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Axisymmetric swirling flow around a vortex breakdown point

Published online by Cambridge University Press:  26 April 2006

Zvi Rusak
Zvi Rusak
Affiliation:
Department of Mechanical Engineering, Aeronautical Engineering and Mechanics, Rensselaer Polytechnic Institute, Troy, NY 12180–3590, USA
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Abstract

The structure of an axisymmetric and inviscid swirling flow around a vortex breakdown point is analysed. The model assumes that a free axisymmetric bubble surface is developed in the flow with a stagnation point at its nose. The classical Squire-Long equation for the stream function ψ(x,y) (where y = r2/2) is transformed into a free boundary problem for the solution of y(x, ψ). The development of the flow is studied in three regions: the approaching flow ahead of the bubble, around the bubble nose and around the separated bubble surface. Asymptotic expansions are constructed to describe the flow ahead of and behind the stagnation point in terms of the radial distance from the vortex axis and from the bubble surface, respectively. In the intermediate region around the stagnation point, the flow is approximated by an asymptotic series of similarity terms that match the expansions in the other regions. The analysis results in two possible matching processes. Analytical expressions are given for the leading term of the intermediate expansion for each of these processes. The first solution describes a swirling flow around a constant-pressure bubble surface, over which the flow is stagnant. The second solution represents a swirling flow around a pressure-varying bubble surface, where the flow expands along the bubble nose. In both solutions, the bubble nose has a parabolic shape, and both exist only when H’ > 0 (where H’ is the derivative at the vortex centre of the total head H with the stream function ψ, and can be determined from the inlet conditions). This result is shown to be equivalent to Brown & Lopez's (1990) criterion for vortex breakdown. Good agreement is found in the region around the stagnation point between the pressure-varying bubble solution and available experimental data for axisymmetric vortex breakdown.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 323 , 25 September 1996 , pp. 79 - 105
Copyright
© 1996 Cambridge University Press

