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 a thin elastic plate. We report a direct measurement of the nonlinear time scale $T_{\mathit{NL}}$ 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 ($T_d\gg T_{\mathit{NL}}\gg T$). 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.

Figure 1

(Squares) Measured dissipative time $T_d$ versus $k/2\pi =1/\lambda$. (Dashed line) Lorentzian fit $T_d={(0.73+0.025{(k/2\pi )}^2)}^{-1}$ used in the following. (Insert) $T_d\omega$ versus $k/2\pi$ (semilog scale).

Figure 2

Schematics of the plate (figured by the vertical rectangular box) of size $2\times 1$ m${}^2$. (Left) Example of the Gabor wavelets used in the analysis. The width of the Gaussian envelop is $\sigma =15$ cm and $k/2\pi =15{\mathrm{m}}^{-1}$. (Right) Collection of positions of the center of the wavelets $\mathbf{R}$ used in our analysis. $\mathbf{k}$ is chosen horizontal. The bottom black dot corresponds to the position of the electromagnetic shaker.

Figure 3

Modulus $|{\mathcal{A}}^c\left(t\right)|$ (red squares, experimental results; blue solid lines, analytical predictions) and real part $|{\mathcal{A}}^c\left(t\right)|$ (red circles, experimental results; blue dashed lines, analytical predictions) of the compensated correlation functions ${\mathcal{A}}_{\mathbf{k},\mathbf{R},\mathbf{R}+\Delta \mathbf{R}}^c\left(t\right)$. No adjustable parameter is used. (a) $k/\left(2\pi \right)=10.6{\mathrm{m}}^{-1}$, $\Delta \mathbf{R}=0$; (b) $k/\left(2\pi \right)=20.2{\mathrm{m}}^{-1}$, $\Delta \mathbf{R}=0$; (c) $k/\left(2\pi \right)=20.2{\mathrm{m}}^{-1}$, $\Delta \mathbf{R}=0.27\mathrm{m}$.

Figure 4

Example of correlations at $k=16.0$ m${}^{-1}$. The real part is shown as the oscillating thin red line and the modulus as the thick black line. (a) ${\mathcal{A}}_{\mathbf{k},\mathbf{R},\mathbf{R}}^c\left(\tau \right)$ for $\mathbf{R}$ at the center of the plate. The vertical dashed lines show the times $2nL/v_g$ for $n=-2$ to 2. (b) ${\mathcal{A}}_{\mathbf{k},\mathbf{R},{\mathbf{R}}^{\prime }}^c\left(\tau \right)$ for $\Delta R=0.27$ m. The vertical dashed lines show the times $(2nL+\Delta R)/v_g$ for $n=-2$ to 2. (c) ${\mathcal{C}}_{\mathbf{k},\mathbf{R},\mathbf{R}}^c\left(\tau \right)$ for $\mathbf{R}$ at the center of the plate. The vertical dashed lines show the times $(2n+1)L/v_g$ for $n=-3$ to 2. In the three cases, the correlations have been normalized by ${\mathcal{A}}_{\mathbf{k},\mathbf{R},\mathbf{R}}\left(0\right)$.

Figure 5

Modulus of the correlations at the same position $\mathbf{R}={\mathbf{R}}^{\prime }$ as a function of time and wave vector. Colors are log coded. (a) Autocorrelations $|{\mathcal{A}}^c|$. Green (light gray) dashed lines, flight times for $2n$ rebounds $t_n^{\mathrm{fly}}=2nL/\left(v_g\left(k\right)\right)$; black dashed line, flight time over the width of the wave packet. (b) Cross correlations $|{\mathcal{C}}^c|$. Green (light gray) dashed line, flight times for $2n-1$ rebounds $t_n^{\mathrm{fly}}=(2n-1)L/\left(v_g\left(k\right)\right)$. At each value of $k$, the correlations have been normalized by ${\mathcal{A}}_{\mathbf{k},\mathbf{R},\mathbf{R}}^c\left(0\right)$ so that the modulus of the correlation is between 0 and 1.

Figure 6

(a) Color plot of $|{\mathcal{A}}_{\mathbf{k},\mathbf{R},{\mathbf{R}}^{\prime }}^c\left(t\right)|/|{\mathcal{A}}_{\mathbf{k},\mathbf{R},\mathbf{R}}^c\left(0\right)|$ versus time $t$ and position ${\mathbf{R}}^{\prime }$. Wave number $k/2\pi =13{\mathrm{m}}^{-1}$; the position $\mathbf{R}$ is at center of the plate. Colors are log coded. (b) Cross correlation $|{\mathcal{C}}_{\mathbf{k},\mathbf{R},{\mathbf{R}}^{\prime }}^c\left(t\right)|/|{\mathcal{A}}_{\mathbf{k},\mathbf{R},\mathbf{R}}^c\left(0\right)|$ (c) Combination of the two functions ${\mathcal{A}}^c$ and ${\mathcal{C}}^c$: $\mathcal{T}=\sqrt{|{\mathcal{A}}^c{|}^2+{\left|{\mathcal{C}}^c\right|}^2}$.

