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Self-similar, time-dependent flows with a free surface

Published online by Cambridge University Press:  29 March 2006

Michael S. Longuet-Higgins
Michael S. Longuet-Higgins
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge and Institute of Oceanographic Sciences, Wormley, Godalming, England
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Abstract

A simple derivation is given of the parabolic flow first described by John (1953) in semi-Lagrangian form. It is shown that the scale of the flow decreases like t−3, and the free surface contracts about a point which lies one-third of the way from the vertex of the parabola to the focus.

The flow is an exact limiting form of either a Dirichlet ellipse or hyperbola, as the time t tends to infinity.

Two other self-similar flows, in three dimensions, are derived. In one, the free surface is a paraboloid of revolution, which contracts like t−2 about a point lying one-quarter the distance from the vertex to the focus. In the other, the flow is non-axisymmetric, and the free surface contracts like t−5.

The parabolic flow is shown to be one of a general class of self-similar flows in the plane, described by rational functions of degree n. The parabola corresponds to n = 2. When n = 3 there are two new flows. In one of these the scale varies as t12/7 and the free surface has the appearance of a trough filling up. In the other, the free surface resembles flow round the end of a rigid wall; the scale varies as t−4·17.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 73 , Issue 4 , 24 February 1976 , pp. 603 - 620
Copyright
© 1976 Cambridge University Press

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