Focusing of concentric free-surface waves

TL;DR

This paper investigates the behavior of concentric gravito-capillary waves on free surfaces, revealing how viscosity and nonlinearity influence wave focusing and interfacial oscillations, with implications for modeling complex fluid dynamics.

Contribution

It introduces a modal approach to analyze wave focusing in viscous and inviscid liquids, emphasizing the importance of nonlinear effects and the impact of viscosity on wave oscillations.

Findings

Viscosity intensifies oscillations at the symmetry axis.

Nonlinear theory accurately models large interfacial oscillations.

Wave focusing leads to significant interfacial dynamics near the axis.

Abstract

Gravito-capillary waves at free-surfaces are ubiquitous in several natural and industrial processes involving quiescent liquid pools bounded by cylindrical walls. These waves emanate from the relaxation of initial interface distortions, which often take the form of a cavity (depression) centred on the symmetry axis of the container. These surface waves reflect from the container walls leading to a radially inward propagating wave-train converging (focussing) onto the symmetry axis. Under the inviscid approximation and for sufficiently shallow cavities, the relaxation is well-described by the linearised potential-flow equations. Naturally, adding viscosity to such a system introduces viscous dissipation that enervates energy and dampens the oscillations at the symmetry axis. However, for viscous liquids and deeper cavities, these equations are qualitatively inaccurate. In this study, we elucidate a modal approach to study the initial-value problem for concentric gravito-capillary waves generated on a free-surface for inviscid as well as viscous liquids. For a sufficiently deep cavity, the inward focusing of waves results in large interfacial oscillations at the axis, necessitating a second-order nonlinear theory. We demonstrate that this theory effectively models the interfacial behavior and highlights the crucial role of nonlinearity near the symmetry axis. Contrary to expectations, the addition of slight viscosity further intensifies the oscillations at the symmetry axis. This finding underscores the limitations of the potential flow model and suggests avenues for more accurate modelling of such complex free-surface flows.