On the turbulence models and turbulent Schmidt number in simulating stratified flows

Transcription

1 Sh, Z., Chen, J., and Chen, Q On he urbulence models and urbulen Schmd number n smulang srafed flows, Acceped by Journal of Buldng Performance Smulaon. On he urbulence models and urbulen Schmd number n smulang srafed flows Zhu Sh a, Jun Chen a, Qngyan Chen b,a a School of Mechancal Engneerng, Purdue Unversy, Wes Lafayee, IN 47907, USA b School of Envronmenal Scence and Engneerng, Tanjn Unversy, Tanjn 0007, Chna Jun Chen (Correspondng Auhor) Emal: junchen@purdue.edu; Tel.: +1 (765) Absrac Srafed flows are prevalen n ndoor and oudoor envronmens. To predc hese flows, hs nvesgaon evaluaed he performance of seven urbulence models by comparng he smulaon resuls wh he expermenal daa of boh wealy and srongly srafed jes. The models esed ncluded sx Reynolds-Averaged Naver-Soes (RANS) models and one Large Eddy Smulaon (LES) model. The velocy, urbulen nec energy, and Reynolds sress dsrbuons were examned. For he wealy srafed je, all seven models could predc well he mean velocy, bu for he srongly srafed je, he RSM and LES models overpredced he velocy n he unsable srafcaon regon. The SST model was he bes. Ths paper also nroduced a new dynamc urbulen Schmd number model whch can mprove he predcon of densy dsrbuon. In addon, hs nvesgaon analysed he compung coss of he models as well as he vorcy and enranmen raos predced by he models. Keywords: CFD; urbulen srafed flow; urbulence models; dynamc urbulen Schmd number model; enranmen Nomenclaure Concenraon of speces j Cj D D g Gb G I p Pj Re R r 1/, r 1/ S Dameer of je nozzle, characersc lengh scale Turbulen eddy dffusvy Gravaonal acceleraon n drecon The generaon of urbulence nec energy due o buoyancy The generaon of urbulence nec energy due o he mean velocy gradens Turbulen nensy Pressure Sress producon Reynolds number Rchardson number The half-wdh of je n sable and unsable regons Source erm of scalar 1

2 Sc,, u U x Y Turbulen Schmd number 0 Y C Tme Velocy magnude n drecon Je nal velocy, characersc velocy Coordnae n drecon Dsspaon of due o urbulence Dsspaon of due o urbulence Densy Scalar componen Knec energy per un mass Turbulen dsspaon rae Specfcaon dsspaon rae Coeffcen of effecve dffuson of scalar,eff Dynamc vscosy C j Turbulen dynamc vscosy Schmd number Shear sress Change n varable Non-dmensonal densy dfference Turbulen nemac vscosy Vorcy n drecon 1 Inroducon Undersandng srafed flows ha have slghly dfferen denses plays a sgnfcan role n ndoor aerodynamcs (Sørensen and Weschler 00), envronmenal and geophyscal flow dynamcs (Blocen e al. 007), and oher engneerng applcaons. In an ndoor envronmen, for example, an arcraf cabn or buldng, srafed flow develops when ar wh dfferen emperaures s suppled no he enclosed space hrough he HVAC sysem (Lau 00). Mxng of fresh waer wh seawaer n he esuary and neracon of warm and cold ocean currens are oher examples of srafed flows n naure. In chemcal plans, dfferen soluons are mxed, wh srafcaon by dfferen denses or emperaures. Sudyng he nerplay of urbulence and srafcaon s of een neres for ganng an undersandng of he mxng dynamcs needed for opmal desgn of ndoor envronmens, accurae predcon of geophyscal dynamcs, and opmal qualy of chemcal producs. In pas years, a varey of expermenal sudes have been done on srafed flows. Kneller e al. (1999) employed Laser-Doppler Anemomery (LDA) o predc he behavor of srafed gravy currens and obaned wo-dmensonal nformaon on urbulen flow feld and energy dsrbuon. Banes (001) adoped a parcle racng mehod o measure he flow

