Harmonic to subharmonic transition of the Faraday instability in miscible fluids

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 $L_{\mathrm{sat}}$ [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 $e_p^b$ of the flow, which increases only after the transition as a signature of irreversible mixing. While the gain in $e_p^b$ 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.

Figure 1

Mathieu diagram: stability curves (black) separate stable regions (gray) from harmonic and subharmonic unstable regions. The marginal stability curve is displayed as a dashed red curve. The various initial conditions, delimited by crosses $(\times$; from $\theta =0$ to $\theta =\pi /2)$, are discussed in the text.

Figure 2

Run F1A01 at $\omega t=76$. (a) Vertical profiles in the nonpenalized domain: horizontal average of the concentration ${\langle C\rangle }_H$, variance ${\langle c^2\rangle }_H$, and energies ${\langle u_i^2\rangle }_H$ (normalized by $\max {\langle u_z^2\rangle }_H)$, with total concentration $C(x=y=0,z)$. (b) Snapshot of the 3D concentration field.

Figure 3

(a) Mixing zone size $L\left(t\right)$ for runs F1A01 (black), F1A02 (dark green), F1A1 (red), and F2A01 (light blue); horizontal dashed lines correspond to the prediction (1). Vertical dashed lines indicate the separation between the harmonic and subharmonic regimes. Inset: Fourier transform of $\dot{L}$ as function of the normalized frequency for run F1A01. (b) Snapshot of the 3D concentration field of run F1A01 at four different times: colorbar same as Fig. 2.

Figure 4

Effects of initial conditions for the configuration of run F1A02 (green) reported here as a reference. Larger $L_0$ for run F1A02L0 (red) and smaller $k_{\mathrm{peak}}$ for run F1A02k5 (dashed black): horizontal and vertical lines as in Fig. 3. The shaded region corresponds to the inset, where $L\left(t\right)$ is plotted for run F1A02L0 as function of $\omega t/\left(2\pi \right)$: square symbols delimit the periods.

Figure 5

Global anisotropy for runs F1A01 (black), F1A1 (red), and F2A01 (light blue); vertical dashed lines indicate the separation between the harmonic and subharmonic regimes. (a) Vertical anisotropy of the velocity field $b_{33}$. (b) Zoom for run F1A1 as a function of $\omega t/\left(2\pi \right)$. (c) Concentration field anisotropy ${sin}^2\gamma$.

Figure 6

Turbulent quantities for runs F1A01 (black) and F2A01 (light blue). (a) Froude number Fr; on the right $y$ axis, Reynolds number ${\text{Re}}_L$ for run F1A01. (b) Eddy diffusivity ${\kappa }_t$.

Figure 7

Spherically averaged spectra for run F1A01. (a) $E_{cc}$ at four different times during the subharmonic regime, corresponding to various extrema of $L\left(t\right)$; vertical dashed lines indicate the Ozmidov wave number $k_O$. (b) $E_{cc}$ during one period in the asymptotic state. (c) Compensated spectra $\overline{E_{cc}}\left(k\right)$ and $\overline{E_{uu}}\left(k\right)$, along with the Ozmidov $k_O$ and Kolmogorov $k_{\eta }$ wave numbers; the period average is done for $\omega t\in [107;114]$ corresponding to the spectra of panel (b).

Figure 8

(a) Spatial anisotropy of the concentration field ${sin}^2\gamma \left(k\right)$ for run F1A01: period-averaged (black), and the same four times as Fig. 7. (b) Budget terms of the equations for $\overline{E_{cc}}$ (black) and $\overline{E_{uu}}$ (blue) for run F1A01: $-$ nonlinear transfers, $--$ dissipation, and $-·$ production.

Figure 9

Potential energy and mixing efficiency for run F1A01. (a) Total, background, and available potential energies $e_p,e_p^b$, and $e_p^a$. First inset: p.d.f. at increasing time: $t=0,\omega t=40$, and $\omega t\simeq 69$ when $L\left(t\right)$ is maximal. Second inset: unsorted $L\left(t\right)$, sorted $\tilde{L}\left(t\right)$. (b) Instantaneous mixing efficiencies ${\eta }_{PE}$ (blue) and ${\eta }_b$ (black), along with the cumulative mixing efficiency ${\eta }_b^c$ (red).

Figure 10

(a) Stability diagram for the multiple-frequency case: $b=0$ (black), $b=0.5$ (run F1A01b05, blue), and $b=1$ (run F1A01b1, red). Arrows indicate the new extent of the first harmonic tongue. (b) $L\left(t\right)$ for runs F1A01b05 and F1A01b1. (c) $e_p,e_p^b$, and $e_p^a$ for run F1A01b1. Inset: sorted $\tilde{L}$ and unsorted $L$ mixing zone sizes.

Figure 11

Time evolutions for increasing $F$, corresponding to the DNS runs of Table 1. (a) Fr; (b) $L\left(t\right)/L_{\mathrm{sat}}$.