Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances

Cover......Page 1
CISM International Centre for Mechanical Sciences 523......Page 3
Asymptotic Methods in Fluid Mechanics......Page 4
ISBN 9783709104071......Page 5
PREFACE......Page 6
Table of Contents......Page 8
1 Introduction......Page 10
2.1 Asymptotic expansion of the flow potential (regular expansion)......Page 12
2.2 Local expansion at leading/trailing edge......Page 14
2.3 Matching procedure......Page 16
3 Flow between rotating discs - Ekman layers......Page 18
3.3 Boundary layers......Page 20
3.4 Matching......Page 21
4 Model equation for fully developed turbulent channel flow......Page 23
4.1 Defect Layer......Page 25
4.2 Viscous wall layer......Page 26
4.3 Matching......Page 27
4.4 Turbulence asymptotics......Page 28
5 Conclusions......Page 29
Bibliography......Page 30
1 Introduction......Page 32
2 Infinite Logarithmic Expansions: Simple Pipe Flow......Page 33
3.1 Oxygen Transport From Capillaries to Skeletal Muscle......Page 42
3.2 A Nonlinear Elliptic Problem......Page 49
4 Slow Viscous Flow Over a Cylinder......Page 51
4.1 Summing Logarithmic Expansions: A Linear Biharmonic Problem......Page 62
4.2 A Convection-Diffusion Problem......Page 65
5 The Fundamental Neumann Eigenvalue in a Planar Domain with Localized Traps......Page 69
Bibliography......Page 76
1 Introduction......Page 80
1.1 Stieltjes Integral......Page 81
1.2 Fourier Integral......Page 82
1.3 Airy Functions......Page 83
1.4 WKB Solutions and Stokes Phenomenon......Page 84
2.1 Forced Harmonic Oscillator......Page 86
2.2 Balanced Flow and Slow Manifold......Page 88
2.3 Waves in a Variable Medium......Page 89
3 Borel Summation: Forced Nonlinear Harmonic Oscillator......Page 92
3.1 Case (a)......Page 93
3.2 Case(b): Outer expansion......Page 99
4.1 Korteweg-de Vries equation......Page 103
4.2 Linear spectrum......Page 104
4.3 Reformulation as a dynamical system......Page 106
4.4 Case (1)......Page 107
4.5 Case (2)......Page 109
4.6 Case (3)......Page 110
5.1 Formulation and outer expansion......Page 112
5.2 Inner expansion and Borel summation......Page 113
5.4 Higher-order terms......Page 117
6.1 Internal waves......Page 120
6.3 Inner expansion and Borel summation......Page 123
6.4 Weak coupling approximation......Page 126
Bibliography......Page 128
1 Introduction......Page 130
2 Generalized Solitary Waves and the 5KdV......Page 131
2.1 Initial Asymptotic Analysis and Late Terms......Page 132
2.2 Optimal Truncation and Stokes Smoothing......Page 133
Bibliography......Page 135
1 Overview......Page 136
2.1 Pure fluid dynamics......Page 137
2.2 Equations of state......Page 142
2.3 The influence of gravity......Page 143
2.4 The effects of Earth’s rotation......Page 145
2.5 Adiabatic motions and the concept of potential temperature......Page 149
2.6 Summarizing the equations......Page 150
3 Introduction to multiple scales asymptotics......Page 151
3.1 Exact solutions for the linear oscillator......Page 152
3.2 Dimensionless representation and small parameters......Page 159
3.3 Regular perturbation analysis for small mass and damping......Page 168
3.4 Multiple scales analysis......Page 175
3.5 Some comments and a question......Page 179
4.1 Universal parameters and distinguished limits......Page 180
4.2 Nondimensionalization and general multiple scales ansatz......Page 183
4.3 Scaled governing equations and general multiple scales expansion scheme......Page 184
5.1 Asymptotic expansion scheme......Page 186
5.2 Some preliminaries......Page 188
5.3 Expansions of the governing equations......Page 189
5.4 Summary of the leading-order balances......Page 192