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References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics, pp. 543–545. Cambridge University Press.
Benjamin, T. B. 1962 Theory of the vortex breakdown phenomenon. J. Fluid Mech. 14, 593–629.Google Scholar
Beran, P. S. 1994 The time-asymptotic behavior of vortex breakdown in tubes. Computers Fluids 23, 913–937.Google Scholar
Beran, P. S. & Culick, F. E. C. 1992 The role of non-uniqueness in the development of vortex breakdown in tubes. J. Fluid Mech. 242, 491–527.Google Scholar
Berger, S. A. & Erlebacher, G. 1995 Vortex breakdown incipience: theoretical considerations. Phys. Fluids 7, 972–982.Google Scholar
Bragg, S. L. & Hawthorne, W. R. 1950 Some exact solutions of the flow through annular cascade actuator discs. J. Aero. Sci. 17, 243–249.Google Scholar
Brewer, M. 1991 Numerische Losung der Navier-Stokes Gleichungen fur reidimensionale inkompressible instationare Stromungen zur des Wirbelaufplatzens. PhD thesis, RWTH Aachen, Germany.
Brown, G. L. & Lopez, J. M. 1990 Axisymmetric vortex breakdown. Part 2. Physical mechanisms. J. Fluid Mech. 221, 553–576.Google Scholar
Bruecker, Ch. 1995 Development and structural changes of vortex breakdown. AIAA Paper 95-2305.
Bruecker, Ch. & Althaus, W. 1992 Study of vortex breakdown by particle tracking velocimetry (PTV), Part I: Bubble type vortex breakdown modes. Exps. Fluids 13, 339–349.Google Scholar
Bruecker, Ch. & Althaus, W. 1995 Study of vortex breakdown by particle tracking velocimetry (PTV), Part 3: Time-dependent structure and development of breakdown modes. Exps. Fluids 18, 174–186.CrossRefGoogle Scholar
Escudier, M. P. 1984 Observations of the flow produced in a cylindrical container by a rotating endwall. Exps. Fluids 2, 189–196.Google Scholar
Escudier, M. P. 1988 Vortex breakdown: observations and explanations. Prog. Aerospace Sci. 25, 189–229.Google Scholar
Escudier, M. P. & Keller, J. J. 1983 Vortex breakdown: a two stage transition. AGARD CP 342, pp. 251–258.
Faler, J. H. & Leibovich, S. 1977 Disrupted states of vortex flow and vortex breakdown. Phys. Fluids 20, 1385–1400.Google Scholar
Faler, J. H. & Leibovich, S. 1978 An experimental map of the internal structure of a vortex breakdown. J. Fluid Mech. 86, 313–335.Google Scholar
Garg, A. K. & Leibovich, S. 1979 Spectral characteristics of vortex breakdown. Phys. Fluids 22, 2053–2064.Google Scholar
Goldshtik, M. & Hussain, F. 1992 Inviscid vortical flows with internal stagnation zones. bull. Am. Phys. Soc. Div. Fluid Dynamics Meeting, vol. 37, no. 8, p. 1707.Google Scholar
Grabowski, W. J. & Berger, S. A. 1975 Solutions of the Navier-Stokes equations for vortex breakdown. J. Fluid Mech. 75, 525–544 (and Corrigendum 76, 1976, 829).Google Scholar
Hall, M. G. 1972 Vortex breakdown. Ann. Rev. Fluid Mech. 4, 195–217.Google Scholar
Keller, J. J., Egli, W. & Exley, J. 1985 Force- and loss-free transitions between flow states. Z. Angew. Math. Phys. 36, 854–889.Google Scholar
Kribus, A. & Leibovich, S. 1994 Instability of strongly nonlinear waves in vortex flows. J. Fluid Mech. 269, 247–264.Google Scholar
Leibovich, S. 1978 The structure of vortex breakdown. Ann. Rev. Fluid Mech. 10, 221–246.Google Scholar
Leibovich, S. 1984 Vortex stability and breakdown: survey and extension. AIAA J. 22. 1192–1206.Google Scholar
Leibovich, S. & Kribus, A. 1990 Large amplitude wavetrains and solitary waves in vortices. J. Fluid Mech. 216, 459–504.Google Scholar
Leibovich, S. & Stewartson, K. 1983 A sufficient condition for the instability of columnar vortices. J. Fluid Mech. 126, 335–356.Google Scholar
Lessen, H., Singh, P. J. & Paillet, F. 1974 The stability of a trailing line vortex. Part 1. Inviscid theory. J. Fluid Mech. 63, 753–763.Google Scholar
Long, R. R. 1953 Steady motion around a symmetrical obstacle moving along the axis of a rotating liquid. J. Met. 10, 197–203.Google Scholar
Lopez, J. M. 1990 Axisymmetric vortex breakdown. Part 1. Confined swirling flow. J. Fluid Mech. 221, 533–552.Google Scholar
Lopez, J. M. 1994 On the bifurcation structure of axisymmetric vortex breakdown in a constricted pipe. Phys. Fluids 6, 3683–3693.Google Scholar
Lundgren, T. S. & Ashurst, W. T. 1989 Area varying waves on curved vortex tubes with application to vortex breakdown. J. Fluid Mech. 200, 283–307.Google Scholar
Marshall, J. S. 1991 A general theory of curved vortices with circular cross-section and variable core area. J. Fluid Mech. 229, 311–338.Google Scholar
Randall, J. D. & Leibovich, S. 1973 The critical state: a trapped wave model of vortex breakdown. J. Fluid Mech. 53, 481–493.Google Scholar
Salas, M. D. & Kuruvila, G. 1989 Vortex breakdown simulation: a circumspect steady of the steady laminar, axisymmetric model. Computers Fluids 17, 247–262.Google Scholar
Sarpkaya, T. 1971 On stationary and travelling vortex breakdown. J. Fluid Mech. 45, 549–559.Google Scholar
Sarpkaya, T. 1974 Effect of adverse pressure gradient on vortex breakdown. AIAA J. 12, 602–607.Google Scholar
Sarpkaya, T. 1995 Vortex breakdown and turbulence. AIAA Paper 95-0433.
Spall, R. E. & Gatski, T. B. 1991 A computational study of the topology of vortex breakdown. Proc. R. Soc. Lond. A 435, 321–337.Google Scholar
Spall, R. E., Gatski, T. B. & Ash, R. L. 1990 The structure and dynamics of bubble type vortex breakdown. Proc. R. Soc. Lond. A 429, 613–637.Google Scholar
Spall, R. E., Gatski, T. B. & Grosch, C. E. 1987 A criterion for vortex breakdown. Phys. Fluids 30, 3434–3440.Google Scholar
Squire, H. B. 1956 Rotating fluids. In Surveys in Mechanics (ed. G. K. Batchelor & R. M. Davies), pp. 139–161. Cambridge University Press.
Trigub, V. N. 1985 The problem of breakdown of a line vortex. Prikl. Math. Mech. 49, 166–171.Google Scholar
Uchida, S., Nakamura, Y. & Ohsawa, M. 1985 Experiments on the axisymmetric vortex breakdown in a swirling air flow. Trans. Japan Soc. Aeronaut. Space Sci. 27, 206–216.Google Scholar