:ra\; - -q+-*=*i-,:j:l*/*4-ffi;D r. -*"n**.*-{ril*___*iuLe*__,aAtqV bv&, vr U"-^i1-4.-*r- - -- -hoa,4 {,{f* tt* ft+"* 3*{ d*Y*t \-r "6 +tr*. * $l*..L =,Q2L€-s-d/L--z/s *+ ,,\ tlj f,n,aO *I *At-3!4-ag**-LI-"""*4* v,utx-= frt t- lg i 0 = ro*f t{A- * ,fu--} 1 -} -'t - Yl^U"'<-AW--A\=\,,-h"-=Lu =$Ly?*=€_-rl^r\-d*r_',S * g{lru-$&*'Yg--&= *+nn-*\g -- &+;=i->n^at4-"J u,o 1e6L;fuo*o, **\beFTGUIE 3'I3canrporicoo of Pckcuillc rurlbeB for verlnr dnct crse :cct*rs racl R.ttq& nudcr ir rmhdby rbe hydtoulk dhrtsler. lYtnlry*n dau taht fralt, Slrlh 4t d Lfut (t97E 'lZ = A* .:i *t{*,i-*3-.i:A*aAM-U*.--A"e {i}q dt$t"it ! ia&Z crz.znn-sSe4"4aJ*: (,F /r)v+? ( c t1\'/+4* I qo -a l'L-t/ tL'l lg-Yt*€guilibrium rangeinertial subrange- ****J&v1!rrr,v,;+=;ffitF'ig. lZ,l2 A rypicsl wavcnumbcr tpcururn obseped in rhc ocean. Dlotted on a loB-lo8' scah'Thc unrr ol s is rrbirrary, end thc dou rcpresent hypothc*ical dala.l0-rKr1'rn/\***,a{4t* qJ"*r*:,s--d:Fq.vu" '221rl rt - IEG6Fd lt Vriahr cf rb diffirior of tddrr irrdf il rdl vsirffcritt laoldr ndr. Erfsr hle dr tuo furtdl a, d (l98ll.(|F dirE oibl, o{|rlf lry|'lVry stollo xtnrrr.--o*4Srongrdtm--*--. _\arao^',"(,,IIIIII?TIIIIt,T,74IiIillFliill drrr:{{ieshErdt il9lra};p{ r strongtrrcflttl.r -- , , Fhlgtataffi tn ly'l +85 hrrosdtrmf;f, tntr"l*o.o -lnnslni er -Cl r \Lti*- *Jdi iF "--" rA rr-qr\ C"-.^ /%:J**rrliltI-***'*",liv bd trvt b.*t. 8f (tuiy. tr.r&Fc. ir|'d(//ttu.|&rrdr I I ll.l].Ho,@!llq'&.lit*rb" !g,tu',ri.o**i -ur*.*Ftg 6.13 Thc Moody chart for pipc frioion with smooth rnd rough walls. (Fron Ref.8, by permisstm {the ASME,I(^ \ LYz Ah=h.-L.hr=-Ah--ioP+Lzl-r -\Y ) D2g/ 11/4.hr= sr6(+.]"-:y1[pvD/ D 2ehr * Vl'75(recall hr oc V for laminar flow)Other useful relationshipsPower law fit to velocity profile:/\mu f Yl--l- | Y:fo-ru*u* \ro /m- m(Re)rr 1 r tt*"max _ ^_ln^o" +Baa --uKrV L,,,\-lt- _ tl+ 1.33f L:: ) -u -u*TABLE IO.I EXPONENTS FORPOWER-LAW EQUATION ANDRAIIO OF MEAN TO MAXIMUM VELOCITYRe-+ +xtd z.gxtd t.txrd ltxtt' g.zxtfrm+ I -l 1' I I6.0 6.6 7.0 8.8 10.0V1r*,-- 0.79t 0.802 0.817 0.850 0.865souRcE: Schlichting (36). Uscd with permission of the McGraw-llill Companies.JU yw 6t-aWw | 4$L{u-?=0.99 when U =3.5 )6*=f['- tv=f(l -r'hrff )0.664./R."5* 1.720x r/R.,0);=, o-f(,- t)t*= rft -f')f P-d,lSo,6.= H:2.59eIAulTr=lt^l:wlrtwJr*F)puf"(0)Fter-WvvUrtf -f@<<lfor Re, >> 1,-rr{T,[3]FlGlrlE, il{Thc Edu iolutbn to. t|ta irt.pldc bo$d[y Inrr: {r, nuftriol !d{ir of Eq, (4,{j} (t)@lltiu d | - s/U ddt .rt dmntr b!, Lilptf,ur {1943)_cf :L o(J'2',0.664 etr{xD( .- dx 1.328=i-/ r:-T-u 1rle,D_lou'rTABLE +TNunericrl solution of lhe Blasius flat-platerclatio& Eq. (4-45)fl'l.|f'(4r t'lq)0.00.1o2030.4OJ0.6o.70i0.9t.0'l,ll2l.l1.4I.f,t.61.7t.E1.9LO7tL6Lil3.03.21.43.61.84.04.24.44,04.tt0.0 0.0 0.469fi0.m235 0.9696 0469560.fix,:r9 0.09391 0.4693r0.02113 0.146t o45E6l0.03755 0.1tr6r O.6rE0.05864 0234zJ 0465030.{4439 0.2$58 0.4{51730.17474 0.32653 0.457180.1496? 0.371% 0.451190.lsrll 0.41672 0.44353.0.23?18 0.46ffi3 0.4343t0.2E1Zt 050354 0.423370.33366 054525 0.41057039021 058559 0.3959EO.4fi??l O.