Low-dimensional models of coherent structures in turbulence

Low-dimensional models of coherent structures in turbulence For fluid flow one has a well-accepted mathematical model: the Navier-Stokes equations. Why, then, is the problem of turbulence so intractable? One major difficulty is that the equations appear insoluble in any reasonable sense. (A direct numerical simulation certainly yields a “solution”, but it provides little understanding of the process per se .) However, three developments are beginning to bear fruit: (1) The discovery, by experimental fluid mechanicians, of coherent structures in certain fully developed turbulent flows; (2) the suggestion, by Ruelle, Takens and others, that strange attractors and other ideas from dynamical systems theory might play a role in the analysis of the governing equations, and (3) the introduction of the statistical technique of Karhunen-Loève or proper orthogonal decomposition, by Lumley in the case of turbulence. Drawing on work on modeling the dynamics of coherent structures in turbulent flows done over the past ten years, and concentrating on the near-wall region of the fully developed boundary layer, we describe how these three threads can be drawn together to weave low-dimensional models which yield new qualitative understanding. We focus on low wave number phenomena of turbulence generation, appealing to simple, conventional modeling of inertial range transport and energy dissipation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Physics Reports Elsevier

Low-dimensional models of coherent structures in turbulence

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Abstract

For fluid flow one has a well-accepted mathematical model: the Navier-Stokes equations. Why, then, is the problem of turbulence so intractable? One major difficulty is that the equations appear insoluble in any reasonable sense. (A direct numerical simulation certainly yields a “solution”, but it provides little understanding of the process per se .) However, three developments are beginning to bear fruit: (1) The discovery, by experimental fluid mechanicians, of coherent structures in certain fully developed turbulent flows; (2) the suggestion, by Ruelle, Takens and others, that strange attractors and other ideas from dynamical systems theory might play a role in the analysis of the governing equations, and (3) the introduction of the statistical technique of Karhunen-Loève or proper orthogonal decomposition, by Lumley in the case of turbulence. Drawing on work on modeling the dynamics of coherent structures in turbulent flows done over the past ten years, and concentrating on the near-wall region of the fully developed boundary layer, we describe how these three threads can be drawn together to weave low-dimensional models which yield new qualitative understanding. We focus on low wave number phenomena of turbulence generation, appealing to simple, conventional modeling of inertial range transport and energy dissipation.

Journal

Physics ReportsElsevier

Published: Aug 1, 1997

References

  • Turbulence, coherent structures, and low dimensional models
    Berkooz, G.
  • The proper orthogonal decomposition in the analysis of turbulent flows
    Berkooz, G.; Holmes, P.; Lumley, J.L.
  • Suppression of bursting
    Coller, B.D.; Holmes, P.
  • Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields
    Guckenheimer, J.; Holmes, P.
  • A mathematical example displaying the features of turbulence
    Hopf, E.
  • A low dimensional Galerkin method for the three-dimensional flow around a circular cylinder
    Noack, B.R.; Eckelmann, H.
  • Lagrangian and Eulerian view of the bursting period
    Podvin, B.; Gibson, J.; Berkooz, G.; Lumley, J.
  • On the structure of dynamical systems describing the evolution of coherent structures in a convective boundary layer
    Rempfer, D.
  • Coherent motions in the turbulent boundary layer
    Robinson, S.K.
  • Coherence and chaos in a model of turbulent boundary layer
    Zhou, X.; Sirovich, L.

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