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Vortex intensification and collapse of the Lissajous-elliptic ring: single- and multi-filament Biot-Savart simulations and visiometrics

Published online by Cambridge University Press:  26 April 2006

Victor M. Fernandez ,
Norman J. Zabusky  and
Vladimir M. Gryanik
Victor M. Fernandez
Affiliation:
Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ 08855, USA
Norman J. Zabusky
Affiliation:
Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ 08855, USA
Vladimir M. Gryanik
Affiliation:
Institute of Atmospheric Physics, Russian Academy of Sciences, 109017 Moscow, Russia
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Abstract

The collapsing ‘Lissajous-elliptic’ (LE) vortex ring is examined via quantifications of Single- and multi-filament Biot-Savart numerical simulations. In the single-filament simulations, parametric studies show simple relationships between the collapse boundary and the impulse and energy invariants. Collapse becomes non-monotonic in time, for a sufficiently small initial core ‘radius’. Self-similar, singular-like behaviour of the off-filament strain-rate growth has been observed in a small interval, just prior to core overlapping. The computation of the strain-rate eigenvalues and vortex stretching in a diagnostics box surrounding the collapse region yields patterns observed previously in continuum simulations. New diagnostics are presented, including line densities of the energy and the linear and angular momentum, all of which approach zero in the collapse region of the ring. These diagnostics may provide critical parameters for initiating surgery in a topology-changing algorithm. Our multi-filament simulations exhibit layer-like vortex regions and a ‘torus’-shaped vortex stretching pattern observed previously in continuum periodic-domain simulations of vortex reconnection. Quantifications in a cross-section of the collapse region indicate that the circulation tends to concentrate in the head or frontside of the convecting dipolar structure. This is also the location of the incipient ‘bridge’ which is evolving from the weak filaments that have been convected from the initially outer-vortex regions. The formation of this smaller scale vortex structure exhibits the largest vorticity amplification in the variable-core model simulations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 299 , 25 September 1995 , pp. 289 - 331
Copyright
© 1995 Cambridge University Press

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References

Almgren, A. S., Buttke, T. & Colella, P. 1994 A fast adaptive vortex method in three dimensions. J. Comput. Phys. 113, 177–200.Google Scholar
Anderson, C. & Greengard, C. 1985 On vortex methods. SIAM J. Numer. Anal. 22, 413–440.Google Scholar
Anderson, C. & Greengard, C. 1989 The vortex merger problem at infinite Reynolds number. Commun. Pure Appl. Maths 42, 1123–1139.Google Scholar
Arms, R. J. & Hama, F. R. 1965 Localized-Induction concept on a curved vortex and motion of an elliptic vortex ring. Phys. Fluids 8, 553–559.Google Scholar
Ashurst, W. T. & Meiron, D. I. 1987 Numerical study of vortex reconnection. Phys. Rev. Lett. 58, 1632–1635.Google Scholar
Ashurst, W. T., Kerstein, A. R., Kerr, R. M. & Gibson, C. H. 1987 Alignment of vorticity and scalar gradient with strain rate in simulated Navier-Stokes turbulence. Phys. Fluids 30, 2343–2353.Google Scholar
Beale, J. T., Kato, T. & Majda, A. 1984 Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66.Google Scholar
Beale, J. T. & Majda, A. 1982 Vortex methods. I: Convergence in three dimensions. Maths Comput. 39, 1–27.Google Scholar
Beale, J. T. & Majda, A. 1985 High order accurate vortex methods with explicit velocity kernels. J. Comput. Phys. 58, 188–208.Google Scholar
Boratav, O. N. & Pelz, R. B. 1994 Direct numerical simulation of transition to turbulence from a high-symmetry initial condition. Phys. Fluids A 6, 2757–2784.Google Scholar
Boratav, O. N., Pelz, R. B. & Zabusky, N. J. 1992 Reconnection in orthogonally interacting vortex tubes: Direct numerical simulations and quantifications. Phys. Fluids A 4, 581–605.Google Scholar
Chen, S.-Y. & Shan, X. 1992 High-Resolution turbulent simulations using the Connection Machine-2. Computers Phys. 6, 643–646.Google Scholar
Chorin, A. J. 1990a Vortex filaments and turbulence theory. In Topological Fluid Dynamics, Proc. IUTAM Symp., Cambridge, August 1989 (ed. H. K. Moffatt & A. Tsinober), pp. 607–616. Cambridge University Press.
