A Kolmogorov spectrum for strongly vibrating plates

TL;DR

This paper demonstrates that strongly vibrating elastic plates exhibit an energy cascade with a Kolmogorov spectrum similar to fluid turbulence, highlighting analogous structures and energy transfer mechanisms.

Contribution

It reveals that elastic plates in a nonlinear regime follow a Kolmogorov spectrum, establishing a new connection between elastic vibrations and turbulence theory.

Findings

Elastic plates show a K41 energy spectrum in the nonlinear regime.

Presence of large and small eddies analogous to turbulence structures.

Constant energy dissipation rate observed across scales.

Abstract

In fluid turbulence, energy is transferred from a scale to another by an energy cascade that depends only on the energy dissipation rate. It leads by dimensional arguments to the Kolmogorov 1941 (K41) spectrum. Remarkably the normal modes of vibrations in elastic plates manifests an energy cascade with the same K41 spectrum in the fully non-linear regime. Moreover, the elastic deformations present large "eddies" together with a myriad of small "crumpling eddies", such that folds, developable cones, and more complex stretching structures, in close analogy with spots, swirls, vortices and other structures in hydrodynamic turbulence. We characterize the energy cascade, the validity of the constant energy dissipation rate over the scales and the role of intermittency via the correlation functions.