Boundary Lyer Theory

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can I)r rct.ni~rrvl in inrotnl~rrssil~lt~ flow :IISO. I)ut, the inf~cr~~rctnt~ion wit11 reference to atli:~l):~t.ic: t:omprcwion vcnsr>s l,o I)(: vnlitl. It is now possil)lc 1.0 conclutle that frictiot~al hc~11, nncl IICRI, (IIIC 1.0 (:ompr~~sion :Ire in~port~nnt. for the c:n.lcnlnt.iorr of the temperature fivltl whrn tltr I'rw-st,rrn.~n vc4oc:ity 11, is so lnrgc t.l~n(, t,lro ntlinhntic t,crnl,er;~ture int:rtx:~sr is ol' l,ltc S:IIII(~ orb nf m;~p~iI~t~lt~ :IS I,lrc prt?srril)c~l I~~:~r~pt>rnt,~trc (lifT
276 XtI. Thermal boundary layers in laniinar flow Whrn sprcial solr~tions are consitlerctl thcn, in most cases, one or more of the tlimensionlrss groups will disappear as the problem will only seldom be of this most general nature. As srrn from eqn. (12 27) the temperature field and, hence, the coefficicnt of heat transfer tlcpcnd on the Eckert number only when the tcmperature cliKcrences arc large (50 to 100° C or 100 to 200° F) and whcn, simultaneously, the vclocitics arc very large and of the order of the velocity of sound. With moderate vclocitirs the ten~peraturc and vclocity fields depend on the Eckert number when temperaturc differences arc small (several degrees). Further, even with moderate vrlocitics, the buoyancy forccs in eqn. (12.21) caused by temperature differences arc small comparcd with the inertia and friction forces. In such cases the problem ceases to dcpend on thc Grashof number. Such flows are called forced flows. Iience, for forced wnvrdion N, = f (R, P) (forced convection) . The Gmshof nwnbcr becomes important only at very small velocities of flow, particularly if the motion is caused by buoyancy forces, such as in the stream which riscs along a heatcd vert.ica1 plate. Such flows are callctl natural, and we refer to thc prohlcrn as one in durn1 co?~vection. In wch cases thc flow becomes indepcntlrnt of the Rcynoltls numbcr, and N, = f (G, P) (natural convection) . Examples of problcms in forced flow are given in Sccs. c to g of the present chapter; cxan~ples of problems in natural convection are contained in Sec. h. (I. Exnct solr~tions for the problem of temperature clistributinn in a viscow flow We shall now proceed to solve several particular problems of temperature distribution. The examples to be discussed will be sclected from the Iarge number of possible cases on the ground of mathematical simplicity. We shall begin by dis- cussing several cases of exact solutions, as given by H. Schlichting [loll, just as we have begun with the discussion of examples of exact solutions of the equations of flow with friction in Chap. V. For the case of incompressible two-dimensional flow with constant properties the system of equations for the velocity and temperature distxibution in steady flow along a horizontal z, z-plane we obtain from eqns. (12.17) to (12.19): d. Exact aolutiona for the problenl of temperature distribution in a viscoua flow Fig. 12.4. Velocity and tcmpcraturo T-T, distribuLion in Couette flow. a) Velocity distribution. b) Temperature distribution with heat generated by friction when the temperatures of both walls arc equal. c) Tcmperatrlre distribaLion with hc~t gcncnted by Friction for tho cast when Lhc lower wall iu non-conducting where 1. Coucttc flow. A particularly simple exact solution of t,llis systcm is obtained for Couette pow, i. e. for the case of flow between two parallel flat walls of which one is at rest, the othcr moving with a constant vclocity U1 in its own plnnc, l'ig. 12.4. The solution of the equations of motion in the absence of a prcssurc gratlic~~t in the x-direction is A very simple solution for the temperature distribution is obtained when it is pos- tulated that the temperature is constant dong the wall, the boundary conditions being 277 y=O: T=T,; y=h: l'=T,. (12.35a) In this case thc dissipation function reduces to the simplc expression @ = (a~/ay)~, and the equation for temperature distribution becomes consequently With thc boundary conditions (12.35a), thc above equation has a solution which is independent of x. Since, with v = 0, the term v aT/ay on thc Icft-hand sitlc also vanishes, all the convective terms on the left-hand side of eqn. (12.34) become cqual to zero. The resulting temperature distribution is, thercfore, due solely to the gencration of heat through friction and to conduction in the transverse direction. Prom eqn. (12.35b) we obtain d2T k-=-p dy2 (:)', (12.35~) and, substituting dultly, we have 'UIC solution of this cquntion which sntisfics conditions (12.36n) is
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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
Page 98 and 99: 174 TX. Exnct ~olut,ionu of tho sto
Page 100 and 101: 178 IX, Exact solutions of tlrc ste
Page 102 and 103: and t,llc? clnsh now clet~otos tlil
Page 104 and 105: 1 86 10 0.8 0.6 0.4 0.2 0 -0.2 -0.4
Page 106 and 107: ccnt,rred nt, (m -1- 1, n). 1'110 t
Page 108 and 109: 194 IX. Jqxnct eolutions of thc stc
Page 110 and 111: O~l:cr ehnpw: Second-order cITcetls
Page 112 and 113: 202 X. Approximate rncl.hoda for st
Page 114 and 115: 206 X. Approxitnnte rnct.l~otls for
Page 116 and 117: 210 X. Approximate mcthod~ for shad
Page 118 and 119: 214 X. Approximato mothotls for sha
Page 120 and 121: 218 X. Approximntn met,hods for shn
Page 122 and 123: 222 X. Approximate methods for stea
Page 124 and 125: 220 XI. Axinlly symniobricnl nnd th