Figure 7

Color plot of $\mathcal{T}$ (see previous figure) versus time $t$ and positions ${\mathbf{R}}^{\prime }$. Wave number $k/2\pi =13{\mathrm{m}}^{-1}$; the position $\mathbf{R}$ is the center of the plate. Colors are log coded. The forcing power is $P_0$, $7P_0$, $27P_0$, and $59P_0$ [from (a) to (d)].

Figure 8

Superimposition of the moduli $|{\mathcal{A}}_{\mathbf{k},\mathbf{R},{\mathbf{R}}^{\prime }}^c\left(\tau \right)|$ (normalized by $|{\mathcal{A}}_{\mathbf{k},\mathbf{R},\mathbf{R}}^c\left(0\right)|$ at the center) at a given $k$ and for a collection of positions ${\mathbf{R}}^{\prime }$ (excluding positions too close to the border for which the wavelet can be truncated). $\mathbf{R}$ is chosen at the center of the plate. The thick line corresponds to $\mathbf{R}={\mathbf{R}}^{\prime }$. Black thin lines, ${\mathcal{A}}_{\mathbf{k},\mathbf{R},{\mathbf{R}}^{\prime }}^c\left(\tau \right)$; cyan (light gray) thin lines, ${\mathcal{C}}_{\mathbf{k},\mathbf{R},{\mathbf{R}}^{\prime }}^c\left(\tau \right)$. (Insert) Squares correspond to the value of the local maximum. The line is an exponential fit of the decay.

Figure 9

Evolution of the nonlinear time scale as a function of the wave number for different forcing. Symbols are experimental data for different values of the forcing intensity (intensity increasing downward): $P_0$, $7P_0$, $27P_0$, $59P_0$, $100P_0$, $149P_0$, and $207P_0$. This color chart for injected powers is used through this section for other figures.

Figure 10

Rescaled nonlinear time $T_{\mathit{NL}}{P}^{0.5}$ as a function of the rescaled wave vector $k{P}^{-1/6}$ for different injected power $P$.

Figure 11

Comparison of the nonlinear time scale with other relevant time scales. Symbols are experimental data for $T_{\mathit{NL}}$ at different values of the forcing intensity. Upper dashed line, twice the dissipative time $T_d$ measured by the decay of energy in the unforced case (see Fig. 1). Middle dashed line, $2/\delta {\omega }_{\mathrm{sep}}=L/\left(\pi kc\right)$. Lower dashed line, $1/\omega$. The forcing is operated at 30 Hz and is shown with the vertical dash-dotted line.

Figure 12

Ratios between different characteristic times as a function of $k$ for various values of the injected power: (a) $\omega T_{\mathit{NL}}$; (b) $2T_d/T_{\mathit{NL}}$; (c) $T_{\mathit{NL}}\delta {\omega }_{\mathrm{sep}}/2$.

Figure 13

(a) Symbols, measured $T_{\mathit{NL}}$. Dashed line, $2/\delta {\omega }_{\mathrm{sep}}$. Red solid squares, crossover wave number $k_c$ given by $T_{\mathit{NL}}\left(k_c\right)=2/\delta {\omega }_{\mathrm{sep}}\left(k_c\right)$. (b) Solid lines, stationary power spectrum of the velocity extracted along the dispersion relation and for $k_x=0$ (see text). Forcing is increased from bottom to top. Plain squares, crossover frequency $f_c$ reported from (a). (c) Synthetic spectrum incorporating the nonlinear widening of the modes (see text). The width of the modes is $1/\left(2\pi T_{\mathit{NL}}\right)$ interpolated as a function of frequency from (a). Curves are vertically shifted for clarity in (b) and (c).

Figure 14

(a) Symbols, ${(\Delta \omega /k)}^2$ as a function of ${\left(T_{\mathit{NL}}k\right)}^{-2}$ for different injected powers. Dashed line (eye guide), $0.6x+25$. (b)$A{\left(T_{\mathit{NL}}k\right)}^{-2}+B$ (pentagrams) and ${(\Delta \omega /k)}^2$ (squares) plotted as a function of $k$ for one given injected power $P=149P_0$. $A=0.6$, $B=25$. Solid line, value of $B$.