Geometry and topology of turbulence in active nematics

TL;DR

This paper presents a theoretical and numerical study of low Reynolds number turbulence in active nematic fluids, revealing vortex distributions, developing a mean-field theory, and linking flow geometry to topological defect proliferation.

Contribution

It introduces a mean-field theory for active turbulence, connecting flow structures to topological defects, and predicts vortex size distributions in active nematics.

Findings

Vortices in active nematics have exponentially distributed areas.

The mean-field theory allows analytical calculation of spectral densities and correlation functions.

The flow geometry is closely linked to the proliferation of topological defects.

Abstract

The problem of low Reynolds number turbulence in active nematic fluids is theoretically addressed. Using numerical simulations I demonstrate that an incompressible turbulent flow, in two-dimensional active nematics, consists of an ensemble of vortices whose areas are exponentially distributed within a range of scales. Building on this evidence, I construct a mean-field theory of active turbulence by which several measurable quantities, including the spectral densities and the correlation functions, can be analytically calculated. Because of the profound connection between the flow geometry and the topological properties of the nematic director, the theory sheds light on the mechanisms leading to the proliferation of topological defects in active nematics and provides a number of testable predictions. A hypothesis, inspired by Onsager's statistical hydrodynamics, is finally introduced to account for the equilibrium probability distribution of the vortex sizes.