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Numerical approximation of the 3D hydrostatic Navier-Stokes system with free surface. (English) Zbl 1434.65147

Summary: In this paper we propose a stable and robust strategy to approximate the 3D incompressible hydrostatic Euler and Navier-Stokes systems with free surface. Compared to shallow water approximation of the Navier-Stokes system, 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.
Compared to previous works of some of the authors, the three dimensional case is studied in this paper. We show that the model admits a kinetic interpretation including the vertical exchanges terms, 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 (well-balancing, positivity of the water depth, …) are discussed. We validate the model and the discrete scheme with some numerical academic examples (3D non stationary analytical solutions) and illustrate the capability of the discrete model to reproduce realistic tsunami waves propagation, tsunami runup and complex 3D hydrodynamics in a raceway.

MSC:

65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76M12 Finite volume methods applied to problems in fluid mechanics
35L65 Hyperbolic conservation laws
35Q30 Navier-Stokes equations
35Q35 PDEs in connection with fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q31 Euler equations
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35R35 Free boundary problems for PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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References:

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