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On the drag of turbulent vortex rings

Published online by Cambridge University Press:  06 August 2012

L. Gan*
Affiliation:
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
J. R. Dawson
Affiliation:
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
T. B. Nickels
Affiliation:
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
*
Email address for correspondence: [email protected]

Abstract

In this paper the pressure field during the early development of turbulent vortex rings at two Reynolds numbers is determined using temporally resolved two-dimensional and stereoscopic particle image velocimetry (PIV). The pressure gradient terms are obtained by solving the incompressible Euler equation so that the drag coefficients of the vortex rings can be evaluated. Maxworthy (J. Fluid Mech., vol. 64, 1974, pp. 227–239) and Glezer & Coles (J. Fluid Mech., vol. 211, 1990, pp. 243–283) each developed models to describe the long-term physics of turbulent vortex rings: the former developed a semi-empirical model which permits loss of impulse via the shedding of vorticity into the wake whereas the latter developed a similarity model based on invariance of the hydrodynamic impulse. Maxworthy’s model implies that a significant correction to the similarity solution is required to account for the drag on the vortex ring bubble. We show that during the early development of the turbulent vortex rings the drag is very small and the similarity scaling can basically be retained.

Type
Papers
Copyright
Copyright © Cambridge University Press 2012

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References

1. Cater, J. E., Soria, J. & Lim, T. T. 2004 The interaction of the piston vortex with a piston-generated vortex ring. J. Fluid Mech. 499, 327343.CrossRefGoogle Scholar
2. Christensen, K. T. & Adrian, R. J. 2002 Measurement of instantaneous Eulerian acceleration fields by particle image accelerometry: method and accuracy. Exp. Fluids 33 (6), 759769.CrossRefGoogle Scholar
3. Dabiri, J. O. 2009 Optimal vortex formation as a unifying principle in biological propulsion. Annu. Rev. Fluid Mech. 41, 1733.CrossRefGoogle Scholar
4. Didden, N. 1979 On the formation of vortex rings: rolling-up and production of circulation. Z. Angew. Math. Phys. 30, 101116.CrossRefGoogle Scholar
5. Einstein, A., Lorentz, H., Minkowski, H. & Weyl, H. 1952 The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity. Dover.Google Scholar
6. Ellinton, C. P. 1999 The novel aerodynamics of insect flight: applications to micro-air vehicles. J. Expl Biol. 202, 34393448.CrossRefGoogle Scholar
7. Gan, L. & Nickels, T. B. 2010 An experimental study of turbulent vortex rings during their early development. J. Fluid Mech. 649, 467496.CrossRefGoogle Scholar
8. Gan, L., Nickels, T. B. & Dawson, J. R. 2011 An experimental study of a turbulent vortex ring: a three-dimensional representation. Exp. Fluids 51 (6), 14931507.CrossRefGoogle Scholar
9. Glezer, A. & Coles, D. 1990 An experimental study of turbulent vortex ring. J. Fluid Mech. 211, 243283.CrossRefGoogle Scholar
10. Johnson, G. M. 1970 Researches on the propagation and decay of vortex rings. Rep. ARL 70-0093. Aerospace Research Labs., Wright-Patterson AFB.Google Scholar
11. Johnson, G. M. 1971 An empirical model of turbulent vortex rings. AIAA J. 9, 763764.CrossRefGoogle Scholar
12. Lim, T. T. & Nickels, T. B. 1995 Vortex rings. In Fluid Vortices (ed. Green, S. I. ). Kluwer.Google Scholar
13. Linden, P. F. & Turner, J. S. 2004 Optimal vortex rings and aquatic propulsion mechanisms. Proc. R. Soc. Lond. B 271, 647653.CrossRefGoogle ScholarPubMed
14. Liu, X. & Katz, J. 2006 Instantaneous pressure and material acceleration measurements using a four-exposure PIV system. Exp. Fluids 41 (2), 227240.CrossRefGoogle Scholar
15. Maxworthy, T. 1972 The structure and stability of vortex rings. J. Fluid Mech. 51, 1532.CrossRefGoogle Scholar
16. Maxworthy, T. 1974 Turbulent vortex rings. J. Fluid Mech. 64, 227239.CrossRefGoogle Scholar
17. Maxworthy, T. 1977 Some experimental studies of vortex rings. J. Fluid Mech. 81, 465495.CrossRefGoogle Scholar
18. Paxson, D. E., Wernet, M. P. & John, W. T. 2007 Experimental investigation of unsteady thrust augmentation using a speaker-driven jet. AIAA J. 45, 607614.CrossRefGoogle Scholar
19. Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
20. Shariff, K. & Leonard, A. 1992 Vortex rings. Annu. Rev. Fluid Mech. 24, 235279.CrossRefGoogle Scholar