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.
We define a generalized vortex to have azimuthal velocity proportional to a power of radiusr−n. The properties of the steady laminar boundary layer generated by such a vortex over a fixed coaxial disk of radiusaare examined. Though the boundary-layer thickness is zero a t the edge of the disk, reversals of the radial component of velocity u must occur, so that an extra boundary condition is needed at any interior boundary radiusrEto make the structure unique. Numerical integrations of the unsteady governing equations were carried out forn= − 1, 0, ½ and 1. Whenn= 0 and − 1 solutions of the self-similar equations are known for an infinite disk. Assuming terminal similarity to fix the boundary conditions atr=rEwhenur> 0, a consistent solution was found which agrees with those of the self-similar equations whenrEis small. However, ifn= ½ and 1, no similarity solutions are known, although the terminal structure forn= 1 was deduced earlier by the present authors. From the numerical integration forn= ½, we are able to deduce the limit structure forr→ 0 by using a combination of analytic and numerical techniques with the proviso of a consistent self-similar form asrE→ 0. The structure is then analogous to a ladder consisting of an infinite number of regions where viscosity may be neglected, each separated by much thinner viscous transitional regions playing the role of the rungs. This structure appears to be characteristic of all generalized vortices for which 0.1217 <n< 1.
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