The instability of axial-symmetric gravity-capillary waves generated by a vertically-oscillating sphere

Abstract

When a floating sphere is forced to oscillate vertically, axial-symmetric outgoing ring waves are generally expected to be produced. Laboratory experiments, however, show that when the oscillation amplitude of the sphere exceeds a threshold value, the axial-symmetric waves abruptly transfigure into asymmetric waves. This problem is related to interfacial instability phenomena widely seen in lab model tests such as sloshing, ship model wakes measurements, etc. Despite its fundamental importance, the mechanism that governs the occurrence of this phenomenon is still unknown. The objective of this thesis is to understand the mechanism of this instability phenomenon using theoretical analysis and direct numerical simulations. We first theoretically 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 progressive radial cross-wave. Unlike standing cross-waves, progressive cross-wave loses energy during propagation by overcoming the work done by surface tension at the waterline and through wave radiation to far field. We then theoretically investigate the problem of subharmonic resonant interaction of progressive ring wave with progressive cross-wave. We derive the nonlinear spatial-temporal evolution equation governing the motion of cross-wave by use of the average Lagrangian method. In addition to energy-input terms from the interaction with 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 oscillation amplitude beyond which the cross-wave becomes unstable and grows with time by taking energy from the ring wave. The theoretical prediction of the characteristic features of generated radial cross-waves agrees well with experimental observations, but the threshold value of oscillation amplitude is about 50% smaller. We finally investigate the instability of finite-amplitude progressive ring waves by direct numerical simulations. The analysis employs the transition matrix (TM) approach and uses a quadratic boundary-element method (QBEM) for computation of the fully-nonlinear wave dynamics. When the nonlinear ring wave effects and viscous effects are accounted for, the predicted threshold value of sphere oscillation amplitude matches the experimental data excellently.

In the case of relatively small-amplitude oscillations, the growth rates and shape of the unstable modes from the TM-QBEM computation agree well with the weakly nonlinear theoretical analysis we developed. This further confirms that the fundamental mechanism of the instability is associated with the triad resonance of the progressive ring wave with its subharmonic progressive radial cross-waves. The dependence of threshold value and growth rate of unstable modes on the physical parameters (such as oscillation frequency and amplitude of the body, initial phase of the disturbance) is also investigated and quantified.