8. TURBULENCE MODELLING IN CFD SPRING 2007

Transcription

1 8. TRBLENE MODELLING IN FD SRING 7 8. Trblence model for general-prpoe FD 8. Lnear edd-vco model 8. Non-lnear edd-vco model 8.4 Dfferenal re model 8.5 Implemenaon of rblence model n FD 8. Trblence Model For General-rpoe FD Trblence model for general-prpoe FD m be frame-nvaran.e. ndependen of an parclar coordnae em and hence m be expreed n enor form. Th rle o mpler model of bondar-laer pe e.g. mxng-lengh model. Trblen flo are comped eher b olvng he Renold-averaged Naver-Soe eqaon h able model for rblen flxe or b compng he flcang qane drecl. The man approache are mmared belo. Renold-Averaged Naver-Soe RANS Model Lnear edd-vco model EVM ame ha he devaorc rblen re proporonal o he mean ran; e an edd vco conrced from rblence calar all one oher, deermned b olvng ranpor eqaon. Non-lnear edd-vco model NLEVM ame ha he rblen re a non-lnear fncon of mean ran and vorc; e coeffcen conrced from rblence calar all one oher, deermned b olvng ranpor eqaon; mmc repone of rblence o ceran mporan pe of ran. Dfferenal re model DSM aa Renold-re ranpor model RSTM or econd-order clore SO; olve ranpor eqaon for all rblen flxe. ompaon of flcang qane Large-edd mlaon LES compe me-varng flo, b model b-grd-cale moon. Drec nmercal mlaon DNS no modellng; reolve he malle cale of he flo. FD 8- Davd Aple

2 8. Lnear Edd-Vco Model 8.. General Form Sre-ran conve relaon:, x x The edd vco derved from rblen qane ch a he rblen nec energ and dpaon rae. Thee qane are hemelve deermned b olvng calarranpor eqaon ee belo. A pcal hear re and normal re are gven b V v From hee he oher re componen are eal dedced b npecon/cclc permaon. General ommen a phcal proper of he fld and can be meared; a hpohecal proper of he flo and m be modelled. vare h poon. A hgh Renold nmber, hrogho mch of he flo. Advanage Ea o mplemen n vco olver. Exra vco ad abl. Some heorecal fondaon n mple hear flo. Dadvanage Lle rblence phc; n parclar, anorop and hor effec are negleced. Trblen ranpor of momenm deermned b a ngle calar, o a mo one Renold re v can be repreened accrael; ch model are qeonable n complex flo. Mo edd-vco model n general-prpoe FD code are of he -eqaon pe;.e. calar-ranpor eqaon are olved for rblen cale. The commone pe are - and - model, for hch pecfcaon are gven belo. FD 8- Davd Aple

3 8.. - Model Edd vco: Scalar-ranpor eqaon non-conervave form: D D D D rae of dffon prodcon dpaon change Dffve and are relaed o he edd vco va randl nmber :, and he rae of prodcon of rblen nec energ per n ma 4 In he andard - model Lander and Spaldng, 974 he coeffcen ae he vale.9,.9,.44,.,. 5 Oher mporan varan nclde RNG - Yaho e al., 99 and lo-re model ch a Lander and Sharma 974, Lam and Bremhor 98, and Len and Lechzner 99. Modfcaon are emploed n lo-re model o ncorporae effec of moleclar vco. Specfcall,, and are mlpled b vco-dependen facor f, f and f repecvel, and an addonal orce erm S ma be reqred n he eqaon. Some model noabl Lander and Sharma, 974 olve for he homogeneo dpaon rae ~ hch vanhe a old bondare and relaed o b ~ / D, D 6 Th conen h he heorecal near-all behavor, / Model nomnall eqal o dmenon of /me, or freqenc. omeme non a he pecfc dpaon rae and ha Edd vco: 7 FD 8- Davd Aple

