Comprehensive multifractal analysis of turbulent velocity using the wavelet leaders

Abstract

Abstract. The multifractal framework relates the scaling properties of turbulence to its local regularity properties through a statistical description as a collection of local singularities. The multifractal properties are moreover linked to the multiplicative cascade process that creates the peculiar properties of turbulence such as intermittency. A comprehensive estimation of the multifractal properties of turbulence from data analysis, using a tool valid for all kind of singularities (including oscillating singularities) and mathematically well-founded, is thus of first importance in order to extract a reliable information on the underlying physical processes. The wavelet leaders yield a new multifractal formalism which meets all these requests. This paper aims at describing it and at applying it to experimental turbulent velocity data. After a detailed discussion of the practical use of the wavelet leader based multifractal formalism, the following questions are carefully investigated: (1) What is the dependence of multifractal properties on the Reynolds number? (2) Are oscillating singularities present in turbulent velocity data? (3) Which multifractal model does correctly account for the observed multifractal properties? Results from several data set analysis are used to discuss the dependence of the computed multifractal properties on the Reynolds number but also to assess their common or universal component. An exact though partial answer (no oscillating singularities are detected) to the issue of the presence of oscillating singularities is provided for the first time. Eventually an accurate parameterization with cumulants exponents up to order 4 confirms that the log-normal model (with c2 = −0.025±0.002) correctly accounts for the universal multifractal properties of turbulent velocity.

Function Spaces Vs. Scaling Functions: Tools for Image Classification

“…The simplest setting for these extension is supplied by Besov spaces; they offer the additional advantage of yielding a wavelet reformulation of the scaling function, which is simple and robust, see [1,3,4,34]. In order to define these spaces, we need to recall the notion of wavelet bases.…”

Multi-level segment analysis: definition and application in turbulent systems

“…There are several different wavelet-based methods, for example, wavelettransform-modulus-maxima (WTMM) (Muzy et al, 1991;Mallat and Hwang, 1992;Muzy et al, 1993) and wavelet-leader (WL) (Jaffard et al, 2005;Wendt et al, 2007;Lashermes et al, 2008). We consider here only WL.…”

“…where Jaffard et al, 2005;Lashermes et al, 2008;Wendt et al, 2007). The expected scaling behavior can be expressed as…”

Large deviation spectra based on wavelet leaders

“…In particular, these new spaces are compared with the S ν spaces, previously introduced using wavelet coefficients. The main difference is that now the profile includes an increasing and a decreasing part, and is therefore more realistic for most multifractal models, for which the decreasing part can prove crucial for identification, or model selection, see [28] for instance.…”