Boundary Lyer Theory

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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)
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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,
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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
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~~
Page 312 and 313: 602 XX. T~rrbulcnt flow thro~tgh pi
Page 314 and 315: 606 XX. 'l't~rhl~lc~rit flow thro~~
Page 316: GI0 XX. Turbulent flow throngh pip
Page 319 and 320: Icror~~ ()I(, phj.sic*:~l ~)oint. o
Page 321 and 322: 620 XX. Tnrbnlcnt flow tlvough pipe
Page 323 and 324: dimensions Fig. 20.26. 1tc.sist.anc
Page 325 and 326: 628 XX. T~lrbnlmt flow through pipe
Page 327 and 328: 632 XX. Turhulcnt flow through pipc
Page 329 and 330: 636 XXI. Td~nlcnt houndnry layers a
Page 331 and 332: 610 XXI. Ttrrbulent bor~ndnry layer
Page 333 and 334: 644 XXI. Turbulent boundary laycrs
Page 335 and 336: 048 XXI. Turbulent boundary layers
Page 337 and 338: 652 XXI. Turt)~~lcnt houndnry layer
Page 339 and 340: 656 XXI. Turbulent boundnry layers
Page 341 and 342: 660 XX I. Tl~rhlent bo~~ndary layer
Page 343: 664 XXI. Turbulent boundary layers
Page 347 and 348: 672 XXll. 'l'llc incon~prcssiblc tu
Page 349 and 350: 070 XXII. Tlic incompreaaibto lnrl)
Page 351 and 352: FRO XXII. Tho incornprcs~iblo turbu
Page 353 and 354: 684 XXII. 'Yhc incon~prcssible Lnrb
Page 355 and 356: 688 XXJI. The incon~prcssible t,urb
Page 357 and 358: 002 XXI1. Tho inro~nprc~sil~lo tt~r
Page 359 and 360: is prol)nt~t,iot~nl to the rnclius.
Page 361 and 362: lf$8] Mucsm:tnn, 11.: Z~~snrntncnhn
Page 363 and 364: 70-t XX I1 I. 'Ihrbulcnt bonrwlnry
Page 365 and 366: 708 XXIII. Turbulent bonndnry Iaycr
Page 367 and 368: 712 XXI 11. 'I'~~rl~~~lemt bo~~ntln
Page 369 and 370: 716 XXIII. 'I'urbr~lcnt bor~ntlnry
Page 371 and 372: 720 XXIII. Turbulent boundary layer
Page 373 and 374: 724 XXlll. 'rur1)ulcnt boundnry Iny
Page 375 and 376: [04] Smith, P.D.: An integral predi
Page 377 and 378: 732 XXIV. Frcc trtrbt~lent flows; j
Page 379 and 380: 700 XXIV. Prrc tr~rb~tlent flowa; j
Page 381 and 382: Neglecting u12, we obtain XXIV. Prc
Page 383 and 384: 744 XXIV. lhc L~trl)ulcnt Ilow~; jo
Page 385 and 386: 748 XXIV. Free turbulent flowa; jet
Page 387 and 388: 762 XXIV. Frcc td)ulcnt flows; jcta
Page 389 and 390: 756 XXIV. Prrc torbulent flows; jet
Page 391 and 392: 760 XXV. DotcrminnLion of profilo d
Page 393 and 394: 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
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
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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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