An exact result in strong wave turbulence of thin elastic plates

Abstract

An exact result concerning the energy transfers between non-linear waves of thin elastic plate is derived. Following Kolmogorov's original ideas in hydrodynamical turbulence, but applied to the Föppl-von Kármán equation for thin plates, the corresponding Kármán-Howarth-Monin relation and an equivalent of the $\frac{4}{5}$-Kolmogorov's law is derived. A third-order structure function involving increments of the amplitude, velocity and the Airy stress function of a plate, is proven to be equal to $-\varepsilon\, \ell$, where $\ell$ is a length scale in the inertial range at which the increments are evaluated and $\varepsilon$ the energy dissipation rate. Numerical data confirm this law. In addition, a useful definition of the energy fluxes in Fourier space is introduced and proven numerically to be flat in the inertial range. The exact results derived in this Letter are valid for both, weak and strong wave-turbulence. They could be used as a theoretical benchmark of new wave-turbulence theories and to develop further analogies with hydrodynamical turbulence.