An analytical study of flatness and intermittency through Riemann's non-differentiable functions

TL;DR

This paper analytically compares flatness and intermittency measures in turbulence using generalizations of Riemann's non-differentiable functions, revealing their dependence on regularity and drawing analogies with viscous effects in turbulent flows.

Contribution

It provides an analytical framework to compare physical and Fourier-based measures of intermittency using mathematical generalizations of Riemann's function, linking regularity to turbulence properties.

Findings

Flatness and intermittency measures depend strongly on regularity.

Analytical comparisons reveal differences between physical and Fourier approaches.

Analogies established between mathematical functions and viscous effects in turbulence.

Abstract

In the study of turbulence, intermittency is a measure of how much Kolmogorov's theory of 1941 deviates from experiments. It is quantified with the flatness of the velocity of the fluid, usually based on structure functions in the physical space. However, it can also be defined with Fourier high-pass filters. Experimental and numerical simulations suggest that the two approaches do not always give the same results. Our purpose is to compare them from the analytical point of view of functions. We do that by studying generalizations of Riemann's non-differentiable function, yielding computations that are related to some classical problems in Fourier analysis. The conclusion is that the result strongly depends on regularity. To visualize this, we establish an analogy between these generalizations and the influence of viscosity in turbulent flows. This article is motivated by the mathematical works on the multifractal formalism and the discovery of Riemann's non-differentiable function as a trajectory of polygonal vortex filaments.