Capillary Wave Turbulence Research Articles

Direct Numerical Simulations of Capillary Wave Turbulence

This work presents direct numerical simulations of capillary wave turbulence solving the full three-dimensional Navier-Stokes equations of a two-phase flow. When the interface is locally forced at large scales, a statistical stationary state appears after few forcing periods. Smaller wave scales are generated by nonlinear interactions, and the wave height spectrum is found to obey a power law in both wave number and frequency, in good agreement with weak turbulence theory. By estimation of the mean energy flux from the dissipated power, the Kolmogorov-Zakharov constant is evaluated and found to be compatible with the exact theoretical value. The time scale separation between linear, nonlinear interaction, and dissipative times is also observed. These numerical results confirm the validity of the weak turbulence approach to quantify out-of equilibrium wave statistics.

Energy flux measurement from the dissipated energy in capillary wave turbulence.

We study experimentally the influence of dissipation on stationary capillary wave turbulence on the surface of a liquid by changing its viscosity. We observe that the frequency power-law scaling of the capillary spectrum departs significantly from its theoretical value when the dissipation is increased. The energy dissipated by capillary waves is also measured and found to increase nonlinearly with the mean power injected within the liquid. Here we propose an experimental estimation of the energy flux at every scale of the capillary cascade. The latter is found to be nonconstant through the scales. For fluids of low enough viscosity, we found that both capillary spectrum scalings with the frequency and the newly defined mean energy flux are in good agreement with wave turbulence theory. The Kolmogorov-Zakharov constant is then experimentally estimated and compared to its theoretical value.

Space-time-resolved capillary wave turbulence

We report experiments on the full space- and time-resolved statistics of capillary wave turbulence at the air-water interface. The three-dimensional shape of the free interface is measured as a function of time by using the optical method of diffusing light photography associated with a fast camera. Linear and nonlinear dispersion relations are extracted from the spatiotemporal power spectrum of wave amplitude. When wave turbulence regime is reached, we observe power-law spectra both in frequency and in wave number whose exponents are found to agree with the predictions of capillary wave turbulence theory. Finally, the temporal dynamics of the spatial energy spectrum highlight the occurrence of stochastic bursts transferring wave energy through the spatial scales.

Microscale Capillary Wave Turbulence Excited by High Frequency Vibration

Low frequency (O(10 Hz-10 kHz)) vibration excitation of capillary waves has been extensively studied for nearly two centuries. Such waves appear at the excitation frequency or at rational multiples of the excitation frequency through nonlinear coupling as a result of the finite displacement of the wave, most often at one-half the excitation frequency in so-called Faraday waves and twice this frequency in superharmonic waves. Less understood, however, are the dynamics of capillary waves driven by high-frequency vibration (>O(100 kHz)) and small interface length scales, an arrangement ideal for a broad variety of applications, from nebulizers for pulmonary drug delivery to complex nanoparticle synthesis. In the few studies conducted to date, a marked departure from the predictions of classical Faraday wave theory has been shown, with the appearance of broadband capillary wave generation from 100 Hz to the excitation frequency and beyond, without a clear explanation. We show that weak wave turbulence is the dominant mechanism in the behavior of the system, as evident from wave height frequency spectra that closely follow the Rayleigh-Jeans spectral response η ≈ ω(-17/12) as a consequence of a period-halving, weakly turbulent cascade that appears within a 1 mm water drop whether driven by thickness-mode or surface acoustic Rayleigh wave excitation. However, such a cascade is one-way, from low to high frequencies. The mechanism of exciting the cascade with high-frequency acoustic waves is an acoustic streaming-driven turbulent jet in the fluid bulk, driving the fundamental capillary wave resonance through the well-known coupling between bulk flow and surface waves. Unlike capillary waves, turbulent acoustic streaming can exhibit subharmonic cascades from high to low frequencies; here it appears from the excitation frequency all the way to the fundamental modes of the capillary wave at some four orders of magnitude in frequency less than the excitation frequency, enabling the capillary weakly turbulent wave cascade to form from the fundamental capillary wave upward.

