Computation of turbulent buoyant ows in enclosures with low-reynolds-number k-x models

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1 Inernaional Journal of Hea and Fluid Flow 0 (1999) 17±184 Compuaion of urbulen buoyan ows in enclosures wih low-reynolds-number k-x models Shia-Hui Peng a,b, Lars Davidson a, * a Thermo and Fluid Dynamics, Chalmers Universiy of Technology, S Gohenburg, Sweden b Work Organizaion and Technology, Naional Insiue for Working Life, S Solna, Sweden Received 5 November 1997; acceped Ocober 1998 Absrac This work deals wih he compuaion of urbulen buoyan convecion ows wih hermal srai caion using he low-reynoldsnumber (LRN) k-x model. When applying he k-e model o buoyancy-driven caviy ows induced by di erenially heaed side walls, a problem commonly encounered a moderae Rayleigh numbers (Ra ˆ ±10 1 ) is ha he model is no capable of giving gridindependen predicions owing o he ransiion regime along he verical walls. I was found ha he buoyancy source erm for he urbulence kineic energy, G k, exhibis srong grid sensiiviy, as his erm is modelled wih he Sandard Gradien Di usion Hypohesis (SGDH). By inroducing a damping funcion ino his erm, he above grid-dependence problem is eliminaed and, addiionally, he modelled G k renders correc asympoic behavior near he verical wall. The mechanism held in he k-x model for describing he onse of ransiion is analyzed. The presen approach is simple for pracical use and gives reasonable predicions. Ó 1999 Elsevier Science Inc. All righs reserved. Keywords: Turbulen naural convecion; Caviy ow; Transiion regime; Grid sensiiviy; LRN model; k-x model Noaion A aspec raio for caviy, A ˆ H/W c l, c k, c rx model consans c 1, c, c 3 model consans c f wall fricion coe cien, c f ˆ (H/u T )(ov/ox) w E heigh of oule f l, f k, f 1, f damping funcions of urbulence models h in heigh of inle H heigh of compuaional domain k urbulen kineic energy N ks shear producion of k normalized by he dissipaion erm N kb buoyancy producion of k normalized by he dissipaion erm Nu local Nussel number, (ot/ox) w H/DT p pressure Pr Prandl number, m/a Ra Rayleigh number, Pr[gb(T h T c )H 3 /m ] R urbulen Reynolds number T emperaure T h, T c emperaures a heaed and cooled walls, respecively u T velociy scale for caviy ow, [gb(t h T c )H] 1= u i mean velociy componens in he x i direcion u 0, v 0 ucuaing velociies in he x and y direcions, respecively * Corresponding auhor. lada@fd.chalmers.se. W widh of compuaional domain x i Caresian space coordinaes (i ˆ 1, ) Greek a hermal di usiviy b hermal expansion coe cien DT emperaure di erence, DT ˆ (T h T c ) e dissipaion rae of k e in dissipaion rae of k a inle l dynamic molecular viscosiy l urbulen dynamic viscosiy m kinemaic viscosiy, l/q m urbulen kinemaic viscosiy, l /q q densiy of uid r k, r z, r T model consans s urbulen ime scale x speci c dissipaion rae of k f similariy variable 1. Inroducion Non-isohermal urbulen convecion ows are encounered in many engineering applicaions, including building venilaion, solar energy collecors, cooling of nuclear reacors and maerial processing. This ype of ow is eiher buoyancy-affeced or purely driven by buoyancy due o hermal srai caion. The well-known buoyancy-driven ow is he naural convecion in an enclosure wih wo di erenially heaed verical walls, i.e. he so-called ``caviy ows'' X/99/$ ± see fron maer Ó 1999 Elsevier Science Inc. All righs reserved. PII: S X ( 9 8 )

2 S.-H. Peng, L. Davidson / In. J. Hea and Fluid Flow 0 (1999) 17± Naural convecion ows in enclosures usually possess wo disinc paerns: he boundary layers along he walls and he encircled recirculaing moion in he core. In cases wih heaed and cooled verical walls, he laminar air ow a a local Rayleigh number larger han 10 9 may be promoed o urbulen ransiion in he verical boundary layer (Incropera and DeWi, 1990). The ransiion from laminar o urbulence causes an increase in he convecive hea ransfer on wall surfaces. In he direc numerical simulaions (DNS) for air ow in a square caviy, Paolucci and Chenoweh (1989) deeced ha, a he criical Rayleigh number beween 10 8 and 10 8, he ow undergoes a Hopf bifurcaion o a periodic unseady ow. A slighly larger Rayleigh number will make he ow display Tollmien±Schliching-like waves in he verical boundary layer. A furher increase of he Rayleigh number will lead o he high dimensional urbulen sae. Henkes and Le QueÂre (1996) revealed numerically ha hree-dimensional perurbaions are less sable han wo-dimensional perurbaions for air- lled caviy ows and give a lower criical Rayleigh number for ransiion onse. They also showed ha he hree-dimensional insabiliy has a combined hermal and hydrodynamic naure. The ransiional ow is essenially of a low-reynolds/rayleigh-number ype, which is unseady and no reproducible. I is of pracical imporance o reasonably predic he onse of ransiion, for example, in applicaions of crysal growh and cooling of nuclear reacors. I is well-known, however, ha he physical ransiion phenomenon iself is no racable wih a Reynolds-averaging model, since all he specra e ecs are los in he ime-averaging process, and he wo-equaion models are capable of disinguishing only he magniude and an average frequency of perurbaions ha fall in a speci c range of frequencies of inducing insabiliy. Neverheless, he LRN varians of he wo-equaion models have been widely applied in recen years for simulaing ransiional ows, paricularly for predicing he onse of by-pass ransiion. A moderae degree of success in such applicaions has been repored (Savill, 1996). An imporan feaure of urbulen naural convecion ows is ha urbulence relies signi canly on he hermal srai caion. A sable srai caion usually dampens urbulence. For naural convecion ows in a caviy a moderae Rayleigh numbers, urbulence is fully developed only in some regions along he verical walls (he upper par along he heaed wall and he lower par along he cooled wall), and i decays away from he walls. This hus requires he urbulence models used in he compuaion o be able o appropriaely capure he urbulence evoluion saring from laminar sae so as o obain reliable hea ransfer predicions. The srong ineracion beween he hermal