5.5 First-order solvability condition / existence of ∇ξp(3)......Page 194
5.6 Classical formulation of the QG-theory and PV transport......Page 195
Appendix......Page 196
Bibliography......Page 204
1 Introduction......Page 206
2 Governing equations and scaling......Page 207
2.1 The friction law......Page 209
2.2 External forces......Page 210
2.3 Scaling assumptions......Page 211
3.1 Shallow water approximation......Page 212
3.2 Multiple Scales Expansion......Page 213
3.3 Uniformly valid differential equation......Page 218
3.4 The equations for the slowly varying variables......Page 220
3.5 Initially fully developed flow......Page 222
Bibliography......Page 227
1 Introduction......Page 230
2.1 Preliminaries......Page 232
2.2 Leading-Order Approximation......Page 233
2.3 Second-Order Outer Problem......Page 235
2.4 Can Classical Small-Defect Theory Describe Boundary Layer Separation?......Page 236
3 Moderately Large Velocity Defect......Page 238
3.1 Intermediate Layer......Page 239
3.2 Outer Defect Region: Quasi-Equilibrium Flows and Non- Uniqueness......Page 240
4 Large Velocity Deficit......Page 244
4.1 Outer Wake Region......Page 245
4.2 Inner Wake Region......Page 246
4.3 Numerical Solution of the Leading-Order Outer-Wake Problem......Page 247
4.4 Marginal Separation......Page 248
5 Conclusions and Outlook......Page 252
Bibliography......Page 253
1 Introduction......Page 256
2.1 Prandtl Boundary Layer equations......Page 257
2.2 Blowing Velocity Induced by the Boundary Layer......Page 258
2.3 Self Similar Solutions of Prandtl Equations......Page 259
2.4 Von K´arm´an Equation Integral Relation......Page 260
3.1 Separation......Page 262
3.2 The Problem of the Influence of Downstream on Upstream......Page 265
4.1 Inverse Boundary Layer......Page 266
4.2 Some Explanations of the Upstream Influence Problem......Page 267
5.2 Interactive Boundary Layer......Page 273
5.3 Link between Interactive Boundary Layer and Triple Deck......Page 274
5.4 Reduced Navier Stokes Prandtl, RNSP equations......Page 276
5.5 Coupling the Solvers......Page 277
6.1 Some Numerical Examples of IBL......Page 282
6.2 Example in Internal Flows: Axi- symmetrical Flows, symmetrical and non symmetrical 2D Flows......Page 285
6.3 Some other Numerical Examples......Page 288
7 Conclusion......Page 291
Bibliography......Page 292
1 Introduction......Page 296
2.1 The potential flow......Page 299
2.2 The wake flow......Page 300
2.3 Numerical solution......Page 301
2.4 Analysis of the wake singularity......Page 303
3 Local Interaction at the Trailing-Edge......Page 306
3.1 Main deck......Page 308
3.2 The upper deck......Page 309
3.3 The lower deck......Page 310
3.4 The local behavior of the lower deck velocity field near the trailing-edge......Page 314
3.5 Resolving the pressure discontinuity on main deck scales......Page 315
4 Summary and Conclusions......Page 318
Bibliography......Page 319
1 Introduction......Page 320
2.1 Formulation of the Problem......Page 327
2.2 The flow upstream of the interaction region......Page 329
2.3 Inspection analysis of the interaction process......Page 331
2.4 Triple-Deck Model......Page 336
3 Marginal Separation Theory......Page 350
3.1 Statement of the Problem. Inviscid-Flow Region......Page 351
3.2 Boundary Layer......Page 354
3.3 Viscous-Inviscid Interaction......Page 386
Bibliography......Page 416
1 Problem formulation......Page 418
2 Results......Page 420
Bibliography......Page 422
1 Fundamental Equations......Page 424
2 Finite Time Blow-up and its Self-Similarity......Page 426
3 Numerical Solution......Page 427
Bibliography......Page 429