ra j l q i aAt qV bv vr 4 ffi r U i1 4 D r n ril iuLe hoa 4 f tt ft 3 d Y t r 6 tr l L Q 2L s d L z s aO I At tlj f n 3 4 ag LItA W Yl U S g lru 4 v utx frt A h Lu A t lg i 0 ro f t fu 1 Ly rl r d r Yg nn g i n at4 J u o 1e6L fuo o b e FTGUIE 3 I3 canrporicoo of Pckcuillc rurlbeB for verlnr dnct crse cct rs racl by rbe hydtoulk dhrtsler lYtnlry n dau taht fralt Slrlh 4t d Lfut nudcr R ttq t97E l ir rmhd Z A i t i A 3 i A aAM U e i q dt t it i a Z e4 4aJ crz znn sS F r v c t1 L l lg Yt 4 I qo a l L t t ffi J v1 rrr v guilibrium range inertial subrange t l0 r Kr1 loB lo8 scah F ig lZ l2 A rypicsl wavcnumbcrtpcururn obseped in rhc ocean Dlotted on a Thc unrr ol s is rrbirrary end thc dou rcpresenthypothc icaldala rn a 4t qJ r s d Fq vu 221rl rt IEG 6 I I I I I I F dirE oibl o rlf Vry stollo xtnrrr o 4 T I Srongrdtm I Fliill drrr ieshErdt il9lra I I hrrosdtrmf lnnsln f tntr l o o 7 4 I i l l Fd lt Vriahr cf rb diffirior ritt laoldr ndr Erfsr ara r strongtrrcflttl Fhlgtata ffi tn ly l 85 i o t T p r I lry l of tddrr irrdf il rdl vsirffc hle dr tuo furtdl a d l98ll i CJl d ir er Lti F ir A rr qr C J rrl il tI li v bd trvt b t 8f tuiy d ttu rrdr tr r Fc I I ir ll l Ho l lq lit rb g tu ri o i Ftg 6 13 Thc Moody chart for pipc frioion with smoothrnd rough walls Fron Ref 8 by permisstm the ur ASME I Y hr Ah ioP Lzl r LYz Ah h L D2g 11 4 hr sr6 y1 pvD D 2e hr Vl 75 recall hr ocV for laminar flow Otherusefulrelationships Powerlaw fit to velocityprofile m u l f Yl u u ro Y fo r m m Re rr 1 l nr tot m a x B aa uKr V tu u L l tl 1 33fL EQUATIONAND TABLE IO I EXPONENTSFORPOWER LAW RAIIO OF MEAN TO MAXIMUM VELOCITY Re m V1r xtd I 6 0 0 79t z gxtd t txrd l 6 6 0 802 1 7 0 0 817 ltxtt I 8 8 0 850 souRcE Schlichting 36 Uscd with permission of the McGraw llill Companies g zxtfr I 10 0 0 865 JU yw aWw 6t 4 L u 0 99 whenU 3 5 r hrff 6 f tv f l o f f f rft t t P d l 5 1 720 x r R 0 664 0 R So 6 H 2 59 e I Aul Tr lt l wl puf 0 Jr F rtw D D 0 664 e cf L o J 2 dx 1 328 i ur T tr x Fter W v 1rle lou r rtf f v U l for Re 1 TABLE T Nunericrl solution of lhe Blasius flat plate rclatio Eq 4 45 fl l f 4r t lq 0 0 0 1 o2 03 0 4 OJ 0 0 0 m235 0 fix r9 0 02113 0 03755 0 05864 0 0 0 9696 0 09391 0 146t 0 1tr6r 0234zJ 0 469fi 046956 0 4693r o45E6l O 6rE 046503 0 6 o 7 0i 0 9 t 0 0 4439 0 17474 0 1496 0 lsrll 0 23 18 0 2E1Zt 0 33366 039021 O 4fi l 0 51503 0 5 296 0 65430 o 72 r 0 80644 0 88680 t 05495 t23t53 t 4r42 r 60328 7e551 f 99058 2 18711 23E559 25 5 t 2 7tf3E8 2 98355 3 ilt318 3 3E329 3 58325 0 2 58 0 32653 0 371 0 41672 0 46ffi3 050354 054525 058559 O tU39 o ffir l 0 696 0 0 72943 0 76106 0 79 m 0 Et669 0 86330 0 90107 0 93060 0 95288 0 96905 0 98 l 0 9nq 0 99N 0 99594 0 957n 0 99882 0 99 10 r 99970 0 99et6 0 4 5173 0 45718 0 45119 0 44353 0 4343t 0 42337 0 41057 0 3959E 03 1969 03619 0342 9 032195 03XX5 O nvs 0 5ft7 0 21058 0 16756 0 12861 0 095rt 0 O6nl O UCr37 0 03054 0 01933 0 0tt76 0 Uf5C7 0 m386 0 0020E 0 0010t 0 ffn54 l l l2 l l 1 4 I f t 6 1 7 t E 1 9 LO 7t L6 Lil 3 0 3 2 1 4 3 6 1 8 4 0 4 2 4 4 4 0 4 tt rr T 3 FlGlrlE il Thc Edu iolutbn to t ta irt pldc bo d y Inrr r nuftriol lltiu d s U ddt rt dmntr b Lilptf ur 1943 d ir of Eq 4 j t Approximatesolution TurbulentBoundary Layer Re 3 X 106 for a flat plate boundarylayer Re rit 500 000 cr d0 2dx as was done for the approximatelaminar flat plate boundary layeranalysis solve by expressingc1 c1 6 and 0 e 6 and integrate i e assumelog law valid acrossentireturbulentboundary layer u I yu ln B neglect laminar sub layer and velocity defectregion UKV aty 6 u IJ U 1 6u ln B UKV 112 n Ib l 2 t or rll2 2l l l cr n trrl nfneo J r 2 441 power law fit ar 2 02 Reu 1 6 Next evaluate ci 6 de gigfr alou dx TJ dxdU can use log law or more simply a power law fit rl Y u a ote cannotbe ed to obtaincr 6 I t r stnce rw I oo e o e a 72 r rtpu pu r g 72 Reu t u dx or 6 16Re l t x 6 x6l7 almostlinear r vf 1 Ur 027 RtYt 031 7 Cr L R ij 6 i e much faster growth rate than laminar boundarylayer Alternate forms given in text dependingon experimental information and power law fit used etc i e dependenton Re range Someadditionalrelationsgiven in texts for largerRe are as follows Total shear stress coefficient Cr i Local shear stress coefficient 1700 455 log oR tt Re 107 Re T2 cr lslogRe 6s z t cr ztogRe for Avcragc shar strm ccfficient completely turbulent boundary lays Combination of laminar and turuulcnt boundary layer o ffi ff 106 tot lo5 loe rtltry Finally a compositeformula that takesinto accountboth the initial laminar boundary layer with translationat Recn 500 000 and subsequentturbulentboundarylayer rD vf 074 1700 R ij R 105 Re loi uS t t VU F 1 …
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