The first-digit frequencies in data of turbulent flows

Abstract

Considering the first significant digits (noted image) in data sets of dissipation for turbulent flows, the probability to find a given number (image or 2 or …9) would be 1/9 for a uniform distribution. Instead the probability closely follows Newcomb–Benford’s law, namely image. 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 provided in appendix is such that part of the presented results can easily be replicated.