Boundary Lyer Theory

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wliic:h rrscm1)lrs t,Ile pl~cnomrna now a sucldcn incrcnsc in cross-scct.ionn1 nrcn nntl somctitncs tlrscrild as (hrnott's loss. A thcorct,icnl cnlculxtlion bnscd on I'mntf(,l's tni sing length hy potlicsis sllowcltl that t,hc vclorit.y dist.ribution i~chnvcs in tho s:lrnr way ns in n t:ircwla.r \wkc (witlt.11 - r1/", centre-line vrlocity -~-~1:'). \\'l~cn $1, jet, of fini1.r wiclt.11 rmrrgrs it1t.o n ~~nifor~n st-ream, the uniform vclocity tlist.ril)nt,ion I)rcwi~rs l.r:~nsfortnrt1 I1c.n.r t,lic n1out.h of tho nozzlc into the prccctling prolilc. 'l'lic caso in hnntl wn.s sl,tijlietl by A. M. J
762 XXIV. Frcc td)ulcnt flows; jcta and wnltcs The first attempt to describe the circumstances of a wall jet by theory was under- taken by M. B. Glauert [16]. Thc former was considerably improved by E. A. Eiehel- brrnner et al. [13]. The semi-empirical theory succeeded for the first time in predicting the separation of a wall jet. Subsequently, J. S. Gartshore and B. (2. Newman [I41 established an integral-momentum method which was based on very extensive mea- surements. Thesc includcd wall jcts witll injection. The calculation made it possible to determine the numerical value of the momentum coefficient that is necessary to avoid separation of the wall jet. Further expcrimental results can bc found in the papers by P. Bmdshaw and M T. Gee [4] as well as of V. IZrulta and S. Rskinnzi [23]. The account by P. Thomas [51] describes expcrirncnts concerning thc mixing of a turbulent,, two-dimensional jet boundetl by a wall on one side with nn external flow on the other. Two-dimensional jete on highly convex, curved walls exhibit the wcll-lrnown C:nn.nrln cffrct, that is the adherence of the jet over wide tlistances along the wall in the flow directiofi. Expcrimental and t,heoretkal investigations into the pattern ereatcd by a plane jet flowing nlong the contour of a circular cylinder have been cn.rrird out, by -1. Gersten [Is]. 17. A. Dvorak [lo] deals with the calcnlat.ion of turbu- Irnt I)o~~ncla~ry hyors on highly convex, curved wall^, pixying special attontion t,o wall jcte flowing along cnrvctl walls. Wall jets are employed in practice for boundary layer cont,rol and in film cooling; compn.rc a,lso H.G. Ncwrnan [25a.], A. hTcl,rnl [24c, 24tlI anti I). W. Young [ROa]. Thrre-rlirnrwionnl unll jrls with a finite ratio of the two sides have been recently stutlirtl expcrimcntnlly Ijy 1'. M. Sforzr~ and G. Ilt.rl)st [42], hy 1%. G. Nrwmnn ct, nl. [25J, by N. V.C. Swnrny nnd U 11. Gowtl:~ 1431, as wcll as by N.V.C. Swarny and 1'. Bandyopatfhyay [44] 'rliese measurcmcnts revealcd a very fast ratc of spreading of the jet in the spanwise direction and the existence of a very different fictitious origin for the growth of the width of the jet In the parallel as opposed to the normal wall direction. e. Dill'usion of tempernlure in free turl~ulent flow r 7 I hc process of turbulent mixing causcs a tlransfcr of the proprrt,ics of the fluid in a tlirot:lion at right angles to tho main stream. On the onc hand the mixing motion enusrs ~nowc~tlicm to flow awny from tho tnriin sl,rc:tm, on the otllcr I~:rntl, p:rrt.iolcs srtspcntlt:d in the Iluitl (1lo:cting particles of dust, chcmical ndtlitivcs) arc directed into the stream, and in atltlit,ion there is a transfer of heat, that is a diKusion of a tempcrature field. The intensity of the transfer of a given property in turbulent motion is asually itcscribctl I)y a suital~le coefficient. Denoting the coefficient for momcntum tmnsfcr by A, ant1 that for heat I)y A,, we can define them (sccScc.XX11Ia) k1.y writing Ilere 11, antl r, T tlertofc morncntum a,rltl hcat per unit mass, respectively, and t and q clcnotfe the flux of momentum and heat (= quantity of heat per unit area ant1 tinic) rcspeclively. In t,his conncxion 14 and T denote temporal means. Since the mechanisms for the transfer of momeritnm antl heat are not iclent,ical the values of A, and A, arc, generally speaking, different. However, according to Prantltl's mixing-length theory the mechanisms of the transfer of ~nomcntnn~ :~,ntl 11r:~t it1 free turbulent flows are itlent~ical which means that A, and A,, are nss~ltnr~l c:cl11:11 to each other. The messuretncnt pcrformctl by A. 1i'nge and V. M. I~:rlkncr 1.50) in the wake behind a row of heat,ctl bars have shown that t,hc tcmprmtrrrc prolilo is witlcr than the velocity profilc antl that, by way of approxirnntion, wc mn.y assrtritc A, = 2 A,. This rcsull agrccs with Cr. I. T:~ylor's tlicory which was t1isc:ussctl in Scc. XIXc, and according to which brlrl)rtlcnt mixing ~not~ion c:~,tlscs :LII c.xc:ltnngc: of vorticity rather than momentum. The problem of t,he tliIFusiorl of tctnprratnrc in free turbulcnt flows was also consitlcrctl I)y It. 1tcicha.rtlt 1301, who tn:~.tl(> I)ot.l~ tlicorctical antl cxpcrirnont,al contribrtl,ions. 'J'he thcorctic.:d work is closoly rol:~lctl to t.ltat tlcscribed in thc prccecling scction. First, empirical rclat~ions have bco~l tlrtlucctl for the temperature profile from expcrimcntal rcsults in the same way as was tlonc previously for the velocity (nlomentum) distsribution, hypothcscs on turbulcnt flow having been avoided. On the basis of an argument which we sl~all omit. I~cre, Rcichardt succeeded in Jcriving a rcn~arlrnble relation bctwccti l,hc t~ctnpc~ral.urc and the velocity distribution. This is given by T = (ufn:)Ar'Aq . (24.60) Tmnz Hers, the subscript mas refers to the n~aximnni valucs, and the sc:llt:s for 11. :111(1 7' must I)c so rcrrattgctl ns Lo rcntlnr Lhc poinl,s li)r WII~CII TC r - 0 n~~tl 7' . - 0 t.oi~t(.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
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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
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 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: 748 XXIV. Free turbulent flowa; jet
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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