Observation of Thermal Equilibrium in Capillary Wave Turbulence

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.

Impact of the Dissipation on the Nonlinear Interactions and Turbulence of Gravity-Capillary Waves

Gravity-capillary waves at the water surface are an obvious example illustrating wave propagation in the laboratory, and also nonlinear wave phenomena such as wave interactions or wave turbulence. However, at high-enough frequencies or small scales (i.e., the frequencies typically above 4 Hz or wavelengths below 10 cm), the viscous dissipation cannot be neglected, which complicates experimental, theoretical, and numerical approaches. In this review, we first derive, from the fundamental principles, the features of the gravity-capillary waves. We then discuss the origin and the magnitude of the viscous wave. dissipation in the laboratory and under field conditions. We then show that the significant level of dissipation has important consequences on nonlinear effects involving waves. The nonlinearity level quantified by the wave steepness must be large enough to overcome the viscous dissipation. Specifically, using water as fluid in the field and in the laboratory, nonlinear wave interactions and wave turbulence occur most of the time in a non-weakly nonlinear regime, when the waves are in the capillary or gravity-capillary range.

Constrained Reversible System for Navier-Stokes Turbulence

Following a Gallavotti's conjecture, stationary states of Navier-Stokes fluids are proposed to be described equivalently by alternative equations besides the Navier-Stokes equation itself. We discuss a model system symmetric under time reversal based on the Navier-Stokes equations constrained to keep the enstrophy constant. It is demonstrated through highly resolved numerical experiments that the reversible model evolves to a stationary state which reproduces quite accurately all statistical observables relevant for the physics of turbulence extracted by direct numerical simulations (DNS) at different Reynolds numbers. The possibility of using reversible models to mimic turbulence dynamics is of practical importance for the coarse-grained version of Navier-Stokes equations, as used in large-eddy simulations. Furthermore, the reversible model appears mathematically simpler, since enstrophy is bounded to be constant for every Reynolds number. Finally, the theoretical interest in the context of statistical mechanics is briefly discussed.

Low-frequency spectra of bending wave turbulence

We study experimentally the dynamics of long waves among turbulent bending waves in a thin elastic plate set into vibration by a monochromatic forcing at a frequency f_{o}. This frequency is chosen large compared with the characteristic frequencies of bending waves. As a consequence, a range of conservative scales without energy flux on average exists for frequencies f<f_{o}. Within this range, we report a flat power density spectrum for the orthogonal velocity, corresponding to energy equipartition between modes. Thus, the average energy per mode β^{-1}-analogous to a temperature-fully characterizes the large-scale turbulent wave field. We present an expression for β as a function of the forcing frequency, amplitude, and of the plate characteristics.

Near-Field Probe of Thermal Fluctuations of a Hemispherical Bubble Surface

We report measurements of resonant thermal capillary oscillations of a hemispherical liquid gas interface obtained using a half bubble deposited on a solid substrate. The thermal motion of the hemispherical interface is investigated using an atomic force microscope cantilever that probes the amplitude of vibrations of this interface versus frequency. The spectrum of such nanoscale thermal oscillations of the bubble surface presents several resonance peaks and reveals that the contact line of the hemispherical bubble is pinned on the substrate. The analysis of these peaks allows us to measure the surface viscosity of the bubble interface. Minute amounts of impurities are responsible for altering the rheology of the pure water surface.

Saturation of the Inverse Cascade in Surface Gravity-Wave Turbulence

We report on the observation of surface gravity-wave turbulence at scales larger than the forcing ones in a large basin. In addition to the downscale transfer usually reported in gravity-wave turbulence, an upscale transfer is observed, interpreted as the inverse cascade of weak turbulence theory. A steady state is achieved when the inverse cascade reaches a scale in between the forcing wavelength and the basin size, but far from the latter. This inverse cascade saturation, which depends on the wave steepness, is probably due to the emergence of nonlinear dissipative structures such as sharp-crested waves.

Wave Turbulence on the Surface of a Fluid in a High-Gravity Environment

We report on the observation of gravity-capillary wave turbulence on the surface of a fluid in a high-gravity environment. By using a large-diameter centrifuge, the effective gravity acceleration is tuned up to 20 times Earth's gravity. The transition frequency between the gravity and capillary regimes is thus increased up to one decade as predicted theoretically. A frequency power-law wave spectrum is observed in each regime and is found to be independent of the gravity level and of the wave steepness. While the timescale separation required by weak turbulence is well verified experimentally regardless of the gravity level, the nonlinear and dissipation timescales are found to be independent of the scale, as a result of the finite size effects of the system (large-scale container modes) that are not taken currently into account theoretically.

Surface hydrodynamics of viscoelastic fluids and soft solids: Surfing bulk rheology on capillary and Rayleigh waves

From the recent advent of the new soft-micro technologies, the hydrodynamic theory of surface modes propagating on viscoelastic bodies has reinvigorated this field of technology with interesting predictions and new possible applications, so recovering its scientific interest very limited at birth to the academic scope. Today, a myriad of soft small objects, deformable meso- and micro-structures, and macroscopically viscoelastic bodies fabricated from colloids and polymers are already available in the materials catalogue. Thus, one can envisage a constellation of new soft objects fabricated by-design with a functional dynamics based on the mechanical interplay of the viscoelastic material with the medium through their interfaces. In this review, we recapitulate the field from its birth and theoretical foundation in the latest 1980s up today, through its flourishing in the 90s from the prediction of extraordinary Rayleigh modes in coexistence with ordinary capillary waves on the surface of viscoelastic fluids, a fact first confirmed in experiments by Dominique Langevin and me with soft gels [Monroy and Langevin, Phys. Rev. Lett. 81, 3167 (1998)]. With this observational discovery at sight, we not only settled the theory previously formulated a few years before, but mainly opened a new field of applications with soft materials where the mechanical interplay between surface and bulk motions matters. Also, new unpublished results from surface wave experiments performed with soft colloids are reported in this contribution, in which the analytic methods of wave surfing synthetized together with the concept of coexisting capillary-shear modes are claimed as an integrated tool to insightfully scrutinize the bulk rheology of soft solids and viscoelastic fluids. This dedicatory to the figure of Dominique Langevin includes an appraisal of the relevant theoretical aspects of the surface hydrodynamics of viscoelastic fluids, and the coverage of the most important experimental results obtained during the three decades of research on this field.