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Nested contour dynamics models for axisymmetric vortex rings and vortex wakes

Published online by Cambridge University Press:  01 May 2014

Clara O’Farrell
Control and Dynamical Systems, California Institute of Technology, Pasadena CA 91125, USA
John O. Dabiri
Graduate Aeronautical Laboratories and Bioengineering, California Institute of Technology, Pasadena CA 91125, USA
E-mail address:


Inviscid models for vortex rings and dipoles are constructed using nested patches of vorticity. These models constitute more realistic approximations to experimental vortex rings and dipoles than the single-contour models of Norbury and Pierrehumbert, and nested contour dynamics algorithms allow their simulation with low computational cost. In two dimensions, nested-contour models for the analytical Lamb dipole are constructed. In the axisymmetric case, a family of models for vortex rings generated by a piston–cylinder apparatus at different stroke ratios is constructed from experimental data. The perturbation response of this family is considered by the introduction of a small region of vorticity at the rear of the vortex, which mimics the addition of circulation to a growing vortex ring by a feeding shear layer. Model vortex rings are found to either accept the additional circulation or shed vorticity into a tail, depending on the perturbation size. A change in the behaviour of the model vortex rings is identified at a stroke ratio of three, when it is found that the maximum relative perturbation size vortex rings can accept becomes approximately constant. We hypothesise that this change in response is related to pinch-off, and that pinch-off might be understood and predicted based on the perturbation responses of model vortex rings. In particular, we suggest that a perturbation response-based framework can be useful in understanding vortex formation in biological flows.

