Third-order analysis of pseudopotential lattice Boltzmann model for multiphase flow

TL;DR

This paper performs a third-order Chapman-Enskog analysis of the pseudopotential lattice Boltzmann model for multiphase flow, revealing anisotropic and isotropic effects on the model's accuracy and stability, and proposing a scheme to improve coexistence density and surface tension predictions.

Contribution

First third-order analysis of the pseudopotential lattice Boltzmann model, identifying key anisotropic and isotropic terms affecting accuracy and stability, and proposing a scheme for independent adjustment of coexistence densities and surface tension.

Findings

Anisotropic term causes droplet shape distortion, which can be mitigated.

Considering the isotropic term yields accurate pressure tensor and coexistence densities.

A new scheme allows independent tuning of coexistence densities and surface tension.

Abstract

In this work, a third-order Chapman-Enskog analysis of the multiple-relaxation-time (MRT) pseudopotential lattice Boltzmann (LB) model for multiphase flow is performed for the first time. The leading terms on the interaction force, consisting of an anisotropic and an isotropic term, are successfully identified in the third-order macroscopic equation recovered by the lattice Boltzmann equation (LBE), and then new mathematical insights into the pseudopotential LB model are provided. For the third-order anisotropic term, numerical tests show that it can cause the stationary droplet to become out-of-round, which suggests the isotropic property of the LBE needs to be seriously considered in the pseudopotential LB model. By adopting the classical equilibrium moment or setting the so-called "magic" parameter to 1/12, the anisotropic term can be eliminated, which is found from the present third-order analysis and also validated numerically. As for the third-order isotropic term, when and only when it is considered, accurate continuum form pressure tensor can be definitely obtained, by which the predicted coexistence densities always agree well with the numerical results. Compared with this continuum form pressure tensor, the classical discrete form pressure tensor is accurate only when the isotropic term is a specific one. At last, in the framework of the present third-order analysis, a consistent scheme for third-order additional term is proposed, which can be used to independently adjust the coexistence densities and surface tension. Numerical tests are subsequently carried out to validate the present scheme.