ASSESSMENT OF BUOYANCY-CORRECTED TURBULENCE MODELS FOR THERMAL PLUMES

Transcription

1 Engineering Applicaions of Compuaional Fluid Mechanics Vol. 7, No., pp (1) ASSESSMENT OF BUOYANCY-CORRECTED TURBULENCE MODELS FOR THERMAL PLUMES Raesh Kumar and Anupam Dewan * * Deparmen of Applied Mechanics, Indian Insiue of Technology Delhi, New Delhi 1116, India adewan@am.iid.erne.in (Corresponding Auhor) ABSTRACT: A compuaional invesigaion of hermal plume is imporan because such flows are encounered in various indusrial applicaions. The invesigaion is also imporan because hermal plume can be considered as a es case for modeling of fire which can help designers and safey engineers o develop prevenive measures and fire safey sysems. The presen sudy primarily focuses on he invesigaion of hermal buoyan plume in he self-similar region. In he presen sudy, an assessmen of hree buoyancy-correced urbulence models, namely -ω, he sandard - model and he RNG - model, has been conduced for a hermal buoyan plume. Modificaions o he urbulence models have been made o accoun for he effec of buoyancy on he producion and dissipaion of urbulen ineic energy and hese modificaions are based on he simple gradien-diffusion hypohesis and generalized gradiendiffusion hypohesis. The model based on he simple gradien diffusion hypohesis is shown o under-predic conribuion o he generaion of urbulence ineic energy due o buoyancy. A comparison wih he experimenal measuremens repored in he lieraure shows ha he generalized gradien-diffusion hypohesis along wih boh urbulence models correcly predics he mean flow field, emperaure field and spread raes. The resuls of he presen simulaions using he RNG - model wih he generalized gradien-diffusion hypohesis are shown o be in good agreemen wih he corresponding experimenal resuls repored in he lieraure for hermal plumes. Keywords: plume, buoyancy, RANS, urbulence modelling, naural convecion 1. INTRODUCTION Thermal plumes have been he subec of research due o heir echnological and environmenal imporance in many physical processes, such as spread of smoe and oxic gases from fires, release of gases form volcanic erupions and indusrial sacs. Lie oher free-shear flows urbulen plume is highly unsable and undergoes ransiion o urbulen flow even a a small value of Grashof number (Dewan, 11). A urbulen flow comprises a wide specrum of ime and lengh scales of moion. Buoyancy plays a significan role in he physics of urbulence. I affecs he producion and dissipaion of urbulence ineic energy of he flow. Therefore he accuracy of compuaional simulaion of hermal plume fundamenally depends on how well he effec of buoyancy on urbulence is modeled. Direc numerical simulaion (DNS) and oher approaches o urbulence modelling (large eddy simulaion, Reynolds-sress ranspor model) are currenly available and hese valuable ools can offer a deeper insigh of he flow field. However, hese ools require expensive compuaional resources. Therefore wo-equaion urbulence models are sill he mos widely used o simulae indusrial problems. In paricular he - model is he mos exensively used and validaed model due o is simpliciy, robusness and compuaional sabiliy. However, he - model requires modificaions o model he effecs of buoyancy on he producion and dissipaion of urbulence in a hermal plume. The effec of buoyancy is usually incorporaed by adding a source erm in he ranspor equaions for urbulence ineic energy and dissipaion. Iniially many researchers used he sandard gradiendiffusion hypohesis (SGDH) o model he effec of buoyancy on urbulence (e.g., Maraos e al., 198; Nam and Bill, 199; Flecher e al., 1994). The SGDH ends o under-esimae he spread rae of verical hermal plumes and over-esimae he spread rae of horizonal, sably-sraified flows. Daly and Harlow (197) proposed a general gradien-diffusion hypohesis (GGDH) o model he effec of buoyancy on urbulence. In order o overcome he problem wih SGDH, a buoyancy source erm based on he GGDH has proved o be a good opion in recen years (Van Maele and Merci, 6; Chung and Devaud, 8). Van Maele and Merci (6) applied boh SGDH and GGDH o model he effec of buoyancy using he sandard - model and realizable - model. The prediced mean flow and urbulen flow Received: 11 Apr. 1; Revised: Nov. 1; Acceped: 1 Jan. 1 9

