Numerical simulation for the conserved Allen–Cahn phase field model of two-phase incompressible flows by an efficient dimension splitting method

Abstract

A second-order, stable, and efficient numerical method is proposed for the simulation of high-dimensional two-phase incompressible flow. The model considers the coupled system of incompressible Navier–Stokes equations with variable density and the conserved Allen–Cahn equations. To save storage and computing costs, a dimension splitting scheme is proposed, which combines a second-order pressure correction method with a splitting effect and an alternating direction implicit method to split a two- or three-dimensional problem into a series of one-dimensional sub-problems that can be solved in parallel. By reconstructing the time derivative terms, the variable coefficient velocity equations are transformed into constant coefficient linear equations. In addition, a second-order stabilization term is added to enhance stability. Finally, a post-processing method is proposed which can preserve both the maximum bound principle and the mass conservation. For space discretization, the weighted essentially non-oscillatory scheme and central difference scheme are used on the marker and cell grid, which can effectively avoid numerical oscillation caused by high Reynolds numbers. Numerical experiments verify the convergence, stability, parallel efficiency, and effectiveness of post-processing, and demonstrate some simulations of two-phase incompressible flows.