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

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 thirdorder 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.

Fluid–fluid interactions in pseudopotential lattice Boltzmann models: Effects of model schemes and fluid properties

“…Two improvements to the MRT forcing model are analyzed and discussed in the following sections. They are referred to in this paper as Li–Luo 57 and Huang–Wu 44 models. In the Li–Luo method, the value of ϵ in the mechanical stability condition is tuned using σ .…”

“…The Huang–Wu 44 method, unlike the Li–Luo methods, combines thermodynamic consistency adjustment and surface tension adjustment into a single term. Thermodynamic consistency is tuned using the sum of k 1 and k 2 and surface tension is tuned using k 1 only, as follows 44 : where σ c is the surface tension coefficient. Huang and Wu 44 investigated setting σ c from 0.1 to 2.0.…”

“…Setting the magic parameter to 1/12 to minimize third‐order errors gives the best results for multiphase simulations 44 . However, setting the magic parameter to even lower values has been reported.…”

Heat Transfer Enhancement Technique of PCMs and Its Lattice Boltzmann Modeling

“…where s 0 , s e , s ε , s j , s q , and s p are the parameters related to relaxation time, which could be chosen as described in Ref. [134]. The discrete force term in the momentum space F m x; t ðÞ is given by [135,136]:…”

High-order lattice-Boltzmann model for the Cahn-Hilliard equation

“…All of the existing LBE models for the CHE and ACE are based on second-order Chapman-Enskog analysis. As indicated in [26,27], the high-order effects are necessary for the pseudo-potential LBE. This suggests that the high-order effects may be also important for the LBE for the phase-field equation.…”