and hydrodynamic insabiliies requires a paricular e or o describe he non-linear growh of disurbances ha lead he ow from laminar o urbulence. Over he pas en years, exensive sudies have been made on he buoyancy-driven ows in enclosures hrough experimens and numerical echniques, because his ype of ow is of ineres in boh fundamenal urbulence research and engineering applicaions. Experimenal work, by e.g. Cheesewrigh e al. (1986) and Giel and Schmid (1986), has ofen been used for he validaion of numerical compuaions, which range from DNS, LES and second-momen closures o wo-equaion eddy viscosiy models, see e.g. Paolucci and Chenoweh (1989), Bergsrom and Huang (1997), Davidson (1990a, b) and Henkes e al. (1991) and Heindel e al. (1994). Among he woequaion eddy viscosiy models, he k-e model and is LRN varians have been commonly used. Ince and Launder (1989) inroduced he Generalized Gradien Di usion Hypohesis (GGDH) o model he hea ux vecor. Togeher wih he Yap correcion in he e-equaion, his approach is able o give saisfacory resuls. Hanjalic and Vasic (1993) proposed an algebraic hea ux model (AFM), which showed a reasonable behavior. A brief overview of previous work on he compuaions of urbulen naural convecion in wo-dimensional enclosures can be found in, e.g., Hanjalic and Vasic (1993). In general, he inclusion of low-reynolds-number modi caions in he k-e model improves he predicion of wall hea ransfer, as saed by Heindel e al. (1994). However, some problems have also been repored in a comparaive sudy using he resuls from a workshop conduced by Henkes and Hoogendoorn (1995). Among ohers, he soluion was found o be grid-dependen when using he k-e model wih he isoropic eddy di usiviy concep (i.e. SGDH): he ransiion onse along he non-adiabaic verical wall is delayed as a resul of re ning grid, and evenually reurning an unrealisic laminar soluion. Wilcox (1994) developed an LRN k-x model for ransiion simulaion. Unlike he approach usually used for by-pass ransiion by di usion of free-sream urbulence ino he ow, a mehod called numerical roughness srip was inroduced o rigger ransiion a a speci ed locaion. This mehod is essenially a poin ransiion approach in which he locaion of ransiion onse needs o be empirically esablished. This model was applied o a series of ransiional boundary layer ows, and realisic simulaions were repored. To improve he predicions for inernal low-reynolds-number recirculaing ows, a modi ed form of his model was proposed by Peng e al. (1997). Boh Wilcox's model and he modi ed model preserve he mechanism for ransiion simulaion of isohermal boundary layer ows as described by Wilcox (1994). For urbulen naural convecion ows, however, he behavior of hese wo models remains unclear. I herefore provides moivaion o invesigae he performance of hese models for ows in which he urbulence is promoed from laminar sae by boh shear and hermal srai caion. In his work, he wo aforemenioned LRN k-x models are used for predicing hermally srai ed urbulen ows in enclosures. The in uence of hermal srai caion on he urbulence evoluion is invesigaed. Emphasis is pu on he predicion of ransiion onse in he boundary layer along he verical heaed/cooled wall. The behavior of he LRN k-x model is analyzed for predicing ransiion. By using a damping funcion for he buoyan source erm in he k-equaion, ogeher wih he simple SGDH approach, he grid-dependence problem for ransiion predicion is removed, and he grid-independen soluion can be asympoically achieved wih successively re ned grids. The presen approach has been applied o a buoyancy-driven naural convecion ow a Ra ˆ in a recangular caviy and o a mixed convecion ow in a square enclosure wih unsable hermal srai caion. For comparison, wo LRN k-e models by Abe e al. (1994) and by Lam and Bremhors (1981) are also used. The resuls have been compared wih experimenal daa.. Turbulence models.1. Model equaions To describe he evoluion of he urbulence eld and de ne he urbulen scales, an equaion for urbulence energy is ofen used when using Reynolds-averaging wo-equaion models. For seady ows, he ranspor equaion for urbulen kineic energy, k, is wrien as o qu j k ˆ P k P k o l l ok r k S k ; 1

3 174 S.-H. Peng, L. Davidson / In. J. Hea and Fluid Flow 0 (1999) 17±184 where he righ-hand side erms are he producion, he dissipaion, he di usion and an addiional source erm, respecively. In he conex of wo-equaion urbulence modelling, an exra equaion is required o consruc he eddy viscosiy, l. In analogy o he k-equaion, his equaion is cas ino a general form as o qu j z ˆ P z P z o l l r z oz S z : The righ-hand side of Eq. () is ermed in a way analogous o hose in Eq. (1). In his sudy, wo ypes of LRN models were used, i.e. he k-e model and he k-x model. Eq. () hen urns ou o be a ranspor equaion for e or x. The eddy viscosiy in he k-e model is de ned as qk l ˆ c l f l e : 3 In he k-x model, i is expressed as qk l ˆ c l f l x : 4 The producion erm in he k-equaion, P k, is modelled by assuming ha he urbulen Reynolds sresses are in alignmen wih he srain rae ensor, S ij ˆ (ou i /ou j +ou j /ou i )/. This gives ou i ou i P k ˆ l S ij ˆ l ox ou j oui : 5 j ox i The wo LRN k-e models used in his work are he Abe± Kondoh±Nagano (AKN) model (1994) and he Lam±Bremhors (LB) model (1981), and he LRN k-x models are he Wilcox (WX) model (1994) and a model by Peng, Davidson and Holmberg (PDH) (1997). The erms appearing in Eqs. (1) and () are given in Table 1. The source erm in he k-equaion, S k, is due o he buoyancy, which represens he exchange beween poenial energy and urbulen kineic energy. This erm is associaed wih he urbulen hea ux in he verical direcion, reading G k ˆ qgbu 0 jt 0 d j : 6 Several sophisicaed opions can be used o model his erm, e.g. he GGDH model by Ince and Launder (1989) and he AFM approach by Hanjalic and Vasic (1993). For is simpliciy, he SGDH model remains mos widely used in engineering applicaions, giving G k ˆ gb l r T ot d j : 7a The GGDH model represens he verical hea ux by including a horizonal emperaure gradien in he presence of shear. I reads G k ˆ c h gb 1 x l S j ot 3 d jqk : 7b Depending on he hermal srai caion, he buoyan source erm in