3 feaures of dense flud down genle slopes no a densy-srafed envronmen and derved a quanave model o descrbe urbulen downslope flows no srafed envronmens. Dalzel e al. (007) used he synhec schleren echnque and Parcle Image Velocmery (PIV) o measure he densy and velocy felds, respecvely, n whch he densy feld measuremen was used no only o predc he srafcaon bu also o correc he normal errors relaed o he refracve ndex varaons n velocy feld measuremen. Xu and Chen (01) conduced smulaneous measuremens of velocy and densy felds n a horzonally nroduced srafed je by combnng PIV and Planar Laser-Induced Fluorescence (PLIF) echnques. The measuremens revealed he srucures of a srafed je n boh sable srafcaon and unsable srafcaon regons. The aforemenoned expermenal sudes and smlar ones explored varous expermenal echnques for nvesgang he srucures of srafed flows. Moreover, wh he developmen of compuaonal resources, compuaonal flud dynamcs (CFD) has become ncreasngly popular n nvesgang he fluds problem. Among he dfferen echnques, Reynolds-Averaged Naver-Soes (RANS) has receved he mos applcaons by adopng dfferen urbulen models o close he equaons. Two-equaon urbulence models have been manly used n he smulang of srafed flows. Lu e al. (008) suded he performances of RNG model and sandard model n smulang snglesded naural venlaon drven by srafcaon effec, whch s due o emperaure dfference. Ther comparson wh expermenal daa concluded ha RNG model performed beer han sandard model n predcng such flow. J e al.(008) adoped he model (Wlcox 1988) o nvesgae naurally venlaed double-sn facades(dsfs) wh Venean blnds nsde he facade cavy. The resuls demonsraed ha Venean blnds could enhance he buoyancy-drven naural venlaon of he facade cavy. Cropper e al. (010) developed a CFD model o smulae he arflow and emperaures around human body usng SST model (Mener 1994). Ths model was furher coupled wh a hermal comfor model, whch was able o predc human hermal comfor n varous envronmenal condons. Venayagamoorhy e al. (00) esed he performance of he sandard model n sably srafed flows, usng daa from drec numercal smulaon (DNS). Ther resuls showed ha he buoyancy parameer C was a very sensve parameer for srafed flows. Besdes he wo-equaon models, Spall (1998) adoped he Reynolds Sress Model (RSM), a seven-equaon model, o nvesgae he naural srafcaon phenomenon n cylndrcal hermal sorage ans, showng ha he RSM model can gve a more accurae predcon of he hermoclne hcness han he model. These numercal smulaons provded dealed nformaon concernng srafed flows, whch was complmenary o he expermenal resuls. However, here has sll been no sysemac evaluaon of he performance of dfferen urbulen models a dfferen urbulence and srafcaon levels, ndcaed by Reynolds numbers and Rchardson numbers, respecvely. Ths s parcularly mporan n he ransonal or developng regon of he flow snce mos of he prevous sudes have been focused on fully-developed regons where many urbulence models have been proven o funcon well n unsrafed flows. Furhermore, n he smulaons of flows wh densy srafcaon, a ey parameer n predcng densy dsrbuon s he urbulen Schmd number ( Sc ). He e al. (1999) denfed he sgnfcan effec of he urbulen Schmd number on he speces spreadng rae n je-ncrossflows. The auhors also concluded ha Sc should be a varable n je-n-crossflows based on a sem-emprcal analyss. Tomnaga and Sahopoulos (007) dscovered ha he opmal

4 urbulen Schmd number depended on local flow characerscs and recommended a dynamc deermnaon of Sc accordng o local flow srucure. Snce Sc has a large mpac on he speces ransfer n smulang srafed flows, adopng such a dynamc model s more reasonable han usng a consan urbulen Schmd number n smulang srafed flows, as n mos exsng pracces. Therefore, here are hree-fold objecves n hs sudy: (1) To sysemacally evaluae he performances of mos prevalen models n smulang srafed flows; () To furher nvesgae he mpac of he urbulen Schmd number on smulang srafed jes, and o develop a dynamc urbulen Schmd number model based on local flow srucure; () To sudy he enranmen effec and vorcy n srafed jes. Ths paper repors our effor n he nvesgaon. Research Mehod Ths secon descrbes he mos prevalen urbulence models used for predcng srafed jes, he expermenal daa used for valdang he models, he numercal algorhm used n solvng he urbulence model, and he dynamc Schmd number model developed o mprove he performance of he urbulence models..1 Turbulence models for srafed flows Srafed flow wh a small densy dfference can be descrbed by connuy equaon ( u ) 0, (=1,,) (1) x momenum equaon u j u j p u j u g, (=1,,) () j x x j xx and speces (scalar) ranspor equaon Cj J j ( Cj) u, (=1,,) () x x The deals of he modelng for he equaons are shown n Table 1. In a RANS smulaon, a specfc flow varable s decomposed no mean componens and flucuang componens: uu u,, where u and u are he mean and flucuang velocy componens, and and are he mean and flucuang scalar componens. The mean componens are solved from he RANS equaons. On he oher sde, n LES, he flow varables are flered by a low-pass flerng operaon wh a chosen fler wdh (correspondng o he grd spacng used n he compuaon). As a resul, he large eddes are solved from flered Naver-Soes equaons, and he nfluence of he unresolved (sub-grd scale, SGS) eddes s descrbed by SGS models. Ths nvesgaon used he followng prevalen urbulen models: he sandard model (Launder and Spaldng 197), RNG model (Yaho and Orszag 1986), realzable model (Shh, e al. 1994), sandard model (Wlcox, D.C. 1998), SST model (Mener 1994), and RSM model (Gbson and Launder 1978; Launder 1989; Launder and Reece 1975). Snce LES has ofen been beleved o yeld a more accurae predcon han RANS, LES has also been examned usng he Smagornsy-Llly model (Smagornsy 196). The ranspor equaons for any mean parameer n he urbulence models can be expressed n a general form (Whe 1991; Paanar 1980): 4

