NUMERICAL SIMULATION OF A LIQUID SODIUM TURBULENT FLOW OVER A BACKWARD FACING STEP WITH A FOUR PARAMETER LOGARITHMIC TURBULENCE MODEL

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1 h European Conference on Compuaional Mechanics (ECCM ) 7h European Conference on Compuaional Fluid Dynamics (ECFD 7) 11 June, Glasgow, UK NUMERICAL SIMULATION OF A LIQUID SODIUM TURBULENT FLOW OVER A BACKWARD FACING STEP WITH A FOUR PARAMETER LOGARITHMIC TURBULENCE MODEL R. Da Vià 1,, A. Chierici,, L. Chirco, and S. Manservisi 4, Universiy of Bologna, via dei Colli 1, 41, Bologna, Ialy; 1 robero.davia@unibo.i, andrea.chierici4@unibo.i, leonardo.chirco@unibo.i, 4 sandro.manservisi@unibo.i. Key words: Liquid Meals, Mixed Convecion, Buoyancy, Four parameer urbulence model Absrac. In recen years a grea ineres has grown around liquid meals. These fluids are characerized by much higher hermal conduciviy, if compared wih sandard fluids like air and waer and can be used in applicaions where large hea fluxes are presen while being subjeced o small emperaure gradiens. In he presen paper we simulae a urbulen flow of liquid sodium, wih a Prandl number equal o.88, over a verical backward-facing sep. A uniform hea flux is applied on he verical wall nex o he change of cross secion. Reynolds sresses and urbulen hea flux are modeled wih a four logarihmic parameer urbulence model. We invesigae he cases of purely forced convecion, where he emperaure field is jus a passive scalar, and of mixed convecion, where emperaure has an impac on he fluid behavior hrough a buoyancy erm ha is inroduced in he momenum equaion wih he Boussinesq approximaion. The resuls are repored for various values of he Richardson number, i.e. Ri= for he purely forced convecion and Ri> for he mixed convecion case, and compared wih daa coming from Direc Numerical Simulaions ha are available in lieraure. 1 INTRODUCTION Liquid meals are becoming an ineresing ype of fluids for engineering applicaions [1]. From he physical properies poin of view hese fluids have low viscosiy and high hermal diffusiviy, which lead o values of molecular Prandl number much smaller han one. A ambien pressure lead, bismuh, sodium and lihium can be kep in liquid phase on a wide emperaure range. This characerisic allows scieniss and engineers o use liquid meals in applicaions where huge amouns of hea ransfer occur wihou he need of using pressurized sysems. As poined ou in numerous works, from he compuaional poin of view, differen and more sophisicaed means are required in order o accuraely simulae urbulen hea ransfer involving liquid meals [,, 4]. Direc Numerical Simulaion (dns) have been performed wih he purpose of providing reference daa o be used for