4 Scalar-ranpor eqaon: D D D D * 8 The dffve of and are relaed o he edd-vco:, The orgnal - model a ha of Wlcox 988a h coeffcen ang he vale * 9 5,,,., The model a frher developed b Wlcox 998 n h boo, h he coeffcen becomng fncon of he rblen Renold nmber. Mener 994 deved a hear-re-ranpor SST model. The model, hch expreed n - form, blend he - model hch allegedl peror n he near-all regon, h he - model hch le enve o he level of rblence n he free ream. All model of - pe ffer from a problemac all bondar condon a Behavor of Lnear Edd-Vco Model n Smple Shear In mple hear flo he hear re v The hree normal ree are predced o be eqal: v herea, n pracce, here conderable anorop; e.g. n he log-la regon: : v :. :.4 :.6 v Acall, n mple hear flo, h no a problem, nce onl he graden of he hear re v pla a dnamcall-gnfcan role n he mean-momenm eqaon. Hoever, a arnng of more ero problem n complex flo. FD 8-4 Davd Aple

5 8. Non-Lnear Edd-Vco Model 8.. General Form The re-ran relaonhp for lnear edd-vco model gve for he devaorc Renold re.e. bracng he race: Dvdng b and rng / gve We defne he LHS of a he anorop enor a ; he dmenonle and racele form of he Renold re: a For he RHS of, he mmerc and anmmerc par of he mean-veloc graden are called he mean ran and mean vorc enor, repecvel: S, Thee can be made non-dmenonal ng he rblen mecale /. ng loer cae for he non-dmenonal form: S, Eqaon can hen be ren n he mpler form a or, a 4 Hence, he conve relaon for lnear edd-vco model mpl a: anorop enor proporonal o dmenonle mean ran The man dea of non-lnear edd-vco model o generale h o a non-lnear relaonhp beeen he anorop enor and he mean ran and vorc: a NL, 5 Addonal non-lnear componen canno be compleel arbrar, b m be mmerc and racele. For example a qadrac NLEVM m be of he form a 6 { } I { } I here {.} denoe a race and I he den marx: { M } race M M, I 7 We hall ee belo ha an approprae choce of he coeffcen, and allo he model o reprodce he correc anorop n mple hear. Theor baed on he ale-hamlon Theorem ho ha he mo general relaonhp FD 8-5 Davd Aple

6 nvolve en ndependen enor bae and nclde erm p o he 5 h poer n and : a T, 8 here all T are lnearl-ndependen, mmerc, racele, econd-ran enor prodc of and. One poble choce of bae b b no mean he onl one Lnear: Qadrac: bc: Qarc: Qnc: T T { } I T T { } I 4 5 { } 6 T { } I T T { } { } I { } I T T T { } { } Exerce. rove ha all hee bae are mmerc and racele. Sho ha bae T 5 T vanh n -d ncompreble flo. The fr bae correpond o a lnear edd-vco model and he nex hree o he qadrac exenon n eqaon 6. T 5, T 7, T 8, T 9 conan mlple of earler bae and hence cold be replaced b mpler form; hoever, he bae choen here enre ha he vanh n -d ncompreble flo. A nmber of roe have been aen n devng ch NLEVM, ncldng: amng he form of he ere expanon o qadrac or cbc order and mpl calbrang agan mporan flo e.g. Spezale, 987; raf, Lander and Sga, 996; mplfng a dfferenal re model b an explc olon e.g. Spezale and Ga, 99 or b cceve approxmaon e.g. Aple and Lechzner, 998; renormalaon grop mehod e.g. Rbnen and Baron, 99; drec neracon approxmaon e.g Yohzaa, 987. In devng ch NLEVM, model developer have ogh o ncorporae ch phcallgnfcan propere a realabl: pove normal ree ach Scharz neqal 9 FD 8-6 Davd Aple

