Multifractal asymptotic modeling of the probability density function of velocity increments in turbulence

Abstract

Moments and probability density functions (PDF) of (absolute value) velocity increments ∣v(x+l)−v(x)∣ in turbulence are linked by simple integral relations. It is shown that the steepest descent method can be applied to evaluate the integrals if the moments (the absolute value structure functions) obey multifractal scaling laws of the type 〈∣v(x+l)−v(x)∣n〉=Anlζn. A double asymptotic relation then relates the moments to the PDF. The dominant (exponential) terms of the asymptotic relation naturally yield the Legendre transform that is at the core of the Parisi–Frisch model of inertial-range intermittency. Using the asymptotic relation, the PDF can be reconstructed from the multifractal exponent spectrum ζn and the statistics of large scale moments. On the basis of experimental results, it is shown that moments are quantitatively represented by multifractal scaling laws and large scale Gaussian (or quasi-Gaussian) statistics. The large scale at which the statistics are Gaussian (or quasi-Gaussian) is determined from inertial-range data alone and is of the order of the integral scale for Taylor-scale Reynolds numbers Rλ in the range (300–2200). This representation of moments together with the double asymptotic relations is able to reconstruct quantitatively the experimental inertial-range PDF. Analytic expressions (She-Lévêque and Log-normal) of scaling exponents are both shown to lead to reconstructed PDF with systematic deviations from experiment.