnsticity. p. 73-112. 

Vol. 11 (1971). ed. by Chin-Shun Yih 

Yao, 1'11.. nnrl '~RII, W.V.: IIydrodynamic~ of ~\vinlmin~ MI~R rind CO(.~CPIIIIU. 1). I ti:!. 

Fung, Y.C.: A survey of the blood flow prohlenl. p. 65-130. 

Sichel, M.: Two-dimensional shock structure in transonic and hyperuonic flow. p. I01 -- 207. 

Vol. 12 (1972), ed. by Chin-Shun Yih 

Harper, J.F.: The motion of bubbles and drops through liquids. p. 69- 129. 

Gerrnnin, P.: Shock waves, jump relations and structure. p. 131 -104. 

Liu, V.C.: Interplanetary gas dynamics. p. 195-237. 

Vol. 13 (3973), ed. by Chia-Shun Yih 

Veronis, G.: Large scale occan circulation. p. 2-02. 

Wehausen, J.V.: Wave resistance of ships. p. 03-246. 

Kuo, H.L.: Dynamics of qunsigeoatrophic flows and instability theory. p. 248-300. 

Vol. I4 (1974), ed. by Chin-Shun Yih 

Stewartson, I


A I). Progress hr Aeronautical Sciences, Pcrgamon Press, London 

(only contributions to fluid mechanics listed) 

Vol. I (lSIil), ed. by A. Forri. D. Kiichernnnn nnd L.H.G. Sterne 

Mmkell, E.C.: On t.lm principlw of acrodynnrnic design. p. 1-7. 

Legendrc, R.: Celcrtl den profils d'nuhcs pour turbomachines t.ranuuoniques. p. 8-25. 

Fennin, M.: l,n thkorie den 6couletnent~ b potent.icl I~omogPno ctses applications an cnlcul den ~ 

en rhgimc ~uperaonique. p. 20- 103. 

C R 

Becker, E.: IrtstationRre Grcnzschiclltcn hinter Verdicl~tungsat~sscn ontl Expansicna\vcllen. 

p. 104- 173. 

Goldworthy. 1P.A.: On 1110 tlynamicn of ioni7.cd gns. 1). 174 --205. 

~nrron, C.l1.E., and Randall, D.C.: The theory of sonic bnngs. p. 238-274. 

Vol. I1 (1962), ed. by A. Forri, D. I

Clementa, lt.R., and Mad, D.J.: The rcprescntntion of sheet8 of velocity by discrete vorlices. 

p. 129-146. 

Chuc, S.Jg.: Prrsst~re probes for fluid n~rasnrcmentn. p. 147-223. 

Tanner, M.: Reduction of bnse drag. p. 360-384. 

Cnrricre, P., Siricix, M., and Delory, J.: MBthodes de calcul dcs Bcoulcment turhulents d6coll(?c~ 

ct suprsonique. p. 385-429. 

Vol. X VII (1976/77), ctl. hy I>. ICiichemann 

Broadbent, E.G.: I'lows with heat addidion. p. 03-107. 

Jones, I). S. : Thc mnthcmnticnl theory of noise ~hiolding. p. 149 -220. 

Broarlhcnt. 1C.G.: Noisc shickling for aircraft.. p. 231-268. 

Glms, J.J.: Shock waves on earth nnd in spncc. p. 260-286. 

Tanetla, S.: Visunl study of unsteady acpnratcd flows around bodies. p. 287-348. 

A 4. Advaneen in Aeronnntical Sciences 

I Vol. I ond Vol. II. T'roccrdings of the First Tntcrnationnl Congrcss in Acronn~~tical 9 CICtlCPR, ' 

Mndrid, 8 -13 Srptetnbcr, 1958. I'crgamon Press, London, 1959. 

Vol. III and Vol. I V Procredings of tho Second Tnkrnntional Congrens in t\cronautical Sciences, 

Zi~rich, 12 - 16 Septenrhcr, IWO. Pergan~on Press, I,ondon, 1962. 

Vol. V: Proceedings of tho ThirdCongrms of the lntornntional Council of the Aeronnnticnl I 'I rlcnces, ' 

Stockl~oln~, 27 -31 Augrwt, 1962. Spartan 13ooks. Wnuhington J).C., 1964. 

Vol. VI Proreedings of the Fourth Congress of the 1ntcrnat.ional Councrl of the Acron~utiral 

Sciencw, l'nris, 24-28 August, 1964. Sparlnn Books, Wnshington D.C., 1065. 

Two Volnnie~ Aer~spn~ePr~rccdings 1966: Proceedings of the Fifth Congrcas of the International 

Coimcil of tho Acronnutirnl Scicnccs, London, 12-10 ScpIamhc.r, 1066, cd. by Tho Royal 

Arronnut.irnl So(-irLy nnd McMillnn, Imldon, 1067. 

R. Ilnedhooks, Collected Pnpern, Applied Mechnnies Congresses 

Princeton University Serie~ on High Spccd Acrodynnmics and Jet Propulsion, ~rinceton University 

Press, I955 -1964, Vol. 1 to XI1 

Vol. 1 (1965), cd. by 1P.D. Rosnini: Thermodynamics and physics of matter. 

Vol. 11 (1.9*56), rtl. by 11. Lcwi~. R.N. Pcnac, 11,s. Tnylor: Con~hustion processes. 

Vol. 111 (1958). cd. by H.W. R~nn~nns: Funrlan~entals of gns dynan~ica. 

Vol. I V (l964), ctl. by F.K. Moore: Theory of lnrnil~nr flows. 

(=onlril~ulions h?/: 

Moore, l7.lC.: 1ntrotluct.ion. 

1,ngerslrorn. P. A. : 1mni11nr llow l.hcory. 

Mngcr, A.: TI~reo-ditt~ct~nional Intninnr boundary Iaycrs. 

Rott,, N.: Theory of tin~c-dcpcntlcnt laminar Bows. 

Moorn, F.IC.: Flypersonic boundary laycr theory. 

Ostrnrh. S.: 1,ntninar flow with body forces. 

Shen, S.F.: Skl)ilit.y of Inminnr flows. 

Vol. V (19.59), ed. by C.C. Lin: Turbnlcnt flows and heat transfer. 

Contrbvtionn 6!y: 

Drytlen, 11.L.: 1'rnnsit.ion frotn laminar to t,urbnlrrlt flow. 

