Wave Breaking in Undular Bores with Shear Flows

Malek Abid,Maria Bjørnestad,Henrik Kalisch,Christian Kharif,Mats Brun

doi:10.1007/s42286-020-00046-6

Abstract

It is well known that weak hydraulic jumps and bores develop a growing number of surface oscillations behind the bore front. Defining the bore strength as the ratio of the head of the undular bore to the undisturbed depth, it was found in the classic work of Favre (Ondes de Translation. Dunod, Paris, 1935) that the regime of laminar flow is demarcated from the regime of partially turbulent flows by a sharply defined value 0.281. This critical bore strength is characterized by the eventual breaking of the leading wave of the bore front. Compared to the flow depth in the wave flume, the waves developing behind the bore front are long and of small amplitude, and it can be shown that the situation can be described approximately using the well known Kortweg–de Vries equation. In the present contribution, it is shown that if a shear flow is incorporated into the KdV equation, and a kinematic breaking criterion is used to test whether the waves are spilling, then the critical bore strength can be found theoretically within an error of less than ten percent.

Highlights

A river bore is an upriver-propagating transition between different flow depths which is generally caused by tidal forces
The main purpose of the current paper is to demonstrate that the critical ratio found by Favre [21] can be predicted using fairly simple nonlinear model equations such as the KdV equation in connection with a kinematic breaking criterion which defines the onset of breaking as the point when the horizontal component of the particle velocity exceeds the crest velocity
Sharpening the kinematic criterion if used as a numerical switch makes sense as the numerical dissipation needs time to have an effect on the waves

Summary

Introduction

A river bore is an upriver-propagating transition between different flow depths which is generally caused by tidal forces. To study an undular bore in the context of the KdV equation, one needs to be able to pose a boundary-value problem where the incident-free surface level is imposed at, say, the left end of the domain, and the undisturbed level is imposed at the right end of the domain. (see Fig. 1, left panel). Such a model has been developed for instance in [14,42].

The Experiments of Favre

The KdV Equation in the Presence of Shear Flows

Non-Dimensionalization and Numerical Discretization

Simulations

22. Measurements were made at 600 depths

Discussion

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