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Structure and stability of hollow vortex equilibria

Published online by Cambridge University Press:  01 December 2011

Stefan G. Llewellyn Smith and
Darren G. Crowdy
Stefan G. Llewellyn Smith
Affiliation:
Institut de Mécanique des Fluides de Toulouse, UMR CNRS/INPT/UPS 5502, Allée Camille Soula, 31400 Toulouse, FranceDepartment of Mechanical and Aerospace Engineering, Jacobs School of Engineering, UCSD, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA
Darren G. Crowdy*
Affiliation:
Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2AZ, UK
*
†Email address for correspondence: sgls@ucsd.edu
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Abstract

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This paper considers the structure and linear stability of two-dimensional hollow vortex equilibria. Equilibrium solutions for a single hollow vortex in linear and nonlinear straining flows are derived in analytical form using free streamline theory. The linear stability properties of this solution class are then determined numerically and a new type of resonance-induced displacement instability is identified. It is found to be a consequence of the fact that one of the shape distortion modes of a circular hollow vortex has the same frequency as one of the modes corresponding to displacement of the vortex centroid. The instability is observed in the case of an isolated hollow vortex situated in straining flow of order three. We also revisit the hollow vortex row solution due to Baker, Saffman & Sheffield (J. Fluid Mech., vol. 74, 1976, p. 1469), and since it is currently lacking in the literature, we present a full linear stability analysis of this solution using Floquet analysis.

JFM classification

Vortex Flows: Vortex dynamicsVortex Flows: Vortex instability

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 691 , 25 January 2012, pp. 178 - 200
Copyright
Copyright © Cambridge University Press 2011

References

1. Ardalan, K., Meiron, D. I. & Pullin, D. I. 1995 Steady compressible vortex flows: the hollow-core vortex array. J. Fluid Mech. 301, 1–17.CrossRefGoogle Scholar
2. Baker, G. R. 1980 Energetics of a linear array of hollow vortices of finite cross-section. J. Fluid Mech. 99, 97–100.CrossRefGoogle Scholar
3. Baker, G. R., Saffman, P. G. & Sheffield, J. S. 1976 Structure of a linear array of hollow vortices of finite cross-section. J. Fluid Mech. 74, 1469–1476.CrossRefGoogle Scholar
4. Burbea, J. 1981 On patches of uniform vorticity in a plane of irrotational flow. Arch. Rat. Mech. Anal. 77, 349–358.CrossRefGoogle Scholar
5. Crowdy, D. G. 1999 Circulation-induced shape deformations of drops and bubbles: exact two-dimensional models. Phys. Fluids 11, 2836–2845.CrossRefGoogle Scholar
6. Crowdy, D. G. & Green, C. C. 2011 Analytical solutions for von Kármán streets of hollow vortices. Phys. Fluids (in press).CrossRefGoogle Scholar
7. Deconinck, B. & Kutz, J. N. 2006 Computing spectra of linear operators using the Floquet–Fourier–Hill method. J. Comput. Phys. 219, 296–313.CrossRefGoogle Scholar
8. Dhanak, M. R. 1992 Stability of a regular polygon of finite vortices. J. Fluid Mech. 234, 297–316.CrossRefGoogle Scholar
9. Dritschel, D. G. 1985 The stability and energetics of corotating uniform vortices. J. Fluid Mech. 157, 95–134.CrossRefGoogle Scholar
10. Hill, D. J. 1998 Part I. Vortex dynamics in wake models. Part II. Wave generation. PhD thesis, California Institute of Technology.Google Scholar
11. Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.CrossRefGoogle Scholar
12. Kamm, J. R. 1987 Shape and stability of two-dimensional uniform vorticity regions. PhD thesis, California Institute of Technology.Google Scholar
13. Leppington, F. G. 2006 The field due to a pair of line vortices in a compressible fluid. J. Fluid Mech. 559, 45–55.CrossRefGoogle Scholar
14. Luzzatto-Fegiz, P. & Williamson, C. H. K. 2010 Stability of conservative flows and new steady-fluid solutions from bifurcation diagrams exploiting a variational argument. Phys. Rev. Lett. 104, 044504.CrossRefGoogle ScholarPubMed
15. MacKay, R. S. & Saffman, P. G. 1986 Stability of water waves. Proc. R. Soc. Lond. A 406, 115–125.Google Scholar
16. 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
17. Moore, D. W. & Pullin, D. I. 1987 The compressible vortex pair. J. Fluid Mech. 185, 171–204.CrossRefGoogle Scholar
18. Moore, D. W. & Saffman, P. G. 1971 Structure of a line vortex in an imposed strain. In Aircraft Wake Turbulence and its Detection (ed. Olsen, J. A., Goldburg, A & Rogers, M. ), pp. 339–354. Plenum.CrossRefGoogle Scholar
19. Moore, D. W., Saffman, P. G. & Tanveer, S. 1988 The calculation of some Batchelor flows: the Sadovskii vortex and rotational corner flow. Phys. Fluids 31, 978–990.CrossRefGoogle Scholar
20. Pocklington, H. C. 1895 The configuration of a pair of equal and opposite hollow straight vortices of finite cross-section, moving steadily through fluid. Proc. Camb. Phil. Soc. 8, 178–187.Google Scholar
21. Sadovskii, V. S. 1971 Vortex regions in a potential stream with a jump of Bernoulli’s constant at the boundary. Appl. Math. Mech. 35, 729–735.CrossRefGoogle Scholar
22. Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
23. Saffman, P. G. & Schatzman, J. C. 1981 Properties of a vortex street of finite vortices. SIAM J. Sci. Stat. Comput. 2, 285–295.CrossRefGoogle Scholar
24. Saffman, P. G. & Szeto, R. 1981 Structure of a linear array of uniform vortices. Stud. Appl. Math. 65, 223–248.CrossRefGoogle Scholar
25. Saffman, P. G. & Tanveer, S. 1984 Prandtl–Batchelor flow past a flat plate with a forward facing flap. J. Fluid Mech. 143, 351–365.CrossRefGoogle Scholar
26. Sedov, L. I. 1965 Two-Dimensional Problems in Hydrodynamics and Aerodynamics. Wiley.CrossRefGoogle Scholar
27. Telib, H. & Zannetti, L. 2011 Hollow wakes past arbitrarily shaped obstacles. J. Fluid Mech. 669, 214–224.CrossRefGoogle Scholar
28. Thomson, J. J. 1883 A Treatise on the Motion of Vortex Rings. Macmillan.Google Scholar
29. Wegmann, R. & Crowdy, D. 2000 Shapes of two-dimensional bubbles deformed by circulation. Nonlinearity 13, 2131–2141.CrossRefGoogle Scholar