Bidirectional energy cascade in surface capillary waves

Statistical Equilibrium of Large Scales in Three-Dimensional Hydrodynamic Turbulence

We investigate experimentally three-dimensional (3D) hydrodynamic turbulence at scales larger than the forcing scale. We manage to perform a scale separation between the forcing scale and the container size by injecting energy into the fluid using centimetric magnetic particles. We measure the statistics of the fluid velocity field at scales larger than the forcing scale (energy spectra, velocity distributions, and energy flux spectrum). In particular, we show that the large-scale dynamics are in statistical equilibrium and can be described with an effective temperature, although not isolated from the turbulent Kolmogorov cascade. In the large-scale domain, the energy flux is zero on average but exhibits intense temporal fluctuations. Our Letter paves the way to use equilibrium statistical mechanics to describe the large-scale properties of 3D turbulent flows.

Observation of Thermal Equilibrium in Capillary Wave Turbulence

We investigate capillary wave turbulence at scales larger than the forcing one. At such scales, our measurements show that the surface waves dynamics is the one of a thermal equilibrium state in which the effective temperature is related to the injected power. We characterize this evolution with a scaling law and report the statistical properties of the large-scale surface elevation depending on this effective temperature.

Frequency dispersion of small-amplitude capillary waves in viscous fluids

This work presents a detailed study of the dispersion of capillary waves with small amplitude in viscous fluids using an analytically derived solution to the initial value problem of a small-amplitude capillary wave as well as direct numerical simulation. A rational parametrization for the dispersion of capillary waves in the underdamped regime is proposed, including predictions for the wave number of critical damping based on a harmonic-oscillator model. The scaling resulting from this parametrization leads to a self-similar solution of the frequency dispersion of capillary waves that covers the entire underdamped regime, which allows an accurate evaluation of the frequency at a given wave number, irrespective of the fluid properties. This similarity also reveals characteristic features of capillary waves, for instance that critical damping occurs when the characteristic time scales of dispersive and dissipative mechanisms are balanced. In addition, the presented results suggest that the widely adopted hydrodynamic theory for damped capillary waves does not accurately predict the dispersion when viscous damping is significant, and an alternative definition of the damping rate, which provides consistent accuracy in the underdamped regime, is presented.