Layer-averaged Euler and Navier–Stokes equations

Bristeau, Marie-Odile; Guichard, Cindy; Martino, Bernard Di; Sainte-Marie, Jacques

doi:10.4310/cms.2017.v15.n5.a3

Cited by 15 publications

(16 citation statements)

References 23 publications

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“…Moreover, only the Euler equations were considered in the present paper. Another field of investigation consists in extending this approach to the approximation of the Navier-Stokes equations by taking into account viscous terms as it was studied in the hydrostatic case in [14].…”

Section: Resultsmentioning

confidence: 99%

A hierarchy of dispersive layer-averaged approximations of Euler equations for free surface flows

Fernández-Nieto

Parisot

Penel

et al. 2018

Communications in Mathematical Sciences

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In geophysics, the shallow water model is a good approximation of the incompressible Navier-Stokes system with free surface and it is widely used for its mathematical structure and its computational efficiency. However, applications of this model are restricted by two approximations under which it was derived, namely the hydrostatic pressure and the vertical averaging. Each approximation has been addressed separately in the literature: the first one was overcome by taking into account the hydrodynamic pressure (e.g. the non-hydrostatic or the Green-Naghdi models); the second one by proposing a multilayer version of the shallow water model. In the present paper, a hierarchy of new models is derived with a layerwise approach incorporating non-hydrostatic effects to approximate the Euler equations. To assess these models, we use a rigorous derivation process based on a Galerkin-type approximation along the vertical axis of the velocity field and the pressure, it is also proven that all of them satisfy an energy equality. In addition, we analyse the linear dispersion relation of these models and prove that the latter relations converge to the dispersion relation for the Euler equations when the number of layers goes to infinity.

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Section: Resultsmentioning

confidence: 99%

A hierarchy of dispersive layer-averaged approximations of Euler equations for free surface flows

Fernández-Nieto

Parisot

Penel

et al. 2018

Communications in Mathematical Sciences

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“…The layer-averaging process for the 2d hydrostatic Euler and Navier-Stokes systems is precisely described in the paper [18] with a general rheology, the reader can refer to it. In the following, we present a Galerkin type approximation of the 3d Euler system also leading to a layer-averaged version of the Euler system, the obtained model reduces to [18] in the 2d context. Using the notations (15), let us consider the space P N,t 0,h of piecewise constant functions defined by…”

Section: The Layer-averaged Euler Systemmentioning

confidence: 99%

“…Notice that, compared to the advection and pressure terms, the discretization of the viscous terms raises less difficulties and we propose a stable scheme that will be extended to more general rheology terms [18] and more completely analyzed in a forthcoming paper.…”

Section: The Discrete Layer-averaged Navier-stokes Systemmentioning

confidence: 99%

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

et al. 2019

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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.

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“…the free surface remains a monovaluated function in space. Several generalizations of the model have been proposed for variable density flow [3], for hydrostatic Navier-Stokes equations [5] and for non hydrostatic flows [10]. In this context, the solution of the Riemann problem (1)-(2) appears as an important step in a better understanding of this now widely used family of free surface fluid models.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the Riemann Problem for a Shallow Water Model with Two Velocities

Aguillon¹,

Audusse²,

Godlewski³

et al. 2018

SIAM J. Math. Anal.

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Some shallow water type models describing the vertical profile of the horizontal velocity with several degrees of freedom have been recently proposed. The question addressed in the current work is the hyperbolicity of a shallow water model with two velocities. The model is written in a nonconservative form and the analysis of its eigenstructure shows the possibility that two eigenvalues coincide. A definition of the nonconservative product is given which enables us to analyse the resonance and coalescence of waves. Eventually, we prove the well-posedness of the two dimensional Riemann problem with initial condition constant by half-plane.

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Layer-averaged Euler and Navier–Stokes equations

Cited by 15 publications

References 23 publications

A hierarchy of dispersive layer-averaged approximations of Euler equations for free surface flows

A hierarchy of dispersive layer-averaged approximations of Euler equations for free surface flows

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

Analysis of the Riemann Problem for a Shallow Water Model with Two Velocities

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