Srhnl)nunr, G.R.: Turbulent flow. 

Bibliography 789 

Lin, C.C.: Statistical theories of turbnlence. 

Yachter, M., and Mayer, E.: Conduction 01- heat. 

Deissler, R.G., and Sabersky, R.H.: Convective heat trnnsfcr and friction in flow of liquids. 

VR~ Driest, E.R.: Convective heat transfer in gases. 

Yuan, S.W.: Cooling by protective fluid films. 

Penner, S.S.: Physical bunis of thermal radintion. 

Hottcl, H.C.: Engineering calculations of rndiant hcnt exchange. 

Vol. VI (1954), ed. by W.R. Sears: General theory of high spced aerodynamics. 

Vol. VlI (I%57), a d by A.F. Donovan and 1I.R. Lnwrcncc: Acroclynnmic componontn of air.. 

craft at high speech. 

Vol. Vlll(1961), ed. by A.F. Donovan, H.R. Lawrence, F.E. Goddnrd, and R.R. Gilruth: 

High speed problems of aircraft and experimentnl methods. 

Vol. IX (1!?54), ed. by R.W. Ladenburg, B. Lewis, R.N. Pease, and H.S. Tnylor: Physical 

rncnsnrementm in gas dynamics and combustion. 

Vol. X (1964), ed. by W.R. Hawthorne: Aerodynamics of turbincs and compressors. 

Vol. XI (1960), cd. by W. R. Hawthorne and W.T. Olson: Ilcsign and perforrnancc of gas 

turbine power plnnta. 

Vol. XII (ISc59), cd. by O.B. Lancaster: Jet propulsion engines. 

Ilnndbucl~ der Phynik, ctl. hy S. IPliiggc, Springer Vorlng, Ik:rli~~/(l;iit,t~i~~l/IIt~itIeII~org 

Vol. VIIIII (I%!?), Stromungsmechnnik I 

Oswntitmh, I

B 2. Collected Works 

Prandtl, L.: (:ean~nn~clto Abhnndlungen zur angcwandten Mechanik, Hydro- und Aerodynamik. 

3 Volumes, etl. by W. Tollmien, H. Schlichting, and H. Gortler. Springer Verlng, 1901. 

von Khrmhn, Th.: Collected works of Theodore von Khrmhn. 4 Volumeu (1902-1951). Butterworth, 

London, 1056; Supplement Volume (1952-1963), von Kkrrnhn Inut,itutc Jthode St. 

Gcni.se, Belgium, 1075. 

Taylor, G.1.: The scientific papers of Sir (hoTTrey Ingrnm Taylor. 4 Volnmes, ccl. by G.K. 

Batchelor. Cambridge University Press, 1W8-1971. 

Taylor, G.1.: Surveys in mechanics. The '2.1. Taylor 70th anniversary volume, ed. by G.K. 

13atchelor, and R.M. Daies. Can~bridgc, 1986. 

B 3. Applied Mechnnirs Congrcr~es 

Uortler, H., and Tvlln~ien, W. (ed.): Fiinfzig .hhre C.ro~~zscl~icl~tforscl~ung. Eine Fcst.schrift in 

Originnlbeitriigen. Vieweg, Braunschweig. 1955, 499 pp. 

Giirtler, It. (od.): (:rct~zscl~icl~tforschung. IUTAM-Symposium, Freiburg/Breisgnu. 1957. Springer 

Verlng, 1958, 41 1 pp. 

MQcnnique de In Tnrhule~~ce, Marseille, 28 August-2 September 1961. Colloqrtes Internationn~lx 

du Centre Nntionnl de la HBcl~crche Scicr~tifique, No. 108, Paris, 1902, 470 pp. 

Proceedings of thr 10th International Congrese of Applied Mechanics, Stresa, Italy, September 

1900, ed. by F. Rolla and W.F. Koiter. Elsevier Publishing Co., AmsterdnmJNew York, 

1962, 370 pp. 

Proceedings of the llth International Congrrm of Applied Mechanics, Miinchen, Germany, 

August 1964, ed. by H. Cortler. Springer Verlag, Berlin, 1906, 1190 pp. 

Proceedings of t.1~ 12th Int,ornnt,ional Congress of Applied Mechnnics, Stanford University, Cal.. 

USA, August 1968, ed. by M. IIethnyi nnd W.G. Vinccnti. Springer Verlag, Berlin, 1969, 

420 pp. 

Proceadingu of tho 13th Intcrnntionnl Congrom of Applied Mcchnnics, Moskan University, Augrtut 

11172, cd. hy lC. Bocltc~r nntl (;.I

792 Bihliography 

13ird, R.R., Slewnrt, W.l

704 Bibliogrnphy Bibliography 706 

White, F.M.: Viscous fluid flow. McOrnw-Hill. New York, 1974. 

Walz, A.: Striimungs- nnd Temperat~~rgrenzscl~ichten. Brnun-Vcrln . Knrlsruhe, 1966. English 

trenslntion: Ihundnry lnycrs of flow nnd tmnpernture, by H. ,J. &or, MIT Presn, Cnmbridgc, 

Mass., USA, 1969. 

D. Bench-mark publications (in chronologicnl order) 

Prnndtl, I,.: Uher IWi~nigkeit~hewegung bei schr kleiner Rcibung. Vcrhnndlungen IIIrd Intern. 

Mnth. Kongrem Heidelberg 1004, 484-491 (1904), Teubner, I~ipzig, 1905. English trana- 

lation: NACA Menlo No. 452 (1928). Reprinted in: Vicr Abhnndlungen zur Hydro- und Aero- 

dynamik, Giittingcn, 1927; Coll. Workn, Vol. 11, 575-584. 

Blnaiun, 1-1.: Grcnzsc11icht.en in I~liissigltcitcn mil, klcincr Rcibung. Dins. Cottingcn 1907. Z. Mnth. 

11. I'hya. 56, 1-37 (1908). Englinh tmnnlntion: NACA TM 1256. 

Boltze. E.: Orcnzscl~ichten nn I~otntionskiirpern. Dins. Gothgen 1908. 

IIierncnz. K.: Dic Grcnzschicht an cincm in den glrichfiirn~igcn Flussigkeitsst.rom eingctm~chtrn 

gcrndcn Kreiszylinder. Diss. Cot.t,ingen 181 I . Dingl. I'olytechn. J. 28, 321 -410 (191 1). 

