Boundary Lyer Theory

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106 V. JCxact solutions of the Navicr-Stokm cq~~ntiona b. Othrr exact, solutions 107 This gives or, tlcfining a Reynolds number based on thc radius and tip vclocity, R'o R = - - and int.ro~lttcing thc nnmerical vnll~c - 2 zG'(0) = 3.87, wc obtain finally Fig. 6.14 shows n plot of this equation, curve (I), and compares it with mcasuremcnta 1391. For RcYnolcls numbcrs up to about R = 3 x LOS there is cxcellcnt agreement hnt,vecn tltoory nnd exporimcnt. At highor Raynolds numbers the flow bccorncs turbulent, an11 tho respective case is considered in Chap. XXI. Curves (2) and (1) in Fig. 6.14 arc ohtainnl from thc turbulent flow thcory. Oldor mcasuremcnts, carried out hy G. Kernpf [lG] and W. Schmidb [31], show tolerable ugrrwnrnt with throrotiral resalts. Prior tn Lrsr aol~t~ions, I). Riahoachinsky [2Gj. 1271 estaldiahcd cmpiricnl fonnulac for the turning mon~cet of rotating disks wllich werc hmcd on vcry carcful mcasurrments. Those formulae showed very good sorrcmcnt with the thcorctical equations discovcred suhscqucntly. -0- - The quantity of liquid which is pumpcd outwards N a result of thc centrifuging nctdon on tho one sidc of a disk of radins R is o NACA Report No. 7g.3 v 0.48 lo 1.69 Kernpf 0 NSchmidt Fig. 5.14. Turning mo- ment on a rohting dink; crlrvc (1) from eqn. (LM), hmimr; eurvea (2) and (3) from eqns. (21.30) and (21.33). tw- bulenl Calculation shows t,hnt Q = 0-885 n R2 (11 = 0.885 n Rn (1, R-lI2 . (5.67) ,I I he q~tanf.ity of fluid flowing towards thc disk in the axial dircct,iorl is of cqrlsl ~napnitutlc. Jt is, filrther, wortlly or no(,c tht tJtc pressure tlini:rrt~cc ovcr t,hn 1:~yrr carried by the (lislr is of tho orrior e r1 a), i. c., vcry small for s~unll vi~co~it~ics. TIIC prcssnm (Iist,ril~ntion tlcpcntls only on tho tlist,nrlc:o from th(1 wall, rinrl (.II(w is r~o riuli:r,l ~~rcss~trc grntlicwt,. A generalisctl fnrnl of tl~e prccetling problem has becn stutlictl 11y M. G, Itogers and G. N. Lance [28] who assumed that the hid moves with an nnnllIar Vl,IO(*i(,V antl thc sccond boilntlnry contlition for tho function G(() mttat, Ito rrplncwl t)y C(m) = s . In this conncxion a comparison should hc mndo with thr caso ofrotating flow ovrr n fixrd disk given in Scc. XTn. Nnmcrical ~olutions for rotatio~~ ilk tl\c s:nnc srnsc (s > 0) can be found in [20]. Whcn the rotations arc in opposite scnsps (s < 0). physically meaningful solutions can bc obtained for s < - 0 2 only iTunifc)rrn suction :it right, :i~~gIrs to the dislr is n(lniittc(1. The prol~lem of a rotating dislr in a housing is discussed in Chap. XXl. It, is particularly tiotcwor(hy that the solutior~ for tlrc rotating disk as wcll as 1.llc solutions obtained for the flow with stagnation are, in the first place, exact solutions of the Nnvicr-Stokes cquntions anti, in the sccond, that thcy are of a houi~drcry-la?/rr (me of vcry small viscosity t,hese solntions show that tho influence of viscosit.y rxl.rntls over a vcry small lnycr in tllc ~~cighl~ot~rl~ootl of. the solid wnll, ~ 1 1 c ~ t . c : : ~ ~ ill 1,llc wl~olo of 1.l1c rcmnining region t.hc flow is, j)rnct,ic;llly spcalring, i(lrnt.i(~:ll \vit,l~ (.he corrcspontling itlcal (potcnti:~l) cnsc. '~hcsc cxamplcs show Surthor l.l~;~t the b~~nnilary hyer has a thickness of the order iv . The one-dimensional examples of flow discussed previonsly display tho snmc bonntlnry-layer character. In this conr~cx-ior~ the rcatlcr may wish to conwit a pnpcr by G. I
