Zeitschrift für Physik B Condensed Matter

, Volume 100, Issue 3, pp 461–468 | Cite as

On a Fourier space master equation for Navier-Stokes turbulence

  • Heinz-Peter Breuer
  • Francesco Petruccione
  • Frithjof Weber


Three-dimensional homogeneous isotropic turbulence is formulated in terms of a discrete stochastic process in Fourier space. The time-dependent joint probability distribution of the stochastic Fourier modes is governed by a multivariate master equation. It is demonstrated that the characteristic functional of the stochastic process obeys the Hopf functional equation. As a first application of the method stochastic simulations of the Burgers's turbulence model are performed and shown to yield the expected energy spectrum.


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  1. 1.
    Monin,A.S., Yaglom, A.M.: Statistical Fluid Mechanics: Mechanics of Turbulence Vol. 2. Cambridge: The MIT Press 1981Google Scholar
  2. 2.
    Me Comb, W.D.: The Theory of Fluid Turbulence. Oxford: Clarendon 1990Google Scholar
  3. 3.
    Hopf, E.: J. Rat. Mech. Anal. 1, 87 (1952)Google Scholar
  4. 4.
    Binder, K.: Monte Carlo Methods in Statistical Physics. Berlin: Springer 1979Google Scholar
  5. 5.
    Breuer, H.P., Honerkamp, J., Petruccione, F.: Computational Polymer Science 1, 233 (1991)Google Scholar
  6. 6.
    Breuer, H.P., Petruccione, F.: Continuum Mech. Thermodyn. in pressGoogle Scholar
  7. 7.
    Bräunl, T.: Parallele Programmierung. Braunschweig/Wiesbaden: Vieweg 1993Google Scholar
  8. 8.
    Breuer, H.P., Petruccione, F.: Phys. Rev. E 50, 2795 (1994)Google Scholar
  9. 9.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. New York: Springer 1988Google Scholar
  10. 10.
    Herring, J.R., Orszag, S.A., Kraichnan, R.H., Fox, D.G.: J. Fluid Mech. 66, 417 (1974)Google Scholar
  11. 11.
    Stanišić, M.M.: The Mathematical Theory of Turbulence. New York: Springer 1985Google Scholar
  12. 12.
    Hopf, E.: Proc. Symp. Appl. Math. 13, 157 (1952)Google Scholar
  13. 13.
    Hopf, E.: Proc. Symp. Appl. Math. 7, 41 (1957)Google Scholar
  14. 14.
    Breuer, H.P., Petruccione, F.: J. Phys. A: Math. Gen. 26, 7563 (1993)Google Scholar
  15. 15.
    Breuer, H.P., Petruccione, F.: Phys. Rev. E 47, 1803 (1993)Google Scholar
  16. 16.
    van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. Amsterdam: Elsevier 1992Google Scholar
  17. 17.
    Burgers, J.M. in: Ehlers, J., Hepp, K. and Weidenmüller, H. A. (ed.): Statistical Methods and Turbulence, Lecture Notes in Physics Vol. 12. Berlin: Springer 1972Google Scholar
  18. 18.
    Gillespie, T.D.: Markov Processes. San Diego: Academic Press 1992Google Scholar
  19. 19.
    Honerkamp, J.: Stochastic Dynamical Systems. New York: VCH 1993Google Scholar
  20. 20.
    Saffman, P.G. in: Zabusky, N.J. (ed.): Topics in Nonlinear Physics. Berlin: Springer 1968Google Scholar
  21. 21.
    Jeng, D.T., Meecham, W.C.: Phys. Fluids 15, 504 (1972)Google Scholar
  22. 22.
    Kraichnan, R.H.: Phys. Fluids 10, 1417 (1967)Google Scholar
  23. 23.
    Batchelor, G.K.: Phys. Fluids Suppl. II, 233 (1969)Google Scholar
  24. 24.
    Whitham, G.B.: Linear and nonlinear waves. New York: John Wiley & Sons 1974Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Heinz-Peter Breuer
    • 1
  • Francesco Petruccione
    • 1
  • Frithjof Weber
    • 1
  1. 1.Fakultät für PhysikAlbert-Ludwigs-Universität FreiburgFreiburgGermany

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