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Near-critical hydraulic flows in two-layer fluids

Published online by Cambridge University Press:  07 March 2007

ALFRED KLUWICK ,
STEFAN SCHEICHL  and
EDWARD A. COX
ALFRED KLUWICK
Affiliation:
Institute of Fluid Dynamics and Heat Transfer, Vienna University of Technology, A-1040 Vienna, Austria
STEFAN SCHEICHL
Affiliation:
Institute of Fluid Dynamics and Heat Transfer, Vienna University of Technology, A-1040 Vienna, Austria
EDWARD A. COX
Affiliation:
School of Mathematical Sciences, University College Dublin, Dublin, Ireland
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Abstract

This paper deals with the propagation of nearly resonant gravity waves in two-layer flows over a bottom topography assuming that both fluids are incompressible and inviscid. Evolution equations are derived for weakly nonlinear surface-layer and internal-layer waves in the hydraulic limit of infinite wavelength. Special emphasis is placed on the flow regime where the quadratic nonlinear parameter associated with internal-layer waves is small or vanishes. For example, this is the case for all possible density ratios if the velocities in both layers are equal and if the interface height is close to one-half the total fluid-layer height. The waves then exhibit so-called mixed nonlinearity leading in turn to the formation of positive and negative hydraulic jumps. Considerations based on a model equation for the internal dissipative–dispersive structure of hydraulic jumps indicate that the admissibility of discontinuities in this regime depends strongly on the relative magnitudes of dispersion and dissipation. Surprisingly, these admissible hydraulic jumps may violate the wave-speed-ordering relationship which requires that the upstream wave speed does not exceed the propagation speed of the discontinuity. An important example is provided by the inviscid hydraulic jump, which has been known for some time, although its non-classical nature, in that it transmits rather than absorbs waves, has apparently not been recognized before.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 575 , March 2007 , pp. 187 - 219
Copyright
Copyright © Cambridge University Press 2007

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