Self-Similarity in Fully Developed Homogeneous Isotropic Turbulence Using the Lyapunov Analysis

TL;DR

This paper uses Lyapunov analysis to derive self-similar velocity correlation functions and statistical properties in fully developed isotropic turbulence, providing insights into turbulence structure and behavior.

Contribution

It introduces a novel Lyapunov-based closure for the von Karman-Howarth equation and derives self-similar solutions for turbulence statistics.

Findings

Correlation functions match known turbulence properties

Velocity difference statistics are consistent with experimental data

Steady-state solutions support the self-similarity hypothesis

Abstract

In this work, we calculate the self-similar longitudinal velocity correlation function and the statistical properties of velocity difference using the results of the Lyapunov analysis of the fully developed isotropic homogeneous turbulence just presented by the author in a previous work (arXiv:0911.1463). There, a closure of the von Karman-Howarth equation is proposed and the statistics of velocity difference is determined through a specific analysis of the Fourier-transformed Navier-Stokes equations. The correlation functions correspond to steady-state solutions of the von Karman-Howarth equation under the self-similarity hypothesis introduced by von Karman. These solutions are numerically determined with the statistics of velocity difference. The obtained results adequately describe the several properties of the fully developed isotropic turbulence.