7 8.. bc Edd-Vco Model The preferred level of modellng a he nver of Mancheer a cbc edd vco model, hch can be ren n he form a f { } I { } { } { } { } I 4 Noe he follong propere ome of hch ll be developed frher belo and on he example hee. A cbc re-ran relaonhp he mnmm order h a lea he ame nmber of ndependen coeffcen a he anorop enor.e. 5. In h cae ll be precel 5 f e ame ee v belo and noe ha he and erm are enorall mlar o he lnear erm ee v belo. The fr erm on he RHS correpond o a lnear edd-vco model. The varo non-lnear erm evoe enve o pecfc pe of ran: he qadrac,, erm evoe env o anorop; he cbc and erm evoe env o crvare; he cbc 4 erm evoe env o rl. v The and erm are enorall proporonal o he lnear erm; hoever he or raher her dfference provde a env o crvare, o have been ep dnc. v The and 4 erm vanh n -d ncompreble flo. v Theor and expermen ndcae ha pre roaon generae no rblence. Th mple ha ogh o be, a lea n he lm S. { } I A an example of ch a model e ce he raf e al. 996 model n hch coeffcen are fncon of he mean-ran nvaran and rblen Renold nmber:.75.[ exp.6e ] /.5 R / R f exp[ ], R 9 4 ~ here S S S,, ~ max S, The coeffcen of he non-lnear erm are n he preen noaon:,,.4,.4,.4 f,,, 4 4, 4,, 8 f Non-lnear bl no boh enor prodc and ran-dependen coeffcen noabl. The model compleed b ranpor eqaon for and ~. Mean ran and vorc are non-dmenonaled ng ~ raher han. FD 8-7 Davd Aple

8 FD 8-8 Davd Aple 8.. General ropere of Non-Lnear Edd-Vco Model -d Incompreble Flo The non-lnear combnaon of and have parclarl mple form n -d ncompreble flo. In ch a flo:, Incomprebl and he mmer and anmmer propere of and, redce hee o, From hee e fnd, 4 ROERTY In -d ncompreble flo: } { } { I I I I 5 here I dag,,. In parclar, ang enor prodc of or h marce hoe hrd ro and hrd colmn are all zero ha he ame effec a mlplcaon b he calar { } or } { repecvel. ROERTY } { a a 6 Moreover, n -d ncompreble flo he qadrac erm do no conrbe o he prodcon of rblen nec energ. roof. S a x

9 No a nce anmmerc, hl ncomprebl mple or S S. Hence, a S a { a} Th re for an ncompreble flo, b, n he -d cae, mlplng b, ang he race and ng he rel 5 fond ha he conrbon of he qadrac erm o {a}. ROERTY In -d ncompreble flo he - and 4-relaed erm of he non-lnear expanon vanh. roof. Sbe he rel 5 for and no. arclar Tpe of Sran The non-lnear conve relaonhp allo he model o mmc he repone of rblence o parclar mporan pe of ran. ROERTY 4 The qadrac erm eld rblence anorop n mple hear: v here 7 Th ma be dedced b bng he rel 4 no, nong ha, hl A an example he fgre rgh ho applcaon of he Aple and Lechzner 998 model o compng he Renold ree n channel flo vv -v FD 8-9 Davd Aple

10 ROERTY 5 The and -relaed cbc erm eld he correc env o crvare. V In crved hear flo,,, here R c rad of crvare. From 4, R R { } { } here, R Rc R Rc Hence, { } { } R Rc Inpecon of he prodcon erm n he reranpor eqaon Secon 7.4 ho ha crvare ablng redcng rblence f ncreae n he drecon aa from he cenre of crvare /R > and deablng ncreang rblence f decreae n he drecon aa from he cenre of crvare /R <. In he conve relaon he repone correc f and are boh pove. c 'able' crvare redcng rblence 'nable' crvare ncreang rblence ROERTY 6 In -d flo, he 4 -relaed erm evoe he correc env o rl. W FD 8- Davd Aple