Prnndtl, I,.: Der Luftnidcrotand von Kugeln. Nnchr. Ges. Wiss. Giitt,ingcn, Mat,h. Phys. Klnasc, 

177---I90 (1914); Coll. Works, Vol. 11, 597-608. 

von Khrnm&n, Th.: Ubcr laminare untl turbulcnte Rcibung. ZAMM I, 233-262 (1921). NACA 

'I'M 1092 (1946). 

Pohlhnusm, K.: Zur nriherungsweisen Intrgrntion der Diffcrcnt~inlgleicl~u~~g der laminaren Grenzachicht. 

ZAMM 1, 252-308 (1!)21). 

Prnndtl, L.: Ucmrrkungen iiber die IPutatehung tler T~~rbnlcnz. ZAMM I, 431 -436 (1921); Coll. 

Works, Vol. 11, 687-690. 

Tietjens. 0.: Beitriige zur Eut,stehung dcr T~~rhulcnz. Disa. Giittingen 1922. ZAMM ti, 200-217 

(1925). 

Burgers, J.M.: Tho motion of n flnid in the boundnry layer nlong n plnne smooth surface. Proc. 

First Intern. Congress p\ppl. Mcch., Tklft, 113- 128 ( 1924). 

Betz, A,: 15in Vcrfnhren znr direkkn Errnit,t,lrnlg des Profiln.idcrstnndcs. ZFM 16. 42 (1925). 

Prnndtl, L.: Uher die nrtsgcbildctc Turbulenz. ZAMM .5, 136-130 (1925); Vcrhnndlungen 11. 

1nt.crn. Kongrcsn Angew. Mcchnnik. Zurich, 02-78 (1926); Coll. Workn, Vol. 11, 714-718. 

Tollmien, W.: Bercchnung turbulent,er A~~shrcitung~vorgnr~ge. ZAMM 6, 468-478 (1926). 

Prnndtl, I,.: The generation of rorticcn in fluids of smnll viscosity. 15th Wilbur \Vright Memorial 

I~cture, London 1927. J. Roy. Aero. Soc. dl, 720 (1927). Sce also: Oie Entstcl~ung \.on 

Wirbeln in einer Fliisoigkcit mit klciner Reib~n~g. Z. lil~~gtcchn. Motorluftscl~. 18, 489 -4!)(i 

(1927); Coll. Workn, Vol. 11, 75%-777. 

Tollmien, W.: Uber dic Entstehung dw Turbulenz. I. Mit,teilung Nnchr. Geu. Wiss. Giittingen. 

Mat.11. Phys. Klnsse, 21--44 (1929). NACA TM 609 (1031). 

Schlichting, H:: Ubcr das cbene Windschattenproblem. Dim. Giittingen 1930. 1ng.-Arch. 1, 

633-571 (1930). 

Nikurndnc, J.: GcsctzmiiBigkciten der turbulenten Stromung in glntten Rohrcn. Forsell.-Arb. 

1ng.-Wen. Heft 350 (1$)32). 

Taylor, (:.I.: The t,ransport of vorticity and heat through fluida in turbulent mot,ion. Appendix 

by A. Fngc and V.M. Fnlkncr. Proc. Roy. Soc. 135, 085-705 (1032). 

PrnncltJ, 1,:: Newre l5rgrhnisve der 'I'r~rbnlcnzforscl,rrng. Z. VI>t 77, 105-114 (1933); Coll. Workn, 

Vol. 11, 81!J -845. 

Nikr~rntlse, J.: Striimungsgesrt,zc in rnuhcn Rohrcn. Forsch.-Arb. 1ng.-Wes. Heft 361 (1933). 

Schlirhting, H.: Zur EntRtchung dcr Turhulenz hei dcr Plnt,tet~stijr~nnn~. Nnchr. Ges. Wiae. 

Giittingen, Math. L'hys. Klasec, 182-203 (1933); sq also ZAMM Id, 171-174 (1933). 

I'rnntl1.l. r.. : Tho ~nrrl~nnics of viscous fluids. In \V. IT. i)urnn:l (ed.) i\crodynnmicn Theory. Vd. IIT. 

Springer Vcrlng, 34 208 (1935). 

'r~ll~l~icn. W. : Ein nilgemcines Kritcriom der Inst,nbilitiit laminarcr Genchn.indigkeitRverteilungen. 

Nnchr. Ocs. Wiss. Giitliugcn, Mnlh. Phyn. Klnssc, Fnchgruppc I, 1, 7:)-114 (1!)35). 

Srl~lic.hting, fl.: tZ~~~~rlitt~tlc~~vertrrilr~r~g rind 1Snwgicbilnnz der klci~len Storungen bei tlcr Platbnnlriimung. 

NncI~r. (:cs. C\lisu. Ciittingcn, Mnth. Phys. Klnsue, Fnchgruppe 1, 1, 47--78 (1938). 

Busemann, A.: Gaastromung mit lnminarer Grcnzschicht entlnng einer Plntte. ZAMM IS, 23---26 

(1935). 

Joncs, B.M.: Flight experiments on the boundnry lnycr. Firnt Wright llrothcrs' Memorinl 1,ccturo 

1937. .I. Acro. Sci. 5, 81 - 101 (1938). 

von KQrmQn, Tit., and Tsien, 1I.S.: Boundary lnycr in comprcasiblo fluids. J. Aero. Sci. 5. 

227-232 (1938). See also: Th. von Iihrmhn: Report on thc Voltn Congrcs8, Rome, 1035. 

Giirtler. 11.: llber einc drcidimensionalo Instnhilitiit lan~inarer Grcnzschichten an konknvcn 

Wiindcn. Nnchr. Gca. Wim. Oottingcn, Mnth. Phys. Iilnesc, New Scrim 2, No. 1 (1940). 

Schubnuer, G.B., and Skrnmstnd, H.K.: Lnn~innr bonndnry lnycr oscillntionn nntl st,nbilit,y of 

lnn~innr flow. .J. Aero. Sci. 14, 09-78 (1!)47). NACA Ilcp. 1109 (11148). 

Tollmien, W.: Asyn~ptotischc lntcgration dcr Stijru~igsdiKcrent~ialgIcichung cbcncr In~ninnrn 

Striimungen bei hohcn Reynolds-Zahlen. ZAMM 25/27, 33-50 nnd 70--83 (1!)47). 

