Consistent and conservative scheme for incompressible two-phase flows using the conservative Allen-Cahn model

In the present work, we consider the conservative Allen-Cahn model and applied it to two-phase flows in a consistent and conservative manner. The consistent formulation is proposed, where the conservative Allen-Cahn equation is reformulated in a conservative form using an auxiliary variable. As a result, the consistency analysis is performed and the resulting two-phase model honors the consistency of reduction, the consistency of mass conservation and the consistency of mass and momentum transport, which are important to reproduce the physical momentum and kinetic energy transport, to achieve mass and momentum conservation, and to satisfy the energy law of the two-phase system. A consistent and conservative scheme is developed, and its properties are carefully analyzed and validated. In order to honor the maximum principle of the conservative Allen-Cahn model, we proposed a boundedness mapping algorithm, which preserves the properties of consistency and conservation of the scheme. The applications of the consistent formulation and the proposed scheme to realistic two-phase flows show that they are accurate, robust and effective for complicated two-phase problems. The applicability of the consistent formulation and consistency analysis to multiphase flows and to the improved Cahn-Hilliard model is discussed. (C) 2020 Elsevier Inc. All rights reserved.