he k-equaion eiher enhances urbulence (as a producion erm) or dampens urbulence (as a desrucion erm). For convenience, his erm is ermed here buoyancy producion in cases of boh sable and unsable hermal srai caion, unless oherwise saed. An exra erm in he PDH model is included in he x- equaion. This erm is he urbulen cross-di usion erm (Peng e al., 1997) and is expressed as l C x ˆ c ok ox rx : 8 k The buoyancy source erm in he e- or x-equaion can readily be derived from dimensional analyses, as shown in Table 1. Noe ha he speci c dissipaion rae, x, is a reciprocal of he urbulen ime scale, i.e. x 1/s. Eqs. (3) and (4) indicae ha x e/k. This suggess ha Dx/D ˆ (1/k)De/D (x/k)dk/ D. Using his relaion, he buoyancy erm in he x-equaion can be shown o have he form of c e3 1 x=k G k. In accordance wih Rodi's argumens (Rodi, 1984), c e3 is close o uniy in he verical boundary layer and close o 0 in he horizonal boundary layer. Henkes e al. (1991) proposed using c e3 ˆ anh jv=uj o saisfy his argumen for compuing urbulen caviy ows. Oher consan values, ranging from 0.8 o 1.5, have been used for c e3 by di eren researchers, see e.g. Hanjalic and Vasic (1993) and Murakami e al. (1996). Markaos e al. (198, 1984) saed ha here is evidence ha he buoyancy e ec should be re eced only in he k-equaion. Heindel e al. (1994) also deeced a negligible e ec of consan c e3 in he calculaions of caviy ows. Based on he relaion of c x3 c e3 1, he e ec of c x3 was invesigaed in his work by employing values from 0.3 o 1.0 for caviy ows. Indeed, he predicion appears o be insensiive o his consan. As a consequence, in he presen calculaions, c x3 is se o zero for boh he WX and PDH models. The closure consans are summarized in Table. The damping funcions, f l, f k, f 1 and f, for di eren models are given in Appendix A... Transiion Regime A a moderaely high Rayleigh number, he naural convecion boundary layer ow along a non-adiabaic verical wall of an enclosure, e.g. he heaed wall, usually undergoes hree sages: he laminar ow near he lower lef corner, and he subsequen ransiional and urbulen ows. The ransiion onse can be observed hrough he convecive hea ransfer over he wall surface, in which a sudden jump occurs. The LRN k-e model, as well as he sandard k-e model, usually predics a delayed ransiion regime when re ning he grid. Using he sandard k-e model, by riggering ransiion in he boundary layer wih a prescribed amoun of kineic energy, Henkes and Hoogendoorn (1995) found ha his grid-dependence problem can be removed. This approach, however, overpredics he average wall hea ransfer. Moreover, inroducing prescribed urbulence ino a con ned sysem migh be unrealisic. Using he LRN k-x models in his sudy, delayed ransiion predicions were also deeced by successively re ning he grid from o A possible ex- Table 1 Terms in urbulence models used in his work Models LB model AKN model WX model PDH model P k qe qe c k f k q x k c k f k q x k S k G k G k G k G k P z c 1 f 1 (e/k) P k c 1 f 1 (e/k) P k c 1 f 1 (x/k) P k c 1 f 1 (x/k) P k P z c f q e /k c f q e /k c q x c q x S z c 3 (e/k) G k c 3 (e/k) G k c 3 (x/k) G k c 3 (x/k) G k + C x

4 S.-H. Peng, L. Davidson / In. J. Hea and Fluid Flow 0 (1999) 17± Table Model consans Consans c l c k c 1 c c 3 r k r z r T c rx LB model 0.09 ± anhjv/uj ± AKN model 0.09 ± anhjv/uj ± WX model ± PDH model planaion on his uncerainy may be aribued o he mechanism for predicing ransiion onse preserved in he k-x model, which is originally subjeced o forced convecion boundary layer ows. Comparing o isohermal ows dominaed by shear or pressure gradien, buoyancy-driven ows possess one furher urbulence evoluion mechanism due o hermal srai caion as represened by he buoyancy-relaed erm, G k, in he k- equaion. The unknown correlaion in G k re ecs he ineracion beween he ucuaing velociy and emperaure elds. If G k is modelled, say, wih he SGDH or he GGDH approach, his erm can be wrien in a general form G k ˆ gbv 0 T 0 ˆ m F x; ou i ; ot ; 9 where F is a funcion of x and he gradiens of velociies and emperaure. For an incompressible, wo-dimensional naural convecion boundary layer ow along a verical heaed wall, in analogy o Wilcox's analysis (Wilcox, 1994), he ne producion per uni dissipaion erm for k and x, N k and N x, can be wrien as, respecively, N k ˆ clf l c k f k # ov=ox F x x 1; 10 N x ˆ clf # l ov=ox F c 1 f 1 c 3 1: 11 c x x Since he x-equaion has a well-behaved soluion as k ˆ 0, he ampli caion or reducion of k and x in magniude can hus be deermined wih he sign changes of N k and N x. To ensure ransiion o occur from laminar o urbulence, k mus be ampli ed earlier han x (ransiion will oherwise never occur). I is hus necessary o have N k > N x P 0. Consequenly, his requires, as R! 0. 1 c 1f 1 ov=ox c k f k c x 1 c k f k c 3 c F x > 0: 1 For isohermal boundary layer ows, F ˆ 0, Eq. (1) is hus sais ed by simply aking c 1 f 1 c k f k < c as R! 0: 13 Eq. (13) holds rue in boh he WX model and he PDH model. For naural convecion boundary layer ows, his condiion is no longer su cien because he urbulen hea ux conribues o he ne producions. As argued above, he model consan c 3 has been se o zero. To rigger ransiion, he ampli caion of k depends on boh he normalized producion due o shear, N ks ˆ C Nk [(ov/ox)/x] wih C Nk ˆ c l f l /c k f k, and he normalized producion due o buoyancy, N kb ˆ C Nk F/ x. For isohermal boundary layer ows (F ˆ 0), Wilcox (1994) showed ha N k and N x increase linearly wih local Reynolds number along he surface. Through an argumen of model consans, k is forced o grow saring from laminar ow a he minimum criical Reynolds number (Re cr ) a which he Tollmien±Schliching waves begin o form in he Blasius boundary layer. For naural convecion boundary layer ows along he heaed/cooled verical walls in an enclosure, DNS resuls by Paolucci and Chenoweh (1989) and by Le QueÂre (199) reveal ha Tollmien±Schliching-like waves occur a a Rayleigh number larger han This seems o sugges ha he free convecion boundary layer ow possesses behavior similar o he Blasius boundary layer, of causing ransiion o urbulence. However, he mechanism held in he k-x model for ransiion simulaion is quaniaively