Turbulent kinetic energy in an approximate Riemann solver

Turbulent kinetic energy (TKE) is taken into account in the approximate HLLC Riemann solver. The Euler equations are supplemented with a hyperbolic equation for TKE, and turbulent pressure is taken into account in the momentum and energy balance equations. The Jacobian of this system of equations and its eigenvalues are found, which are used to modify the HLLC solver. The validity of TKE allowance in the modified HLLC Riemann solver is verified by solving Sod’s problem. It is shown that the scheme is unstable at high turbulent pressure if turbulence is ignored in the computation of characteristic velocities.