Multifractal PDF analysis for intermittent systems

The formula for probability density functions (PDFs) has been extended to include PDF for energy dissipation rates in addition to other PDFs such as for velocity fluctuations, velocity derivatives, fluid particle accelerations, energy transfer rates, etc, and it is shown that the formula actually explains various PDFs extracted from direct numerical simulations and experiments performed in a wind tunnel. It is also shown that the formula with appropriate zooming increment corresponding to experimental situation gives a new route to obtain the scaling exponents of velocity structure function, including intermittency exponent, out of PDFs of velocity fluctuations.Comment: 10 pages, 5 figure

Harmonious Representation of PDF's reflecting Large Deviations

The framework of multifractal analysis (MFA) is distilled to the most sophisticated one. Within this transparent framework, it is shown that the harmonious representation of MFA utilizing two distinct Tsallis distribution functions, one for the tail part of probability density function (PDF) and the other for its center part, explains the recently observed PDF's of turbulence in the highest accuracy superior to the analyses based on other models such as the log-normal model and the $p$ model.Comment: 11 pages, 2 figure

Multifractal Analysis of Various PDF in Turbulence based on Generalized Statistics: A Way to Tangles in Superfluid He

By means of the multifractal analysis (MFA), the expressions of the probability density functions (PDFs) are unified in a compact analytical formula which is valid for various quantities in turbulence. It is shown that the formula can explain precisely the experimentally observed PDFs both on log and linear scales. The PDF consists of two parts, i.e., the {\it tail} part and the {\it center} part. The structure of the tail part of the PDFs, determined mostly by the intermittency exponent, represents the intermittent large deviations that is a manifestation of the multifractal distribution of singularities in physical space due to the scale invariance of the Navier-Stokes equation for large Reynolds number. On the other hand, the structure of the center part represents small deviations violating the scale invariance due to thermal fluctuations and/or observation error.Comment: 10 pages and 5 figure

PDF of Velocity Fluctuation in Turbulence by a Statistics based on Generalized Entropy

An analytical formula for the probability density function (PDF) of the velocity fluctuation in fully-developed turbulence is derived, non-perturbatively, by assuming that its underlying statistics is the one based on the generalized measures of entropy, the R\'{e}nyi entropy or the Tsallis-Havrda-Charvat (THC) entropy. The parameters appeared in the PDF, including the index $q$ which appears in the measures of the R\'{e}nyi entropy or of the THC entropy are determined self-consistently with the help of the observed value $\mu$ of the intermittency exponent. The derived PDF explains quite well the experimentally observed density functions.Comment: 10 pages, 2 figure

Analysis of Velocity Derivatives in Turbulence based on Generalized Statistics

A theoretical formula for the probability density function (PDF) of velocity derivatives in a fully developed turbulent flow is derived with the multifractal aspect based on the generalized measures of entropy, i.e., the extensive Renyi entropy or the non-extensive Tsallis entropy, and is used, successfully, to analyze the PDF's observed in the direct numerical simulation (DNS) conducted by Gotoh et al.. The minimum length scale r_d/eta in the longitudinal (transverse) inertial range of the DNS is estimated to be r_d^L/eta = 1.716 (r_d^T/eta = 2.180) in the unit of the Kolmogorov scale eta.Comment: 6 pages, 1 figur