Numerical approximation of the 3d hydrostatic Navier-Stokes system with free surface

In this paper we propose a stable and robust strategy to approximate incompressible hydrostatic Euler and Navier-Stokes systems with free surface. The idea is to use a Galerkin type approximation of the velocity field with piecewise constant basis functions in order to obtain an accurate description of the vertical profile of the horizontal velocity. Such a strategy has several advantages. It allows • to rewrite the Navier-Stokes equations under the form of a system of conservation laws with source terms, • the easy handling of the free surface, which does not require moving meshes, • the possibility to take advantage of robust and accurate numerical techniques developed in extensive amount for Shallow Water type systems. We show that the model admits a kinetic interpretation, and we use this result to formulate a robust finite volume scheme for its numerical approximation. All the aspects of the discrete scheme (fluxes, boundary conditions,. . . ) are completely described and the stability properties of the proposed numerical scheme are discussed. We illustrate the capability of the model and of the discrete scheme with some numerical academic and realistic examples.