A consistent and conservative Phase-Field method for compressible multiphase flows with shocks

Abstract

A key challenge with interface-capturing methods for compressible multiphase flows is that the numerical diffusion of upwind schemes causes material interfaces to continuously diffuse over time, possibly in a non-uniform fashion. In the present study, a consistent and conservative Phase-Field model for compressible multiphase flows is derived to control mixing between different fluids at material interfaces due to numerical diffusion. The model is general as it admits an arbitrary number of phases and has no prior assumption of the formulation of the Phase-Field mechanism. Necessary physical constraints to formulate the Phase-Field mechanism are obtained by analyzing physical principles, including the second law of thermodynamics. To numerically solve the model, two consistency requirements and a general numerical strategy, called reduction consistent formulation, are proposed. The proposed approach is consistent, conservative, bound/positivity preserving for the volume fraction/mass, and prevents spurious errors at interfaces. Various two-phase compressible flow problems including shocks and interfaces are performed to verify the analysis and demonstrate the efficacy of the method. Finally, temperature equilibrium and the incompressible limit are also discussed.