The periphcrnl vclocit,y vanishes evcrywhcrc. Int,rodnring (his form into the Navier-St,oltes cqllnLiolls writ.lnl~ ill I~oln,r rc~nrtlinnlcs, cqn. (:l.:l(i), nntl rlin~in;~l.i~~g prmsllrc from t.hc cql~nt,iollu ill t,llo r nlltl $ clircrtions, we obtain t.hc following ordinary tlilhrrnl.i:ll cq11a1.io11 for I('(4): 'I'lle co~~st,:rnt, I( clrnotrs the radinl prrssurc gr:~.clic.nt, ;tl, 1.11~ ~~alls, Ii =- --(I/@) (i)l~/ar) (r"v2), where \ve llave 17 0 fore - n anti+ - -a. ns wc:ll as E" - 0 ford) -7 0. The so111lio11 ofrq11. (5.58) ,,.as givrn by (;. lIalncl [ll],'l'l~c f11nrtion 1.' ran be cxprrssrtl rxplicit.ly ns an ellil)l.ic fnnction of$. \Vc sllnll now briefly sltrt,ch t.l~c ch:~rnrt.cr, of the solution refrailling from cliscussi11g the (let,nils of t.lre derivation. The grqh in Fig. 5.15 shows n family of vclocit,y profiles for a tollvcrgc~~t, and a clivrrgcnt chnnnrl for diITerent. IZ.cynoltls nnn~bers plot,tcd on the I)asis of the Il~~r~lerinaI raIor~l:itionn pcrrorrncd by 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
Page 14 and 15: even in fluitls wit,lt vcry srnall
Page 16 and 17: 10 I. Or~llinr of lluicl rnot,ion w
Page 18 and 19: The vclwil y IL at some point i11 t
Page 20 and 21: Fig. 1.6. Firld of flow of oil nho~
Page 22 and 23: 22 I. Outliric of fluid motion with
Page 24 and 25: 2 (i TI. O~~tlittr of Imun~lnry-lsy
Page 26 and 27: Fie. 2.511 Fig. 2 .5~ Fig. 2.5b Fig
Page 28 and 29: S - point orscpnrnt.ion T'ig. 2.12.
Page 30 and 31: 3 8 \ Y?\ If. 011tli11e of boundary
Page 32 and 33: 42 11. 011tli11e of Iw~~ntlnry-Inyr
Page 34 and 35: [:j2a] I?o~cilhrntl, I,.: Thr forn~
Page 36 and 37: 50 111. l)crivnt,ion of thc cquntio
Page 38 and 39: Fig. 3.3. Tmcd clintor1,ion of flui
Page 40 and 41: of flltitl is strcsscd in thrcc nlu
Page 42 and 43: 1:i~ 3.8. Qltnsintzt.ia cotnprrssio
Page 44 and 45: 66 111. 1)wivntion of the eqnntions
Page 46 and 47: CHAPTER I V General properties of t
Page 48 and 49: 74 TV. Gencrol proprtic~ of tho Nov
Page 50 and 51: Ilcre v, c, :tntl k tlrnol,c 1.I1t:
Page 52 and 53: conclil.ion (4.14) plays n part wl~
Page 54 and 55: 86 V. 1':xact solul ions of tlir N:
Page 56 and 57: 90 V. ICxnct sol~ttion~ of tho Nnvi
Page 58 and 59: 94 V. Exnct sohltiono ol' tho Nnvic
Page 60 and 61: Table 5.1. Functions occrtrring in
Page 62 and 63: 11. Flow taenr n rotnting disk. A f
Page 66 and 67: cnu bo npplirtl nt mont, nn fnr RS
Page 68 and 69: 114 VI. Vcry slow motion where r2 =
Page 70 and 71: 1 I8 VT. Very slow motion r. 'l'l~o
Page 72 and 73: 122 VI. Very rrlow motion ext.endcd
Page 74 and 75: 126 VT. \'cry slow rnot.ion Part B.
Page 76 and 77: 130 VII. Boundnry-layor cquntionzl
Page 78 and 79: '1. Skin friction \VlIat1 t.11~ I)o
Page 80 and 81: 138 VII. no~~ndnry-layer cqnntions
Page 82 and 83: 142 VII. Hor~nclnry-lnyrr rq~~ntii~
Page 84 and 85: 146 VII. I~o~~ntlnr~ Inyrr cq~~ntio
Page 86 and 87: CIIAFTER VIII Gencral propertiee of
Page 88 and 89: 164 VIII. Ccnernl propertic8 of Lhe
Page 90 and 91: 158 VIII. General ppropcrtics of th
Page 92 and 93: VIII. General propertien of the bou
Page 94 and 95: 'I'his c.quat,ion Lmnsforms into t1
Page 96 and 97: 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
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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
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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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