Prediction Accuracy of Non-Linear k–ε Models for ...

The 22nd Annual International Conference on Mechanical Engineering-ISME2014 22-24 April, 2014, Mech. Eng. Dept., Faculty of Eng., Shahid Chamran University, Ahvaz, Iran.

ISME2014-0673

Prediction Accuracy of Non-Linear k–ε Models for Pollutant Dispersion around a Model Building Farzad Bazdidi-Tehrani1*, Akbar Mohammadi-Ahmar2, Mohsen Kiamansouri3 1 Professor (*corresponding author); [email protected] M.Sc. Student; [email protected] 3 Ph.D. Candidate; [email protected] School of Mechanical Engineering, Iran University of Science and Technology, Tehran 16846-13114, Iran. 2

Abstract A numerical simulation was conducted to assess prediction accuracy of various non-linear models for estimating concentration field around a cubical model building with a stack vent located on its roof. The results of these models were compared with SKE model and wind-tunnel data. Three nonlinear models, namely, Craft et al., Lien et al. and Rubinstein and Barton models were investigated where Craft et al. and Lien et al. models are cubic and Rubinstein and Barton model is quadratic. All the computations were performed by the use of the self-developed object-oriented C++ programming in OpenFOAM CFD package, which contains applications and utilities for finite volume solvers. Among the various models studied here, results of the SKE model for the concentration field were unfavorable, because it cannot reproduce the basic flow structure, such as the reverse flow on the roof. On the contrary, the non-linear models were able to predict the concentration field better than the SKE model due to inclusion of the quadratic and cubic terms. It can be said that concentrations predicted by all CFD models were less diffusive than those of the experiment, although the non-linear k–ε models have reduced this difference. Keywords: Pollutant dispersion, Numerical simulation, OpenFOAM, Non-linear k-ε models, Model building

Introduction Computational Fluid Dynamics (CFD) is increasingly explored and used to predict wind flow and pollutant dispersion around buildings. The flowfield and dispersion around a model building consist of very complex phenomena such as impingement, stagnation, separation, reattachment, recirculation, favorable and adverse pressure gradients and streamline curvature. In turbulent flows, dispersion can be seen as the combination of the molecular, convective and turbulent mass transport, where the first one is often negligibly small compared with the others. Eddy-viscosity models based on the isotropic Boussinesq hypothesis, for example the standard k–ε of Jones and Launder [1] are suffering from numerous weaknesses, including an inability to resolve normal stress anisotropy, incorrect or insufficient sensitivity to system rotation, swirl, and streamline curvature, secondary strains and

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overproduction of turbulent kinetic energy at impingement zones. The excessive values of turbulent kinetic energy are caused by overestimation of the turbulence production term, which is due to the use of the isotropic eddy viscosity concept in a highly anisotropic flow field. In another word, since the viscosity is a scalar; each component of the stress influences its strain to the same amount. In the non-linear k–ε models, additional nonlinear terms are added into the stress-strain relation, making the Reynolds stresses a more general function of all strain components and therefore, disadvantages of the standard k–ε are improved by them. The Non-linear models were first introduced by Lumley [2]. The first applicable quadratic non-linear model was proposed by Speziale [3] based on the principle of material frame indifference, realizability and necessary invariance requirements, which contains the Oldroyd convective derivative of the mean strain rates. Another non-linear quadratic model has been derived by Rubinstein and Barton [4] based on the renormalization-group approach by evaluating the Reynolds stresses to the second order in the ε expansion of the Yakhot-Orszag theory [5].In addition, in order to account for streamline curvature and flow rotation, cubic models are required. Craft et al. [6] proposed a new non-linear model for low-Reynolds number turbulent and transitional flows in which strain and vorticity tensors to cubic level are maintained. The Lien et al. model [7] is a cubic extension of the high-Re model of Shih et al. [8] to low-Re conditions. The application of these non-linear turbulence models for simulating the wind flow around buildings have not been investigated considerably. Wright and Easom [9] presented a comparison of various non-linear turbulence models against full-scale measurements (Instead of windtunnel scale) to simulate the wind flow around a full-scale building. Moreover, Bazdidi-Tehrani et al. [10] and BazdidiTehrani and Jadidi [11] investigated the effect of sub-grid scale turbulent Schmidt number and various sub-grid scale models in the LES approach for the flow field and pollutant dispersion around a cubical model building. However the previous studies considered only the flow field around buildings and the pollutant dispersion simulation using nonlinear turbulence models has not been studied yet. The purpose of this study is to confirm the accuracy of various non-linear k–ε models in modeling dispersion near and around

a simple model building and to clarify the mechanism of the discrepancy in relation to the linear k–ε computations.