quesionable when F 6ˆ 0, as shown in Eqs. (10) and (11). Two uncerain poins arise. Firs, he addiion of he buoyancy erm in he k- equaion could no ensure ha k is ampli ed before x as F < 0 for sable hermal srai caion where ot/oy > 0. Second, because he buoyancy erm inerferes wih he ne producion of urbulence energy, N k, i canno guaranee ha k sars o grow a he criical Ra number where he secondary insabiliy occurs, e.g. Ra cr 10 8 for D caviy ows as revealed in DNS. This is essenial for accuraely predicing he locaion of ransiion onse. The k-x model has an imporan feaure o enable an analyical observaion on he ow change-over from one sae o anoher as ransiion occurs. Tha is, he x-equaion uncouples from he k-equaion as k ˆ 0, giving a nonrivial laminar soluion for x. To furher reveal how he ransiion-onse occurs in response o he model behavior a a cerain Rayleigh number, a more comprehensive analysis can be made by a similariy ransformaion of Eqs. (10) and (11). I should be noed ha he analysis is sared from laminar sae and ended up wih he change-over of he ow sae where he local producion of k su cienly overwhelms is local dissipaion and ransiion occurs. According o Henkes (1990), he quaniies H, DT ˆ (T h T ref ) and u T ˆ (gbdth) 1= are appropriae scalings for y, T and v, respecively. Here, T ref is a reference emperaure (e.g. T ref ˆ T c for caviy ow). Furher, a similariy variable, f, is used for he following ransformaion of Eqs. (10) and (11) (Whie, 1974; Cebeci and Khaab, 1975) and f ˆ u 1=4 T x: 14 4m Hy Noe ha x is normal o he verical wall, and y is he sreamwise direcion along he wall. Wih his similariy variable and he above scalings, v and T are hen wrien in erms of f, i.e. v ˆ u T V f ; T ˆ DT H f : By means of he asympoic soluion of x, x / 1=x as x! 0, x can be expressed as

5 176 S.-H. Peng, L. Davidson / In. J. Hea and Fluid Flow 0 (1999) 17±184 1= x ˆ u T X f : 17 4Hy The SGDH approach in Eq. (7a), which gives F ˆ =@y, is used o model he buoyancy erm G k. Seing c 3 ˆ 0 and using Eqs. (14)±(17) in Eqs. (10) and (11) yield, respecively, N k ˆ clf l Ra 1= y 1= ov =of c k f k Pr H X f # oh=of 1; 18 r T X N x ˆ clf l c 1 f 1 Ra 1= y # 1= ov =of 1: 19 c Pr H X A dramaic ampli caion in k (sars as N k changes is sign o posiive) indicaes he onse of ransiion, whereafer he ampli caion of x conrols he widh of ransiion. Noe ha he urbulence energy sars o grow as he rs erm in Eq. (18) reaches uniy. Somewhere afer his poin, he eddy viscosiy suddenly increases and he ransiion from laminar o urbulence occurs. A he onse posiion of ransiion, his suggess 1= Ra y 1= k 1 k ˆ a 0 a 0 > 1 ; 0 Pr H r where k 1 ˆ c lf l ov =of ; k ˆ clf l c k f k X c k f k r T foh=of X : 1 In Eqs. (0) and (1), k 1 and k are funcions of he similariy variable, f. As he ransiion occurs, he posiion (heigh) of ransiion onse, y r, can hus be esimaed by solving Eq. (0) for (y/h) r. This gives y r ˆ Pr a0 k H: Ra k 1 Noe ha a 0 ˆ (N ks + N kb ) max is he maximum oal producion per uni dissipaion as ransiion occurs. Eq. () shows ha he heigh of laminar-urbulen ransiion along a verical heaed wall decreases wih an increasing Rayleigh number. For buoyan ows in a square caviy, Henkes (1990) used Chien's LRN k-e model (Chien, 198) o numerically reveal he dependence of y r on Ra. I was shown ha y r ended o be zero as he Rayleigh number was up o Eq. () suggess ha an appropriae predicion of y r depends on he model behavior of simulaing k 1, k and a 0 when he ow approaches ransiional sae from laminar. Usually, k 1 reaches is maximum value somewhere near he velociy peak in he boundary layer, where jk j is ofen raher small due o near-wall large x, having jk j 1 < a 0. To ensure he occurrence of ransiion, wo essenial condiions should be emphasized. Firs, he ne producion N x should no reach zero earlier han N k. Second, he value of a 0 should be raised o a cerain level in he boundary layer. Wih unsable hermal srai caion, he buoyan source erm in he k-equaion usually conribues o he enhancemen of he level of a 0. For sable hermal srai caion, by conras, his erm is ofen negaive as modelled wih he SGDH. To raise a 0 o be a posiive value and much larger han uniy, i hus relies solely on he shear producion, whose maximum value ofen arises in he near-wall boundary layer whereas he buoyancy producion is usually negligibly small (Davidson, 199). This, however, by no means suggess a negligible role played by he buoyancy-relaed erm. In he neighboring ouer region of he boundary layer his erm can be relaively high wih negaive values, behaving as a energy desrucion erm. In conras o he by-pass ransiion where he freesream urbulence is diffused ino he boundary layer, in his case he urbulence energy generaed by shear in he boundary layer may be diverged o he ouer region in which he energy desrucion is reinforced by he buoyan source erm. Such a modelling feaure may deer a 0 from reaching a desired level o rigger ransiion. The model will consequenly predic a delayed ransiion or even an unrealisic laminar soluion. Special care mus hen be aken of he buoyan source erm in he ouer region neighboring o he boundary layer. Indeed, here is evidence ha he aforemenioned grid-dependence of he ransiion predicion is relaed o he buoyancy-generaed source. This can be veri ed by dropping he buoyancy erm in he k-equaion. I was found ha his exclusion can rid he model of he grid-dependen behavior, bu he hea ransfer is largely over-prediced as compared wih experimens and he urbulence energy in he core region is no su cienly dampened as desired. This implies ha a simple exclusion of G k is no accepable. Since he buoyan source erm ends o suppress urbulence in sable hermal srai caion, an inclusion of his erm is hus preferable. On he oher hand, G k mus hen be appropriaely modelled, because he ampli caion of he local k producion in he near-wall boundary layer is associaed wih his erm. If he urbulence in he neighboring ouer region is over-suppressed (hrough G k wih negaive sign), he ampli caion of he near-wall shear producion may be slowed down by di usion of urbulence energy owards he ouer region so as o compensae for he sabilizaion here. This will consequenly