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On two-dimensional inviscid flow in a waterfall

Published online by Cambridge University Press:  28 March 2006

N. S. Clarke
N. S. Clarke
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
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Abstract

This paper is concerned with the two-dimensional flow in a free waterfall, falling under the influence of gravity, the fluid being considered to be incompressible and inviscid. A parameter ε, such that 2/ε is the Froude number based on conditions far upstream, is defined and considered to be small. A flowline co-ordinate system is used to overcome the difficulty that the boundary geometry is not known in advance. An asymptotic expansion based on ε is constructed as an approximation valid upstream and near the edge, but singular far downstream. Another asymptotic expansion, based upon the thinness of the fall, is constructed as an approximation valid far downstream, but failing to satisfy the conditions upstream. The two expansions are then matched to give a solution covering the whole flow field. The shapes of the free streamlines are shown for a number of values of ε for which the solutions are seemingly valid.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 22 , Issue 2 , June 1965 , pp. 359 - 369
Copyright
© 1965 Cambridge University Press

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References

Erdelyi, A. 1961 An expansion procedure for singular perturbations. Atti. Accad. Sci. Tornio, 95, 1.Google Scholar
Kaplun, S. & Lagerström, P. A. 1957 Asymptotic expansions of Navier-Stokes solutions for small Reynolds number. J. Math. Mech. 6, 585.Google Scholar
Keller, J. B. & Weitz, M. L. 1957 Theory of thin jets. Proceedings of the ninth International Congress of Applied Mechanics, Brussels.
Southwell, R. & Vaisey, G. 1946 Fluid motions characterized by free streamlines Phil. Trans. A, 240, 117.Google Scholar
Woods, L. C. 1961 The Theory of Subsonic Plane Flow. Cambridge University Press.