An ALE-based Numerical Scheme with a Two-Dimensional Riemann Solver for Two-Phase Flows with Non-Equilibrium Pressure

Abstract

In this work, a homogeneous two-phase flow model in two-dimensional space is proposed for systems without a sharp interface between phases. The model allows the two phases to coexist within the same domain with distinct pressures, while their local proportions are represented by volume fractions. Within an arbitrary Lagrangian-Eulerian (ALE) framework, a two-dimensional Riemann solver is developed to construct the numerical scheme with novel formulations for numerical fluxes and nodal velocities, ensuring consistency on moving control volumes. Furthermore, a new moving-mesh strategy is designed, in which the evolution of the volume fraction is obtained directly from the geometric motion of the mesh and mass conservation, thereby avoiding solving the non-conservative convective equation with source terms. This enables the incorporation of physical drag models and finite relaxation coefficients, allowing the gradual equilibration of velocity and pressure between the two phases to be calculated. Based on the volume-fraction formulation, a discrete representation of the pressure work term is derived. Finally, a series of numerical experiments are presented to demonstrate the accuracy, stability, and physical consistency of the proposed model and numerical scheme.