Mnnglcr, W.: Z~~snrnmenhnng zwincheu chencn rind rot.nt.ior~nsyn~lnrt.riscl~cn Orcnz~cl~ichtcn in 

komprcasiblcn Fluwigkciten. ZAMM 28, 97-103 (1948). 

TruckcnbrodL, E.: Ein Qnndroturverfnhrcn zur Bercchnung (lor Inminnrcn untl tnrhulcnten 

Ileihungrrschicht bci rhener uud rotntionssyr~~nlctriscl~cr Strii~n~~ng. 1ng.-Arch. 20, 21 1-228 

(1952). 

Dryden, H. I,.: Fifty ycnrs of houndnry layer thcory and cxperimcnt. Scicncca 121.375 ---380 (IMR). 

Schlichting, H.: Application of boundnry lnyer thcory in t.nrbon~nchincry. J. llarric lhg. 81, 

543 -MI (1959). 

Kcatin, J.: Tho ~ITcct ol free-ntmnm t.~~rl~ulcncc on bent. t,rnnnfor rnh. Atlvnncocl iu Iforrf. 't'rnnnfor 

3, 1-32 (lfl66). 

~rarlnhnw, P.: 'I'hc untlcrat,nntling nntl ~)rodict.ion of t~~rl~ulcnt flow. 6l.h Ilc>y~~olcln-I'r1tr1cll.1 IA:~.(.III.c. 

.I. Roy. Aero. 80c. 76, 403-418; acc nlso J)(:LIt Jb. 1972, 51 -82. 

Schlichting, 11.: Recent progrcss in boundnry lnyrr resenrch. 3(il.l1 Wright Ilrothcrs' Mrnmrinl 

IecLurc 1973. AIAA -7. 12, 427-440 (1!)74). 

Smith, A.M.O.: IIigh lilt nerodynnmics. 37th Wright Brothcrn' Mrmorinl lect,urc 1!)74. .I. Airrrnft. 

12, 501 -530 (1975). 

Schlicht,ing, 11.: An nccount of the scientific life of 1,urlwig Prnndt,l. Invited hct.urc prtz~cnlcd n(. 

the Symponium on Plow Scpnrntion of the A(:AIED Fluid Dynamics I'nncl at (:fittingon, 

May 27 to 30, 1975. ZFW 23, 297-316 (1975). 

Tani, J.: History of boundary layer thcory. Ann. ltevicw Fluid Mcch. 9, 87- 111 (1977). 

E. Ludwig Prnndtl Mcrnnrinl Lectures (sincc 1957) 

Betz, A.: Lchren einer fiinfzigjnhrigcn Striin~u~~g~forschung. ZFW ii, 97- 105 (1057). 

Drydcn. L.: Gcgcnwnrt.sprohlcmc dcr L~~ftlnl~rtlorscl~u~~g. Z1W 6, 217 --233 (19fiS). 

Roy, M.: Ubcr die llildung von Wirbelzoncn in Striitnungcn rnit gcringcr ZLl~igltcit. ZlpW 7, 

217-227 (1959). 

SchmidL, E.: Thcrtnischc tt~~ftric~tsntriil~lrlr~gcn nnd Wnrrnoiihcrgnng. ZFW 8, 273-284 (1960). 

Lightldl, M. J.: A technique for rendering npproxirnnte sol~~tionn to phyuirnl prohlrn~s uniformly 

valid. ZFW 9, 267--275 (1961). 

Tolltnicn, W.: Aspektc tlrr Stromungsphysilt 1902. ZIW 10, 403-41:) (1902). 

SGII~.~. It.: I)ir Aufgal)c tlcs Mnthrmatilccrs in drr Arroclynnmik. ZFW 11. 349 357 (In(;:)) 

Ackeret, .I.: ,\naentlungen drr Acrodjnsmik it11 Ihu-rnen. ZF\V 13, 109- 122 (1965) 

Busrmnnn, A,: Minilnnlprohlernt. tler Ldt untl Rnnmfnhrt. ZIW 13. 401 -41 1 (1965). 

Schlirhting, 11.: Einigr nrucre ISrgrltnirmr nrtR der r\~rodynnn~ilz dc4 'I'rngfliigrls. I)(;l,R .TI,. I!)G(;, 

1 l -32 (l9ti7).

Oswatitach, I

B11~1inrll. 1). M. 5!12 

I111ssn1n1in. K. 218,223,989, 

390, 406, 500. 500, 545 

Riiyiiktiir, A. It. :I21 

('ntriet, C. I?. 141, 148, 258. 

2liO 

Vnt~i+rr, P. 088, 0!)8 

(':wler, J. C. 31%. 373 

(::try, A. hl. 524, 545 

(htlicrnll, 1). 1 IO 

(:nza(w, Bl. 1). 420, 445 

('cbcci,'r. 188, 1!)8.515,545, 

5!)4, (i72, (i!)8, 724, 792 

t'er111:~k. .I. I

H:I:IR, 11. 776, 777 

Ilnnsr. 1). 628, K12 

~I:I~I(~~III~IIII, It.. 485 

1Iii111111cr\in, (:. 535. 547 

1I:Lgcll. (:. 12, 2:1, 87. I I0 

I~:I~III~.III:~IIII. tl. 187, I!)!) 

l1:1ll. A. A. 485 

II:III, M. (:. (i!)2 

11:1t11a, 12. It,. 235, 2Gl, 536, 

553, 623, (iW. W7 

Iln~llcl, (:. 73, 82, !)0. 108, 

lo!), 110, lG8. 281, 5!)4 

1Ia111ilto11, ti. 11. 54t; 

Ifn~~~n~an, .I. 443 

I~~IIII:I~I. I). R1. 242, 261 

~I:IIISCV. A. (:. 254, 261. 7!)2 

I~:III%PII, k1. 40,45, 141, 142, 

453, 474, li3(i, Mi6 

II;III~~SCIIC, \v. :

