A physically consistent and numerically robust k-ϵ model for computing turbulent flows with shock waves

High-speed turbulent flows with shock waves pose significant challenge in terms of physical modeling and numerical accuracy. In this paper, we develop a k-ϵ turbulence model for canonical shock-turbulence interaction, based on the physical ideas of shock-unsteadiness damping. The shock-unsteadiness k-ϵ model proposed earlier by Sinha et al. (Phys. Fluids, 15(8), 2003) has model parameters that are functions of upstream Mach number. Our main objective is to eliminate the dependence of the model parameters on upstream flow quantities, and thus overcome difficulties in numerical implementation in complex flow applications. For the turbulent kinetic energy, this is achieved by proposing a new model for the shock-unsteadiness source term that is physically and dimensionally consistent with the earlier formulation. An alternate form of the dissipation rate equation is also used to eliminate upstream dependence of the model parameter in the production term. The proposed model is employed to compute the canonical interaction of a normal shock with homogeneous isotropic turbulence. Large numerical errors at the shock wave are eliminated by employing a transformation of the turbulence variables that result in a conservative form of the governing equations. The new model thus gives physically consistent and numerically accurate results at shock waves. The model predictions for turbulence kinetic energy and its dissipation rate are found to match DNS data over a range of Mach numbers.