Exact result in strong wave turbulence of thin elastic plates

Abstract

An exact result concerning the energy transfers between nonlinear waves of a 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 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 -ɛℓ, where ℓ is a length scale in the inertial range at which the increments are evaluated and ɛ 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 Rapid Communication 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.

Strong turbulence for vibrating plates: Emergence of a Kolmogorov spectrum

“…In the bottom inset of Fig. 4, we present the local slope of the structure functions defined as ξ p ( ) = Moreover, we have verified the exact result derived by two of us (G.D. and G.K.) for thin elastic plates that is valid for both weak and strong wave turbulence [12]. This result is equivalent to the only exact result of hydrodynamic turbulence, known as the 4/5 Kolmogorov law of turbulence.…”

“…In the inset of Fig. 3, the stretching spectra for different forcing amplitude are shown to be consistent with the equipartition law (12), for the scales where the dissipation is negligible.…”

“…In Ref. [12], an exact result for turbulence of thin elastic plates was found. It provides a Kármán-Howarth-Monin-type relationship for the energy flux,…”

“…Here, the nonlinearity of the equation comes from the geometry and is cubic, in contrast to the usual quadratic advection nonlinearity in fluids. However, such system has recently revealed a strong analogy with hydrodynamic turbulence through the derivation of an exact law equivalent to the Kolmogorov's 4/5 law [12]. For finite plate thickness, the small-scale dynamics are always dominated by elastic waves, and wave-turbulence regimes have been predicted theoretically, as well as observed numerically [13] and experimentally [14,15].…”

A Kolmogorov spectrum for strongly vibrating plates

“…In this paper, we precisely address such singular limit in the case of vibrating elastic plates. This system has indeed recently revealed a strong analogy with hydrodynamic turbulence through the derivation of an exact law for a two-point correlation that is the equivalent of the Kolmogorov's 4/5-law [17].…”