Empirical Eigenfunctions and Low Dimensional Systems

  • Lawrence Sirovich
Conference paper


In the course of this lecture I hope to cover a wide and diverse range of topics which are relevant to the study of near chaotic, chaotic and turbulent flows. A thread which joins these topics is the Karhunen-Loève (K-L) procedure for the generation of the empirical eigenfunctions. Lumley introduced this procedure into turbulence theory1 and suggested that it might be used to unambiguously extract coherent structures in a turbulent flow. This idea will be one of the topics to be touched on later.


Nusselt Number Coherent Structure Ginzburg Landau Equation Inertial Manifold High Rayleigh Number 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Lumley, J.L., The structure of inhomogeneous turbulent flows, In: Atmospheric Turbulence and Radio Wave Propagation (A.M. Yaglom and V.I. Tatarski, eds.) 166–178, Moscow: Nauka, (1967).Google Scholar
  2. 2.
    Lumley, J.L., Stochastic Tools in Turbulence, Academic Press, N.Y., (1970).zbMATHGoogle Scholar
  3. 3.
    Lumley, J.L., Coherent structures in turbulence, In: Transition and Turbulence, (R.E. Meyer, ed.), 215–242, Academic Press, N.Y., (1981).Google Scholar
  4. 4.
    Preisendorfer, R.W. Principal Component Analysis in Meteorology and Oceanography, Elsevier (1988)Google Scholar
  5. 5.
    Ash, R.B. and M.F. Gardner, Topics in Stochastic Processes, Academic Press, NY, (1975).zbMATHGoogle Scholar
  6. 6.
    Devijver, P.A. and J. Kittler Pattern Recognition: A Statistical Approach Prentice/Hall International (1982).Google Scholar
  7. 7.
    Breuer, K. and L. Sirovich The use of the Karhunen-Loève procedure for the calculation of linear eigenfunctions, Jour. Comp. Phys.(to appear 1991).Google Scholar
  8. 8.
    Kim, J., P. Moin and R. Moser, Turbulence statistics in fully developed channel flow at low Reynolds number, Jour. Flu. Mech., 177, 133–166 (1987).zbMATHCrossRefGoogle Scholar
  9. 9.
    Drazin, P.G. and W.H. Reid, Hydrodynamic Stability, Cambridge University Press, (1981).Google Scholar
  10. 10.
    Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, (Oxford University Press). (1961).Google Scholar
  11. 11.
    L. Sirovich, S. Balachandar and M. Maxey, Simulations of turbulent thermal convection, Phys. Fluids A, 1, 1911–1914, (1989).CrossRefGoogle Scholar
  12. 12.
    S. Balachandar, M. Maxey and L. Sirovich, Direct numerical simulation of high Rayleigh number turbulent thermal convection, Jour. Sci. Comp., 4, No. 2, (1989).Google Scholar
  13. 13.
    Malkus, W.V.R., Discrete transitions in turbulent convection, Proc. Roy. Soc. of London, Ser. A, 225, 185–212 (1954).MathSciNetCrossRefGoogle Scholar
  14. 14.
    Priestly, C.H.B., Turbulent Heat Transfer in the Lower Atmosphere, Univ. Chicago Press (1959).Google Scholar
  15. 15.
    Castaing, B., G. Gunaratne, F. Heslot, L. Kadanoff, A. Libschaber, S. Thomae, X-Z. Wu, S. Zaleski and G. Zanetti, Scaling of hard thermal turbulence in Rayleigh-Bénard Convection, (submitted for publication).Google Scholar
  16. 16.
    Heslot, F., B. Castaing and A. Libschaber, Transitions in helium gas,, Phys. Rev. A36, 5870–5873 (1987).Google Scholar
  17. 17.
    Landau, L.D., Turbulence, Doklady AN SSR, 44, 339–342, (1944).Google Scholar
  18. 18.
    Constantin P., C. Foiaş, O.P. Manley and R. Temam, Determining modes and fractal dimension of turbulent flows, J. Fluid Mech. 150, 427–440 (1985).MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Bergé, P., Y. Pomeau and C. Vidal, Order in Chaos, John Wiley and Sons (1984).Google Scholar
  20. 20.
    Schuster, G.S., Deterministic Chaos: An Introduction, Physik-Verlag, Weinheim, FRG, (1984).zbMATHGoogle Scholar
  21. 21.
    Kaplan, J.L. and J.A. Yorke, Chaotic behavior of in multi-dimensional difference equations In Lecture Notes in Math, 730, 204, Springer Verlag (1978).MathSciNetCrossRefGoogle Scholar
  22. 22.
    Wolf, A., J. B. Swift, H.L. Swinney and J.A. Vastano, Determining Lyaponuv exponents from a time series, Physica 16D, p. 285–317, (1985).MathSciNetGoogle Scholar
  23. 23.
    Keefe, L., Comparison of calculated and predicted forms for Lyapunov spectral of Navier-Stokes Equations, Bull. Am. Phys. Soc., 34, 2296 (1989).Google Scholar
  24. 24.
    Deane, A. and L. Sirovich, A computational study of Rayleigh-Bénard convection Part 1: Rayleigh number dependence Jour. Flu. Mech. 222, 231 (1990).MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sirovich, L. and A. Deane, A computational study of Rayleigh-Bénard convection Part 2: dimension considerations, Jour. Flu. Mech. 222, 251 (1990).MathSciNetCrossRefGoogle Scholar
  26. 26.
    Sirovich, L., M. Maxey and H. Tarman, An Eigenfunction Analysis of Turbulent Thermal Convection, Post-conference Proceedings (editor, B. Launder) Springer (1988).Google Scholar
  27. 27.
    Sirovich, L., M. Maxey and H. Tarman, Analysis of turbulent thermal convection, Proc. 6th Symposium on Turbulent Shear Flow (1987).Google Scholar
