Mixing induced by Faraday surface waves

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.

💡 Research Summary

This paper investigates how Faraday surface waves, generated by vertical sinusoidal shaking, enhance mixing across a miscible density interface with a small contrast. The authors combine laboratory experiments and direct numerical simulations (DNS) to explore the dynamics from linear wave motion to strongly nonlinear wave breaking. A rectangular tank (94.6 cm × 11 cm × 67 cm) is filled with a two‑layer water system: fresh water (density ≈ 998 kg m⁻³) on top of salt water (density up to 1100 kg m⁻³). The initial interface is made thin (1–2 cm) by a bottom diffuser. The tank is mounted on a hexapod that imposes a vertical displacement Z = a cos(ωt) with ω = 20 rad s⁻¹. By varying the forcing amplitude a, the dimensionless forcing parameter F = a ω²/g is scanned from 0.12 to 0.5, while the initial interface depth h_init ranges from 2 cm (shallow) to 27 cm (deep).

High‑speed imaging (26 fps, 16‑bit) captures the free surface and the internal interface. A segmentation pipeline converts image intensity to a volume‑fraction field f(x,y,t), then extracts the free‑surface curve ζ_surf and the internal interface ζ_mix via binary thresholding (triangle method for the free surface, mean‑value method for the internal layer). The mean separation h_mixed(t) = ⟨ζ_surf − ζ_mix⟩ is used as a proxy for mixing thickness.

The DNS solves the incompressible Navier–Stokes equations for two‑phase flow together with a transport equation for the volume fraction. Surface tension, viscosity, and molecular diffusion (D_mix ≈ 1.3 × 10⁻⁹ m² s⁻¹) are included. The gravity term is modulated as g(t) = −