delay he ransiion onse, or even canno susain he urbulence evoluion in he boundary layer, and reaining he ow in laminar sae. To remove his undesired behavior induced due o hermal srai caion, measures mus be aken o conrol he performance of he buoyan source erm, G k. I should be noed ha more sophisicaed models, e.g., he GGDH model (Ince and Launder, 1989) and he AFM approach (Hanjalic and Vasic, 1993), can be used o approximae his erm. These wo models were applied in he original work o he same caviy ow as considered in his work. There was no grid-dependence problem repored as such ideni ed here when using LRN woequaion models ogeher wih he SGDH approach. The raionale now seems clear: models ha approximae G k in such a way ha he normalized producion (and hus a 0 ) in he boundary layer keeps growing can remove he above problem and lead he ow from laminar o urbulence. Indeed, Davidson (199) showed ha G k is even posiive in he ouer region when using he GGDH model while i is negligibly small in he near-wall region comparing o oher erms. Using he AFM, Hanjalic and Vasic (1993) saed ha G k is usually less han one-hird of he shear producion in he ouer layer. This implies ha he shear producion iself in he ouer layer has he capabiliy o compensae for he energy desrucion wihou exra urbulence di usion from he inner region. We believe ha such modelling feaures inheren in he GGDH and AFM approaches may, o some exen, bring abou he success in avoiding he grid-dependence problem as in he presen consideraion. Alhough hese wo approaches are no included in he following compuaions, i can be shown ha he predicions given by he presen simple approach are acually very similar o he resuls produced by he GGDH and AFM mehods. Since he grid-dependence problem is encounered when using he SGDH, which has so far gained he greaes populariy in pracical applicaions for is simpliciy, he SGDH model is hus he main consideraion o overcome he problem inheren in his approach. For his purpose, a pracical way is o use a damping funcion ha is able o appropriaely deer

6 S.-H. Peng, L. Davidson / In. J. Hea and Fluid Flow 0 (1999) 17± he buoyan source erm in he ouer region from absorbing oo much energy powered by he near-wall shear producion. Furhermore, he mehod of using a damping funcion for G k is suppored by an asympoic analysis near he verical wall: using he SGDH approach, Eq. (7a), gives rise o an incorrec asympoic behavior for he modelled G k. Near he verical wall, v 0 / x, T 0 / x (cf. So and Sommer, 1994). The exac buoyancy source erm is hus proporional o x as x! 0. Since l / x 3 for he PDH model, and l / x 4 for he WX model, wih ot =oy / x 0 as x! 0, he modelled G k erm hen has a relaion of G k / x 3 for he former and of G k / x 4 for he laer. Noe ha he GGDH approach also renders incorrec asympoic behavior near he verical wall. The devised damping funcion, f G, is employed in he PDH model, having f G / 1=x as x 0. Noe ha he use of a damping funcion will give an incorrec asympoic behavior for G k near a horizonal wall (for models wih l / y 3 ), where v 0 / y as y 0, bu his is of no signi can consequence since he ow along he verical walls is dominan in an overall fashion for ows induced by heaed/cooled verical walls. The role played by he damping funcion is mainly o consrain he urbulence desrucion due o buoyancy and hus o suppress he urbulence energy di usion from he boundary layer o he neighboring ouer region so as o avoid delayed predicion of ransiion onse as analyzed above, and addiionally, o obain correc asympoic behavior for G k near he verical wall. This funcion was devised as ( #) f G ˆ 1 exp R 3 c g 1 10 ; 3 R 3:5 where c g ˆ 1, which is numerically opimized. By using his damping funcion ogeher wih he PDH model (hereafer PDH+D), as shown in he nex secion, he grid-dependence problem can be eliminaed. The buoyancy source erm in he k- equaion is hus modelled as G k ˆ gbf G l r T ot d j : 4 The buoyancy producion (normalized by he dissipaion erm), N kb, is ofen raher small in he boundary layer. The ransiion onse from laminar o urbulence herefore relies srongly on he shear producion (hrough a 0 and k 1 ). I should be poined ou ha he above analysis can also be expeced o be applicable for he WX model and for LRN k-e models by ransforming he e-equaion ino an x-equaion wih he relaion of e kx (see Peng, 1998). The proposed approach is applied, however, only o he PDH model in his sudy for veri caion. 3. Soluion procedure The above LRN models have been applied o wo nonisohermal convecion ows: he urbulen buoyancy-driven ow in a recangular caviy wih A ˆ 5, for which he experimen was made by Cheesewrigh e al. (1986); and he urbulen mixed convecion ow in a square enclosure wih a heaed boom wall, as in he Blay e al. experimen (Blay e al., 199). All he compuaions have been performed on he basis of he Boussinesq assumpion. The air properies are evaluaed a a reference emperaure: he densiy is calculaed from he gas law, and he dynamic viscosiy is calculaed by using he Suherland formula. The nie volume mehod is used o discreize he parial di erenial equaions on a collocaed grid (Davidson and Farhanieh, 199). The QUICK scheme is employed for he convecion erms in he momenum equaions and in he hermal energy equaion, and he convecion erm in he urbulence ranspor equaions is discreized by using he hybrid upwind-cenral scheme. In order o verify ha he low numerical accuracy in he urbulence-convecion discreizaion has no conaminaed he predicion of ransiion, a secondorder accurae bounded scheme by van Leer (van Leer, 1974) was also employed o discreize he urbulence-convecion erm. The resuling predicions, however, were found o be nearly idenical o hose obained wih he hybrid scheme, and he model evenually renders a laminar soluion wih re ning grid for he caviy ow when no using he damping funcion in G k. This seems o imply ha he grid-dependence problem in he presen ransiion predicion is no rooed in he numerical accuracy of he urbulence-convecion scheme. To accoun for he velociy-pressure coupling, he SIMPLEC algorihm is applied. The resuling algebraic equaions for velociy componens and urbulence quaniies are solved