Laminar Boundary-layer Theory: a 20th Century Paradox?

Boundary-layer theory is crucial in understanding why certain phenomena occur. We start by reviewing steady and unsteady separation from the viewpoint of classical non-interactive boundary-layer theory. Next, interactive boundary-layer theory is introduced in the context of unsteady separation. This discussion leads onto a consideration of largeReynolds-number asymptotic instability theory. We emphasise that a key aspect of boundary-layer theory is the development of singularities in solutions of the governing equations. This feature, when combined with the pervasiveness of instabilities, often forces smaller and smaller scales to be considered. Such a cascade of scales can limit the quantitative usefulness of solutions. We also note that classical boundarylayer theory may not always be the large-Reynolds-number limit of the Navier-Stokes equations. This is because of the possible amplification of short-scale modes, which are initially exponentially small, by a Rayleigh instability mechanism.