11 8.4 Dfferenal Sre Modellng Dfferenal re model aa Renold-re ranpor model or econd-order clore olve a eparae calar-ranpor eqaon for each re componen : D d F 8 D For a dervaon ee he core noe for he Bondar Laer modle. Sch model, n prncple, conan mch more rblence phc becae he rae-of-change, advecon and prodcon erm are exac. The neare hng o a andard model a hgh-re clore baed on ha of Lander e al. 975 and Gbon and Lander 978. Term Name and role Model F D D RATE OF HANGE me dervave advecon Tranpor h he mean flo. RODTION mean ran Generaon of rblence energ from he mean flo. RODTION bod force Generaon of rblence energ b bod force. EXAT EXAT EXAT n prncple F f f d DIFFSION Spaal redrbon. d l l l Φ RESSRE-STRAIN Redrbon of rblence energ beeen componen. ~ n n ~ l l ~ n n, ~ n n f f / 4 / n / ε DISSIATION Removal of rblence energ b vco Tpcal vale of he conan are:.8,.6,.5,. 9 FD 8- Davd Aple

12 Energ n Trblen Flcaon In mple hear flo here / he onl non-zero mean-veloc graden he prodcon erm of he normal ree are: v, Hence, prodcon of rblence energ predomnanl feed he componen. Energ hen ranferred o flcaon n he cro-ream drecon b he redrbve effec of prere flcaon. A mall cale local graden are ffcenl large for vco o dpae rblen energ. There a connal energ cacade from he energ enerng he rblence a he large cale of he flo, hogh hear nable connall prodcng edde a maller cale, nl lmael energ removed b vco. RODTION ADVETION b mean flo v REDISTRIBTION b prere flcaon DISSIATION b vco FD 8- Davd Aple

13 The re-ranpor eqaon m be pplemened b a mean of pecfng b on ranpor eqaon, or one for a relaed qan ch a. pcall A ggeed b he able, he mo gnfcan erm reqrng modellng he prereran correlaon hch formed, n pracce, b he average prodc of prere flcaon and flcang veloc graden. Th erm racele.e. he m of he dagonal erm and acceped role o promoe orop hence he form of model for and neceang a all-correcon erm. Near all h oropng endenc m be over-rdden,. Where bod force are preen e.g. n boan or roang flo addonal prodcon erm m be nclded. General Aemen of DSM For: Inclde more rblence phc han edd-vco model. Advecon and prodcon erm energ-n erm are exac and do no need modellng. Agan: Model are ver complex and man mporan erm parclarl he redrbon and dpaon erm reqre modellng. Model are ver expenve compaonall 6 re-ranpor eqaon n dmenon and end o be nmercall nable onl he mall moleclar vco conrbe o an or of graden dffon erm. Oher DSM of Inere Spezale e al. 99 non-lnear formlaon, elmnang all-correcon erm; raf 998 lo-re DSM, aempng o elmnae all-dependen parameer; Jarl and Hanal 995 lo-re DSM admng anoropc dpaon; Wlcox 988b lo-re DSM, h raher han a addonal rblen calar. Excellen reference for developmen n Renold-re ranpor modellng can be fond n Lander 989 and Hanal 994. FD 8- Davd Aple