© 2014 Cambridge University Press 

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Afanasyev, Y. D. 2006 Formation of vortex dipoles. Phys. Fluids 18, 037103.CrossRefGoogle Scholar
Albrecht, T. R., Elcrat, A. R. & Miller, K. G. 2011 Steady vortex dipoles with general profile functions. J. Fluid Mech. 670, 8595.CrossRefGoogle Scholar
Aref, J. & Vainchtein, D. L. 1998 Point vortices exhibit asymmetric equilibria. Nature 392, 769770.CrossRefGoogle Scholar
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. Germain, P. & Nayroles, B.), pp. 828. Springer.CrossRefGoogle Scholar
Boyd, J. P. & Ma, H. 1990 Numerical study of elliptical modons using a spectral method. J. Fluid Mech. 221, 597611.CrossRefGoogle Scholar
Chaplygin, S. A. 1903 One case of vortex motion in a fluid. Trans. Phys. Sect. Imperial Moscow Soc. Friends Nat. Sci. 11 (2), 1114.Google Scholar
Corcos, G. M. & Lin, S. J. 1984 The mixing layer: deterministic models of a turbulent flow. Part 2. The origin of the three-dimensional motion. J. Fluid Mech. 139, 6795.CrossRefGoogle Scholar
Corcos, G. M., Sherman, F. S. & Lin, S. J. 1984 The mixing layer: deterministic models of a turbulent flow. Part 1. Introduction and the two-dimensional flow. J. Fluid Mech. 139, 2965.CrossRefGoogle Scholar
Dabiri, J. O., Colin, S. P., Katija, K. & Costello, J. H. 2010 A wake-based correlate of swimming performance and foraging behaviour in seven co-occurring jellyfish species. J. Expl Biol. 213, 12171225.CrossRefGoogle ScholarPubMed
Dabiri, J. O. & Gharib, M. 2004 Delay of vortex ring pinch-off by an imposed bulk counterflow. Phys. Fluids 16 (L), 2830.CrossRefGoogle Scholar
Dabiri, J. O. & Gharib, M. 2005a The role of optimal vortex formation in biological fluid transport. Proc. R. Soc. B 272, 15571560.CrossRefGoogle ScholarPubMed
Dabiri, J. O. & Gharib, M. 2005b Starting flow through nozzles with temporally variable exit diameter. J. Fluid Mech. 538, 111136.CrossRefGoogle Scholar
Deem, G. S. & Zabusky, N. J. 1978 Vortex waves: stationary ‘V states’, interactions, recurrence and breaking. Phys. Rev. Lett. 40, 859862.CrossRefGoogle Scholar
Dickinson, M. H., Farley, C. T., Full, R. J., Koehl, M. A. R., Kram, R. & Lehman, S. 2000 How animals move: an integrative view. Science 288 (5463), 100106.CrossRefGoogle Scholar
Domenichini, F. 2011 Three-dimensional impulsive vortex formation from slender orifices. J. Fluid Mech. 666, 506520.CrossRefGoogle Scholar
Dritschel, D. G. 1988a Contour surgery: a topological reconnection scheme for extended integrations using contour dynamics. J. Comput. Phys. 77, 240266.CrossRefGoogle Scholar
Dritschel, D. G. 1988b The repeated filamentation of two-dimensional vorticity interfaces. J. Fluid Mech. 194, 511547.CrossRefGoogle Scholar
Dritschel, D. G. 1989 Contour dynamics and contour surgery: numerical algorithms for extended high-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows. Comput. Phys. Rep. 10, 79146.CrossRefGoogle Scholar
Flór, J. B. & van Heijst, G. J. F. 1994 An experimental study of dipolar vortices in a stratified fluid. J. Fluid Mech. 279, 101133.CrossRefGoogle Scholar
Fukumoto, Y. & Kaplanski, F. 2008 Global time evolution of an axisymmetric vortex ring at low Reynolds numbers. Phys. Fluids 20, 053103.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, 205222.CrossRefGoogle Scholar
Gharib, M., Rambod, E., Kheradvar, A., Sahn, D. J. & Dabiri, J. O. 2006 Optimal vortex formation as an index for cardiac health. Proc. Natl Acad. Sci. USA 103, 63056308.CrossRefGoogle ScholarPubMed
Gharib, M., Rambod, E. & Shariff, K. 1998 A universal time scale for vortex ring formation. J. Fluid Mech. 360, 121140.CrossRefGoogle Scholar
Hill, M. J. M. 1894 On a spherical vortex. Phil. Trans. R. Soc. Lond. A 185, 213245.CrossRefGoogle Scholar
Jacobs, P. A. & Pullin, D. I. 1989 Multiple-contour-dynamic simulation of eddy scales in the plane shear layer. J. Fluid Mech. 199, 89124.CrossRefGoogle 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
Kelvin, L. 1880 Vortex statics. Phil. Mag. 10, 97109.Google Scholar
Khvoles, R., Berson, D. & Kizner, Z. 2005 The structure and evolution of elliptical barotropic modons. J. Fluid Mech. 530, 130.CrossRefGoogle Scholar
Kizner, Z. & Khvoles, R. 2004 Two variations on the theme of Lamb–Chaplygin: supersmooth dipoles and rotating multipoles. Regular Chaotic Dyn. 4, 509518.CrossRefGoogle Scholar