Linkc, W. 173,200,514,049 

List, I$. 11. 75fi 

Liu, T. Y, 550 

Livhgnotl, .I. N. 13, 308, 

30!), 321, 522, 3!)1, 403 

Ihytl, J . H. 650 

1,obb. Jt. l

Scl~crb:trl,l~, I

Vnrlwti, 11. ,J. 315. 324 

Vnn, I. I(. 372. 376 

Vmat~ln Itatn, V. 257, 263, 

302. 326, fi87. 701 

VnLqa, V. N. 3172, 3176 

Vrrollrb, 13. 644. OW 

Viktori~~, I. 223, 224 

\Vill~~~:rrll~. \\I. \\I. 570, 577, 

(i!f5, 728 

W~~SOII. I

805 811bjrct lntlrx 

209, 354, 385, 637, 673 

tlissipnt,ion (3, 74, 207, 705 

-- fl~nction 267, 705 

ctinlort.ion 54, 57 

tlisLr~rbn~rc~c, nrt.ifirinl 477 

- cquntio~~ see Orr.So~i~~~~rrfrlcl rquntion 

- -, nnt,ur;~l 45!), 477 

, spiral 530 

- , t11rc:c-di~~~c~~sional 460, 481, 525 

~lisl.r~rlw~~ccs. ~~~otl~otl of stl~;tll 457 

tlr;\g 2, 5, lli. 20, 25, 27, 2!), 114, 176, 202, 

7 I , 7 7 7 .vccrrIsofc~r~~~clrag; 

A~II fric4io11 

-, :rcrnfoil 22. 7(i7; scc olso :wroli,il 

- irc~lar yli~lcr 7 4 4 s w rllso 

c~,lir~dcr, circular 

-, flat, plate 2(i, 138, 637,641,644, Ii53, 716; 

sen n1.w fltrt. plate 

- , ~wl.or vrh:cla 35 

- , pressrlrc? 758 

--, prolile 758, 764 

- rerlr~rt,io~~ Ii30 

- - , t,ot.nl 758 

tlyc cxl)cri~rwnt. 38, 449 

finilr* clilliw~~ccs ', 187, 194, 671 

- olc~~~cnl.n 672 

liirst. Law of tl~crn~orly:~n~~~ics 265 

llap li89 

lI:il, pI:~.lr 1 24. 26, 32, 40, 135, 139, 156, 175, 

201, 214, 250, 2!)2, 2!15, 392, 333, 383,443, 

4531, 465, 468, M6, 07!), 707, 716 

- -, oscill:it.ing 93, 432, 4314 

- --, ror~gl~ fi52, 720 

- --, ynwctl 250 

I~lctt,r~er's rotor 380 

I~:I~PII-l'oine~tillo flow 11, 85, 280, 512; .we 

ir1,w pil)e flow, liwli~lnr 

heat-ror~rl~~r:tior~ nqnntion 157; see rclso 

l'ol~ricr cqunt.ion 

-- flux 275, 703, 706 

-, frict,ionnl sce fri~t~io~d l~ent 

-- t~rar~sfrr 3, 2G.5, 286, 2!)6, 315, 514, li8!), 

70'2, 707; ace nl~o co~~vcct,ion; tl~rrn~nl 

11ot1ncl:rt~y 111.yor 

- -- :~ualogy 286, 707 

- - . rough snrf:~cc 712 

Ilclc-Shnw flow 123 

I~III~II:L~ ncrofoil 382. 45fi, 502. 573 

- - IlO\V 3, 1 I 

s111)laycr 563, (iO3, 708 

I,:~pl;~ct:'s rqr~n.liotr 10 

Inm of t,hc \vdI 640, Ii43; ncr nlso ~~nivrrnnl 

vrlocity clist.rib~~tio~~ Inw 

lift 16, 23, Xi, 43, 394 

-- , III:I~~I~IIIII 2, 35, 43. 380, 687 

Lin'n n~ol hod 41 1, 432 

Iovnl xt.t~In. ~winr.il~lc: or 58 

I,ortl I(:aylr~igl~'~ cywil,io~t 4(i2 

- tlrcoro~r~s 4li3, 4M 

loss roclliricnt. fcnnrntlrs) Mi:!. 77 1 

I~rlicqter rotor 254 

1~it'lllcll7. flow 

flow 

!IS, 194; See R ~ ~t~~gllfL(~iol1 

O 

11ylrndir:dly sn~ootl~ rcgin~c ,me ror~gl~~~esn 

height,, crilical 

hydraulic: dinn~rtcr GI2 

hyrlrar~lics 1 

I~ytlrost~nt~ic sl,ress 51

Mnnglor'u trnnsfornintion 245 

nmsn conscrvnt,ion 47, :)!I9 

nicnn motion 557 

~netltotl of indires 14 

niinin~otn s~~ction see suction 

von Miucs trn~~sfort~~~~t~i~)~~ 

157 

mixing aocflicicnk sec wldy cor?fIicicwt2s 

-. lengtlr 3, 57!), 582. 604, 715, 730, 751 

-- --, niodificd 731 

-- - throry 582 

n~ornent see torqno 

nio~nentnn~ eqnilion 175, 201, 20(i, 441). 758 

- intcrpral eqntrl,ion 158, 160, 201, 20(i,:353, 

355, 392, 671, 672, li75, 678, 723, 704 

-, kinemiltic 182. 252 

- method 677, 678 

- thi(.knms 141, 160, 177, 202, 209, 353, 

354, 385, 637, 673, 764 

--, trnnufcr of 40 

nmtion, erpntion of 47 

NACA norofoil ROO, 502, (i!)l, 707, 771 

Nnvicr-Stokm equntion 1. 44,47,04, 70,84, 

320, 561 

.. - - in con~prtwil~lc flow (M 

nentrnl utnl)ility rnrvr 4lil, 4li!), 470, 47 1, 

472, 479, 4!)2, 493. 507. 530, 534 

Newt.oninn flnid scr flnitl, Newtonian 

Newton's law of friction 7, 26 

- Srrond Lnw 48 

no dip contlit ion 5, 20, 72 

nnincricd rnrtl~otl 187, 219 

Nuuurlt nnnibcr 275. 296, 708 

orwin c~rrrcnt~ 5 I3 

Orr-Sotnrncrfeld rquntion 459, 400, 462 

onnillnlions see tlisturhnnms, tnctliotl of 

sn~nll; pcriotlio flow; I,ountl:iry Inycr, 

lwioclic 

oncillogrnn~ (t,~lrl)nlent. tlow) 452, 477, 4!)1 

Ouccn's in~provcmcnt 1 I5 

11"rnflox S/!C d'Alrn~l~rrG'u ~ ~ R ~ ~ I ~ O X 

I1i:clct nnrnl~nr 273 

perfect gas 10, (M, 267, 271, 327, :$!)!I, 705 

pcriotlic flow 41 1. 428, 432 

jwrt~trl~~~liot~ 413 

pipe llow .we rrlso inlett flow; nnnulrrr effect; 

rcsinl.nnce cocflioicnt,; rcsistnnce forlnul:r 

- --, curved (i26 

- , inlct 92, 241, 560 

.~. -. , len~inxr 1 1, 12, 85. 92 

- - , nonnhirtly !)2, (Z!) 

