Advertisement

Turbulent Bursts, Inertial Sets and Symmetry-Breaking Homoclinic Cycles in Periodic Navier-Stokes Flows

  • Basil Nicolaenko
  • Zhen-Su She
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 55)

Abstract

We investigate bursting regimes of two-dimensional Kolmogorov flows. We link these dynamics with symmetry-breaking heteroclinic connections which generate persistent homoclinic cycles. Small-scale turbulent dynamics prevail in a neighborhood of these heteroclinic connections, while large-scale dynamics are associated to hyperbolic tori. These intermittent turbulent regimes are a prime example of dynamics on an inertial set (or exponential attractor) of the Navier-Stokes equations.

Keywords

Maximal Subgroup Stable Manifold Global Attractor Heteroclinic Cycle Laminar Regime 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematics Series, Vol. 68 (Springer, Berlin, 1988).zbMATHGoogle Scholar
  2. [2]
    C. Foias, G. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, 73 (1988), pp. 309–353.MathSciNetzbMATHGoogle Scholar
  3. [3]
    P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Appl. Math. Sciences, no. 70 (Springer, New York, 1988).Google Scholar
  4. [4]
    A. Eden, C. Foias, B. Nicolaenko and R. Temam, Inertial sets for dissipative evolution equations, monograph in preparation.Google Scholar
  5. [5]
    A. Eden, C. Foias, B. Nicolaenko and R. Temam, Inertial sets for dissipative evolution equations, IMA Preprint no. 694 (1990).Google Scholar
  6. [6]
    Meshalkin, L.D. & Sinai, Ya. G., J. Appl. Math., (PMM), 25, 1700 (1961).zbMATHGoogle Scholar
  7. [7]
    Nepomnyachtchyi, A.A., Prikl. Math. Makh., 40 (5), 886 (1976).Google Scholar
  8. [8]
    Sivashinsky, G.I., Physica, 17D, 243 (1985).MathSciNetGoogle Scholar
  9. [9]
    She, Z.-S, Proc. on Current trends in turbulence research, AIAA, 1988.Google Scholar
  10. [10]
    She, Z.-S., Phys. Lett. A, 124, 161 (1987).MathSciNetCrossRefGoogle Scholar
  11. [11]
    Nicolaenko, B. and She, Z.-S., Temporal Intermittency and Turbulence Production in the Kolmogorov Flow, in Topological Fluid Mechanics, Cambridge Univ. Press, 1990, pp. 256–277.Google Scholar
  12. [12]
    Constantin, P. and Foias, C., Navier Stokes Equations, Univ. of Chicago Lectures in Mathematics, 1989, pp. 256–277.Google Scholar
  13. [13]
    Kevrekidis, I., Nicolaenko, B. and Scovel, C., Back in the Saddle Again: A Computer Assisted Study of the Kuramoto-Sivashinsky Equation, SIAM J. Appl. Math., Vol. 90, 3 pages 760–790 (1990).MathSciNetCrossRefGoogle Scholar
  14. [14]
    Armbruster, D. Guckenheimer, J. & Holmes, Ph.,, Physics D., (1989).Google Scholar
  15. [15]
    Melbourne, I., Chossat, P. and Golubitsky, M., Heteroclinic Cycles involving Periodic Solutions in Mode Interactions with O(2) Symmetry, to appear; Also, Armbruster, D. and Chossat, P., Heteroclinic cycles in Mode Interaction with O(3) Symmetry, to appear.Google Scholar
  16. [16]
    Guckenheimer, J., Square Symmetry in Binary Convection, to appear.Google Scholar
  17. [17]
    Golubitsky, M., Stewart, I., Schaeffer, D.G., Singularities and Groups in Bifurcation Theory, Volume II, Springer-Verlag Ed., 1988.Google Scholar
  18. [18]
    Aubry, N., Holmes, P., Lumley, J.L. & Stone, E., J. Fluid Mech., 192, 112 (1987).MathSciNetGoogle Scholar
  19. [19]
    Newell, A., Rand, D., Physics Letters, A (1988).Google Scholar
  20. [20]
    Eden, A., Foias, C. Nicolaenko, B. and She, Z.S., Exponential Attractors and their Relevance to Fluid Dynamics Systems, submitted to Physica D.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1993

Authors and Affiliations

  • Basil Nicolaenko
    • 1
    • 2
  • Zhen-Su She
    • 3
  1. 1.Department of MathematicsArizona State UniversityTempeUSA
  2. 2.Center for Nonlinear StudiesLos Alamos National LaboratoryLos AlamosUSA
  3. 3.Applied and Computational MathematicsPrinceton UniversityPrincetonUSA

Personalised recommendations