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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2512.11191 |
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| _version_ | 1866911340401000448 |
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| author | Guan, Xin |
| author_facet | Guan, Xin |
| contents | We present a numerical study of three-dimensional gravity-capillary standing waves by using cubic and quintic truncated Hamiltonian formulations and the Craig-Sulem expansion of the Dirichlet-Neumann operator (DNO). The resulting models are treated as triply periodic boundary-value problems and solved via a spatio-temporal collocation method without executing initial-value calculations. This approach avoids the numerical stiffness associated with surface tension and numerical instabilities arising from time integration. We reduce the number of unknowns significantly by exploiting the spatio-temporal symmetries for three types of symmetric standing waves. Comparisons with existing asymptotic and numerical results illustrate excellent agreement between the models and the full potential-flow formulation. We investigate typical bifurcations and standing waves that feature square, hexagonal, and more complex flower-like patterns under the three-wave resonance. These solutions are generalisations of the classical Wilton ripples. Temporal simulations of the computed three-dimensional standing waves exhibit perfect periodicity and reveal an instability mechanism based on the reported oblique instability in two-dimensional standing waves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_11191 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Three-dimensional gravity-capillary standing waves: computation, resonance and instability Guan, Xin Fluid Dynamics We present a numerical study of three-dimensional gravity-capillary standing waves by using cubic and quintic truncated Hamiltonian formulations and the Craig-Sulem expansion of the Dirichlet-Neumann operator (DNO). The resulting models are treated as triply periodic boundary-value problems and solved via a spatio-temporal collocation method without executing initial-value calculations. This approach avoids the numerical stiffness associated with surface tension and numerical instabilities arising from time integration. We reduce the number of unknowns significantly by exploiting the spatio-temporal symmetries for three types of symmetric standing waves. Comparisons with existing asymptotic and numerical results illustrate excellent agreement between the models and the full potential-flow formulation. We investigate typical bifurcations and standing waves that feature square, hexagonal, and more complex flower-like patterns under the three-wave resonance. These solutions are generalisations of the classical Wilton ripples. Temporal simulations of the computed three-dimensional standing waves exhibit perfect periodicity and reveal an instability mechanism based on the reported oblique instability in two-dimensional standing waves. |
| title | Three-dimensional gravity-capillary standing waves: computation, resonance and instability |
| topic | Fluid Dynamics |
| url | https://arxiv.org/abs/2512.11191 |