Eddy viscosity turbulence models for compressible mixing

The K - L and K - ϵ turbulence models are used to simulate the turbulent mixing induced by the Rayleigh-Taylor and Richtmyer-Meshkov instabilities. The models contain additional source terms for the turbulence kinetic energy which depend on the type of the instability. A new criterion based on ratio of the averaged flow and turbulence time scales is introduced for differentiating between the two types of instabilities. The original formulation of the turbulence kinetic energy source present in the K - ϵ model is modified in order to accurately capture the evolution of the Richtmyer-Meshkov instability in both heavy/light and light/heavy configurations. Additional constraints are imposed to the models in order to prevent non-physical solutions when strong gradients are present in the flow. Three test problems are considered and the performance of the turbulence models is assessed by comparing their solutions with the results obtained by high resolution Implicit Large Eddy Simulations (ILES). First, the classical Rayleigh-Taylor and Richtmyer-Meshkov problems are solved. A new approach for initializing the turbulence models in proposed for the Rayleigh-Taylor problem. It is found that both turbulence models describe successfully the self similar growth of the Rayleigh-Taylor and Richtmyer-Meshkov instabilities and can predict accurately the spatial distribution of the fluid concentrations and of the turbulent kinetic energy. The last problem involves the mixing induced at two planar interfaces by multiple shock reflections and refractions. The turbulence models estimate correctly the evolution of the mixing and of the total kinetic energy in the mixing zones. The transport equations of the turbulence models are solved numerically and the influence of the numerical schemes on the results is investigated. It is concluded that the numerical schemes do not have an important influence on the results in the case of the classical Rayleigh-Taylor problem (provided that grid convergence has been achieved and the turbulence models have been initialized using the method proposed here). However, in the presence of shocks (such as in the case of the Richtmyer-Meshkov instability), the HLLC Riemann solver should be used together with a reconstruction scheme of third or higher order of accuracy.