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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 by 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 flux-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 alternative of using a semi-Lagrangian discretization for the divergence operators in both the advection term and the pressure equation. The proposed discretization avoids the need to use a flux-correction to the semi-Lagrangian advection term as mass conservation is achieved through consistent discretization. Additionally, avoiding the flux-correction improves the accuracy while reducing the computational cost of the advection term semi-Lagrangian discretization. 

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 by 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 flux-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 alternative of using a semi-Lagrangian discretization for the divergence operators in both the advection term and the pressure equation. The proposed discretization avoids the need to use a flux-correction to the semi-Lagrangian advection term as mass conservation is achieved through the consistent discretization. Additionally, avoiding the flux-correction improves the accuracy while reducing the computational cost of the advection term semi-Lagrangian discretization.