A phase-field based semi-Lagrangian mesh-free lattice Boltzmann method for ternary fluid flows

Numerical simulation of ternary flow is still a challenging problem due to the presence of complex interfacial dynamics and irregular geometric domains. Conventional lattice Boltzmann methods (LBM) are limited by their dependence on structured grids and the strict coupling between spatial and temporal discretizations, which reduces their flexibility in handling such complex systems. This paper presents a novel generalized semi-Lagrangian meshfree lattice Boltzmann method (SL-M-LBM) for ternary fluid flows by using a two-component phase-field model. Our approach fundamentally decouples the discretization by integrating a semi-Lagrangian streaming algorithm with moving least squares (MLS) reconstruction. This enables entirely mesh-free simulations with flexible, non-uniform node distributions and independent control of time stepping, thereby enhancing numerical stability and allowing for local refinement in complex domains. To solve the hydrodynamic equations for incompressible flows and numerically capture interface evolution, our approach creates a Lattice Boltzmann (LB) model with two LB equations (LBEs) for phase-field evolution and a single LB equation (LBE) for hydrodynamics. A wide range of benchmark tests are conducted to evaluate the model, such as Zalesak's rotating disk, diagonal translation, two circular interfaces deformation under shear flow, binary and ternary Rayleigh–Taylor instability, compound droplet passing through a capillary throat under pressure-driven, interaction between a droplet and a rising bubble under gravity, and finally liquid lens spreading. The simulation results indicate that the SL-M-LBM performs effectively for many flow conditions with accurate interface tracking and good mass conservation ability even with non-uniform node distributions. Furthermore, the model can handle irregular geometric domains, high density ratios, and highlights its potential for versatile and efficient simulation of complex multiphase systems containing three immiscible fluids.