Boundary Lyer Theory

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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.
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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
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10 I. Or~llinr of lluicl rnot,ion w
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The vclwil y IL at some point i11 t
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Fig. 1.6. Firld of flow of oil nho~
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22 I. Outliric of fluid motion with
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2 (i TI. O~~tlittr of Imun~lnry-lsy
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Fie. 2.511 Fig. 2 .5~ Fig. 2.5b Fig
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S - point orscpnrnt.ion T'ig. 2.12.
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3 8 \ Y?\ If. 011tli11e of boundary
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42 11. 011tli11e of Iw~~ntlnry-Inyr
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[:j2a] I?o~cilhrntl, I,.: Thr forn~
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50 111. l)crivnt,ion of thc cquntio
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Fig. 3.3. Tmcd clintor1,ion of flui
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of flltitl is strcsscd in thrcc nlu
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1:i~ 3.8. Qltnsintzt.ia cotnprrssio
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66 111. 1)wivntion of the eqnntions
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CHAPTER I V General properties of t
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74 TV. Gencrol proprtic~ of tho Nov
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Ilcre v, c, :tntl k tlrnol,c 1.I1t:
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conclil.ion (4.14) plays n part wl~
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86 V. 1':xact solul ions of tlir N:
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90 V. ICxnct sol~ttion~ of tho Nnvi
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94 V. Exnct sohltiono ol' tho Nnvic
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Table 5.1. Functions occrtrring in
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11. Flow taenr n rotnting disk. A f
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106 V. JCxact solutions of the Navi
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cnu bo npplirtl nt mont, nn fnr RS
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114 VI. Vcry slow motion where r2 =
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1 I8 VT. Very slow motion r. 'l'l~o
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122 VI. Very rrlow motion ext.endcd
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126 VT. \'cry slow rnot.ion Part B.
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130 VII. Boundnry-layor cquntionzl
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'1. Skin friction \VlIat1 t.11~ I)o
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138 VII. no~~ndnry-layer cqnntions
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142 VII. Hor~nclnry-lnyrr rq~~ntii~
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146 VII. I~o~~ntlnr~ Inyrr cq~~ntio
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CIIAFTER VIII Gencral propertiee of
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164 VIII. Ccnernl propertic8 of Lhe
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158 VIII. General ppropcrtics of th
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VIII. General propertien of the bou
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'I'his c.quat,ion Lmnsforms into t1
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170 TX. 1Sxact uolutions of t.ho st
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174 TX. Exnct ~olut,ionu of tho sto
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178 IX, Exact solutions of tlrc ste
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and t,llc? clnsh now clet~otos tlil
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1 86 10 0.8 0.6 0.4 0.2 0 -0.2 -0.4
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ccnt,rred nt, (m -1- 1, n). 1'110 t
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194 IX. Jqxnct eolutions of thc stc
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O~l:cr ehnpw: Second-order cITcetls
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202 X. Approximate rncl.hoda for st
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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,
Page 212 and 213: 402 XIV. Dounclery-layer control 1.
Page 214 and 215: Ni\('A 'rM !I74 (1!)41). 1861 Sinli
Page 216 and 217: 110 XV. Non-~teldy bortndnry layers
Page 218 and 219: 414 XV. Non-st.cncly hour~tlnry Iny
Page 220 and 221: 418 XV. Non-slmrly l~orrnclnry Inyr
Page 222 and 223: 422 XV. Non-st,cndy hoixnd~ry Inycr
Page 224 and 225: 420 XV. Non-nl,mdy Fig. 15.58 Fig.
Page 226 and 227: XV. Non-utcncly boouclary lnyers wh
Page 228 and 229: 434 XV. Non-stcxcly boundary layers
Page 230 and 231: 438 XV. Non-atcndy boundnry inyer~
Page 232 and 233: 442 XV. Norl-st,aady boundary Isycr
Page 234 and 235: 440 XV. Non-st,rndy ho~~nclnry laye
Page 236 and 237: 450 XVI. Origin of tnrhulence I pyi
Page 238 and 239: XVT. Origin of turbulcncc I n. Some
Page 240 and 241: 458 XVI. Origin of turbulence I b.
Page 242 and 243: 462 XVI. Origin of tnrb~rlmcc 1 num
Page 244 and 245: 466 XVi. Origin of t.r~rhr~lrncc 1
Page 246 and 247: 470 XVI. Origin of h~rlwlcnce T I R
Page 248 and 249: 474 XVI. Origin of l,t~rl~~tlct~rr
Page 250 and 251: 478 XVI. Origin of I.rtrb~tlmcr 1 t
Page 252 and 253: 482 XVT. Origin of tmbulcncc I c. E
Page 254 and 255: 486 XVI. Origin of k~rbulenre I 148
Page 256 and 257: 490 XVII. Origin ol t,urbulencc 11
Page 258 and 259: 404 XVII. Oriein of tnrbnlencc TI F
Page 260 and 261: 498 XVII. Origin of turbnlmco II t,
Page 264 and 265: c. Efict of ~urtintl on trnt~nition
Page 266 and 267: 510 XVl I. Origin of Idmlrnro I I '
Page 268 and 269: 514 , XVII. Origin of turbulence 11
Page 270 and 271: 518 XVI I. Origin of turhr~lcncc 11
Page 272 and 273: on the bnsis of t.hc tl~corct,ical
Page 274 and 275: 626 XVII. Origin oI t.11r011lonco I
Page 276 and 277: 630 XVII. Origin ot t.~~rhr~lrncc J
Page 278 and 279: XVII. Origin of t.urh~~lencc TI 7 !
Page 280 and 281: 638 XVII. Origin of t~~~rbnlc~~rr I
Page 282 and 283: \vi(,l~ a (-ylit~(lw envcrv(1 wiI.1
Page 284 and 285: 646 XVII. Origin of turbulence TI F
Page 286 and 287: 550 X\'ll. Origin of t,11rh111rnrc
Page 288 and 289: 554 XVII. Origin of t,rlrbuloncc TI
Page 290 and 291: 558 XVITI. Fundamentals of tr~rbule
Page 292 and 293: 562 XVlI1. R~nclnmont.nIu of tur1)u
Page 294 and 295: The corrrlnt,ion co~ffiricnt~ y~ ra
Page 296 and 297: 670 XV111. I'nntlntnrt~t~nlrr of tt
Page 298 and 299: 574 XVIII. 1'undnnient.alu of tnrl~
Page 300 and 301: CHAPTER XIX Theoretical assumptions
Page 302 and 303: 682 XIX. Thcoroticnl wsltmptionu fo
Page 304 and 305: 586 XIX. Throretical aaotin~ptiona
Page 306 and 307: 5l)O XIX. Tl~rorrt,icnl nmumptions
Page 308 and 309: 594 XIX. 'rheoreticnl nssornptions
Page 310 and 311: 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.
Page 361 and 362:
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
Page 377 and 378:
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
Page 391 and 392:
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
Page 401 and 402:
A. Reviews organized in serial publ
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784 Bibliography Sherman, I?. S., I
Page 405 and 406:
Clementa, lt.R., and Mad, D.J.: The
Page 407 and 408:
792 Bihliography 13ird, R.R., Slewn
Page 409 and 410:
Oswatitach, I
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H:I:IR, 11. 776, 777 Ilnnsr. 1). 62
Page 413 and 414:
Scl~crb:trl,l~, I
Page 415 and 416:
805 811bjrct lntlrx 209, 354, 385,
Page 417 and 418:
812 Snhjcct lntlcx - vorl.irrs 526,
Page 419:
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