Moreover, the turbulent eddy viscosity is given by: t  C 

Governing equations The time averaged governing equations consist of continuity, momentum, concentration and turbulent equations of k–ε which are presented below. Continuity equation: u i 0 x i

(1)

Momentum equations:





u i u i P  u iu j  2u i u j  Bi    t x i x i x j x 2j

(2)

Concentration transport equation and k–ε equations: c c  2c  u j D  u ' c 2 t x j x j x j j

Dk   Dt x i

D   Dt x i

(3)

(9)

In the standard k–ε model, the coefficient C  is usually set a constant and is equal to 0.09. Quadratic and Cubic Non-linear k–ε Models Quadratic and Cubic terms in non-linear k–ε models were provided for the better description of anisotropic turbulence structures and streamline curvature in the numerical prediction of pollutant dispersion around a model building. In this paper, two different cubic non-linear k–ε models i.e. Craft et al. [6] (hereafter CLS model) & Lien et al. [7] (LCL model) and one quadratic non-linear k–ε model i.e. Rubinstein and Barton [4] (RB model) and also the standard k–ε model [1] (SKE model) have been studied. The coefficients C1  C 7 and C  for non-linear k–ε models have been illustrated in "Table 1". Table 1. Coefficients in the non-linear turbulence models

   k    t    Pk    k  x i   

    t  

k2

RB model[4]

CLS model[6]

0.0845

 1.2  min  ,0.09 1  3.5 

0.228

0.4

C

(4)

C1

    2   C  1 Pk  C  2  k k  x i 

(5)

C2

 2   ij  2 t S ij 3 k

(6)

Since the standard k–ε model does not take into account the anisotropy, this shortcoming can be overcome to some extent by introducing a non-linear expression in the constitutive equation as follows [8]: u i' u 'j k



  2  ij  2 t S ij  C 1 t 3 k

 C 2 t Ωik S kj 

C 4t

k 2

1   S ik S kj  3 S kl S kl  ij     1    Ω jk S ki   C 3 t Ωik Ω jk  Ωlk Ωlk  ij  3  

C 5t

(7)

Where the strain rate tensor S ij and vorticity tensor Ωij are defined by: S ij 

1  u i u j   2  x j x i

 u j 1  u  , Ωij   i    2  x j x i 

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   

0.4

15 c

(8)

0.188

1.04

19 c

C4

0

2 80c

1000  S 3 80c 2

C5

0

0

0

C6

0

2 40c

2 16c 

C7

0

2 40c

2 16c 



Another important feature of non-linear models is the dependence of eddy viscosity coefficient C  on the dimensionless strain rate and vorticity invariants which are defined as: S 

S ki Ωlj  S kj Ωli  S kl  

k  2  Ω Ω S  S il Ωlm Ω mj  S lm Ω mn Ω nl ij  2  il lm mj 3   k k C 6t 2 S ij S kl S kl  C 7t 2 S ij Ω kl Ω kl

0.048

1000  S 3

In the standard k–ε model, Reynolds stress tensors are evaluated by the linear constitutive equation, k

3 c 1000  S 3

C3

u i' u 'j

LCL model[7] 2/3 1.25  S  0.9

k

2S ij S ij

Ω

k

2Ωij Ωij  ,     max S ,Ω 

(10)

Computational domain and boundary conditions The present computational domain, on the basis of the experimental setup of Li and Meroney [12], is depicted in "Figure 1". The computational domain dimensions are 26H b  x   6H b  y  13H b  z  in the streamwise, vertical and lateral directions respectively. The computational grids used in the present study are the same as those in our previous work. Bazdidi-Tehrani et al. [13] investigated several grid resolution assessment techniques on prediction accuracy of large eddy simulation (LES) of dispersion around an isolated cubic building. The grid discretization is illustrated in "Figure 2". All the geometry and flow parameters have been summarized in "Table 2". For RANS simulations, the mean velocity profile

and the turbulence quantities at the inlet are required. The vertical distributions of the longitudinal velocity and turbulent intensity at the inflow were set based on the wind tunnel experiment of Li and Meroney [12] as shown in "Figure 1". The vertical profile of the mean streamwise velocity at the inlet u in  y  approximately obeys a power law with the exponent of 0.19 i.e. u in  y  U

y     

0.19

(11)

