Single-wave-number representation of nonlinear energy spectrum in elastic-wave turbulence of the Föppl–von Kármán equation: Energy decomposition analysis and energy budget

A single-wave-number representation of a nonlinear energy spectrum, i.e., a stretching-energy spectrum, is found in elastic-wave turbulence governed by the Föppl–von Kármán (FvK) equation. The representation enables energy decomposition analysis in the wave-number space and analytical expressions of detailed energy budgets in the nonlinear interactions. We numerically solved the FvK equation and observed the following facts. Kinetic energy and bending energy are comparable with each other at large wave numbers as the weak turbulence theory suggests. On the other hand, stretching energy is larger than the bending energy at small wave numbers, i.e., the nonlinearity is relatively strong. The strong correlation between a mode $a_{\mathit{k}}$ and its companion mode $a_{-\mathit{k}}$ is observed at the small wave numbers. The energy is input into the wave field through stretching-energy transfer at the small wave numbers, and dissipated through the quartic part of kinetic-energy transfer at the large wave numbers. Total-energy flux consistent with energy conservation is calculated directly by using the analytical expression of the total-energy transfer, and the forward energy cascade is observed clearly.

Figure 1

Spectra of the total energy $\mathcal{E}$, of the quadratic energy ${\mathcal{E}}^{\left(2\right)}$, of the kinetic energy $\mathcal{K}$, of the potential energy $\mathcal{V}$, of the bending energy ${\mathcal{V}}_{\mathrm{b}}$, of and the stretching energy ${\mathcal{V}}_{\mathrm{s}}$.

Figure 2

Correlation between $a_{\mathit{k}}$ and $a_{-\mathit{k}}$. ($\mathit{k}=(k_x,0)$ and $k_x\in [10\pi ,256\pi ]$.)

Figure 3

Transfers of the total energy, of the quadratic and quartic parts of the kinetic energy, and of the bending and stretching energies. The inset shows the enlargement at the large wave numbers.

Figure 4

Flux of the total energy, and pseudo-fluxes of the quadratic and quartic parts of the kinetic energy, and of the bending and stretching energies.