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Determining the stability of steady two-dimensional flows through imperfect velocity-impulse diagrams

Published online by Cambridge University Press:  13 July 2012

P. Luzzatto-Fegiz  and
C. H. K. Williamson
P. Luzzatto-Fegiz
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA
C. H. K. Williamson
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA
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Abstract

In 1875, Lord Kelvin stated an energy-based argument for equilibrium and stability in conservative flows. The possibility of building an implementation of Kelvin’s argument, based on the construction of a simple bifurcation diagram, has been the subject of debate in the past. In this paper, we build on work from dynamical systems theory, and show that an essential requirement for constructing a meaningful bifurcation diagram is that families of solutions must be accessed through isovortical (i.e. vorticity-preserving), incompressible rearrangements. We show that, when this is the case, turning points in fluid impulse are linked to changes in the number of the positive-energy modes associated with the equilibria (and therefore in the number of modes likely to be linearly unstable). In addition, the shape of a velocity-impulse diagram, for a family of solutions, determines whether a positive-energy mode is lost or gained at the turning point. Further to this, we detect bifurcations to new solution families by calculating steady flows that have been made ‘imperfect’ through the introduction of asymmetries in the vorticity field. The resulting stability approach, which employs ‘imperfect velocity-impulse’ (IVI) diagrams, can be used to determine the number of positive-energy (likely unstable) modes for each equilibrium flow belonging to a family of steady states. As an illustration of our approach, we construct IVI diagrams for several two-dimensional flows, including elliptical vortices, opposite-signed vortex pairs (of both rotating and translating type), single and double vortex rows, as well as gravity waves. By also considering an example involving the Chaplygin–Lamb dipole, we illustrate how the stability of a specific flow may be determined, by embedding it within a properly constructed solution family. The stability data from our IVI diagrams agree precisely with results in the literature. To the best of our knowledge, for a few of the flows considered here, our work yields the first available stability boundaries. Further to this, for several of the flows that we examine, the IVI diagram methodology leads us to the discovery of new families of steady flows, which exhibit lower symmetry.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Mathematical Foundations: Variational methods Vortex Flows: Vortex instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 706 , 10 September 2012 , pp. 323 - 350
Copyright
Copyright © Cambridge University Press 2012

