Incompressible flows have much lower flow speeds to compare with the sound speed and called low Mach number flows. So far, we have simulated incompressible flows by using semi- implicit solvers, in which we iteratively compute the linear equation derived from the pressure Poisson equation. In the case of incompressible gas-liquid two-phase flows, the sparse matrix would be very stiff with non-zero elements changing largely from the gas density to the liquid density at the interface. Even if we use a sophisticated Krylov sub-space iteration solver with the multi-grid preconditioner, the convergence becomes poor for large-scale problems. We have developed a weakly compressible gas-liquid two-phase solver based on fully-explicit time integration of compressible Navier-Stokes equation. For several benchmark problems, the results of the weakly compressible solver are in good agreement with those of the semi-implicit incompressible solver. Since the weakly compressible solver uses a local stencil access, it is possible to implement AMR (Adaptive Mesh Refinement) method with less programming cost. Fine meshes are assigned near the gas-liquid interface and we efficiently compute a liquid film generated by a spoon and a growing soap bubble with the AMR two-phase flow solver.
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