Chorin, A. J. 1990b Hairpin removal in vortex interactions. J. Comput. Phys. 91, 1–21.Google Scholar
Chorin, A. J. 1993 Hairpin removal in vortex interactions II. J. Comput. Phys. 107, 1–9.Google Scholar
Constantin, P. 1994 Geometric statistics in turbulence. SIAM Rev. 36, 73–98.Google Scholar
Crow, S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8, 2172–2179.Google Scholar
Dhanak, M. R. & Bernardinis, B. de 1981 The evolution of an elliptic vortex ring. J. Fluid Mech. 109, 189–216.Google Scholar
Douady, S. & Couder, Y. 1993 On the dynamical structures observed in 3d turbulence. In Proc. Workshop on Turbulence in Extended Systems, Les Houches (ed. R. Benzi, C. Basdevant, S. Ciliberto), pp. 3–17. Nova Science Commack, NY.
Douady, S., Couder, Y. & Brachet, M. E. 1991 Direct observation of the intermittency of intense vorticity filaments in turbulence. Phys. Rev. Lett. 67, 983–986.Google Scholar
Fernandez, V. M. 1994 Vortex intensification and collapse of the Lissajous-Elliptic ring: Biot-Savart simulations and visiometrics. PhD thesis, Rutgers University, New Brunswick, New Jersey.
Fernandez, V. M., Zabusky, N. J. & Gryanik, V. M. 1994 Near-singular collapse and local intensification of a “Lissajous-elliptic” vortex ring: Non-monotonic behavior and zero-approaching local energy densities. Phys. Fluids A 6, 2242–2244.Google Scholar
Fohl, T. & Turner, J. S. 1975 Colliding vortex rings. Phys. Fluids 18, 433–436.Google Scholar
Fukuyu, A. & Arai, T. 1991 Singularity formation in three-dimensional inviscid flow. Fluid Dyn. Res. 7, 229–240.Google Scholar
Greengard, C. 1986 Convergence of the vortex filament method. Maths Comput. 47, 387–398.Google Scholar
Hon, T.-L. & Walker, J. D. A. 1988 Evolution of hairpin vortices in a shear flow. NASA Tech. Memo. 100858, ICOMP-88-9.
Hussain, F. & Husain, H. S. 1989 Elliptic jets. Part 1. Characteristics of unexcited and excited jets. J. Fluid Mech. 208, 257–320.Google Scholar
Husain, H. S. & Hussain, F. 1993 Elliptic jets. Part 3. Dynamics of preferred mode coherent structure. J. Fluid Mech. 248, 315–361.Google Scholar
Inoue, O. 1988 Simulation of a vortex ring. AIAA Paper 88-3571-CP.
Jiménez, J. 1992 Kinematic alignment effects in turbulent flows. Phys. Fluids A 4, 652–654.Google Scholar
Jimenez, J., Wray, A. A., Safeman, P. G. & Rogallo, R. S. 1993 The structure of intense vorticity in isotropic turbulence. J. Fluid Mech. 255, 65–90.Google Scholar
Johnston, R. T. & Sullivan, J. P. 1994 A flow environment for studying vortex interactions. Exps. Fluids 18, 131.Google Scholar
Kerr, R. M. 1985 Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence. J. Fluid Mech. 153, 31–58.Google Scholar
Kerr, R. M. 1987 Histograms of helicity and strain in numerical turbulence. Phys. Rev. Lett. 59, 783–786.Google Scholar
Kerr, R. M. 1993 Evidence for a singularity of the three-dimensional incompressible Euler equations. Phys. Fluids A 5, 1725–1746.Google Scholar
Kerr, R. M. & Husain, F. 1989 Simulation of vortex reconnection. Physica ([A-Z]) 37, 474–484.Google Scholar
Kida, S. & Takaoka, M. 1987 Bridging in vortex reconnection. Phys. Fluids 30, 2911–2914.Google Scholar
Kida, S. & Takaoka, M. 1988 Reconnection of vortex tubes. Fluid Dyn. Res. 3, 257–261.Google Scholar
Kida, S. & Takaoka, M. 1994 Vortex reconnection. Ann. Rev. Fluid Mech. 26, 169–189.Google Scholar
