**Author details**

410 Numerical Simulation – From Theory to Industry

**7.3. Results** 

**8. Conclusion** 

**7.2. Fujiwara and Arakawa model** 

Linear part of this model has the following form:

The model suggested by Fujiwara and Arakawa (1995) for the pressure-dilatation correlation takes into account the compressible part (dilatational) of the kinetic energy ( <sup>2</sup>

where <sup>1</sup> *C .*  0 3 and *ε* is the turbulent dissipation rate. *ij b* is the Reynolds anisotropy tensor.

Comparison between RDT and Fujiwara and Arakawa (1995) model results concerning the pressure-dilatation correlation term Π*d* are represented in Figures 17(a), (b), (c). As one can remark from these results, that the contribution of Π*d* obtained by this model is negligible. Except in the case of the intermediate regime (*Mg*0 = 12), the Fujiwara and Arakawa model

**Figure 17.** Evolution of the pressure dilatation correlation term Π*d* (a) case A0 (*Mg*0 = 1), (b) case A4 (*Mg*0 = 12) and (c) case A10 (*Mg*0 = 66.7). : RDT results, : Fujiwara and Arakawa model.

Rapid distortion theory (RDT) is a computationally viable option for examining linear compressible flow physics in the absence of inertial effects. Evolution of compressible homogeneous turbulence has been described completely by finding numerical solutions obtained by solving linear double correlations spectra evolution. Numerical integration of these equations has been carried out using a second-order simple and accurate scheme. This numerical method has proved more stable and faster and allows in particular to obtain accurately the asymptotic behavior of the turbulence parameters (for large values of *St*) characteristic of equilibrium states. In this chapter, RDT code developed by authors solves linearized equations for compressible homogeneous shear flows (Riahi & Lili, 2011). It has been validated by comparing RDT results with direct numerical simulation (DNS) of Simone et al. (1997) and Sarkar (1995) for various values of initial gradient Mach number *Mg*0 which is the key parameter controlling the level of compressibility. The study of the behavior of the

does not represent an appreciable discrepancies with the RDT results.

*<sup>q</sup> <sup>U</sup> <sup>C</sup> <sup>ρ</sup>q b*

2 <sup>1</sup> <sup>2</sup> <sup>Π</sup> , *<sup>d</sup> <sup>i</sup> d ij*

2

*q x*

*j*

*<sup>d</sup> q* ).

Mohamed Riahi and Taieb Lili

*Faculté des Sciences de Tunis, Département de Physique, Laboratoire de Mécanique des Fluides, Campus Universitaire, Manar II, Tunis, Tunisia* 