2 Engineering Applicaions of Compuaional Fluid Mechanics Vol. 7, No. (1) characerisics obained using he realizable - model in associaion wih he GGDH buoyancy modificaion showed significan improvemen as compared o hose obained using he SGDH. However sill significan differences beween he compuaional resuls and experimenal resuls of hermal plume exis and hese need o be addressed. Models based on he SGDH fail o correcly predic he spread rae, mean flow properies and urbulen flow properies primarily because hey do no include he cross-sream densiy variaion. The inclusion of cross-sream densiy variaion in he models based on he GGDH has a srong effec on he compuaional predicions. There is negligible influence of buoyancy on he dissipaion equaion when SGDH concep is used for he predicion of urbulence ineic energy due o buoyancy. However, when urbulence ineic energy due o buoyancy is calculaed by more accurae approximaion (GGDH), he influence of buoyancy is significanly increased on he dissipaion equaion due o an increased value of urbulence producion. In he sandard - model he eddy viscosiy is calculaed from he single urbulen lengh scale, herefore he compued urbulen diffusion occurs only a a single lengh scale, whereas in fac urbulen diffusion is affeced by all scales of moion presen in he flow. The RNG - model is based on he renormalizing group mehod in which he ranspor equaion is normalized o accoun for he effec of smaller scales of urbulence. This feaure maes he RNG - model more realisic and applicable for wider class of flows han he sandard - model. To he bes of our nowledge he RNG - model along wih buoyancy modificaion by GGDH and -ω model along wih buoyancy modificaions have no been used so far o model a hermal plume. In he presen paper, he efficacies of wo-equaion urbulence models (namely, he -ω, sandard - and RNG - models) have been invesigaed in deail. In order o accomplish he effec of buoyancy, a source erm has been added o he ranspor equaions for, and ω. The source erms in he ranspor equaions of, and ω have been modeled by boh SGDH and GGDH. Flow in a hermal plume is characerized by large densiy variaions; herefore Favre-averaged Navier-Soes equaions have been used in he presen paper. Low Mach number flow is considered and herefore i is assumed o be wealy compressible, which means he densiy can only be varied by emperaure and no by pressure. Mean densiy has been calculaed by using he ideal gas law. A finie-volume mehod based commercial code FLUENT 6..6 has been used for all simulaions. A user defined funcion (UDF) is developed for incorporaing he buoyancy erm in he momenum equaion. Furher, he buoyancy source erms modeled by using SGDH and GGDH have also been included in he UDF. In he presen paper, he compuaional resuls of axisymmeric plume have been compared wih he experimenal resuls of. The paper has been organized as follows. In secion he governing equaions and assumpions are presened. In secion we presen urbulence models along wih modificaions o he urbulence model o accoun for he effecs of urbulence. We presen compuaional schemes, grid generaion, and boundary condiions in secion 4. The resuls of he presen simulaion and comparison wih experimenal measuremens repored in he lieraure have been presened in secion 5.. GOVERNING EQUATIONS Cerain assumpions have been made o sudy he urbulen plume, which simplify he full Navier- Soes equaions, coninuiy equaion and energy equaion. The assumpions made are: The flow is axisymmeric and seady. In addiion, he low- Mach-number (LMN) version of Navier-Soes equaions has been adaped in he presen wor, raher han using he Boussinesq approximaion, whereby he effec of densiy differences is modeled by adding a source erm in he momenum equaion. The low-mach-number version of he Favre-averaged Navier-soes equaions can be expressed as (Van Maele and Merci, 6) x ρu = (1) ( ) ( ) 1 ( ρ ) P uu = + τ ρ i i uu i + ρ ρ gi x xi x x x Pr x ( ρ h uh ) = ρ uh () () The Einsein convenion of summaion has been adoped. Here P 1 denoes he hydrodynamic pressure. In he low-mach number version of Navier-Soes equaions, he densiy is calculaed from he sae equaion, P = ρ RT where, P 4

