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Gravity–capillary waves in reduced models for wave–structure interactions

Published online by Cambridge University Press:  13 March 2020

Sean Jamshidi Open the ORCID record for Sean Jamshidi [Opens in a new window]  and
Philippe H. Trinh Open the ORCID record for Philippe H. Trinh [Opens in a new window]
Sean Jamshidi
Affiliation:
Department of Mathematics, University College London, LondonWC1E 6BT, UK
Philippe H. Trinh*
Affiliation:
Department of Mathematical Sciences, University of Bath, BathBA2 7AY, UK
*
†Email address for correspondence: p.trinh@bath.ac.uk
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Abstract

This paper is concerned with steady-state subcritical gravity–capillary waves that are produced by potential flow past a wave-making body. Such flows are characterised by two non-dimensional parameters: the Froude number, $F$, and the inverse Bond number, $T$. When the size of the wave-making body is formally small, there are two qualitatively different flow regimes and thus a single bifurcation curve in the $(F,T)$ plane. If, however, the size of the obstruction is of order one, then, in the limit $F,T\rightarrow 0$, Trinh & Chapman (J. Fluid Mech., vol. 724, 2013, pp. 392–424) have shown that the bifurcation curve widens into a band, within which there are four new flow regimes. Here, we use results from exponential asymptotics to show how, in this low-speed limit, the water-wave equations can be asymptotically reduced to a single differential equation, which we solve numerically to confirm one of the new classes of waves. The issue of numerically solving the full set of gravity–capillary equations for potential flow is also discussed.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Capillary waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 890 , 10 May 2020 , A18
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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