Artificial compressibility Godunov fluxes for variable density incompressible flows

In this work we present and compare three Riemann solvers for the artificial compressibility perturbation of the 1D variable density incompressible Euler equations. The goal is to devise an artificial compressibility flux formulation to be used in Finite Volume or discontinuous Galerkin discretizations of the variable density incompressible Navier–Stokes equations. Starting from the constant density case, two Riemann solvers taking into account density jumps at fluid interfaces are first proposed. By enforcing the divergence free constraint in the continuity equation, these approximate Riemann solvers deal with density as a purely advected quantity. Secondly, by retaining the conservative form of the continuity equation, the exact Riemann solver is derived. The variable density solution is fully coupled with velocity and pressure unknowns. The Riemann solvers are compared and analysed in terms of robustness on harsh 1D Riemann problems. The extension to multidimensional problems is described. The effectiveness of the exact Riemann solver is demonstrated in the context of an high-order accurate discontinuous Galerkin discretization of variable density incompressible flow problems. We numerically validate the implementation considering the Kovasznay test case and the Rayleigh–Taylor instability problem.