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A model for the pinch-off process of the leading vortex ring in a starting jet

Published online by Cambridge University Press:  21 May 2010

L. GAO  and
S. C. M. YU
L. GAO*
Affiliation:
Thermal and Fluids Engineering Division, School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore639798
S. C. M. YU
Affiliation:
Thermal and Fluids Engineering Division, School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore639798
*
†Email address for correspondence: GA0001EI@ntu.edu.sg
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Abstract

Modifications have been made to an analytical model proposed by Shusser & Gharib (J. Fluid Mech., vol. 416, 2000b, pp. 173–185) for the vortex ring formation and pinch-off process in a starting jet. Compared with previous models, the present investigation distinguishes the leading vortex ring from its trailing jet so as to consider the details of the pinch-off process in terms of the properties of the leading vortex ring, which are determined by considering the flux of kinetic energy, impulse and circulation from the trailing jet into the leading vortex ring by convective transportation. A two-stage process has been identified before the complete separation of the leading vortex ring from its trailing jet. The first stage involves the growth of the leading vortex ring by absorbing all the ejected fluid from the nozzle until certain optimum size is achieved. The second stage is characterized by the appreciable translational velocity of the leading vortex ring followed by a trailing jet. The leading vortex ring is approximated as a Norbury vortex ring with growing characteristic core radius ϵ such that dimensionless energy α, as well as its translational velocity and penetration depth, can be estimated. By applying the Kelvin–Benjamin variational principle, the pinch-off process is signified by two time scales, i.e. the formation number, which indicates the onset of the pinch-off process, and the separation time, which corresponds to the time when the leading vortex ring becomes physically separated from the trailing jet and is therefore referred to as the end of the pinch-off process. The effect of nozzle geometry, i.e. a straight nozzle or a converging nozzle, has also been taken into account by using different descriptions of the growth of the trailing jet. The prediction of the formation number and the characteristics of the vortex ring are found to be in good agreement with previous experimental results on starting jets with straight nozzles and converging nozzles, respectively.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 656 , 10 August 2010 , pp. 205 - 222
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Allen, J. J. & Naitoh, T. 2005 Experimental study of the production of vortex rings using a variable diameter orifice. Phys. Fluids 17, 061701.CrossRefGoogle Scholar
Didden, N. 1979 On the formation of vortex rings: rolling-up and production of circulation. J. Appl. Math. Phys. 30, 101–116.Google Scholar
Fraenkel, L. E. 1972 Examples of steady vortex rings of small cross-section in an ideal fluid. J. Fluid Mech. 51, 119–135.CrossRefGoogle Scholar
Gao, L., Yu, S. C. M., Ai, J. J. & Law, A. W. K. 2008 Circulation and energy of the leading vortex ring in a gravity-driven starting jet. Phys. Fluids 20, 093604.CrossRefGoogle Scholar
Gharib, M., Rambod, E. & Shariff, K. 1998 A universal time scale for vortex ring formation. J. Fluid Mech. 360, 121–140.CrossRefGoogle Scholar
Heeg, R. S. & Riley, N. 1997 Simulations of the formation of an axisymmetric vortex ring. J. Fluid Mech. 339, 199–211.CrossRefGoogle Scholar
Hettel, M., Wetzel, F., Habisreuther, P. & Bockhorn, H. 2007 Numerical verification of the similarity laws for the formation of laminar vortex rings. J. Fluid Mech. 590, 35–60.CrossRefGoogle Scholar
Krueger, P. 2005 An over-pressure correction to the slug model for vortex ring circulation. J. Fluid Mech. 545, 427–443.CrossRefGoogle Scholar
Liess, C. 1978 Experimentelle Untersuchung des Lebenslaufes von Ringwirbeln. Tech. Rep. Max-Planck-Institut für Strömungsforschung, Göttingen.Google Scholar
Lim, T. T. & Nickels, T. B. 1995 Vortex rings. In Vortices in Fluid Flows (ed. Green, S. I.). Kluwer.Google Scholar
Maxworthy, T. 1977 Some experimental studies of vortex rings. J. Fluid Mech. 81, 465–495.CrossRefGoogle Scholar
Mohseni, K. & Gharib, M. 1998 A model for universal time of vortex ring formation. Phys. Fluids 10, 2436–2438.CrossRefGoogle Scholar
Mohseni, K., Ran, H. & Colonius, T. 2001 Numerical experiments on vortex ring formation. J. Fluid Mech. 430, 267–282.CrossRefGoogle Scholar
Norbury, J. 1973 A family of steady vortex rings. J. Fluid Mech. 57, 417–443CrossRefGoogle Scholar
Olcay, A. & Krueger, P. 2008 Measurement of ambient fluid entrainment during laminar vortex ring formation. Exp. Fluids 44, 235–247.CrossRefGoogle Scholar
Pawlak, G., Cruz, C. M., Bazan, C. M. & Hrdy, P. G. 2007 Experimental characterization of starting jet dynamics. Fluid Dyn. Res. 39, 711–730.CrossRefGoogle Scholar
Rosenfeld, M., Rambod, E. & Gharib, M. 1998 Circulation and formation number of laminar vortex rings. J. Fluid Mech. 376, 297–318.CrossRefGoogle Scholar
Saffman, P. 1978 The number of waves on unstable vortex rings. J. Fluid Mech. 84, 625–639.CrossRefGoogle Scholar
Sau, R. & Mahesh, K. 2007 Passive scalar mixing in vortex rings. J. Fluid Mech. 582, 449–461.CrossRefGoogle Scholar
Schram, C. & Riethmuller, M. L. 2001 Vortex ring evolution in an impulsively started jet using digital particle image velocimetry and continuous wavelet analysis. Meas. Sci. Technol. 12, 1413–1421.Google Scholar
Shariff, K. & Leonard, A. 1992 Vortex rings. Ann. Rev. Fluid Mech. 24, 235–279.Google Scholar
Shusser, M. & Gharib, M. 2000 a Energy and velocity of a forming vortex ring. Phys. Fluids 12, 618–621.CrossRefGoogle Scholar
Shusser, M. & Gharib, M. 2000 b A model for vortex ring formation in a starting buoyant plume. J. Fluid Mech. 416, 173–185.CrossRefGoogle Scholar
Shusser, M., Rosenfeld, M., Dabiri, J. O. & Gharib, M. 2006 Effect of time-dependent piston velocity program on vortex ring formation in a piston/cylinder arrangement. Phys. Fluids 18, 033601.CrossRefGoogle Scholar
Turner, J. S. 1962 The ‘starting plume’ in neutral surroundings. J. Fluid Mech. 13, 356–368.CrossRefGoogle Scholar
Weigand, A. & Gharib, M. 1997 On the evolution of laminar vortex rings. Exp. Fluids 22, 447–457.CrossRefGoogle Scholar
Yu, S. C. M., Ai, J. J. & Law, A. W. K. 2007 Vortex formation process in gravity-driven starting jets. Exp. Fluids 42, 783–797.CrossRefGoogle Scholar
Zhao, W., Frankel, S. H. & Mongeau, L. G. 2000 Effects of trailing jet instability on vortex ring formation. Phys. Fluids 12, 589–621.CrossRefGoogle Scholar