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 on the strongly nonlinear regime described in a previous Letter by Yokoyama and Takaoka. At small forcing, a weakly nonlinear regime is compatible with the weak turbulence theory when the dissipation is localized at high wave number. When the forcing intensity is increased, a strongly nonlinear regime emerges: singular structures dominate the dynamics at large scales 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.

Figure 1

Spectra of the plate deformation (multiplied by $k^4$). Forcing intensity is increasing from the bottom curve to the top curve. Top figure: experiment (forcing magnitude in arb. units: 0.5, 0.75, 1, 1.25, 1.5 from bottom to top). Inset: numerics with Lorentz dissipation mimicking the experimental dissipation (the forcing magnitude is increasing by a factor of 2 between each curve from bottom to top). The dashed line is a $1/k$ decay. Bottom figure: numerical simulations with transparent dissipation. The left vertical line marks the forcing wave number, and the right vertical line shows the lower wave number of the dissipation range. The lower dashed line stands for the theoretical prediction for weakly nonlinear wave turbulence [4]. The forcing magnitude is increasing by a factor of 2 between each curve from bottom to top. Inset: spectrum rescaled by $P$.

Figure 2

Snapshots of the deformation for numerical simulations with transparent dissipation. Top: smallest forcing amplitude. Bottom: largest forcing amplitude.

Figure 3

Magnitude of the gradient of the deformation. Numerical simulations with transparent dissipation: Top: smallest forcing amplitude. Center: largest forcing amplitude. Bottom: comparison of the experiment at strongest forcing (left) and numerical simulation with Lorentz dissipation (right). The simulation picture has been truncated to the size of the experimental picture.

Figure 4

Flatness of gradient increments [Eq. (5)]. Main figure: numerics with transparent dissipation. Inset: experiment. The forcing magnitude increases from bottom to top.