Kida, S., Takaoka, M. & Hussain, F. 1991 Collision of two vortex rings. J. Fluid Mech. 230, 583–646.Google Scholar
Klein, R. & Knio, O. M. 1995 Asymptotic vorticity structure and numerical simulation of slender vortex filaments. J. Fluid Mech. 284, 275–321.Google Scholar
Klein, R. & Majda, A. J. 1991 Self-stretching of perturbed vortex filaments II. Structure of solutions. Physica ([A-Z]) 53, 267–294.Google Scholar
Knio, O. M. & Ghoniem, A. F. 1990 Numerical study of a three-dimensional vortex method. J. Comput. Phys. 86, 75–106.Google Scholar
Leonard, A. 1980 Vortex methods for flow simulation. J. Comput. Phys. 37, 289–335.Google Scholar
Leonard, A. 1985 Computing three-dimensional incompressible flows with vortex elements. Ann. Rev. Fluid Mech. 17, 523–559.Google Scholar
Lim, T. T. & Nickels, T. B. 1992 Instability and reconnection in the head-on collision of two vortex rings. Nature 357, 225–227.Google Scholar
Liu, C. H., Tavantzis, J. & Ting, L. 1986 Numerical studies of motion and decay of vortex filaments. AIAA J. 24, 1290–1297.Google Scholar
Maxworthy, T. 1972 The structure and stability of vortex rings. J. Fluid Mech. 51, 15–32.Google Scholar
Meiron, D. I., Shelley, M. J., Ashurst, W. T. & Orszag, S. A. 1988 Numerical studies of vortex reconnection. In Proc. Workshop on Mathematical Aspects of Vortex Dynamics, Leesburg, Virginia (ed. R. E. Caflisch), pp. 183–194. SIAM.
Malander, M. V. & Hussain, F. 1989 Cross-linking of two antiparallel vortex tubes. Phys. Fluids A 1, 633–636.Google Scholar
Melander, M. V. & Zabusky, N. J. 1988 Interaction and apparent reconnection of 3d vortex tubes via direct numerical simulations. Fluid Dyn. Res. 3, 247–250.Google Scholar
Moore, D. W. 1972 Finite amplitude waves on aircraft trailing vortices. Aero. Q. 23, 307–314.Google Scholar
Oshima, Y. & Asaka, S. 1977 Interaction of two vortex rings along parallel axes in air. J. Phys. Soc. Japan 42, 708–713.Google Scholar
Oshima, Y. & Izutsu, N. 1988 Cross-linking of two vortex rings. Phys. Fluids 31, 2401–2403.Google Scholar
Oshima, Y., Izutsu, N., Oshima, K. & Hussain, A. K. M. F. 1988 Bifurcation of an elliptic vortex ring. Fluid Dyn. Res. 3, 133–139.Google Scholar
Oshima, Y., Noguchi, T. & Oshima, K. 1986 Numerical study of interaction of two vortex rings. Fluid Dyn. Res. 1, 215–227.Google Scholar
Perlman, M. 1985 On the accuracy of vortex methods. J. Comput. Phys. 59, 200–223.Google Scholar
Ponce, G. 1985 Remarks on a paper by J. T. Beale, T. Kato and A. Majda. Commun. Math. Phys. 98, 349–353.Google Scholar
Pumir, A. & Kerr, R. M. 1987 Numerical simulation of interacting vortex tubes. Phys. Rev. Lett. 58, 1636–1639.Google Scholar
Pumir, A. & Siggia, E. D. 1987 Vortex dynamics and the existence of solutions to the Navier-Stokes equations. Phys. Fluids 30, 1606–1626.Google Scholar
Pumir, A. & Siggia, E. 1989 Simulations of incipient singularities in the 3-D Euler equations. Physica ([A-Z]) 37, 539–541.Google Scholar
Pumir, A. & Siggia, E. 1990 Collapsing solutions to the 3-D Euler equations. Phys. Fluids A 2, 220–241.Google Scholar
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Ann. Rev. Fluid Mech. 23, 601–639.Google Scholar
Saffman, P. G. 1970 The velocity of viscous rings. Stud. Appl. Maths 49, 371–380.Google Scholar
Schatzle, P. R. 1987 An experimental study of fusion of vortex rings. PhD thesis, California Institute of Technology, Pasadena, California.