3 Engineering Applicaions of Compuaional Fluid Mechanics Vol. 7, No. (1) denoes he hermodynamic pressure and is equal o he amospheric pressure. Treaing hermodynamic pressure as consan is a common pracice. We have used he SGDH o model he urbulen enhalpy flux, ρ uh as h ρuh = σ x (4) Here he urbulen Prandl number σ is aen equal o.85.. TURBULENCE MODELS Unnown urbulen sresses are modeled wih he help of urbulence modeling in which urbulen sresses are expressed in erms of he mean flow properies. In order o model urbulen sresses, Boussinesq hypohesis is used o express urbulen sresses in erms of he mean flow properies: u u i u1 ρ uu i = + δ ρ δ i i x xi x1 (5) where denoes he urbulence ineic energy, 1 = uu 1 1, δ he Kronecer dela, and, i he urbulen viscosiy which needs o be modeled. In he sandard - model he urbulen viscosiy is modeled as = (6) ρc Here C denoes he model consan and he urbulence dissipaion rae. In he presen paper, we have considered wo urbulence models, namely he sandard - model and RNG - model. We will discuss heir deails and he mahemaical formulaions in he subsequen secion..1 Sandard - model The sandard - model, proposed by Jones and Launder (197) is he simples, wo-equaion model which is exensively used for modelling a variey of pracical engineering flows. In his model, he urbulence ineic energy and is dissipaion are considered as ranspor properies. Therefore, in order o describe urbulence, he wo ranspor variables, and, are compued from heir respecive ranspor equaions. The ranspor equaions of and in ensor form may be expressed as ( ρ u ) = + + P ρ + S x x σ x ( ρ u ) = + + C1 P x x σ x Cρ + S (7) (8) In equaions (7) and (8), denoes he molecular viscosiy and P he producion of urbulence due o mean velociy gradien: ( ) i i P= ρ uu u x (9) The effec of buoyancy on he producion of urbulence ineic energy is aen ino accoun by adding a source erm S o he ranspor equaion for. Similarly a source erm S is added o he ranspor equaion for he urbulence dissipaion rae o accoun for he effec of buoyancy on he urbulence dissipaion rae.. Renormalized group - model The renormalized group (RNG) - model was developed by Yaho e al. (199). This model was derived by using he renormalized group heory and a saisical echnique o renormalize he Navier-Soes equaions so ha i could accoun for he smaller scales of urbulence. The RNG - model is similar o he sandard - model wih some modificaions in he ranspor equaion for he dissipaion rae. There is an analyical formulaion of urbulen Prandl number, whereas in he sandard - model, a consan user specified value of urbulen Prandl number is used. The main difference beween he sandard - model and he RNG - model is he addiional erm in he ranspor equaion of dissipaion rae, which is discussed in deail below. The ranspor equaions of RNG - model are ( ρ ) + ( ρu ) = α eff x x x + P ρ + S ( ρ ) + ( ρu ) = α eff x x x * + C1 P Cρ + S (1) (11) 41

4 Engineering Applicaions of Compuaional Fluid Mechanics Vol. 7, No. (1) where α and α denoe he inverse Prandl numbers for urbulence ineic energy and dissipaion rae, respecively. S and S denoe he source erms which accoun for he effec of buoyancy. The expressions of hese source erms, S and S, will be discussed in deail in he subsequen secion on buoyancy correced - models. The effec of addiional erm in he ranspor equaion is accouned for by aing a differen expression for he model consan C. The effecive urbulen viscosiy is modeled by using a differenial equaion as ρ ˆ = 1.7 ˆ υ 1 υ d + wih eff C υ d ˆ υ (1) ˆ υ = and C υ 1 Equaion (1) is inegraed o have an accurae descripion of he effecive urbulen ranspor which varies wih he eddy scale. This helps he model o beer capure he effec of low Reynolds number and near wall regions. In he ranspor equaion of dissipaion equaion he erm C * is expressed as: C ( 1 ηη) C η 1+ βη * = C + (1) wih η = S where S denoes he magniude of he mean srain rae, η = 4.8 and β = 1. The model consans C 1 = 1.4 and C = 1.68 are used.. Buoyancy correced - models In his secion, he modificaions in he ranspor equaions of