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In this study, a Reynolds stress closure, including the Pantano and Sarkar model of the mean part of the pressure-strain correlation is used for the computation of compressible homogeneous at high-speed shear flow. Several studies concerning the compressible homogeneous shear flow show that the changes of the turbulence structures are principally due to the structural compressibility effects which significantly affect the pressure field and then the pressure-strain correlation. Eventually, this term appears as the main term responsible for the changes in the magnitude of the Reynolds stress anisotropies. The structure of the gradient Mach number is similar to that of turbulence, therefore this parameter may be appropriate to study the changes in turbulence structures that arise from structural compressibility effects. Thus, the incompressible model of the pressure strain correlation and its corrected form by using the turbulent Mach turbulent only, fail to correctly evaluate the compressibility effects at high shear flow. An extension of the widely used incompressible Launder, Reece and Rodi model on compressible homogeneous shear flow is the major aim of the present work. From this extension, the standard coefficients C i become a function of the extra compressibility parameters (the turbulent Mach number M t and the gradient Mach number M g ) through the Pantano and Sarkar model. Application of the model on compressible homogeneous shear flow by considering various initial conditions shows reasonable agreement with the DNS results of Simone et al. and Sarkar. The observed trend of the dramatic increase in the normal Reynolds stress anisotropies, the significant decrease in the Reynolds shear stress anisotropy and the increase of the turbulent kinetic energy amplification rate with increasing the gradient Mach number are well predicted by the model. The ability of the model to predict the equilibrium states for the flow in cases A 1 to A 4 from DNS results of Sarkar is examined, the results appear to be very encouraging. Thus, both parameters M t and M g should be used to model significant structural compressibility effects at high-speed shear flow.

Previous studies of compressible flows carried out in the past few years have shown that the pressure-strain is the main indicator of the structural compressibility effects. Undoubtedly, this terms plays a key role toward strongly changing magnitude of the turbulent Reynolds stress anisotropy. On the other hand, the incompressible models of the pressure-strain correlation have not correctly predicted compressible turbulence at high speed shear flow. Consequently, a correction of these models is needed for precise prediction of compressibility effects. In the present work, a compressibility correction of the widely used incompressible Launder Reece and Rodi model making their standard coefficients dependent on the turbulent and convective Mach numbers is proposed. The ability of the model to predict the developed mixing layers in different cases from experiments of Goebel and Dutton is examined. The predicted results with the proposed model are compared with DNS and experimental data and those obtained by the compressible model of Adumitroiae et al. and the original LRR model. The results show that the essential compressibility effects on mixing layers are well captured by the proposed model.

The study of the phenomenon of compressibility for modeling to second order has been made by several authors, and they concluded that models of the pressure-strain are not able to predict the structural evolution of the Reynolds stress. In particular studies and Simone Sarkar et al., a wide range of initial values of the parameters of the problem are covered. The observation of Sarkar was confirmed by the study of Simone et al. (1997,“The Effect of Compressibility on Turbulent Shear Flow: A Rapid Distortion Theory and Direct Numerical Simulation Study,” J. Fluid Mech., 330, p. 307;“Etude Théorique et Simulation Numérique de la Turbulence Compressible en Présence de Cisaillement où de Variation de Volume à Grande Échelle” thése, École Centrale de Lyon, France). We will then use the data provided by the direct simulations of Simone to discuss the implications for modeling to second order. The performance of different variants of the modeling results will be compared with DNS results.

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