- --, osr:illntjing 436 

- --, st.nbi1it.y of 542 

- . , st.nrl, of tnot.ion !)2 

- - - , tmnd itin :)!I, 44!) 

-- --, turhnlcnt 13, 39, 85, 449, 544, 59(i 

I'il.ot tmverse niethod 758 

plntc ace flat plate 

---, rongl~ (i5 

-- t.I~ern~otnctcr 286, 333 

pint of inflexion 132, I65 

-- - -- witcrion 463, 4!10, 514 

- - i~int.nbility 41i2 

.- - tannnit,ion 462 

I'oincnillc llow see flagen-I'oiscr~ille; see nlao 

pipe flow, I:w~innr; 

~~olyn~cr KIO 

pl,ct~ti;il flow 71, !)ti, 128 

power law (117-tlt) R!W, IiOO, 637, 648 

I'rnntltl IIIIIIII)~~ 26!), 273, 274, 283, 289, 330 

- --, ttn-hnlcnt 706, 708 

Prtu~tltl-Scl~licl,Li,~g forrnuln 641 

I'rnntltl'n pip rcuiatnncc law 01 1 

pross~~rc 51, GI 

- diulrilntt.ion 2, 20, 21, 22, 40, 114, 117, 

122, 49!), 504, 770 

-- drag see fortn drag 

- drop 12, 37, 92, 241, 596, 612 

-- grnrlicnt 33, 132, 206, 340, 456, 463, 48!), 

lili8; see nlso wetlgc 

-, t,l~ert~~otly~~~it~~it: 51, GI, 03 

pri~~cipal axcu 57 

prolilc dmg sect ctmg 

propeller 6!)4 

propcrl.irs (L:rbloa of) 8, 9, 269, 662 

])rotlr~~nion we rougl~ne~ 

rnrcfnc:tion wnvc aer! expnnuion fnn 

Il:iylcigl~'u eqnntion see Lord Ikyloigli's 

cqnat.ion 

-- ll~rorcn~~ see J~wd Jtnyloigl~'~ thcorcrnn 

-- prohlcm me Stokcn's first prohlcm 

rc:il gnu 327 

rocovcry fnclor 355. 713, 714 

rcfcrrncc tr~npcrnture 715, 716 

rcsisltrnce see drag 

- cocfficicnt. (of pip) 12,86,507,607,012, 

613, 617 

- fimnuln (RIa.Ui11s'n) 597, 60, 61 1 

- - , nnivcrunl 609 

rcvcrnc flow 2,25,28,85, 108, I2 ; see aho 

ucpnrntion 

rcvc?ruil~lc proc:csu 62 

Itcynolds nurnbcr 12, 14, 72, 128, 150, 772 

- -, criticnl 3. 461, 480, 514, 573 

- - , - (ncrofoil) 490, 500, 502 

- -, - I(cylir~tlcr, upliere) 

- -. - , ( IMI~) 37.80.450 

173 

- - effoct on loss coefficient 772 

-- -, retlucctl 117 

- st.re.wcs 559, 703 

lteynolcls'o analogy 286, 706, 707 

Subjrct Index 

- --, extended 709, 710 

- dye expcritnent 12 

-- cq~tnlion of Iubrir:~t,ion 121 

- principle of siniilnrity 12, 70 

llichartlson nutnbcr 512 

Il,ich:irtlson'u nnnul:ir effect 438 

rigid-l~orly rnt:&ion 55, 56, 57 

rotnting body 242, 005 

- flow 225 

r~llg~lll~~~ 530, 619, 624, 652, 712, 723 

-, tirltnisnil~le li57, 660 

--, tlistril)rrt~cl 540, (iR2 

-- clcn~cnt, 537, 655, (M!) 

- hctor (i 1 (i, 152 

- height,. rriticnl 537, 663 

-, l~ydrnnlicnlly sn~ooth 016, (I50 

-, intmn~ectinto rnngc (transition rcgitnc) 

537. 617. 622, 650, 713 

-, rrlntive 015, 652 

-, stnntl:~rtl 623 

slat 380 

din ti 

ulippcr (of hmring) 11 7 

slit 6H!) . solidit,y rnt,io 7110; see nlso c:nswtlc flow 

sound velocity scc Mnt.11 IIIIIIIII(-~ 

specific lieat 269 

spectr~itn see freq~~onc-y spcctru~n 

qil~cro 17, 10, 21, 25.4'2, I I:!, 237. 243. :)20, 

42 1 

spots aec tnrl~nlc?nt npotu 

sqnnlinmn 557 

st.nl)ility rquntion aec! Orr-So~~i~~~~~rf~~lcl 

I~II:~. 

tion 

- - , frictionlc~u 462 

-- , li~nit of 460, 4!)7, 502 

-, neutral 4G6; am nlso neutral stnliilit,y 

cnrvc 

- t.l~cory 3, 451; 

stngnntion en( l~nlpy 3533 

sand ro~tghncss 615, 623, 654, 663 

- ---, eqt~ivnlrnt r,23,(i54; ncertlaoro~~gl~nrsn 

Srhlic:rcn !ric.t.nrc 320, 3(iO, 363, 364, 3(i5, 

:w :IW 

Scliobnucr-Skm1114tatt cxpcritnent 470 

sccontlnry flow 102, 226, 230, 248,428,431, 

432, ($12, 613, 626, 644, 657 

- flow, tl~rce-tliti~cl~uiot~nl 100, Ili5, 250 

- -, two-dinicnsiot,al 33, 35, !)5, 0!), 156, 

165, 214, 250, 252 

- lonpcrttt.~~ro 208, :I% 

st,~intlnrtl ~01lgllllt~~U sec mugl~nrsn 

1 1 n r ~ ~ i v tonfirti r i li72 

Stnn1.011 IIIIIII~J~~ (is!). 708 

self-sitnilw solutions see bonntlnry layrr, 

siniilnr nntl aclf-uiniilnr solutions 

sen~i-sitnilar uolutions 415 

ucpnriition 2, 25, 28, 33, 43, 131, 152, 172, 

215, 220, 243, 253, 254,258, 259, 382,378, 

417. (Xi!), 674, 687, 769 

--, prevcnt.io~~ sec honntl:rry-lnycr control 

sl~tipc f:lct.or 208, BO(i, 4!l2, 675, 678, (i79 

- .. , ~notlific~l 674 

slic?xrin,q s1.1.c~~ (nt wdl) 26, 134, 138, 143, 

147, 202, 20!), 600, 037, (i54, 670; see nlso 

skin friction 

ship 054, 062 

sl~ook tul~c 4:%0 

- wnve 358, 3(i0. 3(il, :3li3, 314, 365, 368, 

X!), 43!) 

