Development of a Shear Instability in Nodal Zones of a Standing Internal Wave

Abstract

The results of experimentally investigating the initial stage of development of shear instability of the interface between two immiscible fluids relatively oscillating during the parametric excitation of standing internal waves are presented. Three stages of distortion of the sinusoidal wave profile are distinguished: the formation of short secondary waves, their breaking, and transition to large-scale vortex formations. It is shown that in the nodal zones of a standing wave quasi-stationary wave perturbations start to develop at wave steepnesses Γ ∼ 0.08-0.13 and critical Reynolds numbers of the laminar boundary layer R ∼ 90-300. The experimental data are compared with the classical Kelvin-Helmholtz theory.

The subcritical transition to turbulence of Faraday waves in miscible fluids

“…The results reveal the subcritical nature of the wavebreaking detected roughly for wave steepnesses ka ∼ 0.75. These values are consistently larger than ka ∼ 0.4 reported by Kalinichenko (2005) but associated with the earlier 'blurring' stage of the instability. Remarkably, both theories indicate that this should correspond to a parametric resonance subharmonic instability developing when the primary wave amplitude reaches a critical value.…”

“…Also, one cannot exclude the possibility that the unstable modes from the higher parametric bands (PR2, for instance) play a role in the destabilization process of the primary wave. More precisely it may explain the blurring of the interface already observed by Kalinichenko (2005) well before the wavebreaking at ka ∼ 0.4. These PR bands are, however, expected to be seriously damped by the viscosity in addition to having a lower growth rate.…”

“…The local analysis emphasizes the importance of the shear in the breakdown process of the primary wave. Although the works of Thorpe (1968) and Kalinichenko (2005) have early recognized this aspect, the nature of the instability -either Kelvin-Helmholtz (KH) or parametric resonance (PR) type -has not been fully identified in this Faraday wave context. The results of Kelly (1965) and Khenner et al (1999) do not apply to our specific configuration where the acceleration oscillates at frequency ω while the shear velocity is subharmonic with frequency ω/2.…”

“…More precisely, several phases can be identified for the instability with a 'blurring' of the interface preceding its 'roll-up'. Kalinichenko (2005) has also observed and investigated experimentally the breaking process of a Faraday wave between b) Time series images from the camera zooming on one wavelength and presenting two oscillation periods of the primary wave. This illustrates the different stages of the wavebreaking, with first a 'blurring' of the interface at the node followed by a 'roll-up'.…”

“…The Rayleigh-Taylor type instability does not seem to play a role in the process as the acceleration induced by the primary wave displacement is not sufficient to invert the gravity. Thorpe (1968) and Kalinichenko (2005) suggest instead that a sort of Kelvin-Helmholtz instability, 'although not in a simple form', is at work. Due to the strong time dependence of this configuration, evaluating locally the Richardson number at the node cannot be sufficient to assess the importance of the shear instability.…”