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On direct methods in water-wave theory

Published online by Cambridge University Press:  21 April 2006

Jonathan J. Shields  and
William C. Webster
Jonathan J. Shields
Affiliation:
Department of Naval Architecture and Offshore Engineering, University of California at Berkeley, CA 94720, USA
William C. Webster
Affiliation:
Department of Naval Architecture and Offshore Engineering, University of California at Berkeley, CA 94720, USA
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Abstract

Model equations for three-dimensional, inviscid flow between two arbitrary, time-varying material surfaces are derived using a ‘direct’ or variational approach due to Kantorovich. This approach results in a hierarchy of approximate theories, each of a higher level of spatial approximation and complexity. It can be shown that the equations are equivalent in substance to ‘the theory of directed fluid sheets’ of Green & Naghdi (1974, 1976).

The theory can be used to study the propagation of long waves in water of finite depth and, as such, competes with theories derived using the classical Rayleigh–Boussinesq perturbation methods. In order to demonstrate that there is an advantage to the present approach, we compare predictions for steady, two-dimensional waves over a horizontal bottom. Numerical solutions indicate that the direct theory converges more rapidly than the perturbation theories. Also, the equations of the higher-order direct theories contain singularities related to waves of limiting height, and indeed such waves can be predicted with relative accuracy. Finally, the range of applicability of the direct theory is far greater: waves as short as three times the water depth can be modelled. This is essentially a deep-water condition, well beyond the range of convergence of the Rayleigh–Boussinesq approach.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 197 , December 1988 , pp. 171 - 199
Copyright
© 1988 Cambridge University Press

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