14 8.5 Implemenaon of Trblence Model n FD 8.5. Tranpor Eqaon The mplemenaon of a rblence model n FD reqre: a mean of pecfng he rblen ree, b eher: a conve relaon edd-vco model, or ndvdal ranpor eqaon dfferenal re model; he olon of addonal calar-ranpor eqaon. Specal onderaon for he Mean Flo Eqaon repreen a rblen flx of -momenm n he x drecon, b onl a par of h can be reaed mplcl a a dffon-le erm. e.g. for he eqaon hrogh a face normal o he drecon: V v non lnear erm dffve par ranferred o orce The non-dffve par of he flx ranferred o he orce erm and reaed explcl.e. held conan for ha eraon. Neverhele, ll reaed n a conervave fahon;.e. ored o on a cell face o ha he mean momenm lo b one cell eqal o ha ganed b neghbor. The lac of a rblen vco n dfferenal re model can lead o nmercal nabl. Th can be addreed b he e of effecve vcoe ee belo. Specal onderaon for he Trblence Eqaon The are all orce-domnaed;.e. he mo gnfcan erm are prodcon, redrbon and dpaon; h omeme ed a an exce for a lo-order advecon cheme. Varable ch a and m be non-negave. Th demand: care n dcreng he orce erm ee belo; e of an ncondonall-bonded advecon cheme. Sorce-Term Lnearaon For Non-Negave Qane The general dcreed calar-ranpor eqaon for a conrol volme cenred on node aφ afφ F b φ F For abl one reqre To enre non-negave φ one reqre, n addon, FD 8-4 Davd Aple

15 b Yo hold, b npecon of he and ranpor eqaon, be able o denf ho he orce erm lneared n h a, h one pove par and one negave par, he laer preferabl proporonal o he ranpored varable, or. If b < for a qan ch a or hch ala non-negave e.g. de o ranfer of non-lnear par of he advecon erm or non-dffve flxe o he orce erm hen, o enre ha he varable doen become negave, he orce erm hold be rearranged a b * φ φ b here * denoe he crren vale of a varable Wall Bondar ondon A all he no-lp bondar condon apple, o ha boh mean and flcang veloce vanh. A hgh Renold nmber h preen hree problem: here are ver large flo graden; all-normal flcaon are ppreed.e. elecvel damped; vco and rblen ree are of comparable magnde. There are o man a of handlng h n rblen flo: lo-renold-nmber rblence model reolve he flo rgh p o he all h a ver fne grd and vco modfcaon o he rblence eqaon o enre he correc near-all raher han log-laer behavor; all fncon e a coarer grd and ame heorecal profle beeen he near-all node and he bondar. Lo-Renold-Nmber Trblence Model Am o reolve he flo rgh p o he bondar. Have o nclde effec of moleclar vco n he coeffcen of he edd-vco formla and or ranpor eqaon. Tr o enre he heorecal near-all behavor:, ~ ~ conan, Fll reolon of he flo reqre he near-all node o af, here, / Th can be ver compaonall demandng, parclarl for hgh-peed flo. FD 8-5 Davd Aple

16 Hgh-Renold-Nmber Trblence Model Brdge he near-all regon h all fncon;.e. ame profle baed on bondar-laer heor beeen near-all node and bondar. OK f near eqlbrm e.g. lol-developng bondar laer, b dodg n hghl noneqlbrm regon parclarl near mpngemen, eparaon or reaachmen pon. conrol volme near-all node p τ p amed veloc profle The near-all node hold deall be placed n he regon < < 5 range 5-5 generall accepable. Th mean ha nmercal mehe canno be arbrarl refned cloe o old bondare. In he fne-volme mehod, varo qane are reqred from he all-fncon approach. Vale ma be fxed on he all elf or b forcng a vale a he near-all node. Varable Wall bondar condon Reqred from all fncon Mean veloc,v,w relave veloc a he all Wall hear re, a he all; zero flx fxed a near-all node The mean of dervng hee qane are e o belo. ell-averaged prodcon and dpaon Vale a he near-all node Mean-Veloc Eqaon: Wall Shear Sre The frcon veloc defned n erm of he all hear re: If he near-all node le n he logarhmc regon hen ln E, here bcrp denoe he near-all node. Gven he vale of h cold be olved eravel for and hence he all re. Hoever, a beer approach hen he rblence clearl far from eqlbrm e.g. near eparaon or reaachmen pon o emae an eqvalen frcon veloc from he rblen nec energ: / 4 / and negrae he mean-veloc profle amng an edd vco. If e adop he log-la veron: FD 8-6 Davd Aple