(tU39 03'19690.51503 o.ffir{l 03619)0.5&296 0.696?0 0342/,90.65430 0.72943 032195o.72$r 0.76106 03XX50.80644 0.79(m O.nvs0.88680 0.Et669 0.?5ft7t.05495 0.86330 0.21058t23t53 0.90107 0.16756t.4r42 0.93060 0.12861r.60328 0.95288 0.095rt|.7e551 0.96905 0.O6nlf .99058 0.98&l? O.UCr372.18711 0.9nq/ 0.0305423E559 0.99N' 0.0193325'#'5{t 0.99594 0.0t t762.7tf3E8 0.957n 0.Uf5C72.98355 0.99882 0.m3863.ilt318 0.99*10 0.0020E3.3E329 {r.99970 0.0010t3.58325 0.99et6 0.ffn54Approximate solution Turbulent Boundary-LayerRe, - 3 X 106 for a flat plate boundary layerRe.rit - 500,000cr_d02dxas was done for the approximate laminar flat plateboundary-layer analysis, solve by expressing c1: c1(6) and0 - e(6) and integrate, i.e.assume log-law valid across entire turbulent boundary-layeru I. yu-:-=-ln--- +BUKVaty - 6, u: IJU 1, 6u*_:_=:ln--- _BUKVneglect laminar sub layerand velocity defect regiont 'rll2(2lor l-l -2.441\cr /ar 2.02 Reu-1/6Next, evaluate/ ,,112\-n..Ib'l'\2 )[ ( n .trrl*nfneo[;J rpower-law fitci (6)de-gigfr-aloudx dxdU\ TJ)= r* = "rtpu'-pu'::= *r"'#can use log-law or more simply a power law fitrl - (Y\''' I *ote: can not beu [a./ | ;ed to obtain cr (6)t.-r I stnce rw * ooe=*o-e(a) )72\Reu-t'u -g.72+dxor 6 = .16Re*l/ t i.e.' much fasterx growth rate thanlaminar6 * x6l7 almost linear boundary layer.027vf - --r RtYt/-1 '031 '7Ur = -- ,, =:Cr(L)' R.ij' 6Alternate forms given in text depending on experimentalinformation and power-law fit used, etc. (i.e., dependent onRe range.)Some additional relations given in texts for larger Re are asfollows:Totalshear-stresscoefficientLocalshear-stresscoefficientCr=.455 - 1700(log,o R", )''tt Re,i = cr(.lslogRe, -.T2)cr = (ztogRe* -.6s)-z'tRe > 107-'---106 lo5 *.rtltry tot loeFinally, a composite formula that takes into account boththe initial laminar boundary-layer (with translation atRecn: 500,000) and subsequent turbulent boundary layer. ^ .074 1700rD vf : -------;-;;-' R"ij' R.,Avcragc shar strm ccfficient forcompletely turbulent boundary lays-\-Combination of laminar and\ turuulcnt boundary layer$o="ffi-ff105<Re<loi€.<^uSt/ t",---'-^+ )VU-*-,* tJLJ-0"tt-*-{^"\pt,+t1-.**\ r"!r\*Fr^: td1.-^-S1,*J-L^* vL oqf\r*(') ""*? -a+-f i<- f* ry^ l.n,,3,.T""-l!\ <l=*J q Ar\&-*+r(4(r\LJ&I^-.-IA.r^,. -Q \4-t \...*...-t-".;4.- *'\,\rAr|.*l-,* Lr.$-;.**e*u't0U+ z(-rA4.\ x J- d- .r,rr.1fnr-* (J'o(-A,;-1.';(-r',1\;F,^'-Hr/z\xx z< +,tttA-u/*t*;-\t* +'-- Q\r't)^ ,h^f\^r^b s^^11:4--*vv/n\'f /lt/d\b *v rir'.1+..F:1nrv f *\ ' Or{tlx -t rs-u^* = / U?kS*^^r^1 S.,<.-l-.*:7= r\x^+(f ){{: }ttF= -/:,.{ "(o\* (o) = oIto*\* oY\h;- f*4* " /tshh??"\ -:l 1qJa-tl{ \ *r'A^-:- d1" lti=t) 'o r.,.+ n -- \h.)fi- , V2-r.*,6= *L)'*'t .-3Ot r.,*-g t*-t- = ( t(oL L.2D S'-ft\u\$fur14x +14e = c'-?,tv\tlrg*ra = Jtat\,-b=t-e->o A=-O'?a-t o?o*[?* =r\-hZ n 4,rY*-r'r-"}l\q = r^\L( "+syi*-14,\* t 1 l'-"^ r^,-l'".1',t\r*g4\ -, h4\-\G Artlt^) r"r. a-u lc { \I t*-"1{'L *\+,'l$ (-rLt\s -\ ftt+lX=D >.^3A-r\_tg/^z-L<J\k Xo *-r\r-r.- Xo * rn.r L-A ov*1{.u.\= C.n'.S {***' I ?- \"^ - \ -- O-n = Y' b*"t?fk alr At\. sha*T-r* -r- YYr{,,, + *({,-*f {,,) =Q+= {u1?-\-- lx-,^n"_+7 (n- ,ft" __ A,_\\1 @ L'r,-,- ,,\ T,"r r"lrL.,,.{ c^A*\t" &L-4= (s)"= != (ry:*i"=A^*= (fr.r)"'x-'/= o--Ta X-'|r:'oAa = _bt = q J"^ ut v^,^r4-,,r4- "-,- drs-*--Vt.c""""^t^+\^A *. ( 4- YtX Zo*-qJa-i .,Ll* Aat\cr:L/s i*tr\t\ - r/- {^ -rY\- t <, V\
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