5 u, eff S x x x where represens a specfc varable,,eff source erm. Table 1 summarzes he choces of he coeffcen of effecve dffuson, and S he,eff, S and he correspondng consans n he governng equaons and urbulence modelng equaons used n he curren nvesgaon.. Expermenal daa of srafed jes Snce he urbulence modelng used approxmaons, s essenal o valdae he compuaonal resuls usng expermenal daa. The expermenal daa from a srafed je (Xu and Chen 01) were used as benchmars n he presen sudy o valdae and develop he models. Fgure 1 shows he schemac of he expermen, and four ses of daa were acqured. In wo unsrafed cases ( hgh-re and low-re ), he flud dscharged from he je nozzle had he same densy as he flud n he an. In wo srafed cases ( hgh-r and low-r ), he flud njeced no he an was of hgher densy han he flud n he an, leadng o densy srafcaon. In order o quanfy he degree of srafcaon, he Rchardson number was employed: R Dg /( U ) 0 0 0, where 0, D, U 0 are he characersc densy dfferences, lengh scale, and velocy, respecvely. In he expermen, boh velocy and densy felds were measured wh he combned PIV and PLIF sysem. Wh he velocy and densy daa, Xu and Chen examned averaged parameers, Reynolds sresses, vercal densy flux, urbulen nec energy budge, ec., whn cenral vercal plane. Alhough measuremens were avalable n boh he unsrafed and srafed cases, he curren nvesgaon manly focuses on he numercal calculaons n srafed cases. The deals of hgh-r and low-r cases were summarzed n Table. Average velocy, average densy, urbulen nec energy and Reynolds sress values were examned n he presen sudy. In hs horzonal srafed je, boh sable and unsable srafcaon regons exs, as shown n Fg. 1. Sable srafcaon was formed where d dz 0 and urbulence were weaened by he buoyancy effec. Unsable srafcaon was formed where d dz 0 and urbulence was enhanced by he buoyancy effec. The measuremens enable comparave sudes n boh sable and unsable srafcaon regons.. Numercal smulaons of he srafed jes Our numercal smulaon of he srafed je flow used he followng assumpons: (1) Snce he averaged flow feld was symmerc wh respec o he cenral vercal plane (y=0), half of he doman was used n he RANS smulaons. In LES, he whole doman should be used o resolve he hree dmensonal unseady flow moons. () A sold cylnder was deployed n he an o smulae he exsence of he je nozzle, and he velocy and scalar profles a he je ex were prescrbed as boundary condons. Fg. a shows he dmensons of he compuaon doman, whch s exacly he same sze as n he expermen. Fg. b, c and d presen he mesh of he CFD model for RANS cases. The mesh srucure for LES smulaon was very smlar o ha shown n Fg., bu he grds were much fner, as dealed below. Ths sudy adoped a non-unform grd sze mehod for he meshng. The grd was he fnes a he je nozzle and gradually ncreased from he nozzle. To chec grd ndependence, RANS smulaons (usng he sandard model) of he Low-R case were conduced on hree dfferen grds: 14,990, 41,80, and 811,190 grds, respecvely, represenng coarse, medum, and fne grds. However, dfferen numbers of cells were used on he je axes and cross- (4) 5

6 secons (vercal o axes) of he flud doman. The able n Fg. descrbes he dealed dfferences of he hree grds. Fg. also shows he velocy profles along he cenerlne from hese hree grd sysems. The resuls from he coarse and medum grds showed sgnfcan dfferences, whle he resuls from he medum and fne grds almos collapsed. Ths suggess ha he medum grd led o grd-ndependen resuls and he grd was used n he followng RANS smulaons. Snce LES needs o use he enre doman and ypcally requres fner meshes, he grd ndependence es for LES was conduced separaely n a smlar way. The number of grds for LES was fnalzed a 1,64,10. Ths sudy employed a numercal solver n ANSYS Fluen 14.0 o solve equaons (1) o () o oban he flow and scalar felds. Pressure-velocy was coupled usng he SIMPLE scheme. The second-order upwnd scheme was used o dscreze he momenum, urbulen nec energy, and speces erm. In order o assure accuracy, he second order mplc mehod was employed for he ransen formulaon. Unseady smulaons were adoped wh me sep 0.005s. A sensvy sudy was also done on me sep sze by confrmng ha a smaller me sep dd no change he smulaon resuls a lo. A each me sep, 0 eraons were conduced. Whn each me sep, x, y and z velocy resduals dropped by 4 orders of magnude; a he end -8 of each me sep sze, energy resdual decreased o 10, and scaled speces resdual decreased o 10. For LES smulaons, -second me nerval was used for daa collecng and averagng n obanng averaged values from nsananeous parameers. A he je ex, he velocy and scalar (mass fracon) profles were prescrbed accordng o he expermenal daa. The rgh boundary of he calculaon doman was defned as a pressure-oule, whch served as an oule for he flow. (For he pressure-oule boundary, when gravy was enabled n he calculaon, he ncrease of pressure due o gravy was consdered auomacally.) In he expermen, he opsde was he nerface beween he flud and he amosphere, so he zero-shear (symmery) boundary condon was defned for he op boundary n he presen smulaon. For oher boundares, no-slp boundary condons were specfed..4 Dynamc Schmd number model To smulae he srafed flows usng eddy-vscosy-ype models, he vercal momenum flux and densy flux along he buoyancy drecon were wo ey parameers beng modeled by eddy vscosy and eddy dffusvy D : u1 uu 1 and u D x (5) x In parcular, and D were relaed o urbulen Schmd number Sc D, whch s usually chosen as a consan. Many aforemenoned sudes have shown he defcency of such a smple model. For example, Xu and Chen (01) demonsraed ha densy flux ( ' u ') was no only dependen on densy graden, bu also on velocy graden. The presen sudy proposed a dynamc urbulen Schmd number model ha relaes he local Sc wh a local velocy graden and scalar graden. If one assumes n srafed flows ha momenum flux and densy flux are dependen on velocy graden and densy graden and apples he Taylor expanson,.e., 6

