Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-25T01:49:29.409Z Has data issue: false hasContentIssue false
  • English
  • Français

Modelling of confined vortex rings

Published online by Cambridge University Press:  05 June 2015

Ionut Danaila ,
Felix Kaplanski  and
Sergei Sazhin
Ionut Danaila
Affiliation:
Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, F-76801, Saint-Étienne-du-Rouvray, France
Felix Kaplanski
Affiliation:
Laboratory of Multiphase Media Physics, Tallinn University of Technology, Akad. tee 15A, Tallinn 12618, Estonia
Sergei Sazhin*
Affiliation:
Sir Harry Ricardo Laboratories, School of Computing, Engineering and Mathematics, University of Brighton, Brighton BN2 4GJ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is focused on the investigation of vortex rings evolving in a tube. A new theoretical model for a confined axisymmetric vortex ring is developed. The predictions of this model are shown to be in agreement with available experimental data and numerical simulations. The model combines the viscous vortex ring model, developed by Kaplanski & Rudi (Phys. Fluids, vol. 17, 2005, 087101), with Brasseur’s (PhD thesis, Stanford University) approach to deriving a wall-induced streamfunction correction. Using the power-law assumption for the time variation of the viscous length of the vortex ring, the time variations of the main integral characteristics, circulation, kinetic energy and translational velocity are obtained. Direct numerical simulation (DNS) is used to test the range of applicability of the model and to investigate new physical features of confined vortex rings recently reported in the experimental study by Stewart et al. (Exp. Fluids, vol. 53, 2012, pp. 163–171). The model is shown to lead to a very good approximation of the spatial distribution of the Stokes streamfunction, obtained by DNS. The vortex signature and the time evolution of the energy of the vortex are also accurately predicted by the model. A procedure for fitting the model with realistic vortex rings, obtained by DNS, is suggested. This opens the way to using the model for practical engineering applications.

JFM classification

Mathematical Foundations: General fluid mechanics Vortex Flows: Vortex dynamics Vortex Flows: Vortex Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 774 , 10 July 2015 , pp. 267 - 297
Check for updates
Copyright
© 2015 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

