A mass-conservative semi-implicit volume of fluid method for the Navier–Stokes equations with high order semi-Lagrangian advection scheme

This paper deals with the development of a semi-implicit numerical method for the Navier–Stokes equations using the non-linear volumes of fluid (VOF) approach and a semi-Lagrangian scheme for the discretization of the advection contribution based on a high order reconstruction of the velocity field. The VOF approach guarantees high flexibility and is able to reproduce several phenomena that appear in real scenarios such as free surface flows, pressurized channels and jets. The discrete velocity field from the momentum conservation law is formally inserted into the discrete continuity equation, hence yielding a mildly non-linear system for the unknown hydraulic head which can be solved through a nested Newton-type algorithm. The computation of the non-linear convective diffusion contribution is then based on a high order reconstruction of the velocity field, which is furthermore constrained to exactly recover the original pointwise values of the numerical solution. As a consequence, the mass conservation is fully preserved while providing information about the main velocity field and its high order moments, later employed in the computation of the Lagrangian trajectories needed for the discretization of the convective and diffusive terms. Furthermore, the bottom friction and the tangential stresses can be directly computed from the high order velocity reconstruction. The method is derived in a general form with the only requirement to be structured in the z− direction, so that it applies to the three- and the two-dimensional cases with unstructured grids in the horizontal space. Convergence studies are carried out to demonstrate the accuracy of the reconstruction operator. Finally, the numerical scheme is validated against several benchmarks that include 2Dxz, 2Dxy and 3D non-hydrostatic flows with complex geometry in order to show the flexibility of the proposed algorithm, including a real-world application.