Multifractal Analysis of Various PDF in Turbulence based on Generalized Statistics: A Way to Tangles in Superfluid He

TL;DR

This paper develops a unified multifractal analytical formula for probability density functions in turbulence, accurately explaining experimental PDFs across different scales and linking intermittency and scale invariance to turbulence phenomena.

Contribution

It introduces a compact analytical formula for PDFs in turbulence based on generalized statistics, capturing both tail and center behaviors and explaining experimental observations.

Findings

The formula accurately fits experimental PDFs on log and linear scales.

Tail part reflects intermittency and multifractal singularities.

Center part accounts for thermal fluctuations and measurement errors.

Abstract

By means of the multifractal analysis (MFA), the expressions of the probability density functions (PDFs) are unified in a compact analytical formula which is valid for various quantities in turbulence. It is shown that the formula can explain precisely the experimentally observed PDFs both on log and linear scales. The PDF consists of two parts, i.e., the {\it tail} part and the {\it center} part. The structure of the tail part of the PDFs, determined mostly by the intermittency exponent, represents the intermittent large deviations that is a manifestation of the multifractal distribution of singularities in physical space due to the scale invariance of the Navier-Stokes equation for large Reynolds number. On the other hand, the structure of the center part represents small deviations violating the scale invariance due to thermal fluctuations and/or observation error.