Multi-time, multi-scale correlation functions in turbulence and in turbulent models

Abstract

A multifractal-like representation for multi-time, multi-scale velocity correlation in turbulence and dynamical turbulent models is proposed. The importance of subleading contributions to time correlations is highlighted. The fulfillment of the dynamical constraints due to the equations of motion is thoroughly discussed. The predictions stemming from this representation are tested within the framework of shell models for turbulence.

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“…One obvious limitation of random multiplicative process is the absence of any time dynamics in the field, as one can immediately highlight by considering space-time correlations. Space-time scaling is a crucial and delicate issue when considering multifractal fields for the Navier-Stokes equations [8,9]. It is the aim of this paper to understand how one can exhibits a multifractal field whose space and time scaling is consistent with the scaling constraints imposed by the Navier-Stokes equations.…”

“…Expression (9) has been introduced in [8] and analyzed in details in [9]. We underline that, as a consequence of (9), we can predict the scaling properties of quantities like…”

“…We now want to understand how to define a random process satisfying both multiscaling in space (6) and multiscaling in time (9). In a more general way, we would like to exhibit random multifractal fields with prescribed dynamical scaling.…”

A Gibbs-Like Measure for Single-Time, Multi-Scale Energy Transfer in Stochastic Signals and Shell Model of Turbulence

“…the model is a reliable approximation of velocity evolution in a quasi-Lagrangian reference frame. The model is known to possess realistic multi-time and multi-scale correlation functions, including anomalous inertial range scaling and dissipative anomaly, for a review see [8,5,10]. Typical observable checking single-scale Probability Density Functions (PDF) are given by structure functions:…”

“…The energy cascade process has been often, and fruitfully, described in terms of a multi-step fragmentation process describing the tendencies of inertial range eddies to break in smaller and smaller eddies, following the celebrated Richardson scenario [1]. The spatio-temporal complexity of the fragmentation process has been successfully described by [2] using the multifractal language, which have proved able to reproduce qualitatively and quantitatively single-scale, multi-scale and multi-time multi-scale velocity correlation functions [3,4,5,6]. By using the multifractal language one can assume that the velocity difference on scale r 2 is linked to the velocity difference at scale r 1 ≥ r 2 by the equation δv(r 2 ) = M (r 2 , r 1 )δv(r 1 ),…”