2 evaluaing he accuracy of urbulence models [5]. Four parameer urbulence models have been developed for he evaluaion of hermal field characerisic ime scales [, 7, 8]. Oher urbulence models have been developed o obain increased numerical sabiliy [] or o propose more sophisicaed expressions of urbulen hea flux, in order o ake ino accoun flow anisoropic behavior [1]. Wih he presen paper we deal wih he simulaion of a liquid sodium urbulen flow over a backward facing sep. This is a geomery ha has been exensively sudied by many researchers, from he kinemaic poin of view, as i is a geomery configuraion ha can be found in many engineering applicaions. Many works have recenly been proposed o invesigae he behavior of liquid sodium urbulen flow over such geomery, in paricular o sudy he influence of buoyancy on he flow paern for mixed convecion cases. The flow regime will be denoed hrough he value of he Richardson number Ri which is defined as Ri = gβ T h/ub where g is he modulus of graviy acceleraion, β he hermal expansion coefficien, T a reference emperaure difference, h he sep heigh and U b he bulk velociy. We here repor a brief review on he known lieraure sudies for his ype of flow. A firs dns simulaion for a liquid sodium urbulen flow over a backward facing sep, wih Ri =, is provided in [11], for a paricular geomery where a consan hea flux is applied on he whole wall behind he sep. For he same simulaion case, i.e. geomery configuraion and Reynolds number Re, oher sudies are performed [1, 1]. In [1] a comparison is provided beween he resuls obained from dns simulaion and from he soluion of a Reynolds Averaged Navier Sokes (rans) sysem of equaions closed wih various urbulence models. The considered case is of pure forced convecion. I is shown ha wo equaion hea ransfer urbulence models, coupled wih non linear expressions for Reynolds sresses, allow o improve he predicions of hea flux wihin he re-circulaion zone. In [1] a dns sudy is performed for he cases Ri = and Ri =.8. In paricular, for he mixed convecion case, i.e. Ri =.8, a differen domain configuraion is considered as an adiabaic secion is added behind he heaed wall in order o minimize he influence of he oule on he flow behavior over he heaed wall. Here he effec of buoyancy resuls in a diminished recirculaion lengh wih respec o he case of pure forced convecion. A grea number of simulaions have been performed for sudying he influence of buoyancy for a liquid sodium urbulen flow wih Re = 1 [,, 1, 17, ]. Various Richardson number cases are simulaed, ranging from Ri = o Ri = 1. In [,, 1] dns simulaions are performed for various values of he Richardson number, showing ha recirculaion lengh decreases and hea ransfer increases wihin he recirculaion zone as Ri increases. These sudies provide useful daa for he evaluaion of urbulence models. In [1] Large Eddy Simulaions (les) are performed for higher Reynolds numbers, namely Re h = and Re h = 4. Numerical simulaions wih rans models are repored in [17, ]. In he firs work a comparison is performed beween he resuls obained using a four parameer urbulence model [8, 1,, 1] and a wo equaion urbulence model [], for modeling he Reynolds sresses, and he Kays correlaion for modeling he eddy hermal diffusiviy []. I is shown ha similar resuls are obained using he wo differen models, for ha paricular Reynolds number case. In he laer work a sudy on a wider range of

3 y z g x h E q L in L h L o W Figure 1: Skech of backward facing sep geomery. Richardon number values is performed. Here he rans sysem of equaion is closed by using a wo equaion model and he Kays correlaion. A comparison beween he previously menioned lieraure daa is repored in Table 1, where he backward facing sep geomery is paramerized by using he sep heigh h and he lengh labels for inle secion L in, heaed secion L h, adiabaic secion L a, domain widh W and downsream channel heigh E, as skeched in Fig. 1. We refer o E r as he expansion raio, calculaed as E r = E/(E h). Table 1: Comparison of lieraure sudies on sodium buoyan urbulen flows on backward facing sep, as a funcion of geomerical parameers and of Reynolds and Richardson numbers. L in /h L h /h L a /h E r W/h Re h Mehod Ri [11] dns [1] dns/rans [1] dns -.8 [] dns [] dns dns [1] 1 1 les -. 4 les [17] 4-1 rans -. [] 4-1 rans NUMERICAL METHOD The liquid sodium urbulen flow over a backward facing sep is simulaed using a rans se of equaions. We use he assumpions of incompressible flow and he Boussinesq approximaion in order o accoun for buoyancy forces. The sysem of equaions consiss