7 u1 u 1 u 1 1 f, A1 A B1 B x x x x x x u1 u 1 u 1 u g, C1 C D1 D x x x x x x uu, (7), where he A, B, C, and D were expanson coeffcens (=1, ). When he frs-order approxmaon was employed, u1 u1 uu 1 A1 B1 and u C1 D1 (8) x x x x Then he expresson of Sc led o B1 A1 1 1 A u 1 B1 / u u 1 / x x x x x Sc (9) D u u1 1 C1 D1 / D1 C1 / x x x x x * u Sc hus can be expressed as a funcon of 1 u1 /, or s normalzed erm / ( * * x x x x * x x = D, * u u 1 = 1 U, where D and U 0 are he characersc lengh scale and characersc velocy. 0 * u By usng Taylor s expanson agan and denong 1 /, Sc * * can be expressed as x x Sc h() 01 (10) where he model coeffcens s can be deermned by he expermenal daa, as shown n Fg. 4, for boh hgh-r and low-r cases, as well as dfferen downsream locaons. The fng yelded wo coeffcens: 0=1.57 and 1=-0.46, where he hgher order erms were negleced. Thus, he urbulen Schmd number can be dynamcally expressed as: Sc (11) In he presen sudy, hs dynamc urbulen Schmd number model (DTSN-Model) s appled when he velocy graden and scalar graden fall no he followng range: ( u U )/ ( x D) , he regon where mos mxng processes happened and where he expermenal Sc values were seleced for DTSN-model developmen. The model was mplemened no he RANS models hrough a user-defned funcon. Resuls Fg. 5 shows he velocy conours and sreamlnes for boh wea srafcaon (low-r) and srong srafcaon (hgh-r) cases. The presen sudy evaluaes he performance of sx RANS models and LES n srafed flow, under he wo cases. For each case, frs order momen (mean (6) 7

8 velocy) and second order momens (urbulen nec energy and Reynolds sresses) were compared wh expermenal daa a fully-developed downsream locaons (x=0d for low-r case and x=10d for hgh-r case, respecvely). The urbulen nec energy s TKE 1 uu 1 1 uu u u (1) and a shear Reynolds sress, uu 1, were compared wh he expermenal daa. As suggesed by Brer and Schazmann (007), quanave comparson wh expermenal daa s a good mehod o evaluae he performances of varous urbulence models. Thus MSE (Mean Squared Error) (Lehmann and Casella 1998) was used o descrbe he degree of devaon of predced values from expermenal values n hs sudy. MSE s defned as N 1 MSE ( X p, X m, ), (1) N 1 where X p, s he predced value a -h locaon, X m, s he measured value a -h locaon and N s he number of locaons compared. Furhermore, he proposed DTSN-model was esed by nvesgang he predced scalar dsrbuon..1 Mean velocy In he horzonally nroduced srafed je, s cenerlne devaes from he horzonal drecon due o he buoyancy effec, as demonsraed n Fg. 6. To quanfy he degree of hs devaon, z c was defned such ha c 1 U () x u(,0, x z ), where U() x was he pea value a downsream c locaon x. Meanwhle, o characerze he je expanson, wo half-wdh locaons, z 1/ and z 1/ were defned n sable and unsable regons, respecvely: u x z 1/ u x z 1/ 8 c (,0, ) (,0, ) U (14) The correspondng half-wdhs of he je n sable and unsable regons were defned respecvely: c r 1/ zc z 1/, r 1/ z 1/ zc (15) r/ r, can be uu c vs. 1/ The self-smlary characerscs of a homogenous round je, found n many oher sudes, e.g., Pope (001). Fg. 7 shows he self-smlary curves from he expermenal daa and smulaon resuls. Due o he srafcaon, z coordnae was normalzed as ( z z c ) r1/, where r 1/ ( r 1/ r 1/ ). In he low-r case (Fg. 7a), he smulaon resuls agree well wh he expermenal daa n he sable srafcaon regon, bu n he unsable srafcaon regon, a dscrepancy was observed for ceran urbulence models. Based on he MSE values from Table, he SST model, sandard model and RNG model yelded he bes performances, whle he resuls from he oher models were sll accepable. Ths shows ha when he srafcaon was wea and he urbulence effec was domnan (Re=4,000), hese models could yeld an accurae predcon of he mean velocy. In he hgh-r case (Fg. 7b), he srafed je bends more qucly han n he low-r case. Overall, he performances of he urbulence models were worse n he hgh-r case han n he low-r case. Ths ndcaes ha mos of he esed models wor beer n hgh Reynolds number flows han n low or ransonal Reynolds number flows. The predcon accuracy n he sable

