A pressure-based numerical scheme for compressible-incompressible two-phase flows

This article presents a numerical scheme for two-phase flows consisting of compressible gas and incompressible liquid. Assuming that the gas density is determined by pressure only and that the liquid density is constant, we develop a new set of governing equations for compressible-incompressible two-phase flows, which can be seen as a simplification of the previous unified governing equations. The volume fraction of gas, velocity, and pressure are the primary variables. Given the liquid's usual leading role in the flow motion, a pressure-based method is employed to solve the equations. A fifth-order accurate weighted essentially nonoscillatory (WENO) scheme is applied to discretize the advection terms, and the tangent of hyperbola for interface capturing (THINC) scheme is utilized to capture the interface. The numerical scheme proposed is validated against the generalized Bagnold model and the free drop of a water patch in a closed tank. The results show that the scheme can tackle low Mach number compressible-incompressible two-phase flows. For the one-dimensional Bagnold model, the present numerical model achieves first-order convergence and gives better results than that based on the previous unified governing equations with the same numerical methods. In the simulation of the free drop of a water patch in a closed tank, a similar pressure curve with other reported work is given.