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References

1. Arnol’d, V. I. 1965 Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid. Dokl. Akad. Nauk SSSR 162, 773–777.Google Scholar
2. Arnol’d, V. I. 1966 Sur un principe variationnel pour les écoulements stationnaires des liquides parfaits et ses applications aux problèmes de stabilité non-linéaires. J. Méc. 5, 29–43.Google Scholar
3. Arnol’d, V. I. & Avez, A. 1968 Ergodic Problems of Classical Mechanics. Benjamin.Google Scholar
4. Benjamin, T. B. 1976 The alliance of practical and analytical insights into the nonlinear problems of fluid mechanics. In Applications of Methods of Functional Analysis to Problems in Mechanics, pp. 8–29. Springer.CrossRefGoogle Scholar
5. Casademunt, J. & Jasnow, D. 1991 Defect dynamics in viscous fingering. Phys. Rev. Lett. 67, 3677–3680.CrossRefGoogle ScholarPubMed
6. Cerretelli, C. & Williamson, C. H. K. 2003 A new family of uniform vortices related to vortex configurations before merging. J. Fluid Mech. 493, 219–229.CrossRefGoogle Scholar
7. Chaplygin, S. A. 1903 One case of vortex motion in fluid. Trans. Phys. Sect. Imperial Moscow Soc. Friends of Natural Sciences 11, 11–14.Google Scholar
8. Chen, B. & Saffman, P. G. 1980 Numerical evidence for the existence of new types of gravity waves of permanent form on deep water. Stud. Appl. Maths 62, 1–21.CrossRefGoogle Scholar
9. Deem, G. S. & Zabusky, N. J. 1978 Vortex waves: stationary ‘V states’, interactions, recurrence, and breaking. Phys. Rev. Lett. 40, 859–862.CrossRefGoogle Scholar
10. Delbende, I. & Rossi, M. 2009 The dynamics of a viscous vortex dipole. Phys. Fluids 21, 073605.CrossRefGoogle Scholar
11. Doedel, E. J. 1981 AUTO: a Program for the Automatic Bifurcation Analysis of Autonomous Systems. University of Manitoba.Google Scholar
12. Dritschel, D. G. 1985 The stability and energetics of corotating uniform vortices. J. Fluid Mech. 157, 95–134.CrossRefGoogle Scholar
13. Dritschel, D. G. 1995 A general theory for two-dimensional vortex interactions. J. Fluid Mech. 293, 269–303.CrossRefGoogle Scholar
14. Dritschel, D. G., Scott, R. K. & Reinaud, J. N 2005 The stability of quasi-geostrophic ellipsoidal vortices. J. Fluid Mech. 536, 401–421.CrossRefGoogle Scholar
15. Ehrenstein, U. & Le Dizès, S. 2005 Relationship between corotating vortex-pair equilibria and a single vortex in an external deformation field. Phys. Fluids 17, 074103.CrossRefGoogle Scholar
16. Flierl, G. R. & Morrison, P. J. 2011 Hamiltonian-Dirac simulated annealing: application to the calculation of vortex states. Physica D 240, 212–232.CrossRefGoogle Scholar
17. Fukumoto, Y. & Moffatt, H. K. 2008 Kinematic variational principle for motion of vortex rings. Physica D 237, 2210–2217.CrossRefGoogle Scholar
18. Holm, D. D., Marsden, J. E., Ratiu, T. & Weinstein, A. 1985 Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1–116.CrossRefGoogle Scholar
19. Jiménez, J. 1987 On the linear stability of the inviscid Kármán vortex street. J. Fluid Mech. 178, 177–194.CrossRefGoogle Scholar
20. Kamm, J. R. 1987 Shape and stability of two-dimensional uniform vorticity regions. PhD thesis, California Institute of Technology, Pasadena.Google Scholar
21. Katz, J. 1978 On the number of unstable modes of an equilibrium. Mon. Not. R. Astron. Soc. 183, 765–769.CrossRefGoogle Scholar
22. Kirchhoff, G. 1876 Vorlesungen über Mathematische Physik: Mechanik. Teubner.Google Scholar
23. Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
24. Lanczos, C. 1986 The Variational Principles of Mechanics. Dover.Google Scholar
25. Legras, B. & Dritschel, D. G. 1993 Comparison of the contour surgery and pseudo-spectral methods. J. Comput. Phys. 104, 287–302.CrossRefGoogle Scholar
26. Longuet-Higgins, M. S. 1978 The instabilities of gravity waves of finite amplitude in deep water. 2. Subharmonics. Proc. R. Soc. Lond. A 360, 489–505.Google Scholar
27. Longuet-Higgins, M. S. 1984 New integral relations for gravity waves of finite amplitude. J. Fluid Mech. 149, 205–215.CrossRefGoogle Scholar
28. Longuet-Higgins, M. S. 1986 Bifurcation and instability in gravity waves. Proc. R. Soc. Lond. A. 403, 167–187.Google Scholar
29. Longuet-Higgins, M. S. & Cokelet, E. D. 1976 The deformation of steep surface waves on water. Part 1. A numerical method of computation. Proc. R. Soc. Lond. A. 350, 1–26.Google Scholar
30. Longuet-Higgins, M. S. & Fox, M. J. H. 1978 Theory of the almost-highest wave. Part 2. Matching and analytic extension. J. Fluid Mech. 85, 769–786.CrossRefGoogle Scholar