he wo urbulence models are presened o incorporae he effecs of buoyancy. In boh he sandard and RNG - models he producion of urbulence due o mean shear, P is deermined on he basis of he Boussinesq assumpion as i ( i ) P= ρuu u x u u i u 1 = + δi ρδ i x x i x1 (14) A source erm S is added o he ranspor equaions of urbulence ineic energy, o accoun for he effec of buoyancy. This source erm denoes he producion of urbulence due o buoyancy and is denoed as G. The erm G is usually modeled by he simple gradien-diffusion hypohesis (SGDH) as 1 1 ρ P S = G = ρ g + σ ρ x x (15) However, he modificaion based on he SGDH has a deficiency of under-predicing he spread rae in he case of a verical hermal plume and over-predicing he spread rae in he case of horizonal, sably-sraified flows (Maraos e al., 198; Nam and Bill, 199; Flecher e al., 1994; Shabbir and Taulbee 199). Shabbir and Taulbee (199) have found ha he hea flux perpendicular o he graviy is significanly under-prediced in case of he simple gradien diffusion hypohesis (SGDH). There is significan effec of he erm G on he spread rae of hermal plume (Worhy e al., 1). To overcome he deficiency of he SGDH, a modificaion based on he generalized gradien diffusion hypohesis (GGDH) is used as an alernaive o calculae he producion of urbulence due o buoyancy. The GGDH model was developed from he second-order closure model as 1 ρ P ρ x x G= uu + g σρ (16) I can be seen ha here is a difference in he formulaion of G by he SGDH and GGDH. The expression of G by he GGDH is differen due o he erm ρ x which represens he variaion of densiy in he direcion perpendicular o he graviy vecor. In he SGDH model, he densiy gradien is included only in he direcion of he graviy. Van Maele and Merci (6) suggesed an approximaion of replacing he urbulen normal sress, uu (no summaion here) in Eq. (16) as being equal o he urbulence ineic energy. This approximaion is usified due o he fac ha he urbulen normal sress in he sreamwise direcion is wice ha of he normal sresses in oher direcions (Shabbir and George, 1994). The expression of G in Eqs. (15) & (16) can be reformulaed and simplified by using cerain approximaion as discussed above and he ideal gas law P = ρ RT where P denoes he hermodynamic pressure which is aen as he amospheric pressure in he presen sudy. The 4

5 Engineering Applicaions of Compuaional Fluid Mechanics Vol. 7, No. (1) hermodynamic pressure in he presen sudy is reaed as consan in boh ime and space. ρ ρ T = (17) x T x Anoher approximaion can be made o simplify he source erm G if he reference densiy is 1 chosen such ha P x << ρ g. Now, he producion of urbulence due o buoyancy G can be rewrien as ρ 1 T G = g for SGDH model (18) ρ σ T z ρ T T G= + uw g ρ σt z r for GGDH model (19) The expressions of G given in Eqs. (18) and (19) have been used in he presen sudy. Furher modificaion has been done in he ranspor equaion of dissipaion rae. In Eqs. (8) and (11) a source erm is added o incorporae he effec of buoyancy on he ranspor equaion of dissipaion rae. Rodi (198) suggesed a modified ranspor equaion for dissipaion as ( ρ u ) = + x x σ x + C1 ( P+ G)( 1+ C Rf ) Cρ () where R f denoes he flux Richardson number which is defined as Rf = GP (1) An alernae definiion of Richardson number given by Rodi is R f = G (G + P) which is now aen as he sandard definiion. In he presen sudy, his alernae definiion has been used. Thus he source erm in he ranspor equaion of dissipaion rae can be expressed as S = C1 ( 1 C ) G () where C denoes a buoyancy consan. The value of C in he range of. o does no affec he simulaions, as repored in he lieraure (Maraos e al., 198; Nam and Bill, 199; Yan and Holmsed, 1). In he presen sudy C =.8 has been used (Van Maele and Merci 6). Various source erms have been proposed in he lieraure for his purpose and a deail lis of various expressions of source erm used in he lieraure can be found in Chung and Devaud (8)..4 -ω model wih buoyancy correcion The -ω model used in he presen wor was proposed by Wilcox (1998). This model is based on he ranspor equaion of he urbulen ineic energy () and he specific dissipaion rae (ω). This model incorporaes he modificaions of he low