Boundary Lyer Theory

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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
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McGRAW-I4ILL SERIES IN MECHANICAL E
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vi Contents CIIAPTEII V. Exnct ~olu
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Y Contents A " I XI X 'I'lirorrt.ir
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xvi I'orcworcI Thc result was t.11~
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From Author's Preface to the First
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the aid of thorctical considcmtioti
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even in fluitls wit,lt vcry srnall
Page 16 and 17: 10 I. Or~llinr of lluicl rnot,ion w
Page 18 and 19: The vclwil y IL at some point i11 t
Page 20 and 21: Fig. 1.6. Firld of flow of oil nho~
Page 22 and 23: 22 I. Outliric of fluid motion with
Page 24 and 25: 2 (i TI. O~~tlittr of Imun~lnry-lsy
Page 26 and 27: Fie. 2.511 Fig. 2 .5~ Fig. 2.5b Fig
Page 28 and 29: S - point orscpnrnt.ion T'ig. 2.12.
Page 30 and 31: 3 8 \ Y?\ If. 011tli11e of boundary
Page 32 and 33: 42 11. 011tli11e of Iw~~ntlnry-Inyr
Page 34 and 35: [:j2a] I?o~cilhrntl, I,.: Thr forn~
Page 36 and 37: 50 111. l)crivnt,ion of thc cquntio
Page 38 and 39: Fig. 3.3. Tmcd clintor1,ion of flui
Page 40 and 41: of flltitl is strcsscd in thrcc nlu
Page 42 and 43: 1:i~ 3.8. Qltnsintzt.ia cotnprrssio
Page 44 and 45: 66 111. 1)wivntion of the eqnntions
Page 46 and 47: CHAPTER I V General properties of t
Page 48 and 49: 74 TV. Gencrol proprtic~ of tho Nov
Page 50 and 51: Ilcre v, c, :tntl k tlrnol,c 1.I1t:
Page 52 and 53: conclil.ion (4.14) plays n part wl~
Page 54 and 55: 86 V. 1':xact solul ions of tlir N:
Page 56 and 57: 90 V. ICxnct sol~ttion~ of tho Nnvi
Page 58 and 59: 94 V. Exnct sohltiono ol' tho Nnvic
Page 60 and 61: Table 5.1. Functions occrtrring in
Page 62 and 63: 11. Flow taenr n rotnting disk. A f
Page 64 and 65: 106 V. JCxact solutions of the Navi
Page 68 and 69: 114 VI. Vcry slow motion where r2 =
Page 70 and 71: 1 I8 VT. Very slow motion r. 'l'l~o
Page 72 and 73: 122 VI. Very rrlow motion ext.endcd
Page 74 and 75: 126 VT. \'cry slow rnot.ion Part B.
Page 76 and 77: 130 VII. Boundnry-layor cquntionzl
Page 78 and 79: '1. Skin friction \VlIat1 t.11~ I)o
Page 80 and 81: 138 VII. no~~ndnry-layer cqnntions
Page 82 and 83: 142 VII. Hor~nclnry-lnyrr rq~~ntii~
Page 84 and 85: 146 VII. I~o~~ntlnr~ Inyrr cq~~ntio
Page 86 and 87: CIIAFTER VIII Gencral propertiee of
Page 88 and 89: 164 VIII. Ccnernl propertic8 of Lhe
Page 90 and 91: 158 VIII. General ppropcrtics of th
Page 92 and 93: VIII. General propertien of the bou
Page 94 and 95: 'I'his c.quat,ion Lmnsforms into t1
Page 96 and 97: 170 TX. 1Sxact uolutions of t.ho st
Page 98 and 99: 174 TX. Exnct ~olut,ionu of tho sto
Page 100 and 101: 178 IX, Exact solutions of tlrc ste
Page 102 and 103: and t,llc? clnsh now clet~otos tlil
Page 104 and 105: 1 86 10 0.8 0.6 0.4 0.2 0 -0.2 -0.4
Page 106 and 107: ccnt,rred nt, (m -1- 1, n). 1'110 t
Page 108 and 109: 194 IX. Jqxnct eolutions of thc stc
Page 110 and 111: O~l:cr ehnpw: Second-order cITcetls
Page 112 and 113: 202 X. Approximate rncl.hoda for st
Page 114 and 115: 206 X. Approxitnnte rnct.l~otls for
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210 X. Approximate mcthod~ for shad
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214 X. Approximato mothotls for sha
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218 X. Approximntn met,hods for shn
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222 X. Approximate methods for stea
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220 XI. Axinlly symniobricnl nnd th
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230 XI. Axinlly symmctricnl and thr
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23.1- XI. Axially uynimet.rira1 nnc
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the two wries expnnsicws for sin (%
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242 XI. Axially nymmct.riral and tl
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in cross-flow, tlrprntls only on lh
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250 XI. Axinlly nyrnmet,rirnl nntl
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XI. Axinlly ~,vn~n~et.rir;~l nnrl I
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258 XI. AxhIly syn~rnrtrirnl nncl t
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. . lit 1 h1:lgt.r. '1.: 'l'l~idc I
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206 XIT. l'l~rr~nnl bo1111r11iry ln
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270 XIT. 'I'l~rrtnal boundary layer
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can I)r rct.ni~rrvl in inrotnl~rrss
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if wc: pillf - - 'I1, - (A7'),. 11,
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Rotntirig rli~k: (:II:I~I. V, in pn
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286 X11. 'IY~rrrr~al bo~~ndnry laye
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290 XlI. Tl~rrnmal boundary layers
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with 0,' -. () at, r] - 0 and 0, =