ieraively wih a line-by-line TDMA soluion procedure, and he Srongly Implici Procedure (SIP) is used o solve he pressure-correcion equaion. Seady sae soluion is obained by using under-relaxaion echnique and a false ime sep o ensure a sable soluion procedure. The use of LRN models requires a su cien number of grid poins in he near-wall boundary layer. Special aenion has hus been paid o he grid re nemen, paricularly for compuing he buoyancy-driven caviy ow. Non-uniform grid has been used in all he calculaions, wih he grid clusered in he near-wall region o resolve he wall-damping e ec. The low-reynolds-number modi caions allow he urbulence model o be inegraed direcly owards he wall surface, wihou using wall funcions as a bridge. On he wall surface, he boundary values for u, v and k were se o zero, while he emperaure, as well as he inle condiions in he mixed convecion case, was speci ed in accordance wih experimens. Neumann boundary condiions were used for all he variables a he oule. Wih boh he LB and he AKN k-e models, he wall condiion for e was speci ed as! p o k e w ˆ m : 5 on w This relaion was derived from a balance of he k-equaion in he viscous sublayer by using a Taylor series expansion (To and Humphrey, 1986). By means of a similar balance for he x-equaion, an asympoic expression for x as he wall is approached can be obained (Wilcox, 1994) x ˆ 6m as y! 0: 6 c y Eq. (6) is employed for calculaing x a he rs grid poin close o he wall surface. 4. Resuls and discussion The resuls compued wih various urbulence models are presened in his secion, and are discussed hrough comparison wih experimenal daa Naural convecion in a recangular caviy Various caviy con guraions have been used in previous experimenal and numerical work. Henkes and Hoogendoorn (1995) proposed in a Euroherm Workshop he use of a square enclosure as he benchmark es case. In heir comparison sudy using he workshop resuls, hey showed ha he resuls for caviies wih aspec raios of 1 and 5 (a nearly he same Rayleigh number) are very close only if hey are scaled by he

7 178 S.-H. Peng, L. Davidson / In. J. Hea and Fluid Flow 0 (1999) 17±184 caviy heigh. Hanjalic and Vasic (1993) found ha he predicion is more erroneous for he ow in a all caviy wih an LRN k-e model han for he ow in a square or lower-aspec raio caviy wih he same LRN model. For a direc comparison wih experimenal daa, he ow in a recangular enclosure wih an aspec raio of A ˆ 5 was compued. The Rayleigh number is abou wih T h ˆ 77. and T c ˆ The horizonal walls are assumed o be adiabaic. When using he LRN k-e model, he compuaion needs o be sared wih some iniial urbulence. I was found ha he LB model and he AKN model may give a laminar soluion wih boh and grids unless he iniial values of k and e are speci ed in such a way ha he iniial eddy viscosiy is abou 100 imes higher han he molecular one. This was also observed by Hanjalic and Vasic in heir previous compuaions (Hanjalic, Privae communicaion, 1997). However, such an iniial speci caion does no lead o a urbulen soluion when using a ner grid, e.g , a which he model always evenually reurns a laminar soluion even hough a raher high iniial value was given o k. Wih a grid of 10 10, he AKN model was observed o rs give urbulen soluion, bu his soluion did no susain as furher ieraions were performed. This phenomenon may be aribued o he nonuniqueness of he ransiion, as explained by Henkes e al. (1991), he model has a bifurcaion poin a Ra cr. Wih he LRN k-x models, a very small iniial value, e.g , was speci ed for boh k and x. This iniial speci caion is able o give a urbulen soluion wih a grid re ned o, e.g., Noe ha his in fac implicily ses a raher high iniial urbulen Reynolds number, R ˆ k/mx. I was found ha he WX model and he PDH model also rendered a laminar soluion if a small iniial urbulen Reynolds number was se by specifying eiher a smaller iniial value of k or a larger x. Using boh he WX and PDH models, we also ried o sar he compuaion wih a laminar soluion for u, v and x wih a very small iniial k value (e.g ), bu he soluion remained in laminar sage. This laminar soluion was kep even when a numerical roughness srip, as proposed by Wilcox (1994) o rigger ransiion in forced convecion boundary layer ows, was speci ed around he ransiion regime. This seems o indicae ha he numerical-roughness-srip approach does no work for he presen case. I is desired o aain an asympoically grid-independen soluion by means of re ning grid. The grid used in his sudy was successively re ned from 60 60, 80 80, , 10 10, o All he models urn ou o be srongly sensiive o grid re nemen. Fig. 1 shows ha he prediced onse of ransiion along he verical wall is delayed wih re ned grids. As wih he LRN k-e model, he wo LRN k-x models also give grid-dependen predicions, see Fig. 1(c) and (d), alhough he delay is relaively slow. By re ning he grid furher, i can be expeced ha hese wo models will nally reurn a laminar soluion, as he LRN k-e model does. The shif of ransiion wih re ning grids suggess a delayed ampli caion of local urbulence producion, hough his ampli caion has occurred before dissipaion is ampli ed (on a coarser grid). Re ning he grid seems o increase he ampli caion rae for he dissipaion and evenually, he ampli caion of local urbulence producion may be overwhelmed by he local dissipaion. Moreover, i was found ha all he models used here give reasonable k-disribuions (no shown here) as compared wih he experimenal daa provided ha hey predic he ransiion onse a y r /H ˆ 0.3 ± 0.4 (along he ho wall) wih a cerain grid. However, his posiion does no agree wih he experimenal resul. Bowles and Cheesewrigh's experimen (Bowles and Cheesewrigh, 1989) gave y r /H 0.. An earlier ransiion (y r /H < 0.3) prediced by a model corresponds in general o a lower predicion of he k-peak downsream of he ransiion regime in he boundary layer. The conradicion beween he predicion Fig. 1. Grid-dependence of he soluion: Nussel number, Nu, along he verical wall. (± ± ±) Grid 60 60, (á á á á á á á á á) Grid 80 80, (±±±) Grid 10 10, (- - -) Grid , (D) Experimen (ho wall), (}) Experimen (cold wall). (a) AKN model, (b) LB model, (c) WX model, (d) PDH model.

