Chapter
2.3 Detection of coherent structures
2.5 The turbulent boundary layer
2.6 A preview of things to come
3: Proper orthogonal decomposition
3.1.1 Finite dimensional spaces
3.1.2 Scalar-valued functions
3.1.3 Vector-valued functions
3.1.4 Technical properties of the POD
3.2 On domains and averaging
3.3 Properties of the POD
3.3.1 Span of the empirical basis
3.4.1 Method of snapshots
3.4.2 Relationship to singular value decomposition
3.4.3 On inner products for compressible flows
3.4.4 On using an empirical basis over a parameter range
3.5 Stochastic estimation
3.6 Coherent structures and homogeneity
3.7.3 Rayleigh–Bénard convection
3.8 Appendix: some foundations
3.8.1 Probability measures
3.8.3 Symmetry and invariant subspaces
3.8.4 Spectral decay and approximate compactness
4.2 Some simple PDEs revisited
4.3 The Navier–Stokes equations
4.4 Towards low-dimensional models
5: Balanced proper orthogonal decomposition
5.4 Connections with standard POD
5.4.1 Non-orthogonal projection
5.4.2 Observability Gramian as an inner product
5.4.3 Guaranteed stability
5.5 Extensions of balanced POD
5.5.3 Adjoint-free balancing
5.6.1 Example 1: a non-normal ODE
5.6.2 Example 2: a one-dimensional PDE
5.6.3 Example 3: linearized channel flow
PART TWO: Dynamical systems
6.1 Linearization and invariant manifolds
6.2 Periodic orbits and Poincaré maps
6.3 Structural stability and genericity
6.4 Bifurcations local and global
6.5 Attractors simple and strange
7.1 Equivariant vector fields
7.2 Local bifurcation with symmetry
7.3 Global behavior with symmetry
7.4 An O(2)-equivariant ODE
8: One-dimensional "turbulence''
8.1 Projection onto Fourier modes
8.2 Local bifurcations from u = 0
8.3 The second bifurcation point
8.4 Spatio-temporal chaos
9: Randomly perturbed systems
9.1 An Ornstein–Uhlenbeck process
9.2 Noisy heteroclinic cycles
9.3 Power spectra of homoclinic attractors
PART THREE: The boundary layer
10: Low-dimensional models
10.1 Equations for coherent structures
10.2 The eigenfunction expansion
10.5 Geometrical structure of the model
10.6 Choosing subspaces and domains
10.7.1 The ratio (u1u2) /(uiui)
10.7.2 The mean velocity profile
10.9 Interaction with unresolved modes
11: Behavior of the models
11.1 Backbones for the models
11.5 More modes and instabilities
11.7 Appendix: coefficients
PART FOUR: Other applications and related work
12: Some other fluid problems
The flow and large-scale phenomena of interest
The POD procedure and key results
Choice of basis functions and resulting equations
12.2 The transitional boundary layer
The flow and large-scale phenomena of interest
The POD procedure and key results
Choice of basis functions and resulting equations
12.3 A forced transitional mixing layer
The flow and large-scale phenomena of interest
The POD procedure and key results
Choice of basis functions and resulting equations
12.4 Flows in complex geometries
The flows and large-scale phenomena of interest
The POD procedure and key results
Choice of basis functions and resulting equations
12.5 "Full channel'' wall layer models
The flow and large-scale phenomena of interest
The POD procedure and key results
Choice of basis functions and resulting equations
12.6 Flows in internal combustion engines
A phase-invariant POD for periodically-forced flows
12.7 A miscellany of results: 1995–2011
13: Review: prospects for rigor
13.1 The quality of models
Long-term tracking and inertial manifolds
Localization in physical space
Resolution of coherent structures alone
13.2 A short-time tracking estimate
13.3 Stability, simulations, and statistics
13.4 Spatial localization
13.5 The utility of models
Turbulence, Coherent Structures, Dynamical Systems and Symmetry
( Cambridge Monographs on Mechanics )
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