On a self-sustaining process in shear flows

Abstract

A self-sustaining process conjectured to be generic for wall-bounded shear flows is investigated. The self-sustaining process consists of streamwise rolls that redistribute the mean shear to create streaks that wiggle to maintain the rolls. The process is analyzed and shown to be remarkably insensitive to whether there is no-slip or free-slip at the walls. A low-order model of the process is derived from the Navier–Stokes equations for a sinusoidal shear flow. The model has two unstable steady solutions above a critical Reynolds number, in addition to the stable laminar flow. For some parameter values, there is a second critical Reynolds number at which a homoclinic bifurcation gives rise to a stable periodic solution. This suggests a direct link between unstable steady solutions and almost periodic solutions that have been computed in plane Couette flow. It is argued that this self-sustaining process is responsible for the bifurcation of shear flows at low Reynolds numbers and perhaps also for controlling the near-wall region of turbulent shear flows at higher Reynolds numbers.