Three-dimensional gravity-capillary standing waves: computation, resonance and instability

TL;DR

The paper addresses the computation and analysis of fully nonlinear three-dimensional gravity-capillary standing waves in deep water. It adopts Zakharov's Hamiltonian formulation with the Dirichlet-Neumann operator, truncating the operator to cubic ($n=3$) and quintic ($n=5$) orders to form reduced models that are solved as triply periodic boundary-value problems using a spatio-temporal collocation method. By exploiting extensive spatio-temporal symmetries, the authors reduce the number of unknowns and solve via Newton iterations, validating the approach against full potential-flow results and exploring rich resonance-induced patterns, including Wilton ripples generalizations to 3D. The results reveal multiple bifurcation branches, square/hexagonal/flower-like patterns under three-wave resonance, and an oblique-type instability mechanism, with implications for energy transfer and pattern formation in nonlinear water waves.

Abstract

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.

Figures (35)

• Figure 1: Typical contours of $\eta$ for a Case I standing wave. The rectangles surround by the black dashed and solid lines represent a unit periodic cell and a quadrant on the $(x,y)$-plane, respectively. The shaded region denotes the real computational domain by using \ref{['symmetry_eta']} and \ref{['symmetry_phi']}.
• Figure 2: Typical contours of $\eta$ for a Case II standing wave. The squares surround by the black dashed and solid lines represent a unit periodic cell and a quadrant on the $(x,y)$-plane, respectively. The blue lines stand for the two diagonals $x = \pm y$, and the four red lines denote $x+y = \pm L$ and $x-y = \pm L$. The shaded region denotes the real computational domain by using \ref{['symmetry_eta2']} and \ref{['symmetry_phi2']}.
• Figure 3: Typical contours of $\eta$ for a Case III standing wave. The squares surround by the black dashed and solid lines represent a unit periodic cell and a quadrant on the $(x,y)$-plane, respectively. The blue lines stand for the two diagonals $x = \pm y$. The shaded region denotes the real computational domain by using \ref{['diagonal_eta']} and \ref{['diagonal_phi']}.
• Figure 4: Typical contours of $\eta(x,y,T/4)$ for Case I (a), Case II (b) and Case III (c) standing waves. The rectangle and squares surround by the solid lines represent a quadrant on the $(x,y)$-plane, respectively. The blue lines are the two diagonals $x = \pm y$. The red lines denote $x+y = \pm L$ and $x-y = \pm L$. The green lines stand for $x = \pm L_1/2$ and $y = \pm L_2/2$. The shaded regions denote the real computational domain.
• Figure 5: Schematic of the reduced computational domain by using spatio-temporal symmetries.