An adaptive mesh redistribution method for the incompressible mixture flows using phase-field model

Abstract

A phase field model which describes the motion of mixtures of two incompressible fluids is presented by Liu and Shen [C. Liu, J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Phys. D 179 (2003) 211–228]. The model is based on an energetic variational formulation. In this work, we develop an efficient adaptive mesh method for solving a phase field model for the mixture flow of two incompressible fluids. It is a coupled nonlinear system of Navier–Stokes equations and Allen–Cahn phase equation (phase-field equation) through an extra stress term and the transport term. The numerical strategy is based on the approach proposed by Li et al. [R. Li, T. Tang, P.-W. Zhang, Moving mesh methods in multiple dimensions based on harmonic maps, J. Comput. Phys. 170 (2001) 562–588] to separate the mesh-moving and PDE evolution. In the PDE evolution part, the phase-field equation is numerically solved by a conservative scheme with a Lagrange multiplier, and the coupled incompressible Navier–Stokes equations are solved by the incremental pressure-correction projection scheme based on the semi-staggered grid method. In the mesh-moving part, the mesh points are iteratively redistributed by solving the Euler–Lagrange equations with a parameter-free monitor function. In each iteration, the pressure and the phase are updated on the resulting new grid by a conservative-interpolation formula, while the velocity is re-mapped in a non-conservative approach. A simple method for preserving divergence-free is obtained by projecting the velocity onto the divergence-free space after generating the new mesh at the last iterative step. Numerical experiments are presented to demonstrate the effectiveness of the proposed method for solving the incompressible mixture flows.