9 srafcaon regon was dfferen from he one n he unsable srafcaon regon. In he sable srafcaon regon, all he urbulence models gave accepable predcons of he mean velocy. However, n he unsable srafcaon regon, large dscrepances from he expermenal resuls were observed n he resuls from he RSM model and LES. Alhough he RSM model solved ranspor equaons for Reynolds sresses, whch can be helpful for predcng second order flow characerscs, was defcen n predcng he mean velocy when he srafcaon was srong. The LES resul n he unsable srafcaon regon devaed even more from he expermenal daa han he RSM resul, possbly due o he problem of he Smogrnsy-Llly model for flow n a ransonal regon. As ndcaed by Voe (1996), he coeffcens of he Smagornsy-Llly model are proporonal o he square of he grd scale, and vansh oo slowly when he Reynolds number s low. As a resul, LES wh he Smagornsy-Llly model overpredcs he subgrd eddy-vscosy. Our resuls show once agan ha he predcon of flow feaures n he unsable srafcaon regon was more dffcul han n he sable srafcaon regon. The MSE values showed ha SST model gave he bes mean velocy profle among all he models esed, smlar o he ess for he low-r case. One mporan advanage of he SST model s ha a low Reynolds number correcon can be used o damp he urbulen vscosy n low Reynolds number smulaons.. Turbulen nec energy Fg. 8 shows he predced TKE a a downsream locaon n low-r and hgh-r cases, and Table 4 llusraes he MSE values under varous models. For he low-r case (Fg. 8a), among all he wo-equaon models, he SST model led o he bes resuls. The hree varaons of models also capured he general rend of he TKE profle. However, sandard sgnfcanly underpredced TKE a he core regon of he je (-<z/d<). Compared o he sandard model, he SST model modfed he urbulen vscosy formulaon o accoun for he ranspor effecs of he prncpal urbulen shear sress. Snce he srafed je flow was a ypcal shear sress flow, ha s why he SST model yelded a sgnfcanly beer predcon. The RSM model also led o accepable predcon of TKE. The expermenal resul shows a den around he cener of he je, and he RSM was he only model ha could predc. On he oher hand, LES dd no produce sasfacory resuls as expeced because he SGS model esed n he presen sudy may be he source of he predcon error. Fg. 8b compares he TKE profles predced wh he expermenal daa n he hgh-r case. Due o he srong srafcaon n he hgh-r case, he TKE profle was asymmerc and he pea devaed downwards. Smlar o n he low-r case, he RSM and SST models gave good predcons of he TKE dsrbuon a locaons close o je axs, and SST model gave he bes overall TKE predcons. LES overpredced sgnfcanly he TKE, whch may be arbued o he defcency of he Smagornsy-Llly model n low Reynolds number flows. Overall, he predcons of TKE n hs case were no as accurae as hose n he low-r case, whch was smlar for he mean velocy.. Shear sress Fgure 9 shows he comparson beween shear sresses. From Table 5, for he low-r case, he models ha performed well n predcng TKE, he SST model, and he RSM model, also gave good predcons of uu 1, especally n sable srafcaon regon. The RSM model solves he ranspor equaon for Reynolds sresses, whle oher eddy vscosy models reles on he assumpon ha was soropc, whch s no rue n srafed flows. Thus, he RSM model was beer n predcng Reynolds sresses han mean veloces. Snce he RNG model can ae 9

10 he srafcaon effecs (Moghaddas-Nan e al. 1998) no accoun, also performed well. For shear sress resuls, all predced profles capured he nverse-symmerc characersc. However, he magnude n he unsable srafcaon regon was underpredced compared wh n he sable srafcaon regon. Ths ndcaes ha he smulaons n he unsable srafcaon regon were more dffcul due o he complex physcs of flud n hs regon. For he hgh-r case, he RNG model sll yelded he bes predcon among all hree models, alhough s predcon performance n he unsable regon was much worse han n he sable regon. The SST model and RSM model also underpredced he shear sress n he unsable srafcaon regons. However, hese wo models performed bes when evaluaed by he overall resuls. All he oher RANS models underpredced he sresses. LES overpredced he shear sress n sable srafcaon regon bu underpredced n unsable srafcaon regon..4 Predcons of scalar dsrbuon Secon 5 nroduced a new dynamc urbulen Schmd number model. By applyng o he SST model ha gave he bes predcon of he mean flow characerscs above, hs nvesgaon could evaluae he mpac of he urbulen Schmd number n predcng a scalar, dmensonless densy dfference ( )/. The sandard urbulen Schmd amben number has been conroversal (He e al. 1999; Tomnaga e al. 007). He e al. (1999) suggesed Sc 0. for a je-n-cross flow, whch s very smlar o he srafed je n hs sudy. A consan urbulen Schmd number 0.7 s always recommended n commercal CFD sofware as he defaul value. The presen nvesgaon evaluaes he dfference n choosng hree urbulen Schmd numbers: Sc 0., Sc 0.7, and Sc deermned by DTSN (equaon 1). Fg. 10 and Fg. 11 show normalzed densy dsrbuons a wo dfferen locaons (beween x=10d and x=0d, ndcaed by upsream and downsream, respecvely) n boh he low-r and hgh-r cases. The predced densy dfference dsrbuons n he srafed flows were hghly dependen on he urbulen Schmd number. The varable DTSN-model gave he bes densy dsrbuons, especally n downsreams. One may also noe ha he larger he value of Sc was, he hgher he predced pea densy was. Ths s because he mxng of he wo speces n he srafed flows was nversely dependen on Sc. A lower Sc can dffuse dense flud faser no he amben lgh flud, and hus, lead o a lower pea densy..5 Predcon of vorcy n he srafed jes Sudyng vorcy s mporan for characerzng he local flow srucure. Fg. 1 shows he vorcy conours a he cener vercal plane n he low-r and hgh-r cases predced by he SST model and compares hem o he expermenal daa. In he low-r case, he vorcy dsrbuon was almos anmerc, and he boundary beween negave and posve vorcy was bascally he cenerlne when x/d<15. In he hgh-r case, n conras, he boundary ben downwards wh he ncrease of x/d. The vorcy n he sable srafcaon regon was larger han n he unsable srafcaon regon. Overall, he vorcy dsrbuons n boh he wea and srong srafcaon jes were capured wh accepable accuracy. These resuls show agan ha he SST model can predc he srafed flow characerscs..6 Enranmens n he srafed jes The numercal smulaons also enable us o analyze he enranmen n he srafed jes. The 10