4 hen of he following u =, (1) u + (u )u = 1 ρ P + [ν ( u + u ) T u u ] gβ(t T in ), () T + u T = [α T u T ], () where u, P and T are he mean velociy, pressure and emperaure. The unknown Reynolds sresses ρu u and he urbulen hea flux ρc p u T are modeled wih he following logarihmic four parameer urbulence model [( k +u k = ν + ν ) ] ( k + ν + ν ) k k + P k σ k σ ε e C k µ e Ω, (4) [( Ω +u Ω = ν + ν ) ] ( Ω + ν + ν ) k Ω+ σ ε σ ( ε + ν + ν ) Ω Ω + c ε1 1 P σ ε e k k C µ (c ε f exp 1) e Ω (5) [( k θ α + α ) ] ( k θ + α + α ) k θ k θ + P θ C µ e Ω θ, () +u k θ = Ω θ +u Ω θ = ( + α + α ) σ εθ σ θ [( α + α σ εθ ) ] Ω θ + σ θ ( α + α σ εθ e k θ ) k θ Ω θ + Ω θ Ω θ + c p1 1 P e k θ + c p θ e P k k (c d1 1) C µ e Ω θ c d C µ e Ω. (7) The model is a full logarihmic version of he one repored in []. The sae variables, namely k, Ω, k θ and Ω θ, represen he logarihmic values of urbulen kineic energy k and is specific dissipaion ω, mean squared emperaure flucuaions k θ and heir specific dissipaion rae ω θ. The ineresed reader can find he definiion of model consans and funcions in []. The model has been obained o provide an increased numerical sabiliy wih respec o he original formulaion. The sysem of equaions is solved wih a finie elemen code on a domain discreizaion having a non dimensional wall normal disance y + = δu τ /ν smaller han five on he firs mesh poin near wall boundaries, where δ is he wall disance, u τ he fricion velociy and ν he fluid kinemaic viscosiy. RESULTS Table : Non-dimensional parameers for he classificaion of he sudied case. L in /h L h /h L a /h E r W/h Re h Mehod Ri rans -.8 4

5 k/u b.5 k θ / T.7e- Θ e e e e e e a) b) c) Figure : Non-dimensional values of mean squared velociy flucuaions a), mean squared emperaure flucuaions b) and emperaure difference c) for he case of Richardon number equal o cf 1 Nu a) b) Figure : Skin fricion coefficien c f a) and Nussel number values b) along he heaed wall. The resuls are compared wih dns daa obained from [1]. In he presen work we repor he resuls obained for he simulaion of a backward facing sep case similar o hose sudied in [11, 1, 1]. The geomerical parameers of he simulaed domain are repored in Table, in accordance wih he classificaion proposed 5

6 in Table 1. On he inle secion of he domain we impose a velociy field obained from he simulaion of a fully developed urbulen channel flow having a fricion Reynolds number Re τ. The obained mean velociy leads o a Reynolds number Re h = 485. Homogeneous Neumann boundary condiions are imposed on he inle secion for he resolved urbulence variables, while for emperaure a uniform value equal o C is se. The same emperaure is used as reference emperaure for he evaluaion of he liquid sodium physical properies used in he sysem (1-7) hrough he correlaions provided in [4]. We obain a molecular Prandl number equal o P r =.88. On he wall boundaries we impose no slip boundary condiion for he velociy field, adiabaic boundary condiion for he emperaure field wih he excepion of he wall behind he sep where a uniform hea flux q is imposed. For he urbulence variables we use boundary condiions in accordance wih heir near wall behavior. On he oule secion an ouflow boundary condiion is imposed on he velociy field, while for all he oher variables we se a zero gradien. We sudy he cases of pure forced convecion, i.e. Ri = by seing β =, and he mixed convecion case for Ri =.8 in order o compare he resuls wih he ones obained in [1, 1]. Case Ri = A view of he non-dimensional urbulen kineic energy is repored in Fig. a), where he field is calculaed as k/ub, being U b he bulk velociy on he inle secion. In he same Fig. he flow sreamlines are presened, showing he presence of he wo ypical vorices ha arise behind he sep. The urbulen kineic energy reaches is maximum value in he region near he reaaching poin and hen decreases in sreamwise direcion. On he conrary he mean squared emperaure flucuaions behave, as can be seen in Fig. b). The skin fricion c f = τ w /ρub profile along he heaed wall is shown in Fig. a) and compared wih dns values obained from he profile presened in [1]. By seeing he change of sign in he values of c f we can calculae he reaaching lenghs of he wo vorices ha are presen in he region behind he sep. In paricular, in erms of non-dimensional sream-wise coordinae ỹ = wih ỹ = being he posiion of he sep, we obain he values ỹ 1.85 and ỹ.. The reaaching posiions obained from dns calculaion are repored wih red color in Fig. a). As observed in [1] he model underesimaes he corner eddy and he reaachmen lenghs. A beer agreemen wih dns resuls is obained for he Nussel number values, in paricular in he region behind he reaachmen poin, as can be seen in Fig. b), while for ỹ < 7 he Nussel number values are overesimaed by he four parameer urbulence model. The Nussel number is here calculaed as Nu = qh/(t T in )λ, where λ is he liquid sodium hermal conduciviy calculaed for T = C. The effecs of reaachmen lenghs underesimaion is refleced in he values of non dimensional emperaure incremen Θ, calculaed as Θ = (T T in )/ T. The values of Θ field are repored in Fig. c). Maximum and minimum values of Θ, ogeher wih heir posiion, along he heaed wall are repored in Table and compared wih he relaive dns daa. The maximum value of Θ is obained in he recirculaion area and i is close o he dns value, while is posiion is shifed in upsream direcion due o he underesimaion of he smaller eddy reaachmen poin. The wall emperaure value hen increases in sreamwise direcion afer he bigger

