Intermittency and emergence of coherent structures in wave turbulence of a vibrating plate

We report numerical investigations of wave turbulence in a vibrating plate. The possibility to implement advanced measurement techniques and long time numerical simulations makes this system extremely valuable for wave turbulence studies. The purely 2D character of dynamics of the elastic plate makes it much simpler to handle compared to much more complex 3D physical systems that are typical of geo- and astrophysical issues (ocean surface or internal waves, magnetized plasmas or strongly rotating and/or stratified flows). When the forcing is small the observed wave turbulence is consistent with the predictions of the Weak Turbulent Theory. Here we focus on the case of stronger forcing for which coherent structures can be observed. These structures look similar to the folds and D-cones that are commonly observed for strongly deformed static thin elastic sheets (crumpled paper) except that they evolve dynamically in our forced system. We describe their evolution and show that their emergence is associated with statistical intermittency (lack of self similarity) of strongly nonlinear wave turbulence. This behavior is reminiscent of intermittency in Navier-Stokes turbulence. Experimental data show hints of the weak to strong turbulence transition. However, due to technical limitations and dissipation, the strong nonlinear regime remains out of reach of experiments and therefore has been explored numerically.Comment: accepted for publication in Phys. Rev.

The role of dissipation in flexural wave turbulence: from experimental spectrum to Kolmogorov-Zakharov spectrum

The Weak Turbulence Theory has been applied to waves in thin elastic plates obeying the F\"oppl-Von K\'arm\'an dynamical equations. Subsequent experiments have shown a strong discrepancy between the theoretical predictions and the measurements. Both the dynamical equations and the Weak Turbulence Theory treatment require some restrictive hypotheses. Here a direct numerical simulation of the F\"oppl-Von K\'arm\'an equations is performed and reproduces qualitatively and quantitatively the experimental results when the experimentally measured damping rate of waves $\gamma_\mathbf{k}= a + bk^2$ is used. This confirms that the F\"oppl-Von K\'arm\'an equations are a valid theoretical framework to describe our experiments. When we progressively tune the dissipation so that to localize it at the smallest scales, we observe a gradual transition between the experimental spectrum and the Kolmogorov-Zakharov prediction. Thus it is shown dissipation has a major influence on the scaling properties stationary solutions of weakly non linear wave turbulence.Comment: 10 pages, 11 figure

Wave turbulence buildup in a vibrating plate

We report experimental and numerical results on the buildup of the energy spectrum in wave turbulence of a vibrating thin elastic plate. Three steps are observed: first a short linear stage, then the turbulent spectrum is constructed by the propagation of a front in wave number space and finally a long time saturation due to the action of dissipation. The propagation of a front at the second step is compatible with scaling predictions from the Weak Turbulence Theory.Comment: accepted for publication in European Physical Journal

Transition from wave turbulence to dynamical crumpling in vibrated elastic plates

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.Comment: accepted for publication in Physical Review Letter

Nonlinear dynamics of flexural wave turbulence

The Kolmogorov-Zakharov spectrum predicted by the Weak Turbulence Theory remains elusive for wave turbulence of flexural waves at the surface of an thin elastic plate. We report a direct measurement of the nonlinear timescale $T_{NL}$ related to energy transfer between waves. This time scale is extracted from the space-time measurement of the deformation of the plate by studying the temporal dynamics of wavelet coefficients of the turbulent field. The central hypothesis of the theory is the time scale separation between dissipative time scale, nonlinear time scale and the period of the wave ($T_d>>T_{NL}>>T$). We observe that this scale separation is valid in our system. The discrete modes due to the finite size effects are responsible for the disagreement between observations and theory. A crossover from continuous weak turbulence and discrete turbulence is observed when the nonlinear time scale is of the same order of magnitude as the frequency separation of the discrete modes. The Kolmogorov-Zakharov energy cascade is then strongly altered and is frozen before reaching the dissipative regime expected in the theory.Comment: accepted for publication in Physical Review

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 $f_0$. 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 $\beta^{-1}$ -- analogous to a temperature -- fully characterizes the large-scale turbulent wave field. We present an expression for $\beta$ as a function of the forcing frequency and amplitude, and of the plate characteristics

A direct derivation of the Gent-McWilliams/Redi diffusion tensor from quasi-geostrophic dynamics