sitnilnr nnd self-aimilnr uolnliona 90, 101, 

107, 13li. 151, 152, 164, IMi, 203, 2D3, 300, 

316, 344, 38!), 415,48!), 735, 737, 740, 746, 

75 ; see rrlso honntlnry I~iycr, ni~nilnr nntl 

~tntc see eqnnl.ion of sknte; locnl slnlc 

stor1111 tnrbinc, louses cl~tc to rougl~nrs~ li02 

Shkcs'u tlrng fortnnla (upl~cro) 114 

- liruL problcn~ !)O 

- second problen~ 93 

- hypotliesis (if) 

- Inw of frirtion 2, 7, 48, 58 

- second prol)lr~n !)3 

nl.r~iin 48 

--, mtc of 7, 52. 58 

stmtilicntion 512, 73!) 

stream function 74, 133, 136, 15:I. 157, I(i3 

~t.renn~ing ace arcontla.ry flow 

streiwlinc I~otly 22, 42 

st,rrss 48, 4!) 

-, nppnrcnt or Jlcynolds's 3, 55!), 560, 704 

--, tlcvintoric: 48, 41) 

--, hytlrontntk 50 

- tennor 50 

Slrol~linl nunlbt~r 31 

srIf-~i~nil:w sol~~lionn 

siniilnrity 12, 70, 151, 271, 450, 597; see 

also bonntlary Inyer, si~nilnr solut,ions and 

self-similar solutions 

- in liento t.mndcr 271 

-, von I

812 Snhjcct lntlcx 

- vorl.irrs 526, 527 

-- -- behind bars 741, 744 

tm~~pcrat.~~re, tlill't~sio~r of 752 - - , chnnnel 84, 107, 168, 277, 668 

- licltl am? Ll~crnrnl bountlnry Inyer - --, c$itldor 171, 21(1 

-- rise, :~dinl)ntic 270, 27!), 286, :1R2, 717 - -, jet see jct 

t,l~eorcticnl I~ydrotly~~n~nics 1 - -, lubricntion 117, 121 

I,I~errnnl barrier 3 

.- - , pipe see pipe flow 

- honntln.ry hycr 3, 78, 265, 327, 330, 514, - .- , pinto 142, 205, 385, 454, 039 

702, 712, 713, 754 

-. .- , WWI~O 165 

co11d11c1ivit.y 2Mi, 260, Mi2 

-- -, wing 249, 085, W!), 690 

-- difl~~~ivit:~ 2(i!). 273 

-- grntlicnt 33, 128, 132 

t,l~or~~~otly~~:t~~~ic: 

prcss~~rt! 51, 61, 63 

--- ofpropngntio~~ (ofclist~~rl)n~~rc) 459, 460, 

'l'oll~~~ir~l-Srl~lic:l~ti~~$ 

\VILVCR 45!), 474, 4!)5, 

SRC (rlx~ 0t.r-S0111111t:rfcl11 tq~~iilio~~ 

see n2.w tlist,~~rl)a~~con, 11101.l1otl of sinall - of sonntl aec Mach nnn~bnr 

t,orqr~e 105. 24'2, 647, 649 - thiclc~~ess 356 

hce (of nt.rcs~, tensor) 51 

viscosity 6, 6, 8, 9, 60, 2n9, 328 

tra~n~forn~ation, c*o~nl~rcssiblc tt~rbl~lent flow - , convcraion factors 8 

7 18; ace r~lno I lli~~gtvor(.l~-Slcn.~rtm -, Ici~~e~nntic 7, 8, 9, 269 

trn~~nfor~nnt,ion - ~ncnaurcn~ent 12, 88 

t,ra~~sforn~rcl vnri:r.l)lvs (Tor nun~cricnl t~reLhod) - tnbles 8, 9, 269, 662 

I87 

vortex f lnn~cnt 89 

- for~nnt,ion 2, 19, 25, 28, 425, 427, 525, 

529 

- sliedtling freqt~cncy 91, 173 

- so~~rco 230 

- spirnl 529 

- st.rccb (von JCdr~~~rlln's) 18, 28, 173 

vort.icily 58, 73 

-- trn~~sfer cquntion 73 

-- - thory (G. 1. 'l'nylor) 584, (i08, 755 

\v;iko 25, 175, 234, 729, 733, 741, 758 

- behind bhnt body 738 

-- - c,nsmrle 772 

- - row of bnrs 744 

-- single lmly 73!) 

- - , oiror~lnr 733, 743. 747 

, t~wo-~li~~rc~~nio~~al 175, 733, 745 

~mll, ncliebetic 268, 277, 28(i, 2!)4, 332, 333, 

335, 337, 344, 517, 5l!), 718 

--, 

c11rvrr1 510, 525, 526, 6!)0 

-, flexible 505 

-- jct 750 

\wvc tlrng 76!) 

n;~vclc~~gtl~ 459, 532 

rrctlgc 156, 364; see n1.w si~nilar a ~ self- d 

siwilar solnt.ions 

wi~d (i54 

wi11d-1.1111nrl t,~rrl~~~lc~~ro 572 

wing 24!). 685, fi8!), (i!)O 

-~-, slolt,otl 381 

-- . swept, 253 

--, ynwod 248 

I 

Abbreviations 

The following abbreviat.ions lmve been un(d thro~~gl~o~~t. the book 

AIAA J. - Journnl o/ the American Inslilale of Aeromutics nnrl A.dronmr/ir.~. New 

York, publiahetl aince 1903 (nee JAS nnd JASS) 

ARC = Aeronnuticnl Rc.senrch Cooncil, Jmndon. Publint~r.~ two ~rrir~ of (lot-11- 

~nontq, rnch n~~tnbcrctl ucpnrntnly 

ARC RM - Iteporta nnd Memornndn 

ARC CP - C~lrrent Pnprm 

ARSJ = Jor~rnnl t1111erirn11 Rocket Sorirty 

ASME .- An~rricnn Socidy of hlccl~nnicnl Knginrors. Now York 

- I~eotsche Forsch~~~~gs- r111c1 \'ersi~c~l~s:~~~st;~lt fiir 1,11rt.. 11w1 IZ;it~n~Ltl~rt,, 

Kiiln (nincc I!)(;!)) 

