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Perturbation response and pinch-off of vortex rings and dipoles

Published online by Cambridge University Press:  29 June 2012

Clara O’Farrell  and
John O. Dabiri
Clara O’Farrell*
Affiliation:
Control and Dynamical Systems, California Institute of Technology, Pasadena, CA 91125, USA
John O. Dabiri
Affiliation:
Graduate Aeronautical Laboratories and Bioengineering, California Institute of Technology, Pasadena, CA 91125, USA
*
†Email address for correspondence: [email protected]
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Abstract

The nonlinear perturbation response of two families of vortices, the Norbury family of axisymmetric vortex rings and the Pierrehumbert family of two-dimensional vortex pairs, is considered. Members of both families are subjected to prolate shape perturbations similar to those previously introduced to Hill’s spherical vortex, and their response is computed using contour dynamics algorithms. The response of the entire Norbury family to this class of perturbations is considered, in order to bridge the gap between past observations of the behaviour of thin-cored members of the family and that of Hill’s spherical vortex. The behaviour of the Norbury family is contrasted with the response of the analogous two-dimensional family of Pierrehumbert vortex pairs. It is found that the Norbury family exhibits a change in perturbation response as members of the family with progressively thicker cores are considered. Thin-cored vortices are found to undergo quasi-periodic deformations of the core shape, but detrain no circulation into their wake. In contrast, thicker-cored Norbury vortices are found to detrain excess rotational fluid into a trailing vortex tail. This behaviour is found to be in agreement with previous results for Hill’s spherical vortex, as well as with observations of pinch-off of experimentally generated vortex rings at long formation times. In contrast, the detrainment of circulation that is characteristic of pinch-off is not observed for Pierrehumbert vortex pairs of any core size. These observations are in agreement with recent studies that contrast the formation of vortices in two and three dimensions. We hypothesize that transitions in vortex formation, such as those occurring between wake shedding modes and in vortex pinch-off more generally, might be understood and possibly predicted based on the observed perturbation responses of forming vortex rings or dipoles.

JFM classification

Vortex Flows: Contour dynamics Vortex Flows: Vortex dynamics Vortex Flows: Vortex instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 704 , 10 August 2012 , pp. 280 - 300
Copyright
Copyright © Cambridge University Press 2012

