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Flow past a constriction in a channel: a modal description

Published online by Cambridge University Press:  26 April 2006

G. G. Tomasson  and
W. K. Melville
G. G. Tomasson
Affiliation:
R. M. Parsons Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
W. K. Melville
Affiliation:
R. M. Parsons Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
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Abstract

We consider the waves generated by transcritical flow past a constriction in a channel, or by ships or surface pressure distributions travelling at transcritical speeds. The two-dimensionality of the upstream advancing nonlinear waves, which has been observed both experimentally and numerically by several authors, is described by a modal decomposition of the flow response. We show that the lowest transverse mode may evolve nonlinearly, leading to a two-dimensional response upstream, with the higher transverse modes swept downstream. This description is supported by comparing the initial evolution of the solutions to the corresponding linear and nonlinear problems. Averaging across the channel demonstrates that the three-dimensional problem may be related to the corresponding two-dimensional problem with an additional effective forcing coming from the nonlinear coupling of the higher modes to the lowest two-dimensional mode. This coupling leads to a dependence of the upstream solutions on the channel width as well as the Froude number. Solutions are also obtained for two-layer fluids in which cubic nonlinearity is also important. The inclusion of cubic nonlinearity permits the generation of two-dimensional fronts upstream, and demonstrates that the transition from three- to two-dimensional solutions upstream is not specific to Boussinesq solitary waves.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 232 , November 1991 , pp. 21 - 45
Copyright
© 1991 Cambridge University Press

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