Boundary Lyer Theory

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626 XX. 'rnrb~rlent flow through pip@ h. Flow in cttrvctl pipes nntl rliffuscrs 627 roughness showctl vnlurs of ks/R = 1/40 to 1/20, as calculated from the resistance coefficient wl~icli was, in turn, dctcrmincd with the aid of the mcasnred values of niasn Ilow. It appcnrs, therefore, that rib-like corrugations lead to much higher rcsistances thnn sand roughness of thc name absolute dimension. Extcnsivc experiments on tho increase in the rcsistnnce found in commercial ducts, for example in mine shafts, can be found tlcscribed in a paper by E. IIuebner [26]. Purt.licr tlct.nils concerning t,hc rrsisl.:~ncc offcrcxi to flow by rough wnlls, perticularly t.hosc due to single protrusions, will be givcn in Chnp. XXt in conncxion with t,lic (lis~nssion on t.hc r~sistnncc of flat plntrs. h. Flow in curved pipes ntd diuu~cr~ Carved pipca: 'l'lir prcc.rcling considcrntions conrcrning pipc flow n.rc valid only for sl.r:ligI~t~ pipes. JII curvcd pipes thcrc rxisb a, scronthry flow hcrn~wn t,hc pnrtielcs ncnr t.hc flow axis wllicll 11:tvc a higl~er vclocil,y arc act.ccl upon by n lnrgcr ccnlrifugal force t.hnn t.ltc slower pnrticlcs near the walls. This Icntls to 611e cmcrgcnce of a sccontlnry flow which is tlirectctl ontwxrds in t.hc rcntrr and inwards (i. e. towards t.ht ccr~t~re of rr~rvnt~nrc) nrnr t h wall, Fig. 20.27. r 3 J he inflr~cnrc of rt~rvnt~trr is strongnr in 1nmin:~r than in t,r~rl)~~lrnt, flow. C. AT. \Vllitc [RO] untl ]\.I. Atllnr (21 rnrrictl ont. cxprrimonl~n otl 1n.mirmr Ilow. 'l'l~c! I~~rl~t~lcnl, case was invcsLigntcd c!xprdrnont.nlly by 11. Nippcrt [47] and 11. ltichtcr [5G]. Theoretical cnlculations for the laminar case were carried out, by W.R. Dean [lo] nntl RI. Atllrr 121. '1'11e rl~a.r~ctrcistic tlimrnsionlcss vnrinl)lc, whir11 tlct.crmincs t.1~ inllrwrlcc of r~~rv:~t~~rc it1 /he lmni~mr cose, is the Deem number Fig. 20.27. Flow in n c.nrvrtl pipe, aflrr Prnntltl [52] may be u sd for values of the parameter D exceeding about 102." TThc rcsults of mcnsuremenB urc approximated with a higher dcgrcc of precision by the following empirical equntion, first given by L. Prnndtl 1531 : This cqunI,ion givcs good n.gmcmcnt wit,h cxporirncnLnl rcsulta in 1.h~ r:l,nfio C). M. Whil,c [RI] has Pouncl thal thc rrsisl,nnc:r cocfkir~~t for lr~rhulwrl lloro ill a. wrvrd pipc can hc rcpresentctl by the cquation whosc form indicates rlcnrly that thc ])ran numbrr can no longer srrve as n ~llitnl~lr independent variable. In more recent times, JI. G' Cuming [R] carried out an irivcsti- g,ztion into tho phcnomrnon of scrondarg flow in rurvccl piprs. in wlricl~ JZ -T= 2 D. If the nt~rncri~~d cocffiricl~t, o~~l.