A doubly-localized equilibrium solution of plane Couette flow

Abstract

We present an equilibrium solution of plane Couette flow that is exponentially localized in both the spanwise and streamwise directions. The solution is similar in size and structure to previously computed turbulent spots and localized, chaotically wandering edge states of plane Couette flow. A linear analysis of dominant terms in the Navier-Stokes equations shows how the exponential decay rate and the wall-normal overhang profile of the streamwise tails are governed by the Reynolds number and the dominant spanwise wavenumber. Perturbations of the solution along its leading eigenfunctions cause rapid disruption of the interior roll-streak structure and formation of a turbulent spot, whose growth or decay depends on the Reynolds number and the choice of perturbation.

11 pages, 5 figures. Discussion of symmetries expanded to explain angle-bracket notation and antisymmetries of eigenfunctions. More explicit discussion of the determination of the dominant spanwise wavenumber of the tails

References in corpus (3)

• Computational Study of Turbulent-Laminar Patterns in Couette Flow
• Lower branch coherent states in shear flows: transition and control
• Laminar-turbulent boundary in plane Couette flow

Cited by in corpus (6)

• Streamwise and doubly-localised periodic orbits in plane Poiseuille flow
• Experiments on transient growth of turbulent spots
• Self-sustainment of coherent structures in counter-rotating Taylor-Couette flow
• Localization in a spanwise-extended model of plane Couette flow
• Streamwise decay of localized states in channel flow
• Bifurcation structure of unstable periodic orbits in plane Couette flow with the Smagorinsky model