Observation of Thermocapillary Effect in Gas -Assisted CO2 Laser Cutting of Steel

Investigations of melt’s dynamics have been performed with pyrometer in the gasjet-assisted CO2 laser cutting. Pyrometer consists of 4 sensors and local luminosity of metal melt has been measured precisely on different depth along front of cutting with long dimension resolution about 0.1 mm and with time resolution as short as 0.06 ms. The results of data processing of brightness temperature are reported for laser cutting of mild-steel plate 3 mm, 6 mm and 10 mm thick, that allows obtaining spectra of melt’s surface deformation for different values of cutting velocity and assisted gas pressure. The generation of capillary-wave turbulence following thermo-capillary effect can be observed in the locations, where intensity of radiation of CO2 laser on melt’s surface exceeds 1 MW/cm2. The R-M-S of temperature fluctuations is greater than 10 K in the range of sub-millimeter capillary waves. Thus, an additional mechanism of the anomalous absorption on the front of cutting can compensate the low absorption of the metal in case of 10.6 μm laser in comparison with the absorption of the metal in the near infrared range.

Decay of capillary wave turbulence

We report on the observation of freely decaying capillary wave turbulence on the surface of a fluid. The capillary wave turbulence spectrum decay is found to be self-similar in time with the same power law exponent as the one found in the stationary regime, in agreement with weak turbulence predictions. The amplitude of all Fourier modes are found to decrease exponentially with time at the same damping rate. The longest wavelengths involved in the system are shown to be damped by a viscous surface boundary layer. These long waves play the role of an energy source during the decay that sustains nonlinear interactions to keep capillary waves in a wave turbulent state.

Different Regimes for Water Wave Turbulence

We present an experimental study on gravity capillary wave turbulence in water. By using space-time resolved Fourier transform profilometry, the behavior of the wave energy density |η(k,ω)|(2) in the 3D (k,ω) space is inspected for various forcing frequency bandwidths and forcing amplitudes. Depending on the bandwidth, the gravity spectral slope is found to be either forcing dependent, as classically observed in laboratory experiments, or forcing independent. In the latter case, the wave spectrum is consistent with the Zakharov-Filonenko cascade predicted within wave turbulence theory.

Application of optical pyrometry for on-line monitoring in laser-cutting technologies

The results of measurement of fluctuations of brightness temperature T in the region of exposure to laser radiation of a 3 mm-thick steel plate are presented. The local luminosity along the cut-front was measured using two-color multichannel pyrometer. Cutting trials were carried out with CO2 laser (10.6 μm, 1200 W) and fiber laser (1.07 μm, 1800 W). Special attention was given to the frequency range of temperature fluctuations above frequency of melt overflight, aiming on on-line monitoring applications. It is shown that local fluctuations of T are related to local melt’s surface deformations due to unequal radiation absorption; thus the noise spectrum of T fluctuations reflects turbulent surface deformation, caused by gas jet and capillary waves. It is also shown that the thermo-capillary effect with capillary-wave turbulence generation can be observed in case of exposure to 10.6 μm radiation with a laser intensity of about 1 MW/cm2. The power law of “−7/6” describes the spectrum of the T fluctuation variance in this case of anomalous absorption of radiation, and the standard deviation of T is in excess of 10 K for a frequency of 14 kHz. There is no such effect in case of fiber-laser radiation applying, and the source of the capillary waves is only forced low-frequency deformations of the melt surface. The standard deviation of T does not exceed 3 K on the frequency of 14 kHz, and above, and a power law of the spectrum fluctuation is described by about “−3” in that range.

Classical capillary turbulence on the surface of quantum liquid He-II

Superfluid helium-4 is a unique liquid for the experimental study of capillary wave turbulence because of its very low viscosity. We have studied the influence of the amplitude and spectral characteristics of an excitation force on the behavior of the turbulence cascade of capillary waves in a 30-cm-diam cylindrical cell. The experimental results can be explained in terms of wave turbulence theory (WTT) when the pump amplitude is relatively high. However, a very interesting phenomenon is observed at moderate harmonic surface excitation amplitudes. The turbulence spectrum deviates from the power law form predicted by WTT at high frequencies; a local maximum develops, which can be interpreted as wave energy accumulation. Our estimates show that a special case of wave turbulence was realized in our experiments, namely, a discrete turbulence in which the discreteness of the cell resonant frequencies has a strong effect on the mechanism of the nonlinear interaction.

