A Comparative Study for Compressible Turbulence Models

The incompressible models for the pressure-strain correlation are unable to correctly predict the turbulence flows evolving with significant compressibility. Huang and Fu use a damping function of the turbulent Mach number to modify two numerical coefficients of the incompressible model for the pressure strain developed by Launder, Reece and Rodi. This model predicts the spreading rate and the shear stress behavior in compressible turbulent mixing well. However, the model does not show the well-known compressibility effects on the compressible homogenous shear flow. In the present work, the model of Huang-Fu is revised, all resulting model coefficients become dependent on the turbulent Mach number, the gradient Mach number and the convective Mach number. The proposed model is tested in different compressible turbulent homogeneous shear flow and mixing layers cases. In general, the predicted results from the proposed model are in an acceptable agreement with DNS and experiment data.


Introduction
Many studies carried out in the last decade have shown that the compressibility has important effects on turbulence flows. One of the well-known effects is the reduction of the turbulent kinetic energy redistribution phenomenon. Extensive studies [1]- [6] have been conducted to study the compressible homogeneous shear flow. In this context, the DNS results [1] [3] [5] are selected as the basic documents to understand some physical discrepancies of the compressibility effects on homogeneous turbulent shear flows. The analysis of these DNS results suggests that  [5] indicates that compressibility effects cause a notable reduction in the growth rate of the turbulent kinetic energy. So the turbulence modeling is essential for predicting compressibility effects in agreement with numerical and experimental data.
According to the DNS results [1] [5], the structural compressibility effects cause reduction of the pressure field and then the reduction of the pressure strain correlation, which leads to the dramatic changes in the magnitude of the Reynolds stress anisotropy components. The experiments [7] [8] and the DNS results [9] [10] [11] reached similar conclusions concerning the role of pressure in the developed compressible plane mixing layers. As a consequence, in several studies, the pressure strain modeling has been considered as important attractive research in the second-order closure for the compressible turbulence flows. Different compressibility corrections using turbulent Mach number and gradient Mach number of the incompressible pressure strain model have been proposed by several authors [12] [13] [14] [15] [16]. At compressibility effects levels, the models have well predicted different characteristic parameters in some turbulence configurations. However, they failed in other configurations for which the compressibility is higher.
In this study, a revision of the model of Huang and Fu [12] of the pressure strain correlation is made making all coefficients depend on the turbulent Mach number, the gradient Mach number and the convective Mach number. The ability of Huang et al. [12] and the proposed model to predict compressible homogeneous shear and mixing layers are examined by considering different initial conditions.

Comparison between the predictions models with DNS and experiments data
shows that the proposed model describes better compressibility effects on the turbulence.

Basic Equations
Classically, The basic equations describing the motion of a compressible flow are the Navier-Stokes energy and state equations which can be written as follow for continuity, momentum and energy conservation equations [15] [16]: where, p RT where For the compressible turbulence, the commonly Favre Reynolds stress models is used in this study to describe the Reynolds stress ij i j R u u ρ ρ ′′ ′′ = as follow [16]: where the symbols ij Pr , ij D , ij ϕ , ij ε and ij V , represent turbulent production, turbulent diffusion, pressure strain correlation, turbulent dissipation and the mass flux variation respectively.
Here, the isotropic model for the dissipation tensor ij ε , is used [16]: And the dissipation rate ε , is written as in [3]: where for homogeneous shear flow turbulence The compressible dissipation rate c ε , is determined by the models as [3]: α is a numerical coefficient model, . ( )

Compressible Turbulence Models for the Pressure Strain
where 1 Model of Huang and Fu [12] Huang and Fu [12] use a damping function in compressible mixing layers to modify the LRR model for the pressure strain as follows: ( )

A Proposed Model
The compressibility extension of the incompressible LRR [17] model developed by Huang et al. [12] explicitly involve the turbulent Mach number. As can be seen in Equation15, the model of Huang et al. [12] is based on a compressibility correction of the coefficients C 3 and C 4 , which affect the polynomial linear term of the Reynolds stress and the mean strain rate, the other coefficients, C 2 , which affects the mean strain rate, and C 1 of the return to isotropy model, are conserved as in the LRR model, without any compressibility correction. On the other hand, different analyses have been carried to show the influence of the pressurestrain on the Reynolds stress behavior. Hamba [2] presented a fine analysis for the compressible homogeneous shear flow case, and confirmed that the reduction of the transverse component P 22 of the pressure-strain correlation principally caused the reduction of the transverse Reynolds stress R 22 , which in turn induced a systematic reduction of the shear Reynolds stress, the streamwise component P 11 of the pressure-strain, and then the growth rate of the turbulent kinetic energy. Thus, the compressibility correction of the coefficients C 3 and C 4 seem to be sufficient to capture compressibility effects. In this context, according to Park et al. [14] and Huang et al. [12], in addition to the compressibility correction of the coefficients C 3 and C 4 , the coefficient C 2 should be corrected with compressible parameters, such as M t and M g , or others. One can see that C 2 directly affects the shear component P 12 of the pressure-strain, which has an evident contribution in the transport equation for the Reynolds shear stress, R 12 . On the other hand, the reduction of P 12 , which works as a sink term in the transport equation for R 12 , leads to an increase in the growth rate of the turbulent kinetic energy via the growth of R 12 . This is not suitable using model. So, more attention should be paid to the modeling for P 12 . Khlifi et al. [16] considered an equation of the dilatation fluctuation to modify the incompressible C 2 and the retour to isotropy C 1 -coefficients [16], as follows: Equation (16) [16], came from the isotropic model of the turbulent dissipation rate s c ε ε to distinguish between low-M t and high-M t regimes. But this link seems to be not adaptable to Huang et al. [12] in calculation of the mixing layers. For this modeling, the model [3] for s c ε ε which linked to 2 t M was chosen in this study. Thus, all of the coefficients of the Huang et al. [12] model are expressed as a function of M t , M g and M c . Considering Equations (11) and (12), the proposal coefficients models are summarized in the next section.