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2!)H XI[. Thcrn~al bor~nrlnry Inyrr
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302 XJI. Therninl boundary layers i
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306 XII. Tllcrrnal hor~ndary layers
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310 XJI. 'I'hcrmnl boundnry layers
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314 X11. Thermal boundary laycrs in
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318 XIT. Tl~crnial bonndnry layers
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322 XII. Tl~or~nnl I)onndnry lnyers
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326 X l I. 'l'l~crn~nl hounrlnry ln
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330 X[[1. lmninar bor~ndnry lnyers
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334 XIIT. Lnrninnr 1)oundary laycra
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338 XI I I. Im~linnr I>orirtclt~ry
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342 XI11. Tmninnr Imlndary layeru i
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346 XI 11. T,ntninar I~or~nrjary hy
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350 XIII. 1,mninnr honnclxry lnycrn
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354 XI 11. 1,nminnr h~nrlnry 1:ryrm
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Fig.. l3.16. In 13.18. lain~innr ho
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a Intninn.r l)out~rlary I:~.ycr, bu
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1 I Itc vnriolts c4Twl.s ol'sl~oclt
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370 XIIJ. I~tminar boundary layers
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\'all I)rirst., 15.11..: 'l'hr prol
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CIIAPTER XIV Boundary-layer control
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5. J'rrvrntion of trauaitinn by the
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frorii th; 1c;ulitig rxlge! (l3l:wi
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390 XIV. Ilo~~n~lnry-lnycr contml F
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. - v, (x) =cons/ Fig. 14.14. 1,ant
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invc?st.igai.ctl. In this msc too,
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402 XIV. Dounclery-layer control 1.
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Ni\('A 'rM !I74 (1!)41). 1861 Sinli
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110 XV. Non-~teldy bortndnry layers
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414 XV. Non-st.cncly hour~tlnry Iny
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418 XV. Non-slmrly l~orrnclnry Inyr
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422 XV. Non-st,cndy hoixnd~ry Inycr
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420 XV. Non-nl,mdy Fig. 15.58 Fig.
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XV. Non-utcncly boouclary lnyers wh
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434 XV. Non-stcxcly boundary layers
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438 XV. Non-atcndy boundnry inyer~
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442 XV. Norl-st,aady boundary Isycr
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440 XV. Non-st,rndy ho~~nclnry laye
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450 XVI. Origin of tnrhulence I pyi
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XVT. Origin of turbulcncc I n. Some
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458 XVI. Origin of turbulence I b.
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462 XVI. Origin of tnrb~rlmcc 1 num
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466 XVi. Origin of t.r~rhr~lrncc 1
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470 XVI. Origin of h~rlwlcnce T I R
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474 XVI. Origin of l,t~rl~~tlct~rr
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478 XVI. Origin of I.rtrb~tlmcr 1 t
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482 XVT. Origin of tmbulcncc I c. E
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486 XVI. Origin of k~rbulenre I 148
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490 XVII. Origin ol t,urbulencc 11
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404 XVII. Oriein of tnrbnlencc TI F
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498 XVII. Origin of turbnlmco II t,
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Tlw tlisl.nr~cc Iwt,wccn the point,
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c. Efict of ~urtintl on trnt~nition
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510 XVl I. Origin of Idmlrnro I I '
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514 , XVII. Origin of turbulence 11
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518 XVI I. Origin of turhr~lcncc 11
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on the bnsis of t.hc tl~corct,ical
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626 XVII. Origin oI t.11r011lonco I
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630 XVII. Origin ot t.~~rhr~lrncc J
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XVII. Origin of t.urh~~lencc TI 7 !
