Formation and Instabilities of Coherent Structures in Channel Flows

Abstract

Channel flows are known to pass directly from a laminar, one-dimensional flow to intermittent turbulence as the Reynolds number is increased. In closed domains, the intermittency takes the form of a repeating cycle of four basic steps: the formation of coherent structures consisting of streamwise vortices and streaks, linear instability of these structures, a sudden onset of turbulence, and collapse of the turbulence. This paper presents a discussion of the first two steps, describing the formation of coherent structures and their instabilities. We add a small forcing to the Navier-Stokes equations, which makes the streamwise vortices a steady equilibrium, and study their stability by direct numerical solution of the appropriate linear equations. We find that the vortices are linearly unstable to a secondary mode, which produces the type of three-dimensional vortical structures commonly found in turbulent channel flow. We also report a new type of linear mode for the case of plane Couette flow, which is a periodic but non-sinusoidal function of time. We conclude with an interpretation of our observations in terms of phase space dynamics.Key wordsStabilitycoherent structuresPoiseuille flowCouette flow