11 m enranmen rao s defned as m, where m 0 s he mass of flud dscharged from he je nozzle 0 and m s he mass across a secon perpendcular o he je. Rcou and Spaldng (1960) concluded ha he enranmen rao of a horzonal je could be expressed by usng an emprcal 1 m 1 x formula: 0.( ), where 0 s he densy of he flud dscharged from he nozzle, m0 0 D and 1 s he densy of amben flud. In hs sudy, we employed he SST model o predc he enranmen rao and compared he resuls wh hose of he emprcal formula. As shown n Fg. 1, boh mehods gave a good predcon for he enranmen rao when x/d<0, where he ncrease of he enranmen rao was proporonal o x/d. However, a large dscrepancy was found where x/d>0. Ths was manly due o he confnemen of he flud an, whch decreased he amoun of enranmen. The emprcal formula assumed a perfec free-je. Due o he enranmen, he percenage of flud n he je from he nozzle (called new flud ) decreased as x/d ncreased. When x/d>0, only abou 10% of he flud n he je orgnaed from he nozzle. The enranmen n he hgh-r case was much more complex han n he low-r case. The enranmen rao predced by he SST model was smaller han ha from he emprcal formula. The reason s ha he srong buoyancy effec n he hgh-r je ben he je heavly, whch led o a decrease n he enranmen whle he emprcal formula assumed he buoyancy effec o be neglgble. Therefore, he emprcal formula should no be used for deermnng he je enranmen wh srong srafcaon. The numercal predcon n Fg. 14 shows ha he enranmen rao curve can be dvded no a lnear regon and a nonlnear regon. In he lnear regon, he enranmen rao ncreased wh x/d wh a lnear coeffcen of 0.5, a much smaller value han 0. n he emprcal formula. The enranmen rao n he nonlnear regon ncreased more slowly han n he lnear regon due o he mpngemen of he je a he an wall, whch decreased he enranmen amoun. 4 Dscussons on compuaon coss Ths sudy also evaluaed he compuaon me by hese seven models. All he sx RANS models used he grd number of 41,80. Due o he hgh requremen for he grd resoluon, he LES smulaon used a much larger grd number of 1,64,10, whch s abou four mes ha for RANS smulaons. The hgh-r case was used for comparson. All he smulaons were esed on one node of a Lnux-cluser wh wo.5 GHz Quad-Core AMD 80 processors. The calculaon me for runnng a 1-second nerval ransen smulon wh dfferen models was recorded and ploed n Fg. 15. Among all he models, he RNG model requred he longes compuaon me. The compuaon cos of he SST model was slghly hgher han ha of he sandard model. Neverheless, he compuaon coss of all fve eddy-vscosy models were close. The compuaon cos of he RSM models was abou 5% hgher han he average of he eddy-vscosy models. Ths s undersandable because he RSM model solved seven ranspor equaons for urbulence parameers, whle he eddy-vscosy model solved only wo for urbulence. The LES smulaon requred abou wce as much compuaon me as he RSM and almos hree mes he average compuaon me for he eddyvscosy models. Ths s manly arbued o he much larger number of grds used n he LES han n he RANS smulaons. Noe ha he use of DTSN requres 10% addonal compuaon me for calculang he dynamc Schmd number. 11

