We study the free decay of capillary turbulence on the charged surface of liquid hydrogen. We find that decay begins from the high frequency end of the spectral range, while most of the energy remains localized at low frequencies. The apparent discrepancy with the self-similar theory of nonstationary wave turbulent processes is accounted for in terms of a quasi-adiabatic decay wherein fast nonlinear wave interactions redistribute energy between frequency scales in the presence of finite damping at all frequencies. Numerical calculations based on this idea agree well with experimental data.
Using the FLUIDICS (Fluid Dynamics in Space) experiment in the International Space Station, turbulence of capillary waves at the air-water interface is experimentally investigated in weightlessness. Capillary waves are excited in a spherical container partially filled with water and undergoing sinusoidal or random oscillations. The fluctuations of the interface, recorded with two capacitive probes are analyzed by means of the frequency power spectrum of wave elevation. For high enough forcing amplitudes, we report power-law spectra with exponents close to the prediction of weak wave turbulence…
We report on direct numerical simulation of quasi-one-dimensional bidirectional capillary-wave turbulence. Although nontrivial three-wave and four-wave resonant interactions are absent in this peculiar geometry, we show that an energy transfer between scales still occurs concentrated around the linear dispersion relation that is broadened by nonlinearity. The wave spectrum displays a clear wave number power-law scaling that is found to be in good agreement with the dimensionally prediction for capillary-wave turbulence involving four-wave interactions. The carried out high-order correlation an…
Wave turbulence in quasi-1D geometry is usually not investigated experimentally since low-order resonant wave interactions are theoretically prohibited. Here, we report on the first observation of unidirectional capillary-wave turbulence on the surface of a fluid in a canal. We also show that five-wave interactions are the lowest-order resonant process subsisting at small scales, and are thus probably the one generating such quasi-1D capillary wave turbulence. We show that the wave spectrum is compatible with the corresponding dimensional analysis prediction. The main assumptions of weak turbu…
We study experimentally the dynamics and statistics of capillary waves forced by random steep gravity waves mechanically generated in laboratory. Capillary waves are produced here by gravity waves from nonlinear wave interactions. Using a spatio-temporal measurement of the free-surface, we characterize statistically the random regimes of capillary waves in the spatial and temporal Fourier spaces. For a significant wave steepness ($0.2-0.3$), power-law spectra are observed both in space and time, defining a turbulent regime of capillary waves transferring energy from large scale to small scale.…
We present a link between the theory of deep water waves and that of bubble surface perturbations. Theory correspondence is shown analytically for small wavelengths in the linear regime and investigated numerically in the nonlinear regime. To do so, we develop the second-order spatial perturbation equations for the Rayleigh-Plesset equation and solve them numerically. Our code is publicly available. Studying capillary waves on stable bubbles, we recreate the Kolmogorov-Zakharov spectrum predicted by weak turbulence theory, putting wave turbulence theory to use for bubbles. In this investigatio…
The wave turbulence equation is an effective kinetic equation that describes the dynamics of wave spectrum in weakly nonlinear and dispersive media. Such a kinetic model has been derived by physicists in the sixties, though the well-posedness theory remains open, due to the complexity of resonant interaction kernels. In this paper, we provide a global unique radial strong solution, the first such a result, to the wave turbulence equation for capillary waves.
