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Steady-state harmonic resonance of periodic interfacial waves with free-surface boundary conditions based on the homotopy analysis method

Published online by Cambridge University Press:  21 April 2021

Jiyang Li ,
Zeng Liu Open the ORCID record for Zeng Liu [Opens in a new window] ,
Shijun Liao Open the ORCID record for Shijun Liao [Opens in a new window]  and
Alistair G.L. Borthwick Open the ORCID record for Alistair G.L. Borthwick [Opens in a new window]
Jiyang Li
Affiliation:
School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai200240, PR China
Zeng Liu*
Affiliation:
School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology (HUST), Wuhan, Hubei430074, PR China Hubei Provincial Engineering Research Center of Data Techniques and Supporting Software for Ships (DTSSS), Wuhan, Hubei430074, PR China
Shijun Liao
Affiliation:
School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai200240, PR China State Key Laboratory of Plateau Ecology and Agriculture, Xining810018, PR China School of Hydraulic and Electric Engineering, Qinghai University, Xining810018, PR China
Alistair G.L. Borthwick
Affiliation:
School of Engineering, University of Edinburgh, EdinburghEH9 3FB, UK School of Engineering, Computing and Mathematics, University of Plymouth, PlymouthPL4 8AA, UK
*
†Email address for correspondence: z_liu@hust.edu.cn
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Abstract

We investigate the steady-state harmonic resonance of periodic interfacial gravity waves in a two-layer fluid with free surface. Two independent ‘external’ and ‘internal’ modes with separate linear dispersion relationships exist for this two-layer fluid. Exact harmonic resonance occurs when an external mode and an internal mode share the same phase speed and have an integer ratio of wavelengths. The singularity or small divisor caused by the exactly or nearly resonant component is successfully removed by the homotopy analysis method (HAM). Convergent series solutions are obtained of steady-state interfacial wave groups with harmonic resonance. It is found that steady-state resonant waves form a continuum in parameter space. For finite amplitude interfacial waves, the energy carried by surface waves mirrors that carried by interface waves as the water depth varies. As the upper layer depth increases, energy carried by both surface and interface waves transfers from the shorter resonant component to the longer primary one. The paper utilizes a HAM-based analytical approach to obtain a steady-state, periodic, interfacial wave system with exact- and near-resonant interactions between internal and external modes.

JFM classification

Waves/Free-surface Flows: Waves/Free-surface Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 916 , 10 June 2021 , A58
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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