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

TL;DR

This paper introduces a novel non-hydrostatic shallow water model derived from energy constraints, differing from Green-Naghdi, with analytical solutions demonstrating its properties and accuracy in approximating Euler and Navier-Stokes systems.

Contribution

The paper presents a new derivation of a non-hydrostatic shallow water model using energy constraints, offering an alternative to asymptotic expansions and the Green-Naghdi model.

Findings

The model closely approximates the Euler system with rotational flows.

Analytical solutions validate the model's accuracy and differences from Green-Naghdi.

Comparison shows the model's potential advantages in certain flow regimes.

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 aminimal energy constraint instead of an asymptotic expansion. The model slightly differs from thewell-known Green-Naghdi model and is confronted with stationary andanalytical solutions of the Euler system corresponding to rotationalflows. At the end of the paper, we givetime-dependent analytical solutions for the Euler system that are alsoanalytical solutions for the proposed model but that are not solutionsof the Green-Naghdi model. We also give and compare analytical solutions of thetwo non-hydrostatic shallow water models.