Transition in shear flows. Nonlinear normality versus non-normal linearity

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Abstract
A critique is presented of recent works promoting the concept of non-normal operators and transient growth as the key to understanding transition to turbulence in shear flows. The focus is in particular on a simple model [Baggett et al., Phys. Fluids 7, 883 (1995)] illustrating that view. It is argued that the question of transition is really a question of existence and basin of attraction of nonlinear self-sustaining solutions that have little contact with the non-normal linear problem. An alternative nonlinear point of view [Hamilton et al., J. Fluid Mech. 287, 317 (1995)] that seeks to isolate a self-sustaining nonlinear process, and the critical Reynolds number below which it ceases to exist, is discussed and illustrated by a simple model. Connections with the Navier–Stokes equations and observations are highlighted throughout.