Abstract
In Richardson’s cascade description of turbulence, large vortices break up to form smaller ones, transferring the kinetic energy of the flow from large to small scales. This energy is dissipated at the smallest scales due to viscosity. We study energy cascade in a phenomenological model of vortex breakdown. The model is a binary tree of spring-connected masses, with dampers acting on the lowest level. The masses and stiffnesses between levels change according to a power law. The different levels represent different scales, enabling the definition of “mass wavenumbers.” The eigenvalue distribution of the model exhibits a devil’s staircase self-similarity. The energy spectrum of the model (defined as the energy distribution among the different mass wavenumbers) is derived in the asymptotic limit. A decimation procedure is applied to replace the model with an equivalent chain oscillator. For a significant range of the stiffness decay parameter, the energy spectrum is qualitatively similar to the Kolmogorov spectrum of 3D homogeneous, isotropic turbulence and we find the stiffness parameter for which the energy spectrum has the well-known -,5/3 scaling exponent.
Highlights
Many natural phenomena and engineering processes exhibit energy transfer among a range of different scales
The energy spectrum of the mechanistic model is calculated with different σ values using the equivalent chain oscillator formulation for n = 20 levels (Fig. 7)
8 Conclusions and future work Inspired by Richardson’s eddy hypothesis, we introduced a mechanistic model of turbulence with different mass scales
Summary
Many natural phenomena and engineering processes exhibit energy transfer among a range of different scales. Though energy cascades are usually related to nonlinear phenomena, infinite-dimensional linear systems can contain a wide range of scales and exhibit com-. To answer this question in the affirmative, we present a linear phenomenological model of turbulence inspired by Richardson’s cascade description We show that this hierarchical model exhibits the characteristic spectrum of energy transfer seen in turbulent flows. This new point of view will hopefully stimulate research connecting eigenvalue distributions of infinite-dimensional linear systems with energy transfer and resonance capture in nonlinear systems. Wavenumber, energy spectrum, and energy flux of our phenomenological model analogously to those of turbulent flow. We show how the energy spectrum of the model behaves for different parameters and how this is related to the self-similar structure of the model through the eigenvalues
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