Lyapunov Analysis of Homogeneous Isotropic Turbulence

TL;DR

This paper employs Lyapunov analysis to investigate the energy cascade and velocity fluctuations in homogeneous isotropic turbulence, providing a novel theoretical framework for understanding turbulence dynamics.

Contribution

It introduces a Lyapunov-based approach to analyze turbulence, leading to closure of the von Kármán-Howarth equation and new insights into the energy cascade mechanism.

Findings

Provides a theoretical explanation for the energy cascade process.

Achieves closure of the von Kármán-Howarth equation.

Includes numerical results validating the analysis.

Abstract

The present work studies the isotropic and homogeneous turbulence for incompressible fluids through a specific Lyapunov analysis, assuming that the turbulence is due to the bifurcations associated to the velocity field. The analysis consists in the study of the mechanism of the energy cascade from large to small scales through the Lyapunov analysis of the relative motion between two particles and in the calculation of the velocity fluctuation through the Lyapunov analysis of the local deformation and the Navier-Stokes equations. The analysis provides an explanation for the mechanism of the energy cascade, leads to the closure of the von K\'arm\'an-Howarth equation, and describes the statistics of the velocity difference. Several tests and numerical results are presented.