17 and olve for from e ge / 4 ln E If he rblence ere gennel n eqlbrm, hen old eqal and and 4 old be eqvalen. A beer approach o ame a oal vco moleclar edd hch mache boh he vco eff and log-laer eff lm: max{, } 5 eff here a machng hegh. Smlar negraon o before lead o boh vco blaer and log-la lm,, 6 ln{ }, here e noe ha baed on raher han he nnon. A mlar approach can be appled for rogh-all bondar laer Aple, 7, here a fncon of roghne. A pcal mooh-all vale of 7.7. A far a he compaonal mplemenaon concerned he reqred op for a fnevolme calclaon he all hear re n erm of he mean veloc a he near-all node, p, no vce vera. To h end, 6 convenenl rearranged n erm of an effecve all vco eff,all ch ha p eff, all 7 here p, eff, all, 8 ln{ } Eqaon: ell-averaged rodcon and Dpaon The orce erm of he ranpor eqaon reqre cell-averaged vale of prodcon and dpaon rae. Thee are derved b amng profle for hee qane: v here eff 9 > d > 4 FD 8-7 Davd Aple

18 FD 8-8 Davd Aple here, for mooh all, he machng hegh and offe d are gven n all n b ee Aple, 7: 7.4, 9 4. d Inegraon over a cell ee example hee hen lead o cell average ] ln[ / d av 4 ln d av 4 Eqaon: Bondar ondon on fxed from amed profle eqaon 4 a he near-all node. A parclar vale a a cell cenre can be forced n a fne-volme calclaon b modfng he orce coeffcen: b, here a large nmber e.g.. The marx eqaon for ha cell hen become F F a a φ φ or F F a a a φ φ Snce a large nmber h effecvel force φ o ae he vale. Renold-Sre Eqaon For he Renold ree, one mehod o fx he vale a he near-all node from he nearall vale of and he rcre fncon /, he laer beng derved from he dfferenal re-ranpor eqaon on he ampon of local eqlbrm. For he andard model h gve ee he example hee: v v v v v 4 Wh he vale for,, ec. from he andard model h gve

19 .98, v.48,.654, v When he near-all flo and all-normal drecon are no convenenl algned n he x and drecon repecvel, he acal rcre fncon can be obaned b roaon. Hoever, for -dmenonal and eparang/reaachng flo he flo-orened coordnae em no fxed a pror and can ng rond gnfcanl beeen eraon f he mean veloc mall, mang convergence dffcl o oban. A econd and no m preferred approach Aple, 7 o e cell-averaged prodcon and dpaon n he Renold-re eqaon n he ame manner a he -eqaon, nong ha, n mple hear and n floalgned coordnae: v, v v v, h he rao v / v deermned from 44 a.97. In he all-fncon formlaon, proporonal o he qare of he veloc a he near-all node, o roang from flo-algned coordnae o he acal arean coordnae em doe no cae dconne n he re prodcon here he veloc revere gn; e.g. near eparaon or reaachmen pon Effecve Vco for Dfferenal Sre Model DSM conan no rblen vco and have a repaon for nmercal nabl. An arfcal mean of promong abl o add and brac a graden-dffon erm o he rblen flx: 45 h he fr par averaged beeen nodal vale and he la par dcreed acro a cell face and reaed mplcl; ver mlar o he Rhe-ho algorhm for prere-veloc coplng n he momenm eqaon. The mple choce for he effecve vco 46 A beer choce o mae e of a naral lnage beeen ndvdal ree and he correpondng mean-veloc graden hch are from he acal re-ranpor eqaon. Amng ha he re-ranpor eqaon h no bod force are orce-domnaed hen or, h he bac DSM ho all-reflecon erm, FD 8-9 Davd Aple

20 FD 8- Davd Aple Expand h, denfng he erm hch conan onl or x a follo. For he normal ree : Hence, x Smlarl for he hear ree : hence x Hence, from he re-ranpor eqaon, x x 47 here he effecve vcoe boh for he α componen of momenm are:, 48 Noe ha he effecve vcoe are anoropc, beng lned o parclar normal ree. A more dealed anal can accommodae all-reflecon erm n he prere-ran model, b he exra complex no fed.