31. Love, A. E. H. 1893 On the stability of certain vortex motions. Proc. Lond. Math. Soc. 25, 18–42.CrossRefGoogle Scholar
32. Lowry, B. J. & Steen, P. H. 1995 Capillary surfaces: stability from families of equilibria with application to the liquid bridge. Proc. R. Soc. Lond. A 449, 411–439.Google Scholar
33. Luzzatto-Fegiz, P. & Williamson, C. H. K. 2010 Stability of elliptical vortices from imperfect-velocity-impulse diagrams. Theor. Comput. Fluid Dyn. 24, 181–188.CrossRefGoogle Scholar
34. Luzzatto-Fegiz, P. & Williamson, C. H. K. 2011a An accurate and efficient method for computing uniform vortices. J. Comput. Phys. 230, 6495–6511.CrossRefGoogle Scholar
35. Luzzatto-Fegiz, P. & Williamson, C. H. K. 2011b Resonant instability in two-dimensional vortex arrays. Proc. R. Soc. A 467, 1164–1185.CrossRefGoogle Scholar
36. Luzzatto-Fegiz, P. & Williamson, C. H. K. 2012 Structure and stability of the finite-area Kármán street. Phys. Fluids 24, 066602.CrossRefGoogle Scholar
37. Lynden-Bell, D. & Katz, J. 1981 Isocirculational flows and their Lagrangian and energy principles. Proc. R. Soc. Lond. A 378, 179–205.Google Scholar
38. Maddocks, J. H. 1987 Stability and folds. Arch. Rat. Mech. Anal. 99, 301–328.CrossRefGoogle Scholar
39. Makarov, V. G. & Kizner, Z. 2011 Stability and evolution of uniform-vorticity dipoles. J. Fluid Mech. 672, 307–325.CrossRefGoogle Scholar
40. Mallier, R. & Maslowe, S. A. 1993 A row of counter-rotating vortices. Phys. Fluids A 5, 1074–1075.CrossRefGoogle Scholar
41. Meiron, D. I., Saffman, P. G. & Schatzman, J. C. 1984 The linear two-dimensional stability of inviscid vortex streets of finite-cored vortices. J. Fluid Mech. 147, 187–212.CrossRefGoogle Scholar
42. Meleshko, V. V. & van Heijst, G. J. F. 1994 On Chaplygin’s investigations of two-dimensional vortex structures in an inviscid fluid. J. Fluid Mech. 272, 157–182.CrossRefGoogle Scholar
43. Pierrehumbert, R. T. 1980 A family of steady, translating vortex pairs with distributed vorticity. J. Fluid Mech. 99, 129–144.CrossRefGoogle Scholar
44. Pierrehumbert, R. T. & Widnall, S. E. 1981 The structure of organized vortices in a free shear layer. J. Fluid Mech. 102, 301–313.CrossRefGoogle Scholar
45. Poincaré, H. 1885 Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation. Acta Math. 7, 259–380.CrossRefGoogle Scholar
46. Poston, T. & Stewart, I. 1978 Catastrophe Theory. Dover.Google Scholar
47. Saffman, P. G. 1985 The superharmonic instability of finite-amplitude water waves. J. Fluid Mech. 159, 169–174.CrossRefGoogle Scholar
48. Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
49. Saffman, P. G. & Schatzman, J. C. 1981 Properties of vortex street of finite vortices. SIAM J. Sci. Stat. Comput. 2, 285–295.CrossRefGoogle Scholar
50. Saffman, P. G. & Schatzman, J. C. 1982 Stability of a vortex street of finite vortices. J. Fluid Mech. 117, 171–186.CrossRefGoogle Scholar
51. Saffman, P. G. & Szeto, R. 1980 Equilibrium shapes of a pair of equal uniform vortices. Phys. Fluids 23, 2339–2342.CrossRefGoogle Scholar
52. Saffman, P. G. & Szeto, R. 1981 Structure of a linear array of uniform vortices. Stud. Appl. Maths 65, 223–248.CrossRefGoogle Scholar
53. Saffman, P. G. & Tanveer, S. 1982 The touching pair of equal and opposite uniform vortices. Phys. Fluids 25, 1929–1930.CrossRefGoogle Scholar
54. Stokes, G. G. 1879 Letter to William Thomson. In The Correspondence between Sir George Gabriel Stokes and Sir William Thomson, Baron Kelvin of Largs (ed. D. B. Wilson), vol. 2, pp. 470–471. Cambridge University Press (1990).Google Scholar
55. Stuart, J. T. 1967 On finite amplitude oscillations in laminar mixing layers. J . Fluid Mech. 29, 417–440.CrossRefGoogle Scholar
56. Thomson, W. 1876 Vortex statics. Proc. R. Soc. Edin. 9, 59–73, also Phil. Mag. (1880) 10, 97–109.CrossRefGoogle Scholar
57. Thomson, W. 1880a On maximum and minimum energy in vortex motion. Nature 22, 618–620.Google Scholar
58. Thomson, W. 1880b Vibrations of a columnar vortex. Proc. R. Soc. Edin. 10, 443–450, also Phil. Mag. (1880) 10, 155–168.CrossRefGoogle Scholar
59. Thompson, J. M. T. 1975 Experiments in catastrophe. Nature 254, 392–395.CrossRefGoogle Scholar
60. Thompson, J. M. T. 1979 Stability predictions through a succession of folds. Phil. Trans. R. Soc. Lond. A 292, 1–23.Google Scholar
61. Waite, M. L. & Smolarkiewicz, P. K. 2008 Instability and breakdown of a vertical vortex pair in a strongly stratified fluid. J. Fluid Mech. 606, 239–273.CrossRefGoogle Scholar
62. Zakharov, V. E. 1968 Stability of periodic waves of finite amplitudes on the surface of a deep fluid. Zh. Prikl. Mekh. Tekh. Fiz. 9, 86–94.Google Scholar