Akhmetov, D. G. 2009 Vortex Rings. Springer.Google Scholar
Amick, C. J. & Fraenkel, L. E. 1986 The uniqueness of Hill’s spherical vortex. Arch. Rat. Mech. Anal. 92, 91–119.Google Scholar
Amick, C. J. & Fraenkel, L. E. 1988 The uniqueness of a family of vortex rings. Arch. Rat. Mech. Anal. 3, 207–241.Google Scholar
Batchelor, G. K. 1988 An Introduction to Fluid Dynamics, 7th edn. Cambridge University Press.Google Scholar
Begg, S., Kaplanski, F., Sazhin, S., Hindle, M. & Heikal, M. 2009 Vortex ring-like structures in gasoline fuel sprays under cold-start conditions. Intl J. Engine Res. 10 (4), 195–214.Google Scholar
Brasseur, J. G.1979 Kinematics and dynamics of vortex rings in a tube. PhD thesis, Report JIAA, TR-26, Joint Institute for Aeronautics and Acoustics, Department of Aeronautics and Astronautics, Stanford University.Google Scholar
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, 327–343.Google 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 Fluid entrainment by isolated vortex rings. J. Fluid Mech. 511, 311–331.CrossRefGoogle Scholar
Dabiri, J. O. & Gharib, M. 2005 Starting flow through nozzles with temporally variable exit diameter. J. Fluid Mech. 538, 111–136.Google Scholar
Danaila, I. & Helie, J. 2008 Numerical simulation of the postformation evolution of a laminar vortex ring. Phys. Fluids 20, 073602.CrossRefGoogle Scholar
Danaila, I., Vadean, C. & Danaila, S. 2009 Specified discharge velocity models for the numerical simulation of laminar vortex rings. Theor. Comput. Fluid Dyn. 23, 317–332.Google Scholar
Esteban, M. J. 1983 Nonlinear elliptic problems in strip-like domains: symmetry of positive vortex rings. Nonlinear Anal. Theory Meth. Applics. 7, 365–379.Google Scholar
Fraenkel, L. E. & Berger, M. S. 1974 A global theory of steady vortex rings in an ideal fluid. Acta Mathematica 132, 13–51.CrossRefGoogle Scholar
Fukumoto, Y. 2010 Global evolution of viscous vortex rings. Theor. Comput. Fluid Dyn. 24, 335–347.Google Scholar
Fukumoto, Y. & Kaplanski, F. B. 2008 Global time evolution of an axisymmetric vortex ring at low Reynolds numbers. Phys. Fluids 20, 053103.Google Scholar
Fukumoto, Y. & Moffatt, H. K. 2000 Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity. J. Fluid Mech. 417, 1–45.Google Scholar
Fukumoto, Y. & Moffatt, H. K. 2008 Kinematic variational principle for motion of vortex rings. Physica D 237, 2210–2217.Google 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, 6305–6308.Google Scholar
Gharib, M., Rambod, E. & Shariff, K. 1998 A universal time scale for vortex ring formation. J. Fluid Mech. 360, 121–140.Google Scholar
Hecht, F. 2012 New developments in FreeFem++. J. Numer. Math. 20, 251–266.CrossRefGoogle Scholar
Hecht, F., Pironneau, O., Hyaric, A. Le. & Ohtsuke, K.2007 FreeFem++ (manual). www.freefem.org.Google Scholar
Helmholtz, H. 1958 On integrals of the hydrodynamical equations, which express vortex-motion. Crelle’s J. 55, 485–512.Google Scholar
Hill, M. J. M. 1894 On a spherical vortex. Phil. Trans. R. Soc. Lond. A 185, 213–245.Google Scholar
James, S. & Madnia, C. K. 1996 Direct numerical simulation of a laminar vortex ring. Phys. Fluids 8, 2400–2414.Google Scholar
Kaplanski, F. B., Fukumoto, Y. & Rudi, Y. A. 2012 Reynolds-number effect on vortex ring evolution in a viscous fluid. Phys. Fluids 24, 033101.Google Scholar
Kaplanski, F. B. & Rudi, Y. A. 1999 Dynamics of a viscous vortex ring. Intl J. Fluid Mech. Res. 26, 618–630.Google Scholar
Kaplanski, F. B. & Rudi, Y. A. 2005 A model for the formation of ‘optimal’ vortex rings taking into account viscosity. Phys. Fluids 17, 087101.CrossRefGoogle Scholar
Kaplanski, F., Sazhin, S. S., Begg, S., Fukumoto, Y. & Heikal, M. 2010 Dynamics of vortex rings and spray-induced vortex ring-like structures. Eur. J. Mech. B 29 (3), 208–216.CrossRefGoogle Scholar
Kaplanski, F., Sazhin, S. S., Fukumoto, Y., Begg, S. & Heikal, M. 2009 A generalized vortex ring model. J. Fluid Mech. 622, 233–258.CrossRefGoogle Scholar
Kim, J. & Moin, P. 1985 Application of a fractional step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59, 308–323.Google Scholar
Krieg, M. & Mohseni, K. 2013 On the approximation of the translational velocity of vortex rings. Trans. ASME J. Fluids Engng 135, 124501.Google Scholar
