Gravity–capillary multi-component wave patterns generated by a single-frequency wave-maker oscillation and subsequent resonances

Cross-waves are standing waves with crests perpendicular to a wave-maker; they are subharmonic waves excited by parametric instability. The modulational and chaotic behaviours of nonlinear cross-waves have been studied widely since the 1970s. Most of the previous work has focused on gravity waves where surface tension can be neglected. In this work we study cross-waves that are highly dependent on surface tension as well as gravity. By oscillating a planar wave-maker either vertically or horizontally with frequencies of 25 Hz through 40 Hz at one end of a rectangular basin, two-dimensional multi-component surface patterns are realized. Using the free-surface synthetic Schlieren technique to measure the surface elevations, multi-dimensional Fourier transforms are utilized to track the evolutionary spectrum of the water surface in both the temporal and spatial domains. Wavelet transforms are implemented to show the development of the various frequency components. Three-wave resonances with and without first subharmonics are observed for small nonlinearity. Three-dimensional oblique propagating cross-waves are generated at higher nonlinearity; unlike most previous cross-wave experiments, this staggered pattern propagates far downstream. Experimental evidence shows that two oblique propagating waves form a two-dimensional short-crested pattern, and that the lateral component of the waves develops into parametric sloshing modes corresponding to the width of the tank. Two regimes of nonlinear wave patterns, resonant triads and oblique propagating cross-waves, are delineated.

CFD Simulation of Ship Seakeeping Performance and Slamming Loads in Bi-Directional Cross Wave

Accurate prediction of ship seakeeping performance in complex ocean environment is a fundamental requirement for ship design and actual operation in seaways. In this paper, an unsteady Reynolds-averaged Navier–Stokes (RANS) computational fluid dynamics (CFD) solver with overset grid technique was applied to estimate the seakeeping performance of an S175 containership operating in bi-directional cross waves. The cross wave is reproduced by linear superposition of two orthogonal regular waves in a rectangle numerical wave tank. The ship nonlinear motion responses, bow slamming loads, and green water on deck induced by cross wave with different control parameters such as wave length and wave heading angle are systemically analyzed. The results demonstrate that both vertical and transverse motion responses, as well as slamming pressure of ship induced by cross wave, can be quite large, and they are quite different from those in regular wave. Therefore, ship navigational safety when suffering cross waves should be further concerned.

Subharmonic resonant interaction of a gravity–capillary progressive axially symmetric wave with a radial cross-wave

We theoretically investigate the problem of subharmonic resonant interaction of a progressive (axially symmetric) ring wave with a radial cross-wave in the context of the potential-flow formulation for gravity-capillary waves. The objective is to understand the nonlinear mechanism governing energy transfer from a progressive ring wave to its subharmonic cross-waves through triadic resonant interactions. We first show that for an arbitrary three-dimensional body floating in an unbounded free surface, there exists a set of homogeneous solutions at any frequency in the gravity-capillary wave context. The homogeneous solution depends solely on the mean free-surface slope at the waterline of the body and physically represents a progressive radial cross-wave. Unlike standing cross-waves, a progressive cross-wave loses energy during propagation by overcoming the work done by surface tension at the waterline and through wave radiation. We then consider the subharmonic interaction of a progressive ring wave, which is forced by a radial swelling–contraction deformation of a vertical circular cylinder, with subharmonic cross-waves. We derive the nonlinear spatial–temporal evolution equation governing the motion of the cross-wave by use of the average Lagrangian method. In addition to energy-input terms from the interaction with the forced ring wave, the evolution equation contains a damping term associated with energy loss in cross-wave propagation. We show that the presence of the damping term leads to a non-trivial threshold value of the ring wave steepness (or amplitude) beyond which the cross-wave becomes unstable and grows with time by taking energy from the ring wave. Finally, we extend this analysis to the experimental case of Tatsuno et al. (Rep. Res. Inst. Appl. Mech. Kyushu University, vol. 17, 1969, pp. 195–215) in which asymmetric wave patterns are observed during high-frequency vertical oscillations of a surface-piercing sphere. The theoretical prediction of the threshold value of oscillation amplitude and characteristic features of generated radial cross-waves agrees reasonably well with experimental observations.