The Mathematical Institute, University of Oxford, Eprints Archive

Nonclassical shallow water flows

Edwards, C. and Howison, S. D. and Ockendon, H. and Ockendon, J. R. (2007) Nonclassical shallow water flows. IMA Journal of Applied Mathematics . (In Press)

[img]
Preview
PDF
768Kb

Abstract

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.

Item Type:Article
Uncontrolled Keywords:delta-shock, jet impact, hypercritical flow
Subjects:O - Z > Partial differential equations
D - G > Fluid mechanics
Research Groups:Oxford Centre for Industrial and Applied Mathematics
ID Code:578
Deposited By:Sam Howison
Deposited On:21 Mar 2007
Last Modified:20 Jul 2009 14:22

Repository Staff Only: item control page