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

The multiple-time-scale concept is applied to develop a turbulence model fo r compressible flows. Transport equations for the turbulent kinetic energie s and the energy transfer rates are linked to each domain of the turbulent spectrum. The model coefficients are calibrated, with respect to simple flo ws, by using a new method which takes advantage of the spectral character o f the model. One innovation of this method is to use, as a component, the C G model [V. M. Canuto and I. Goldman, Phys. Rev. Lett. 54, 430 (1985)] whic h gives the large scale spectrum as a function of the instability-generatin g turbulence. Then, the two-time-scale model, with its complete set of coef ficients, has been successfully applied to the simulation of plane mixing l ayers 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 interac ts with a homogeneous turbulence. We first compare model results with exper imental data for a 2.8 Mach number interaction [D. Alem, Ph.D. thesis, Univ ersite de Poitiers, 1995]. The decrease of the integral length scale, predi cted by the linear analysis, is reproduced with the two-time-scale model, w hich, moreover, recovered the rate of reduction measured by Alem. The ampli fication of the turbulence level through the shock wave is also consistent with the measurements. Then, we confront our results with a direct numerica l simulation of the shock-turbulence interaction at M=1.2 [S. Lee , J. Flui d Mech. 251, 533 (1993)]. The spectrum of the turbulence injected in the in flow region of the direct numerical simulation appeared to be far from the freely decaying state. The two-time-scale model, which accounts for the spe ctral nonequilibrium effects, is able to recover the spatial decrease of tu rbulence in the inflow region whereas a single-time-scale model fails. More over, the profiles for the turbulent kinetic energy and its dissipation rat e over all the calculation domain are much better reproduced with the two-t ime-scale model than with the primary k-epsilon model. (C) 1999 American In stitute of Physics. [S1070-6631(99)02412-5].