Mixing induced by Faraday surface waves

TL;DR

The paper investigates Faraday-wave–driven mixing in a two-fluid system with a small density contrast, using experiments and direct numerical simulations to reveal two short-term mixing pathways and a non-monotonic long-term entrainment as the upper layer homogenizes. A linear three-layer inviscid model and Floquet stability analysis identify how depth, diffusion, and two-interface coupling shape natural modes and resonances, while a Mathieu-based treatment captures saturation amplitudes of surface waves and their nonlinear evolution. Energetic analyses connect surface-driven motions to interfacial mixing via changes in potential and kinetic energy, highlighting a downward, turbulence-driven entrainment mechanism and the reducing interfacial barrier. Finally, a POD-based decomposition separates large-scale, wave-driven flow from small-scale turbulence in both velocity and concentration fields, supporting a physically consistent 1D model that includes depth-dependent turbulent diffusivity and a local Richardson number to reproduce the observed evolution. The work provides qualitative agreement with experiments and suggests pathways for scaling to industrial and oceanographic contexts, including improvements to turbulence parametrizations in geophysical models.

Abstract

We investigate how surface waves enhance mixing across the interface between two miscible fluids with a small density contrast. Imposing a vertical, time-periodic acceleration, we excite Faraday waves both experimentally and numerically. In systems with a shallow density gradient, these standing waves advect the interface and can trigger secondary instabilities. When driven beyond the linear regime, large Faraday crests collapse to form cavities, injecting bubbles and lighter fluid deep into the heavier layer. Together, these mechanisms gradually homogenize the upper layer, diminish the interfacial density jump, and drive the interface downward until it decouples from surface forcing. We report a non-monotonic mixing rate -- first increasing as the interfacial energy barrier lowers, then decreasing as less energy is injected into the weakened surface -- revealing a balance between barrier reduction and energy input. Based on these observations, we introduce a one-dimensional model incorporating a turbulent diffusivity coefficient that depends on depth and the internal Richardson number, which captures the qualitative evolution of the system.

Figures (4)

• Figure 29: Roots of equation \ref{['eqn:A6']} as function of $k W/\pi$ for different scenarios: (a) $L=0$, $k H \gg 1$ and varying $h_{\color{black}\mathrm{init}}$; (b) $L=0$, $H=35$ and varying $h_{\color{black}\mathrm{init}}$; and (c) $k H \gg 1$, $h_{\color{black}\mathrm{init}}=10$ and varying $L$.
• Figure 30: Linear stability of a three-layer system with $H = 35~\unit{\centi\meter}$ and: (a,c) $\omega = 20~\unit{\radian\per\second}$; (b,d) $\omega = 10~\unit{\radian\per\second}$. Subfigures (a,b) present the stability charts for $h_{\color{black}\mathrm{init}} = 10~\unit{\centi\meter}$, while (c,d) present the relative amplitude of the eigenmodes for a case with ${\color{black}F} = 0.5$. (Inset: schematic of interfaces obtained from the unstable eigenmodes $v_j$).
• Figure 31: POD of the velocity and concentration fields used for the scale decomposition of the DNS12 data. Subfigure (a) shows the cumulative explained variances as a function of the number of POD modes; Subfigures (b) and (c) show the spatial structure of the first and second POD modes for the concentration field (in colours) and vertical velocity field (in streamlines).
• Figure 32: Top row shows an example of the decomposition of the vertical velocity field (a) into large- (b) and small-scales (c) based on POD analysis. Bottom row shows a similar decomposition of the concentration field (d) into large- (e) and small-scales (f).