Refinement of a previous hypothesis of the Lyapunov analysis of isotropic turbulence

TL;DR

This paper refines a Lyapunov analysis of isotropic turbulence by providing a more rigorous proof of the structure function without relying on Kolmogorov scales, focusing on stochastic variables and correlations.

Contribution

It offers a more rigorous derivation of the velocity difference structure function, removing the dependence on Kolmogorov scales from previous work.

Findings

Derived the same structure function without Kolmogorov scale assumptions

Proposed a stochastic variable decomposition for velocity differences

Confirmed the sufficiency of pair and triple correlations for turbulence statistics

Abstract

The purpose of this brief comunication is to improve a hypothesis of the previous work of the author (de Divitiis, Theor Comput Fluid Dyn, doi:10.1007/s00162-010-0211-9) dealing with the finite--scale Lyapunov analysis of isotropic turbulence. There, the analytical expression of the structure function of the longitudinal velocity difference $\Delta u_r$ is derived through a statistical analysis of the Fourier transformed Navier-Stokes equations, and by means of considerations regarding the scales of the velocity fluctuations, which arise from the Kolmogorov theory. Due to these latter considerations, this Lyapunov analysis seems to need some of the results of the Kolmogorov theory. This work proposes a more rigorous demonstration which leads to the same structure function, without using the Kolmogorov scale. This proof assumes that pair and triple longitudinal correlations are sufficient to determine the statistics of $\Delta u_r$, and adopts a reasonable canonical decomposition of the velocity difference in terms of proper stochastic variables which are adequate to describe the mechanism of kinetic energy cascade.