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Evolution of vortex-surface fields in viscous Taylor–Green and Kida–Pelz flows

Published online by Cambridge University Press:  06 October 2011

Yue Yang  and
D. I. Pullin
Yue Yang*
Affiliation:
Graduate Aerospace Laboratories, 205-45, California Institute of Technology, Pasadena, CA 91125, USA
D. I. Pullin
Affiliation:
Graduate Aerospace Laboratories, 205-45, California Institute of Technology, Pasadena, CA 91125, USA
*
†Email address for correspondence: yy@caltech.edu
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Abstract

In order to investigate continuous vortex dynamics based on a Lagrangian-like formulation, we develop a theoretical framework and a numerical method for computation of the evolution of a vortex-surface field (VSF) in viscous incompressible flows with simple topology and geometry. Equations describing the continuous, timewise evolution of a VSF from an existing VSF at an initial time are first reviewed. Non-uniqueness in this formulation is resolved by the introduction of a pseudo-time and a corresponding pseudo-evolution in which the evolved field is ‘advected’ by frozen vorticity onto a VSF. A weighted essentially non-oscillatory (WENO) method is used to solve the pseudo-evolution equations in pseudo-time, providing a dissipative-like regularization. Vortex surfaces are then extracted as iso-surfaces of the VSFs at different real physical times. The method is applied to two viscous flows with Taylor–Green and Kida–Pelz initial conditions respectively. Results show the collapse of vortex surfaces, vortex reconnection, the formation and roll-up of vortex tubes, vorticity intensification between anti-parallel vortex tubes, and vortex stretching and twisting. A possible scenario for understanding the transition from a smooth laminar flow to turbulent flow in terms of topology of vortex surfaces is discussed.

JFM classification

Mathematical Foundations: Topological fluid dynamics Turbulent Flows: Turbulence theory Vortex Flows: Vortex dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 685 , 25 October 2011 , pp. 146 - 164
Copyright
Copyright © Cambridge University Press 2011