  28. 28.
    Tarman, H. and L. Sirovich, An analysis of turbulent thermal convection, (submitted) (1989).Google Scholar
  29. 29.
    Deane, A., and L. Sirovich, Lyapunov Dimension of Rayleigh-Bénard Convection (The Forum on Chaotic Flows Third Joint ASCE/ASME Mechanics Conference, July 1989) (to appear).Google Scholar
  30. 30.
    Kraichnan, R., Turbulent thermal convection at arbitrary Prandlt number, Phys. Flu. 5, 1374–1389 (1962).CrossRefGoogle Scholar
  31. 31.
    Payne, F.R., and J.L. Lumley, Large eddy structure of the turbulent wake behind a circular cylinder, Phys. Fluids 10, S194–S196, (1967).CrossRefGoogle Scholar
  32. 32.
    Glauser, M.N., S.J. Lieb and W.K. George, Coherent structure in an axisymmetric jet mixing layer, Proc. 5th Symp. Turb. Shear Flow, Cornell University, Springer Verlag (1985).Google Scholar
  33. 33.
    Moser, R.D., P. Moin and A. Leonard, A spectral numerical method for the Navier-Stokes equations with applications to Taylor-Couette flow, J. Comput. Phys. 52, No. 3, 524–544 (1983).MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    K.S. Ball, L. Sirovich and L.R. Keefe, Dynamical eigenfunction decomposition of turbulent channel flow, International Journal for Numerical Methods in Fluids (To appear, 1990).Google Scholar
  35. 35.
    Sirovich, L., Turbulence and the dynamics of coherent structures, Pt. I: Coherent Structures. Quar. Appl. Math., Vol XLV, No. 3, 561–571; Turbulence and the dynamics of coherent structures, Pt. II: Symmetries and transformations, Quar. Appl. Math., Vol. XLV, No. 3, 573–582; Turbulence and the dynamics of coherent structures, Pt. III: Dynamics and scaling, Quar. Appl. Math., Vol. XLV, No. 3, 583–590, (1987).Google Scholar
  36. 36.
    Sirovich, L. and H. Park, Turbulent Thermal Convection in a Finite Domain: Theory, Phys. Flu. A, 2, 1649 (1990).MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Park, H. and L. Sirovich, Turbulent Thermal Convection in a finite domain: Numerical experiments and results,, Phys. Flu. A, 2, 1659 (1990).MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Sirovich, L., M. Kirby and M. Winter, An eigenfunction approach to large scale transitional structures in jet flow, Phys. Fluids A 2 127–136 (1990).CrossRefGoogle Scholar
  39. 39.
    Sirovich, L., and J.D. Rodriguez, Coherent structures and chaos: A model problem, Physics Letters A, 120, 211 (1987).MathSciNetCrossRefGoogle Scholar
  40. 40.
    Sirovich, L., J.D. Rodriguez and B. Knight, Two boundary value problem for Ginzburg Landau equation, Physica D 43, 63 (1990).MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    J.D. Rodriguez and L. Sirovich, Low dimensional dynamics for the complex Ginzburg Landau Equation, Physica D 43, 77 (1990).MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Sirovich, L., Chaotic dynamics of coherent structures, Physica D 37, 126–145 (1989).MathSciNetCrossRefGoogle Scholar
  43. 43.
    Tarman, H., Analysis of turbulent thermal convection, Thesis, Brown University (1989).Google Scholar
  44. 44.
    Aubry, N., P. Holmes, J.L. Lumley, and E. Stone, The dynamics of coherent structures in the wall region of a turbulent boundary layer, J. Flu. Mech. 192, 115–173 (1988).MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Haken, H., Synergetics, 3rd Edition, Springer-Verlag, (1983).Google Scholar
  46. 46.
    van Kampen, N.G., Elimination of Fast Variables, Phys. Rep. 124, 69–160, (1985).MathSciNetCrossRefGoogle Scholar
  47. 47.
    Foias, C., G.R. Sella and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J.D.E. 73, 309–353, (1988).zbMATHCrossRefGoogle Scholar
  48. 48.
    Mallet-Paret, J. and G.R. Sell, Inertial manifolds for reaction-diffusion equations in higher space dimensions, Jour. A.M.S., Volume 1, Number 4, 805–866 (1988).MathSciNetGoogle Scholar
  49. 49.
    Constantin, P., C. Foias, B. Nicolenko and R. Temam, Integral manifolds and inertial manifolds for dissipative partial differential equations, Applied Math. Sci., Springer Verlag (1989).Google Scholar
  50. 50.
    Constantin, P., Remarks on the Navier-Stokes Equations (These proceeding).Google Scholar
  51. 51.
    Sirovich, L., B.W. Knight and J.D. Rodriguez, Optimal low dimensional dynamical approximations, Quar. Appl. Math 48, 535 (1990).MathSciNetzbMATHGoogle Scholar
  52. 52.
    Titi, E., On approximate inertial manifolds to the Navier-Stokes equations, Math. Sci. Inst. Rep. (Cornell), (1989).Google Scholar
  53. 53.
    Foias, D., O.P. Manley and R. Teman, Sur l’interaction des petits et grands tourbillars dans les écoulements turbulents, C.R. Acad. Sci. Paris, Serie I, 305, 497–500, (1987).zbMATHGoogle Scholar
  54. 54.
    Riesz, F. and B. Sz. Nagy, Functional Analysis, Ungar, N.Y., 1955.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1991

Authors and Affiliations

  • Lawrence Sirovich

There are no affiliations available

Personalised recommendations