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This chapter develops proposals for an evaluation coupled second order model of SSG-LRR on an homogeneous turbulence submitted to an inclined shear for high and low stratification. The effect of Shear number on thermal and dynamic turbulent fields of the problem is performed for Shear number fixed at 2, 6, 14 and 20. Two values of Froude number equal to 0.35 and 1.29 are adopted for all numerical simulations corresponding to high and low stratification, respectively. For all simulations, value of angle theta is fixed at θ = π/4 corresponding to the angle between the shear and the vertical gradient of stratification. SSG-LRR model is adopted to compute turbulent parameters of principal component of anisotropy b12, normalized turbulence dissipation ε/KS and the density flux ρu1¯. A good agreement is detected by comparison of findings via model of SSG-LRR with the reported results in the literature by Direct Numerical Simulation of Jacobitz (DNSJ). It is found that the variation of Shear number predict a very strong influence on thermal and dynamic turbulent characteristics. Hence, findings with SSG-LRR model prove the existence of an asymptotic equilibrium states for various thermal and dynamic parameters in particularly for a low stratification.

The present investigation is carried out to reveal Richardson number (Ri) effects on an homogeneous and stratified turbulence under horizontal shear. The problem is simulated via Lagrangian Stochastic model (LSM). Hence, the method of Runge Kutta with fourth order is adopted for the numerical integration of three differential systems under non linear initial conditions of Jacobitz (2002) and Jacobitz et al. (1998). This study is performed for Ri ranging from 0.2 to 3.0. It has been found that computational results by the adopted model (LSM) gave same findings than that of preceding works. It has been shown a global tendency of different parameters governing the problem to equilibrium asymptotic states for various values of Ri. The comparative study between the computations of the present LSM and direct numerical simulation of Jacobitz demonstrates a good agreement for both methods for the ratios of; potential energy Kθ/E and kinetic energy K/E toward the total energy E and the principal component of anisotropy b 12 It has been found that Ri is the most important parameter affecting the thermal and dynamic fields of the flow. Hence, increase Ri conduct to increase the uniform stable stratification and decrease for the uniform mean shear S. It can be concluded that Ri is a main non-dimensional parameter which enable us to understand physical phenomenons produced inside stratified shear flows.