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Nonlinear hydroelastic waves on a linear shear current at finite depth

Published online by Cambridge University Press:  31 July 2019

T. Gao Open the ORCID record for T. Gao [Opens in a new window] ,
Z. Wang Open the ORCID record for Z. Wang [Opens in a new window]  and
P. A. Milewski Open the ORCID record for P. A. Milewski [Opens in a new window]
T. Gao
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
Z. Wang*
Affiliation:
Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China School of Engineering Science, University of Chinese Academy of Sciences, Beijing 100049, China
P. A. Milewski
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
*
†Email address for correspondence: zwang@imech.ac.cn
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Abstract

This work is concerned with waves propagating on water of finite depth with a constant-vorticity current under a deformable flexible sheet. The pressure exerted by the sheet is modelled by using the Cosserat thin shell theory. By means of multi-scale analysis, small amplitude nonlinear modulation equations in several regimes are considered, including the nonlinear Schrödinger equation (NLS) which is used to predict the existence of small-amplitude wavepacket solitary waves in the full Euler equations and to study the modulational instability of quasi-monochromatic wavetrains. Guided by these weakly nonlinear results, fully nonlinear steady and time-dependent computations are performed by employing a conformal mapping technique. Bifurcation mechanisms and typical profiles of solitary waves for different underlying shear currents are presented in detail. It is shown that even when small-amplitude solitary waves are not predicted by the weakly nonlinear theory, we can numerically find large-amplitude solitary waves in the fully nonlinear equations. Time-dependent simulations are carried out to confirm the modulational stability results and illustrate possible outcomes of the nonlinear evolution in unstable cases.

JFM classification

Waves/Free-surface Flows: Shear waves Waves/Free-surface Flows: Solitary waves Waves/Free-surface Flows: Surface gravity waves

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 876 , 10 October 2019 , pp. 55 - 86
Copyright
© 2019 Cambridge University Press 

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