Where U  and  are defined in "Table 2" and the other velocity components were assumed to be zero at the inlet. The turbulent kinetic energy k and turbulent dissipation rate  at the inlet are calculated from velocity u in  y  and turbulent intensity I in  y  as:



k in  y   u in  y   I in  y 



2

1

 in  y   Pk  y   C  2 k in  y 

(12) u in  y  y

(13)

Where C  0.09 . The standard wall functions based on logarithmic law for a smooth wall are used for the building and ground surfaces. Other boundary conditions are shown in "Figure 1". Numerical Method All the numerical computations in the present study were performed by the use of the self-developed object-oriented C++ programming in OpenFOAM CFD package, which contains applications and utilities for finite volume solvers. In this study the PISO (Pressure-Implicit with Splitting of Operators) algorithm is used for pressure-velocity coupling in transient calculations. For the spatial derivatives, all the transport equations are discretized using second-order centraldifferencing schemes. The implicit Euler scheme was used for the time advancement in the governing equations. The turbulent Schmidt number Sct was set to the optimum value of 0.7, which has given the most moderate results in the previous studies [14-15].

Figure 1. (a) Domain and boundary conditions (b) inflow profiles according to experiment [12] Table 2. Geometry and flow parameters according to the experiments (Li and Meroney [12]) Quantity Building height Inlet velocity at the building height Free stream velocity Boundary layer thickness Reynolds number

Symbol

Exhaust Vent diameter Effluent momentum ratio Pollutant release rate Contaminant exit velocity

Ve

Value

Hb

0.05 m

U Hb

3.3 m / s

U

4.5 m / s



0.3 m

Re Hb

d

1.1  104 0.005 m

M

0.19

Qe

12.5  106 m 3 / s 0.637 m / s

Figure 2. Cross-section grids at different plane, (a) at z / H b  0 , (b) at x / H b  0 and (c) at the model building's roof

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Results and Discussion Mean concentration distributions The distribution of K on the centerline of the roof and leeward wall is shown in "Figure 3(a)". In this study, dimensionless concentration K was defined as: (14) c c K 

c0



Qe / H b2U b

The SKE model fails to represent the concentration upwind of the vent, while it overestimates the concentration downwind of the vent in comparison with the experiments and the nonlinear models. However, on the leeward wall, similar distributions of K are predicted by all the models. The differences among the non-linear models compared are not so large in the streamwise direction. The computed concentrations behind the model building in all CFD models are over-predicted in comparison with the experiment. "Figure 3(b)" shows the K distributions on the roof and sidewall of the model building in the lateral direction. On the sidewall, all the CFD results generally under-predict the concentration measured by the experiment, i.e., the diffusion

in the lateral direction is still underestimated. However, in the area around the edge, the CLS result shows a larger concentration than the other models, and it is closer to the experiment. In general, the CLS results are the best and the SKE results are the worst in terms of reproducing the concentration of exhaust on the roof and the walls of the building. "Figure 4" shows the distribution of non-dimensional concentration K at the cross-section in the near wake region of the model building. Results of the three non-linear k–ε models are very similar, but it can be said that the CLS model is more widened in the spanwise direction. Therefore, only the CLS results are shown along with those from SKE and the experimental data. A peak concentration at a little higher than the building height region is common in the results of the experiment and CFD simulations. However, the concentrations in the central region are much higher and the horizontal spreads are smaller in CFD than the experiment. Non-linear models, especially the CLS model show a more diffusive concentration in the horizontal direction in comparison with SKE, and they are closer to the results of the experiment.

Figure 3. Distribution of dimensionless concentration K on the centerline of roof and walls in the (a) streamwise and (b) lateral direction

x / Hb  1

x / Hb  2

x / Hb  3

Figure 4. Contours of dimensionless concentration K in the near wake region, (a) Exp, (b) CLS, (c) SKE

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Scalar fluxes distribution SI system of units is deemed to be used. If necessary use the equivalent value in the other system of units in brackets after the SI system of units. Scalar transport of concentration consists of convection and turbulent diffusion effects, which are expressed by the convection as the mean scalar fluxes u i c and the turbulent diffusion fluxes uic  , respectively. The convection fluxes can be estimated by using mean velocities and mean concentration. The turbulent diffusion fluxes in RANS models are modeled by the gradient diffusion hypothesis: (16) u ' c    t / Sct   c / x i  i

Where  t is eddy viscosity and Sct is turbulent Schmidt number. "Figure 5" compares contours of convection flux u i c and turbulent diffusion flux uic  on the roof obtained by CLS and SKE models. The negative region of uc in CLS is much larger than that in SKE, because the reverse flow on the roof in CLS is stronger than that in SKE. On the other hand, the turbulent diffusion flux u c  for SKE and CLS models are similar, and the values in both models are rather small in comparison with the convection flux.

