Fully Developed Turbulence and the Multifractal Conjecture

Abstract

We review the Parisi-Frisch [1] MultiFractal formalism for Navier-Stokes turbulence with particular emphasis on the issue of statistical fluctuations of the dissipative scale. We do it for both Eulerian and Lagrangian Turbulence. We also show new results concerning the application of the formalism to the case of Shell Models for turbulence. The latter case will allow us to discuss the issue of Reynolds number dependence and the role played by vorticity and vortex filaments in real turbulent flows.

Global geomagnetic responses to the IMF <i>B</i><sub>z</sub> fluctuations during the September/October 2003 high-speed stream intervals

“…Vörös (2000), using the multi-fractal approach on highlatitude geomagnetic fluctuations time series, found an important timescale around 60 min. According to (Benzi and Biferale, 2009), the multi-fractal approach to turbulence is based on the hypotheses that the statistical properties of the turbulent time series do exhibit scaling properties. They stated that this 60 min scale is due to the loading-unloading mechanism, which depends on the geomagnetic activity level.…”

Inertial range Eulerian and Lagrangian statistics from numerical simulations of isotropic turbulence

“…Third, it is important to improve experimental and numerical accuracy to measure small-scales and small-time fluctuations, where highly non trivial physics is developing as shown by the strong enhancement of local intermittency in the dip region τ ∈ [1 : 10]τ η . Such strong enhancement of fluctuations around the viscous scale is due to local fluctuations of the dissipative cut-off, which reflects in to the existence of different viscous effects for different moments and different correlation functions (Frisch & Vergassola (1991);L'vov & Procaccia (1996); Yakhot & Sreenivasan (2004); Benzi & Biferale (2009)). It is well described, within the multifractal theory, by the Paladin-Vulpiani relation (5.8), as shown by the fact that the dip region disappear by removing the viscous fluctuations in (5.10-5.11), see inset of the right side of fig.…”

“…Since those works, they have been extended to describe also velocity gradients (Nelkin (1990); Benzi et al (1991)); multi-scale velocity correlations (Belinicher et al (1998); Benzi et al (1998)) and fluctuations of the Kolmogorov viscous scale (Paladin & Vulpiani (1987); Meneveau (1996); Biferale (2008); Frisch & Vergassola (1991)). Let us notice that other attempts have been made to introduce fluctuations of the Kolmogorov scale in turbulence (L'vov & Procaccia (1996); Yakhot & Sreenivasan (2004), most of them are equivalent or give anyhow results almost undistinguishable from the multifractal approach followed here (Benzi & Biferale (2009)). …”

Cumulative compressibility effects on slow reactive dynamics in turbulent flows

“…where ... now stands for a time and ensemble average and B(r) is a disk of radius r. Within the inertial range, we find that C q μ (r) ∼ r −a q with a q a nonlinear function of q constrained to a 0 = a 1 = 0 [17,18] ( In the inset of Figure 2, we show the scaling of C 2 μ (r) versus r for a particular value of μ = 0 and κ = 0.16; plots for other values look similar.). It is important to remember that each κ > 0 can be associated with a compressibility length (and time) scale, = u rms / (∇ · u) 2 1/2 l κ = u rms / (∇ · u) 2 1/2 where = √ Eu rms = √ E is the root-mean-square velocity.…”