Volume 53, Number 6, November-December 2019
|Page(s)||1981 - 2024|
|Published online||29 November 2019|
Numerical approximation of the 3D hydrostatic Navier–Stokes system with free surface
Research School of Earth Sciences, Australian National University, Canberra, ACT, Australia
2 Inria Paris, 2 Rue Simone Iff, CS 42112, 75589 Paris Cedex 12, France
3 Sorbonne Université, Lab. Jacques-Louis Lions, 4 Place Jussieu, 75252 Paris Cedex 05, France
4 SAUR, Research & Development Department, Technological School, 2 rue de la Bresle, 78310, Maurepas France
5 Université de Paris, Institut de physique du globe de Paris CNRS, F-75005 Paris, France
* Corresponding author: firstname.lastname@example.org
Accepted: 20 June 2019
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.
Mathematics Subject Classification: 65M8 / 65M12 / 76M12 / 35L65 / 35Q30 / 35Q35 / 76D05
Key words: Free surface flows / Navier–Stokes equations / Euler system / free surface / 3D model / hydrostatic assumption / kinetic description / finite volumes
© The authors. Published by EDP Sciences, SMAI 2019
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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