Self-similar formation of an inverse cascade in vibrating elastic plates

Düring, Gustavo; Josserand, Christophe; Rica, Sergio

doi:10.1103/physreve.91.052916

Cited by 12 publications

(19 citation statements)

References 36 publications

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“…However, although it is needed to dissipate at small scale the energy to reach a stationary regime, there is a priori no need to pump the wave action at large scale: it should be automatically done by the dynamics since the wave-action is not conserved. This configuration has been studied recently in great details in [30], showing a self-similar singular dynamics followed by a relaxation process that converges at large scale towards the expected inverse cascade spectrum eq. (102).…”

Section: A Possible Inverse Cascade?mentioning

confidence: 86%

“…Soon after our original work, elastic plate wave turbulence was observed experimentally, showing slightly different spectrum power laws [21,22]. Following these discoveries, numerous works have investigated the dynamics of oscillating elastic plates to explain this discrepancy between numerics-theory and experiments in the turbulent spectra [23,24,25,26,27,28,29], showing eventually that it could be mainly attributed to the particular dissipation of real plates.Vibrating elastic plates have been shown to provide an excellent prototype system to investigate and test different wave turbulence regimes, such as inverse cascade [30], transitory dynamics [29,31,32,33] high forcing [34,35], the breakdown of the WTT or the onset of intermittency [36,37,38] for instance.…”

Section: Introductionmentioning

confidence: 99%

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Wave turbulence theory of elastic plates

Düring

Josserand

Rica

2017

Physica D: Nonlinear Phenomena

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This article presents the complete study of the long-time evolution of random waves of a vibrating thin elastic plate in the limit of small plate deformation so that modes of oscillations interact weakly. According to the wave turbulence theory a nonlinear wave system evolves in longtime creating a slow redistribution of the spectral energy from one mode to another. We derive step by step, following the method of cumulants expansion and multiscale asymptotic perturbations, the kinetic equation for the second order cumulants as well as the second and fourth order renormalization of the dispersion relation of the waves. We characterize the non-equilibrium evolution to an equilibrium wave spectrum, which happens to be the well known Rayleigh-Jeans distribution. Moreover we show the existence of an energy cascade, often called the Kolmogorov-Zakharov spectrum, which happens to be not simply a power law, but a logarithmic correction to the Rayleigh-Jeans distribution. We perform numerical simulations confirming these scenarii, namely the equilibrium relaxation for closed systems and the existence of an energy cascade wave spectrum. Both show a good agreement between theoretical predictions and numerics. We show also some other relevant features of vibrating elastic plates, such as the existence of a self-similar wave action inverse cascade which happens to blow-up in finite time. We discuss the mechanism of the wave breakdown phenomena in elastic plates as well as the limit of strong turbulence which arises as the thickness of the plate vanishes. Finally, we discuss the role of dissipation and the connection with experiments, and the generalization of the wave turbulence theory to elastic shells.

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Section: A Possible Inverse Cascade?mentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Wave turbulence theory of elastic plates

Düring

Josserand

Rica

2017

Physica D: Nonlinear Phenomena

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“…Finally, we propose an expression allowing a full characterization of the low frequency part of the velocities spectrum knowing the injected power and the forcing. The build-up of this equipartition possibly occurs through a transient inverse cascade as reported in [26]. An experimental study of the transient associated with high frequency forcing would be a challenging endeavor to generalize previous studies of direct cascade transients in elastic plates [34].…”

Section: Discussionmentioning

confidence: 91%

“…Secondly, we focus on the low frequency plateau using some scaling arguments. Energy is the unique conserved quantity during nonlinear interactions of bending waves [26]. In contrast to surface gravity waves in fluids, there is no other quantities (as wave action) to sustain an inverse cascade.…”

Section: Discussionmentioning

confidence: 99%

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Low frequency spectra of bending wave turbulence

Miquel,

Naert,

Aumaître

2021

Preprint

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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 0 . This frequency is chosen large compared with the characteristic frequencies of bending waves. As a consequence, a range of conservative scales, without energy flux in average, exists for frequencies f < f 0 . 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 and amplitude, and of the plate characteristics.

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Thermally driven elastic membranes are quasi-linear across all scales

Steinbock

Katzav

2023

J. Phys. A: Math. Theor.

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We study the static and dynamic structure of thermally fluctuating elastic thin sheets by investigating a model known as the overdamped dynamic Föppl-von Kármán equation, in which the Föppl-von Kármán equation from elasticity theory is driven by white noise. The resulting nonlinear equation is governed by a single nondimensional coupling parameter g where large and small values of g correspond to weak and strong nonlinear coupling respectively. By analysing the weak coupling case with ordinary perturbation theory and the strong coupling case with a self-consistent methodology known as the self-consistent expansion, precise analytic predictions for the static and dynamic structure factors are obtained. Importantly, the maximum frequency n max supported by the system plays a role in determining which of three possible classes such sheets belong to: (1) when g ≫ 1, the system is mostly linear with roughness exponent ζ = 1 and dynamic exponent z = 4, (2) when g ≪ 2/n max, the system is extremely nonlinear with roughness exponent ζ = 1/2 and dynamic exponent z = 3, (3) between these regimes, an intermediate behaviour is obtained in which a crossover occurs such that the nonlinear behaviour is observed for small frequencies while the linear behaviour is observed for large frequencies, and thus the large frequency linear tail is found to have a significant impact on the small frequency behaviour of the sheet. Back-of-the-envelope calculations suggest that ultra-thin materials such as graphene lie in this intermediate regime. Despite the existence of these three distinct behaviours, the decay rate of the dynamic structure factor is related to the static structure factor as if the system were completely linear. This quasi-linearity occurs regardless of the size of g and at all length scales. Numerical simulations confirm the existence of the three classes of behaviour and the quasi-linearity of all classes.

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Self-similar formation of an inverse cascade in vibrating elastic plates

Cited by 12 publications

References 36 publications

Wave turbulence theory of elastic plates

Wave turbulence theory of elastic plates

Low frequency spectra of bending wave turbulence

Thermally driven elastic membranes are quasi-linear across all scales

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