12 5 Concluson The nvesgaon led o he followng conclusons: (1) Ths nvesgaon evaluaed he performances of sx RANS models and one LES model n predcng srafed flows. In he wealy srafed flow where he urbulen effec was domnan, all seven models could predc accuraely he mean flow, bu wh large dscrepances n predcng he second-order flow characerscs. Overall, he RNG and SST models performed very well, bu he SST was he bes. The superory of LES was no observed n predcng he second-order flow characerscs. () I was more dffcul for he models o predc srongly srafed flow. All he models could sll predc well he mean flow n he sable srafcaon regon, bu he RSM model and LES overesmaed he velocy n he unsable srafcaon regon. For predcng he secondorder flow characerscs, he RSM, SST and RNG models can be used, and he frs wo yelded he bes overall resuls. The LES wh he sandard Smagornsy model may no be suable for he low Reynolds number ransonal flows n hs sudy. Therefore, LES does no always gve beer predcng resuls han RANS models, alhough usually aes much longer me. () Ths curren paper shows ha urbulen Schmd number has large mpac on scalar dsrbuon predcon. Thus aenon should be pad o n scalar fled smulaon. Ths sudy proposed a new dynamc urbulen Schmd number model based on local velocy graden and densy graden. The model can mprove smulang scalar varables, such as densy dfference dsrbuons n he jes. (4) The compuaon coss of he fve eddy-vscosy models n RANS were comparable, bu he RSM model requred 5% more compung me, and he LES needed hree mes more compung me. The adopon of he dynamc urbulen Schmd number model used an addonal 10% compung me. (5) The CFD models can predc vorcy dsrbuons n he srafed jes. The enranmen rao can be calculaed by he emprcal formula for he wealy srafed je bu no for he srongly srafed je. I s no suggesed o use emprcal formula o predc enranmen rao when srafcaon s srong n je flows. References ANSYS. Inc ANSYS Fluen Theory Gude, release 14.0 [onlne]. Avalable from hp:// [Accessed 1 November 01] Bacon, S "Decadal varably n he ouflow from he Nordc seas o he deep Alanc Ocean." Naure 94 (6696): Blocen, B., T. Sahopoulos, and J. Carmele "CFD smulaon of he amospherc boundary layer: wall funcon problems." Amospherc envronmen 41 (): 8-5. Brer, R., and M. Schazmann. (Ed.) "Bacground and jusfcaon documen o suppor he model evaluaon gudance and proocol: COST acon 7 Qualy assurance and mprovemen of mcroscale meeorologcal models." Hamburg: Unversy of Hamburg Meeorologcal Insue. Cropper, P. C., T. Yang, M. Coo, D. Fala, and R. Yousaf "Couplng a model of human hermoregulaon wh compuaonal flud dynamcs for predcng human envronmen neracon." Journal of Buldng Performance Smulaon (): -4. Dalzel, S. B., M. Carr, J. K. Sveen, and P. A. Daves "Smulaneous synhec schleren and PIV measuremens for nernal solary waves." Measuremen Scence and Technology 18 (): 5. 1

13 Ghasas, N. S., D. A. Shey, and S. H. Franel. 01. "Large eddy smulaon of hermal drven cavy: Evaluaon of sub-grd scale models and flow physcs." Inernaonal Journal of Hea and Mass Transfer 56 (1): Gbson, M. M., and B. E. Launder "Ground effecs on pressure flucuaons n he amospherc boundary layer." Journal of Flud Mechancs 86 (0): He, G., Y. Guo, and A. T. Hsu "The effec of Schmd number on urbulen scalar mxng n a je-n-crossflow." Inernaonal Journal of Hea and Mass Transfer 4 (0): 778. Hll, B. J "Measuremen of local enranmen rae n he nal regon of axsymmerc urbulen ar jes." Journal of Flud Mechancs 51 (04): J, Y., M. J. Coo, V. Hanby, D. G. Infeld, D. L. Loveday, and L. Me "CFD modellng of naurally venlaed double-sn facades wh Venean blnds." Journal of Buldng Performance Smulaon 1 (): Kneller, B. C., S. J. Benne, and W. D. McCaffrey "Velocy srucure, urbulence and flud sresses n expermenal gravy currens." Journal of Geophyscal Research: Oceans ( ) 104 (C): Lau, J., and J. L. Nu. 00. "Measuremen and CFD smulaon of he emperaure srafcaon n an arum usng a floor level ar supply mehod." Indoor and Bul Envronmen 1 (4): Launder, B. E., G. Jr Reece, and W. Rod "Progress n he developmen of a Reynoldssress urbulence closure." Journal of Flud Mechancs 68 (0): Launder, B. E "Second-momen closure: presen and fuure?." Inernaonal Journal of Hea and flud flow 10 (4): Launder, B. E., and D. B. Spaldng "Lecures n mahemacal models of urbulence." London: Academc press. Lehmann, E. L., and G. Casella Theory of pon esmaon.vol. 1. Berln: Sprnger. Lu, X., J. Nu, M. Perno, and P. Heselberg "Numercal smulaon of ner-fla ar crossconamnaon under he condon of sngle-sded naural venlaon." Journal of Buldng Performance Smulaon 1 (): Mener, F. R., M. Kunz, and R. Langry. 00. "Ten years of ndusral experence wh he SST urbulence model." Turbulence, Hea and Mass ransfer 4: Mener, F. R "Two-equaon eddy-vscosy urbulence models for engneerng applcaons." AIAA Journal (8): Moghaddas-Nan, H.R., S. W. Armfeld, and J. Rezes "Smulaon of srafed flow around a square cylnder usng he RNG -epslon urbulence model." In: Proceedngs of he 1h Ausralan Flud Mechancs Conference, Melbourne, Ausrala. Orszag, S. A., V. Yaho, W. S. Flannery, F. Boysan, D. Choudhury, J. Maruzews, and B. Pael "Renormalzaon group modelng and urbulence smulaons." Near-wall Turbulen Flows: Pope, S. B Turbulen Flows. Cambrdge: Cambrdge unversy press. Rcou, F. P., and D. B. Spaldng "Measuremens of enranmen by axsymmercal urbulen jes." Journal of Flud Mechancs 11 (01): 1-. Shh, T-H., W. W. Lou, A. Shabbr, Z. Yang, and J. Zhu "A new -epslon eddy vscosy model for hgh Reynolds number urbulen flows: Model developmen and valdaon." NASA STI/Recon Techncal Repor N 95: Smagornsy, J "General crculaon expermens wh he prmve equaons: I. The basc expermen*." Monhly Weaher Revew 91 ():

