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Small-scale structure of the Taylor–Green vortex

Published online by Cambridge University Press:  20 April 2006

Marc E. Brachet ,
Daniel I. Meiron ,
Steven A. Orszag ,
B. G. Nickel ,
Rudolf H. Morf  and
Uriel Frisch
Marc E. Brachet
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA 02139 Present address: CNRS, Observatoire de Nice, 06-Nice, France.
Daniel I. Meiron
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA 02139
Steven A. Orszag
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA 02139
B. G. Nickel
Affiliation:
University of Guelph, Guelph, Ontario
Rudolf H. Morf
Affiliation:
R.C.A. Laboratories, Zurich, Switzerland
Uriel Frisch
Affiliation:
CNRS, Observatoire de Nice, 06-Nice, France
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Abstract

The dynamics of both the inviscid and viscous Taylor–Green (TG) three-dimensional vortex flows are investigated. This flow is perhaps the simplest system in which one can study the generation of small scales by three-dimensional vortex stretching and the resulting turbulence. The problem is studied by both direct spectral numerical solution of the Navier–Stokes equations (with up to 2563 modes) and by power-series analysis in time.

The inviscid dynamics are strongly influenced by symmetries which confine the flow to an impermeable box with stress-free boundaries. There is an early stage during which the flow is strongly anisotropic with well-organized (laminar) small-scale excitation in the form of vortex sheets located near the walls of this box. The flow is smooth but has complex-space singularities within a distance $\hat{\delta}(t)$ of real (physical) space which give rise to an exponential tail in the energy spectrum. It is found that $\hat{\delta}(t)$ decreases exponentially in time to the limit of our resolution. Indirect evidence is presented that more violent vortex stretching takes place at later times, possibly leading to a real singularity ($\hat{\delta}(t) = 0$) at a finite time. These direct integration results are consistent with new temporal power-series results that extend the Morf, Orszag & Frisch (1980) analysis from order t44 to order t80. Still, convincing evidence for or against the existence of a real singularity will require even more sophisticated analysis. The viscous dynamics (decay) have been studied for Reynolds numbers R (based on an integral scale) up to 3000 and beyond the time tmax at which the maximum energy dissipation is achieved. Early-time, high-R dynamics are essentially inviscid and laminar. The inviscidly formed vortex sheets are observed to roll up and are then subject to instabilities accompanied by reconnection processes which make the flow increasingly chaotic (turbulent) with extended high-vorticity patches appearing away from the impermeable walls. Near tmax the small scales of the flow are nearly isotropic provided that R [gsim ] 1000. Various features characteristic of fully developed turbulence are observed near tmax when R = 3000 and Rλ = 110:

  1. a k−n inertial range in the energy spectrum is obtained with n ≈ 1.6–2.2 (in contrast with a much steeper spectrum at earlier times);

  2. th energy dissipation has considerable spatial intermittency; its spectrum has a k−1+μ inertial range with the codimension μ ≈ 0.3−0.7.

Skewness and flatness results are also presented.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 130 , May 1983 , pp. 411 - 452
Copyright
© 1983 Cambridge University Press

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References

Baker, G. A. 1975 Essentials of Padé Approximants. Academic.
Baumel, R. T., Gammel, J. L. & Nuttall, J. 1981 J. Comp. Appl. Maths 7, 135.
Benettin, G., Galgani, L. & Strelcyn, J. M. 1976 Phys. Rev. A14, 2338.
Caffarelli, L., Kohn, R. & Nierenberg, L. 1982 Communs Pure Appl. Maths 35, 771.
Chang, Y. F., Tabor, M. & Weiss, J. 1982 J. Math. Phys. 23, 531.
Chorin, A. J. 1981 Communs Pure Appl. Maths 34, 853.
Fisher, M. E. & AU-YANG, M. 1979 J. Phys. A: Math. & Gen. 12, 1677.
Fisher, M. E. & AU-YANG, M. 1980 J. Phys. A: Math. & Gen. 13, 1517.
Frisch, U. 1983 Fully developed turbulence and singularities. In Proc. Les Houches Summer School 1981. North-Holland (to appear).
Frisch, U., Pouquet, A., Sulem, P.-L. & Meneguzzi, M. 1983 J. Méc. Théor. Appl. (to appear).
Frisch, U., Sulem, P.-L. & Nelkin, M. 1978 J. Fluid Mech. 87, 719.
Furth, H. P., Killeen, J. & Rosenbluth, M. N. 1963 Phys. Fluids 6, 459.
Grant, H. L., Stewart, R. W. & Moilliet, A. 1962 J. Fluid Mech. 12, 241.
Hénon, M. 1966 C.R. Acad. Sci. Paris 262, 322.
Hunter, D. L. & Baker, JR, G, A. 1979 Phys. Rev. B19, 3808.
Kahane, J. P. 1976 In Turbulence and Navier–Stokes Equation (ed. R. Temam). Lecture Notes in Mathematics, vol. 565, p. 94. Springer.
Kolmogorov, A. N. 1941 C.R. Acad. Sci. USSR 30, 301.
Kolmogorov, A. N. 1962 J. Fluid Mech. 13, 82.
Kuo, A. Y. & Corrsin, S. 1971 J. Fluid Mech. 50, 285.
Mandelbrot, B. 1976 In Turbulence and Navier–Stokes Equation (ed. R. Temam). Lecture Notes in Mathematics, vol. 565, p. 121. Springer.
Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics, vol. 2. M.I.T. Press.
Morf, R. H., Orszag, S. A. & Frisch, U. 1980 Phys. Rev. Lett. 44, 572.
Morf, R. H., Orszag, S. A., Meiron, D. I., Frisch, U. & Meneguzzi, M. 1981 In Proc. 7th Intl Conf. on Numerical Methods in Fluid Dynamics (ed. R. W. MacCormack & W. C. Reynolds). Lecture Notes in Physics, vol. 141, p. 292. Springer.
Orszag, S. A. 1971 Stud. Appl. Maths 50, 293.
Orszag, S. A. 1974 In Proc. Symp. on Computing Methods in Applied Sciences and Engineering (ed. R. Glowinski & J. L. Lions), part II, p. 50. Springer.
Orszag, S. A. 1976 In Computer Science and Scientific Computing (ed. J. Ortega), p. 191. Academic.
Orszag, S. A. 1977 Statistical theory of turbulence. In Fluid Dynamics, Les Houches Summer School 1973 (ed. R. Balian & J.-L. Peube), p. 237. Gordon & Breach.
Saffman, P. G. 1971 Stud. Appl. Maths 50, 377.
Scheffer, V. 1976 In Turbulence and Navier–Stokes Equation (ed. R. Temam). Lecture Notes in Mathematics, vol. 565, p. 174. Springer.
Siggia, E. 1981 J. Fluid Mech. 107, 375.
Sulem, C., Sulem, P.-L. & Frisch, H. 1983 J. Comp. Phys. (to appear).
Taylor, G. I. & Green, A. E., 1937 Proc. R. Soc. Lond. A 158, 499.
Wolibner, W. 1933 Math. Z. 37, 698.