Large-Eddy mulaon and ubgrd-cale Modelng Large-eddy smulaon has araced consderable aenon n he recen years. The foundaon of hs approach s based on he observaon ha he small-scale urbulence srucures are nearly soropc and que unversal n characer whle he largescale srucures of urbulen flows vary consderably. In hs approach, he small-scale urbulence s modeled hrough he subgrd-scale sresses whle he large eddes are drecly calculaed. The averagng process on he grd-scale s oulned n he subsequen secon. The basc conservaon laws are Mass = Lnear Momenum 0, (1 ρ v ρf x =, (2 where v s he velocy vecor, s he sress ensor and f s he body force per un mass. For a Newonan flud ( v v = pδ µ. (ncompressble (3,, In large-eddy smulaon he flow parameers are decomposed as: v = v v, p = p p, =, (4 where bar on he op of a leer sands for he large-scale par and a prme denoes he resdual (subgrd-scale par. Followng Leonard, he large-scale componen of a quany φ s defned as ( = G( x, x φ( x φ x dx, (5 D where G s a flerng funcon and D s he flow doman. A Gaussan fler s frequenly used. Noe ha unlke he radonal eynolds averagng φ φ and φ 0. (6 1 G. Ahmad
Applyng he flerng procedure o equaons (1 and (2, follows ha = 0 (7 ρ v = ρf x, (8 where he subgrd-scale sress ensor s defned as or ( v v v v = ρ (9 = (10 L C wh L = ρv v, ρ( v v v v =, (11 Here C ( v v v v = ρ. (12 s he eynolds sresses, L s he Leonard sresses and C s he cross sresses. The eynolds sress ensor s commonly modeled by usng an eddy vscosy..e., = 2ν D, T 1 D =. (13 2 For eddy vscosy, he magornsky model gven by T 1 2 ( c ( D D 2 ν =, (14 kl kl 1 s frequenly used. Here s he sze of he grd ( = ( 1 2 3 3, and c s he magornsky consan, whch s abou c 0. 21 (0.17 0.24. 2 G. Ahmad
The Leonard sresses need no be modeled and may be calculaed as par of he soluon. The cross sresses was modeled as (eynolds e al. c r ( u u u u = ρc, (15 where c r = 1. 1 was orgnally suggesed. pezale found c r = 1 make he equaon Gallean nvaran as s requred by he Naver-okes equaon. Turbulen Channel Flow ample resuls for drec numercal smulaon for a urbulen channel flow s presened n hs secon. Here s assumed ha he flow n he axal and he ransverse drecons are perodc wh perod of 660 wall uns. The dsance beween he walls s 250 wall uns. The no slp boundary condons a he surface of he plaes a y = ± 125 are mposed. Fgure 1 shows he nsananeous velocy vecor feld a a secon across he channel a = 200 and x = 157. 5. Whle he vecor feld s random, he presence of srucures near he walls could be clearly seen from hs fgure. Fgure 1. Insananeous velocy vecor feld across he channel. Fgure 2a shows anoher sample nsananeous velocy vecor feld n he yzplane a = 100 and x = 236. 25. Whle he vecor flow vares randomly, s seen ha s general feaures are smlar o hose observed n Fgure 1. The srucures near he wall are sll presen bu are parly shfed and modfed. Fgure 2b shows he 3 G. Ahmad
nsananeous velocy feld a a secon very close o he lower plae a y = 3. Ths fgure shows ha near he wall here are roughly perodc hgh speed and low speed sreaks wh markedly dfferen veloces. Fgure 2. Insananeous velocy vecor feld a wo planes n he channel. Fgures 3a and 3b, respecvely, show he conour plos of shor-me mean flud veloces n he sreamwse drecon, u, and normal o he wall, v, n he y-z plane near he upper wall. The shor-me mean flud veloces are obaned by spaal averagng over a dsance of 630 wall uns n he sreamwse drecon for a me duraon of 100 wall uns. Fgure 3a shows ha he mean sreamwse velocy has a roughly perodc varaon n he spanwse drecon wh he dsances of nearby hgh speed (or low speed regons beng abou 100 wall uns. mlar resuls for v -velocy n Fgure 3b shows alernang sreams oward and away from he upper wall. Comparng Fgures 3a and 3b, s observed ha he locaons of hgh-speed axal sreams roughly correspond o he regons ha he flow moves oward he wall, and he low-speed axal sreams, on he average, concde wh he regons ha he flow moves away from he wall. 4 G. Ahmad
(a (b Fgure 3. Varaons of axal and vercal velocy feld near he upper wall. Fgure 4a shows he DN smulaon for dfferen grd resoluon and he expermenal daa of Nederschule e al. for he mean velocy near he wall. Here, s s he nondmensonal dsance from he wall. The heorecal lnear velocy profle n he vscous sublayer and he logarhmc varaons n he neral sublayer are also shown n hs fgure for comparson. DN smulaons were performed for grd szes of 16 64 64, 32 64 64 and 32 128 128. I s observed ha he smulaed mean velocy s n good agreemen wh he daa and he classcal soluons. The smulaed roo-mean-square (M flucuaon veloces are shown n Fgure 4b and are compared wh he earler hgh-resoluon smulaons of Km e al. Fgure 2b shows ha he general agreemen s reasonable. Fgure 2 also shows ha he mean velocy and urbulence nenses do no change apprecably when he grd resoluon s ncreased by a facor of 2 o 8. 5 G. Ahmad
Fgure 4. Comparson of he DN of mean and M velocy felds wh dfferen grd resoluon wh he expermenal daa and earler smulaon resuls. Fgure 5 shows sample conour plos of a channel flow a hgh eynolds number usng large eddy smulaon (LE. The elongaed srucure of flow along he flow drecon can be clearly seen from hs fgure. 6 G. Ahmad
(a (b Fgure 5. Large eddy smulaon (LE of a urbulen channel flow. 7 G. Ahmad