Scientific jonrnnl enlitlccl 

Ingmieur-Arrhiv, 1krli11 iu~cl, sinro 1947, 1hxli11 nrd Ilritlcllwrg 

JAS Journal of /he A~ron(ttrtirrr1 A"j'icncc*. Ncw York, (1!)32 1!)58); 

replnced in I959 by ,].ASS 

J ASS = Jourtutl of Anro/Space Scienrc~, Ncn York (1!)5!) -l!)T,2): 

rcplnced in 19G3 by AIAA .I.

814 

NACA 

NASA 

Proc. Roy. 

I? I\ I< 

USAF 

V Dl 

ZAMM 

ZAMI' 

ZFM 

- Tho Nntionnl Advisory Committor for Arronautica, Wrurhingtan D. C. 

[rrplncctl in 195'3 by NASA (nrc below)] hblishrd three nrrica of 

tlorumenk, ear11 nunihrrrcl srpnrntcly: 

NACA Rep. Rcportn 

NACA TM Trc1niic.d hlr~nornrida 

NACA 'I'N 'I'rcl~nicd Notcn 

- Nntionnl Aeronn11tica ancJ Spacr Adtninintrntion (crenbd in 1'359 in 

rrplnrrrncnt of NACA) 

= Nntio~~nl (inn 'l'11r11inr 15nl:cl1lisl111irrlt, Crrnt Rritnin 

- Office Nationd d'fit~~tlea et do 1Erchorrhc~ A6ronpntinlen, ChBti11on-aou8- 

13ngncux. Frnnce 

- Itoynl Aircraft I'~nt:~l~linl~l~~r~it.(:rrnt I%rit,~~iri 

- Unitctl Stntrn Air Irorcr 

- Vcrein Drutsclirr Ingrnicurc (German Society of Engineera), Dueascidorf. 

Publial~es: Pornchg. 1ng.-Wcs. with its supplement Forachrtngsheft 

(aoo nbovc) 

- Jnhrl~~trl~ rlor \Vissc:nncl~nft~lic.he~i (:cncllnchnft fur I,r~lt.f:iI~rt, 1952 - l!W2; 

fiir I,11ft- rrntl lZ.aunifal~rt,, lf)liJ--1975 (H. J%lenk ard W. SCIIIIIZ, ctls., 

Vicweg, Ilrn~inocliwcig) 

- Zcibchrifl fiir nngntmnrllr: Mnlheninlik sm! Phpik, nasrl. Switzerland 

- Zoitschrifl /iir Flqtechnik und rlfolorla/lschi//ahrt, Munich nnd Ilerlin. 

Germany 

List of most commonly used synlbols 

In order not to depart too drnat~icnlly from the convent.iona normnlly cmyloyrtl in pnpcrn 

on thc subject, it was follnd neccannry 1.0 rtw t.1~ anrnc aymbol to thoto ncvcrsl dilTcrrnt, qrlnlltitics. 

Thus, for vxnmple, 1 clcriotru tho rcniat.anco cocfficicnt of pipe flow, both I:aniin~ir ant1 

turbulent,, and in the theory of stability of ln~ninnr houndnry lnyera it dcnotcs the rvnvrlc~ngl.l~ 

o[n clinturbnnco. Sirnilnrly, k tlenotes t.her~nal concluctivity in the theory of t,hcrn~nl bo~~ncl:try 

Iayera, nntl llic Iicight of n protubcrnnce in thc discussion of the infludncc of roughncw on 

turbulent flow. 

'I'l~r follorving in a lint of ~yn~boln n~ost romnlonly used in 1,110 book. 

I. General symbols 

.1 -- wcl.tccl nrcn, or fronlnl nrca 

c: -, vc:lorit,y of R~IIIIC~ 

d, L) = dinnictcr 

g E nrcclcrntion (IUC tn grnvily 

h - rhnnnrl wirlt h 

I, 1, - lrng611 

p = prranuro (Torco prr rinil nrm) 

T = 4 e V2 = tlyrian~ic hrncl 

r, 4, 2 = cylindricnl coordinntea 

r, R - rndiun 

a - nirnn velocity (in pipc) 

Urn =. froe-ntrcnni velocity 

U(z) -- velocity in potrnt.ial flow 

u, v, cu = velocity coniponcnta 

72 = tempornl mean of velocity (pipc or bonnrlary Iayrr) 

z, y, z = cnrtcsinn coordinntcs 

V = frcc-strcnn~ velocity 

e = dcnsit.y (111~s per unit volume) 

w = nngulsr velocity 

A, = eddy viscosity 

b = width of jet or wake 

c~ = drag coefficient 

c, = akin-friction coefficient 

c,' = locnl akin-friction coefficient 

D 5 drag force 

lllz -- t51/dz - first nhnpr fnetor of vrl~city j~rofilr

List of most commonly used synibola 

If,, = d,/rl, = uecond ellape factor of velocity profile 

M = (rrlc) = Mach number 

k = l~eiglit of rongl~neaa elemcnt (protuberance) 

ks = lieight of grain for equivalent aand rougl~ness 

K = sl~npc factor of velocity profilc in boundary layer 

1 = mixing length 

R = (VL/v or ridlv or Ud/v) = Reynolds number 

Rr = Riclinrdson nurnbcr 

S = Stronhal nnml~cr 

T = turbnlcncn intensity (also dcgrce or level of turbulence) 

u', v', w' = componcnta of turhulmt, velocity 

.~. .- 

~'2, v'*, u' v' . . . = temporal means of tnrbulent velocities 

U = maxinlutn vclocity at bipc centre 

U, = free stream velocity 

E'* = 4

Boundary Lyer Theory

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