Eulerian-Lagrangian scaling of the Lyapunov exponent in homogeneous turbulence

TL;DR

This paper derives new relations for the Lyapunov exponent in homogeneous turbulence, confirming their validity through simulations and showing how key scales decrease with increasing Reynolds number.

Contribution

It introduces a heuristic derivation of the Lyapunov exponent scaling and related length scales, supported by turbulence simulations.

Findings

Derived sweeping relation and integral length scale scaling.

Confirmed relations and scaling through turbulence simulations.

Observed decrease of length scales and inverse Lyapunov exponent with Reynolds number.

Abstract

We present a heuristic derivation of the maximal Lyapunov exponent $\gamma$ of homogeneous turbulence which yields two new relations, a sweeping relation and the scaling of the uncertainty field's integral length $L_{\Delta}$. These relations and the maximal Lyapunov exponent's scaling that they imply are confirmed by periodic turbulence simulations. As the Reynolds number $Re_\lammbda$ increases, $L_\Delta$ and $\gamma^{-1}$ decrease towards values smaller than the Kolmogorov length and time scales.