Strong turbulence for vibrating plates: Emergence of a Kolmogorov spectrum

Abstract

In fluid turbulence, energy is transferred from one 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. Here we show that the normal modes of vibrations in elastic plates also manifest an energy cascade with the same K41 spectrum in the fully nonlinear regime. In particular, we observe different patterns in the elastic deformations such as folds, developable cones, and even more complex stretching structures, in analogy with spots, swirls, vortices, and other structures in hydrodynamic turbulence. We show that the energy cascade is dominated by the kinetic contribution and that the stretching energy is at thermodynamical equilibrium. We characterize this energy cascade, the validity of the constant energy-dissipation rate over the scales. Finally, we discuss the role of intermittency using the correlation functions that exhibit anomalous exponents.

Finite-time localized singularities as a mechanism for turbulent dissipation

“…A Kolmogorov-like argument is not sufficient to predict such a spectrum, because eqn. (1) has two distinct conservation laws, leading to an extra dimensionless number (the same situation happens in the case of zero thickness elastic plates [17]). By dimensional analysis and according to the numerical observation that the final spectrum does not depend on ǭ one concludes that the spectrum S k ∼ α/g 4 1/3 k −1/3 Remarkably, this final argument is different than for wave turbulence so that our spectrum is not at all of this type [31].…”

“…Beside turbulence, dissipation by singularity events has been proposed in different physical contexts: for instance, it is believed that the formation of steep slope deformations of the sea surface, leading to "white caps", are responsible for energy dissipation [15,16]. Ridges, folds and conical singularities are good candidates for dissipating the energy of strongly vibrating elastic plates [17] and the same is true for the focusing of light in nonlinear media [18] and strong turbulence in plasmas [19].…”

Feynman rules for forced wave turbulence

“…The kinetic equation governs the occupation number n k (t) = a † k (t)a k (t) at late times. 13 The Hamiltonian in terms of complex amplitudes a k was given in (2.2) and the equations of motion were given in (2.1),…”

“…A 3. This includes: elastic plate wave turbulence[8][9][10][11][12][13][14], turbulence in surface gravity waves[15][16][17][18][19], in gravitational waves[20][21][22][23], acoustic turbulence[24], planetary Rossby waves[25], waves on bubbles[26], emergent hydrodynamics[27], in a diatomic chain[28], in plasmas[29,30], optical waves[31][32][33], and rotating waves[34,35], and in FPUT chains[36].…”

Резонансный Отклик Масштабно-Инвариантных Функций Случайного Процесса С Турбулентным Спектром

“…Согласно закону Колмогорова−Обухова, спектральная плотность турбулентных пульсаций зависит от частоты как f −α , где α = 5/3 ( " закон −5/3"). Для различных турбулентных течений экспериментально доказано существование интервалов частот, в которых выполняется закон Колмогорова−Обухова [4][5][6][7][8]. Но не все случайные процессы со степенным поведением спектральной плотности и амплитудных распределений можно свести к турбулентности.…”