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A geometrical Green–Naghdi-type system for dispersive-like waves in prismatic channels

Published online by Cambridge University Press:  26 June 2025

Sergey Gavrilyuk Open the ORCID record for Sergey Gavrilyuk [Opens in a new window]  and
Mario Ricchiuto Open the ORCID record for Mario Ricchiuto [Opens in a new window]
Sergey Gavrilyuk
Affiliation:
Aix Marseille Université, CNRS, IUSTI, UMR 7343, Marseille, France
Mario Ricchiuto*
Affiliation:
INRIA, Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, 200 Avenue de la Vieille Tour, Talence CEDEX 33405, France
*
Corresponding author: Mario Ricchiuto, mario.ricchiuto@inria.fr
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Abstract

We consider two-dimensional (2-D) free surface gravity waves in prismatic channels, including bathymetric variations uniquely in the transverse direction. Starting from the Saint-Venant equations (shallow-water equations) we derive a one-dimensional transverse averaged model describing dispersive effects related solely to variations of the channel topography. These effects have been demonstrated in Chassagne et al. 2019 J. Fluid Mech. 870, 595–616 to be predominant in the propagation of bores with Froude numbers below a critical value of approximately 1.15. The model proposed is fully nonlinear, Galilean invariant, and admits a variational formulation under natural assumptions about the channel geometry. It is endowed with an exact energy conservation law, and admits exact travelling-wave solutions. Our model generalises and improves the linear equations proposed by Chassagne et al. 2019 J. Fluid Mech. 870, 595–616, as well as in Quezada de Luna and Ketcheson, 2021 J. Fluid Mech. 917, A45. The system is recast in two useful forms appropriate for its numerical approximations, whose properties are discussed. Numerical results allow the verification of the implementation of these formulations against analytical solutions, and validation of our model against fully 2-D nonlinear shallow-water simulations, as well as the famous experiments by Treske 1994 J. Hyd. Res. 32, 355–370.

JFM classification

Geophysical and Geological Flows: Shallow water flows Waves/Free-surface Flows: Channel flow Waves/Free-surface Flows: Surface gravity waves

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Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1014 , 10 July 2025 , A5
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© The Author(s), 2025. Published by Cambridge University Press

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