7 Table : Non dimensional emperaure difference along heaed wall. Maximum and minimum values, ogeher wih relaive posiion, are compared wih dns daa. Ri max(θ w ) min(θ w ) val val rans dns v/ub Θ a) b) Figure 4: Non-dimensional velociy a) and emperaure b) profiles on ransverse channel secion a various non-dimensional posiions. The resuls are for Ri = and compared wih dns daa from [1]. eddy reaachmen poin, showing a linear behavior. A more deailed comparison of he resuls wih reference dns daa is given in Fig. 4 where non-dimensional profiles of velociy and emperaure incremen, aken on channel cross secion planes, are repored for various values of sreamwise direcion. We can observe an overall good agreemen wih reference daa, wih he excepion of he emperaure profiles aken on = and =, which suffer from he underesimaion of vorex dimensions. Case Ri =.8 Wih he presence of a buoyancy force he flow field is subjeced o some significan changes, as can be noed from he non-dimensional field of sreamwise velociy componen v/u b ha is repored in Fig. 5 a). The recirculaion zone behind he sep is smaller han he case of pure forced convecion and he clockwise roaing vorex is now deached from he wall. We observe no reaaching poin on he heaed wall. The sreamwise velociy componen is grealy acceleraed by he buoyancy force near he heaed wall. The maximum emperaure incremen is sill found in he recirculaion 7

8 v/u b Θ.7 k θ / T.4e e e e e e e- a) b) c) Figure 5: Non-dimensional values of sreamwise velociy componen a), emperaure difference b) and mean squared emperaure flucuaions c) for he case of Richardon number equal o.8. zone, as can be seen from Fig. 5 b), bu is maximum value is much smaller han his considered for Ri =. Differenly from he case Ri = he maximum value of emperaure flucuaions is reached in he recirculaion area, as shown in Fig. 5 c). The comparison wih dns daa shows ha he velociy profile, near he ho wall, is slighly underesimaed, while a sligh overesimaion is found near he adiabaic wall, as can be seen in Fig. a), where non-dimensional velociy componen v/u b profiles on crosssecion planes are repored for various values of he non-dimensional posiion. We noe ha here is a good maching wih he locaion of he maximum values of v/u b. The non-dimensional emperaure gain, presened in Fig. b), is very close o he reference daa, so he underesimaion of sreamwise velociy componen is no due o he buoyancy force. In Fig. c) he non-dimensional urbulen kineic energy k/ub is compared wih dns daa. The bigges differences beween he values compued here and he reference ones are found in he recirculaion region, where an underesimaion of k is presen near boh adiabaic and heaed walls. For increasing values of he sreamwise coordinae y we observe a beer agreemen wih he dns daa. The underpredicion of sreamwise velociy componen maximum value near he ho wall is refleced in he profile of he skin fricion coefficien, along he heaed wall, which is repored in Fig. 7 a), as a funcion 8

