Preface ......................................................... V
Acknowledgements ............................................. VIII
Abbreviations ................................................ XVII
1 Introduction .................................................. 1
2 Introduction to Singular Perturbation Problems ................ 7
2.1 Regular and Singular Problems ............................. 8
2.1.1 Linear Oscillator ................................... 8
2.1.2 Secular Problem .................................... 11
2.1.3 Singular Problem ................................... 14
2.2 Approximation Methods for Singular Perturbation
Problems ................................................. 15
2.2.1 Method of Matched Asymptotic Expansions ............ 16
2.2.2 Successive Complementary Expansion Method .......... 19
2.2.3 Multiple Scale Method .............................. 20
2.2.4 Poincare-Lighthill's Method ........................ 22
2.2.5 Renormalization Group Method ....................... 24
2.3 Conclusion ............................................... 25
Problems ..................................................... 25
3 Boundary Layer Structure ..................................... 31
3.1 Study of a Second Order Differential Equation ............ 31
3.2 Analysis of each Case .................................... 35
3.3 Conclusion ............................................... 40
Problems ..................................................... 41
4 Asymptotic Expansions ........................................ 43
4.1 Order Functions. Order of a Function ..................... 43
4.1.1 Definition of an Order Function .................... 43
4.1.2 Comparison of Order Functions ...................... 43
4.1.3 Total Ordering ..................................... 44
4.1.4 Order of a Function ................................ 45
4.2 Asymptotic Sequence ...................................... 46
4.2.1 Definition of an Asymptotic Sequence ............... 46
4.2.2 Class of Equivalence ............................... 46
4.2.3 Gauge Functions .................................... 47
4.3 Asymptotic Expansion ..................................... 47
4.3.1 Asymptotic Approximation ........................... 47
4.3.2 Regular Functions .................................. 49
4.3.3 Regular and Generalized Asymptotic Expansions ...... 50
4.3.4 Convergence and Accuracy ........................... 51
4.3.5 Operations on Asymptotic Expansions ................ 54
4.4 Conclusion ............................................... 55
Problems ..................................................... 55
5 Successive Complementary Expansion Method .................... 59
5.1 Method of Matched Asymptotic Expansions .................. 59
5.1.1 Expansion Operator ................................. 59
5.1.2 Outer Expansion - Inner Expansion .................. 60
5.1.3 Asymptotic Matching ................................ 61
5.2 Boundary Layer ........................................... 65
5.2.1 Expansion Operator to a Given Order ................ 65
5.2.2 Significant Approximations ......................... 66
5.3 Intermediate Matching .................................... 67
5.3.1 Kaplun's Extension Theorem ......................... 67
5.3.2 Study of Examples .................................. 67
5.3.3 Rule of Intermediate Matching ...................... 69
5.4 Asymptotic Matching Principle ............................ 71
5.4.1 Van Dyke's Principle ............................... 71
5.4.2 Modified Van Dyke's Principle ...................... 72
5.5 Examples and Counter-Examples ............................ 72
5.5.1 Example 1 .......................................... 72
5.5.2 Example 2 .......................................... 73
5.5.3 Example 3 .......................................... 74
5.5.4 Example 4 .......................................... 75
5.6 Discussion of the Matching Principle ..................... 76
5.6.1 Corrective Boundary Layer .......................... 77
5.6.2 The MVDP from the Overlap Hypothesis ............... 79
5.7 Successive Complementary Expansion Method ................ 81
5.7.1 Principle .......................................... 81
5.7.2 Equivalence of MVDP and of Regular SCEM ............ 84
5.8 Applications of SCEM ..................................... 86
5.8.1 Example 1 .......................................... 86
5.8.2 Example 2 .......................................... 88
5.8.3 Example 3 .......................................... 89
5.9 Conclusion ............................................... 90
Problems ..................................................... 91
6 Ordinary Differential Equations .............................. 99
6.1 Example 1 ................................................ 99
6.1.1 Application of MMAE ............................... 100
6.1.2 Application of SCEM ............................... 102
6.2 Example 2 ............................................... 107
6.2.1 Application of MMAE ............................... 107
6.2.2 Application of SCEM ............................... 109
6.2.3 Identification with MMAE Results .................. 111
6.2.4 Numerical Results ................................. 112
6.3 Example 3 ............................................... 112
6.3.1 Application of MMAE ............................... 112
6.3.2 Application of SCEM ............................... 116
6.3.3 Identification with MMAE Results .................. 118
6.4 Stokes-Oseen's Flow Model ............................... 118
6.4.1 Application of SCEM ............................... 118
6.4.2 Numerical Results ................................. 120
6.5 Terrible Problem' ....................................... 121
6.5.1 Application of SCEM ............................... 122
6.5.2 Numerical Results ................................. 125
6.6 Conclusion .............................................. 125
Problems .................................................... 127
7 High Reynolds Number Flows .................................. 133
7.1 Boundary Layer Theories ................................. 135
7.1.1 Prandtl's Boundary Layer .......................... 135
7.1.2 Triple Deck ....................................... 140
7.2 Analysis of an Integral Method .......................... 148
7.2.1 Integral Method ................................... 148
7.2.2 Direct Mode ....................................... 151
7.2.3 Inverse Mode ...................................... 152
7.2.4 Simultaneous Mode ................................. 153
7.3 Viscous-Inviscid Interaction ............................ 155
7.4 Conclusion .............................................. 157
Problems .................................................... 158
