Boundary Lyer Theory

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710 XXIII. l'urhnlrnt. honntlnry lnycrn in rornprrmihlo flow In order to apply the preccding equation to particular cases it is still necessary to malzc a suitable ass~~mption about the ratio of thc mcan velocity at tho outer edge of the laminar suh-laycr to that in the free stream t. In the particular casc when P = 1, thc I'randtl-Taylor equation (23.18) reduces to Reynoltls's cquntion (23.16). In clcriving tho Prar~tlt~l-Taylor equation (23.18) it was supposed that tllc bountlary laycr coulcl bc sharply divitlcd into a turbulcnt laycr and a laminar sob-layer. In actual fact onc merges intp thc othcr in a continuous way antl it is possible to discern the cxistcnce of an inkrmcdiatc, or I~uCfer layer in which thc mngnit~~rlcs of tlic molccl~lar and tml)ulcnt cxchangc arc comparable. 'S?Il. vo11 I
712 XXI 11. 'I'~~rl~~~lemt bo~~ntlnry lnycrs in comprwsiblc flow b. Rclat,ion bctween velocity and ternpcrnture distrilmlion 7 1 3 Tnhlc 23. I. l'lw conntnntn n nrd 11 for bhc cnlrulntion or the coclficior~t nf hent transfer from cqn. (23.20) nnd of tho rcrovrry htor l'ron~ rrp. (2:1.27), nfler H. Ibichnrdt [73] nrtd J. C. Rottn [El]. 'l'lic tcmpcmturc tlistribution in turbulent boundary layers on flat platcs in the prescncc of an arbitrarily vn.rying, turbulent T'mndtl number, P,, was studied by R. R. von Ilricst 1283 :rnd J. C. Itotta 1811. In thc I;~tt.cr rcfcrcncc it is showti t,l~at only thc vsl~rcs whidi Ll~c turhulcnt I'r:dtl numbcr, P,, assumes c1o.s~ to t,he wall detcrrnine the rate of hcat transfer antl the tempcraturc distribution; conscqucntly, tlic details of the variation of P, away from the wall are less important. The variation of P, with tlist,:incc: from tfl~o w:Jl is hrougl~t to Iw:w only through tho int.crrnctliary or the cp~ntily A wlion in t.hc rcrnnintlcr thc val~~c of P, at the wall is substit~ited. A suitable value for this scems to be Pt = 0.9. J. R. Taylor [98] performed such calculations for boundary layers with variable pressure and temperature along the wall. 2. The transfer of heat from rough mrfaces. tt tins bccn clcmonstratctl in Secs. XXf and XXIc that rough surfaces develop considerably larger values of skin friction in turhulcnt flow than do smooth ones. The samc is true of thc coefficient of heat tmnsfcr. Normally, however, the percentage increase in the rate of hcat tmn.sfcr is smallcr t,l~arl that in skin friction. This is understandable, because a part of the turhulcnt sharing stresscs can bc transmitted to the wall through pressure forcw cxert~tl on protubcranccs; but thcrc exists no analogue for this mcchanism in Iicat, transfer. Expcrimenhl invcstig,ztions on the transfer of heat to a rough ttrbc wcrc carried out, among others, by W. Nunncr [66] and V. F. Dipprey and R. IT. Sabcrsky 1261. 'rh~ latter autliors made mcasurcmenta at different values of trhc I'rancltl nunibor. l'l~coretical conaidcmtions clue to 1). F. Ilipprcy and R. TI. Sal)crsky [26] :13 wwrl 113 to P. It. Owen antl W. It. Thomson [67] arc bnscd on thc hypothesis that the elfcct of roughness on the rncchanism of exchange is confined to the regions locatd in thc proximity of thc wall. Starting with this hypothesis, it is possible to derive an equation which has the same structure as cqn. (23.20), and differs onlg in that the term (P - P,) must be rcplaccd by a quantity, P, which is a function of the Prandtl numlm, P, and of the roughncss. In the particular case when P, = I, we othin I c ' R P N z - --- -- .--;:r _-%._ _ . .. (r)' ipprcy, Sabersky, Owen, 'I?~omsorr ; P, = I). (23.22) I -4- 4 c,' /I (a, k/v; P) Pig. 2:1.4. Tho roughnc~n funct,ion (p 4- 8.5) P-0.44 rts n ft~nr.t,ion of lJ+ k./v for sand FOII~IIIICRR nt vsrioun Prnnclll nu~~~bcrci, from 1110 lncn~r~ro~~~o~~l~ by I). IT lI)ipprc*y 111rt1 It. II. S1\11t!wI 70. '1'110 grnpl~ ol' fho function p togctlicr with tlic expcrimcntd rcsr~lt,~ ovcr Lhc wholc rarlgc of roughncss Reynolds numbcr v, ks/v is shown in Pig. 23.4. Owcn ant1 'l'homson corrclntcd experimental results from various sourccs, including thosc from refs. [25] and [66], and coricludcd that (23.24) Proccdurcs for thc calculatio~~ of heat-transfer ratcs in turbulcnt flows with nonisothermal surfaces havc been worked out by D. U. Spdding [88], and J. Kcstin antl coworkers [36, 46, 461. Extcnsivc mcasuremcnts under such conditions awe performed by W. C. Itcynolcls, W. M. Kays, and S. J. Klinc [77]. 3. Temperature distribution in comprceaiblc flow. In ortlw Lo ut~tlc~rat~ancl the laws which govern the tcmpcraturc distribution in compressible flows, tho rcadcr may wish first to rofcr to tho rolovant consirlcrntions for lnminnr I)o1111tlary lnyors wliicll wcrc advanccd in See. XlItb. Wlicn tlic pressurc remains (:oIIHI.:LI~I~ I I I ~ P = P, = 1, the tcmpcraturc distribution satisfies cqns. (13.12), and cqn. (13.13) in tho general case with hcat transfer, both owing to the evolution of frictional I~cat,. When P + P, $r 1, it is possible to cvaluatc thc recovery t~mpcrat~~irc on an (atliabatic) wall by tlic usc of cqn. (13.19), i. c. by The rccovory factor, r, is somowhat larger in turbulcrlt flow than it was in inrnir~ar
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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~~
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 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: 708 XXIII. Turbulent bonndnry Iaycr
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
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:
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Boundary Lyer Theory

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