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

In this work, a novel discretization of the incompressible Navier-Stokes equations for a gas-liquid flow is developed. Simulations of gas-liquid flows are often performed discretizing time with a predictor -> pressure -> corrector approach and the phase interface is represented by a volume of fluid (VOF) method. Recently, unsplit, geometric VOF methods have been developed that use a semi-Lagrangian discretization of the advection term within the predictor step. A disadvantage of the current methods is that an alternative discretization (e.g.~finite volume or finite difference) is used for the divergence operator in the pressure equation. Due to the inconsistency in discretizations, a correction to the semi-Lagrangian advection term is required to achieve mass conservation, which increases the computational cost and reduces the accuracy. In this work, we explore the idea of using a semi-Lagrangian discretization for the divergence operators in both the advection term and the pressure equation. The proposed discretization avoids the correction to semi-Lagrangian fluxes improving the accuracy. Additionally, this method has the potential to reduce the computational cost of VOF simulations for gas-liquid flows.