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Finite-amplitude steady-state resonant waves in a circular basin

Published online by Cambridge University Press:  31 March 2021

Xiaoyan Yang ,
Frederic Dias Open the ORCID record for Frederic Dias [Opens in a new window] ,
Zeng Liu Open the ORCID record for Zeng Liu [Opens in a new window]  and
Shijun Liao Open the ORCID record for Shijun Liao [Opens in a new window]
Xiaoyan Yang
Affiliation:
Faculty of Maritime and Transportation, Ningbo University, Ningbo315211, PR China School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai200240, PR China
Frederic Dias
Affiliation:
School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland
Zeng Liu
Affiliation:
School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan430074, PR China
Shijun Liao*
Affiliation:
School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai200240, PR China State Key Laboratory of Ocean Engineering, Shanghai200240, PR China State Key Laboratory of Plateau Ecology and Agriculture, School of Hydraulic and Electric Engineering, Qinghai University, Xining810018, PR China
*
†Email address for correspondence: sjliao@sjtu.edu.cn
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Abstract

The steady-state second-harmonic resonance between the fundamental and the second-harmonic modes for waves in a circular basin is investigated by solving the water-wave equations as a nonlinear boundary-value problem. The resulting waves are called (1,2)-waves. The geometry of the basin allows for both travelling waves (TW) and standing waves (SW). A solution procedure based on a homotopy analysis method (HAM) approach is used. In the HAM framework, the mathematical obstacle due to the singularity corresponding to the resonant-wave component can be overcome by adding the resonant term in the initial guess of the velocity potential. Approximate homotopy-series solutions can be obtained for both (1,2)-TW and (1,2)-SW. Two branches of (1,2)-TW and two branches of (1,2)-SW are found. They bifurcate from the trivial solution. For (1,2)-TW, the HAM-based approach is combined with a Galerkin numerical-method-based approach to follow the branches of nonlinear solutions further. The approximate homotopy-series solutions are used as initial guesses for the Galerkin method. As the nonlinearity increases, an increasing number of wave components are involved in the solution.

JFM classification

Waves/Free-surface Flows: Waves/Free-surface Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 915 , 25 May 2021 , A136
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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