Reproducing the Kolmogorov spectrum of turbulence with a hierarchical linear cascade model

Abstract

In Richardson's cascade description of turbulence, large vortices break up to form smaller ones, transferring the kinetic energy towards smaller scales. Energy dissipation occurs at the smallest scales due to viscosity. We study this energy cascade in a phenomenological model of vortex breakdown. The model is a binary tree of decreasing masses connected by softening springs, 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 wavenumber) is derived in the asymptotic limit. A decimation procedure is applied to replace the model with an equivalent chain oscillator. For a range of stiffness 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.