Statistics of Finite Scale Local Lyapunov Exponents in Fully Developed Homogeneous Isotropic Turbulence

Nicola De Divitiis

doi:10.1155/2018/2365602

Abstract

The present work analyzes the statistics of finite scale local Lyapunov exponents of pairs of fluid particles trajectories in fully developed incompressible homogeneous isotropic turbulence. According to the hypothesis of fully developed chaos, this statistics is here analyzed assuming that the entropy associated with the fluid kinematic state is maximum. The distribution of the local Lyapunov exponents results in an unsymmetrical uniform function in a proper interval of variation. From this PDF, we determine the relationship between average and maximum Lyapunov exponents and the longitudinal velocity correlation function. This link, which in turn leads to the closure of von Kármán–Howarth and Corrsin equations, agrees with results of previous works, supporting the proposed PDF calculation, at least for the purposes of the energy cascade main effect estimation. Furthermore, through the property that the Lyapunov vectors tend to align the direction of the maximum growth rate of trajectories distance, we obtain the link between maximum and average Lyapunov exponents in line with the previous results. To validate the proposed theoretical results, we present different numerical simulations whose results justify the hypotheses of the present analysis.

Highlights

The finite scale Lyapunov exponents of the fluid kinematic field are of paramount importance because they (i) describe the turbulent energy cascade phenomenon and (ii) give the fluid viscous dissipation when the length scale goes to zero
It is worth remarking that the adoption of the finite scale Lyapunov exponents in place of the classical ones is justified by the fact that the turbulence is a complex phenomenon involving numerous length scales; each of them is characterized by different properties
In order to justify the plausibility of the previous hypotheses, this section presents one statistical analysis of numerical simulations of a simple differential system representing incompressible fluid kinematics

Summary

Introduction

The finite scale Lyapunov exponents of the fluid kinematic field are of paramount importance because they (i) describe the turbulent energy cascade phenomenon and (ii) give the fluid viscous dissipation when the length scale goes to zero. One of the characteristics of these exponents in turbulence is that their statistics is related to the instantaneous velocity field, whereas they do not depend directly on the time variations of this latter. Such characteristic, which represents the crucial point of this work, is consequence of the fact that the times of variations of the velocity field, which changes according to the Navier–Stokes equations, are much greater than those of the fluid displacements which follow the fluid kinematics [1, 2]. The several perturbations of finite size will vary following nonlinear differential equations out of tangent space; the sole use of classical Lyapunov exponents is not adequate to describe the perturbations behavior associated with the different length scales and the energy cascade

Objectives

Results

Conclusion