~itIc tlhe I )~~C~I~IICSVS is rrltl:tc~t1 by 0.101, the equation gives good agreement with experirnentnl res~~lt~s in the range of K > 30. In the ti~rbulcnt case Ito 1271 hns sl~own thcoreticnlly that the ratio of 1.11~ ~.osintance coefficients, I/,Io, may bc exprcs3ed in terms of t.hc ditncnsionlcs-q vnriahle R (R/r)z. The experiment.al results of Jto [28J can hc represent,ed with sufncirnt i~cc~lracy by the equations mentioned in thc footnote. In flow through a bend or elbow t.herc is not only some loss of enorgy within t,he bend itself, bnt n part of the Ions protl~~ccd by the bent1 hkcs place in the stmight. pipe following it. 15xtensive measurements of the loss cocfficicnts for smooth pipe bends and n correlation of results wore given by 11. Ito 1291. Thcorcticd rrs~~lt.s nrc reported by W. M. Collins et ai. [Sb]. In flow through a radially rotating st,raight pipe, n. secondary flow sirnilnr t,o that found in a curvcd pip is sct, up by the action of n (hriolis forc.n; it. fiirrs risv 1.0 n Inrgc incrcc~sc in resisl,r~ncc. I3xt,c11sivc: ~ncnsurrrncn~a :1.11cl 1.11corct.ic:~l c:t~lcl~l;~l.ions on this subject were carried out by 11. Tto and li. Nanbu 1301. t H. Ito [27] givcs: and These differ somewhat from, hut are in general nprecmcnt with, C. M. IVl~itc'~ cqnnt,iot~ RI)OVC.
628 XX. T~lrbnlmt flow through pipes i. Noa-shady flow throngh a pip 15xttcnsivc mcasurcmcnt~s and tlicorcLica1 calculations on frictional losses in turbulcr~t flow hnve also bcen carried out. hy R. W. Dctra [ll] who includcd curved pips of nonrirculer cross-scclion in his investigations. It is found t,haS the resistranee offered by an cllipt,ic pipe is grcater whcn the major axis of the ellipse lies in the plnnc of CII~VA~JI~C than when it is pcrpcndiculnr to it. - - 11:. l
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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
Page 274 and 275: 626 XVII. Origin oI t.11r011lonco I
Page 276 and 277: 630 XVII. Origin ot t.~~rhr~lrncc J
Page 278 and 279: XVII. Origin of t.urh~~lencc TI 7 !
Page 280 and 281: 638 XVII. Origin of t~~~rbnlc~~rr I
Page 282 and 283: \vi(,l~ a (-ylit~(lw envcrv(1 wiI.1
Page 284 and 285: 646 XVII. Origin of turbulence TI F
Page 286 and 287: 550 X\'ll. Origin of t,11rh111rnrc
Page 288 and 289: 554 XVII. Origin of t,rlrbuloncc TI
Page 290 and 291: 558 XVITI. Fundamentals of tr~rbule
Page 292 and 293: 562 XVlI1. R~nclnmont.nIu of tur1)u
Page 294 and 295: The corrrlnt,ion co~ffiricnt~ y~ ra
Page 296 and 297: 670 XV111. I'nntlntnrt~t~nlrr of tt
Page 298 and 299: 574 XVIII. 1'undnnient.alu of tnrl~
Page 300 and 301: CHAPTER XIX Theoretical assumptions
Page 302 and 303: 682 XIX. Thcoroticnl wsltmptionu fo
Page 304 and 305: 586 XIX. Throretical aaotin~ptiona
Page 306 and 307: 5l)O XIX. Tl~rorrt,icnl nmumptions
Page 308 and 309: 594 XIX. 'rheoreticnl nssornptions
Page 310 and 311: 698 XX. 'I'~~rl~ulrnt flow f,lrro~~
Page 312 and 313: 602 XX. T~rrbulcnt flow thro~tgh pi
Page 314 and 315: 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: dimensions Fig. 20.26. 1tc.sist.anc
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 and 374: 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
Page 409 and 410:
Oswatitach, I
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H:I:IR, 11. 776, 777 Ilnnsr. 1). 62
Page 413 and 414:
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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