A diffuse-interface method for compressible two-phase flows with seven- and six-equation models

Abstract: In this work, a novel phase-field method is proposed for the six- and seven-equation non-equilibrium models for simulating compressible two-phase flows. Such formulations allow for monotonic mixture speed of sound, minimizing artificial wave delay during transmission across an interface. The proposed phase field formulation is constructed from the baseline seven-equation model, and interface-regularization terms are added in divergence form, while maintaining consistency between the partial differential equations without introducing spurious source terms. It admits conservative phasic and mixture entropy transport equations, thus facilitating the construction of discrete conservative schemes. The six-equation formulation is obtained under instantaneous velocity equilibrium. To avoid eigenvector degeneracy of the system of PDEs, the volumetric interface regularization flux is modified to account for a finite amount of conjugate phase, which improves on how phasic density is captured implicitly. Stability of compressible two-phase flow rely on the preservation of the interface-equilibrium conditions, and the preservation of discrete kinetic energy and entropy. A detailed analysis of IEC demonstrates additional requirements on the consistency of flux splittings between the convective and interface-regularization terms for all quantities, as well as the effects on the phasic internal energy flux splittings. A KEEP discretization is proposed and evaluated over a suite of high-density ratio test cases, including interface advection, acoustic wave-induced bubble oscillation, oblique acoustic wave reflection and transmission, and two-phase Taylor-Green vortex flow. Results demonstrate accuracy, stability and robustness for very long time integrations, a desired feature for simulation of turbulent flows and acoustics, since the framework does not rely on the addition of numerical dissipation.