Laboratory experiments on wave turbulence

This review paper is devoted to a presentation of recent progress in wave turbulence. I first present the context and state of the art of this field of research both experimentally and theoretically. I then focus on the case of wave turbulence on the surface of a fluid, and I discuss the main results obtained by our group: caracterization of the gravity and capillary wave turbulence regimes, the first observation of intermittency in wave turbulence, the occurrence of strong fluctuations of injected power in the fluid, the observation of a pure capillary wave turbulence in low gravity environment and the observation of magnetic wave turbulence on the surface of a ferrofluid. Finally, open questions in wave turbulence are discussed.

Symmetry Induced Four-Wave Capillary Wave Turbulence

We report theoretical and experimental results on 4-wave capillary wave turbulence. A system consisting of two immiscible and incompressible fluids of the same density can be written in a Hamiltonian way for the conjugated pair (eta, Psi). Adding the symmetry z --> -z, the set of capillary waves display a Kolmogorov-Zakharov spectrum k(-4) in wave vector space and f(-8/3) in the frequency domain. The wave system is studied experimentally with two immiscible fluids of almost equal densities (water and silicon oil) where the capillary surface waves are excited by a low-frequency random forcing. The probability density function of the local wave amplitude shows a quasi-Gaussian behavior and the power spectral density is shows a power-law behavior in frequency with a slope of -2.75. Theoretical and experimental results are in fairly good agreement with each other.

Study of high-frequency edge of turbulent cascade on the surface of He-II

We present the results of recent experimental study of capillary wave turbulence on the surface of superfluid 4He. Capillary waves on the charged liquid surface were excited by an external electric force and detected by means of an optical technique. It was observed for the first time that a local maximum in wave amplitudes at frequencies of the order of the viscous cut-off frequency can be formed in the case where the wave system is excited by a low-frequency periodical driving force. We suppose that the reason for this wave energy condensation could be in detuning of the nonlinearly generated waves from the resonant frequency of the cell of finite size, i.e. the mechanism considered earlier in the frozen turbulence theory. It was found also that the decrease of wave amplitudes within a viscous damping region observed in the case of excitation by a periodical force is much steeper than that observed in the case of noisy driving force.

Wave Turbulence on the Surface of a Ferrofluid in a Magnetic Field

We report the observation of wave turbulence on the surface of a ferrofluid mechanically forced and submitted to a static normal magnetic field. We show that magnetic surface waves arise only above a critical field. The power spectrum of their amplitudes displays a frequency-power law leading to the observation of a magnetic wave turbulence regime which is experimentally shown to involve a 4-wave interaction process. The existence of the regimes of gravity, magnetic and capillary wave turbulence is reported in the phase space parameters as well as a triple point of coexistence of these three regimes. Most of these features are understood using dimensional analysis or the dispersion relation of the ferrohydrodynamic surface waves.

Discreteness and resolution effects in rapidly rotating turbulence

Rotating turbulence is characterized by the nondimensional Rossby number Ro, which is a measure of the strength of the Coriolis term relative to that of the nonlinear term. For rapid rotation (Ro-->0) , nonlinear interactions between inertial waves are weak, and the theoretical approaches used for other weak (wave) turbulence problems can be applied. The important interactions in rotating turbulence at small Ro become those between modes satisfying the resonant and near-resonant conditions. Often, discussions comparing theoretical results and numerical simulations are questioned because of a speculated problem regarding the discreteness of the modes in finite numerical domains versus continuous modes in unbounded continuous theoretical domains. This argument finds its origin in a previous study of capillary waves, for which resonant interactions have a very particular property that is not shared by inertial waves. This possible restriction on numerical simulations of rotating turbulence to moderate Ro has never been quantified. In this paper, we inquire whether the discreteness effects observed in capillary wave turbulence are also present in inertial wave turbulence at small Ro. We investigate how the discreteness effects can affect the setup and interpretation of studies of rapidly rotating turbulence in finite domains. In addition, we investigate how the resolution of finite numerical domains can affect the different types of nonlinear interactions relevant for rotating inertial wave turbulence theories. We focus on Rossby numbers ranging from 0 to 1 and on periodic domains due to their relevance to direct numerical simulations of turbulence. We find that discreteness effects are present for the system of inertial waves for Rossby numbers comfortably smaller than those used in the most recent numerical simulations of rotating turbulence. We use a kinematic model of the cascade of energy via selected types of resonant and near-resonant interactions to determine the threshold of Ro below which discreteness effects become important enough to render an energy cascade impossible.