Reynolds number effec, effec of compressibiliy and shear flow spreading. The ranspor equaions for and ω are: ( ρ u ) = + + P x x σ x βρ ω+ G ω ( ρω u ) = + x x σ ω x ω α P βρω Sω + + () (4) where P denoes he producion of urbulence due o shear. G and S ω denoe he source erms in and ω equaions, respecively ρ 1 T G = g for SGDH model (5) ρ σ T z ρ T T G= + uw g ρ σt z r for GGDH model (6) S ω ω = (( α + 1 ) C( G 1) ) (7) The model consans are given by β = 9, α = 5 9, β = 75, σ =, σ ω = and C = 1 The urbulen viscosiy is relaed o he urbulen ineic energy () and he specific dissipaion rae (ω) as = (8) ρ ω 4. COMPUTATIONAL METHOD A commercial CFD code FLUENT 6..6 based on he finie volume mehod has been used for he presen simulaions. The buoyancy modificaions o he ranspor equaions for and are implemened in FLUENT 6..6 by using he 4

6 Engineering Applicaions of Compuaional Fluid Mechanics Vol. 7, No. (1) user-defined-funcions (UDFs). Boh he modificaions based on SGDH and GGDH have been incorporaed in UDFs. A source erm (ρ ρ )g is added o he momenum equaion o accoun for he effec of buoyancy forces and densiy differences. This source erm in he momenum equaion is also incorporaed in UDF. All he convecive erms in he momenum, energy and urbulence equaions have been discreized using he second-order upwind scheme. A second-order accurae cenraldifference scheme has been used for he diffusion erms in all he governing equaions. The SIMPLE algorihm has been used for he pressure-velociy coupling. All he simulaions were run unil he scaled residuals for he momenum, urbulence and energy equaions were less han 1-6. In he presen sudy, an axisymmeric hermal plume was simulaed using a recangular domain of m in he radial direcion and. m in he axial direcion. Devenish e al. (1) used he maximum rise heigh of he plume as D (here D denoes he diameer of he source) which ensures ha he axial heigh of. m in he compuaional domain is sufficien o have fully developed flow and he ambien condiions a he upper par of he domain. The whole recangular compuaional domain was modeled and meshed using GAMBIT.4.6. The schemaic of he compuaional domain along wih differen boundary condiions has been shown in Fig. 1. The compuaional domain conained 4 recangular cells, 4 in he radial direcion and 1 in he axial direcion. The nodes were clusered near he source in he radial as well as axial direcions. Velociy a he source was specified as.67 m/s in he axial direcion wih a small value of urbulen inensiy (.5%) and he urbulen lengh scale equal o D /15. Temperaure a he source was specified as 57 K. A he boom of he compuaional domain near he source, he wall boundary condiion was specified. A he remaining boundary, saic pressure has been specified o allow flow ino and ou of he compuaional domain. An ambien pressure of 115 Pa was used for he calculaions. Bacflow emperaure of K was specified a he saic pressure boundaries. For he grid independence sudy, hree differen grids were used in he presen sudy, ermed A, B and C grids. The grid A conained 4x1 cells, wih 1 equally spaced cells a he source, remaining cells in he radial direcion and 1 in he axial direcion. Similarly grids B and C conained 8x and 16x4 cells, respecively. Grid independence sudy showed ha less han.5% of difference of mean axial velociy a he cenerline was observed beween he compued resuls using grids A and B. Fig. 1 Axis Velociy and Temperaure Pressure Oule Schemaic of compuaional domain wih boundary condiions. 5. RESULTS AND DISCUSSION The experimenal daa of George e al. (1977) was employed for assessing he performance of he buoyancy-correced urbulence models discussed in he previous secion. They performed measuremens in a plume wih a quiescen ambien air mainained a a consan emperaure of K. The diameer of he hea source used was 6.5 cm. In heir experimens he consan hea source emperaure was mainained a 57 K and he exi velociy of 67 cm/s a he hea source was calculaed from he measured hea flux. The exi condiions a he hea source correspond o Reynolds number of 87 and densimeric Froude number of 1.4. They concluded on he basis of he experimenal measuremens ha he laminar flow behavior was no deeced beyond source diameers in he sreamwise direcion. 5.1 Mean flow resuls The presen compuaions have been obained by using he sandard - model, -ω model and RNG - model along wih SDGH and GGDH modificaions. In he presen sudy, he following Gaussian profiles obained experimenally by have been used for comparison ( ) ( η ) 1/ 1/ WF z =.4 exp 58η (9) gβ TF z 9.4 exp 68 Pressure Oule Solid wall / 5/ = () here, W denoes he mean verical velociy, F he buoyancy added a he source of he hermal plume, z he verical disance, η he similariy 44