She, Z.-S., Jackson, E. & Orszag, S. A. 1990a Intermittent vortex structures in homogeneous isotropic turbulence. Nature 344, 226–228.Google Scholar
She, Z.-S., Jackson, E. & Orszag, S. A. 1990b Vortex structure and dynamics in turbulence. Computer Meth. Appl. Mech. Engng 80, 173–183.Google Scholar
She, Z.-S., Jackson, E. & Orszag, S. A. 1991 Structure and dynamics of homogeneous turbulence: Models and simulations. Proc. R. Soc. Lond. A 434, 101–124.Google Scholar
Shelley, M. J. & Meiron, D. I. 1991 Vortex reconnection and smoothness of the Euler equations. Lectures in Applied Mathematics, vol. 26, pp. 647–677. Springer.
Shelley, M. J., Meiron, D. I. & Orszag, S. A. 1993 Dynamical aspects of vortex reconnection of perturbed anti-parallel vortex tubes. J. Fluid Mech. 246, 613–652.Google Scholar
Siggia, E. D. 1981 Numerical study of small-scale intermittency in three-dimensional turbulence. J. Fluid Mech. 107, 375–406.Google Scholar
Siggia, E. D. 1985 Collapse and amplification of a vortex filament. Phys. Fluids 28, 794–805.Google Scholar
Siggia, E. D. & Pumir, A. 1985 Incipient singularities in the Navier-Stokes equations. Phys. Rev. Lett. 55, 1749–1752.Google Scholar
Smith, G. B. 1992 Turbulent cascade to small scales during the off-axis collision of two vortex rings. MS thesis, Rutgers University, Graduate Program in Mechanical and Aerospace Engineering.
Smith, G. B. & Wei, T. 1994 Small-scale structure in colliding off-axis vortex rings. J. Fluid Mech. 259, 281–290.Google Scholar
Ting, L. & Klein, R. 1991 Viscous Vortical Flows. Lecture Notes in Physics, vol. 374, section 2.4, pp. 103–115. Springer.
Vincent, A. & Meneguzzi, M. 1991 The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech. 225, 1–20.Google Scholar
Vincent, A. & Meneguzzi, M. 1994 The dynamics of vorticity tubes in homogeneous turbulence. J. Fluid Mech. 258, 245–254.Google Scholar
Waele, A. T. M. M. de & Aarts, R. G. K. M. 1994 Route to vortex reconnection. Phys. Rev. Lett. 72, 482–485.Google Scholar
Widnall, S. E., Bliss, D. B. & Tsai, C.-Y. 1974 The instability of short waves on a vortex ring. J. Fluid Mech. 66, 35–47.Google Scholar
Widnall, S. E. & Sullivan, J. P. 1973 On the stability of vortex rings. Proc. R. Soc. Lond. A 332, 335–353.Google Scholar
Winckelmans, G. S. 1989 Topics in vortex methods for the computation of three- and two-dimensional incompressible unsteady flows. PhD thesis, California Institute of Technology, Pasadena, California.
Winckelmans, G. S. & Leonard, A. 1993 Contributions to vortex particle methods for the computation of three-dimensional incompressible unsteady flows. J. Comput. Phys. 109, 247–273.Google Scholar
Zabusky, N. J., Boratav, O. N., Pelz, R. B., Gao, M., Silver, D. & Cooper, S. P. 1991 Emergence of coherent patterns of vortex stretching during reconnection: A scattering paradigm. Phys. Rev. Lett. 67, 2469–2472.Google Scholar
Zabusky, N. J. & Melander, M. V. 1989 Three-dimensional vortex tube reconnection: Morphology for orthogonally-offset tubes. Physica ([A-Z]) 37, 555–562.Google Scholar
Zabusky, N. J., Silver, D. & Pelz, R. 1993 Visiometrics, Juxtaposition and modeling. Phys. Today 46, 24–31.Google Scholar
Zakharov, V. E. 1988 Wave collapse. Sov. Phys. Usp. 31, 672–674.Google Scholar