A consistent and conservative Phase-Field method for compressible multiphase flows with the six-equation model

Abstract

In the present study, the consistent and conservative Phase-Field method is extended to the six-equation model for compressible multiphase flows. Based solely on the conservation laws and the second law of thermodynamics, the six-equation model with the Phase-Field mechanism is first derived. In addition to satisfying Galilean invariance and consistency of reduction, the model is general to admit an arbitrary number of phases with different formulations of the Phase-Field mechanism. The isobaric closure by the pressure relaxation and the incompressible limit of the proposed model are analyzed. The derivation and analysis identify additional terms arising from the Phase-Field mechanism, which are absent from previous studies. The consistent and conservative numerical approach is adapted to the proposed six-equation model with moderate modifications, retaining the interfacial equilibrium condition, conservation, and flexibility and robustness of incorporating different formulations of the Phase-Field mechanism with bound preservation. As the new component of solving the six-equation model, both the pressure and pressure-temperature relaxations are theoretically analyzed in a general multiphase setup, with a proof of the existence and uniqueness of a thermodynamically admissible solution for these two relaxations. Various two-phase compressible flow benchmarks are performed to demonstrate the method, and good agreement with exact solutions is achieved.