Consistent and conservative numerical approach with Phase-Field mechanism for compressible N-phase flows

Multiphase flows including shocks are ubiquitous in natural phenomena and engineering processes, while reliable numerical approaches to accurately simulate problems with an arbitrary number of phases are still in demand. Simply repeating numerical approaches for two-phase flows to individual phases of N-phase flows produces unphysical behaviors that are not observed in two-phase simulations, such as fictitious phases, local voids, or overfilling, which are sources of numerical instability. To resolve the issue, a consistent and conservative approach is developed to solve the N-phase Euler/Phase-Field model. The proposed method is general in the sense that it admits N different phases and is not restricted to a specific Phase-Field formulation. Not only is the volume fraction bounded and the mass non-negative, the volume fractions also sum up to unity, preventing the production of fictitious phases, local voids, or overfilling. Kinematic, mechanical, and thermal equilibria are maintained across material interfaces. High-order implementation of the approach is straightforward. The Phase-Field mechanism produces controllable interface thickness, thus effectively preventing the numerical mixing of different phases due to numerical diffusion in shock-capturing schemes. Various numerical examples are provided to demonstrate the approach.