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Resonant generation of finite-amplitude waves by the flow of a uniformly stratified fluid over topography

Published online by Cambridge University Press:  26 April 2006

Roger Grimshaw  and
Yi Zengxin
Roger Grimshaw
Affiliation:
School of Mathematics, University of New South Wales, Australia
Yi Zengxin
Affiliation:
National Research Centre for Marine Environment Forecasts, Hai Dian Division, Beijing, China
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Abstract

The forced Korteweg-de Vries equation is now established as the canonical equation to describe resonant, or critical, flow over topography. However, when the fluid is uniformly and weakly stratified, this equation degenerates in that the quadratic nonlinear term is absent. This anomalous, but important, case requires an alternative theory which is the purpose of this paper. We derive a new evolution equation to describe this case which, while having some similarities to the forced Korteweg-de Vries equation, contains two important differences. First, a topography of amplitude α now produces a finite-amplitude response, whereas in the canonical forced Korteweg-de Vries equation, the response scales with α½. Secondly, the maximum amplitude the fluid flow response can achieve is limited by wave breaking, whose onset is characterized by an incipient flow reversal. Various numerical solutions of the new evolution equation are presented spanning a parameter space defined by a resonance detuning parameter, the topographic amplitude and a parameter measuring the strength of the stratification.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 229 , August 1991 , pp. 603 - 628
Copyright
© 1991 Cambridge University Press

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References

Abramovich, G. W. 1963 The Theory of Turbulent Jets, pp. 444–474. The MIT Press.
Andreopoulos, J., Praturi, A & Rodi, W. 1986 Experiments on vertical plane buoyant jets in shallow water. J. Fluid Mech. 168, 305–336.Google Scholar
Brenner, G. & Laghi, M. 1987 Weitere Untersuchungen über zweidimensionale Blasenplumes. Diplomarbeit in Fluiddynamik, ETH Zürich.
Bulson, P. S. 1963 Large scale bubble breakwater experiments. Dock and Harbour Authority. London, Vol. XII, No. 516, Oct. 1963, pp. 191–197.
Bulson, P. S. 1968 The theory and design of bubble breakwaters. Proc. 11th Conf. Coastal Eng, London, vol. 2, pp. 995–1015.
Fanneløp, T. K. & Sjøen, K. 1980 Hydrodynamics of underwater blowouts. AIAA 8th Aerospace Sciences Meeting, January 14–16, Pasadena, California, AIAA paper 80–0219.
Goossens, L. K. 1979 Reservoir destratification with bubble columns. Thesis, Delft University Press.
Hirschberg, S. 1985 Theoretische und experimentelle Studien über zweidimensionale Blasenplumes. Diplomarbeit in Fluiddynamik, ETH Zürich.
Hüsler, S. 1988 Untersuchungen über einen nach oben gerichteten, zweidimensionalen Freistrahl im Wasser. Diplomarbeit in Fluiddynamik, ETH Zürich.
Iamandi, C. & Rouse, H. 1969 Jet-induced circulation and diffusion. J. Hydraul. Div. ASCE 94, (HY 2), 589–601.Google Scholar
Jirka, G. H. 1982 Turbulent buoyant jets in shallow fluid layers. In Turbulent Buoyant Jets and Plumes (ed. W. Rodi). Pergamon.
Jirka, G. H. & Harleman, D. R. F. 1979 Stability and mixing of a vertical plane buoyant jet in confined depths. J. Fluid Mech. 94, 275–304.Google Scholar
Launder, B. E. & Rodi, W. 1983 The turbulent wall jet—Measurements and modeling. Ann. Rev. Fluid Mech. 15, 429–459.Google Scholar
Leitch, A. M. & Baines, W. D. 1989 Liquid volume flux in a weak bubble plume. J. Fluid Mech. 205, 77–98.Google Scholar
Mcguirk, J. & Rodi, W. 1977 A mathematical model for a vertical jet discharging into a shallow lake. Proc. 17th IAHR-Congress, Paper A72.
Milgram, J. H. 1983 Mean flow in round bubble plumes. J. Fluid Mech. 133, 345–376.Google Scholar
Milgram, J. H. & Burgess, J. J. 1984 Measurements of the surface flow above round bubble plumes. Appl. Ocean Res. 6, 40–44.Google Scholar
Sjøen, K. 1983 Modelling of bubble plumes from subsea blowouts. Dissertation, Norwegian Institute of Technology, University of Trondheim, Division of Aero- and Gas Dynamics.
Taylor, G. I. 1955 The action of a surface current, used as a breakwater.. Proc. R. Soc. Lond. A 231, 466–478.Google Scholar
Tekeli, S. & Maxwell, W. H. C. 1978 Behaviour of air bubble screens. Rep. UILU-ENG-74-2019, Dept. Civil Engng, University of Illinois, Urbana-Champaign.
Topham, D. R. 1975a Hydrodynamics of an oil well blowout. Beaufort Sea Tech. Rep. 33.Google Scholar
Topham, D. R. 1975b Measurements of deepset bubble plumes. Institute of Ocean Sciences, Sidney B.C. (unpublished).