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

Abstract

We consider 2D free surface gravity waves in prismatic channels with bathymetric variations uniquely in the transverse direction. Starting from the Saint-Venant equations (shallow water equations) we derive a 1D transverse averaged model describing dispersive effects solely related to variations of the channel topography. These effects have been demonstrated in Chassagne et al. JFM 2019 to be predominant in the propagation of bores with Froude numbers below a critical value of about 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 generalizes and improves the linear equations proposed by Chassagne et al. JFM 2019, as well as Quezada de Luna and Ketcheson JFM 2021. The system is recast in two useful forms appropriate for its numerical approximations, whose properties are discussed. Numerical results allow to verify against analytical solutions the implementation of these formulations, and validate our model against fully 2D nonlinear shallow water simulations, as well as the famous experiments by Treske J. Hyd. Res. 1994.