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On generalized-vortex boundary layers

Published online by Cambridge University Press:  29 March 2006

R. J. Belcher ,
O. R. Burggraf  and
K. Stewartson
R. J. Belcher
Affiliation:
Department of Aeronautical and Astronautical Engineering, The Ohio State University Present address: Vought Aeronautics Company, P.O. Box 5907, Dallas, Texas 75222.
O. R. Burggraf
Affiliation:
Department of Aeronautical and Astronautical Engineering, The Ohio State University
K. Stewartson
Affiliation:
Department of Aeronautical and Astronautical Engineering, The Ohio State University Present address: Department of Mathematics, University College, London, Gower Street, London, W.C.1.
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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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 52 , Issue 4 , 25 April 1972 , pp. 753 - 780
Copyright
© 1972 Cambridge University Press

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