Hostname: page-component-7bb8b95d7b-qxsvm Total loading time: 0 Render date: 2024-09-13T13:08:01.388Z Has data issue: false hasContentIssue false
  • English
  • Français

Formation of an orifice-generated vortex ring

Published online by Cambridge University Press:  26 February 2021

Raphaël Limbourg Open the ORCID record for Raphaël Limbourg [Opens in a new window]  and
Jovan Nedić Open the ORCID record for Jovan Nedić [Opens in a new window]
Raphaël Limbourg
Affiliation:
Department of Mechanical Engineering, McGill University, Montréal, QCH3A 0C3, Canada
Jovan Nedić*
Affiliation:
Department of Mechanical Engineering, McGill University, Montréal, QCH3A 0C3, Canada
*
†Email address for correspondence: jovan.nedic@mcgill.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

The formation of orifice-generated vortex rings, at a Reynolds number of $5300$ and for a tube-to-orifice diameter ratio of $2.0$, is experimentally investigated for stroke-to-diameter ratios of $0.5(0.5)5.0$. A significant increase is observed in the production of the total invariants of the motion, namely the circulation $\varGamma$, the hydrodynamic impulse $I$ and the kinetic energy $E$, compared with the equivalent nozzle-generated vortex rings. The formation number, as defined by Gharib et al. (J. Fluid Mech., vol. 360, 1998, pp. 121–140), is found to be approximately $2.0$. By measuring the kinematics and the invariants of the ring for increasing stroke ratios, a limiting process in the ring formation is observed, which allows us to define the critical parameters and time scales in the vortex formation process. In particular, it was shown that the ring circulation, impulse, and energy do not reach their asymptotic state at the same non-dimensional time and stroke ratio, hence these two terms cannot be used interchangeably. The stroke ratio required to produce a ring with maximum energy is defined as the ‘optimal stroke ratio’, which is found to be around $4$. The non-dimensional time at which the ring reaches this state, termed the ‘optimal formation time’, is found to be approximately 6–7. The non-dimensional vortex ring numbers $\alpha =E/\rho ^{1/2}\varGamma ^{3/2}I^{1/2}$, $\beta =\varGamma /\rho ^{-1/3}I^{1/3}U^{2/3}$ and , are measured to be $0.33$, $1.8$ and $1.9$, respectively, consistent with previous experimental, numerical and analytical work, suggesting these numbers to be universal for all isolated vortex rings.

JFM classification

Vortex Flows: Vortex dynamics Wakes/Jets: Jets
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 913 , 25 April 2021 , A29
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

