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A class of exact, time-dependent, free-surface flows

Published online by Cambridge University Press:  29 March 2006

M. S. Longuet-Higgins
M. S. Longuet-Higgins
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, and National Institute of Oceanography, Wormley, Surrey
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Abstract

Attention is drawn to a class of inviscid irrotational flows which satisfy the conditions at a time-dependent free surface exactly. The flows are related to the ellipsoids of Dirichlet (1860).

Depending on a parameter P, the cross-section may take the form of a variable ellipse (P < 0), a hyperbola (P > 0) or a pair of parallel lines (P = 0). The elliptical case was investigated both theoretically and experimentally by Taylor (1960). The hyperbolic case (P > 0) is remarkable in that the flow develops a singularity when the angle between the asymptotes approaches a right-angle. It is suggested that this solution represents a possible instability near the crest of a standing gravity wave of large amplitude.

In the intermediate case (P = 0) the solution describes an open-channel flow in which the fluid filaments are stretched uniformly in a horizontal direction. The latter flow is demonstrated experimentally.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 55 , Issue 3 , 10 October 1972 , pp. 529 - 543
Copyright
© 1972 Cambridge University Press

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References

Byrd, P. F. & Friedman, M. D. 1954 Handbook of Elliptic Integrals for Engineers and Physicists. Springer.
Dirichlet, G. L. 1860 Untersuchungen über ein Problem der Hydrodynamik. Abh. Kön. Gest. Wiss. Göttingen, 8, 3–42.Google Scholar
Forchheimer, P. 1930 Hydraulik. Leipzig: Teubuer.
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Longuet-Higgins, M. S. 1969 The exact hydrodynamical description of a slosh. Proc. NATO Advanced Study Institute on Topics in Geophys. Fluid Dyn., p. 44. Univ. Coll. N. Wales, Bangor.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1961 The changes in amplitude of short gravity waves on steady non-uniform currents. J. Fluid Mech. 10, 529–549.Google Scholar
Penney, W. G. & Price, A. T. 1952 Finite periodic stationary gravity waves in a perfect fluid. Phil. Trans. Roy. Soc. A 244, 254–284.Google Scholar
Taylor, G. I. 1953 An experimental study of standing waves. Proc. Roy. Soc. A 218, 44–59.Google Scholar
Taylor, G. I. 1960 Formation of thin flat sheets of water. Proc. Roy. Soc. A 259, 1–17.Google Scholar
Taylor, G. I. 1962 Standing waves on a contracting or expanding current. J. Fluid Mech. 13, 182–192.Google Scholar