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.

Figure 1

Evolution of the Fourier power spectrum $E^{\eta }\left(k\right)$ of the deformation $\eta$ of the plate (integrated over the directions of the wave vector $\mathbf{k}$) as a function of the forcing intensity. (a) Experiment. The average forcing power is $P=[1,4,9,16,25,36,50]$ (arb. unit) from bottom to top. The power is proportional to the square of the force magnitude (at a given excitation frequency). (b) Numerical simulations. The input power is $P=[1,3.5,14,60,245,910,3760,11100]$ (arb. unit) from bottom to top and thus evolves on a much larger range than the experiment. The dissipation is also quite different (see text). The vertical dashed line corresponds to the wave number over which the dissipation operates. The forcing operates at $k/2\pi =2.5{\mathrm{m}}^{-1}$ (that corresponds to 30 Hz for linear waves as in the experiment). Black squares show the change of behavior of the spectrum between structure dominated regime (at low $k$) and weak turbulence at high $k$.

Figure 2

Snapshots of the deformation of the plate for various forcing intensities (numerical simulation). The corresponding run number is displayed in the title. The input power is increasing with the run number and the relative values $(1,14,245,3760)$ from (a) to (d), respectively.

Figure 3

Scaling properties of the spectrum of the deformation. The color code is the same as in Fig. 1. (a) spectra normalized by $k^3/P^{1/3}$ to compare to the prediction of the WTT that predicts a stationary spectrum $E^{\eta }\left(k\right)\propto P^{1/3}\frac{{log}^{1/3}(k^{*}/k)}{k^3}$. The dashed line correspond to a fitted ${log}^{1/3}(k^{*}/k)$ with $k^{*}/2\pi =90{\mathrm{m}}^{-1}$. The insert is a zoom of the logarithmic region in semilogarithmic scale. (b) Spectrum normalized by $k^3/P$. This normalization was chosen to illustrate the linear scaling with $P$ of the low-frequency part of the spectrum. Black squares show the transition as in Fig. 1.

Figure 4

Sketch of the different regimes encountered in the spectrum of the motion as the injected power increases. The different numerical runs labeled from 1 to 8 (see Table 1) are represented by vertical dotted lines. The largest scales and the smallest scales are dominated by the forcing and the dissipation, respectively. The intermediate scales exhibit a Weak Turbulence (WT) scaling for weak forcing, or are dominated by coherent structures (CS) for very large forcing. At intermediate forcing, the large scales are populated by the coherent structure while the small scales follow a WT dynamics. The transition observed in the spectra represented in Figs. 1 and 3 is denoted by plain squares. They correspond to the change from the the strongly nonlinear regime $[E^{\eta }\left(k\right)\propto P{k}^{-5}]$ to the KZ spectrum $[E^{\eta }\left(k\right)\propto P^{1/3}\frac{{log}^{1/3}(k^{*}/k)}{k^3}]$. Thus, the transition wave number $k_c$ is expected to follow roughly $k_c\propto P^{1/3}$, discarding logarithmic corrections.

Figure 5

Examples of the spectrum $E^{\eta }(k,\omega )$ obtained from the numerical simulations, for run1 (weakest forcing), run 6, run 7, and run 8 (strongest) ($P=1$, 910, 3760, and 11 100, from (a) to (d), respectively). For a better contrast the displayed quantity is actually the normalized spectrum $k^4{E}^{\eta }(k,\omega )$.

Figure 6

Cuts of the $k,\omega$ spectrum for two given values of $k$ [(a) $k=20\pi$ and (b) $k=80\pi {\mathrm{m}}^{-1}]$ and for all the values of the forcing magnitude. The linear values of the linear wave frequency at the given values of $k$ are shown with vertical dashed lines.

Figure 7

Snapshots of both sorts of curvature $L$ and $G$ at various forcing intensities in the DNS. Top line, $L$; bottom line, $G$. (a), (b) Run 1 (weak forcing), (c), (d) run 6 (intermediate forcing), (e), (f) run 8 (strongest forcing). The input power are, respectively, $P=1$, 910, and 11 100.

Figure 8

PDF of $L_0$ (left) and $G_0$ (right) at various forcing intensities. In each figure: from top to bottom, runs 1 to 8 (the curves have been shifted vertically by a factor 4 for clarity). Left: the dashed line is a Gaussian distribution.

Figure 9

Distribution of the velocity of the strong events of $\left|L\right|$ at the three highest forcing intensities (runs 6 to 8, blue, red, and green, respectively). The dashed line is a Gaussian distribution. The corresponding forcing powers are, respectively, $P=910$, 3760, and 11 100.

Figure 10

Cuts of the $(k,\omega )$ spectrum at various values of $k/2\pi =15,20,25,30,35,40,45{\mathrm{m}}^{-1}$ for run 8 (strongest forcing). Main figure: $E^{\eta }(k,\omega )/E(k,0)$ as a function of $\omega /k$. The dashed line is a Gaussian shape of width 20 m/s. Insert: $E^{\eta }(k,\omega )/E(k,0)$ as a function of $\omega$

Figure 11

Decay of the space-time spectrum at zero frequency $k^{6}E(k,\omega =0)/\int \int E^{\eta }(k,\omega )dkd\omega$ for all simulations. The normalization of the spectrum means that the deformation has been normalized to be of variance 1. The dashed line shows a decay $k{.}^{-2/3}$. The forcing increases from bottom to top.

Figure 12

Examples of principal curvature $L$ of the low pass filtered deformation. Top line (a), (b), (c), weakest forcing (run 1, $P=1$). Bottom line (d), (e), (f), strongest forcing (run 8, $P=11100$). (a), (d) Curvature of the deformation. (b), (e) Curvature of the deformation after smoothing over 5 pixels (1.3 cm). (c), (f) Curvature of the deformation after smoothing over 32 pixels (8.3 cm). The images display only a portion of the full plate.

Figure 13

PDF of both curvature $L$ and $G$ as a function of the smoothing for run5 [(a), (b) $P=245]$ and run1 [(c), (d) $P=1]$. (a), (c) $L$, the dashed line shows a Gaussian distribution; (b), (d) $G$. In each subfigure, the length scale is increasing from top to bottom and the curves have been shifted vertically for clarity. The smoothing scales are from bottom to top: 0.05, 0.14, 0.29, 0.43, 0.57, 0.86, 1.1, 1.7, 2.3, 3.4, 4.6, 6.9, 9.2, 13, and 18 cm.

Figure 14

Evolution of the flatness of (a) $L$ and (b) $G$ with scale.

Figure 15

Evolution with scale of the Gaussian curvature of the smoothed images for experiments. (a) Higher forcing (runG, $P=50$). (b) Smallest forcing, which is not affected by the noise (run C, $P=9$).

Figure 16

Evolution with scale of the flatness (a) and hyperflatness (b) of the curvature for the experiment for runs E, F, G (colors blue, red, and black, respectively, input power $P=25$, 36, and 50, respectively).