8 S.-H. Peng, L. Davidson / In. J. Hea and Fluid Flow 0 (1999) 17± and he experimen may possibly be due o he hea loss hrough he horizonal walls in he experimen, as saed by Cheesewrigh e al. (1986). The hea loss may have a eced he measured posiion of ransiion onse along he verical wall. When using he PDH + D model, he compued convecive hea ransfer along he heaed verical wall is given in Fig. (a), and he k-disribuion a he mid-secion (y ˆ H/) is shown in Fig. (b). By re ning he grid successively from o , an asympoically grid-independen soluion is achieved. This indicaes ha he use of a damping funcion, f G, in G k does preven he model from being grid-dependen in ransiion predicion. Again, he lower k-peak is relaed o he prediced locaion of ransiion regime, which is however in reasonable agreemen wih he predicion using he AFM approach by Hanjalic and Vasic (1993). When using he PDH+D model, all he resuls shown below were obained wih a grid of , where he soluion can be regarded as being gridindependen, as disclosed in Fig.. Fig.. Predicions using he PDH + D model wih re ned grids. (á á á á á á á á á) Grid 80 80, (- - -) Grid 10 10, (±±±) Grid , (± ± ±) Grid (a) Nussel number, Nu, along he verical wall. (b) Turbulence kineic energy, k, a secion y ˆ H/. Fig. 3(a)±(c) gives he disribuions, compued wih he PDH+D model, for he mean velociy, emperaure and urbulen shear sress a y ˆ H/. Also shown in Fig. 3(d) is he fricion coe cien along he verical walls. As seen, he resuls are in saisfacory agreemen wih he experimenal daa. The in uence of he consan c g in he damping funcion f G was also invesigaed. I was found ha he predicion of ransiion onse is raher sensiive o he value of f G. A smaller c g, i.e. a larger f G, makes he model predic a laer ransiion. This observaion con rms he above analysis ha he buoyan source erm, as modelled wih he SGDH, ends o slow down he urbulence evoluion in he boundary layer and delay he ransiion onse. To invesigae he di eren roles played by he shear producion and he buoyancy producion in he k-equaion, i is desirable o compare heir respecive conribuion o he urbulence evoluion in he boundary layer as ransiion occurs. The budge for he k-equaion in he PDH + D model is given in Fig. 4 a locaion y ˆ 0.3H, which corresponds o he prediced onse posiion of he ransiion regime along he heaed wall. The k-budge a y ˆ H/ was also invesigaed (no shown here), indicaing ha he balance of he k-equaion in he boundary layer downsream of he ransiion regime is susained mainly by he shear producion, he dissipaion and he di usion. Moving o he locaion of ransiion onse (a y ˆ 0.3H), as shown in Fig. 4, he convecion erm becomes also signi can in he balance, because he urbulence energy grows rapidly as approaching o his sage. In he ouer region away from he boundary layer, all he erms become so small ha i is di cul o disinguish heir separae funcions in riggering ransiion a he poslaminar sage. As argued above, one of he essenial condiions o rigger laminar-urbulen ransiion is ha he oal producion per uni dissipaion erm for k due o shear and buoyancy, i.e. (N ks + N kb ), mus grow over uniy before ransiion occurs. Moreover, he urbulen kineic energy mus be ampli ed earlier han is dissipaion. Insead of using he direc budge comparison, a reasonable way is hen o compare he dissipaion-erm-normalized producions, N ks and N kb, near he posiion where ransiion occurs. In Fig. 5(a) and (b), such a comparison is made for he PDH model. Because his model is sensiive o he grid densiy, he disribuions obained wih wo grids, and , are used o observe he in uence of re ning grid. Wih a grid, he model predics a ransiion onse a y 0.38H and gives reasonable predicions for he oher variables. A his grid level, he normalized producion due o shear, N ks, grows in he nearwall layer up o a maximum value of abou 5. (a x/w ˆ , f ˆ 1.54), whereas N kb is abou This gives a 0 ˆ (N ks + N kb ) max 5. as ransiion sars. In he neighboring ouer region, by conras, N kb is negaively dominan over N ks and is absolue value is much larger han he maximum N ks in he near-wall layer. If he maximum numerical value of k 1 in he boundary layer ( ) and he corresponding value of k here ( ) are used in Eq. (), i gives y r 0.38H (Pr ˆ 0.7 for air). This is idenical wih he prediced locaion of ransiion onse. When he grid is re ned o , he normalized producion, N kb, a he posiion y 0.38H, where he ransiion occurs for a grid, decreases rapidly o large negaive values in he ouer region, and N ks is brough down wih a maximum value of abou 1.9 in he inner layer a x/w ˆ 0.035, see Fig. 5(a). Noe ha no ransiion occurs a his posiion wih a grid. Insead, he prediced ransiion shifs o y r 0.63H wih his grid, as shown in Fig. 1(d). The resul in Fig. 5 suggess ha a ner grid may over-amplify he buoyan source erm in he neighboring ouer region and bring down he shear

9 180 S.-H. Peng, L. Davidson / In. J. Hea and Fluid Flow 0 (1999) 17±184 Fig. 3. Predicions using he PDH+D model. The line is for he predicion and he symbols for he experimenal daa. (D) Ho wall, (}) Cold wall. producion in he boundary layer. As a resul, he ransiion onse is delayed. On he one hand, he urbulence growh in he boundary layer relies solely on he shear producion. The use of damping funcion f l will in general delay he ransiion onse, as shown by Eqs. (1) and () where jk j 1 near he verical wall. On he oher hand, he urbulence energy is di used from he nearwall layer o he neighboring ouer region, and is ``absorbed'' owing o he sable hermal srai caion and he sabilizaion funcion of he horizonal boundary layers. An appropriae equilibrium beween he near-wall producion ampli caion and he ouer-region energy desrucion is hus needed o make he model reasonably represen he ransiion onse. The resul in Fig. 5 indicaes ha he SGDH as used in he PDH model has over-ampli ed he buoyan source erm, G k, in he neighboring ouer region before ransiion occurs. This over-esimaion is so large (negaively) ha he capabiliy of he shear producion in he boundary layer is no large enough as a resource o di use energy owards he ouer region in order o balance he energy desrucion here (due o mainly he buoyancy-relaed erm and parly he dissipaion erm). This is re eced by he fac ha he normalized buoyan producion in he ouer region (wih negaive value) is much larger han he normalized shear producion in he boundary layer (wih posiive value). Since he energy desrucion in he ouer region canno be ``sauraed'' by di usion of urbulence energy from he near-wall layer, he urbulence evoluion in he boundary layer will no go furher. Consequenly, he value of a 0 in Eq. () is deerred from growing o a cerain level o rigger ransiion a a desired locaion. I is ineresing o noe ha he over-ampli caion of G k is reinforced for re ning grid. The ner he grid is, he more is he over-esimaion of G k and hus he more he shear producion is reduced. Wih a su cienly re ned grid, he shear producion in he near-wall boundary layer is reduced o a level a which he ampli caion rae of urbulence energy becomes slower han ha of he dissipaion rae. The near-wall shear producion is evenually no able o susain a coninuous ampli caion of k so as o bring a changeover of he ow sae. By conras, he ampli caion of dissipaion is coninued. As long as he local dissipaion overwhelms he local producion, he k-equaion goes o a new balance wih a very large x and a very small k. The model evenually reurns a laminar soluion, and no ransiion will arise a all. In Fig. 6(a) and (b), he normalized producions are compared for he PDH+D model a he ransiion onse locaion Fig. 4. Conribuions of di eren erms in he k-equaion for he PDH+D model a y ˆ 0.3H. (± ± ±) Convecion, (±±±) Shear producion, (á á á á á á á á á) Buoyancy producion, (- - -) Dissipaion, (- -) Di usion.

