Turbulence, Coherent Structures, Dynamical Systems and Symmetry ( Cambridge Monographs on Mechanics )

Publication series ：Cambridge Monographs on Mechanics

Author： Philip Holmes; John L. Lumley; Gal Berkooz

Publisher： Cambridge University Press‎

Publication year： 1996

E-ISBN: 9780511888670

P-ISBN(Paperback): 9780521551427

Subject： O357.5 turbulence (turbulence)

Keyword：物理学

Language： ENG

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Turbulence, Coherent Structures, Dynamical Systems and Symmetry

Description

For turbulent flows at relatively low speeds there exists an excellent mathematical model in the incompressible Navier–Stokes equations. Why then is the 'problem of turbulence' so difficult? One reason is that these nonlinear partial differential equations appear to be insoluble, except through numerical simulations, which offer useful approximations but little direct understanding. Three recent developments offer new hope. First, the discovery by experimentalists of coherent structures in certain turbulent flows. Secondly, the suggestion that strange attractors and other ideas from finite-dimensional dynamical systems theory might play a role in the analysis of the governing equations. And, finally, the introduction of the Karhunen-Loève or proper orthogonal decomposition. This book introduces these developments and describes how they may be combined to create low-dimensional models of turbulence, resolving only the coherent structures. This book will interest engineers, especially in the aerospace, chemical, civil, environmental and geophysical areas, as well as physicists and applied mathematicians concerned with turbulence.

Chapter

1.2 Low-dimensional models

1.3 The contents of this book

1.4 Notation and mathematical jargon

2 Coherent structures

2.1 Introduction

2.2 Flows with coherent structures

2.3 Detection of coherent structures

2.4 The mixing layer

2.5 The turbulent boundary layer

2.6 A preview of things to come

3 Proper orthogonal decomposition

3.1Introduction

3.2 On domains and averaging

3.3 Properties of the POD

3.3.1 Span of the empirical basis

3.3.2 Optimality

3.3.3 Symmetry

3.3.4 Attractors

3.4 Further results

3.5 Stochastic estimation

3.6 Coherent structures and homogeneity

3.7 Some applications

3.7.1 Wall bounded flows

3.7.2 Free shear flows

3.7.3 Rayleigh-Benard convection

3.7.4 Model problems

3.8 Appendix: some foundations

3.8.1 Probability measures

3.8.2 Compactness of 91

3.8.3 Symmetry and invariant subspaces

3.8.4 Spectral decay and approximate compactness

4 Galerkin projection

4.1 Introduction

4.2 Some simple PDEs revisited

4.3 The Navier-Stokes equations

4.4 Towards low-dimensional models

Part two: Dynamical systems

5 Qualitative theory

5.1 Linearisation and invariant manifolds

5.2 Periodic orbits and Poincare maps

5.3 Structural stability and genericity

5.4 Bifurcations local and global

5.5 Attractors simple and strange

6 Symmetry

6.1 Equivariant vector fields

6.2 Local bifurcation with symmetry

6.3 Global behavior with symmetry

6.4 An O(2)-equivariant ODE

7 One-dimensional "turbulence"

7.1 Projection onto Fourier modes

7.2 Local bifurcations from u = 0

7.3 The second bifurcation point

7.4 Spatio-temporal chaos

8 Randomly perturbed systems

8.1 An Ornstein-Uhlenbeck process

8.2 Noisy heteroclinic cycles

8.3 Power spectra of homoclinic attractors

8.4 Symmetry breaking

Part three: The boundary layer

9 Low-dimensional models

9.1 Equations for coherent structures

9.2 The eigenfunction expansion

9.3 Symmetries

9.4 Galerkin projection

9.5 Geometrical structure of the model

9.6 Choosing subspaces and domains

9.7 The energy budget

9.7.1 The ratio {u1u2)/{uiui)

9.7.2 The mean velocity profile

9.8 Non-linear feedback

9.9 Interaction with unresolved modes

10 Behavior of the models

10.1 Backbones for the models

10.2 Heteroclinic cycles

10.3 Bursts and sweeps

10.4 The pressure term

10.5 More modes and instabilities

10.6 A tentative summary

10.7 Appendix: coefficients

Part four: Other applications and related work

11 Some other fluid problems

11.1 The circular jet

11.2 The transitional boundary layer

11.3 A forced transitional mixing layer

11.4 Flows in complex geometries

11.5"Full channel" wall layer models

11.6 Discussion

12.Review: prospects for rigor

12.1 The quality of models

12.2 A short-time tracking estimate

12.3 Stability, simulations, and statistics

12.4 Spatial localisation

12.5 The utility of models

Bibliography

Index

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