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Evaluation of Various Turbulence Models in Predicting Airflow And Zhang, Z., Zhang, W., Zhai, Z., and Chen, Q. 2007. “Evaluation of various turbulence models in predicting airflow and turbulence in enclosed environments by CFD: Part-2: comparison with experimental data from literature,” HVAC&R Research, 13(6). 1 Evaluation of Various Turbulence Models in Predicting Airflow and 2 Turbulence in Enclosed Environments by CFD: Part-2: Comparison 3 with Experimental Data from Literature 4 5 Zhao Zhang Wei Zhang Zhiqiang Zhai Qingyan Chen* 6 Student Member ASHRAE Member ASHRAE Member ASHRAE Fellow ASHRAE 7 8 Numerous turbulence models have been developed in the past decades, and many of them may be 9 used in predicting airflows and turbuence in enclosed environments. It is important to evaluate 10 the generality and robustness of the turbulence models for various indoor airflow senarios. This 11 study evaluated the performance of eight turbulence models potentially suitable for indoor 12 airflow in terms of accuracy and computing cost. These models cover a wide range of 13 computational fluid dyanmics (CFD) approaches including Reynolds averaged Navier-Stokes 14 (RANS) modeling, hybrid RANS and large eddy simulation (or detached eddy simulation, DES), 15 and large eddy simulation (LES). The RANS turbulence models tested include the indoor zero- 16 equation model, three two-equation models (the RNG k-ε, low Reynolds number k-ε, and SST k- 17 ω models), a three-equation model ( vf2 − model), and a Reynolds stress model (RSM). The 18 investigation tested these models for representative airflows in enclosed environments, such as 19 force convection and mixed convection in ventilated spaces, natural convection with medium 20 temperature gradient in a tall cavity, and natural convection with large temperature gradient in 21 a model fire room. The predicted air velocity, air temperature, Reynolds stresses, and turbulent 22 heat fluxes by the models were compared against the experimental data from the literature. The 23 study also compared the computing time used by each model for all the cases. The results reveal 24 that LES provides the most detailed flow features while the computing time is much higher than 25 RANS models and the accuracy may not always be the highest. Among the RANS models studied, 26 the RNG k-ε and a modified vf2 − model have the best overall performance over four cases 27 studied. Meanwhile, the other models have superior performance only in some particular cases. 28 While each turbulence model has good accuracy in certain flow categories, each flow type 29 favors different turbulence models. Therefore, we summarize both the performance of each 30 partcular model in different flows and the best suited turbulence models for each flow category 31 in the conclusions and recommendations. 