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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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References

Anderson, O. L.1966 Numerical solutions of the compressible boundary-layer equations for rotating axisymmetric flows. Ph.D. thesis, Hartford Graduate Center, Rensselaer Polytechnic Institute.
Belcher, R.1970 The structure of a laminar boundary layer under a generalized vortex. Ph.D. thesis, Ohio State University.
Bödewadt, U. T. 1940 Die Drehströmung über festem Grunde. Z. angew. Math. Mech. 20, 241.Google Scholar
Brown, S. & Stewartson, K.1969 Laminar separation. Ann. Rev. Fluid Mech. vol. 1.
Burgcraf, O. R., Stewartson, K. & Belcher, R.1971 Boundary layer induced by a potential vortex. Phys. Fluids 14, 1821–1833.Google Scholar
Cooke, J. C.1966 Numerical solution of Taylor's swirl atomizer problem. R.A.E. Tech. Rep. no. 66128.Google Scholar
Crank, J. & Nicholson, P.1947 A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Proc. Cam. Phil. Soc. 43, 50.Google Scholar
Goldshtik, M. A.1960 A paradoxical solution of the Navier-Stokes equations. J. Appl. Math. Mech. 24, 913.Google Scholar
Goldstein, S.1965 On backward boundary layers and flow in converging passages. J. Fluid Mech. 21, 33.Google Scholar
Hall, M. O.1969 The boundary layer over an impulsive started flat plate. Proc. Roy. Soc. A 310, 401.Google Scholar
Kendall, J. M.1962 Experimental study of a compressible viscous vortex. Jet Prop. Lab., Pasadena, Calif., Tech. Rep. no. 32-290.Google Scholar
King, W. S. & Lewellen, W. S.1964 Boundary-layer similarity solutions for rotating flows with and without magnetic interaction. Phys. Fluids, 7, 1674.Google Scholar
Kuo, H. L.1971 Axisymmetric flows in the boundary layer of a maintained vortex. J. Atmos. Sci. 28, 20.Google Scholar
Mcleod, J. B.1971 The existence of axially symmetric flow above a rotating disk. Proc. Roy. Soc. A 324, 391.Google Scholar
Mack, L.1962 The laminar boundary layer on a disk of finite radius in a rotating flow. Part 1. Jet Prop. Lab., Pasadena, Calif., Tech. Rep. no. 32-224.Google Scholar
Pearson, C. E.1965 Numerical solutions for the time-dependent viscous flow between two rotating coaxial disks. J. Fluid Mech. 21, 623.Google Scholar
Rayleigh, Lord1916 On the dynamics of revolving fluids. Proc. Roy. Soc. A 93, 148.Google Scholar
Reyhner, T. A. & Flügge-Lotz, I. 1968 The interaction of a shock wave with a laminar boundary layer. Int. J. Non-lin. Mech. 3, 173.Google Scholar
Riehl, H.1954 Tropical Meteorology. McGraw-Hill.
Rott, N. & Lewellen, W. S.1966 Boundary layers and their interactions in rotating flows. In Progress in Aeronautical Science (ed. D. Küchemann), vol. 7, p. 111. Pergamon.
Schlichting, H.1968 Boundary-Layer Theory (6th edn.). McGraw-Hill.
Serrin, J.1972 The swirling vortex. Phil. Trans. Roy. Soc. A 270, 325ndash;360.Google Scholar
Stewartson, K.1958 On rotating laminar boundary layers. Boundary-Layer Research, Symp., Freiburg, p. 59. Springer.