Volumetric theory of intermittency in fully developed turbulence

TL;DR

This paper develops a rigorous volumetric framework to describe intermittency in turbulence, linking active regions of velocity fields to multifractal spectra and energy cascade concentration, with implications for classical turbulence theories.

Contribution

It introduces a new family of volumetric flatness factors and a dimension function that systematically recover intermittency corrections and multifractal formalism in turbulence.

Findings

Active regions carry most energy at each scale.

Random fields tend to produce classical K41 spectrum.

Intermittency deviations are quantifiable at finite scales.

Abstract

This study introduces a new family of volumetric flatness factors which give a rigorous parametric description of the phenomenon of intermittency in fully developed turbulent flows. These quantities gather information about the most "active" part of a velocity field at each scale $\ell$, and allows one to define a dimension function $p \to D_p$ that recovers intermittency correction to the structure exponents $\zeta_p$ in an explicit way. In particular, the predictions of the Frisch-Parisi multifractal formalism can be recovered in a systematic and rigorous way. Within this framework we identify active regions that carry the most energetic part of a velocity field at a given scale $\ell$. A threshold for what constitutes to be active is defined explicitly. Active regions have proven to be experimentally observable in our previous joint work \cite{Ph-paper}, and shown to capture concentration of the energy cascade as $\ell \to 0$, in \cite{CS2014}. We present several examples of fields which exhibit arbitrary multifractal spectrum within theoretically permitted limitations. At the same time we demonstrate with the use of a probabilistic argument that a random field is expected to produce the classical K41 spectrum in the limit $\ell\to 0$. Intermittent deviations from K41 theory are estimated at any finite scale also. Lastly, we present a detailed information-theoretic analysis of the introduced objects. In particular, we quantify concentration of a given source-field in terms of the volume factors, thresholds, and active regions.