Wave turbulence in vibrating plates: The effect of damping

Humbert, Thomas; Cadot, Olivier; Düring, Gustavo; Josserand, Christophe; Rica, Sergio; Touzé, Cyril

doi:10.1209/0295-5075/102/30002

Cited by 51 publications

(89 citation statements)

References 20 publications

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“…Furthermore the measurement of the exponent of the injected power is also different from that predicted by WTT [10]. Recent work on water waves and vibrated plates suggest that wideband dissipation is most likely responsible for the latter observation [11][12][13]. Another experiment also suggest that several regimes of wave turbulence of water wave may exist depending on the intensity and frequency of the forcing [14].…”

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confidence: 89%

Nonlocal Resonances in Weak Turbulence of Gravity-Capillary Waves

Aubourg

Mordant

2015

Phys. Rev. Lett.

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We report a laboratory investigation of weak turbulence of water surface waves in the gravitycapillary crossover. By using time-space resolved profilometry and a bicoherence analysis, we observe that the nonlinear processes involve 3-wave resonant interactions. By studying the solutions of the resonance conditions we show that the nonlinear interaction is dominantly 1D and involves collinear wave vectors. Furthermore taking into account the spectral widening due to weak nonlinearity explains that nonlocal interactions are possible between a gravity wave and high frequency capillary ones. We observe also that nonlinear 3-wave coupling is possible among gravity waves and we raise the question of the relevance of this mechanism for oceanic waves.A large ensemble of nonlinear waves can exchange energy and develop a turbulent state. The statistic properties of such wave turbulence have been described theoretically for weak nonlinearity in the framework of the Weak Turbulence Theory (WTT). In this theory, only resonant waves are able to exchange significant amounts of energy over long times due to the weak nonlinear coupling. The predicted phenomenology of the stationary statistical states resembles that of fluid turbulence: energy is injected at large scales and cascades down scale to wavelengths at which dissipation takes over and absorbs energy into heat. A major difference with fluid turbulence is that analytical predictions for the stationary spectra (and other statistical quantities) can be derived for weak wave turbulence For isotropic systems, the predicted energy spectrum E(k) has the following expressionwhere k = |k| is the wavenumber, P the energy flux, C a dimensional constant that can be calculated and α the spectral exponent. N is the number of waves taking part in the resonances. Usually N − 1 corresponds to the order of the nonlinear coupling term of the wave equation (N = 3 for quadratic nonlinearities, N = 4 for cubic ones,...). The waves have then to satisfy the resonance conditions such as k 1 = k 2 + k 3 and ω 1 = ω 2 + ω 3 (for 3-wave interaction). In some cases, these resonances conditions do not have solutions. This is the case in particular for gravity waves at the surface of water. The dispersion relation (for infinite depth) is ω = √ gk and its negative curvature does not allow for solutions of the resonance conditions for 3 waves. Thus the resonances are expected to involve 4 waves. At small wavelengths, water waves are capillary waves for which the dispersion relation is ω = γ ρ 1/2 k 3/2 whose curvature allows for 3-wave resonances (γ is the surface tension and ρ the density). The predicted spectra for water waves are thus expected to be E(k) ∝ P 1/2 k −7/4 for capillary waves and E(k) ∝ P 1/3 k −5/2 [2] for gravity waves. Laboratory experiments largely fail to reproduce this predictions, in particular for the gravity waves. In large or small waves tanks, the spectral exponent of the gravity waves is seen to vary strongly with the forcing intensity and to be close to the WTT predictions a...

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confidence: 89%

Nonlocal Resonances in Weak Turbulence of Gravity-Capillary Waves

Aubourg

Mordant

2015

Phys. Rev. Lett.

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“…The statistical steadiness ∂ E(k)/∂t = 0 should be rigorously verified. Furthermore, the energy injected into the system is not necessarily in strict accordance with the energy flux that cascades in the inertial subrange [14]. In laboratory experiments of surface waves, the energy flux is estimated indirectly by the energy decay rate after switching off the energy input or by the dissipation spectrum.…”

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confidence: 99%

Single-wave-number representation of nonlinear energy spectrum in elastic-wave turbulence of the Föppl–von Kármán equation: Energy decomposition analysis and energy budget

Yokoyama

Takaoka

2014

Phys. Rev. E

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A single-wavenumber representation of nonlinear energy spectrum, i.e., stretching energy spectrum is found in elastic-wave turbulence governed by the Föppl-von Kármán (FvK) equation. The representation enables energy decomposition analysis in the wavenumber space, and analytical expressions of detailed energy budget in the nonlinear interactions are obtained for the first time in wave turbulence systems. We numerically solved the FvK equation and observed the following facts. Kinetic and bending energies are comparable with each other at large wavenumbers as the weak turbulence theory suggests. On the other hand, the stretching energy is larger than the bending energy at small wavenumbers, i.e., the nonlinearity is relatively strong. The strong correlation between a mode a k and its companion mode a −k is observed at the small wavenumbers. Energy transfer shows that the energy is input into the wave field through stretching-energy transfer at the small wavenumbers, and dissipated through the quartic part of kinetic-energy transfer at the large wavenumbers. A total-energy flux consistent with the energy conservation is calculated directly by using the analytical expression of the total-energy transfer, and the forward energy cascade is observed clearly.

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“…Such discrepancy between the WTT and the experiments is mostly due to the real dissipation that is present at every scale so that no truly transparent window is present [13,14].…”

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Transition from Wave Turbulence to Dynamical Crumpling in Vibrated Elastic Plates

et al. 2013

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We study the dynamical regime of wave turbulence of a vibrated thin elastic plate based on experimental and numerical observations. We focus our study to the strongly non linear regime described in a previous letter by N. Yokoyama & M. Takaoka. At small forcing, a weakly non linear regime is compatible with the Weak Turbulence Theory when the dissipation is localized at high wavenumber. When the forcing intensity is increased, a strongly non linear regime emerges: singular structures dominate the dynamics at large scale whereas at small scales the weak turbulence is still present. A turbulence of singular structures, with folds and D-cones, develops that alters significantly the energy spectra and causes the emergence of intermittency.

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Wave turbulence in vibrating plates: The effect of damping

Cited by 51 publications

References 20 publications

Nonlocal Resonances in Weak Turbulence of Gravity-Capillary Waves

Nonlocal Resonances in Weak Turbulence of Gravity-Capillary Waves

Single-wave-number representation of nonlinear energy spectrum in elastic-wave turbulence of the Föppl–von Kármán equation: Energy decomposition analysis and energy budget

Transition from Wave Turbulence to Dynamical Crumpling in Vibrated Elastic Plates

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