The transport induced by ocean mesoscale eddies remains unresolved in most state-of-the-art climate models and needs to be parameterized instead. The natural scale separation between the forcing and the emergent turbulent flow calls for a diffusive parameterization, where the eddy-induced fluxes are related to the large-scale gradients by a diffusion tensor. The standard parameterization scheme in climate modeling consists in adopting the Gent-McWilliams/Redi (GM/R) form for the diffusion tensor, initially put forward based on physical intuition and educated guesses before being put on firm analytical footing using thickness-weighted average (TWA). In the present contribution we provide a direct derivation of this diffusion tensor from the quasi-geostrophic (QG) dynamics of a horizontally homogeneous three-dimensional patch of ocean hosting a large-scale vertically-sheared zonal flow on the beta plane. While less general than the TWA approach, the present QG framework leads to rigorous constraints on the diffusion tensor. First, there is no diapycnal diffusivity arising in the QG GM/R tensor for low viscosity and small-scale diffusivities. The diffusion tensor then involves only two vertically dependent coefficients, namely the GM transport coefficient $K_{GM}(z)$ and the Redi diffusivity $K_R(z)$. Secondly, as already identified by previous authors the vertical structures of the two coefficients are related by the so-called Taylor-Bretherton relation. Finally, while the two coefficients generically differ in the interior of the water column, we show that they are equal to one another near the surface and near the bottom of the domain for low-enough dissipative coefficients. We illustrate these findings by numerically simulating the QG dynamics of a horizontally homogeneous patch of ocean hosting a vertically sheared zonal current resembling the Antarctic Circumpolar Current

Nouvelles techniques pratiques pour la modélisation du comportement dynamique des systèmes eau-structure

RÉSUMÉ L’´etude des comportements dynamique et sismique des ouvrages hydrauliques est, comme pour les structures conventionnelles, primordiale afin d’assurer la protection des vies humaines. Elle a aussi pour objectif de limiter les dommages structuraux que peut engendrer un tremblement de terre et d’´eviter le cas de rupture ou d’effondrement. Ces structures particulieres subissent non seulement les d´eplacements impos´ees par les secousses sismiques, mais aussi ceux induits par les forces hydrodynamiques g´en´er´ees par l’interaction fluide-structure. Cette th`ese passe en revue les diff´erentes m´ethodes complexes et simplifi´ees existantes permettant l’analyse dynamique d’ouvrages hydrauliques. En ce qui concerne les m´ethodes complexes,une attention particuli`ere est consacr´ee aux difficult´es li´ees `a leur utilisation. Parmi celles-ci, nous insisterons sur la mod´elisation des conditions aux fronti`eres transmettantes ermettant de simuler num´eriquement l’effet de la g´eom´etrie semi-infinie du r´eservoir. En d´eveloppant une proc´edure pour estimer l’erreur qu’induisent les conditions existantes, nous montrons que celles-ci peuvent avoir un effet tr`es important sur le comportement dynamique des structures en contact avec de l’eau. Pour les besoins des ing´enieurs-praticiens, des m´ethodes simplifi´ees sont n´ecessaires afin de v´erifier le comportement dynamique des ouvrages en contact avec de l’eau. La revue des m´ethodes simplifi´ees existantes montre que celles-ci sont bas´ees sur de nombreuses simplifications qui peuvent affecter la qualit´e des r´esultats. L’un des objectifs de cette th`ese a ´et´e de d´evelopper des m´ethodes simplifi´ees plus performantes que celles existantes. Une premi`ere m´ethode a ´et´e mise au point pour la r´ealisation d’une analyse spectrale de ces ouvrages. Pour son d´eveloppement, il a ´et´e n´ecessaire de proposer une m´ethodologie pour le calcul pr´ecis de la p´eriode fondamental d’un syst`eme eau-structure. Nous montrons que cette nouvelle proc´edure est facilement programmable, avec un temps de calcul instantan´e, et que celle-ci donne d’excellents r´esultats lorsque compar´ee `a des m´ethodes complexes.----------ABSTRACT The dynamic or seismic behavior of hydraulic structures is, as for conventional structures, essential to assure protection of human lives. These types of analyses also aim at limiting structural damage caused by an earthquake to prevent rupture or collapse of the structure. The particularity of these hydraulic structures is that not only the internal displacements are caused by the earthquake, but also by the hydrodynamic loads resulting from fluid-structure interaction. This thesis reviews the existing complex and simplified methods to perform such dynamic analysis for hydraulic structures. For the complex existing methods, attention is placed on the difficulties arising from their use. Particularly, interest is given in this work on the use of transmitting boundary conditions to simulate the semi infinity of reservoirs. A procedure has been developed to estimate the error that these boundary conditions can introduce in finite element dynamic analysis. Depending on their formulation and location, we showed that they can considerably affect the response of such fluid-structure systems. For practical engineering applications, simplified procedures are still needed to evaluate the dynamic behavior of structures in contact with water. A review of the existing simplified procedures showed that these methods are based on numerous simplifications that can affect the prediction of the dynamic behavior of such systems. One of the main objectives of this thesis has been to develop new simplified methods that are more accurate than those existing. First, a new spectral analysis method has been proposed. Expressions for the fundamental frequency of fluid-structure systems, key parameter of spectral analysis, have been developed. We show that this new technique can easily be implemented in a spreadsheet or program, and that its calculation time is near instantaneous. When compared to more complex analytical or numerical method, this new procedure yields excellent prediction of the dynamic behavior of fluid-structure systems