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. P. Germain & B. Nayroles), pp. 8–29. Springer.CrossRefGoogle Scholar
Dabiri, J.O. 2009 Optimal vortex formation as a unifying principle in biological propulsion. Annu. Rev. Fluid Mech. 41 (1), 17–33.CrossRefGoogle Scholar
Dabiri, J.O. & Gharib, M. 2004 Delay of vortex ring Pinchoff by an imposed bulk counterflow. Phys. Fluids 16 (4), L28–L30.CrossRefGoogle Scholar
Dabiri, J.O. & Gharib, M. 2005 Starting flow through nozzles with temporally variable exit diameter. J. Fluid Mech. 538, 111–136.CrossRefGoogle Scholar
Danaila, I. & Hélie, J. 2008 Numerical simulation of the postformation evolution of a laminar vortex ring. Phys. Fluids 20 (7), 073602.CrossRefGoogle Scholar
Didden, N. 1979 On the formation of vortex rings: rolling-up and production of circulation. Z. Angew. Mech. Phys. 30, 101–116.CrossRefGoogle Scholar
Fraenkel, L.E. 1972 Examples of steady vortex rings of small cross-section in an ideal fluid. J. Fluid Mech. 51 (1), 119–135.CrossRefGoogle Scholar
Friedman, A. & Turkington, B. 1981 Vortex rings: existence and asymptotic estimates. Trans. Am. Math. Soc. 268 (1), 1–37.CrossRefGoogle Scholar
Gao, L., Guo, H.-F. & Yu, S.C.M. 2020 A general definition of formation time for starting jets and forced plumes at low Richardson number. J. Fluid Mech. 886, A6.CrossRefGoogle Scholar
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
Gao, L. & Yu, S.C.M. 2012 Development of the trailing shear layer in a starting jet during pinch-off. J. Fluid Mech. 700, 382–405.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 (9), 093604.CrossRefGoogle Scholar
Gharib, M., Rambod, E., Kheradvar, A., Sahn, D.J. & Dabiri, J.O. 2006 Optimal vortex formation as an index of cardiac health. Proc. Natl Acad. Sci. USA 103 (16), 6305–6308.CrossRefGoogle ScholarPubMed
Gharib, M., Rambod, E. & Shariff, K. 1998 A universal time scale for vortex ring formation. J. Fluid Mech. 360, 121–140.CrossRefGoogle Scholar
Glezer, A. & Amitay, M. 2002 Synthetic jets. Annu. Rev. Fluid Mech. 34 (1), 503–529.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
James, S. & Madnia, C.K. 1996 Direct numerical simulation of a laminar vortex ring. Phys. Fluids 8 (9), 2400–2414.CrossRefGoogle Scholar
Kaplanski, F. & Rudi, Y.A. 2005 A model for the formation of ‘optimal’ vortex rings taking into account viscosity. Phys. Fluids 17 (8), 087101.CrossRefGoogle Scholar
Krieg, M. & Mohseni, K. 2013 a Modelling circulation, impulse and kinetic energy of starting jets with non-zero radial velocity. J. Fluid Mech. 719, 488–526.CrossRefGoogle Scholar
Krieg, M. & Mohseni, K. 2013 b On approximating the translational velocity of vortex rings. J. Fluids Engng 135 (12), 124501.CrossRefGoogle Scholar
Krueger, P.S., Dabiri, J.O. & Gharib, M. 2006 The formation number of vortex rings formed in uniform background co-flow. J. Fluid Mech. 556, 147–166.CrossRefGoogle Scholar
Krueger, P.S. & Gharib, M. 2003 The significance of vortex ring formation to the impulse and thrust of a starting jet. Phys. Fluids 15 (5), 1271–1281.CrossRefGoogle Scholar
Krueger, P.S., Moslemi, A.A., Nichols, J.T., Bartol, I.K. & Stewart, W.J. 2008 Vortex rings in bio-inspired and biological jet propulsion. Adv. Sci. Technol. 58, 237–246.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Lawson, J.M. & Dawson, J.R. 2013 The formation of turbulent vortex rings by synthetic jets. Phys. Fluids 25 (10), 105113.CrossRefGoogle Scholar
Linden, P.F. & Turner, J.S. 2001 The formation of ‘optimal’ vortex rings, and the efficiency of propulsion devices. J. Fluid Mech. 427, 61–72.CrossRefGoogle Scholar
Mohseni, K. & Gharib, M. 1998 A model for universal time scale of vortex ring formation. Phys. Fluids 10 (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
Nitsche, M. & Krasny, R. 1994 A numerical study of vortex ring formation at the edge of a circular tube. J. Fluid Mech. 276, 139–161.CrossRefGoogle Scholar
Norbury, J. 1973 A family of steady vortex rings. J. Fluid Mech. 57 (3), 417–431.CrossRefGoogle Scholar
Olver, P.J. 1982 A nonlinear Hamiltonian structure for the Euler equations. J. Math. Anal. Appl. 89 (1), 233–250.CrossRefGoogle Scholar
Pullin, D.I. 1978 The large-scale structure of unsteady self-similar rolled-up vortex sheets. J. Fluid Mech. 88 (3), 401–430.CrossRefGoogle Scholar
Roberts, P.H. 1972 A Hamiltonian theory for weakly interacting vortices. Mathematika 19 (2), 169–179.CrossRefGoogle Scholar
Rosenfeld, M., Katija, K. & Dabiri, J.O. 2009 Circulation generation and vortex ring formation by conic nozzles. J. Fluids Engng 131 (9), 091204.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.G. 1978 The number of waves on unstable vortex rings. J. Fluid Mech. 84 (4), 625–639.CrossRefGoogle Scholar
Saffman, P.G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Shariff, K. & Leonard, A. 1992 Vortex rings. Annu. Rev. Fluid Mech. 24 (1), 235–279.CrossRefGoogle Scholar
Shusser, M., Gharib, M. & Mohseni, K. 1999 A new model for inviscid vortex ring formation. In 30th Fluid Dynamics Conference. AIAA.CrossRefGoogle Scholar
Yu, S.C.M., Law, A.W.K. & Ai, J.J. 2007 Vortex formation process in gravity-driven starting jets. Exp. Fluids 42 (5), 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 (3), 589–596.CrossRefGoogle Scholar