Lyapunov spectrum of forced homogeneous isotropic turbulent flows

TL;DR

This paper computes the Lyapunov spectrum of forced homogeneous isotropic turbulence, revealing how chaotic dynamics relate to flow structures and how perturbation responses localize with increasing Reynolds number.

Contribution

It provides the first direct computation of the Lyapunov spectrum and attractor dimension scaling for 3D homogeneous isotropic turbulence, linking chaos to flow features.

Findings

Chaotic response aligns with large velocity gradients at low Reynolds numbers.

Flow response to perturbations becomes more localized as Reynolds number increases.

Energy spectrum of Gram-Schmidt vectors is nearly independent of Lyapunov index.

Abstract

In order to better understand deviations from equilibrium in turbulent flows, it is meaningful to characterize the dynamics rather than the statistics of turbulence. To this end, the Lyapunov theory provides a useful description of turbulence through the study of the perturbation dynamics. In this work, the Lyapunov spectrum of forced homogeneous isotropic turbulent flows is computed. Using the Lyapunov exponents of a flow at different Reynolds numbers, the scaling of the dimension of the chaotic attractor for a three-dimensional homogeneous isotropic flow (HIT) is obtained for the first time through direct computation. The obtained Gram-Schmidt vectors (GSV) are analyzed. For the range of conditions studied, it was found that the chaotic response of the flow coincides with regions of large velocity gradients at lower Reynolds numbers and enstrophy at higher Reynolds numbers, but does not coincide with regions of large kinetic energy. Further, the response of the flow to perturbations is more and more localized as the Reynolds number increases. Finally, the energy spectrum of the GSV is computed and is shown to be almost insensitive to the Lyapunov index.