uc

u c 

The result of CLS for wc shows that there are two sharp peaks in the area above and near the stack position, but in SKE, peak values are just adjacent to the stack. On the other hand, the peak values of wc for both model are identical. By contrast, in SKE, large values of flux almost a little are widely spread in the lateral directions on the roof. Conclusions In this study, the proficiency of three non-linear models compared with SKE for predicting concentration field around a cubical model building with vent emission in the neutral turbulent boundary layer was examined. The main findings may be summarized as follows:  Among the various models studied here, results of the SKE model for the concentration field were unfavorable, because it cannot reproduce the basic flow structure, such as the reverse flow on the roof. By contrast, the non-linear models because of the inclusion of the quadratic and cubic terms were able to predict the concentration field better than the SKE model.  It can be said that concentrations predicted by all the CFD models were less diffusive than those of the experiment, although the non-linear k–ε models have reduced this difference.  By comparing the scalar fluxes distribution, we can say that the turbulent diffusion for SKE and CLS models are similar, and the values in both models are rather small in comparison with the convection flux.

wc

wc

Figure 5. Contours of convection flux u i c and turbulent diffusion flux uic  on the roof obtained by (a) CLS, (b) SKE

References [1] Jones, W. P., Launder, B. E., Spalding, 1972. “The prediction of laminarization with a two-equation model of turbulence”. Int. J. Heat Mass Transfer, 15, pp. 301-314.

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[2] Lumley, J. L., 1970. “Toward a turbulent constitutive relation”. J. Fluid Mech., 41, part 2, pp. 413-434. [3] Speziale, C. G., 1987. “On non-linear k-I and k-e models of turbulence”. J. Fluid Mech., 178, pp. 459-475.

[4] Rubinstein, R., Barton, J. M., 1990. “Nonlinear Reynolds stress models and the renormalization group”. Phys. Fluids A 8, pp. 1472-1476. [5] Yakhot, V., Orszag, S. A., Thangam, S., Gatski, T. B., Speziale, C. G., 1992. “Development of turbulence models for shear flows by a double expansion technique”. Phys. Fluids A 4, pp. 1510–1520. [6] Craft, T. J., Launder, B. E., Suga, K. 1996. “Development and application of a cubic eddy-viscosity model of turbulence”. Int. J. Heat Fluid Flow, 17(2), pp. 108-115. [7] Lien, F. S., Chen, W. L., Leschziner, M. A., 1996. LowReynolds-number eddy-viscosity modelling based on non-linear stress-strain/vorticity relations, in: W. Rodi (Ed.), Engineering Turbulence Modelling and Experiments 3, Elsevier, Amsterdam, pp. 91-100. [8] Shih, T. H., Zhu, J., Lumley, J. L., 1993. “A realizable Reynolds stress algebraic equation model”. NASA Tech Memo, 105993, August, pp. 1-34. [9] Wright, N. G., Easom, G. J., 2003. “Non-linear turbulence model results for flow over a building at full-scale”. Appl. Math Model, 27(12), pp. 1013-1033. [10] Bazdidi-Tehrani, F, Jadidi, M, Khalili, H, Karami, M., 2011. “Investigation of effect of subgrid scale turbulent Schmidt number on pollutant dispersion”, In: Proceedings of the 22nd International Symposium on Transport Phenomena, Delft University. Delft, the Netherlands. [11] Bazdidi-Tehrani, F., Jadidi, M., 2013. "Large eddy simulation of dispersion around an isolated cubic building: evaluation of localized dynamic kSGS-equation sub-grid scale model". Environ. Fluid Mech., pp. 1–25. [12] Li, W. W., Meroney, R. N., 1983. “Gas dispersion near a cubical model building. Part I. Mean concentration measurements”. J. Wind Eng. Ind. Aerodyn. 12, pp.15– 33. [13] Bazdidi-Tehrani, F., Ghafouri, A., Jadidi, M., 2013. “Grid resolution assessment in large eddy simulation of dispersion around an isolated cubic building”. J. Wind Eng. Ind. Aerodyn. 121, pp. 1–15. [14] Tominaga, Y., Stathopoulos, T., 2007. “Turbulent Schmidt numbers for CFD analysis with various types of flow field”. Atmos. Environ. 41, pp. 8091–8099. [15] Tominaga, Y., Stathopoulos, T., 2010. “Numerical simulation of dispersion around an isolated cubic building: model evaluation of RANS and LES”. Build. Environ. 45, pp. 2231–2239.

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