A Mass and Momentum-Conservative Semi-Implicit Finite Volume Scheme for Complex Non-Hydrostatic Free Surface Flows

In this article, a novel mass and momentum conservative semi-implicit method is presented for the numerical solution of the incompressible free-surface Navier–Stokes equations. This method can be seen as an extension of the semi-implicit mass-conservative scheme presented by Casulli. The domain is covered by the fluid, by potential solid obstacles, and by the surrounding void via a scalar volume fraction function for each phase, according to the so-called diffuse interface approach. The semi-implicit finite volume discretization of the mass and momentum equations leads to a mildly nonlinear system for the pressure. The nonlinearity on the diagonal of the system stems from the nonlinear definition of the volume, while the remaining linear part of the pressure system is symmetric and at least positive semi-definite. Hence, the pressure can be efficiently obtained with the family of nested Newton-type techniques recently introduced and analyzed by Brugnano and Casulli. The time step size is only limited by the flow speed and eventually by the velocity of moving rigid obstacles contained in the computational domain, and not by the gravity wave speed. Therefore, the method is efficient also for low Froude number flows. Moreover the scheme is formulated to be locally and globally conservative: for this reason it fits well in the presence of shock waves, too. In the special case of only one grid cell in vertical direction, the proposed scheme automatically reduces to a mass and momentum conservative discretization of the shallow water equations. The proposed method is first validated against the exact solution of a set of one-dimensional Riemann problems for inviscid flows. Then, some computational results are shown for non-hydrostatic flow problems and for a simple fluid-structure interaction problem.