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

Vortex Convection Manifold Vorticity Actual Element 

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References

  1. [1]
    R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematics Series, Vol. 68 (Springer, Berlin, 1988).MATHGoogle Scholar
  2. [2]
    C. Foias, G. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, 73 (1988), pp. 309–353.MathSciNetMATHGoogle 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).MATHGoogle 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

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