Compared with the lattice Boltzmann equation (LBE) model based on incompressible phase field theory, the LBE based on quasi-incompressible phase field theory has the advantage of local mass conservation. However, previous quasi-incompressible phase-field-based LBE model does not satisfy the well-balance property, resulting in spurious velocities in the vicinity of interface and density profiles inconsistent with thermodynamics.
To address this difficulty, a novel LBE model is developed based on the quasi-incompressible phase-field theory. First, numerical artifacts in the original LBE for the Cahn-Hilliard are analyzed. Based on this analysis, the equilibrium distribution function and source term are reformulated to remove the numerical artifacts, which makes the new LBE realize the well-balance property at discrete level.
The performance of the proposed LBE model is tested by simulating a number of typical two-phase systems. The numerical results of the planar interface and static droplet problems demonstrate that the present model can remove spurious velocities and achieve well-balanced state. Numerical results of the layered Poiseuille flow demonstrate the accuracy of the present model in simulating dynamic two-phase flow problems. The well-balance properties of the LBE model with two different formulations of surface tension ($\boldsymbol{F}_s=-\phi \nabla \mu$ and $\boldsymbol{F}_s=\mu \boldsymbol{\nabla} \phi$) are also investigated. It is found that the formulation of $\boldsymbol{F}_s=\mu \boldsymbol{\nabla} \phi$ cannot eliminate the spurious velocities, while the formulation of $\boldsymbol{F}_s=\mu \boldsymbol{\nabla} \phi$ can achieve the well-balance state. The influences of viscosity formulations of the fluid mixture are also compared. Particularly, four mixing rules are considered. It is shown that the use of step mixing rule gives more accurate results for the layered Poiseuille flow. Finally, we compare the performance of the present quasi-incompressible LBE model and the original fully incompressible LBE model by simulating the phase separation problem, and the results show that the present model can ensure the local mass conservation, while the fully incompressible LBE can yield quite different predictions.