A semi-Lagrangian meshfree lattice Boltzmann method for incompressible two-phase flows

Abstract

A new numerical method, which is based on the coupling between semi-Lagrangian meshfree method and two-phase lattice Boltzmann (LB) model, is developed for incompressible two-phase flows. Two LB equations, one is for solving the Cahn-Hilliard equation for capturing interface and the other is for solving the two-phase Navier-Stokes equations for flow field, are involved. In view of the pure convection property of the LB equations, the semi-Lagrangian meshfree method we proposed before is used to solve them. This is a fully meshfree method that inherits all the distinct advantages of meshfree method and traditional LB method. Firstly, it generalizes the LB method to the meshfree case and breaks through the limitation of uniform mesh. Secondly, it eliminates the coupling of temporal and spatial step sizes and thus is easier to meeting specified accuracy requirements. Thirdly, it gives full play to the advantages of LB model in interface capture and parallel computation. Finally, it can solve two-phase flow problems in complex geometric domains by using non-uniform spatial discretization, and maintain high accuracy and stability. Three benchmark tests show the new method possesses an excellent mass conservation ability on both uniform and non-uniform node arrangements. Subsequently, the method is applied to several two-phase flow problems by using non-uniform meshfree nodes. The results agree very well with the analytical or reference solutions even for the rather extreme simulation cases. The good performance of this method suggests it is a powerful tool for solving two-phase flows.