10 S.-H. Peng, L. Davidson / In. J. Hea and Fluid Flow 0 (1999) 17± Fig. 5. Near-verical-wall normalized producions for he PDH model a prediced ransiion onse locaion for grid (y r 0.38H). (a) Grid-dependence of N ks and N kb a y ˆ 0.38H. (±±±) N ks (10 10), (á á á á á á á á á) N kb (10 10), (± ± ±) N ks ( ), (- - -) N kb ( ). (b) Numerical values for k 1 and k a y ˆ 0.38H wih a grid. (y r 0.3H) prediced by his model on a grid. Noe ha his locaion is a grid-independen predicion. Fig. 6(a) shows ha jn kb j is signi canly reduced in he neighboring ouer region of he boundary layer by means of he damping funcion. The maximum normalized shear producion occurs a x/w (f 1.77) wih (N ks ) max ˆ 4.55, whereas N kb ˆ This gives a , where he numerical value for k 1 is found o be and k ˆ , see Fig. 6(b). Eq. () hus gives y r 0.3H as does he model. Unlike wih he PDH model, he value of jn kb j max in he ouer region becomes smaller han (N ks ) max in he near-wall boundary layer wih he PDH+D model. This implies ha he shear producion has an exra capabiliy o enhance urbulence in he boundary layer besides o di use energy owards he neighboring ouer region so as o saurae he urbulence desrucion here. The resuls in Fig. 6 indicae ha a reasonable Fig. 6. Near-verical-wall normalized producions for he PDH + D model a prediced ransiion onse locaion (y r 0.3H). (a) Disribuions of N ks and N kb a y ˆ 0.3H. (b) Numerical values for k 1 and k a y ˆ 0.3H. approximaion for he buoyan source erm can be achieved if he normalized buoyancy producion in he ouer region is suppressed below he normalized shear producion in he nearwall boundary layer. Under his condiion, he model becomes insensiive o he grid re nemen in predicing ransiion onse, whose locaion depends now on he degree of damping of he buoyancy erm. A srong damping of his erm means ha a large exra capabiliy reains in he near-wall shear producion o enhance urbulence in he boundary layer, and hus riggering an earlier ransiion, and vice versa. This is consisen wih he previous analysis. For a naural laminar convecion boundary layer ow along a verical a plae, k 1 is a similariy parameer whose sreamwise dependence vanishes. Assuming here exiss a nearwall sreamwise-independen maximum value for k 1 nex o he ransiion regime in he upsream laminar sage where jk j is usually much smaller han a 0, a condiion for he oal

11 18 S.-H. Peng, L. Davidson / In. J. Hea and Fluid Flow 0 (1999) 17±184 normalized producion a 0 o rigger ransiion a he desired locaion (y r /H) is hen a 0 ˆ N ks N kb max Ra 1= y r k 1 : 7 Pr H Eq. (7) appears o be a necessary condiion for a wo-equaion LRN model o correcly predic he onse locaion of ransiion regime in he naural convecion boundary layer ow along a verical heaed/cooled wall. 4.. Mixed convecion in a square caviy For he caviy ow wih sable hermal srai caion, he foregoing compuaion showed ha he SGDH performs reasonably well in conjuncion wih a damping funcion by which he grid-dependence problem inheren in his approach is removed for ransiion predicion in he boundary layer. The presen proposal is essenially moivaed for dealing wih naural convecion ows, in which ransiional boundary layers exis and cause problems in numerical simulaions. This, however, should no consrain is relevance o oher inernal urbulen buoyan ows. To furher verify he performance of he model for oher hermally srai ed ows, a mixed convecion ow is considered in his subsecion. The predicion was compared wih he experimenal daa by Blay e al. (199). In his case, he air ow is induced ino a square caviy (H W ˆ 1.04 m 1.04 m) hrough a slo (h in ˆ m) under he op wall and exhaused hrough an opening (E ˆ 0.04 m) on he opposie side above he boom wall. The supplied air has he same emperaure as ha on he surfaces of he side and op walls, T in ˆ T w ˆ 15 C. The boom wall was heaed o a consan surface emperaure of 35.5 C. Unlike in he closed caviy ow, an unsable hermal srai caion is se up near he horizonal walls in his case. The supply condiions a he inle are: u in ˆ 0.57 m/s, v in ˆ 0 and k in ˆ (m/s) (Blay e al., 199). The dissipaion rae of k is given by e in ˆ c l k 3= in /(0.5 h in). The inle x value is esimaed by using he relaion of x e/k. A non-uniform grid was used o perform he compuaion. Fig. 7 shows he resuls comparing wih he experimenal daa a he verical and horizonal mid-secions (a y ˆ H/ and x ˆ W/, respecively). The LRN k-e models used in his work generally give less saisfacory predicions han he LRN k-x models do. The mean velociy is over-prediced by he AKN model, and he LB model over-esimaes boh he emperaure and he urbulence energy. None of he LRN k-e models gives overall reasonable predicions for he urbulen kineic energy. All he models over-predic he mean velociy near he wall, wih he resul from he PDH model closes o he experimens. The PDH model produces saisfacory predicions for k in near-wall regions. The urbulence energy in he recirculaing region prediced by he WX model agrees well wih he experimenal daa, alhough his model signi canly under-predics he near-wall k level. As shown in Fig. 7, inroducing he damping funcion f G ino he PDH model (he PDH+D model) produces nearly same disribuions for he mean quaniies as he PDH model does. The urbulen kineic energy is, however, slighly overesimaed in he near-wall region. In general, he predicion obained wih he PDH+D model is saisfacory and very similar o ha compued wih he PDH model. Fig. 7. Comparison of di eren models for he mixed convecion ow in a square caviy. (- - -) AKN model, (- -) LB model, (á á á á á á á á á) WX model, (± ± ±) PDH model, (±±±) PDH + D model, (s) Experimen. (a) Mean velociy a y ˆ H/. (b) Mean velociy a x ˆ W/. (c) Mean emperaure a y ˆ H/. (d) Mean emperaure a x ˆ W/. (e) Turbulence energy a y ˆ H/. (f) Turbulence energy a x ˆ W/.