A three-dimensional phase-field lattice Boltzmann method for incompressible two-components flows

Abstract

In this paper, a lattice Boltzmann model for the coupled Allen–Cahn–Navier–Stokes equations in three dimensions is presented. Two equations are solved: one for the fluid velocity and one for the order parameter. Both are written within the general multiple-relaxation-time framework, where all the equilibrium and forcing terms are described by using the full set of Hermite polynomials. The resultant practical implementation is compact. The gradient of the order parameter can be computed by the non-local finite differences or the local central moments. The latter suffers from grid-scale oscillations. The very good accuracy properties are demonstrated against nine well-consolidated benchmark tests. Specifically, two groups of tests are tackled. In the former, the velocity field is superimposed. Hence, only the equation for the evolution of the order parameter is solved. These numerical experiments demonstrate the ability of the proposed scheme to capture the correct evolution of the interface. In the latter, two immiscible fluids are considered and the two equations are solved. Simulations of the vertical penetration of a wedge-shaped body, two- and three-dimensional Rayleigh–Taylor instability prove that two-fluids systems can be successfully simulated by our approach.