Session E14: Waves: Faraday Waves

Faraday Waves revisited

We revisit the theoretical description of Faraday waves. We show that, consistently with experiments, the relation of dispersion is not that of free unforced waves; the forcing amplitude and the viscosity play a significant role in the dispersion relation. We then determine the instability thresholds and the wavenumber selection in cases of both short and long waves. We also show that, depending on the depth, the instability leading to the formation can be either supercritical or subcritical, as shown by experimental observations.   

Streaming patterns in Faraday waves

Wave patterns in the Faraday instability have been studied for decades. Besides the rich wave dynamics observed at the interface, Faraday waves hide elusive flow patterns in the bulk --the streaming patterns-- that have not been studied in detail until now. We analyse these streaming flows by conducting experiments in a Faraday-wave setup. To visualize the flows, we perform stroboscopic measurements: tracers are used to generate both trajectory maps and to probe the streaming velocity field via PIV. We identify three types of patterns that can coexist under identical Faraday waves. Next we propose a three-dimensional model that explains streaming flows in quasi-inviscid fluids. We show that the streaming inside the fluid arises from a complex coupling between the bulk and the boundary layers. This coupling can be taken into account by applying modified boundary conditions in a three-dimensional Navier-Stokes formulation for the streaming in the bulk. Numerical simulations based on this theoretical framework show good agreement with experimental results. Simulations reveal that the variety of experimental patterns is linked to the boundary condition at the top interface, which may be strongly affected by the presence of contaminants along the surface.   

Hysteretic Faraday waves

We study with numerical simulations the two-dimensional Faraday waves in two immiscible incompressible fluids when the lower fluid layer is shallow. After the appearance of the well known subharmonic stationary waves, a further instability is observed while the control parameter passes a secondary threshold. A new state then arises, composed of stationary waves with about twice the original pattern amplitude [1],[2]. The bifurcation presents hysteresis: there exists a finite range of the control parameter in which both states are stable. By means of a simple stress balance, we show that a change of the shear stress can explain this hysteresis [1]. Our predictions based on this model are in agreement with our numerical results.\\\\ $[1]$ X. Li, Z. Yu and S. Liao, Phys. Rev. E 92, 033014 (2015)\\ $[2]$ N. P\'erinet, C. Falc\'on, J. Chergui, D. Juric and S. Shin, Phys. Rev. E 93, 063114 (2016)   

Observation of Phillips's spectrum in Faraday waves

We consider the problem of wave turbulence generated by singularities from an experimental point of view. We study a system of Faraday waves interacting with waves generated by a wave-maker driven with a random forcing. We measure the temporal fluctuations of the surface wave amplitude at a given location and we show that for a wide range of forcing parameters the surface height displays a power-law spectra that greatly differs from the one predicted by the WT theory. In the capillary region the power spectrum turns out to be proportional to $f^{-5}$, which we believe is due to singularities moving across the system.   

Galaxy-like organization of floaters at the air-water interface of Faraday waves

The fluid properties mismatch across an air-liquid interface allows to trap particles at it. These particles are called floaters and appear in nature at different scales: plankton, organic residues, and garbage, all relevant for the oceanic ecosystem. In static systems they tend to attract or repel each other, depending on their wetting properties and buoyancy [1]. When they are subjected to a flow, such as surface waves, they may drift and form structures at the interface [2]. In a recent work using PIV on Faraday waves, we have measured a streaming flow that emerges inside the bulk, leading to a slow circulation of fluid particles across the liquid [3]. The flow is mainly generated by the viscous shearing at the walls of the container. Our new experiments show that this flow has a remarkable effect on the drift of small hydrophilic particles (floaters), which leads to a rare arrangement of the floaters that resemble rotating galaxies. The forcing amplitude determines the galaxy shape, controlling the number and the length of its arms as well as its rotation velocity.\\ $[1]$ Vella D. and Mahadevan L., Am. J. Phys. 73, 814 (2005)\\ $[2]$ Sanli C. et al., Phys. Rev. E 89 (5), 053011 (2014)\\ $[3]$ P\'{e}rinet N. et al., http://arxiv.org/abs/1603.07353 (2016)\\   

Wave - fluid particle interaction in the Faraday waves

Faraday waves are parametrically excited perturbations that appear on a liquid surface when the latter is vertically vibrated. Recently it has been discovered that: 1) such wave field can be described as a disordered lattice made of localised oscillating excitations, termed oscillons, 2) the horizontal motion of fluid particles on the water surface reproduces in detail the motion of fluid in two-dimensional turbulence.\\ Here we report experimental measurements of the motion of both entities using Particle Image Velocimetry and Particle Tracking Velocimetry techniques. Those techniques allow to measure Lagrangian and Eulerian features of the oscillon motion and compare them with those of the fluid motion. A strong coupling is uncovered between the erratic motion of the waves and the turbulent agitation of the fluid particles. Both motions show Brownian-type dispersion and the r.m.s velocity of oscillons is directly related to the r.m.s. velocity of the fluid particles in a broad range of vertical accelerations. These results offer new perspectives for predicting surface fluid transport from the knowledge of the wave fields and vice versa. In particular, the broadening of the wave spectra at high wave amplitude can be predicted if the 2D turbulence energy is known.