21 Some Reference for Indvdal Trblence Model Aple, D.D., 7, FD calclaon of rblen flo h arbrar all roghne, Flo, Trblence and ombon, 78, Aple, D.D. and Lechzner, M.A., 998, A ne lo-renold-nmber nonlnear oeqaon rblence model for complex flo, In. J. Hea Fld Flo, 9, 9-. raf, T.J., 998, Developmen n a lo-renold-nmber econd-momen clore and applcaon o eparang and reaachng flo, In. J. Hea Fld Flo, 9, raf, T.J., Lander, B.E. and Sga, K., 996, Developmen and applcaon of a cbc eddvco model of rblence, In. J. Hea Fld Flo, 7, 8-5. Ga, T.B. and Spezale,.G., 99, On explc algebrac re model for complex rblen flo, J. Fld Mech., 54, Gbon, M.M. and Lander, B.E., 978, Grond effec on prere flcaon n he amopherc bondar laer, J. Fld Mech., 86, Hanal, K., 994, Advanced rblence clore model: a ve of crren a and fre propec, In. J. Hea Fld Flo, 5, 78-. Jarl, S. and Hanal, K., 995, A econd-momen clore for non-eqlbrm and eparang hgh- and lo-re-nmber flo, roc. h Smp. Trblen Shear Flo, ennlvana Sae nver. Lam,.K.G. and Bremhor, K.A., 98, Modfed form of he -e model for predcng all rblence, Jornal of Fld Engneerng,, Lander, B.E., 989, Second-Momen lore and e n modellng rblen ndral flo, In. J. Nmer. Meh. Fld, 9, Lander, B.E., Reece, G.J. and Rod, W., 975, rogre n he developmen of a Renoldre rblence clore, J. Fld Mech., 68, Lander, B.E. and Sharma, B.I., 974, Applcaon of he energ-dpaon model of rblence o he calclaon of flo near a pnnng dc, Leer n Hea and Ma Tranfer,, -8. Lander, B.E. and Spaldng, D.B., 974, The nmercal compaon of rblen flo, omper Meh. Appl. Mech. Eng.,, Len, F-S. and Lechzner, M.A., 99, Second-momen modellng of recrclang flo h a non-orhogonal collocaed fne-volme algorhm, n Trblen Shear Flo 8 Mnch, 99, Sprnger-Verlag. Mener, F.R., 994, To-eqaon edd-vco rblence model for engneerng applcaon, AIAA J.,, Rbnen, R. and Baron, J.M., 99, Nonlnear Renold re model and he renormalaon grop, h. Fld A,, Spezale,.G., 987, On nonlnear K-l and K- model of rblence, J. Fld Mech., 78, Spezale,.G., Sarar, S. and Ga, T.B., 99, Modellng he prere-ran correlaon of rblence: an nvaran dnamcal em approach, J. Fld Mech., 7, Wlcox, D.., 988, Reaemen of he cale-deermnng eqaon for advanced rblence model, AIAA J., 6, 99-. Wlcox, D.., 988, Ml-cale model for rblen flo, AIAA Jornal, 6, -. Wlcox, D.., 998, Trblence modellng for FD, nd Edon, DW Indre. Yaho, V., Orzag, S.A., Thangam, S., Ga, T.B. and Spezale,.G., 99, Developmen of rblence model for hear flo b a doble expanon echnqe, h. Fld A, 7, 5. Yohzaa, A., 987, Sacal anal of he dervaon of he Renold re from edd-vco repreenaon, h. Fld, 7, FD 8- Davd Aple