Reduced-Order Modelling for Flow Control

Cover......Page 1
Reduced-Order Modelling for Flow Control......Page 5
ISBN 9783709107577......Page 8
FOREWORD......Page 6
PREFACE......Page 10
Table of Contents......Page 12
1 Introduction......Page 14
2.1.1.1 General points......Page 15
2.1.1.2 Terminology......Page 16
2.1.2 Linearized framework......Page 18
2.1.3 Different types of problems......Page 21
2.2.1 Similarity transformations......Page 22
2.2.2 Controllability and observability......Page 23
2.2.2.1 Controllability......Page 24
2.2.2.2 Observability......Page 26
2.2.2.3 Duality......Page 27
2.2.2.4 Balanced truncation......Page 29
2.3.1 Need for reduced-order modeling......Page 31
2.3.2 Overview of model-reduction methods......Page 32
3.1.1 Abstract description......Page 35
3.1.2 III-posed optimization problem and choice of the cost functional......Page 36
3.2.1 Introduction of the Lagrange multiplier......Page 38
3.2.2 Derivation of the optimality system......Page 39
3.2.3 Numerical resolution......Page 43
3.3 Optimization methods......Page 47
3.3.1 Functional gradients through sensitivities......Page 48
3.4 Differentiation and discretization......Page 50
4.1 Choice of the cost functional......Page 52
4.2 Original problem and Lagrange multipliers......Page 54
4.3.1 Direct problem......Page 55
4.3.2 Adjoint problem......Page 56
4.3.3 Optimality conditions......Page 57
4.4 Riccati equation......Page 58
5 Optimal growth perturbation......Page 61
5.1 Variational formulation......Page 62
5.1.1 Original problem, inner products and Lagrangian formulation......Page 63
5.1.2.2 Determination......Page 64
5.1.2.3 Determination......Page 65
5.2 Formulation based on matrix exponential......Page 67
6 Linearized Burgers equation......Page 70
6.1 Problem formulation and Lagrangian-based approach......Page 71
6.2.2 Adjoint equations......Page 73
6.2.3 Optimality conditions......Page 76
7.1 Formulation and optimality system......Page 77
7.2.1 Numerical parameters and space-time discretization......Page 79
7.2.2 Optimization procedure......Page 81
8 Conclusion......Page 83
Bibliography......Page 86
1 Introduction......Page 90
2 Global flow stability analysis......Page 92
3 Finite Element Method discretization of the global flow stability problem......Page 93
4 Numerical techniques to solution of the eigenvalue problem......Page 96
4.1 Solution of Algebraic Eigenvalue Problem......Page 97
4.2 Subspace iteration method......Page 98
4.3 Preconditioning......Page 99
4.4 Eigensolution via system identification......Page 101
5.1 Steady flow global stability......Page 102
5.2 Time–averaged flow stability......Page 105
6.1 General philosophy of flow control for stabilization......Page 107
7.1 Mean-field correction......Page 110
7.2 Hybrid model employing stability modes......Page 113
7.3 Continuous mode interpolation......Page 114
7.4 Mean field correction and Galerkin Model for NACA-0012 flow......Page 115
8 Summary and perspectives - 3D flow stability analysis......Page 117
Bibliography......Page 120
1 Introduction......Page 124
2.1 Problem formulation......Page 125
2.2 Traditional Galerkin method......Page 127
2.3 Galerkin expansion......Page 129
2.4 Galerkin system......Page 130
2.5 Non-orthogonal modes......Page 133
2.6 POD models......Page 134
3.1 Principles of balance equations......Page 137
3.2 Modal balance equations......Page 138
3.3 Finite-time thermodynamics as closure model......Page 142
4.1 Challenges and modeling principles......Page 144
Shift mode......Page 146
Deformable oscillatory modes......Page 150
4.3 Modeling natural dynamics......Page 151
Boundary actuation......Page 154
5 Conclusions and Outlook......Page 156
Bibliography......Page 157
1 Introduction......Page 164
2.1 The Cylinder Wake......Page 166
2.2 The Actuated Cylinder Wake Configuration......Page 167
2.3 Dominant Coherent Structures of the Natural Flow......Page 169
2.4 A High Lift Configuration......Page 171
3 Low Order Galerkin Models: Some Added Concepts......Page 172
3.1 The Constitutive First Principles Model......Page 173
3.2 The Galerkin Modeling Framework......Page 174
3.3 A Simple Example of an Utter Failure......Page 175
Model structure inconsistency......Page 176
Inconsistency with moving boundaries......Page 177
3.4 The Triple Reynolds Decomposition (TRD)......Page 178
3.5 Harmonic Modes and Harmonic Expansions......Page 179
