Turbulence on a Fractal Fourier set

TL;DR

This study explores how constraining Fourier modes on a fractal set affects turbulence, revealing modifications in energy spectra and intermittency, and providing insights into the energy transfer mechanisms in turbulent flows.

Contribution

It introduces a novel fractal Fourier set methodology to analyze turbulence, preserving symmetries while varying the fractal dimension to study effects on energy transfer and intermittency.

Findings

Energy cascade persists despite mode reduction.

Deviations from Kolmogorov scaling observed.

Intermittency becomes quasi-Gaussian at high mode reduction.

Abstract

A novel investigation of the nature of intermittency in incompressible, homogeneous and isotropic turbulence is performed by a numerical study of the Navier-Stokes equations constrained on a fractal Fourier set. The robustness of the energy transfer and of the vortex stretching mechanisms is tested by changing the fractal dimension, D, from the original three dimensional case to a strongly decimated system with D=2.5, where only about $3\%$ of the Fourier modes interact. This is a unique methodology to probe the statistical properties of the turbulent energy cascade, without breaking any of the original symmetries of the equations. While the direct energy cascade persists, deviations from the Kolmogorov scaling are observed in the kinetic energy spectra. A model in terms of a correction with a linear dependency on the co-dimension of the fractal set, $E(k) \sim k^{-5/3 + 3-D}$, explains the results. At small scales, the intermittency of the vorticity field is observed to be quasi-singular as a function of the fractal mode reduction, leading to an almost Gaussian statistics already at $D \sim 2.98$. These effects must be connected to a genuine modification in the triad-to-triad nonlinear energy transfer mechanism.