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The first-digit frequencies in data of turbulent flows

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  • Biau, Damien

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 a 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 provided in appendix is such that part of the presented results can easily be replicated.

Suggested Citation

  • Biau, Damien, 2015. "The first-digit frequencies in data of turbulent flows," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 440(C), pages 147-154.
    • Handle: RePEc:eee:phsmap:v:440:y:2015:i:c:p:147-154
    DOI: 10.1016/j.physa.2015.08.016
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    1. A. Pocheau, 2006. "The significant digit law: a paradigm of statistical scale symmetries," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 49(4), pages 491-511, February.
    2. Fewster, R. M., 2009. "A Simple Explanation of Benford's Law," The American Statistician, American Statistical Association, vol. 63(1), pages 26-32.
    3. Pietronero, L. & Tosatti, E. & Tosatti, V. & Vespignani, A., 2001. "Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 293(1), pages 297-304.
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