The first-digit frequencies in data of turbulent flows

TL;DR

This paper investigates the distribution of first significant digits in turbulent flow data, finding that they follow Newcomb-Benford's law in homogeneous conditions and linking deviations to intermittent turbulence events.

Contribution

It demonstrates the scale invariance of first-digit distributions in turbulent flows and connects deviations to turbulence intermittency, supported by numerical simulations and entropy analysis.

Findings

First-digit frequencies follow Newcomb-Benford's law in homogeneous turbulence.

Deviations from the law are linked to intermittent turbulence events.

Scale invariance relates to equilibrium turbulent states with inertial ranges.

Abstract

Considering the first significant digits (noted d) in data sets of dissipation for turbulent flows, the probability to find a given number (d=1 or 2 or... 9) would be 1/9 for an uniform distribution. Instead the probability closely follows Newcomb-Benford's law, namely P(d)=log(1+1/d). The discrepancies between Newcomb-Benford's law and first-digits frequencies in turbulent data are analysed through Shannon's entropy. The data sets are obtained with direct numerical simulations for two types of fluid flow: an isotropic case initialized with a Taylor-Green vortex and a channel flow. Results are in agreement with Newcomb-Benford's law in nearly homogeneous cases and the discrepancies are related to intermittent events. Thus the scale invariance for the first significant digits, which supports Newcomb-Benford's law, seems to be related to an equilibrium turbulent state, namely with a significant inertial range. A matlab/octave program is provided in appendix in such that part of the presented results can easily be replicated.