Statistics of finite scale local Lyapunov exponents in fully developed homogeneous isotropic turbulence

TL;DR

This paper investigates the statistical properties of finite scale local Lyapunov exponents in fully developed turbulence, linking them to velocity correlations and energy cascade dynamics through theoretical analysis and numerical validation.

Contribution

It introduces a maximum entropy-based PDF for local Lyapunov exponents in turbulence, deriving key relationships and closure models validated by simulations.

Findings

The distribution of local Lyapunov exponents is an unsymmetrical uniform function.

Derived relationships between average and maximum Lyapunov exponents.

Numerical simulations support the theoretical PDF and derived relationships.

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 to the fluid kinematic state is maximum. The distribution of the local Lyapunov exponents results to be 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 K\`arm\`an-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.