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638 XVII. Origin of t~~~rbnlc~~rr I
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\vi(,l~ a (-ylit~(lw envcrv(1 wiI.1
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646 XVII. Origin of turbulence TI F
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550 X\'ll. Origin of t,11rh111rnrc
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554 XVII. Origin of t,rlrbuloncc TI
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558 XVITI. Fundamentals of tr~rbule
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562 XVlI1. R~nclnmont.nIu of tur1)u
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The corrrlnt,ion co~ffiricnt~ y~ ra
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670 XV111. I'nntlntnrt~t~nlrr of tt
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574 XVIII. 1'undnnient.alu of tnrl~
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CHAPTER XIX Theoretical assumptions
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682 XIX. Thcoroticnl wsltmptionu fo
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586 XIX. Throretical aaotin~ptiona
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5l)O XIX. Tl~rorrt,icnl nmumptions
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594 XIX. 'rheoreticnl nssornptions
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698 XX. 'I'~~rl~ulrnt flow f,lrro~~
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602 XX. T~rrbulcnt flow thro~tgh pi
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606 XX. 'l't~rhl~lc~rit flow thro~~
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GI0 XX. Turbulent flow throngh pip
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Icror~~ ()I(, phj.sic*:~l ~)oint. o
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620 XX. Tnrbnlcnt flow tlvough pipe
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dimensions Fig. 20.26. 1tc.sist.anc
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628 XX. T~lrbnlmt flow through pipe
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632 XX. Turhulcnt flow through pipc
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636 XXI. Td~nlcnt houndnry layers a
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610 XXI. Ttrrbulent bor~ndnry layer
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644 XXI. Turbulent boundary laycrs
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048 XXI. Turbulent boundary layers
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652 XXI. Turt)~~lcnt houndnry layer
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656 XXI. Turbulent boundnry layers
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660 XX I. Tl~rhlent bo~~ndary layer
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664 XXI. Turbulent boundary layers
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CHAPTER XXII The incompreesible tur
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672 XXll. 'l'llc incon~prcssiblc tu
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070 XXII. Tlic incompreaaibto lnrl)
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FRO XXII. Tho incornprcs~iblo turbu
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684 XXII. 'Yhc incon~prcssible Lnrb
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688 XXJI. The incon~prcssible t,urb
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002 XXI1. Tho inro~nprc~sil~lo tt~r
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is prol)nt~t,iot~nl to the rnclius.
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lf$8] Mucsm:tnn, 11.: Z~~snrntncnhn
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70-t XX I1 I. 'Ihrbulcnt bonrwlnry
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708 XXIII. Turbulent bonndnry Iaycr
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712 XXI 11. 'I'~~rl~~~lemt bo~~ntln
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716 XXIII. 'I'urbr~lcnt bor~ntlnry
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720 XXIII. Turbulent boundary layer
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724 XXlll. 'rur1)ulcnt boundnry Iny
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[04] Smith, P.D.: An integral predi
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732 XXIV. Frcc trtrbt~lent flows; j
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700 XXIV. Prrc tr~rb~tlent flowa; j
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Neglecting u12, we obtain XXIV. Prc
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744 XXIV. lhc L~trl)ulcnt Ilow~; jo
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748 XXIV. Free turbulent flowa; jet
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762 XXIV. Frcc td)ulcnt flows; jcta
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756 XXIV. Prrc torbulent flows; jet
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760 XXV. DotcrminnLion of profilo d
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764 XXV. I)ot.crtninntiotl of profi
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768 XXV. Determination of profile d
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XXV. 1)obrlninalion of proRlo drag
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776 XXV. 1)ctormination of profile
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A. Reviews organized in serial publ
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784 Bibliography Sherman, I?. S., I
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Clementa, lt.R., and Mad, D.J.: The
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792 Bihliography 13ird, R.R., Slewn
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Oswatitach, I
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H:I:IR, 11. 776, 777 Ilnnsr. 1). 62
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Scl~crb:trl,l~, I
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805 811bjrct lntlrx 209, 354, 385,
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812 Snhjcct lntlcx - vorl.irrs 526,
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List of most commonly used synibola
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