The problem of describing transition in wall bounded shear flows such as channel and boundary layer flows is an important and old problem in Hydrodynamic Stability. Classical linear hydrodynamic stability theories provides predictions that are at odds with most experiments where ``natural'' transition occurs. However, in the past decade it has become recognized that a new analysis of the linearized Navier-Stokes equations yields much more satisfactory answers. This analysis is known as related to so-called non-normal transient growth or psuedo-spectral analysis.
In this talk, we will review this new hydrodynamic stability theory from the point of view of quantifying uncertainty. The linearized Navier-Stokes equations in strongly sheared flows exhibit remarkable sensitivity to dynamical perturbations and external forcing or noise. We will argue that the right kind stability analysis must take this uncertainty explicitly into account, and that in such geometries, transition is not a question only of stability, but robustness as well. We point out connections with modern Robust Control Theory, were analysis of uncertainty effects on stability has been heavily studied.
We show how this analysis brings out the ubiquitous coherent structures of stream-wise vortices and streaks as fundamental to the dynamics of the linearized NS equations. We will also illustrate how distributed wall roughness can act as a generator of flow disturbances which then initiate transition scenarios. We discuss the implications of this analysis to flow control.