Nonclassical shallow water flows

This paper deals with violent discontinuities in shallow water flows with large Froude number $F$. On a horizontal base, the paradigm problem is that of the impact of two fluid layers in situations where the flow can be modelled as two smooth regions joined by a singularity in the flow field. Within the framework of shallow water theory we show that, over a certain timescale, this discontinuity may be described by a delta-shock, which is a weak solution of the underlying conservation laws in which the depth and mass and momentum fluxes have both delta function and step functioncomponents. We also make some conjectures about how this model evolves from the traditional model for jet impacts in which a spout is emitted. For flows on a sloping base, we show that for flow with an aspect ratio of \emph{O}($F^{-2}$) on a base with an \emph{O(1)} or larger slope, the governing equations admit a new type of discontinuous solution that is also modelled as a delta-shock. The physical manifestation of this discontinuity is a small `tube' of fluid bounding the flow. The delta-shock conditions for this flow are derived and solved for a point source on an inclined plane. This latter delta-shock framework also sheds light on the evolution of the layer impact on a horizontal base

KP solitons in shallow water

The main purpose of the paper is to provide a survey of our recent studies on soliton solutions of the Kadomtsev-Petviashvili (KP) equation. The classification is based on the far-field patterns of the solutions which consist of a finite number of line-solitons. Each soliton solution is then defined by a point of the totally non-negative Grassmann variety which can be parametrized by a unique derangement of the symmetric group of permutations. Our study also includes certain numerical stability problems of those soliton solutions. Numerical simulations of the initial value problems indicate that certain class of initial waves asymptotically approach to these exact solutions of the KP equation. We then discuss an application of our theory to the Mach reflection problem in shallow water. This problem describes the resonant interaction of solitary waves appearing in the reflection of an obliquely incident wave onto a vertical wall, and it predicts an extra-ordinary four-fold amplification of the wave at the wall. There are several numerical studies confirming the prediction, but all indicate disagreements with the KP theory. Contrary to those previous numerical studies, we find that the KP theory actually provides an excellent model to describe the Mach reflection phenomena when the higher order corrections are included to the quasi-two dimensional approximation. We also present laboratory experiments of the Mach reflection recently carried out by Yeh and his colleagues, and show how precisely the KP theory predicts this wave behavior.Comment: 50 pages, 25 figure

On the excitation of edge waves on beaches

The excitation of standing edge waves of frequency ½ω by a normally incident wave train of frequency ω has been discussed previously (Guza & Davis 1974; Guza & Inman 1975; Guza & Bowen 1976) on the basis of shallow-water theory. Here the problem is formulated in the full water-wave theory without making the shallow-water approximation and solved for beach angles β = π/2N, where N is an integer. The work confirms the shallow-water results in the limit N » 1, shows the effect of larger beach angles and allows a more complete discussion of some aspects of the problem

SWASHES: a compilation of Shallow Water Analytic Solutions for Hydraulic and Environmental Studies

Numerous codes are being developed to solve Shallow Water equations. Because there are used in hydraulic and environmental studies, their capability to simulate properly flow dynamics is critical to guarantee infrastructure and human safety. While validating these codes is an important issue, code validations are currently restricted because analytic solutions to the Shallow Water equations are rare and have been published on an individual basis over a period of more than five decades. This article aims at making analytic solutions to the Shallow Water equations easily available to code developers and users. It compiles a significant number of analytic solutions to the Shallow Water equations that are currently scattered through the literature of various scientific disciplines. The analytic solutions are described in a unified formalism to make a consistent set of test cases. These analytic solutions encompass a wide variety of flow conditions (supercritical, subcritical, shock, etc.), in 1 or 2 space dimensions, with or without rain and soil friction, for transitory flow or steady state. The corresponding source codes are made available to the community (http://www.univ-orleans.fr/mapmo/soft/SWASHES), so that users of Shallow Water-based models can easily find an adaptable benchmark library to validate their numerical methods.Comment: 40 pages There are some errors in the published version. This is a corrected versio