Capillary waves are perhaps the simplest example to consider for an introduction to wave turbulence. Since the first paper by Zakharov and Filonenko (1967), capillary wave turbulence has been the subject of many studies but a didactic derivation of the kinetic equation is still lacking. It is the objective of this paper to present such a derivation in absence of gravity and in the approximation of deep water. We use the Eulerian method and a Taylor expansion around the equilibrium elevation for the velocity potential to derive the kinetic equation. The use of directional polarities for three-w…
The bubble size distribution below a breaking wave is of paramount interest when quantifying mass exchanges between the atmosphere and oceans. Mass fluxes at the interface are driven by bubbles that are small compared to the Hinze scale $d_h$, the critical size below which bubbles are stable, even though individually these are negligible in volume. Combining experimental and numerical approaches, we report a power law scaling 3/2 of the diameter of small bubble size distribution, for sufficiently large separation of scales between the injection size and the Hinze scale. From an analysis of ind…
The statistical properties of a large number of weakly nonlinear waves can be described in the framework of the Weak Turbulence Theory. The theory is based on the hypothesis of an asymptotically large system. In experiments, the systems have a finite size and the predictions of the theory may not apply because of the presence of discrete modes rather than a continuum of free waves. Our study focusses on the case of waves at the surface of water at scales close to the gravity-capillarity crossover (of order 1~cm). Wave turbulence has peculiar properties in this regime because 1D resonant intera…
The last decade has seen a significant increase in the number of studies devoted to wave turbulence. Many deal with water waves, as modeling of ocean waves has historically motivated the development of weak turbulence theory, which adresses the dynamics of a random ensemble of weakly nonlinear waves in interaction. Recent advances in experiments have shown that this theoretical picture is too idealized to capture experimental observations. While gravity dominates much of the oceanic spectrum, waves observed in the laboratory are in fact gravity-capillary waves, due to the restricted size of wa…
Direct numerical simulation of threedimensional chaotic motion of a dielectric liquid with a free surface under the action of external horizontal electric field is carried out. The numerical model takes into account the effects of surface tension, viscosity, and external isotropic random forcing acting at large scales. A transition from dispersive capillary wave turbulence to quasi-isotropic nondispersive EHD surface turbulence with increase of the external electric field strength is numerically observed for the first time. At the regime of developed EHD wave turbulence, the total electrical e…
The turbulence of capillary waves on the surface of a ferrofluid with a high permeability in a horizontal magnetic field is considered in the framework of a one-dimensional weakly nonlinear model. In the limit of a strong magnetic field, the surface waves under study can propagate without distortions along or against the direction of external field, i.e., similar to Alfvén waves in a perfectly conducting fluid. The interaction of counter-propagating nonlinear waves leads to the development of wave turbulence on the surface of the liquid. The computational data show that the spectrum of turbule…
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 the Earth 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 experimentall…
{Three-wave resonant interactions constitute an essential nonlinear mechanism coupling capillary surface waves. In a previous work, Haudin et al. [Phys. Rev E 93, 043110 (2016)], we have characterized experimentally the generation by this mechanism of a daughter wave, whose amplitude saturates due to the viscous dissipation. Here, we show experimentally the generation of a daughter wave verifying the resonant conditions, but not the dispersion relation.} By modeling the response of the free surface at the lowest nonlinear order, we explain this observation as a forced interaction. {The bandwid…
It is shown that the second-order moment of helicity distribution (the Levich-Tsinober invariant of inviscid hydrodynamics) can be preserved as an adiabatic invariant in magnetohydrodynamics. The distributed chaos approach and inertial range phenomenology have been developed for the magnetohydrodynamic processes dominated by this adiabatic invariant, and some consequences of the theoretical consideration have been compared with results of direct numerical simulations and measurements in the solar wind. Helical scaling in interstellar MHD turbulence has been also briefly discussed in this conte…
Processes of propagation and interaction of nonlinear gravity-capillary waves on the free surface of a deep non-conducting liquid with high dielectric constant under the action of a tangential electric field are numerically simulated. The computational method is based on the time-dependent conformal transformation of the region occupied by the fluid into a half-plane. In the limit of a strong electric field, when the gravitational and capillary forces are negligibly small, there exists an exact analytical solution of the electro-hydrodynamic equations describing propagation without distortions…
A direct numerical simulation of the interaction of plane capillary waves on the surface of a liquid dielectric in an external tangential electric field taking into account viscous forces has been performed. It has been shown that the interaction of counter-propagating nonlinear waves can generate a direct energy cascade. In the quasistationary energy dissipation regime, probability density functions for angles of inclination of the boundary tend to a Gaussian distribution and the shape of the boundary becomes complex and chaotic. The spectrum of the surface perturbations in this regime is des…
This report summarises the results for the numerical simulation of thermocapillary convection in a two-fluid system with a deformable interface. An explicit technique with 3rd order Runge-Kutta method in time, 2nd order ENO for the advection terms, and 2nd order central-differencing for the diffusion terms are employed in the momentum equations for simulating the flow on a staggered grid using a Marker and Cell method. An energy equation is solved numerically and a level-set method is used to implicitly capture the interface. A constant contact angle condition is assumed between the end walls …
We discuss the impact of dissipation on the development of the energy spectrum in wave turbulence of gravity surface waves with emphasis on the effect of surface contamination. We performed experiments in the Coriolis facility which is a 13-m diameter wave tank. We took care of cleaning surface contamination as well as possible considering that the surface of water exceeds 100~m$^2$. We observe that for the cleanest condition the frequency energy spectrum shows a power law decay extending up to the gravity capillary crossover (14 Hz) with a spectral exponent that is increasing with the forcing…
