Surface tension modeling plays a significant role in diffuse-interface simulations of liquid–gas two-phase flows. However, widely used surface tension formulations in phase-field models often produce unphysical force even at flat interfaces under equilibrium conditions, leading to spurious velocity fields and degraded numerical fidelity. The objective of this study is to identify an optimal combination of surface tension force formulation and discretization schemes that eliminates such unphysical force and velocity while maintaining computational locality. Several surface tension formulations are systematically examined using three-dimensional flat liquid films and spherical droplets under equilibrium conditions. Special attention is given to curvature-based formulations, which theoretically satisfy the requirement that the surface tension force vanishes at flat interfaces. Although these formulations can be written in mathematically equivalent forms, their numerical behavior differs significantly after discretization. The distributions of surface tension forces are analyzed to clarify how discretization errors arise and how they depend on the choice of formulation. Furthermore, the impact of finite difference schemes on computational locality and numerical accuracy is investigated. It is shown that evaluating the gradient of the order parameter using an isotropic difference scheme while computing the curvature term with a central difference scheme provides an effective balance between accuracy and locality, significantly reducing communication overhead in parallel computations. The proposed formulation and discretization strategy are further validated through phase-field lattice Boltzmann simulations of a three-dimensional static droplet. The results demonstrate a substantial reduction in spurious velocity around the droplet compared with conventional formulations. These findings provide practical and physically consistent guidelines for surface tension modeling in phase-field lattice Boltzmann simulations, particularly for large-scale and parallel computations of multiphase flows.
Sugimoto et al. (Thu,) studied this question.