Boundary Lyer Theory

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[40] fioffninnn, E.: Ikr WRrmciibrrgeng bei drr Stromung im Itohr. X. Ges. K5ltr-Ind. 44, nn -107 - II!I:~I \- .,. [41] Johnson, 1).S.: Vclocsity, temprmtl~rc, and hmt transfer mcnsuren~cnts in a t~~rbulcnt bountlnry lnyrr do~~r~trrxtn of n ntepwise d~scontinuity in wall tempcrntc~rc. J. Appl. Mcch. 24 ..' 2 - X .. 1 ,-..,.,. lOTr7I (421 .Johnnori, D.S. : Vclocit,y and l~niperat~urc fluctuation rncnaurcmei~ts in a t~~rbl~lcnt houndary laycr downntrcnrn of a step\visc discontinuity in wall tenlporat~tre. Trn~ls. ASME ,I. Appl. Mecdi. 26, 32.5--33G (I 9.59). [43] von KB.rn16.n. Th.: '1'hc ptoblctn of rcsi~tancc in cotnprc~niblc fluirls. Volta Congress Rome 19JR. 222 -277; see also Coll. Works Ill, 17!)-221. 1441 von Khnhn, '1'11.: 'rho analogy bet.wccw lluid friction and hcnt t,ranafcr. Trnns. ASME 61, 70R--7 10 (I!):)!)); RCP 111~0 (:011. Work8 111, 3.56-X7. 14.51 Koatin, .I., and Hicltartlson, P.D.: Heat tmnufer ncrom turbnlent incompressible boundary Inyera. Int.. J. Ilcat anti Mnun 'I'rnnufor 6, 147-189 (1963). 1461 Kcstin, J., nnd Ricl~arclson, 1'.1).: Wiirmeiibcrbrngung in turbulenten (:ren7*lcllichten. Foruchp;. 1ng.-WCR. 29, 93-- 104 (1963). [46a] Kcstin, J., and Perscn. L.N.: The Imnafer of hcnt across a turbulent boundary layer at very high l'riindtl numbera. lnt. .J. I-lmt and Mass Transfer 5, 355-371 (1962). 1471 KistJer, A.L.:' Fluct.unt.ion rneasnrctiicti~ in a supcrsonic tnrbulcnt boundary layer. Phys. @ Iui& 2, 2!)0 - 296 (1969). 1481 Klobnnoff, Y.S.: Chnrackristic8of turbulence in n boundary layerwith zeropressurcgmtlient. NACA TN 3178 (1954); TR 1247, 1135-1163 (1955). [49] Kovnuenny, L.S.O.: The hot-wire nncmometer in ~npersonic flow. JAS 17,565-673 (1950). [50] Kutateladze, S.S., and Leont'ev, A.I.: Turbulent boundary layer in compressible gesee. I'ransl. by D.U. Spnlding. Edward Arnold Publishera Ltd., London, 1904. [6l] Lilley, G.M.: An npproximation solution of the turbulent boundary lnycr equation in incompressible and compressible flow. Coll. Aero. Cranfield Rep. 134 (1960). 1621 Licpmann, H.W., and Goddard, F.E.: Note on the Mach number effect upon the skin friction of rough sitrfncca. JAS 24, 7R4 (1957). [R3] Lohb, R.K., Winklcr, EM., and Porsh, J.: ICxpcrirncntnl invratigntionoftr~rbr~lettt bormdary layern in hypcrnonic flow. NAVOJtD Rep. 3880 (1955). [54] Ludwieg, li.: Ein Gerat zur Measnng der Wandschubspannung turbulcnter Reibungsschichten. 1ng.-Arch. 17, 207-218 (1949). [65] Ludwicg, 11.: 13cntiminung des Verhrilt.niaaea der A~tstauachkoeffizienten fiir Warme and Inipula bci tnr1)ulenten Grcn7achichten. ZFW 4, 73-81 (1966). [RO] Mack, L.M.: An experimental investigntion of the temperature recovery-factor. Jet Propuluion Laboratnry, Calif. IIIRL. TCCIIII., Ynundcnn, Rep. 20-80 (1054). 1571 Mnger, 8.: Trnnalbrmntion of tlic compresuible turbulent boundary layer. JAS 25,305-31 1 (1958). [BR] Matting. P.W., Chapman, D.R., Nyholtn, J.R., and tho me^, A.G.: Turbulent skin friction at high Mach nnmbcrr and Reynold~ numbers in air and helium. NASA TI2 R-82 (1961). [59] Mchffcrty, O.11.. irntl Hehbor, RE.: 'rho cffect of ndverae proasure gradienb on the characteristic~ of turhulcnt honndary lnyers in supcrsonic atrcams. JASS 29, 1-10, 18 (1962). [60] Meier, H.U.: Experirncntellc und thcorctische Unterauchungen von turbulenten Grenzschichten bei Uherschnlistrii~n~~ng. Mitt. MPI Stromungsfonchg. u. Aerodyn. Vernuchsanat. Nr. 49, 1 - 136, (1070); Dim. Brnrtnschweig 1970. . [61] Mcier, H.U., Lee, R.E., and Voisinet, I2.L.P.: Vergleichsmessungen mit einer Danberg- Teniperaturaonde und einer kombinierten Druck-Tempcraturaonde in turbulenten Grenzachichtcn bci Obcrsclinllotro~n~tng. ZPW. 22, 1-10 (1974). [62] Moicr, H.U., Voiainct, R.L.P.; and Gatea, D.P.: Temperature diatributiona using the law of the wall for cornprcauible flow with variable turbulent Prandtl numbers. AIAA 7th Fluid and Plnnma Dynamics Conf.. Palo Alto, Calif. 