A filtering approach for the conservative Allen–Cahn equation solved by the lattice Boltzmann method and a numerical study of the interface thickness

Abstract

The numerical solution to the conservative Allen–Cahn (AC) equation obtained by the lattice Boltzmann method (LBM) is problematic due to the creation of an unphysical phase in the bulk region. In this study, we propose an approach to suppress this unexpected behaviour by filtering the phase-field function. In the AC equation simulations, the phase-field function typically oscillates in the bulk region and overshoots or undershoots near the interface. The aim of the filtering approach is to create a smooth interface profile through several filtering iterations and improve the calculation accuracy of the interfacial normal. We demonstrate that the filtering process effectively suppresses the creation of an unphysical phase through Zalesak’s disc rotation and single vortex tests. The filtering process improves the accuracy of the calculated interface tension under the same thickness for a stationary droplet benchmark test. A comparative study with the Cahn-Hilliard (CH) equation is conducted for a single rising bubble flow. The characteristics of both equations are discussed. Furthermore, the filtered conservative AC equation is found to preserve the shape of the rising bubble well.