Asymptotic Analysis and Boundary Layers (eBook)

Preface 5
Acknowledgements 8
Contents 9
Abbreviations 15
1 Introduction 16
2 Introduction to Singular Perturbation Problems 22
2.1 Regular and Singular Problems 23
2.2 Approximation Methods for Singular Perturbation Problems 30
2.3 Conclusion 40
Problems 40
3 Boundary Layer Structure 45
3.1 Study of a Second Order Differential Equation 45
3.2 Analysis of each Case 49
3.3 Conclusion 54
Problems 55
4 Asymptotic Expansions 57
4.1 Order Functions. Order of a Function 57
4.2 Asymptotic Sequence 60
4.3 Asymptotic Expansion 61
4.4 Conclusion 69
Problems 69
5 Successive Complementary Expansion Method 72
5.1 Method of Matched Asymptotic Expansions 72
5.2 Boundary Layer 78
5.3 Intermediate Matching 80
5.4 Asymptotic Matching Principle 84
5.5 Examples and Counter-Examples 85
5.6 Discussion of the Matching Principle 89
5.7 Successive Complementary Expansion Method 94
5.8 Applications of SCEM 99
5.9 Conclusion 103
Problems 104
6 Ordinary Differential Equations 112
6.1 Example 1 112
6.2 Example 2 120
6.3 Example 3 125
6.4 Stokes-Oseen’s Flow Model 131
6.5 Terrible Problem 134
6.6 Conclusion 138
Problems 140
7 High Reynolds Number Flows 145
7.1 Boundary Layer Theories 147
7.2 Analysis of an Integral Method 160
7.3 Viscous-Inviscid Interaction 167
7.4 Conclusion 169
Problems 170
8 Interactive Boundary Layer 180
8.1 Application of SCEM 181
8.2 First Order Interactive Boundary Layer 184
8.3 Second Order Interactive Boundary Layer 186
8.4 Displacement Effect 188
8.5 Reduced Model for an Irrotational External Flow 189
8.6 Conclusion 191
Problems 192
9 Applications of Interactive Boundary Layer Models 195
9.1 Calculation of a Flow with Separation 196
9.2 Application to Aerodynamic Flows 200
9.3 Influence of a Rotational External Flow 205
9.4 Conclusion 221
Problems 221
10 Regular Forms of Interactive Boundary Layer 224
10.1 Second Order Boundary Layer Model 224
10.2 Triple Deck Model 230
10.3 Summary of Approximations of Navier- Stokes Equations 235
10.4 Conclusion 235
Problems 236
11 Turbulent Boundary Layer 245
11.1 Results of the Standard Asymptotic Analysis 245
11.2 Application of SCEM 251
11.3 Interactive Boundary Layer 257
11.4 Approximation of the Boundary Layer: Velocity Profile 262
11.5 Conclusion 268
Problems 268
12 Channel Flow 274
12.1 Formulation of the problem 274
12.2 Uniformly Valid Approximation 277
12.3 IBL Model for the Lower Wall 279
12.4 Global IBL Model 281
12.5 Numerical Solution 282
12.6 Application of the Global IBL model 286
12.7 Conclusion 302
Problems 302
13 Conclusion 307
I Navier-Stokes Equations 309
II Elements of Two-Dimensional Linearized Aerodynamics 311
II.1 Thickness Problem (Non Lifting Case) 312
II.2 Zero-Thickness Problem (Lifting Case) 313
III Solutions of the Upper Deck of the Triple Deck Theory 315
III.1 Two-Dimensional Flow 315
III.2 Three-Dimensional Flow 318
IV Second Order Triple Deck Theory 324
IV.1 Main Results 324
IV.2 Global Model for the Main Deck and the Lower Deck 330
V Behaviour of an Asymptotic Expansion 332
V.1 Formulation of the Problem 332
V.2 Study of the Gauge Functions 333
V.3 Study of the Outer Expansion 335
Solutions of Problems 337
References 423
Author index 430
Subject index 431