Highly optimized transitions to turbulence

Abstract

We study the Navier-Stokes equations in three dimensional plane Couette flow geometry subject to stream-wise constant initial conditions and perturbations. The resulting two dimensional/three component (2D/3C) model has no bifurcations and is globally (non-linearly) stable for all Reynolds numbers R, yet has a total transient energy amplification that scales like R/sup 3/. These transients also have the particular dynamic flow structures known to play a central role in wall bounded shear flow transition and turbulence. This suggests a highly optimized tolerance (HOT) model of shear flow turbulence, where streamlining eliminates generic bifurcation cascade transitions that occur in bluff body flows, resulting in a flow which is stable to arbitrary changes in Reynolds number but highly fragile in amplifying arbitrarily small perturbations. This result indicates that transition and turbulence in special streamlined geometries is not a problem of linear or nonlinear instability, but rather a problem of robustness.