Simulation of Compressible Homogeneous Shear Flow
For compressible homogeneous shear flow, the mean velocity gradient is given by: where, S is the mean shear rate.
The Favre averaged basic second order model equations are: As we can see the above models are parameterized by turbulent Mach number, such a parameter is described by the transport equation as follow [16]: where Models [3] are chosen for the dilatational terms: Results and Discussion In this section, the fourth-order Runge-Kutta numerical scheme is used to numerically solve the averaged transport Equations (19), (20) and (21)  and Fu [12] and with the DNS [5] results, the detailed of the models are listed in Table 1.     All the cases of DNS [5] correspond to different initial conditions listed in Table 2. We consider two DNS [5] cases: A1 and A4 which correspond respectively      This is due to the approach modeling of the model 1 [14]. The results for case A 1 are shown in Figure 3, Model 1 and model 2 results are in disagreement with the DNS data for the normal stress anisotropies b 11 and b 22 , but satisfy the shear stress component b 12 . For high compressibility as in case A 4 , the DNS data show a strong anisotropy changes in Reynolds stress magnitude. Figure 4 shows the predicted results for case the models 1 and 2 are still unable to predict correct behavior of the normal Reynolds stress anisotropies. As can be seen in Figure 3 and Figure

Simulation of Compressible Mixing Layers
The flow is governed by the averaged Navier-Stokes equations associated to those describe the energy, the Reynolds stress and the turbulent dissipation. The simplest of resulting continuity, momentum and energy equation for stationary mixing layers can be written as: The Reynolds stress is solutions of the follow equation: In the above mentioned transport equations, different terms should be modeled, the gradient diffusion hypothesis is used to represent: -The turbulent heat flux [19]: -The diffusion term [19]: Results and Discussion In this study, computation of two free streams of a fully developed compressible mixing layer (see Figure 7) is examined. The flows are characterized typically by the parameters r U U = , are respectively the density and velocity ratios, the experiment conditions of Goebel et al. [7] are listed in Table 3.
The Equations (23)-(29) are solved using a finite difference scheme. The grid of computational physical domain which is rectangular box defined by the set of point (x, y) has 6666x41 points. The initial profiles for s ε , ρ and T % which are not available in the experiment of Goebel et al. [7] are generated as: -The initial profile of the turbulent dissipation is determined from the turbulent viscosity model. Figure 7. Turbulent mixing layer.
-The state equation of perfect gas is used to determine the initial profile of the density.
The values of the constants models used in the present simulation are: According to Sarkar [5], homogeneous shear flow is closely related to the mixing layers, this allows M g to be connected to M c . Thus, the coefficients C i of the Huang et al. [12] is expressed as a function of the turbulent Mach number and the convective Mach number as in Table 4: The normalized stream mean velocity The calculated velocity profiles with the models 1 and 2 are in reasonable Table 4. Numerical coefficients of the pressure strain model. Model Model 1: Huang et al. [12] 3.        Figure 12. It can be seen that the model1does not reproduce the decrease of these turbulent quantities the reduction of this term with increasing M c is slightly than in DNS results [6] [11]. However, the pressure strain reduction which is the main responsible for the reduction of production term and of the shear layer growth rate appear to be accurately captured by the proposal model 2. Therefore, the convective Mach number is concluded to be important in addition with the turbulent Mach number for modeling the pressure strain in turbulent mixing layers.

Conclusion
In this study, the model of Huang and Fu gives results which do not reflect anisotropy, it gives very poor predictions of the changes in the normal Reynoldsstress anisotropy magnitude but predicts reasonable behavior of shear stress anisotropy and the ratio time scale. A revision of this model of Hung has been proposed to reflect compressibility effects. Application of model 2 to predict compressible homogeneous shear flow shows satisfactory agreement with available DNS [5]. Model 2 appears to be able to predict accurately the structural compressibility effects on homogeneous shear flow as the significant decrease in the magnitude of the Reynolds shear stress and the reduction of the pressures strain components with increasing initial values of the gradient Mach number. Also, model 2 successfully predicts the changes in the compressible mixing layers. Therefore, a priori, blending between compressible models is found to be an important issue in the modeling of the pressure-strain correlation. World Journal of Mechanics .