Semi-Lagrangian Pressure Solver for Accurate, Consistent, and Conservative Volume-of-Fluid Simulations

Unsplit, geometric volume-of-fluid techniques are state-of-the-art numerical methods for capturing the interface between gas-liquid flows and use a semi-Lagrangian (SL) method to advect discontinuous quantities near the interface. Although highly successful, the method requires a nonphysical SL flux correction to achieve conservation of mass, momentum, and any other quantities advected with the SL discretization, which hinders the accuracy and adds computational cost. The issue arises due to an inconsistency between the SL discretization of advection terms and the finite volume discretization commonly used in the pressure equation. In this work, we propose an updated scheme that uses the SL method to consistently discretize the divergence operator in both the advection terms and the pressure equation. This ensures that the velocity field is divergence-free under a SL discretization at the next time step. This work aims to quantify gains in computational efficiency and accuracy compared to previous methods.