32 33 INTRODUCTION 34 The companion paper (Zhai et al., 2007) reviewed the recent development and applications 35 of computational fluid dynamics (CFD) approaches and turbulence models for predicting air 36 motion in enclosed spaces. The review identified eight prevalent and/or recently proposed 37 turbulence models for indoor airflow prediction. These models include: the indoor zero-equation 38 model (0-eq.) by Chen and Xu (1998), the RNG k-ε model by Yakhot and Orszag (1986), a low 39 Reynolds number k-ε model (LRN-LS) by Launder and Sharma (1974), the SST k-ω model 40 (SST) by Menter (1994), a modified v2f model (v2f-dav) by Davidson et al. (2003), a Reynolds * Zhao Zhang is a PhD candidate, Wei Zhang is an affiliate, and Qingyan Chen is a professor in the School of Mechanical Engineering, Purdue University, West Lafayette, IN. Zhiqiang Zhai is an assistant professor in the Department of Civil, Environmental & Architectural Engineering, University of Colorado, Boulder, CO. 1 stress model (RSM-IP) by Gibson and Launder (1978), the large eddy simulation (LES) with a 2 dynamic subgrid scale model (LES-Dyn) (Germano et al. 1991 and Lilly 1992), and the detached 3 eddy simulation (DES-SA) by Shur et al. (1999). This paper evaluates and compares the selected 4 turbulence models for several indoor benchmark cases that represent the primary flow 5 mechanism of air movement in enclosed environments. 6 All these turbulence model equations mentioned above can be written in a general form as: ∂φ ∂φ ∂⎡⎤ ∂φ 7 ρ+ρ−uSj,eff⎢⎥ Γφ =φ (1) ∂∂∂∂txxxjj⎣⎦⎢⎥ j 8 where φ represents variables, Γφ,eff the effective diffusion coefficient, and Sφ the source term of 9 an equation. Table 1 briefly summarizes the mathematical expressions of the eight turbulence 10 models selected. In Table 1, ui is the velocity component in i direction, T the air temperature, k 11 the kinetic energy of turbulence, ε the dissipation rate of turbulent kinetic energy, and ω the 12 specific dissipation rate of turbulent kinetic energy. P the air pressure, H the air enthalpy, μt the 13 eddy viscosity, Gφ the turbulence production for φ, and S the rate of the strain. The other 14 coefficients are case-specific and only some important ones are introduced here. 15 For the 0-eq. model, V is the velocity magnitude and l is the wall distance. The GB is the * * 16 buoyancy production term for the RNG k-ε model. For LRN-LS model; the fμ , Cε1 , Cε 2 are the 17 three modified coefficients (i.e., damping functions) to the standard k-ε model; and D and E are 18 two additional terms. These five major modifications in the LRN model are responsible for 19 improving model performance near the wall. In SST model the Y is the dissipation term in the k 20 and ω equations. The F1 and F2 are blending functions that control the switch between the 21 transformed k- ε model and the standard k-ω model. The Dω is produced from the transformed k- 22 ε model. So it vanishes in the k-ω mode when the blending function F1 is unit. In v2f-dav model 23 (Davidson et al., 2003), the v′2 is the fluctuating velocity normal to the nearest wall. The f is part 24 of the v′2 source term that accounts for non-local blocking of the wall normal stress. The f is 25 implicitly expressed by an elliptical partial differential equation. So the scalar f can be in 26 principle solved by the same partial-differential-equation solver as for the other variables. Note 27 that the T in v2f-dav