Boundary Layer Induced by a Potential Vortex

A numerical computation of the laminar boundary layer on a fixed circular disk of radius a whose axis is concentric with that of a vortex having circulation Γ is described. The computations were started at the edge of the disk and continued inward toward the axis until the properties of the terminal flow became evident. A two-layer asymptotic expansion was formulated for the solution of the boundary-layer equations near the axis, and the terminal-flow properties revealed by the analysis are shown to be in excellent agreement with the numerical results. The structure of the terminal boundary layer consists of an inner layer next to the surface with thickness O(ν/Γ)1/2r in which the flow is primarily radial, and an outer layer with thickness O(ν/Γ)1/2a of predominantly inviscid nature in which the flow recovers to the external potential vortex. The mass flux in the outer layer does not vanish as r→0, indicating that the boundary layer must erupt from the surface at r=0in the manner envisioned by Moore.