Chaotic measures as an alternative to spectral measures for analysing turbulent flow

TL;DR

This paper reviews how chaotic measures, like Lyapunov exponents, serve as effective tools alongside spectral measures for analyzing turbulent flows, especially in small-scale simulations and phase transition studies.

Contribution

It demonstrates the viability of chaotic measures in turbulence analysis through DNS studies and highlights their advantages over spectral measures in certain scenarios.

Findings

Chaotic measures accurately quantify turbulence at small scales.

Chaotic measures reveal phase transition behavior in turbulent systems.

DNS confirms the relation between Lyapunov exponent and Reynolds number.

Abstract

Turbulence has associated chaotic features. In the past couple of decades there has been growing interest in the study of these features as an alternative means of understanding turbulent systems. Our own input to this effort has been in contributing to the initial studies of chaos in Eulerian flow using direct numerical simulation (DNS). In this review we discuss the progress achieved in the turbulence community in understanding chaotic measures including our own work. A central relation between turbulence and chaos is one by Ruelle that connects the maximum Lyapunov exponent and the Reynolds number. The first DNS studies, ours amongst them, in obtaining this relation has shown the viability of chaotic simulation studies of Eulerian flow. Such chaotic measures and associated simulation methodology provides an alternative means to probe turbulent flow. Building on this, we have analyzed the finite time Lyapunov exponent (FTLE) and studied its fluctuations, and found that chaotic measures could be quantified accurately even at small simulation box sizes where for comparative sizes spectral measures would be inconclusive. We further highlight applications of chaotic measures in analyzing phase transition behavior in turbulent flow and two dimensional thin layer turbulent systems. This work has shown chaotic measures are an excellent tool that can be used alongside spectral measures in studying turbulent flow.