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2. STRCTRE OF A TRBLENT BONDARY LAYER SPRING Shar tr and friction vlocit 2.2 Lngth and vlocit cal 2.3 Innr lar 2.4 Outr lar 2.5 Ovrlap lar th log la 2.6 Vicou ublar 2.7 Limit of th variou rgion
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2. STRCTRE OF A TRBLENT BONDARY LAYER SPRING Shar tr and friction vlocit 2.2 Lngth and vlocit cal 2.3 Innr lar 2.4 Outr lar 2.5 Ovrlap lar th log la 2.6 Vicou ublar 2.7 Limit of th variou rgion 2.8 Vlocit-dfct lar: Col La of th Wa 2.9 Effct of roughn Exampl Th anali i applicabl to a flat-plat boundar lar or full-dvlopd pip or channl flo. Firt conidr mooth all. 2.1 Shar Str and Friction Vlocit Th har tr ( rat of tranport of momntum pr unit ara in th poitiv dirction i uv (1 Th vicou part vari from bing th ol tranportr of momntum at th all to a ngligibl fraction of th total tr in th outr part of a turbulnt boundar lar. For 0.1, i approximatl contant (h? and qual to it valu at th all: Thi i th contant-tr lar. A ha dimnion of [dnit] [vlocit] 2, it i poibl to dfin an important vlocit cal th friction vlocit, u b 2 u (2 or u / (3 2.2 Lngth and Vlocit Scal Wall nit Vr clo to th all th mot important caling paramtr ar: inmatic vicoit ; all har tr τ. Th charactritic vlocit and lngth cal ar: friction vlocit: u / u (4 τ vicou lngth cal: (5 u u ( δ Turbulnt Boundar Lar 2-1 David Apl From th can form non-dimnional vlocit and hight in all unit: u, u (6 i a ort of local Rnold numbr. It valu i a maur of th rlativ importanc of vicou and turbulnt tranport at diffrnt ditanc from th all. Boundar-Lar nit At larg th dirct ffct of vicoit on momntum tranport i mall and hight can b pcifid a a fraction of th boundar-lar dpth : (7 Th quantit u R i calld th friction Rnold numbr and i a global paramtr of th boundar lar. (8 Sinc full-dvlopd boundar-lar flo i compltl pcifid b,,,, and u, dimnional anali (6 variabl, 3 indpndnt dimnion ild a functional rlationhip btn dimnionl group, convnintl tan a f (, u i.. f (, (9 Almot all boundar-lar anali i bad upon th mooth ovrlap of th limiting ca innr lar ( 0 and outr lar (» Innr Lar (Prandtl, 1925 Dimnional paramtr,,,, but not. Dimnional anali (5 paramtr, 3 indpndnt dimnion 2 indpndnt dimnionl group, convnintl tan a / u and u /. Thn hav th la of th all: f ( (10 f i xpctd to b a univral function; i.. indpndnt of th xtrnal flo. According to Pop (2000, th innr lar corrpond roughl to / ovr hich th har tr i approximatl contant. 0.1, or th rgion Turbulnt Boundar Lar 2-2 David Apl 2.4 Outr Lar (Von Kármán, 1930 Dimnional paramtr,,,, but not. Dimnional anali (5 paramtr, 3 indpndnt dimnion 2 indpndnt dimnionl group, convnintl tan a, u Thn on ha th vlocit-dfct la: f o ( u (11 nli f hich i xpctd to b univral, f o ( ill var ith th particular flo. 2.5 Ovrlap Lar th Log La A notd b C.B. Millian (1937 th innr and outr lar can onl ovrlap moothl if th ovrlap-rgion vlocit profil i logarithmic. Outr lar: f ( Innr lar: f ( Introducing u /, o that ( f ( f ( o o, and adding: For a function f of th product to b th um of parat function of and, f mut b logarithmic. Thi can b provd formall b diffrntiating uccivl ith rpct to ach variabl, a follo. Diffrntiat rt : ( 0 f Diffrntiat rt η: 0 f ( ( f ( f ( f ( d df ( d d Hnc, df contant d Thi contant i convntionall rittn a 1/, hr df 1 d hich intgrat to giv 1 f ln B, B anothr contant. ( 0.41, i von Kármán contant. Turbulnt Boundar Lar 2-3 David Apl Hnc hav th log-la vlocit profil: 1 ln B or, quivalntl, 1 ln E (12 (13 Not. (1 Thr i om variation btn ourc, but tpical valu of th contant ar 0.41 (1/ 2.44 and B 5.0 (E (2 Excpt in trong advr prur gradint (.g. in a diffur th logarithmic vlocit profil i a good approximation acro much of th har lar. Thi obrvation turn out to xtrml uful in driving friction formula Sction 3. (3 In th log la rgion, u or contant u Thi i oftn ud a an altrnativ tarting point for th drivation of th log la. 2.6 Vicou Sublar Vr clo to th all, turbulnt fluctuation