Turbulent patterns in wall-bounded flows: a Turing instability?

TL;DR

This paper proposes that the transition to patterned turbulence in wall-bounded flows can be explained by a Turing instability in a reaction-diffusion model derived from classical turbulence models, revealing intrinsic pattern wavelengths.

Contribution

It introduces a reaction-diffusion framework to model turbulence transition, linking pattern formation to Turing instability, and identifies the significance of diffusivity ratios in turbulent patterning.

Findings

Turing instability causes pattern formation in turbulent flows.

A reduced model clarifies the parameter space for pattern emergence.

The pattern wavelength has an intrinsic significance in the model.

Abstract

In their way to/from turbulence, plane wall-bounded flows display an interesting transitional regime where laminar and turbulent oblique bands alternate, the origin of which is still mysterious. In line with Barkley's recent work about the pipe flow transition involving reaction-diffusion concepts, we consider plane Couette flow in the same perspective and transform Waleffe's classical four-variable model of self-sustaining process into a reaction-diffusion model. We show that, upon fulfillment of a condition on the relative diffusivities of its variables, the featureless turbulent regime becomes unstable against patterning as the result of a Turing instability. A reduced two-variable model helps us to delineate the appropriate region of parameter space. An {\it intrinsic} status is therefore given to the pattern's wavelength for the first time. Virtues and limitations of the model are discussed, calling for a microscopic support of the phenomenological approach.