Flow between a stationary and a rotating disk shrouded by a co-rotating cylinder

Boundary layers on stationary and rotating disks have received much attention since von Kármán' s [Z. Angew. Math. Mech. 1, 233 (1921)] and Bödewadfs [Z. Angew. Math. Mech. 20, 241 (1940)] studies of the cases witfa disks of infinite radius. Theoretical treatments have focused on similarity treatments leading to conflicting ideas about existence and uniqueness, and where self-similar somtions exist, whether they are physically realizable. The coupling between the boundary layer flows and the interior flow between them, while being of practical importance in a variety of situations such as turbomachinery and ocean circulations, is not well understood. Here, a numerical treatment of the axisymmetric Navier-Stokes equations, together with some experiments for the case of finite stationary and rotating disks bounded by a co-rotating sidewall is presented. We show that in the long tune limit, solutions are steady and essentially self-similar. Yet the transients are not. In particular, axisymmetric waves propagate in die stationary disk boundary layer when the vortex lines entering the boundary layer develop inflection points, and there are subsequent eruptions of vortical flow out of the boundary layer deep into the interior at large Reynolds numbers.