An energy-consistent depth-averaged Euler system: Derivation  and properties

Marie-Odile Bristeau; Anne Mangeney; Jacques Sainte-Marie; Nicolas Seguin

doi:10.3934/dcdsb.2015.20.961

Article Contents

2015, Volume 20, Issue 4: 961-988. Doi: 10.3934/dcdsb.2015.20.961

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An energy-consistent depth-averaged Euler system: Derivation and properties

1.
INRIA Roquencourt, B.P. 105, 78153 Le Chesnay Cedex
2.
University Paris Diderot, Sorbonne Paris Cité, Institut de Physique du Globe de Paris, Seismology group, 1 rue Jussieu, 75005 Paris
3.
CEREMA, 134 rue de Beauvais, F-60280 Margny-Lès-Compiègne
4.
Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris

Received: April 2014

Revised: September 2014

Published: June 2015

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Abstract

In this paper, we present an original derivation process of a non-hydrostatic shallow water-type model which aims at approximating the incompressible Euler and Navier-Stokes systems with free surface. The closure relations are obtained by a minimal energy constraint instead of an asymptotic expansion. The model slightly differs from the well-known Green-Naghdi model and is confronted with stationary and analytical solutions of the Euler system corresponding to rotational flows. At the end of the paper, we give time-dependent analytical solutions for the Euler system that are also analytical solutions for the proposed model but that are not solutions of the Green-Naghdi model. We also give and compare analytical solutions of the two non-hydrostatic shallow water models.

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Mathematics Subject Classification: 35L60, 35Q30, 76B15, 76D05.

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