 | Turbulence, coherent structures, dynamical systems and symmetry / Ph.Holmes et al. - 2nd ed. - Cambridge; New York: Cambridge University Press, 2012. - xvi, 386 p.: ill. - (Cambridge monographs on mechanics). - Ref.: p.364-381. - Ind.: p.382-386. - ISBN 978-1-107-00825-0
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Preface to the first edition page .............................. ix
Preface to the second edition ................................ xiii
Acknowledgements ............................................... xv
PART ONE TURBULENCE ............................................. 1
1 Introduction ................................................. 3
1.1 Turbulence .............................................. 3
1.2 Low-dimensional models .................................. 5
1.3 The contents of this book ............................... 8
1.4 Notation and mathematical jargon ....................... 11
2 Coherent structures ......................................... 17
2.1 Introduction ........................................... 17
2.2 Flows with coherent structures ......................... 21
2.3 Detection of coherent structures ....................... 32
2.4 The mixing layer ....................................... 35
2.5 The turbulent boundary layer ........................... 50
2.6 A preview of things to come ............................ 65
3 Proper orthogonal decomposition ............................. 68
3.1 Introduction ........................................... 69
3.2 On domains and averaging ............................... 73
3.3 Properties of the POD .................................. 74
3.4 Further results ........................................ 86
3.5 Stochastic estimation .................................. 91
3.6 Coherent structures and homogeneity .................... 93
3.7 Some applications ...................................... 96
3.8 Appendix: some foundations ............................ 100
4 Galerkin projection ........................................ 106
4.1 Introduction .......................................... 106
4.2 Some simple PDEs revisited ............................ 110
4.3 The Navier-Stokes equations ........................... 116
4.4 Towards low-dimensional models ........................ 121
5 Balanced proper orthogonal decomposition ................... 130
5.1 Balanced truncation ................................... 131
5.2 Balanced POD .......................................... 133
5.3 Output projection ..................................... 136
5.4 Connections with standard POD ......................... 137
5.5 Extensions of balanced POD ............................ 139
5.6 Some examples ......................................... 143
PART TWO DYNAMICAL SYSTEMS ................................... 153
6 Qualitative theory ......................................... 155
6.1 Linearization and invariant manifolds ................. 156
6.2 Periodic orbits and Poincare maps ..................... 162
6.3 Structural stability and genericity ................... 165
6.4 Bifurcations local and global ......................... 168
6.5 Attractors simple and strange ......................... 179
7 Symmetry ................................................... 190
7.1 Equivariant vector fields ............................. 190
7.2 Local bifurcation with symmetry ....................... 194
7.3 Global behavior with symmetry ......................... 195
7.4 An 0(2)-equivariantODE ................................ 202
7.5 Traveling modes ....................................... 211
8 One-dimensional "turbulence" ............................... 214
8.1 Projection onto Fourier modes ......................... 215
8.2 Local bifurcations from и = 0 ......................... 217
8.3 The second bifurcation point .......................... 220
8.4 Spatio-temporal chaos ................................. 226
9 Randomly perturbed systems ................................. 236
9.1 An Ornstein-Uhlenbeck process ......................... 237
9.2 Noisy heteroclinic cycles ............................. 240
9.3 Power spectra of homoclinic attractors ................ 247
9.4 Symmetry breaking ..................................... 249
PART THREE THE BOUNDARY LAYER ................................ 253
10 Low-dimensional models ..................................... 255
10.1 Equations for coherent structures ..................... 256
10.2 The eigenfunction expansion ........................... 259
10.3 Symmetries ............................................ 260
10.4 Galerkin projection ................................... 262
10.5 Geometrical structure of the model .................... 269
10.6 Choosing subspaces and domains ........................ 272
10.7 The energy budget ..................................... 275
10.8 Nonlinear feedback .................................... 281
10.9 Interaction with unresolved modes ..................... 285
11 Behavior of the models ..................................... 289
11.1 Backbones for the models .............................. 290
11.2 Heteroclinic cycles ................................... 293
11.3 Bursts and sweeps ..................................... 297
11.4 The pressure term ..................................... 299
11.5 More modes and instabilities .......................... 303
11.6 A tentative summary ................................... 307
11.7 Appendix: coefficients ................................ 312
PART FOUR OTHER APPLICATIONS AND RELATED WORK ................ 315
12 Some other fluid problems .................................. 317
12.1 The circular jet ...................................... 317
12.2 The transitional boundary layer ....................... 321
12.3 A forced transitional mixing layer .................... 326
12.4 Flows in complex geometries ........................... 328
12.5 "Full channel" wall layer models ...................... 331
12.6 Flows in internal combustion engines .................. 335
12.7 A miscellany of results: 1995-2011 .................... 341
12.8 Discussion ............................................ 342
13 Review: prospects for rigor ................................ 345
13.1 The quality of models ................................. 345
13.2 A short-time tracking estimate ........................ 349
13.3 Stability, simulations, and statistics ................ 352
13.4 Spatial localization .................................. 356
13.5 The utility of models ................................. 360
References ................................................. 364
Index ......................................................... 382
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