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The approach to self-similarity of the solutions of the shallow-water equations representing gravity-current releases

Published online by Cambridge University Press:  20 April 2006

R. E. Grundy  and
James W. Rottman
R. E. Grundy
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews KY16 9SS
James W. Rottman
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW Present address: Meteorology & Assessment Division, Environmental Sciences Research Laboratory, US Environmental Protection Agency, Research Triangle Park, North Carolina 27711.
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Abstract

Known similarity solutions of the shallow-water equations representing the motion of constant-volume gravity currents are studied in both plane and axisymmetric geometries. It is found that these solutions are linearly stable to small correspondingly symmetric perturbations and that they constitute the large-time limits of the solutions of the initial-value problem. Furthermore, the analysis reveals that the similarity solution is approached in an oscillatory manner. Two initial-value problems are solved numerically using finite differences and in each case the approach to the similarity solution is compared with the analytic predictions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 156 , July 1985 , pp. 39 - 53
Copyright
© 1985 Cambridge University Press

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