3.6 The Harmonically Dominated Galerkin System......Page 184
3.7 Interim Comments......Page 187
Dynamic Power Balancing: NSE Definitions......Page 188
Dynamic Power Balancing: Galerkin System Definitions......Page 189
3.9 Closing Comments......Page 192
4.1 The Need for a Mean Field Model: An NSE Perspective......Page 193
4.2 Simple Galerkin-Reynolds Mean Field Models......Page 195
5.1 The Need for Subgrid Models......Page 198
An eddy viscosity estimate of TSC......Page 201
Energy Balance......Page 202
The Galerkin Subgrid Terms......Page 203
An analogy between mean field and subgrid representations......Page 205
Is the subgrid model simply a calibration method?......Page 206
6 Mode Deformation and Models on Nonlinear Manifolds......Page 207
Poor resolution of the flow field by the Galerkin expansion......Page 208
Distorted phase predictions......Page 209
Extended mode sets......Page 210
Offline mode set adaptation......Page 211
6.2 Deformable modes and accurate low order models......Page 212
6.3 Computational Aspects of Parametrized Mode Sets......Page 214
The dynamics of operating point parametrization......Page 215
The form and computation of deformable modes......Page 216
Parametrized modes vs. center and inertial manifold models......Page 218
Analytic construction of the parametrized expansion set......Page 219
7.1 A Galerkin Modeling Conundrum and Existing Solutions......Page 221
Actuation modes......Page 222
Lagrangian- ulerian methods and deformable grids......Page 224
The augmented domain and velocity field......Page 225
Admissible embedding in a canonical computational domain......Page 226
An auxiliary flow......Page 227
A “spring analogy” construction of S: Basic steps......Page 228
The “spring analogy” construction of S: Cautionary comments......Page 229
The construction of S: Slip conditions and actuation modes......Page 230
Comments & alternative guidelines......Page 231
Example: A continuously deforming cylinder......Page 232
The construction of actuation modes......Page 234
The Galerkin approximation in deforming geometries......Page 236
The Galerkin Dynamical System......Page 237
7.6 Closing Comments......Page 238
8 Feedback Design......Page 239
8.1 Volume Force Actuation of the Cylinder Wake......Page 240
8.2 Direct and Indirect Design Objectives......Page 241
3. Model based feedback design should maintain the flow within the validity envelope of the reduced order model......Page 242
8.3 Modeling Periodic Actuation......Page 243
The forcing terms: A Direct Galerkin Projection......Page 244
The forcing terms: A phasor equation derivation......Page 245
An imbalance in (87).......Page 246
Phase averaging requirement in phasor models actuation.......Page 247
8.5 Performance Limitations......Page 248
Stability of the operating point: An amplitude perspective.......Page 249
Stability of the operating point: A phase perspective.......Page 250
9 Concluding Remarks......Page 251
Bibliography......Page 252
1 Introduction......Page 266
2 Wake stabilization benchmark......Page 270
3.1 Short Time POD - SPOD......Page 273
3.2 Double POD - DPOD......Page 276
4 Artificial neural network system identification to develop a numerical plant model......Page 279
4.1 Input Layer......Page 283
4.5 Training the ANN......Page 284
5 Feedback control wake stabilization results......Page 285
5.1 Single Mode Feedback - SISO......Page 287
5.2 Two Mode Feedback - MISO......Page 291
6 Key enablers / strategy......Page 293
6.1 Integration of Experiments, Modelling and Simulation......Page 294
6.2 Applicability to Other Flow Fields......Page 295
Bibliography......Page 298
1 Introduction......Page 300
2 Linear systems......Page 302
3 Methods in closed-loop control......Page 308
Classical extremum seeking control......Page 309
Slope seeking control......Page 312
Increase of bandwidth......Page 313
3.2 Linear and robust control......Page 315
3.3 Model predictive control......Page 320
MPC formulation for linear unconstrained problems......Page 322
MPC formulation for constrained and /or nonlinear problems......Page 324
4 Applications......Page 325
4.1 Two-dimensional bluff body - slope seeking......Page 326
4.2 Two-dimensional bluff body - robust control......Page 329
4.3 MPC, energy-based and LPV-control of the cylinder wake......Page 332
5 Conclusions......Page 336
Bibliography......Page 338