A unified forcing scheme for the single relaxation lattice Boltzmann method

Abstract

The lattice Boltzmann method (LBM) has been successfully used to simulate fluid flow phenomena, which involves incorporating a force term. To the best of the authors' knowledge, no study has extensively investigated the effect of the discretization of the Boltzmann transport equation on the force term. The current study focuses on closing this gap by performing a parametric discretization of the collision and force term. To ensure consistent behavior during the discretization, the moments of both the force term and equilibrium distribution function are compared for the continuous and discrete spaces. The recovered momentum equation using the Chapman-Enskog analysis has an additional term proportional to the square of the external forces. Hence, a novel parametric unified formulation of momentum forcing scheme is proposed to compensate for the additional term. The novel unified forcing scheme is able to improve and predict the error of several previously published forcing schemes. Furthermore. the unified forcing scheme is validated for single- and multi-phase problems. The natural convection in a cavity and the stationary droplet problems are used to benchmark the general proposed discretization and the unified forcing scheme for single-phase and multiphase systems, respectively. The results show that the unified proposed forcing scheme agrees with the reference results of natural convection in a cavity. For multiphase flow, the unified proposed scheme predicts a high-density ratio second after Kupershtokh et al.’s and Shan and Chen’s approaches, which in their turn do not recover the correct momentum equation. The unified proposed approach predicts a slightly larger radius. This indicates that the mean-field potential function should be improved to increase the density ratio and to predict a correct radius while retaining the correct momentum and continuity equations.