Topological Entropy of Two Dimensional Turbulence

TL;DR

This paper presents an exact Eulerian formula for the topological entropy in 2D turbulence, enabling efficient estimation of material line stretching without particle tracking, validated by numerical simulations.

Contribution

Introduces a novel Eulerian approach to compute topological entropy in 2D turbulence, bypassing the need for Lagrangian particle tracking.

Findings

Eulerian formula accurately estimates stretching rates

Excellent agreement with Lagrangian measurements

Applicable across various Reynolds numbers

Abstract

Deformation of material lines drives transport and dissipation in many industrial and natural flows. Here we report an exact Eulerian formula for the stretching rate of a material line, also known as the topological entropy, in a prototype two-dimensional fluid. The only requirement is a distribution of eigenvalues of the strain rate tensor and their decorrelation time. This eliminates the need for Lagrangian tracking in experimental turbulence where particle trajectories are entangled, and thus poorly resolved. Numerical simulations reveal an excellent agreement between our Eulerian estimate and the stretching rate of a Lagrangian material line, over a wide range of Reynolds number.