Eliminating Flux Corrections in Geometric VOF with Consistent Discretization

Unsplit, geometric volume-of-fluid techniques are state-of-the-art numerical methods for capturing and advecting the interface between gas-liquid flows. The geometric discretization, known as the semi-Lagrangian (SL), can be used not only to advect the interface but also to consistently advect any other discontinuous quantities near the interface, such as momentum. However, an inconsistency arises in the projection step used to solve for pressure which is typically discretized with the finite volume (FV). This results in a velocity field that is not ensured to be divergence-free in the SL sense on the subsequent time step, leading to mass conservation errors if not addressed. Flux corrections have been used to artificially adjust the SL fluxes to the alleviate this discrepancy. These flux corrections add computational cost and may reduce simulation accuracy due to their non-physical nature. To avoid the need for flux corrections, we have discretized the pressure correction equation with the SL, which leads to a non-canonical form of the pressure equation, ∇·(u^* - Δt/ρ ∇p) = 0, where u^* is the intermediate velocity before the pressure projection. This formulation arise due to a non-linearity present within the SL discretization itself. In this work, we explore different iterative methods to solve this pressure equation and compare performance across several test cases.