Elservier, Physics Reports 287, 1997. р. 337-384
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; B) 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 goveing equations, and C) the introduction of the statistical technique of Karhunen-Loeve 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.
Introduction
Turbulence and coherent structures
The dynamical systems paradigm
The proper orthogonal decomposition
Derivation of empirical eigenfunctions
Optimality
Symmetries in the POD
Approximating attractors
Representation of boundary layer flows
Symmetries of the empirical eigenfunctions
The modal expansion
Projection of the Navier-Stokes equations and modeling
Reynolds decomposition
The mean flow
Losses to neglected modes and normalization
Galerkin projection
The pressure term
Structure and some solutions of the models
Choice of truncation
Symmetries
Behavior of the models
Modeling of other open flows
A circular jet
A transitional boundary layer
A forced transitional mixing layer
Two flows in complex geometries
Discussion
Symmetry: translations, reflections, and O(2)-equivariance
Equivariant ODEs and heteroclinic cycles O(2) and normal forms
The Kuramoto-Sivashinsky equation
Galerkin projection
Bifurcation and center manifold reduction
Perturbed heteroclinic cycles, timing and experimental observations
The noisy connection
Noise and the boundary layer
Conclusion
References
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; B) 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 goveing equations, and C) the introduction of the statistical technique of Karhunen-Loeve 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.
Introduction
Turbulence and coherent structures
The dynamical systems paradigm
The proper orthogonal decomposition
Derivation of empirical eigenfunctions
Optimality
Symmetries in the POD
Approximating attractors
Representation of boundary layer flows
Symmetries of the empirical eigenfunctions
The modal expansion
Projection of the Navier-Stokes equations and modeling
Reynolds decomposition
The mean flow
Losses to neglected modes and normalization
Galerkin projection
The pressure term
Structure and some solutions of the models
Choice of truncation
Symmetries
Behavior of the models
Modeling of other open flows
A circular jet
A transitional boundary layer
A forced transitional mixing layer
Two flows in complex geometries
Discussion
Symmetry: translations, reflections, and O(2)-equivariance
Equivariant ODEs and heteroclinic cycles O(2) and normal forms
The Kuramoto-Sivashinsky equation
Galerkin projection
Bifurcation and center manifold reduction
Perturbed heteroclinic cycles, timing and experimental observations
The noisy connection
Noise and the boundary layer
Conclusion
References