The first-digit frequencies in data of turbulent flows

• Analysis of first significant digit distributions in data sets of turbulent flows.
• Conformance measured with Shannon’s entropy.
• Results in agreement with Newcomb–Benford’s law.
• The discrepancy is related with the phenomenon of intermittency in turbulence.
• A matlab program is provided in appendix.

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