Page 126 and 127: 230 XI. Axinlly symmctricnl and thr
Page 128 and 129: 23.1- XI. Axially uynimet.rira1 nnc
Page 130 and 131: the two wries expnnsicws for sin (%
Page 132 and 133: 242 XI. Axially nymmct.riral and tl
Page 134 and 135: in cross-flow, tlrprntls only on lh
Page 136 and 137: 250 XI. Axinlly nyrnmet,rirnl nntl
Page 138 and 139: XI. Axinlly ~,vn~n~et.rir;~l nnrl I
Page 140 and 141: 258 XI. AxhIly syn~rnrtrirnl nncl t
Page 142 and 143: . . lit 1 h1:lgt.r. '1.: 'l'l~idc I
Page 144 and 145: 206 XIT. l'l~rr~nnl bo1111r11iry ln
Page 146 and 147: 270 XIT. 'I'l~rrtnal boundary layer
Page 150 and 151: if wc: pillf - - 'I1, - (A7'),. 11,
Page 152 and 153: Rotntirig rli~k: (:II:I~I. V, in pn
Page 154 and 155: 286 X11. 'IY~rrrr~al bo~~ndnry laye
Page 156 and 157: 290 XlI. Tl~rrnmal boundary layers
Page 158 and 159: with 0,' -. () at, r] - 0 and 0, =
Page 160 and 161: 2!)H XI[. Thcrn~al bor~nrlnry Inyrr
Page 162 and 163: 302 XJI. Therninl boundary layers i
Page 164 and 165: 306 XII. Tllcrrnal hor~ndary layers
Page 166 and 167: 310 XJI. 'I'hcrmnl boundnry layers
Page 168 and 169: 314 X11. Thermal boundary laycrs in
Page 170 and 171: 318 XIT. Tl~crnial bonndnry layers
Page 172 and 173: 322 XII. Tl~or~nnl I)onndnry lnyers
Page 174 and 175: 326 X l I. 'l'l~crn~nl hounrlnry ln
Page 176 and 177: 330 X[[1. lmninar bor~ndnry lnyers
Page 178 and 179: 334 XIIT. Lnrninnr 1)oundary laycra
Page 180 and 181: 338 XI I I. Im~linnr I>orirtclt~ry
Page 182 and 183: 342 XI11. Tmninnr Imlndary layeru i
Page 184 and 185: 346 XI 11. T,ntninar I~or~nrjary hy
Page 186 and 187: 350 XIII. 1,mninnr honnclxry lnycrn
Page 188 and 189: 354 XI 11. 1,nminnr h~nrlnry 1:ryrm
Page 190 and 191: Fig.. l3.16. In 13.18. lain~innr ho
Page 192 and 193: a Intninn.r l)out~rlary I:~.ycr, bu
Page 194 and 195: 1 I Itc vnriolts c4Twl.s ol'sl~oclt
Page 196 and 197: 370 XIIJ. I~tminar boundary layers
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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
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048 XXI. Turbulent boundary layers
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652 XXI. Turt)~~lcnt houndnry layer
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656 XXI. Turbulent boundnry layers
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660 XX I. Tl~rhlent bo~~ndary layer
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664 XXI. Turbulent boundary layers
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CHAPTER XXII The incompreesible tur
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672 XXll. 'l'llc incon~prcssiblc tu
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070 XXII. Tlic incompreaaibto lnrl)
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FRO XXII. Tho incornprcs~iblo turbu
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684 XXII. 'Yhc incon~prcssible Lnrb
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688 XXJI. The incon~prcssible t,urb
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002 XXI1. Tho inro~nprc~sil~lo tt~r
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is prol)nt~t,iot~nl to the rnclius.
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lf$8] Mucsm:tnn, 11.: Z~~snrntncnhn
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70-t XX I1 I. 'Ihrbulcnt bonrwlnry
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708 XXIII. Turbulent bonndnry Iaycr
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712 XXI 11. 'I'~~rl~~~lemt bo~~ntln
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716 XXIII. 'I'urbr~lcnt bor~ntlnry
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720 XXIII. Turbulent boundary layer
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724 XXlll. 'rur1)ulcnt boundnry Iny
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[04] Smith, P.D.: An integral predi
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732 XXIV. Frcc trtrbt~lent flows; j
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700 XXIV. Prrc tr~rb~tlent flowa; j
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Neglecting u12, we obtain XXIV. Prc
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744 XXIV. lhc L~trl)ulcnt Ilow~; jo
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748 XXIV. Free turbulent flowa; jet
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762 XXIV. Frcc td)ulcnt flows; jcta
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756 XXIV. Prrc torbulent flows; jet
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760 XXV. DotcrminnLion of profilo d
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764 XXV. I)ot.crtninntiotl of profi
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768 XXV. Determination of profile d
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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
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Oswatitach, I
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H:I:IR, 11. 776, 777 Ilnnsr. 1). 62
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Scl~crb:trl,l~, I
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805 811bjrct lntlrx 209, 354, 385,
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812 Snhjcct lntlcx - vorl.irrs 526,
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List of most commonly used synibola
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Boundary Lyer Theory

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