Fluctuations of Energy Flux in Wave Turbulence

We report that the power driving gravity and capillary wave turbulence in a statistically stationary regime displays fluctuations much stronger than its mean value. We show that its probability density function (PDF) has a most probable value close to zero and involves two asymmetric roughly exponential tails. We understand the qualitative features of the PDF using a simple Langevin-type model.

Correlation dimension analysis and capillary wave turbulence in Dragon-Wash phenomena

This paper describes the evolution of surface capillary waves of deep water excited by gradually increasing the lateral external force at a single frequency. The vertical velocities of the water surface are measured by using a Polytec Laser Vibrometer with a thin layer of aluminium powder scattering on the surface to reflect the laser beam. Nonlinear interaction processes result in a stationary Fourier spectrum of the vertical surface velocities (the same as the surface elevation), i.e. Iω ∼ ω−3.5. The observed spectrum can be interpreted as a wave-turbulent Kolmogorov spectrum for the case of 'narrowband pumping' for a direct cascade of energy. Correlation dimension analysis of the whole development process reveals four distinct stages during the wave structure development and identifies the wave turbulence stage.

Turbulence of Capillary Waves on the Surface of Quantum Liquids

The techniques developed earlier for study of turbulence on the surface of liquid hydrogen was used for generation of nonlinear waves on the charged surface of liquid 4He. This permitted us to observe for the first time capillary turbulence on the surface of liquid 4He in normal and superfluid state, and to expand in a few times the relative width of the inertial interval, where the Kolmogorov-Zakharov spectrum of turbulence was observed.

Observation of Gravity-Capillary Wave Turbulence

We report the observation of the crossover between gravity and capillary wave turbulence on the surface of mercury. The probability density functions of the turbulent wave height are found to be asymmetric and thus non-Gaussian. The surface wave height displays power-law spectra in both regimes. In the capillary region, the exponent is in fair agreement with weak turbulence theory. In the gravity region, it depends on the forcing parameters. This can be related to the finite size of the container. In addition, the scaling of those spectra with the mean energy flux is found in disagreement with weak turbulence theory for both regimes.

Decay of capillary turbulence on the surface of a viscous liquid

Decay of the turbulence of capillary waves on the surface of a real liquid is studied in the presence of the viscous damping of the waves at all frequencies after stepwise removal of external pumping. The investigation is performed using two different models: the weak turbulence approximation and the local turbulence model in which the energy redistribution over frequencies is described by the polynomial expression in the wave-occupation number. It is shown that the decay of turbulence in the viscous liquid proceeds self-similarly and begins at high frequencies. In the decay process, the frequency distribution of the energy of waves is close to the stationary form Eω ∼ ω−3/2 in a wide frequency range below the boundary frequency of the inertial range during a relatively long time after removal of the external force. The calculation results agree qualitatively with the results of the experiments on capillary turbulence on the charged surface of liquid hydrogen.

Effects of discretization of the spectrum in water-wave turbulence

The effects of discretization of the spectrum on the evolution of weak turbulence of surface gravity waves are investigated by direct numerical simulations. Discretization of the spectrum is implemented by imposing periodic boundary conditions on the horizontal plane. Initial wave fields are constructed so that they have the JONSWAP spectrum, and the nonlinear energy transfer among component waves is estimated by following their evolution deterministically according to the primitive governing equations. It is found that the discretized system can produce the same nonlinear energy transfer as that predicted theoretically for a continuous spectrum, even when the distribution of component waves on the wavenumber plane is so sparse that the corresponding area of the water surface on the horizontal plane is just 4 λ p×4 λ p, with λ p being the wavelength corresponding to the peak of the spectrum, provided that the ensemble averaging is taken over a sufficient number of realizations. The regime of “frozen” turbulence, which is known to appear in the weak turbulence of capillary waves with a discretized spectrum, is not observed in the case of gravity waves. This difference in the effects of discretization of the spectrum on the weak turbulence of capillary waves and that of gravity waves is discussed in relation to the number of trios or quartets of component waves which satisfy the conditions for quasi-resonant nonlinear interactions.