A consistent and conservative diffuse-domain lattice Boltzmann method for multiphase flows in complex geometries

Abstract: Modeling and simulation of multiphase flows in complex geomerties are challenging due to the complexity in describing the interface topology changes among different phases and the difficulty in implementing the boundary conditions on the irregular solid surface. In this work, we first developed a diffuse-domain (DD) based phase-field model for multiphase flows in complex geometries. In this model, the irregular fluid region is embedded into a larger and regular domain by introducing a smooth characteristic function. Then, the reduction-consistent and conservative phase-field equation for the multiphase field and the consistent and conservative Navier-Stokes equations for the flow field are reformulated as the diffuse-domain based consistent and conservative (DD-CC) equations where some additional source terms are added to reflect the effects of boundary conditions. In this case, there is no need to directly treat the complex boundary conditions on the irregular solid surface, and additionally, based on a matched asymptotic analysis, it is also shown that the DD-CC equations can converge to the original governing equations as the interface width parameter tends to zero. Furthermore, to solve the DD-CC equations, we proposed a novel and simple lattice Boltzmann (LB) method with a Hermite-moment-based collision matrix which can not only keep consistent and conservation properties, but also improve the numerical stability with a flexible parameter. With the help of the direct Taylor expansion, the macroscopic DD-CC equations can be recovered correctly from the present LB method. Finally, to test the capacity of LB method, several benchmarks and complex problems are considered, and the numerical results show that the present LB method is accurate and efficient for the multiphase flows in complex geomerties.