A semi-Lagrangian element-free Galerkin method for incompressible two-phase flow problems

A novel meshfree method is developed for simulating incompressible two-phase flows by combining the advantages of the semi-Lagrangian (SL) method and the element-free Galerkin (EFG) method. The proposed SL-EFG method inherits good stability from the SL method, and its meshfree framework greatly simplifies the backward tracing and function interpolation procedures in conventional SL implementations. The incompressible two-phase flow system is governed by the coupled Navier–Stokes equations and Level-Set (LS) equations, including the LS reinitialization equation for interface correction. A temporal decoupling strategy based on the fractional step algorithm is constructed within the SL framework to decouple velocity and pressure variables, which successfully circumvents the Ladyzhenskaya–Babuška–Brezzi (LBB) constraint and yields stable pressure fields. In addition, the pure-convection LS and reinitialization equations are transformed into ordinary differential equations via SL discretization, which further simplifies the numerical solution process without complex spatial discretization. Benefiting from the unconditional stability of the SL method for convection terms, the high-precision approximation capability of the EFG method for diffusion terms, the velocity–pressure decoupling effect of the fractional step technique, and the improved mass conservation performance of the coupled SL-LS scheme for interface capturing, the proposed SL-EFG method provides an effective and promising alternative for numerical simulation of incompressible two-phase flows. A series of numerical examples are conducted to comprehensively validate the interface capturing performance and mass conservation property of the presented method. The numerical results demonstrate that the SL-EFG method possesses good numerical stability and high interfacial resolution in simulating complex incompressible two-phase flow problems.