1974, AIAA Paper No. 74-696 (1974). 16.31 Mcicr, H.11., and Jtotta, J.C.: Tempcrature di~tfibntions in supersonic turbulent boundary Inyers. AIAA J. 9, 2149-21R6 (1971). I641 Meior. H.U.: 1nvcstignt.ion of the hcat tranafcr tncclianism in auperuonic turbulent boundary Inycrs. Wiirme- und Sbffiihcrtragnng 8, 169-165 (1975). [c,:',] Morkovin, M. V.: Effect* of comprraaibility on turbulent flows. Colloqucs Jnt. CNRS No. IOR, 3fi7- 3R0, MCcirniquo dc In turbulcn~~, Marseille, 1962. [66] N~tnncr, W.: Wiirtneiibergang und I)rnckabfalI in rauhen Rohrcn. VDI-Porxcli. 455 (1956). 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). [69] Perscn, L.N.: A note on the bnnic cquntions of tr~rbulcnt, boundary laycn nntl tho hcat transfer throng11 such laycrs. ZFW 15, 31 1-314 (1967). [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. [71] Reichardt, H.: Dio Wiirmciihcrtragnng it1 Cnrb~~lonbn Rcil~~tngsscllicl~te~l. ZAMM 20, 297.- 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 (1944). [73] Reichnrdt, 11.: Ocr Eilifluu~ der wnndnahcn St,rijmung nt~f tlcn trrrbnlcnten Wiirmciibcrnng. Rcp. Max-Plnnck-Inst, fiir St~riit~tungsforscl~~~ng No. 3, 1---li3 (1!)60). [74] fZeichardt, H.: Die Grundlngen des turbulctrten Wiirmeiibcrgn~lges. Arch. Wiirtnotc!chn. 2, 129- 142 (1061). [75] Reahotko, E., and Tucker, M.: Approxirnate calculation of the compre~~ihle tllrbulent boundary layer with heat transfer and arbitrary preaanre grdicnt. NACA TN 4164 (19.571. (761 Reynolds, 0.: 011 tho extent and nct,ion of tho heating surface for rtcaln hnilcrs. h"~. Munchester Lit. Phil. Soc. 14, 7-12 (1874). [77] Reynolds, W.C., Kays, Mr.M., and Kline, S. J.: Hcat trnnufcrin the turbulcnt inconlpr~~ible boundary Iayor. I. Constant wall temperature. NASA Mcmo. 12-1-58 W (1968); 11. Stop wnll tcmpernt,ure distribution. NASA Mcmo. 12 -2-68 W (1958); 111. Arbitrary wall temperatnrc and heat flux. NASA Mcmo. 12-3-58 W (1958); IV. Effect or loc:ntion of trnnsition nnd prcdiction of hont tmnafor in n known tran~ibion ragion. NASA Momo. 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 t. . Rep, 281 (1900). [80] Itotta, J.C.: llemerkung zum Einflnes dcr 1)ichtoschwank~ngcn in turbulcnlotl Urcns- scliichten bei kom~reaaibler Stromung. 1ng.-Arch. 32, 187-190 (1963). - .. . [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). [83] Rotta, J.C.: Eine Bcziehung zwischen den ortlichcn Ibib~rngsbciwerten turbulcnkr Grcnzachichten hci kotnpressiblcr und inkomproasiblcr Stromong. ZFW 18, 195-201 (1!)70). [84] Itott.~, J.C.: POICI'RAN IV - Hechrnprogmmm fiir ~rcnzachichtcn hoi kotnlirrsd~lon ebenen untl achuensymmetrischen Strbmungcn. DLR FB 71 -51, 1-82 (1971). 1851 Rubeain. M.W.: A modified Reynolda analogy for the conlprcssible turbulrnt boundary laycr on a flat plate. NACA TN 2917 (1963). . 