Boundary Lyer Theory

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766 XXV. Determination or profile drag The coefficient of profile drag can be evaluated from the above oqnation, if the momentum tltickness at the trailing edge is known from the boundary-lnyer calculation ant1 if, in addition, the ideal, potential vclocity at thc trailing edge, U1, is known. The latter can be found, for example, from a reading of the static pressure at the trailing edgc. According to a method proposed by H. R. Helmbold 1221 the determination of IJ,/IJ, can also proceed as follows: We begin by evaluating the momentum thickncss at the trailing edge, 8,,/1, from eqn. (22.17) using the value 7~ = 4. This valuc is thcn substiti~tcd into eqn. (25.28), and in thc resulting formula IJ1/U, is raised to the power -1 0.2. 'I'hus this factor can bc approximntcd by the vnluc of unity, because UJU, itsclf (Ides not dilTer much from unity, ant1 the value of the coefficient of profile drag for o m side (R = U,Z/v) can be found from eqn. (25.28) to he i I with The subscript 1 rcfers to tho point of transition nnd thc vnluc of the constant C can bc cletcrmincd from the condition that thc laminar and turbulent momentum thicknesses must be cqi~nl to each other at the point of transitmion, bzt = dztwb = aZfarn. The value of dzlatn can be found from cqn. (10.37). For uniform ~ot.entia1 flow 1vit.h IJ = U, eqn. (25.20) transforms to the corresponding exprcssion for the flnt plate at zero incidence, eqn. (21.11), if, in addition, we put C = 0 for fully tleveloprtl t~~trk~ulettt flow. TC. 'l'rucl~cnbrotlt [G8] tmnsformcd cqn. (25.20) replacing tho potential vclocity distribut,ion by t,hc coordinates of the acrofoil scction thus, evirlent.ly, eCcct.ing a considerable simplificnt.iort. 11. B. Sq~~irc n.nd A. I). Young [64] cvnlr~atcd a number of cxamplcs by tile use of a tliffcrcnt mctltotl. We shall now tlcscribe somc of thern, rcfcrrit~g to Fig. 25.3, wltirh csontains a waumk of thcsc rcsrrlt~. Tltc thickness of tlhe acrofoils was vnricti from dl1 - 0 (flnt platc) to rill -- 0.25 and tho Reynohls nurnl~crs R -- IT, 1/v mnpcd from 10Q,o 108. It is found that the profilc drag is very sensitive to the position of the point of transition from laminar t,o turbnlent flow. This lntter parametm wns vnricd from x,/E = 0 to 0.4. The increase in profile drag with thicltncss is, rssenl~inlly, tluc t,o an incrcnsc in form drng. Fig. 25.4 shows the rclation bctwccn form atttl profile tlrng. Analogolls calculations were performctl by J. Pretsch [40] in rclation to von I
768 XXV. Determination of profile drag c. Losses in thc flow through cilclcad~s 769 Fig. 25.5. Increase in the coefficient of profile clrag plotted in terms of relative thick- ncas, as calculnted by Scholz [58] Totnl or proflla drag cl)tOt - CU form -1- CI metrical cases, applied to rough walls (equivalent sand roughness) as well. From a very large number of calculated examples on aerofoils (two-amensional case) and bodies of revolution, it proved possible to deduce relations to describe the influence of thickness on profile drag. 'Shese are shown plotted in Fig. 25.5. The difference Ac, = c, - c,, denotes the i~crcase in the coefficient of skin friction, related to the wetted surface, as against, its value for rr. flat plate at zero incitlcnce, c,. The curve for the two-dimcnsiorlal case agrees fairly well with the results shown plotted in Pig. 25.3 for the case of a fully turbulent boundary layer (z,/l = 0). In this conncxion the paper by P. S. Granville [lS] may also be consulted. These calculations give an indication about the effect of friction on lift. The displacement of the external streamlines caused by the bountlary layer modifies the pressure distribution on an acrofoil and causes the experimental value to become lower than that givcn by potential theory. This loss of lift was calculated 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
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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
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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
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 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: 764 XXV. I)ot.crtninntiotl of profi
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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