Instability and fragility in Wall Turbulence Transition
The mathematical analysis of transition to turbulence in wall-bounded shear flows has been a long standing problem in hydrodynamic stability since the early twentieth century. The problem is central to understanding skin-friction drag, and its reduction and control by both active and passive mechanisms. While a satisfactory mathematical theory of transition in these flows remains incomplete, substantial progress has been made in the past two decades, and is related to concepts of non-normal growth and pseudo-spectral analysis. I will argue that one key to this progress is to view transition in shear flows not only as a stability problem, but also as a fragility of the underlying dynamics to structural perturbations. I will describe these recent developments which have interesting connections with control theory, and in particular with robust stability problems. The implications of this new theory to control of turbulent skin friction drag will also be outlined.