1861 Schubauer. G.U.. and Tchen, C.M.: Turbdent flow. High Spcocl Aorodyt~trtnics and Jot > Propulsion V, 75-196, Princeton (1959). [87] Seiff, A.: Examination of the existing data on the hcat transfer of turbulent houndnry lavera at su~enonic speeds from the point of view of Reynolds analogy. NACA TN 3284 Heat trnnsfer to a turbnlent stream from a surface with n step-wise tliscontinuity in wall temperature. International dovcloptnents in hcat tranafcr (L'roc. Cod. organized by ASME at Boulder, Coloredo, 1961), Part 11, 439-446. [89] Spalding, D.B., and Chi, S.W.: The drag of n compressible turbulent boudary layer on a smooth flat. plate wit.h and without hcat trnnsfer. JFM 18, 117-143 (1964). [90] Spence, D.A.: Velocity and enthalpy distributions in the compr~lsible turbnlent boundary layer on a flat plato. JFM 8, 368-387 (19GO). [9l] Spcnce, D.A.: Some applicatione of Crocco'a integral for tho turbulent boundary Iayor. Proc. 1960 Heat Transfer Fluid Mech. Inat., Stanford Univ. 62-76 (1960). [92] Spmce, D.A.: The growth of comprwible turbulent boundary layers on isothermal and adtubatic walls. ARC RM 3191 (1961). [93] Stratford, B.S., and Beavers, G.S.: The calculation of tho con~prmaiblc tnrbulcnt bonndnry layer in an arbitrary presaure gradient. A correlation of certain previous mclhods. ARC RM 3207 (3959).
[04] Smith, P.D.: An integral prediction met.hnd for three-dimensional compressible turbulent bou~ldary Inyers. ARC IIM 3730, 1-54 (1074). [BB] 8cln1ltz-Jantler, R.: Hont t,ransfer cnlculrlt,ions in tnrhulcnt boundary layers rtsing integral relations. Acta Mechanicn 21, 301-312 (1075). [!lo] Tnylor, G.I.: Conditions st the swfnco of a hot body exposed to the wind. ARC RM 272 I1RlRL 3 , [07] Tnylor, G.1.: Tlw transport of vorticity and hat through fluids in tnrbulent motion. Appendix hy A. Fagc and V. M. J'alknrr. l'roc. Roy. Soc. 135, 685 (1032); .we also Phil. Tram. A 215, 1 (1015). / Taylor, .J.It.: Tetnpcraturr and heat RIIX tlistriht~tions in incon~nrrosible tr~rbulent eouilibriun~ honntlary iayero. Int. .I. llcnt MILRR 'I'rnnsfcr 15, 2473--i488 (1!)72). Tnclcer, M.: Approxitni~tc t,urbulcnt hor~ntlary layer tlcvclopn~ent in planc? con~pressible flow along t~hrrnmally insulated sr~rfaccu wit.11 application to uupcrso~~ic-t,ur~nrI contour corrcct,ion. NACA TN 2045, 78 (1950). Walz, A.: Niil~er~~ngstheorie fiir kornpressihlc t~~rbulente Grenzncl~ichten. ZAMM-Sontlerheft 36, 50-56 (1050). Walz, A.: Uher Fortsc11rit.k in Niihcrwigsthnorir r~nd I'mxis tler Rercchnung kornprcssibler Ian~ino.rer nnd turbulenter Grenzschicht~en mit Wiirn~eiibergang. ZFW 13, 80-102 (1065). Wilson, It. E. : Turbulent bountlnry layer chamobristics at supersonic npectls - <strong>Theory</strong> sntl experiment. JAB 17, 585-504 (1!)5O). Winltler, 1E.M.: Invcstipntion of flat plate hypersonic turbulent boondary layers with heat transfer. J. Appl. Mech. 83, 323-32!) (1901). Winkler. E.M., and Cha, IM.11.: Investigation of fiat plate hypersonic turbulent boundary layers with heat t,ransfrr at a Mnch nr~nil)cr of 5.2 (U). NAVORL) Rep. 0031 (19.59). Winter, K. G., Roth. J.C., and Slnith, I
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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
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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
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 and 344: 664 XXI. Turbulent boundary layers
Page 345 and 346: CHAPTER XXII The incompreesible tur
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: 724 XXlll. 'rur1)ulcnt boundnry Iny
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
Page 395 and 396: 768 XXV. Determination of profile d
Page 397 and 398: XXV. 1)obrlninalion of proRlo drag
Page 399 and 400: 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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