Krueger, P. S., Dabiri, J. O. & Gharib, M. 2006 The formation number of vortex rings in uniform background coflow. J. Fluid Mech. 556, 147166.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, 12711281.CrossRefGoogle Scholar
Lamb, H. 1895 Hydrodynamics. 2nd edn Cambridge University Press.Google Scholar
Lamb, H. 1906 Hydrodynamics. 3rd edn Cambridge University Press.Google Scholar
Lim, T. T. & Nickels, T. B. 1995 Vortex rings. In Fluid Vortices, Kluwer Academic Publishers.Google Scholar
Lin, S. J. & Corcos, G. M. 1984 The mixing layer: deterministic models of a turbulent flow. Part 3. The effects of plane strain on the dynamics of streamwise vortices. J. Fluid Mech. 141, 139178.CrossRefGoogle Scholar
Linden, P. F. & Turner, J. S. 2001 The formation of ‘optimal’ vortex rings, and the efficiency of propulsive devices. J. Fluid Mech. 427, 6172.CrossRefGoogle Scholar
Linden, P. F. & Turner, J. S. 2004 ‘Optimal’ vortex rings and aquatic propulsion mechanisms. Proc. R. Soc. B 271, 647653.CrossRefGoogle ScholarPubMed
Lugt, H. J. 1995 Vortex Flow in Nature and Technology. Krieger Publishing Company.Google Scholar
Meleshko, V. V. & van Heijst, G. J. F. 1994 On Chaplygin’s investigations of two-dimensional vortex structures in an inviscid fluid. J. Fluid Mech. 272, 157182.CrossRefGoogle Scholar
Mellander, M. V., McWilliams, J. C. & Zabusky, N. J. 1987 Axisymmetrization and vorticitity-gradient intensification of an isolated two-dimensional vortex through filamentation. J. Fluid Mech. 178, 137158.CrossRefGoogle Scholar
Mohseni, K. & Gharib, M. 1998 A model for universal time scale of vortex ring formation. Phys. Fluids 10, 24362438.CrossRefGoogle Scholar
Mohseni, K., Ran, H. & Colonius, T. 2000 Numerical experiments on vortex ring formation. J. Fluid Mech. 430, 267282.CrossRefGoogle Scholar
Norbury, J. 1973 A family of steady vortex rings. J. Fluid Mech. 57, 417431.CrossRefGoogle Scholar
O’Farrell, C. & Dabiri, J. O. 2010 A Lagrangian approach to vortex pinch-off. Chaos 20, 017513.CrossRefGoogle ScholarPubMed
O’Farrell, C. & Dabiri, J. O. 2012 Perturbation response and pinch-off of vortex rings and dipoles. J. Fluid Mech. 704, 280300.CrossRefGoogle Scholar
Pawlak, G., Cruz, C. M., Bazan, C. M. & Hrdy, P. G. 2007 Experimental characterization of starting jet dynamics. Fluid Dyn. Res. 39, 711730.CrossRefGoogle Scholar
Pedrizzetti, G. 2010 Vortex formation out of two-dimensional orifices. J. Fluid Mech. 655, 198216.CrossRefGoogle Scholar
Pierrehumbert, R. T. 1980 A family of steady, translating vortex pairs with distributed vorticity. J. Fluid Mech. 99, 129144.CrossRefGoogle Scholar
Pozrikidis, C. 1986 The nonlinear instability of Hill’s vortex. J. Fluid Mech. 168, 337367.CrossRefGoogle Scholar
Pullin, D. I. 1981 The nonlinear behavour of a constant vorticity layer at a wall. J. Fluid Mech. 108, 401421.CrossRefGoogle Scholar
Pullin, D. I. & Jacobs, P. A. 1986 Inviscid evolution of stretched vortex arrays. J. Fluid Mech. 171, 377406.CrossRefGoogle Scholar
Rayner, J. M. V. 1979 A vortex theory of animal flight. Part 2: the forward flight of birds. J. Fluid Mech. 91, 731763.CrossRefGoogle Scholar
Shariff, K. & Leonard, A. 1992 Vortex rings. Annu. Rev. Fluid Mech. 24, 235279.CrossRefGoogle Scholar
Shariff, K., Leonard, A. & Ferziger, J. H. 2008 A contour dynamics algorithm for axisymmetric flow. J. Comput. Phys. 227, 90449062.CrossRefGoogle Scholar
Shusser, M. & Gharib, M. 2000 Energy and velocity of a forming vortex ring. Phys. Fluids 12, 618621.CrossRefGoogle Scholar
Trieling, R., Santberg, R., van Heijst, G. J. F. & Kizner, Z. 2010 Barotropic elliptical dipoles in a rotating fluid. Theor. Comput. Fluid Dyn. 24, 111115.CrossRefGoogle Scholar
Turner, J. S. 1960 On the intermittent release of smoke from chimneys. Mech. Engng. Sci. 2, 97100.CrossRefGoogle Scholar
van Geffen, J. H. G. M. & van Heijst, G. J. F. 1998 Viscous evolution of 2D dipolar vortices. Fluid Dyn. Res. 22, 191213.CrossRefGoogle Scholar
Weigand, A. & Gharib, M. 1997 On the evolution of laminar vortex rings. Exp. Fluids 22, 447457.CrossRefGoogle Scholar
Ye, Q. Y. & Chu, C. K. 1995 Unsteady evolutions of vortex rings. Phys. Fluids 7, 795801.CrossRefGoogle Scholar
Zabusky, N. J., Hughes, M. H. & Roberts, K. V. 1979 Contour dynamics for the Euler equations in two dimensions. J. Comput. Phys. 30, 96106.CrossRefGoogle Scholar
Zhao, W., Frankel, S. H. & Mongeau, L. G. 2000 Effects of trailing jet instability on vortex ring formation. Phys. Fluids 12, 589621.CrossRefGoogle Scholar
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