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

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. (c) 2007 Elsevier Inc. All rights reserved.