Riemann solvers and pressure gradients in Godunov-type schemes for variable density incompressible flows

Variable density incompressible flows are governed by parabolic equations. The artificial compressibility method makes these equations hyperbolic-type, which means that they can be solved using techniques developed for compressible flows, such as Godunov-type schemes. While the artificial compressibility method is well-established, its application to variable density flows has been largely neglected in the literature. This paper harnesses recent advances in the wider field by applying a more robust Riemann solver and a more easily parallelisable time discretisation to the variable density equations than previously. We also develop a new method for calculating the pressure gradient as part of the second-order reconstruction step. Based on a rearrangement of the momentum equation and an exploitation of the other gradients and source terms, the new pressure gradient calculation automatically captures the pressure gradient discontinuity at the free surface. Benchmark tests demonstrate the improvements gained by this robust Riemann solver and new pressure gradient calculation.