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References

1. Afanasyev, Y. D. 2006 Formation of vortex dipoles. Phys. Fluids 18, 037103.Google Scholar
2. Albrecht, T. R. 2011 Steady vortex dipoles with general profile functions. J. Fluid Mech. 670, 85–95.Google Scholar
3. Bellhouse, B. J. 1972 Fluid mechanics of a model mitral valve and left ventricle. Cardiovascular Res. 6, 199–210.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 (ed. Germain, P. & Nayroles, B. ), pp. 8–28. Springer.Google Scholar
5. Boyd, J. P. & Ma, H. 1990 Numerical study of elliptical modons using a spectral method. J. Fluid Mech. 221, 597–611.Google Scholar
6. Couder, Y. & Basdevant, C. 1986 Experimental and numerical study of vortex couple in two-dimensional flows. J. Fluid Mech. 173, 225–251.Google Scholar
7. Crowdy, D. & Surana, A. 2007 Contour dynamics in complex domains. J. Fluid Mech. 593, 235–254.CrossRefGoogle Scholar
8. Dabiri, J. O. & Gharib, M. 2004 Delay of vortex ring pinch-off by an imposed bulk counterflow. Phys. Fluids 16 (L), 28–30.Google 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. Dickinson, M. H. & Götz, K. G. 1996 The wake dynamics and fly forces of the fruit fly Drosophila melanogaster . J. Expl Biol. 199, 2085–2104.CrossRefGoogle ScholarPubMed
11. Domenichini, F., Pedrizzetti, G. & Baccani, B. 2005 Three-dimensional filling flow into a model left ventricle. J. Fluid Mech. 539, 179–198.Google Scholar
12. Dritschel, D. G. 1988a Contour surgery: a topological reconnection scheme for extended integrations using contour dynamics. J. Comput. Phys. 77, 240–266.CrossRefGoogle Scholar
13. Dritschel, D. G. 1988b The repeated filamentation of two-dimensional vorticity interfaces. J. Fluid Mech. 194, 511–547.Google Scholar
14. Dritschel, D. G. 1995 A general theory for two-dimensional vortex interactions. J. Fluid Mech. 293, 269–303.CrossRefGoogle Scholar
15. Duran-Matute, M., Albagnac, J., Kamp, L. P. J. & van Heijst, G. J. F. 2010 Dynamics and structure of decaying shallow dipolar vortices. Phys. Fluids 22, 116606.Google Scholar
16. Gao, L. & Yu, S. C. M. 2010 A model for the pinch-off process of the leading vortex ring in a starting jet. J. Fluid Mech. 656, 205–222.CrossRefGoogle Scholar
17. van Geffen, J. H. G. M. & van Heijst, G. J. F. 1998 Viscous evolution of 2D dipolar vortices. Fluid. Dyn. Res. 22, 191–213.Google Scholar
18. Gharib, M., Rambod, E. & Shariff, K. 1998 A universal time scale for vortex ring formation. J. Fluid Mech. 360, 121–140.Google Scholar
19. van Heijst, G. J. F. & Flór, J. B. 1989 Dipole formation and collisions in a stratified fluid. Nature 340, 212–215.Google Scholar
20. Kaplansky, F. B. & Rudi, Y. A. 2005 A model for the formation of ‘optimal’ vortex rings taking into account viscosity. Phys. Fluids 17, 087101.Google Scholar
21. Kelvin, Lord 1880a On the vibrations of a columnar vortex. Phil. Mag. 10, 155–168.Google Scholar
22. Kelvin, Lord 1880b Vortex statics. Phil. Mag. 10, 97–109.Google Scholar
23. Kern, S. & Koumoutsakos, P. 2006 Simulations of optimized anguilliform swimming. J. Expl Biol. 209, 4841–4857.CrossRefGoogle ScholarPubMed
24. Khvoles, R., Berson, D. & Kizner, Z. 2005 The structure and evolution of elliptical barotropic modons. J. Fluid Mech. 530, 1–30.Google Scholar
25. Kim, D. & Gharib, M. 2011 Flexibility effects on vortex formation of translating plates. J. Fluid Mech. 677, 255–271.Google Scholar
26. Kizner, Z. & Khvoles, R. 2004 Two variations on the theme of Lamb-Chaplygin: supersmooth dipole and rotating multipoles. Regular Chaotic Dyn. 4, 509–518.Google Scholar
27. Krueger, P. S., Dabiri, J. . O. & Gharib, M. 2006 The formation number of vortex rings in uniform background coflow. J. Fluid Mech. 556, 147–166.Google Scholar
28. Krueger, P. S. & Gharib, M. 2003 The significance of vortex ring formation to the impulse and thrust of a starting jet. Phys. Fluids 15, 1271–1281.CrossRefGoogle Scholar
29. Linden, P. F. & Turner, J. S. 2001 The formation of ‘optimal’ vortex rings, and the efficiency of propulsive devices. J. Fluid Mech. 427, 61–72.Google Scholar
30. Makarov, V. G. & Kizner, Z. 2011 Stability and evolution of uniform-vorticity dipoles. J. Fluid Mech. 672, 307–325.Google Scholar
31. Moffatt, H. K. & Moore, D. W. 1978 The response of Hill’s spherical vortex to a small axisymmetric disturbance. J. Fluid Mech. 168, 337–367.Google Scholar
32. Mohseni, K. & Gharib, M. 1998 A model for universal time scale of vortex ring formation. Phys. Fluids 10, 2436–2438.Google Scholar
33. Nguyen Duc, J.-M. & Sommeria, J. 1998 Experimental characterization of two-dimensional vortex couples. J. Fluid Mech. 192, 1175–1192.Google Scholar
34. Nitsche, M. 2001 Self-similar shedding of vortex rings. J. Fluid Mech. 435, 397–407.Google Scholar
35. Norbury, J. 1973 A family of steady vortex rings. J. Fluid Mech. 57, 417–431.Google Scholar
36. Pawlak, G., Cruz, C. M., Bazan, C. M. & Hrdy, P. G. 2007 Experimental characterization of starting jet dynamics. Fluid Dyn. Res. 39, 711–730.Google Scholar
37. Pedrizzetti, G. 2010 Vortex formation out of two-dimensional orifices. J. Fluid Mech. 655, 198–216.Google Scholar
38. Pierrehumbert, R. T. 1980 A family of steady, translating vortex pairs with distributed vorticity. J. Fluid Mech. 99, 129–144.Google Scholar
39. Pozrikidis, C. 1986 The nonlinear instability of Hill’s vortex. J. Fluid Mech. 168, 337–367.Google Scholar
40. Pullin, D. I. 1991 Contour dynamics methods. Annu. Rev. Fluid Mech. 24, 84–115.Google Scholar
41. Reul, H., Talukder, N. & Muller, W. 1981 Fluid mechanics of the natural mitral valve. J. Biomech. 14, 361–372.Google Scholar
42. Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
43. Saffman, P. G. & Szeto, R. 1980 Equilibrium shapes of a pair of equal uniform vortices. Phys. Fluids 23, 2339–2342.Google Scholar
44. Saffman, P. G. & Tanveer, S. 1982 The touching pair of equal and opposite uniform vortices. Phys. Fluids 25, 1929–1930.Google Scholar
45. Shariff, K., Leonard, A. & Ferziger, J. H. 2008 A contour dynamics algorithm for axisymmetric flow. J. Comput. Phys. 227, 9044–9062.Google Scholar
46. Shusser, M. & Gharib, M. 2000 Energy and velocity of a forming vortex ring. Phys. Fluids 12, 618–621.Google Scholar
47. Trieling, R., Santberg, R., van Heijst, G. J. F. & Kizner, Z. 2010 Barotropic elliptical dipoles in a rotating fluid. Theor. Comput. Fluid Dyn. 24, 111–115.Google Scholar
48. Velasco Fuentes, O. U. & van Heijst, G. J. F. 1994 Experimental study of dipolar vortices on a topographic -plane. J. Fluid Mech. 259, 79–106.Google Scholar
49. Wan, Y.-H. 1988 Variational principles for Hill’s spherical vortex and nearly spherical vortices. Trans. Am. Math. Soc. 308, 299–312.Google Scholar
50. Wieting, D. W. & Stripling, T. E. 1984 Dynamics and fluid dynamics of the mitral valve. In Recent Progress in Mitral Valve Disease (ed. Duran, C., Angell, W. W., Johnson, A. D. & Oury, J. H. ), pp. 13–46. Butterworths.Google Scholar
51. Ye, Q. Y. & Chu, C. K. 1995 Unsteady evolutions of vortex rings. Phys. Fluids 7, 795–801.Google Scholar
52. Zabusky, N. J., Hughes, M. H. & Roberts, K. V. 1979 Contour dynamics for the Euler equations in two dimensions. J. Comput. Phys. 30, 96–106.Google Scholar