Parametrically forced gravity waves in a circular cylinder and finite-time singularity

Abstract

In this paper we present results on parametrically forced gravity waves in a circular cylinder in the limit of large fluid-depth approximation. The phase diagram that shows the stability-forcing-amplitude threshold and the wave-breaking threshold has been determined in the frequency range of existence of the lowest axisymmetric wave mode. The instability is shown to be supercritical for forcing frequencies at and above the natural frequency and subcritical below in a frequency range where the instability and breaking thresholds do not coincide. Above the instability threshold, the growth in wave amplitude is exponential, but with an initial time delay. The wave-amplitude response curve of stationary wave motions exhibits steady-state wave motion, amplitude modulations and bifurcations to other wave modes at frequencies where the parametric instability boundary of the axisymmetric mode overlaps with the neighbouring modes. The amplitude modulations are either on a slow time scale or exhibit period tripling and intermittent period tripling, without wave breaking. In the wave-breaking regime, a finite-time singularity may occur with intense jet formation, a phenomenon demonstrated by others in fluids of high viscosity and large surface tension. Here, this singular behaviour with jet formation is demonstrated for a low viscosity and low kinematic surface tension liquid. The results indicate that the jet is driven by inertial collapse of the cavity created at the wave trough. Therefore, the jet velocity is determined by the wave fluid velocity but depends, in addition, on kinematic surface tension and viscosity as these affect the last stable wave crest shape and the cavity size.

Singular jets during the collapse of drop-impact craters

“…However, the free-surface dynamics are not self-similar and divergent surface tension is not the driving mechanism, as they suggested and was explained by Brenner (2000). See also the follow-up study by Das & Hopfinger (2008) who suggest inertial focusing, but did not have time-resolved imaging to capture the fastest jets. We further verify this by tracking the collapse of the final dimple radius, in Fig.…”

“…It is therefore unclear what role surface tension and viscosity play, as discussed in Thoroddsen et al (2007b) who showed the focusing of capillary waves on a free-falling drop can generate a fine jet. Das & Hopfinger (2008) have investigated the parametrically forced waves in more detail.…”

Numerical simulation of supersquare patterns in Faraday waves

“…In most of these (e.g. Ciliberto & Gollub, 1985;Crawford, Knobloch & Riecke, 1990;Das & Hopfinger, 2008;Batson, Zoueshtiagh & Narayanan, 2013), the interface takes the form of Bessel functions, including axisymmetric modes, while Rajchenbach, Clamond & Leroux (2013) report intriguing patterns of the form of stars and pentagons.…”

“…Douady & Fauve, 1988;Douady, 1990;Simonelli & Gollub, 1989) and cylindrical (e.g. Ciliberto & Gollub, 1985;Das & Hopfinger, 2008;Batson, Zoueshtiagh & Narayanan, 2013;Rajchenbach, Clamond & Leroux, 2013) containers, which have led to important theoretical developments (e.g. Meron, 1987;Silber & Knobloch, 1989;Crawford, Knobloch & Riecke, 1990;Crawford, 1991;Gomes & Stewart, 1994) in the fields of nonlinear physics and pattern formation.…”

Interfacial flow computations using adaptive Eulerian–Lagrangian method for spacecraft applications

“…When the acceleration exceeds a certain threshold level, referred to threshold acceleration, the stable periodic surface waves are transformed to an unstable aperiodic state and show some upward jets in Figure 13(c). Goodridge et al [14] defined the threshold acceleration as the acceleration level where two droplets are detected within 10 s. Das and Hopfinger [10], however, defined it as a jet forming or wave crest pinching off, followed by irregular motion. In this study, droplets or long filament-like jet formations are assumed as criteria for threshold acceleration because droplet formation itself in numerical studies may result in a different criterion according to grid resolution.…”