A two-time-scale turbulence model for compressible flows: Turbulence dominated by mean deformation interaction

Abstract

The multiple-time-scale concept is applied to develop a turbulence model for compressible flows. Transport equations for the turbulent kinetic energies and the energy transfer rates are linked to each domain of the turbulent spectrum. The model coefficients are calibrated, with respect to simple flows, by using a new method which takes advantage of the spectral character of the model. One innovation of this method is to use, as a component, the CG model [V. M. Canuto and I. Goldman, Phys. Rev. Lett. 54, 430 (1985)] which gives the large scale spectrum as a function of the instability-generating turbulence. Then, the two-time-scale model, with its complete set of coefficients, has been successfully applied to the simulation of plane mixing layers and homogeneous shear flows. A significant issue of this work is the study of the behavior of the two-time-scale model when a shock wave interacts with a homogeneous turbulence. We first compare model results with experimental data for a 2.8 Mach number interaction [D. Alem, Ph.D. thesis, Université de Poitiers, 1995]. The decrease of the integral length scale, predicted by the linear analysis, is reproduced with the two-time-scale model, which, moreover, recovered the rate of reduction measured by Alem. The amplification of the turbulence level through the shock wave is also consistent with the measurements. Then, we confront our results with a direct numerical simulation of the shock–turbulence interaction at $\text{M=1.2}$ [S. Lee et al., J. Fluid Mech. 251, 533 (1993)]. The spectrum of the turbulence injected in the inflow region of the direct numerical simulation appeared to be far from the freely decaying state. The two-time-scale model, which accounts for the spectral nonequilibrium effects, is able to recover the spatial decrease of turbulence in the inflow region whereas a single-time-scale model fails. Moreover, the profiles for the turbulent kinetic energy and its dissipation rate over all the calculation domain are much better reproduced with the two-time-scale model than with the primary $\mathrm{k–\epsilon }$ model.