7 Engineering Applicaions of Compuaional Fluid Mechanics Vol. 7, No. (1) variable, g he acceleraion due o graviy, β he coefficien of hermal expansion and T he emperaure difference beween he mean local and he ambien. The correlaions (9) and () were obained experimenally by for he self-similar regime of a hermal plume. In he self-similar region no laeral variaion of similariy variables wih he axial posiion occurs and he flow behavior does no depend on he source condiions. In Fig. he radial disribuion of normalized mean axial velociy is ploed a hree differen axial posiions (z =.75, z = 1.75 and z =.75). The radial profiles of mean axial velociy a z = 1.75 and z =.75 show no significan differences and herefore i can be concluded ha he hermal plume has reached a self-similar sae. The verical mean velociy and mean buoyancy profiles have been ploed using he similariy variables in Figs. and 4. Henceforh, we have compared he flow variables a z = 1.75 m. I is clear from Fig. ha boh he sandard - model and RNG - model wih he SGDH buoyancy modificaion over-predic he mean axial velociy a he cenerline. Furher, he spread rae is underesimaed by boh models when used wih he SGDH buoyancy modificaion. The buoyancy modificaions wih GGDH applied along wih sandard - model produced resuls which were slighly closer o he experimenal resuls of as compared o he presen predicions obained using he SGDH. However, he mean axial velociy a he cenral line is underesimaed and spread rae is overesimaed by GGDH wih he sandard - model. The RNG - model wih buoyancy modificaion and using GGDH predics he axial velociy a he cenerline and spread rae in good agreemen wih he experimenal daa of Shabbir and George (1994). In Table 1 he mean axial velociy a he cenerline and non-dimensional half-widhs have been presened. The nondimensional half-widh is defined as he nondimensional radial disance where he normalized velociy or emperaure is half of is value a he corresponding axial locaions. The radial disance is normalized wih he axial locaion. Shabbir and George (1994) did no quanify he uncerainy in heir measuremens. Therefore we have assumed an uncerainy of 5% in he experimenal daa repored by, which is a general value in a measuremen. In Fig. 5 he fracion of producion of urbulence due o buoyancy o he oal urbulence producion is WF -1/ z / Fig. Fig. Fig W F -1/ z1/ gβ TF -/ z5/ z=.75 z=1.75 z=.75 Normalized profile of mean axial velociy a differen sreamwise locaions Normalized mean axial velociy profile for - based models Normalized mean buoyancy profiles for differen models ploed. I can be observed from Fig. 5 ha in case of models in which he SGDH is applied he fracion of producion of urbulence due o buoyancy wih oal producion of urbulence, a he cenral posiion is quie high bu his value reduces quicly in he radial direcion. In case of he models in which GGDH is applied for he calculaion of he producion of urbulence, his fracion does no reduce quicly in he radial 45

8 Engineering Applicaions of Compuaional Fluid Mechanics Vol. 7, No. (1) Table 1 Comparisons of spreading rae, normalized mean axial velociy and mean buoyancy a cenerline. Nondimensional parameer Shabbir and George (1994).19 ± l W/ 5.11 ± l T/ 5 WF -1/ z 1/.4 ± c.17 gβ TF -/ z 5/ 9.4 ± c.47 G/(G+P) Fig. 5 Fig. 6 SKE- SGDH SKE- GGDH RNG- SGDH RNG- GGDH WF -1/ z / Profiles of fracion of buoyancy producion for differen models. -ω -ω wih SGDH -ω wih GGDH Normalized mean axial velociy profile using -ω based models. direcion. Therefore i can be concluded ha he prediced conribuion of he buoyancy producion of urbulence, G, wih SGDH model is smaller han ha