Krueger, P. S. 2008 Circulation and trajectories of vortex rings formed from tube and orifice openings. Physica D 237, 2218–2222.Google Scholar
Krueger, S. & Gharib, M. 2003 The significance of vortex ring formation on the impulse and thrust of a starting jet. Phys. Fluids 15, 1271–1280.Google Scholar
Lamb, H. 1932 Hydrodynamics. Dover.Google Scholar
Lim, T. T. & Nickels, T. B. 1995 Vortex rings. In Vortices in Fluid Flows (ed. Green, S. I.), pp. 95–153. Kluwer.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.Google Scholar
Mathematica2007 MATHEMATICA Book, version 6.0.0, http://functions.wolfram.com, Wolfram Research Inc.Google Scholar
Michalke, A. 1984 Survey on jet instability theory. Prog. Aerosp. Sci. 21, 159–199.CrossRefGoogle Scholar
Moffatt, H. K. 1988 Generalised vortex rings with and without swirl. Fluid Dyn. Res. 3, 22–30.Google Scholar
Mohseni, K. 2001 Statistical equilibrium theory for axisymmetric flow: Kelvin’s variational principle and an explanation for the vortex ring pinch–off process. Phys. Fluids 13, 1924–1931.CrossRefGoogle Scholar
Mohseni, K. 2006 A formulation for calculating the translational velocity of a vortex ring or pair. Bioinspir. Biomim. 1, S57–S64.CrossRefGoogle ScholarPubMed
Mohseni, K. & Gharib, M. 1998 A model for universal time scale of vortex ring formation. Phys. Fluids 10, 2436–2438.CrossRefGoogle Scholar
Ni, W. M. 1980 On the existence of global vortex rings. J. Anal. Math. 37, 208–247.Google Scholar
Norbury, J. 1972 A steady vortex ring close to Hill’s spherical vortex. Proc. Camb. Phil. Soc. 72, 253–282.Google Scholar
Norbury, J. 1973 A family of steady vortex rings. J. Fluid Mech. 57, 417–431.Google Scholar
Orlandi, P. 1999 Fluid Flow Phenomena: A Numerical Toolkit. Kluwer.Google Scholar
Orlanski, I. 1976 A simple boundary condition for unbounded hyperbolic flows. J. Comput. Phys. 21, 251–269.Google Scholar
Phillips, O. M. 1956 The final period of decay of non-homogeneous turbulence. Proc. Camb. Phil. Soc. 252, 135–151.CrossRefGoogle Scholar
Rai, M. & Moin, P. 1991 Direct simulations of turbulent flow using finite-difference schemes. J. Comput. Phys. 96, 15–53.Google Scholar
Rosenfeld, M., Rambod, E. & Gharib, M. 1998 Circulation and formation number of a laminar vortex ring. J. Fluid Mech. 376, 297–318.Google Scholar
Rott, N. & Cantwell, B. 1993a Vortex drift. Part I. Dynamic interpretation. Phys. Fluids 5, 1443–1450.Google Scholar
Rott, N. & Cantwell, B. 1993b Vortex drift. Part II. The flow potential surrounding a drifting vortical region. Phys. Fluids 5, 1451–1455.Google Scholar
Ruith, M. R., Chen, P. & Meiburg, E. 2004 Development of boundary conditions for direct numerical simulations of three-dimensional vortex breakdown phenomena in semi-infinite domains. Comput. Fluids 33, 1225–1250.Google Scholar
Saffman, P. G. 1970 The velocity of viscous vortex rings. Stud. Appl. Maths 49, 371–380.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Sau, R. & Mahesh, K. 2007 Passive scalar mixing in vortex rings. J. Fluid Mech. 582, 449–461.Google Scholar
Sazhin, S. S. 2014 Droplets and Sprays. Springer.Google Scholar
Shariff, K. & Leonard, A. 1992 Vortex rings. Annu. Rev. Fluid Mech. 24, 235–279.Google Scholar
Shusser, M. & Gharib, M. 2000 Energy and velocity of a forming vortex ring. Phys. Fluids 12, 618–621.Google Scholar
Stewart, K., Niebel, C., Jung, S. & Vlachos, P. 2012 The decay of confined vortex rings. Exp. Fluids 53, 163–171.Google Scholar
Sullivan, J. P. 1973 Study of a vortex ring using a laser Doppler velocimeter. AIAA J. 11, 1384–1389.Google Scholar
Sullivan, I. S., Niemela, J. J., Hershberger, R. E., Bolster, D. & Donnelly, R. J. 2008 Dynamics of thin vortex rings. J. Fluid Mech. 609, 319–347.Google Scholar
Tung, C. & Ting, L. 1967 Motion and decay of a vortex ring. Phys. Fluids 10, 901–910.Google Scholar
Verzicco, R. & Orlandi, P. 1996 A finite-difference scheme for three-dimensional incompressible flow in cylindrical coordinates. J. Comput. Phys. 123, 402–414.CrossRefGoogle Scholar
Weigand, A. & Gharib, M. 1997 On the evolution of laminar vortex rings. Exp. Fluids 22, 447–457.Google Scholar
Zhao, W., Steven, H. F. & Mongeau, L. G. 2000 Effects of trailing jet instability on vortex ring formation. Phys. Fluids 12, 589–596.Google Scholar