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Dispersive dam-break and lock-exchange flows in a two-layer fluid

Published online by Cambridge University Press:  14 January 2011

J. G. ESLER  and
J. D. PEARCE
J. G. ESLER*
Affiliation:
Department of Mathematics, University College London, 25 Gower Street, London WC1E 6BT, UK
J. D. PEARCE
Affiliation:
Department of Mathematics, University College London, 25 Gower Street, London WC1E 6BT, UK
*
†Email address for correspondence: gavin@math.ucl.ac.uk
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Abstract

Dam-break and lock-exchange flows are considered in a Boussinesq two-layer fluid system in a uniform two-dimensional channel. The focus is on inviscid ‘weak’ dam breaks or lock exchanges, for which waves generated from the initial conditions do not break, but instead disperse in a so-called undular bore. The evolution of such flows can be described by the Miyata–Camassa–Choi (MCC) equations. Insight into solutions of the MCC equations is provided by the canonical form of their long wave limit, the two-layer shallow water equations, which can be related to their single-layer counterpart via a surjective map. The nature of this surjective map illustrates that whilst some Riemann-type initial-value problems (dam breaks) are analogous to those in the single-layer problem, others (lock exchanges) are not. Previous descriptions of MCC waves of permanent form (cnoidal and solitary waves) are generalised, including a description of the effects of a regularising surface tension. The wave solutions allow the application of a technique due to El's approach, based on Whitham's modulation theory, which is used to determine key features of the expanding undular bore as a function of the initial conditions. A typical dam-break flow consists of a leftwards-propagating simple rarefaction wave and a rightward-propagating simple undular bore. The leading and trailing edge speeds, leading edge solitary wave amplitude and trailing edge linear wavelength are determined for the undular bore. Lock-exchange flows, for which the initial interface shape crosses the mid-depth of the channel, by contrast, are found to be more complex, and depending on the value of the surface tension parameter may include ‘solibores’ or fronts connecting two distinct regimes of long-wave behaviour. All of the results presented are informed and verified by numerical solutions of the MCC equations.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Ocean processes Geophysical and Geological Flows: Shallow water flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 667 , 25 January 2011 , pp. 555 - 585
Copyright
Copyright © Cambridge University Press 2011

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