14 Sørensen, D. N., and C. J. Weschler. 00. "Modelng-gas phase reacons n ndoor envronmens usng compuaonal flud dynamcs." Amospherc Envronmen 6 (1): Spall, R. E "A numercal sudy of ransen mxed convecon n cylndrcal hermal sorage ans." Inernaonal Journal of Hea and Mass Transfer 41 (1): Tomnaga, Y., and T. Sahopoulos "Turbulen Schmd numbers for CFD analyss wh varous ypes of flowfeld." Amospherc Envronmen 41 (7): Venayagamoorhy, S. K., J. R. Koseff, J. H. Ferzger, and L. H. Shh. 00. "Tesng of RANS urbulence models for srafed flows based on DNS daa. " Sanford: Sanford Unversy envronmenal flud mechancs laboraory. Voe, P. R "Subgrd-scale modellng a low mesh Reynolds number." Theorecal and Compuaonal Flud Dynamcs 8 (): Wlcox, D. C "Mulscale model for urbulen flows." AIAA Journal 6 (11): Wlcox, D. C Turbulence modelng for CFD. Vol.. La Canada, CA: DCW ndusres. Xu, D., and J. Chen. 01. "Expermenal sudy of srafed je by smulaneous measuremens of velocy and densy felds." Expermens n Fluds 5 (1): Yaho, V., and S. A. Orszag "Renormalzaon group analyss of urbulence. I. Basc heory." Journal of Scenfc Compung 1 (1):

15 Table 1. Coeffcens of Equaon (4) Equaon or model,eff S Consans Connuy 1 0 p u j Reynolds Momenum u j x x j x averaged varables C C, Speces C : urbulen Schmd number S C C, (1) Sandard b G G C, G S, S S j S j, Gb g T, CG 1 C C1 1.44, C 1.9, C 0.09, 1.0, 1. T x, -equaon () Realzable b G G CS 1 C C, G S, S S j S j, G b g C1 max 0.4, 5, 1 S, C U *, A0 As, C1 1.44, C 1.9, 1.0 U * SjSj jj 1. T T, x,, () RNG b G G C, G S, S S j S j, G b g T T, x, 15

16 CG C R 1 C (1 / 0) R, S /, , C1, 1.4, C, 1.68, , C, 1.0, (4) Sandard (5) SST 7-equaon (6) RSM uu j G Y G Y Hgh Re : 1, 0 Re R, Low Re : 1 Re R Re, R 6, , G S, G G, Y.0,.0 G Y 1 SF max, a1 G Y P j Gj j j f, Y 1,1 1,, Y, f, ,, F1,1(1 F1 ), 1, G S, G G, F (1 F) G mn( G,10 ),,, ,,1.0,, 1.0,, 1.168, a1 0.1 u u j Pj uju u u x x Y, G j g ju g uj, 16

17 17 j j j u u p x x, j j u u x x LES (7) LES (Smagorns y-llly) 1 0 j u j j j p x x 1 j j j j u u x x, s j j L S S

18 Table Parameers a he je nozzle of hgh-r and low-r cases Table Mean squared errors of mean velocy self-smlary values MSE (Mean Squared Error) Sandard RNG Realzable Sandard SST RSM Low-R Hgh-R LES Table 4 Mean squared errors of urbulen nec energy dsrbuons MSE (Mean Squared Error) Sandard RNG Realzable Sandard SST RSM LES Low-R Hgh-R Table 5 Mean squared errors of shear Reynolds sress dsrbuons MSE (Mean Squared Error) Sandard RNG Realzable Sandard SST RSM Low-R.7 10 Hgh-R Case Low-R Hgh-R Je velocy (mean) U ( m/ s ) Turbulen nensy I u/ U0 6.0%.5% Inal densy dfference 0 s 0.5% 0.5% Reynolds number Re 0 s 0 / 4,000,00 Rchardson number R Dg /( U ) s 0-5 LES

19 Fgure 1 Seup of he srafed flow expermen (Xu and Chen 01) Fgure Compuaonal doman and mesh srucure n he presen sudy 19

20 Fgure The je cenerlne velocy predced by RANS from dfferen grds Fgure 4 Fng curve of expresson (10) 0

21 Fgure 5 Velocy conours and sreamlnes from he smulaon resuls (m/s) 1

22 Fgure 6 Mean velocy profle of a ypcal horzonal srafed je

23 Fgure 7 Self-smlary curves of mean velocy

24 Fgure 8 Turbulen nec energy dsrbuons 4

25 Fgure 9 Shear Reynolds sress uu 1 dsrbuons 5

26 Fgure 10 The normalzed densy dsrbuons a upsream Fgure 11 The normalzed densy dsrbuons a downsream 6

27 Fgure 1 Vorcy (normalzed as DU0) conours n he low-r and hgh-r cases 7

28 Fgure 1 The enranmen rao and he percenage of new flud n he low-r case Fgure 14 The enranmen rao and he percenage of new flud n he hgh-r case 8

29 Fgure 15 The compuaon me needed by dfferen CFD models 9