Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-19T05:32:06.515Z Has data issue: false hasContentIssue false
  • English
  • Français

Solution breakdown in a family of self-similar nearly inviscid axisymmetric vortices

Published online by Cambridge University Press:  26 April 2006

R. Fernandez-Feria ,
J. Fernandez de la mora  and
A. Barrero
R. Fernandez-Feria
Affiliation:
Universidad de Málaga, ETS Ingenieros Industriales, 29013 Málaga, Spain
J. Fernandez de la mora
Affiliation:
Yale University, Mechanical Engineering Dept., New Haven, CT 06520-2159, USA
A. Barrero
Affiliation:
Universidad de Sevilla, ETS Ingenieros Industriales, 41012 Sevilla, Spain
Get access Rights & Permissions [Opens in a new window]

Abstract

Many axisymmetric vortex cores are found to have an external azimuthal velocity v, which diverges with a negative power of the distance r to their axis of symmetry. This singularity can be regularized through a near-axis boundary layer approximation to the Navier-Stokes equations, as first done by Long for the case of a vortex with potential swirl, v∼r−1. The present work considers the more general situation of a family of self-similar inviscid vortices for which v∼rm−2, where m is in the range 0 n< m < 2. This includes Longs Vortex for the case m =1. The corresponding solutions also exhibit self-similar structure, and have the interesting property of losing existence when the ratio of the inviscid near-axis swirl to axial velocity (the swirl parameter) is either larger (when 1m < 2) or smaller (when 0m < 1) than an m-dependent critical value. This behaviour shows that viscosity plays a key role in the existence or lack of existence of these particular nearly inviscid vortices and supports the theory proposed by Hall and others on vortex breakdown. Comparison of both the critical swirl parameter and the viscous core structure for the present family of vortices with several experimental results under conditions near the onset of vortex breakdown show a good agreement for values of m slightly larger than 1. These results differ strongly from those in the highly degenerate case m =1.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 305 , 25 December 1995 , pp. 77 - 91
Copyright
© 1995 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Batchelor, G. K. 1964 Axial flow in trailing line vortices. J. Fluid Mech. 20, 645–658.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics, Section 7.5. Cambridge University Press.
Benjamin, T. B. 1962 Theory of the vortex breakdown phenomenon. J. Fluid Mech. 14, 593.Google Scholar
Beran, P. S. & Culik, 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
Bojarevics, V., Freibergs, Ya., Shilova, E. I. & Shcherbinin, E. V. 1989 Electrically Induced Vortical Flows, Chap. 2. Kluwer.
Buntine, J. D. & Saffman, P. G. 1995 Inviscid swirling flows and vortex breakdown.Proc. R. Soc. Lond. A 449, 139–153.
Burggraf, O. R. & Foster, M. R. 1977 Continuation or breakdown in tornado-like vortices. J. Fluid Mech. 80, 685–703.Google Scholar
Drazin, P. G., Banks, W. H. H. & Zaturska, M. B. 1995 The development of Long's vortex. J. Fluid Mech. 286, 359–377.Google Scholar
Escudier, M. P. 1988 Vortex breakdown: observations and explanations. In Prog. Aero. Sci. 25, 189–229.Google Scholar
Faler, J. H. & Leibovich, S. 1977 Disrupted states of vortex flow and vortex breakdown. Phys. Fluids 20 1385–1400.Google Scholar
Fernandez-Feria, R., Fernandez de la Mora, J. & Barrero, A. 1995 Conically similar swirling flows at high Reynolds numbers. Part 1. One-cell solutions. SIAM J. Appl. Maths (submitted).Google Scholar
Foster, M. R. & Duck, P. W. 1982 The inviscid stability of Long's vortex, Phys. Fluids 25 1715–1718.
