Self Similarity, Critical Points, and Hill’s Spherical Vortex

  • G. A. AllenJr.
Conference paper
Part of the Lecture Notes in Engineering book series (LNENG, volume 13)

Abstract

Hill’s spherical vortex, a classical solution dating from 1894, is examined in a new perspective. The axisymmetric vorticity transport equation is taken to its low Reynolds number limit (no convective term) and cast into a form that is self similar in time. The subsequent equation is solved by separation of variables and shown to have two linearly independent solutions. One of these two solutions satisfies the Navier-Stokes equation exactly for all Reynolds numbers. However, to satisfy boundary conditions for the unbounded problem, both of these solutions as well as irrotational components are required. The results are represented by self similar particle paths. The flow topology is examined in the context of critical points of the self similar particle paths. It is shown that this result undergoes two transitions in topological structure with changing Reynolds number. Also shown is that the creeping flow (low Reynolds number) solution has three possible topological states in the axisymmetric case.

Keywords

Vortex Vorticity Tate Geophysics Topo 

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References

  1. [1]
    G.A. ALLEN, B.J. CANTWELL: Transition and Mixing in Axisymmetric Jets and Vortex Rings. Ph. D. Thesis, Stanford University, Department of Aeronautics and Astronautics Report, SUDAAR541, May 1984; NASA Contractors Report (to be published also).Google Scholar
  2. [2]
    G.K. BATCHELOR: An Introduction to Fluid Dynamics. Cambridge University Press, p. 526, (1967).MATHGoogle Scholar
  3. [3]
    B.J. CANTWELL: Transition in the Axisymmetric Jet. J. Fluid Mech., 104, 369–386 (1980).CrossRefMathSciNetGoogle Scholar
  4. [4]
    B.J. CANTWELL, G.A. ALLEN: Entrainment Diagrams for Viscous Flows. Proceedings Third Symposium on Turbulent Shear FLows, 17.12–17.18 (1981).Google Scholar
  5. [5]
    B.J. CANTWELL, G.A. ALLEN: Transition and Mixing in Axisymmetric Jets and Vortex Rings. Proceedings IUTAM Symposium on Turbulence and Chaotic Phenomena in Fluids, 80–89 (1983).Google Scholar
  6. [6]
    M.J.M. HILL: On a Spherical Voxtex. Phil Trans. A, clxxxv (1894)Google Scholar
  7. [7]
    H. LAMB: Hydrodynamics. Dover Publications, 245–246 (1932).MATHGoogle Scholar
  8. [8]
    L. LANDAU: A New Exact Solution of Navier-Stokes Equations. C.R. Acad. Sci. Dok. 43, 286–288 (1944).MATHGoogle Scholar
  9. [9]
    A.E. PERRY, B.D. FAIRLIE: Critical Points in Flow Patterns. Adv. Geophysics, 18, 299–315 (1974).CrossRefGoogle Scholar
  10. [10]
    C. SOZOU: Development of the Flow Field of a Point Force in an Infinite Fluid. J. Fluid Mech., 91, 541–546 (1979).CrossRefMathSciNetGoogle Scholar
  11. [11]
    H.B. SQUIRE: The Round Laminar Jet. Quart. J. Mech. Appl. Math. 4, 321–329 (1951).MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1985

Authors and Affiliations

  • G. A. AllenJr.

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