Journal of Statistical Physics

, Volume 55, Issue 5–6, pp 1303–1312 | Cite as

Turbulence without strange attractor

  • U. Brosa
Short Communications


It is shown that pipe-flow turbulence consists of transients. The “fractal” dimensions of the dynamical process are thus all zero. Nevertheless, this is compatible with Grassberger-Procaccia analyses suggesting the existence of a high-dimensional strange attractor. The usefulness of the Grassberger-Procaccia method to detect the aging of transients is demonstrated.

Key words

Chaos fractal dimensions intermittency Navier-Stokes equation pipe flow strange attractor turbulence 


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  1. 1.
    P. Grassberger, Information content and predictablity of lumped and distributed dynamical systems, inProceedings of the Conference on Chaos and Related Non-linear Phenomena, Kiryat Anavim, December 1986, preprint University of Wuppertal WU V 87-8.Google Scholar
  2. 2.
    H. G. Schuster,Deterministic Chaos (VCH Verlagsgesellschaft, Weinheim, 1988), Chapter 5.Google Scholar
  3. 3.
    P. Grassberger and I. Procaccia,Physica 9D:189 (1983).Google Scholar
  4. 4.
    H. L. Swinney and J. P. Gollup, eds.,Hydrodynamic Instabilities and the Transition to Turbulence (Springer, New York, 1985).Google Scholar
  5. 5.
    M. Nishioka, S. Ida, and Y. Ichikawa,J. Fluid Mech. 72:731 (1975).Google Scholar
  6. 6.
    L. Boberg and U. Brosa,Z. Naturforsch. 43a:697 (1988).Google Scholar
  7. 7.
    J. Guckenheimer and G. Buzyna,Phys. Rev. Lett. 51:1438 (1983).Google Scholar
  8. 8.
    B. Malraison, P. Atten, P. Berge, and M. Dubois,J. Phys. Lett. (Paris)44:L-897 (1983).Google Scholar
  9. 9.
    M. Giglio, S. Musazzi, and U. Perini,Phys. Rev. Lett. 53:2402 (1984).Google Scholar
  10. 10.
    A. Brandstäter, J. Swift, H. L. Swinney, A. Wolf, J. D. Framer, E. Jen, and P. J. Crutchfield,Phys. Rev. Lett. 51:1442 (1983).Google Scholar
  11. 11.
    A. Brandstäter, H. L. Swinney, and G. T. Chapman, inDimensions and Entropies in Chaotic Systems, G. Mayer-Kress, ed. (Springer, Berlin, 1986), p. 150.Google Scholar
  12. 12.
    M. Sieber,Phys. Lett. A 122:467 (1987).Google Scholar
  13. 13.
    C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang,Spectral Methods in Fluid Mechanics (Springer, New York, 1988).Google Scholar
  14. 14.
    O. Reynolds,Phil. Trans. 174:935 (1883).Google Scholar
  15. 15.
    I. J. Wygnanski and F. H. Champagne,J. Fluid Mech. 59:281 (1973).Google Scholar
  16. 16.
    J. Meseth,Arch. Mech. 26:391 (1974).Google Scholar
  17. 17.
    F. Takens, inDynamical Systems and Turbulence, D. A. Rand and L. S. Young, eds. (Springer, Berlin, 1981), p. 366.Google Scholar
  18. 18.
    O. E. Lanford, inHydrodynamic Instabilities and the Transition to Turbulence, H. L. Swinney and J. P. Gollup, eds. (Springer, New York, 1985).Google Scholar

Copyright information

© Plenum Publishing Corporation 1989

Authors and Affiliations

  • U. Brosa
    • 1
  1. 1.HLRZ c/o KFA Jülich GmbHJülichWest Germany

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