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On generalized-vortex boundary layers
Published online by Cambridge University Press: 29 March 2006
Abstract
We define a generalized vortex to have azimuthal velocity proportional to a power of radius r−n. The properties of the steady laminar boundary layer generated by such a vortex over a fixed coaxial disk of radius a are 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 radius rE to make the structure unique. Numerical integrations of the unsteady governing equations were carried out for n = − 1, 0, ½ and 1. When n = 0 and − 1 solutions of the self-similar equations are known for an infinite disk. Assuming terminal similarity to fix the boundary conditions at r = rE when ur > 0, a consistent solution was found which agrees with those of the self-similar equations when rE is small. However, if n = ½ and 1, no similarity solutions are known, although the terminal structure for n = 1 was deduced earlier by the present authors. From the numerical integration for n = ½, we are able to deduce the limit structure for r → 0 by using a combination of analytic and numerical techniques with the proviso of a consistent self-similar form as rE → 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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- © 1972 Cambridge University Press
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