A multigrid solver for the coupled pressure-temperature equations in an all-Mach solver with VoF

We present a generalisation of the all-Mach solver in Fuster & Popinet (2018) to account for heat diffusion between two different compressible phases. By solving a two-way coupled system of equations for pressure and temperature using a multigrid solver, the current code is shown to increase the robustness and accuracy of the solver with respect to classical explicit discretization schemes. Different test cases are proposed to validate the correct implementation of the thermal effects: an Epstein-Plesset like problem for temperature is shown to compare well with a spectral method solution. The code also reproduces free small amplitude oscillations where analytical solutions showing the transition between isothermal and adiabatic regimes are available. In addition, we show results of a single sonoluminescent bubble (SBSL) in standing waves, where the result of the DNS is compared with that of other methods in the literature. Finally, the collapse of a bubble near a rigid boundary is studied reporting the change of heat flux as a function of the stand-off distance.