Foster, M. R. & Jacqmin, D. 1992 Non-parallel effects in the instability of Long's vortex. J. Fluid Mech. 244 289–306.Google Scholar
Foster, M. R. & Smith, F. T. 1989 Stability of Long's vortex at large flow force. J. Fluid. Mech. 206, 405–432.Google Scholar
Gartshore, I. S. 1962 Recent work in swirling incompressible flow. NRC Can. Aero. Rep. LR-343.
Goldshtik, M. A. 1960 A paradoxical solution of the Navier-Stokes equations. Appl. Mat. Mech.(Sov.) 24, 610–621.Google Scholar
Goldshtik, M. A. & Shtern, V. N. 1990 Collapse in conical viscous flows. J. Fluid Mech. 218, 483–508.Google Scholar
Hall, M. G. 1961 A theory for the core of a leading-edge vortex, J. Fluid Mech. 11, 209–228.
Hall, M. G. 1972 Vortex breakdown. Ann. Rev. Fluid Mech. 4, 195–218.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.
Khorrami, M. R. & Trivedi, P. 1994 The viscous stability analysis of Long's vortex. Phys. Fluids 6, 2623–2630.Google Scholar
Landau, L. 1944 A new exact solution to the Navier-Stokes equations. Dokl. Akad. Nauk SSSR 43, 286–288.Google Scholar
Leibovich, S. 1978 The structure of vortex breakdown. Ann. Rev. Fluid Mech. 10, 221–346.Google Scholar
Leibovich, S. 1984 Vortex stability and breakdown: survey and extension. AIAA J. 22, 1192–1206.Google Scholar
Long, R. L. 1958 Vortex motion in viscous fluid. J. Met. 15, 108–112.Google Scholar
Long, R. L. 1961 A vortex in an infinite viscous fluid. J. Fluid Mech. 11, 611–625.Google Scholar
Mayer, E. W. & Powell, K. G. 1992 Similarity solutions for viscous vortex cores, J. Fluid Mech. 238, 487–507.
Ogawa, A. 1993 Vortex Flows CRC Press.
Paull, R. & Pillow, A. F. 1985a Conically similar viscous flows. Part 2. J. Fluid Mech. 155, 343–358.Google Scholar
Paull, R. & Pillow, A. F. 1985b Conically similar viscous flows. Part. 3. J. Fluid Mech. 155, 359–379.Google Scholar
Pillow, A. F. & Paull, R. 1985 Conically similar viscous flows. Part 1. J. Fluid Mech. 155, 327–341.Google Scholar
Robertson, J. M. 1965 Hydrodynamics in Theory and Application. Prentice-Hall.
Saffman, P. G. 1992 Vortex Dynamics, Section 14.4. Cambridge University Press.
Shtern, V. N. & Hussain, F. 1993 Hysteresis in a swirling jet as a model tornado. Phys Fluids A 5, 2183–2195.Google Scholar
Sozou, C. 1992 On solutions relating to conical vortices over a plane wall. J. Fluid. Mech. 244, 633–644.Google Scholar
Sozou, C., Wilkinson, L. C. & Shtern, V. N. 1994 On conical flows in an infinite fluid. J. Fluid Mech. 276, 261–271.Google Scholar
Spall, R. E., Gatski, T. B. & Grosh, C. E. 1987 A criterion for vortex breakdown. Phys. Fluids. 30, 3434–3440.Google Scholar
Squire, H. B. 1951 The round laminar jet. Q. J. Mech. 4, 321–329.Google Scholar
Squire, H. B. 1952 Some viscous fluid flow problems. 1. Jet emerging from a hole in a plane wall. Phil. Mag. 43, 942–945.Google Scholar
Squire, H. B. 1956 Rotating fluids. In Surveys in Mechanics (ed. G. K. Batchelor & R. M. Davies). Cambridge University Press.
Stewartson, K. & Hall, M. G. 1963 The inner viscous solution for the core of a leading-edge vortex. J. Fluid Mech. 15, 306–318.Google Scholar
Vatistas, G. H., Kozel, V. & Mih, W. C. 1991 A simpler model for concentrated vortices. Exps. Fluids 11, 73–76.Google Scholar
Vatistas, G. H., Lin, S. & Kwok. C. K. 1986 Theoretical and experimental studies on vortex chambers flows. AIAA J. 24, 635–642.Google Scholar
Wang, S. & Rusak, Z. 1995 A theory of the axisymmetric vortex breakdown in a pipe. (Preprint.)
Yih, C. Y., Wu, F., Garg, A. K. & Leibovich, S. 1982 Conical vortices: a class of exact solutions of the Navier Stokes equations. Phys. Fluids 25, 2147–2157.Google Scholar