prediced wih he GGDH. The mean axial velociy has been ploed in Fig. 6 by using he -ω model wih he buoyancy modificaions. I can be observed from Fig. 6 ha he mean axial velociy is underprediced whereas spread rae is overprediced by using -ω model wih no buoyancy modificaions. The resuls from he modificaion by SGDH show improvemen in boh he mean axial velociy and spread rae. There is no noiceable difference in he resuls beween -ω and -ω model wih GGDH modificaion. I can be concluded ha he - based models are beer in capuring flow in hermal plume han he -ω model. This is probably due o he fac ha -ω model performs beer in case of wall-bounded and low Reynolds number flows and no for buoyancy dominaed flows in he absence of wall effecs such as hermal plume. Thus he buoyancy-correced - model is used for subsequen compuaions. The radial variaion of he mean buoyancy is presened in Fig. 4. The non-dimensional halfwidh and mean buoyancy a he cenerline are presened in Table 1. Boh models wih he buoyancy modificaion by he SGDH over-predic he mean buoyancy (Fig. 4). The sandard - model wih he SGDH underpredics he spread rae, whereas, he RNG - model wih he SGDH correcly predics he half-widh. Boh models wih he GGDH modificaion under-predic he mean buoyancy, however, hese profiles are closer o he experimenal correlaion of Shabbir and George (1994) han hose by he SGDH models. I can be seen in Fig. ha he predicions using he RNG - model wih GGDH modificaions are closes o he correlaion of. I can be concluded ha overall he RNG - model along wih he buoyancy modificaions by GGDH produces he correc predicions of he mean axial velociy and buoyancy along wih heir spread raes. 5. Turbulence properies In his secion urbulen normal sresses and shear sresses have been compared using he similariy variables. The correlaions derived by Shabbir and George (1994) experimenally have been used for a comparison of various urbulen properies: / / 1.1+ η w F z = (1) η ( ) 4 / / η 7.6η u F z = () 4 1+ η ( ) / / 6.5η 14.η uwf z = 1+ 4η ( ) () here, u and w denoe flucuaions in he radial and axial direcions, respecively, and η he nondimensional radial locaion. 46

9 Engineering Applicaions of Compuaional Fluid Mechanics Vol. 7, No. (1) The radial variaions of axial and radial urbulen normal sresses have been presened in Figs. 7, 8, 9 and 1. In Fig. 7 he resuls obained by using he sandard - model wih boh SGDH and GGDH modificaions are shown. There is a significan difference beween he experimenal correlaion and he compuaional resuls obained by using he sandard - model. Turbulen axial normal sress obained using he GGDH modificaion, however, is closer o he correlaion of han hose using SGDH. In Fig. 8 he resuls of he RNG - model wih boh modificaions have been shown and i is clear ha he resuls using he GGDH mach wih he correlaion away from he cenral locaion. In Figs. 9 and 1 he urbulen radial normal sresses have been ploed using boh urbulence models and boh SGDH and GGDH buoyancy modificaions. I can be clearly observed ha boh he urbulence models do no capure he anisoropy of hese normal sresses. The values of urbulen axial and radial sresses prediced by boh models are approximaely he same, whereas he values measured by Shabbir and George are quie differen. Van Maele and Merci (6) have also made a similar observaion. The non-dimensional urbulen shear-sresses wwf -/ z/ have been ploed in Figs. 11 and 1. The buoyancy modificaion wih he GGDH incorporaed in boh sandard - model and RNG - model produces resuls which are in good agreemen wih he experimenal daa. The buoyancy modificaion using GGDH wih he RNG - model predics he urbulen shear sresses closes o he correlaion compared o ha by oher models discussed in he presen sudy. The budge of differen erms of he mean momenum equaion (4) a z =.75 m prediced by he RNG - model have been presened in Fig. 1. The similariy variable of each erm in he mean momenum equaion is obained by scaling velociy by F 1/ z -1/ and emperaure by F / z -5/. The radial ranspor and buoyancy erms are relaively significan erms in he budge. This is due o radial spread of hermal plume. As shown in Fig. 1 he buoyancy erm is significan in he momenum equaion and herefore i is imporan o accuraely model he effec of buoyancy in such flows. ( ) W W 1 ruw W + U gβ( T T ) z r r r buoyancy verical convecion radial convecion radial ranspor uu F -/ z/ (4). Fig wwf -/ z/ Profiles of normalized axial urbulen normal sress for sandard - model. Fig. 9 uu F -/ z/ Profiles of normalized radial urbulen normal sress using sandard - model.. Fig Profiles of normalized axial urbulen normal sress for RNG - model Fig. 1 Profiles of normalized radial urbulen normal sress using RNG - model. 47