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References

1. Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
2. Bermejo-Moreno, I. & Pullin, D. I. 2008 On the non-local geometry of turbulence. J. Fluid Mech. 603, 101–135.CrossRefGoogle Scholar
3. Boratav, O. N. & Pelz, R. B. 1994 Direct numerical simulation of transition to turbulence from a high-symmetry initial condition. Phys. Fluids 6, 2757–2784.CrossRefGoogle Scholar
4. Brachet, M. E., Meiron, D. I., Orszag, S. A., Nickel, B. G., Morf, R. H. & Frisch, U. 1983 Small-scale structure of the Taylor–Green vortex. J. Fluid Mech. 130, 411–452.CrossRefGoogle Scholar
5. Chen, S., Eyink, G. L., Wan, M. & Xiao, Z. 2006 Is the Kelvin theorem valid for high Reynolds number turbulence? Phys. Rev. Lett. 97, 144505.CrossRefGoogle ScholarPubMed
6. Constantin, P., Procaccia, I. & Segel, D. 1995 Creation and dynamics of vortex tubes in three-dimensional turbulence. Phys. Rev. E 51, 3207–3222.CrossRefGoogle ScholarPubMed
7. Crandall, M. G. & Lions, P. L. 1983 Viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 277, 1–42.CrossRefGoogle Scholar
8. Fukumoto, Y. & Okulov, V. L. 2005 The velocity field induced by a helical vortex tube. Phys. Fluids 17, 107101.CrossRefGoogle Scholar
9. Ghrist, R. & Komendarczyk, R. 2002 Topological features of inviscid flows. In Introduction to the Geometry and Topology of Fluid Flows, NATO–ASI Series II, vol. 47 , pp. 183–202. Kluwer.Google Scholar
10. Helmholtz, H. 1858 Über Integrale der Hydrodynamischen Gleichungen Welche den Wirbelbewegungen Entsprechen. J. Reine Angew. Math. 55, 25–55.Google Scholar
11. Hussain, F. & Duraisamy, K. 2011 Mechanics of viscous vortex reconnection. Phys. Fluids 23, 021701.CrossRefGoogle Scholar
12. Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.CrossRefGoogle Scholar
13. Jiang, G. S. & Shu, C. W. 1996 Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228.CrossRefGoogle Scholar
14. Jimenez, J., Wray, A. A., Saffman, P. G. & Rogallo, R. S. 1993 The structure of intense vorticity in isotropic turbulence. J. Fluid Mech. 255, 65–90.CrossRefGoogle Scholar
15. Kerr, R. M. 1993 Evidence for a singularity of the three-dimensional, incompressible Euler equations. Phys. Fluids A 5, 1725–1746.CrossRefGoogle Scholar
16. Kida, S. 1985 Three-dimensional periodic flows with high-symmetry. J. Phys. Soc. Japan 54, 2132–2136.CrossRefGoogle Scholar
17. Kida, S. & Takaoka, M. 1994 Vortex reconnection. Annu. Rev. Fluid Mech. 26, 169–189.CrossRefGoogle Scholar
18. Kida, S., Takaoka, M. & Hussain, F. 1991 Collision of two vortex rings. J. Fluid Mech. 230, 583–646.CrossRefGoogle Scholar
19. Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
20. Mingyu, H., Küpper, T. & Masbaum, N. 1997 Computation of invariant tori by the Fourier methods. SIAM J. Sci. Comput. 18, 918–942.CrossRefGoogle Scholar
21. Ohkitani, K. & Constantin, P. 2003 Numerical study of the Eulerian–Lagrangian formulation of the Navier–Stokes equations. Phys. Fluids 15, 3251–3254.CrossRefGoogle Scholar
22. Osher, S. & Fedkiw, R. 2003 Level Set Methods and Dynamic Implicit Surfaces. Springer.CrossRefGoogle Scholar
23. Pullin, D. I. & Saffman, P. G. 1998 Vortex dynamics in turbulence. Annu. Rev. Fluid Mech. 30, 31–51.CrossRefGoogle Scholar
24. Ricca, R. L. 1994 The effect of torsion on the motion of a helical vortex filament. J. Fluid Mech. 273, 241–259.CrossRefGoogle Scholar
25. Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
26. She, Z. S., Jackson, E. & Orszag, S. A. 1990 Intermittent vortex structures in homogeneous isotropic turbulence. Nature 344, 226–228.CrossRefGoogle Scholar
27. Shu, C. W., Don, W. S., Gottlieb, D., Schilling, O. & Jameson, L. 2005 Numerical convergence study of nearly incompressible, inviscid Taylor–Green vortex flow. J. Sci. Comput. 24, 569–595.CrossRefGoogle Scholar
28. Siggia, E. D. 1985 Collapse and amplification of a vortex filament. Phys. Fluids 28, 794–805.CrossRefGoogle Scholar
29. Taylor, G. I. & Green, A. E. 1937 Mechanism of the production of small eddies from large ones. Proc. R. Soc. Lond. A 158, 499–521.Google Scholar
30. Truesdell, C. 1954 The Kinematics of Vorticity. Indiana University Press.Google Scholar
31. Wang, L. 2010 On properties of fluid turbulence along streamlines. J. Fluid Mech. 648, 183–203.CrossRefGoogle Scholar
32. Yang, Y. & Pullin, D. I. 2010 On Lagrangian and vortex-surface fields for flows with Taylor–Green and Kida–Pelz initial conditions. J. Fluid Mech. 661, 446–481.CrossRefGoogle Scholar
33. Yang, Y. & Pullin, D. I. 2011 Geometric study of Lagrangian and Eulerian structures in turbulent channel flow. J. Fluid Mech. 674, 67–92.CrossRefGoogle Scholar
34. Yang, Y., Pullin, D. I. & Bermejo-Moreno, I. 2010 Multi-scale geometric analysis of Lagrangian structures in isotropic turbulence. J. Fluid Mech. 654, 233–270.CrossRefGoogle Scholar