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IMA Journal of Applied Mathematics Advance Access published online on December 24, 2007
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxm064
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© The Author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Non-classical shallow water flows

Carina M. Edwards, S. D. Howison, H. Ockendon and J. R. Ockendon{dagger}

Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, 24–29 St Giles', Oxford OX1 3LB, UK

{dagger} Corresponding author. Email: ock{at}maths.ox.ac.uk

Received on November 24, 2006; Accepted on January 24, 2007

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 time-scale, 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 function components. 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 O(F–2) on a base with an 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

Keywords: delta shock; jet impact; hypercritical flow.


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