An accurate low-Mach scheme for a compressible two-fluid model applied to free-surface flows

Classical approximate Riemann solvers are known to be too much dissipative in the low-Mach number regime. For this reason, since the Mach number in liquids is generally very small, usual upwind schemes may provide inaccurate solutions when applied to the simulation of two-phase flows. In this paper, to circumvent this difficulty while keeping a compressible model for the description of both gas and liquid, an original accurate low-Mach scheme is introduced and theoretically studied. Extending some ideas already used for the gas dynamics system, the proposed scheme is based on a centred formulation for the pressure gradient term in the momentum equation and on the introduction of a stabilising term proportional to the pressure difference between two neighbouring cells. The scheme stability is ensured, and theoretically proved under a convective CFL-like condition, by using a semi-implicit time discretisation algorithm. Finally, the correct asymptotic behaviour of the scheme in the limit of small Mach numbers is assessed on several academic test cases.