9 1 1 1 v/ub Θ k/u b a) b) c) Figure : Non-dimensional velociy a) emperaure b) and urbulen kineic energy c) profiles on ransverse channel secion a various non-dimensional posiions. The resuls are for Ri =.8 and compared wih dns daa from [1]. 5. cf a) b) Nu Figure 7: Skin fricion coefficien c f a) and Nussel number values b) along he heaed wall for he case Ri =.8. The resuls are compared wih dns daa obained from [1]. of non dimensional posiion, and compared wih dns daa from [1]. Since he skin fricion coefficien c f is proporional o he sreamwise velociy componen derivaive along he direcion normal o he wall, if he maximum value of v is smaller han he dns one hen is derivaive v/ x will consecuively be smaller han he one obained from dns compuaions. We can see in Fig. 7 a) ha he c f profile shows he same qualiaive behavior as he reference one, bu wih small values. As already menioned, he skin fricion coefficien, for his mixed convecion case, does no change sign and is always posiive, showing he presence of no reaaching poins. In Fig. 7 b) he Nussel

10 number compued on he heaed wall is presened and compared wih dns daa. We noe a quie good agreemen in he predicion of he locaions of he poins wih Nu maximum and minimum values, while he obained ones are generally smaller han he reference ones. 4 CONCLUSIONS In recen years many sudies have been performed wih he inen o provide reference soluions for he numerical simulaions of low Prandl number fluids like liquid meals. In paricular resuls from Direc Numerical Simulaions of low Pr number fluid urbulen flows are now available in lieraure and hey involve many geomeries like plane channel, circular annulus and backward facing sep. In he presen work we simulaed a urbulen flow of liquid sodium, having a Prandl number equal o.88, over a backward facing sep wih an expansion raio E r = 1.5, by using a four parameer logarihmic urbulence model. A consan hea flux is applied on he wall behind he sep. We invesigaed a urbulen flow having a Reynolds number equal o Re = 485 in boh he regimes of forced and mixed convecion, namely wih a Richardson number Ri = and Ri =.8. The obained resuls are compared wih dns daa represenaive of he same simulaed cases, showing an overall good agreemen. For he case of forced convecion, as already found from oher lieraure sudies, he used urbulence model underesimae he sizes of he wo vorices ha arise behind he sep, while a good agreemen wih he dns resuls is found furher downsream afer he recirculaion zone. For he mixed convecion case he main discrepancies wih reference resuls are found in a slighly difference of sreamwise velociy componen maximum values near he heaed wall, leading o a underesimaion of he skin fricion coefficien on he same geomery side. Fuure sudies will be performed dealing wih a wider range of boh Richardson and Reynolds numbers, in order o improve he urbulence model range of validiy. REFERENCES [1] Heinzel A e al. 17 Energy Technol [] Cheng X and Tak N i Nucl. Eng. Des. 85 [] Cheng X and Tak N Nucl. Eng. Des [4] Grözbach G 1 Nucl. Eng. Des [5] Kawamura H, Abe H and Masuo Y 1 In J. Hea Fluid Fl. 1 7 [] Nagano Y and Shimada M 1 Phys. Fluids [7] Abe K, Kondoh T and Nagano Y 15 In. J. Hea Mass Tran [8] Manservisi S and Menghini F In. J. Hea Mass Tran. 1 [] Da Vià R, Manservisi S and Menghini F 1 In. J. Hea Mass Tran

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