8 Interactive Boundary Layer .................................. 169
8.1 Application of SCEM ..................................... 170
8.1.1 Outer Approximation ............................... 170
8.1.2 Determination of a Uniformly Valid
Approximation ..................................... 171
8.1.3 Gauge for the Pressure ............................ 173
8.2 First Order Interactive Boundary Layer .................. 173
8.2.1 Generalized Boundary Layer Equations .............. 173
8.2.2 Boundary Conditions ............................... 174
8.2.3 Estimate of the Remainders of Equations ........... 175
8.3 Second Order Interactive Boundary Layer ................. 175
8.3.1 Generalized Boundary Layer Equations .............. 175
8.3.2 Boundary Conditions ............................... 176
8.3.3 Estimate of the Remainders of Equations ........... 176
8.4 Displacement Effect ..................................... 177
8.5 Reduced Model for an Irrotational External Flow ......... 178
8.6 Conclusion .............................................. 180
Problems .................................................... 181
9 Applications of Interactive Boundary Layer Models ........... 185
9.1 Calculation of a Flow with Separation ................... 186
9.1.1 Definition of the Flow ............................ 186
9.1.2 Numerical Method .................................. 186
9.1.3 Results ........................................... 188
9.2 Application to Aerodynamic Flows ........................ 190
9.2.1 Flat Plate of Finite Length ....................... 190
9.2.2 Airfoils at High Reynolds Numbers ................. 192
9.3 Influence of a Rotational External Flow ................. 195
9.3.1 Inviscid Flow ..................................... 195
9.3.2 Method of Resolution .............................. 197
9.3.3 Flows Studied ..................................... 200
9.3.4 Results ........................................... 200
9.4 Conclusion .............................................. 211
Problems .................................................... 211
10 Regular Forms of Interactive Boundary Layer ................ 215
10.1 Second Order Boundary Layer Model ..................... 215
10.1.1 Second Order Interactive Boundary Layer
Model .......................................... 217
10.1.2 Van Dyke's Second Order Model .................. 217
10.2 Triple Deck Model ..................................... 221
10.2.1 Flow on a Flat Plate with a Small Hump ......... 221
10.2.2 Regular Expansions ............................. 223
10.3 Summary of Approximations of Navier-Stokes
Equations ............................................. 226
10.4 Conclusion ............................................ 226
Problems ................................................... 227
11 Turbulent Boundary Layer ................................... 237
11.1 Results of the Standard Asymptotic Analysis ........... 237
11.1.1 Averaged Navier-Stokes Equations ............... 237
11.1.2 Scales ......................................... 238
11.1.3 Structure of the Flow .......................... 239
11.2 Application of SCEM ................................... 243
11.2.1 First Approximation ............................ 243
11.2.2 Contribution of the Outer Region of
the Boundary Layer ............................. 243
11.2.3 Contribution of the Inner Region of
the Boundary Layer ............................. 246
11.3 Interactive Boundary Layer ............................ 249
11.3.1 First Order Model .............................. 249
11.3.2 Second Order Model ............................. 250
11.3.3 Global Model ................................... 250
11.3.4 Reduced Model for an Irrotational External
Flow ........................................... 251
11.4 Approximation of the Boundary Layer: Velocity
Profile ............................................... 254
11.4.1 Formulation of the Problem ..................... 254
11.4.2 Turbulence Model ............................... 256
11.4.3 Outer Region ................................... 256
11.4.4 Equation to Solve .............................. 257
11.4.5 Examples of Results ............................ 258
11.5 Conclusion ............................................ 260
Problems ................................................... 260
12 Channel Flow ............................................... 267
12.1 Formulation of the problem ............................ 267
12.2 Uniformly Valid Approximation ......................... 270
12.3 IBL Model for the Lower Wall .......................... 272
12.4 Global IBL Model ...................................... 274
12.5 Numerical Solution .................................... 275
12.5.1 General Method ................................. 275
12.5.2 Simplified Method for the Pressure ............. 277
12.6 Application of the Global IBL model ................... 279
12.6.1 Discussion of the Numerical Procedure .......... 279
12.6.2 Comparisons with Smith's theory ................ 283
12.6.3 Comparison with Navier-Stokes Solutions ........ 290
12.7 Conclusion ............................................ 295
Problems ................................................... 295
13 Conclusion ................................................. 301
Appendices .................................................... 303
I Navier-Stokes Equations ................................... 303
II Elements of Two-Dimensional Linearized Aerodynamics ....... 305
II.1 Thickness Problem (Non Lifting Case) ................ 306
II.2 Zero-Thickness Problem (Lifting Case) ............... 307
III Solutions of the Upper Deck of the Triple Deck Theory ..... 309
III.1 Two-Dimensional Flow ................................ 309
III.2 Three-Dimensional Flow .............................. 312
III.2.1 Zero Perturbations at Infinity .............. 313
III.2.2 Non Zero Cross-Flow Perturbations at
Downstream Infinity ......................... 314
IV Second Order Triple Deck Theory ........................... 319
IV.1 Main Results ........................................ 319
IV.2 Global Model for the Main Deck and the Lower Deck ... 325
V Behaviour of an Asymptotic Expansion ...................... 327
V.1 Formulation of the Problem .......................... 327
V.2 Study of the Gauge Functions ........................ 328
V.3 Study of the Outer Expansion ........................ 330
Solutions of Problems ......................................... 332
References .................................................... 419
Author index .................................................. 427
Subject index ................................................. 428
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