Harmonic to subharmonic transition of the Faraday instability in miscible fluids

Abstract

When a stable stratification between two miscible fluids is excited by a vertical and periodic forcing, a turbulent mixing zone can develop, triggered by the Faraday instability. The mixing zone grows and saturates to a recently predicted final value L sat [Gréa and Ebo Adou, J. Fluid Mech. 837, 293 (2018)] when resonance conditions are no longer fulfilled. Notably, it is expected from the Mathieu stability diagram that the instability may evolve from a harmonic to a subharmonic regime for particular initial conditions. This transition is evidenced here in the full inhomogeneous system using direct numerical simulations with 1024 3 points: the analysis of one-point statistics and spectra reveals that turbulence is greatly enhanced after the transition, while the global anisotropy of both the velocity and concentration fields is significantly reduced. Furthermore, using the concept of sorted density field, we compute the background potential energy e b p of the flow, which increases only after the transition as a signature of irreversible mixing. While the gain in e b p strongly depends on the control parameters of the instability, the cumulative mixing efficiency is more robust. At saturation of the instability, available potential energy is partially released in the flow as background potential energy. Finally, it is shown numerically that for fixed parameters, a multiple-frequency forcing can modify the duration of the harmonic regime without significantly altering the asymptotic state.

The turbulent Faraday instability in miscible fluids

“…For some particular initial conditions, we have evidenced a short harmonic response of the instability, by observing an initial frequency doubling of the mixing zone size L(t), thus confirming recent numerical results (Briard et al 2019). This harmonic regime can only be observed for large enough F and small enough L 0 .…”

“…This phenomenology is crucial since it means that viscosity becomes negligible in the homogeneous framework when the mixing layer is fully developed. This justifies a posteriori why the saturation criterion L sat of (1.1) is well verified in the DNS of Gréa & Ebo Adou (2018) and Briard et al (2019) and in the present laboratory experiments.…”

“…This prediction (1.1) was assessed numerically for a wide range of parameters by DNS: first within a homogeneous framework, and secondly with the full inhomogeneous problem accounting for the two reservoirs of pure fluids (Gréa & Ebo Adou 2018). In both cases, the final size of the mixing layer obtained numerically L end always slightly exceeded the prediction L sat for large forcing terms F. Further confirmation of this criterion was provided in DNS at higher resolution also in the full inhomogeneous system (Briard, Gréa & Gostiaux 2019). In the latter reference, the separation of gravitational potential energy e p into available potential energy e a p and background potential energy e b p was analysed to quantify irreversible mixing.…”

“…The time increment is determined using a third-order low-storage strong-stability-preserving Runge-Kutta scheme, with implicit viscous terms. Periodicity in the vertical inhomogeneous direction is ensured by choosing an initial condition in which the concentration field changes sharply from its light to heavy value at the vertical boundaries, as in Briard et al (2019),…”

“…We characterise the 'harmonic to subharmonic transition' of the instability with the size L tr from which only subharmonic modes subsist. This transition was numerically analysed in Briard et al (2019) and can be detected by a change of frequency of the quantity of interest.…”