Separation regions in two-dimensional high-Reynolds-number flow

Two-dimensional flow inside a closed, rigid container is driven by a spatially invariant source of vorticity. At high Reynolds number an asymptotic structure with an inviscid core and boundary layers emerges. For an internal flow, the slip boundary velocity must in general decelerate in order to be periodic. Hence an adverse pressure gradient acts in the standard boundary-layer equations with the associated danger of flow separation. This problem, considered inside a nearly circular container, is found to be regularized by the shear stress and displacement conditions imposed on the boundary layer. Flow reversal can then occur within the layer and the formation of small separation regions can be studied. Both steady and time-periodic motions are considered. Asymptotic solutions in the large-Reynolds-number limit are constructed for various parameter regimes involving the shape perturbation parameter delta and the frequency of time oscillations controlled by the Womersley number alpha (-1).