Adaptation of Phase-Field methods to compressible multiphase flows

Thanks to the consistency of mass conservation and consistency of mass and momentum transport, a general model is derived to adapt the Phase-Field methods to multiphase compressible flows, without any prior assumptions on the formulation of the Phase-Field methods. The second law of thermodynamics, isolated interfaces under a uniform flow, Galilean invariance, and consistency of reduction are analyzed and illustrate the physical constraints that need to be satisfied by the formulation of the Phase-Field methods. Then, the proposed model is implemented in the two-phase Eulerian system, and a general numerical strategy is developed to tackle the added terms due to the Phase-Field methods. The numerical strategy guarantees the boundedness of the volume fraction and positivity of the mass, independent of the formulation of the Phase-Field methods. As a specific example, the conservative Allen-Cahn Phase-Field formulation is applied, and canonical problems are investigated to demonstrate the effectiveness and improvement.