Numerical approximation of the 3D hydrostatic Navier–Stokes system with free surface

Abstract

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.

How do microalgae perceive light in a high-rate pond? Towards more realistic Lagrangian experiments

“…The impact of the paddle wheel on the fluid was represented as a normalized force applied by the wheel's blades proportional to the square of the velocity at each point. Here, we used a three-dimensional extension [ 39 ] to the model presented by Bernard et al . [ 36 ].…”

Some analytical solutions for validation of free surface flow computational codes

“…These hydrostatic models are very often used for the study of geophysical flows, see Brenier (1999) and Grenier (1999), such as landslides, glaciers or tsunamis (e.g. Fernandez-Nieto et al 2016;Allgeyer et al 2019) for justifications of such models.…”

“…All the graphics presented is this document represent the exact analytical solutions, but we are able to obtain a good approximation of them by using the Freshkiss3D simulation code (Freshkiss3d 2017) by imposing only the geometry of the domain and the corresponding initial and boundary conditions. With these analytical solutions, it is possible to test simulation codes and to obtain convergence curves when increasing the numerical scheme order and the mesh resolution, see for example Allgeyer et al (2019).…”

A two-dimensional method for a family of dispersive shallow water models

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