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Article

Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments

1
School of Computing and Mathematics, Keele University, Staffordshire ST5 5BG, UK
2
Department of Ocean Engineering, Texas A&M University, 727 Ross Street, College Station, TX 77843, USA
3
Naval Architecture and Ocean Engineering College, Dalian Maritime University, Dalian 116026, China
4
School of Mathematics and Statistics, University College Dublin, Belfield, D04 V1W8 Dublin, Ireland
5
School of Mathematical Sciences, Technological University Dublin—City Campus, Grangegorman Lower, D07 ADY7 Dublin, Ireland
*
Author to whom correspondence should be addressed.
Former Affiliation: Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI 48109, USA.
Former Affiliation: State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116023, China; experiments performed at the University of Michigan.
Academic Editor: Alexander I. Dyachenko
Fluids 2021, 6(6), 205; https://doi.org/10.3390/fluids6060205
Received: 6 April 2021 / Revised: 26 May 2021 / Accepted: 27 May 2021 / Published: 1 June 2021
(This article belongs to the Special Issue Recent Advances in Free Surface Hydrodynamics)
In this work we consider the problem of finding the simplest arrangement of resonant deep-water gravity waves in one-dimensional propagation, from three perspectives: Theoretical, numerical and experimental. Theoretically this requires using a normal-form Hamiltonian that focuses on 5-wave resonances. The simplest arrangement is based on a triad of wavevectors K1+K2=K3 (satisfying specific ratios) along with their negatives, corresponding to a scenario of encountering wavepackets, amenable to experiments and numerical simulations. The normal-form equations for these encountering waves in resonance are shown to be non-integrable, but they admit an integrable reduction in a symmetric configuration. Numerical simulations of the governing equations in natural variables using pseudospectral methods require the inclusion of up to 6-wave interactions, which imposes a strong dealiasing cut-off in order to properly resolve the evolving waves. We study the resonance numerically by looking at a target mode in the base triad and showing that the energy transfer to this mode is more efficient when the system is close to satisfying the resonant conditions. We first look at encountering plane waves with base frequencies in the range 1.32–2.35 Hz and steepnesses below 0.1, and show that the time evolution of the target mode’s energy is dramatically changed at the resonance. We then look at a scenario that is closer to experiments: Encountering wavepackets in a 400-m long numerical tank, where the interaction time is reduced with respect to the plane-wave case but the resonance is still observed; by mimicking a probe measurement of surface elevation we obtain efficiencies of up to 10% in frequency space after including near-resonant contributions. Finally, we perform preliminary experiments of encountering wavepackets in a 35-m long tank, which seem to show that the resonance exists physically. The measured efficiencies via probe measurements of surface elevation are relatively small, indicating that a finer search is needed along with longer wave flumes with much larger amplitudes and lower frequency waves. A further analysis of phases generated from probe data via the analytic signal approach (using the Hilbert transform) shows a strong triad phase synchronisation at the resonance, thus providing independent experimental evidence of the resonance. View Full-Text
Keywords: water gravity waves; 5-wave resonances; pseudospectral numerical simulations; water wave tank experiments water gravity waves; 5-wave resonances; pseudospectral numerical simulations; water wave tank experiments
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MDPI and ACS Style

Lucas, D.; Perlin, M.; Liu, D.-Y.; Walsh, S.; Ivanov, R.; Bustamante, M.D. Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments. Fluids 2021, 6, 205. https://doi.org/10.3390/fluids6060205

AMA Style

Lucas D, Perlin M, Liu D-Y, Walsh S, Ivanov R, Bustamante MD. Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments. Fluids. 2021; 6(6):205. https://doi.org/10.3390/fluids6060205

Chicago/Turabian Style

Lucas, Dan, Marc Perlin, Dian-Yong Liu, Shane Walsh, Rossen Ivanov, and Miguel D. Bustamante. 2021. "Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments" Fluids 6, no. 6: 205. https://doi.org/10.3390/fluids6060205

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