Boundary Lyer Theory

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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
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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
Page 178 and 179: 334 XIIT. Lnrninnr 1)oundary laycra
Page 180 and 181: 338 XI I I. Im~linnr I>orirtclt~ry
Page 182 and 183: 342 XI11. Tmninnr Imlndary layeru i
Page 184 and 185: 346 XI 11. T,ntninar I~or~nrjary hy
Page 186 and 187: 350 XIII. 1,mninnr honnclxry lnycrn
Page 188 and 189: 354 XI 11. 1,nminnr h~nrlnry 1:ryrm
Page 190 and 191: Fig.. l3.16. In 13.18. lain~innr ho
Page 192 and 193: a Intninn.r l)out~rlary I:~.ycr, bu
Page 194 and 195: 1 I Itc vnriolts c4Twl.s ol'sl~oclt
Page 196 and 197: 370 XIIJ. I~tminar boundary layers
Page 198 and 199: \'all I)rirst., 15.11..: 'l'hr prol
Page 200 and 201: CIIAPTER XIV Boundary-layer control
Page 202 and 203: 5. J'rrvrntion of trauaitinn by the
Page 204 and 205: frorii th; 1c;ulitig rxlge! (l3l:wi
Page 206 and 207: 390 XIV. Ilo~~n~lnry-lnycr contml F
Page 208 and 209: . - v, (x) =cons/ Fig. 14.14. 1,ant
Page 210 and 211: 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 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 262 and 263: Tlw tlisl.nr~cc Iwt,wccn the point,
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
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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
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
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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
Page 397 and 398:
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
Page 403 and 404:
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
Page 411 and 412:
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:
List of most commonly used synibola
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Boundary Lyer Theory

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