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A new model of shoaling and breaking waves. Part 2. Run-up and two-dimensional waves

Published online by Cambridge University Press:  20 March 2019

G. L. Richard Open the ORCID record for G. L. Richard [Opens in a new window] ,
A. Duran  and
B. Fabrèges
G. L. Richard*
Affiliation:
LAMA, UMR5127, Université de Savoie Mont-Blanc, CNRS, 73376 Le Bourget-du-Lac, France
A. Duran
Affiliation:
Institut Camille Jordan, CNRS UMR 5208, Université Claude Bernard Lyon 1, 69100 Villeurbanne, France
B. Fabrèges
Affiliation:
Institut Camille Jordan, CNRS UMR 5208, Université Claude Bernard Lyon 1, 69100 Villeurbanne, France
*
†Email address for correspondence: [email protected]
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Abstract

We derive a two-dimensional depth-averaged model for coastal waves with both dispersive and dissipative effects. A tensor quantity called enstrophy models the subdepth large-scale turbulence, including its anisotropic character, and is a source of vorticity of the average flow. The small-scale turbulence is modelled through a turbulent-viscosity hypothesis. This fully nonlinear model has equivalent dispersive properties to the Green–Naghdi equations and is treated, both for the optimization of these properties and for the numerical resolution, with the same techniques which are used for the Green–Naghdi system. The model equations are solved with a discontinuous Galerkin discretization based on a decoupling between the hyperbolic and non-hydrostatic parts of the system. The predictions of the model are compared to experimental data in a wide range of physical conditions. Simulations were run in one-dimensional and two-dimensional cases, including run-up and run-down on beaches, non-trivial topographies, wave trains over a bar or propagation around an island or a reef. A very good agreement is reached in every cases, validating the predictive empirical laws for the parameters of the model. These comparisons confirm the efficiency of the present strategy, highlighting the enstrophy as a robust and reliable tool to describe wave breaking even in a two-dimensional context. Compared with existing depth-averaged models, this approach is numerically robust and adds more physical effects without significant increase in numerical complexity.

JFM classification

Geophysical and Geological Flows: Coastal engineering Geophysical and Geological Flows: Shallow water flows Waves/Free-surface Flows: Wave breaking
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 867 , 25 May 2019 , pp. 146 - 194
Copyright
© 2019 Cambridge University Press 

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Richard et al. supplementary movie

Numerical simulation of the laboratory experiments of Swigler (2009) using the 2D model: propagation of a solitary wave on a realistic coastal configuration including a reef and the run-up and run-down phenomena on a beach.

Download Richard et al. supplementary movie(Video)
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