Multifractality of Inverse Statistics of Exit Distances in 3D Fully Developed Turbulence

Abstract

The inverse structure functions of exit distances have been introduced as a novel diagnostic of turbulence which emphasizes the more laminar regions [1-4]. Using Taylor's frozen field hypothesis, we investigate the statistical properties of the exit distances of empirical 3D fully developed turbulence. We find that the probability density functions of exit distances at different velocity thresholds can be approximated by stretched exponentials with exponents varying with the velocity thresholds below a critical threshold. We show that the inverse structure functions exhibit clear extended self-similarity (ESS). The ESS exponents ξ(p,2) for small p (p<3.5) are well captured by the prediction of ξ(p,2)= p/2 obtained by assuming a universal distribution of the exit distances, while the observed deviations for large p's characterize the dependence of these distributions on the velocity thresholds. By applying a box-counting multifractal analysis of the natural measure constructed on the time series of exit distances, we demonstrate the existence of a genuine multifractality, endowed in addition with negative dimensions. Performing the same analysis of reshuffled time series with otherwise identical statistical properties for which multifractality is absent, we show that multifractality can be traced back to non-trivial dependence in the time series of exit times, suggesting a non-trivial organization of weakly-turbulent regions.