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

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.

to appear in Europhysics Letters in a different format

References in corpus (6)

• The Significance of Simple Invariant Solutions in Turbulent Flows
• Computational Study of Turbulent-Laminar Patterns in Couette Flow
• Simplifying the complexity of pipe flow
• From temporal to spatiotemporal dynamics in transitional plane Couette flow
• Modeling transitional plane Couette flow
• Modeling the transition to turbulence in shear flows

Cited by in corpus (11)

• Scale invariance at the onset of turbulence in Couette flow
• Transition to turbulence in wall-bounded flows: Where do we stand?
• On the transition to turbulence of wall-bounded flows in general, and plane Couette flow in particular
• On the nature of laminar-turbulence intermittency in shear flows
• Laminar-Turbulent Patterning in Transitional Flows
• Linear instability of turbulent channel flow
• Patterns in transitional shear turbulence. Part 2: Emergence and optimal wavelength
• Laminar-turbulent patterning in wall-bounded shear flows: a Galerkin model
• Chaotic self-sustaining structure embeded in turbulent-laminar interface
• Parametric pattern selection in a reaction-diffusion model
• Model for transitional turbulence in a planar shear flow