ar dampd out and th all har tr i almot ntirl vicou:, contant hich ild a linar vlocit profil: Stting 2 u and rarranging, (14 Exprimnt ho that th linar vicou ublar corrpond roughl to Limit of th Variou Rgion Pop (2000 giv th folloing rough dlimiting and / valu. Innr lar (roughl / 0.1 vlocit cal on u and, but not on. Outr lar (roughl 50 th dirct ffct of vicoit i ngligibl. Ovrlap rgion - xit at ufficintl high Rnold numbr. In th ovrlap rgion th man-vlocit profil mut b logarithmic. In fact th log la i a good approximation bond th ovrlap rgion. Pop uggt: Turbulnt Boundar Lar 2-4 David Apl Vicou ublar: 5 linar vlocit profil Buffr lar: 5 30 Log la rgion: 30, / 0.3 logarithmic vlocit profil 2.8 Vlocit-Dfct Lar: Col La of th Wa In th outr lar th vlocit profil dviat lightl from th log la, particularl in nonquilibrium boundar lar ith a prur gradint. Col (1956 notd that th dviation or xc vlocit abov th log la had a a-li hap rlativ to th fr tram; i.. f ( log la hr f i om S-hapd function ith f(0 0, f(1 1; popular form ar 2 f ( in f ( 3 2 Thn hav th Col La of th Wa: 1 2 ln B f ( / u hr th dviation from th log la i quantifid b th Col a paramtr. Tpical valu ar: pip flo or channl flo: 0 zro-prur-gradint flat-plat boundar lar: 0.45 (15 In gnral, i a function of prur gradint. 2.9 Effct of Roughn Th minal xprimntal or a don b Prandtl PhD tudnt Johann Niurad, ho maurd th friction factor in pip artificiall roughnd ith dnl-pacd and grain of iz. Th rlativ roughn /D varid from 1/30 to 1/1000. Th influnc of all roughn i charactrid b u /. Hdraulicall Smooth: ( 5 ; i.. l than th vicou ublar dpth In thi rgim roughn ha no ffct on th friction factor or man-vlocit profil. Full Rough: ( 70 Tranfr of momntum to th all i prdominantl b prur drag on roughn lmnt, not vicou tr, and all friction bcom ntiall indpndnt of Rnold numbr for ufficintl larg R. Dimnional anali impli 1 ln B From xprimntal data, B 8.5. Turbulnt Boundar Lar 2-5 David Apl Tranitional Roughn ( 5 70 Both roughn and vicou ffct oprat. (Th limit ar tho of Schlichting. Whit giv 4 and 60 intad, hilt Cbci and Bradha tranition formula blo u 2.25 and 90. An all-ncompaing man-vlocit profil ma b rittn 1 ~ ln B( hr ~ B B B ( 1 ln ( Suitabl intrpolation formula ar: Cbci and Bradha (1977: 1 B ~ (1 B ( B ln, Apl (2007: ~ 1 B B ln( C, 0; hdraulicall mooth ; full rough 0, ln( / 2.25 in, 2 ln(90 / , C ( B B 90 (Both author ud lightl diffrnt valu of B and B from tho ud in th Not. 90 In practic, ar oftn mor intrtd in th rulting friction la ( Sction 3. For pip flo thi i th Colbroo-Whit formula. Th ffct of urfac roughn dpnd on it form a ll a it iz. Th or of Colbroo (1939 and Mood (1944 hlpd to dfin quivalnt and roughn for man commrcial pip matrial. Gophical Flo Prhap th ultimat in rough-all boundar lar i th atmophric boundar lar. In thi ca th man vlocit profil i tpicall rittn (ith th mtorological convntion of z for a vrtical coordinat: u z ln( (16 z0 z 0 i calld th roughn lngth and comparion ith th abov formula, fitting all contant inid th natural logarithm and taing B 8.5, giv z0 / 30. For tpical rural condition z 0 ha a valu of about 0.1 m. Turbulnt Boundar Lar 2-6 David Apl Exampl Qution 1. Conidr airflo at 10 m 1 ovr a flat plat. If th friction Rnold numbr i 1200, calculat (a th friction vlocit; (b th all har tr; (c th dpth of th boundar lar. Aum a Col a paramtr Qution 2. Wind vlociti ovr opn fild r maurd a 5.89 m 1 and 8.83 m 1 at hight of 2 m and 10 m rpctivl. thi data to timat: (a th roughn lngth z 0 ; (b th friction vlocit u ; (c th vlocit at hight 25 m; (d th avrag vlocit ovr a dpth of 25 m. Qution 3. (From Whit, 1994 J. Laufr (1954 pip-flo xprimnt gav th folloing data at R D r/r / hr 0 i th cntrlin vlocit. Find th bt-fit por-la profil of th form 1 n ( / 0 R hr R r i th ditanc from th all. Anr (1 (a u 0.41 m 1 (b 0.20 N m 2 (c 44 mm (2 (a z m (b u 0.75 m 1 (c (z 25 m 10.5 m 1 (d av 8.7 m 1 (3 n 9 Turbulnt Boundar Lar 2-7 David Apl
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