model also represents the turbulence time scale. In RSM model, the φlm is 28 the pressure strain term and requires further modeling. In the present study, a liner pressure strain 29 model by Gibson and Launder (1978) is used. S S 30 In the LES, the over bar represents the filtering. The τij and h j represent the subgrid scale 31 (SGS) stress and heat flux. Lilly’s SGS model (1992) adopts the Boussinesq hypothesis and 32 derives methods to calculate the coefficient Cs in the eddy viscosity expression automatically. 33 The presented DES (Shur et al. 1999) couples the LES with a one-equation RANS model 34 (Spalart and Allmaras, 1992). This one-equation model solves directly a modified eddy viscosity 35 rather than the turbulence kinetic energy as most one-equation models do. The d is the wall 36 distance, the fν1 and fν2 are the damping functions. Due to the space limit of the paper, a more 37 detailed description of these models is not possible. Since many of the models are available in 38 some commercial software, one could also refer to the user manual (e.g., FLUENT, 2005) for 39 detailed model descriptions. 40 1 Table 1. Coefficients and Source Terms for Eq. (1) φ Γφ,eff Sφ Constants and coefficients Reynolds Averaged Navier-Stokes (RANS) methods Reynolds 1 0 filtered u μ + μ −∂∂−ρβ−p/ x g (H H )/C 1u⎛⎞⎛⎞∂∂ij u 1u ∂∂ ij u i t ii opNote : S=+Ω=−⎜⎟⎜⎟ ; variables T μ/σ +μ /σ ij⎜⎟⎜⎟ ij T t T,t SH 2x⎝⎠⎝⎠∂∂ji x 2x ∂∂ ji x for (1)-(6) C μ/σC + μt/σC,t SC (1) 0-eq. − − − μt = CρVll ; C=− 0.03874; wall distance 2 GG− ρε + kT2 ∂ (2) RNG k μ + μt/σk,t kB μ t ==μ≡=βμσCμρ ;Gk t S ;S 2SS ij ij ;G B g i() t / T,t 2 ε ∂xi k-ε ε μ + μt/σε,t CGεε1kε− /kC 2 ρε /k Cε1=1.44, Cε2=1.92, Cμ=0.09, σT,t=0.9, σk,t=1.0, σε,t=1.3, σC,t=1.0 2 1/2 2 ⎛⎞ 2 ⎡ ⎤ ∂k μ t = fμμC ρk/ε ; fμ =−exp 3.4 /() 1 + Ret /50 ; D2=μt ⎜⎟; GGD− ρε + + ⎣ ⎦ ∂x (3) LRN- k μ + μt/σk,t kB ⎝⎠⊥ **2 2 LS ε μ + μ /σ 2 t ε,t CGεε1kε /kC−ρε+ 2 /k E 2uμμ⎛⎞ ∂ E = t//; CC* = ; CC*2=−⎡10.3expRe −⎤ ⎜⎟2 ε11ε εε22⎣ ( t)⎦ ρ∂⎝⎠x⊥ ρk1 * ραGk μ t = ; G=minG,10kk()ρβ kω ;Gω = ; ω ⎡⎤* μ max⎣⎦ 1/αω ,SF21 / a t (4) SST k μ + μt/σk GYkk− * 2 1k∂ ∂ω Yk =ρβ kω ; Yω =ρβω ; D21Fωω=−ρσ()1,2 ; k-ω ω μ + μt/σω GYDω − ωω+ ω ∂∂xxjj 4 * β+it/3 (Re/6) ρk **4/15+ (Re/8)t α= ; Ret = ; β=β∞ 4 ; 1(Re/6)+ t μω 1(Re/8)+ t ⎛⎞22⎛⎞ 22 Cv1 2 ′ Gvk ′ k μ fL−∇ f =⎜⎟ − + C2 +5 ; Tmax,6= ⎜⎟ ; Tk⎜⎟3 ρ kkT ε ρε ⎝⎠ ⎝⎠ Gk − ρε 3/4 k μ + μt/σk,t ⎡⎤3/2 2 k ⎛⎞μ −1/4 ⎧ k ⎫ (5) v2f- 2 LCmax=ε ,C ; 2 ε μ + μ /σ CGεε1kε− /kC 2 ρε /k L ⎢⎥η ⎜⎟ μρt = min⎨ 0.22vT′ ,0.09 ⎬ dav t ε,t ερ 2 ⎣⎦⎢⎥⎝⎠ ⎩⎭ε v′ μ + μt/σk,t S v′2 ⎛⎞2 Cε1=1.4⎜⎟ 1+ 0.05kv / ′ ; Cε2=1.9, C1=1.4, C2=0.3, Cμ=0.22, ⎝⎠ CL=0.23, Cη=70, σk,t=1.0, σε,t=1.3 ⎛⎞∂∂uu Puuuu≡−ρ ''ml + ' ' ; GguTguT=−ρβ '' + ' '; lm⎜⎟ l j m j lm( l l m m ) ⎝⎠∂∂x jjx (6) RSM- '' PG+ + φ -ε uulmμ + μt/σj lm lm lm lm IP ⎛⎞∂∂uu'' 2 ml φlm =+p⎜⎟;εlm= δ lmε ; ⎝⎠∂∂xlmx 3 Large Eddy Simulation (LES) (All variables are filtered) S S τ≡ijuu ij − uu ij; hTuTujjj≡−; 1 0 S (7) LES- −∂∂−∂τ∂p/ xiijj / x ui μ ⎛⎞∂u1∂u 2 Dyn S τ=SSμρδ⎜⎟i +j + τ ;μρ=Δ()C2SS T μ/σT −∂∂h/xjj ij t⎜⎟ kk ij t s ij ij ⎝⎠∂∂xx3ji Detached Eddy Simulation (DES) switches between a RANS (e.g.
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