10 Engineering Applicaions of Compuaional Fluid Mechanics Vol. 7, No. (1) uw F -/ z/ Fig. 11 Profiles of normalized urbulen shear sress using sandard - model. uw F -/ z/ Fig. 1 Profiles of normalized urbulen shear sress using RNG - model. SIMILARITY VARIABLES Fig. 1 Prediced profiles of differen erms of mean momenum equaion using RNG - model. 6. CONCLUSIONS Buoyancy Radial Convecion Axial Convecion Radial Transpor In he presen sudy he performance of he -ω model, sandard - model and RNG - model has been assessed for he buoyan hermal round plume by using suiable modificaions in he ranspor equaions of urbulence ineic energy and dissipaion rae. Boh simple and generalized gradien diffusion hypoheses have been used. These modificaions have been done o incorporae he effec of buoyancy on urbulence. I can be concluded ha he varians of - model are beer han hose of -ω model in capuring hermal plume. I is observed ha he buoyancy modificaion by he SGDH in - based urbulence models does no accuraely capure he effec of buoyancy. The buoyancy modificaion by he GGDH in - based models shows large effec of buoyancy whereas he SGDH modificaion underpredics his behavior. The RNG - model wih he modificaions by he GGDH capures he mean and urbulen flow properies accuraely. The resuls obained by his model are closes o he experimenal daa of compared o hose by oher models discussed in he presen sudy. I can be concluded ha he RNG - model wih he buoyancy modificaions by he GGDH is overall a good model o capure he complex flow and hermal fields in a buoyan hermal round plume. REFERENCES 1. Chung W, Devaud CB (8). Buoyancycorreced models and large eddy simulaion applied o a large axisymmeric helium plume. In. J. Num. Meh. Fluids 58: Daly BJ, Harlow FH (197). Transpor equaions in urbulence. Phys. Fluids 1: Devenish BJ, Rooney GG, Thomson DJ (1). Large-eddy simulaion of a buoyan plume in uniform and Sably Sraified Environmens. J. Fluid Mech. 65: Dewan A (11). Tacling Turbulen Flows in Engineering. Springer. 5. Flecher DF, Ken JH, Ape VB, Green AR (1994). Numerical simulaions of smoe movemen from a pool fire in a venilaed unnel. Fire Saf. J. : George WK, Alper RL, Tamanini F (1977). Turbulence measuremens in an axisymmeric buoyan plume. In. J. Hea Mass Transfer : Jones WP, Launder BE (197). The predicion of laminarizaion wih a -equaion model of urbulence. In. J. Hea and Mass Transfer 15: Maraos NC, Malin MR, Cox G (198). Mahemaical modeling of buoyancy-induced smoe flow in enclosures. In. J. Hea Mass Transfer 5:

11 Engineering Applicaions of Compuaional Fluid Mechanics Vol. 7, No. (1) 9. Nam S, Bill RG (199). Numerical simulaion of hermal plumes. Fire Saf. J. 1: Rodi W (198). Turbulence Models and Their Applicaion in Hydraulics. Ph.D. Thesis, Universiy of Karlsruhe, Germany. 11. Shabbir A, Taulbee DB (199). Evaluaion of urbulence models for predicing buoyan flows. J. of Hea Transfer 11: Shabbir A, George WK (1994). Experimens on a round urbulen buoyan plume. J. Fluid Mech. 75:1. 1. Van Maele K, Merci B (6). Applicaion of wo buoyancy modified - urbulence models o differen ypes of buoyan plumes. Fire Saf. J. 41: Wilcox DC (1998). Turbulence Modeling for CFD. DCW Indusries. 15. Worhy J, Sanderson V, Rubini P (1). Comparison of modified urbulence models for buoyan plumes. Numerical Hea Transfer 9: Yaho V, Orszag SA, Thangam S, Gasi TB, Speziale CG (199). Developmen of urbulence models for shear flows by a double expansion echnique. Physics of Fluids A 4(7): Yan Z, Holmsed G (1). A wo-equaion urbulence models and is applicaion o a buoyan diffusion flame. In. J. of Hea and Mass Transfer 4: