On Accuracy of Lattice Boltzmann Method Coupled with Cahn-Hilliard and Allen-Cahn Equations for Simulation of Multiphase Flows at High-Density Ratios

Abstract

In this work, the accuracy of the multiphase lattice Boltzmann method (LBM) based on the phase-field models, namely the Cahn-Hilliard (C-H) and Allen-Cahn (A-C) equations, are evaluated for simulation of two-phase flow systems with high-density ratios. The mathematical formulation and the schemes used for discretization of the derivatives in the C-H LBM and A-C LBM are presented in a similar notation that makes it easy to implement and compare these two phase-field models. The capability and performance of the C-H LBM and A-C LBM are investigated, specifically at the interface region between the phases, for simulation of flow problems in the two-dimensional (2D) and three-dimensional (3D) frameworks. Herein, the equilibrium state of a droplet and the practical two-phase flow problem of the rising bubble are considered to evaluate the mass conservation capability of the phase-filed models employed at different flow conditions and the obtained results are compared with available numerical and experimental data. The effect of employing different equations proposed in the literature for calculating the relaxation time on the accuracy of the implemented phase-field LBMs in the interfacial region is also studied. The present study shows that the LBM based on the A-C equation (A-C LBM) is advantageous over that based on the C-H equation in dealing with the conservation of the total mass of a two-phase flow system. Also, the results obtained by the A-C LBM is more accurate than those obtained using the C-H LBM in comparison with other numerical results and experimental observations. The present study suggests the A-C LBM as a sufficiently accurate and computationally efficient phase-field model for the simulation of practical two-phase flows to resolve their structures and properties even at high-density ratios.