Part I Fundamentals of the linear stability theory .............. 1
1 Concept of hydrodynamic stability ............................ 3
1.1 Hydrodynamic stability .................................. 3
1.2 Stability of fluid motion in time ....................... 5
1.2.1 Critical parameters for onset of instability ..... 6
1.2.2 Conditional stability ............................ 8
1.2.3 Growth of disturbance energy ..................... 9
References .................................................. 11
Further Reading ............................................. 12
2 Theoretical aspects ......................................... 13
2.1 Formulation of linear hydrodynamic stability
problems ............................................... 13
2.1.1 Spectral formulation of stability ............... 17
2.1.2 Inviscid instability mechanism .................. 18
2.1.3 Viscous instability mechanism ................... 20
2.2 Instability in space ................................... 21
2.3 Gaster's transformation ................................ 23
2.4 Squire theorem ......................................... 23
2.5 Adjoint problem and bi-orthogonality of normal modes ... 25
2.6 Completeness of solutions for the Orr-Sommerfeld and
Squire equations ....................................... 26
References .................................................. 30
Further Reading ............................................. 32
Part II Generic problems ....................................... 33
3 Instability of plane parallel flows ......................... 35
3.1 Plane Couette flow .................................... 35
3.2 Plane Poiseuille flow .................................. 36
3.2.1 Numerical results ............................... 37
3.2.2 Experimental linear stability investigations .... 39
3.3 Method of linear stability calculations ................ 40
Exercises ................................................... 45
References .................................................. 45
Further Reading ............................................. 46
4 Instability of the flat-plate boundary layer ................ 47
4.1 Historical notes ....................................... 47
4.2 Solution of the Orr-Sommerfeld equation for the
boundary layers ........................................ 51
4.3 Nonparallel flow effects ............................... 54
4.3.1 Outline of theoretical approaches to account
for nonparallel effects ......................... 55
4.3.2 Modern view on the place and role of
nonparallel effects in the Blasius boundary
layer ........................................... 60
Exercises ................................................... 64
References .................................................. 64
Further Reading ............................................. 66
5 Instabilities of plane flows over curvilinear surfaces ...... 67
5.1 Influence of curvature on the basic flow ............... 67
5.1.1 Equations of motion in cylindrical
coordinates ..................................... 67
5.1.2 Description of the flow in boundary layers
over curvilinear surfaces ....................... 69
5.2 Hydrodynamic instability at curvilinear surfaces ....... 71
5.2.1 Taylor problem .................................. 73
5.2.2 Dean problem .................................... 76
5.2.3 Görtler problem ................................. 78
Exercises ................................................... 88
References .................................................. 88
Further Reading ............................................. 90
6 Some other basic factors of shear-layer stability ........... 91
6.1 Axial flow symmetry .................................... 91
6.2 Two-dimensional geometry ............................... 94
6.3 Transverse flow periodicity ........................... 102
6.4 Pressure gradients .................................... 106
6.4.1 Stream wise pressure gradient .................. 107
6.4.2 Transverse pressure gradient ................... 108
6.5 Heat transfer ......................................... 112
6.6 Fluid suction ......................................... 114
6.7 Compliant boundaries .................................. 115
6.8 Dusty flow ............................................ 120
Exercises .................................................. 123
References ................................................. 124
Further Reading ............................................ 126
7 Instability of separated boundary layers ................... 129
7.1 Basic features of separated flow instability .......... 129
7.2 Linear instability at flow separation in plane
configurations ........................................ 130
7.2.1 Waveform ....................................... 132
7.2.2 Growth rates ................................... 133
7.2.3 Propagation velocities ......................... 135
7.3 Instability of axisymmetric separation bubbles ........ 136
7.4 Flow instability at separation of a swept-wing
boundary layer ........................................ 137
7.5 Outline of approaches for linear stability
calculations in separation regions .................... 139
Exercises .................................................. 140
References ................................................. 141
Further Reading ............................................ 143
Part III Special topics on linear stability ................... 147
8 Linear wave packets of instability waves ................... 149
8.1 Classification of wave packets" ....................... 149
8.2 Group velocity ........................................ 150
8.3 Experimental investigations of wave packets ........... 154
Exercises .................................................. 157
References ................................................. 157
Further Reading ............................................ 158
9 Transient disturbances in shear flows ...................... 159
9.1 Lift-up effect ........................................ 159
9.2 Growth of optimal disturbances ........................ 161
9.3 Experimental studies .................................. 168
Exercises .................................................. 174
References ................................................. 174
Further Reading ............................................ 176
10 Excitation of shear flow disturbances ...................... 177
10.1 Receptivity problem ................................... 177
10.2 Localized and distributed generation of laminar flow
disturbances .......................................... 179
10.3 Methodology of receptivity studies .................... 180
10.3.1 Theoretical approaches ......................... 181
10.3.2 Experimental strategy .......................... 184
10.4 Receptivity of two-dimensional boundary layers ........ 185
10.4.1 Leading-edge receptivity ....................... 185
10.4.2 Excitation of instability waves at local
boundary-layer non-uniformities ................ 186
10.4.3 Receptivity to localized unsteady
disturbances ................................... 187
10.4.4 Excitation of instability waves at laminar
boundary-layer separation ...................... 189
10.5 Receptivity of a swept-wing boundary layer ............ 190
10.6 Excitation of the Gцrtier vortices .................... 191
10.7 Excitation of streaky structures ...................... 194
10.7.1 Localized generation of streaks ................ 194
10.7.2 Distributed generation of streaks .............. 197
References ................................................. 199
Further Reading ............................................ 201
11 Secondary instabilities of shear layers .................... 207
11.1 Secondary instability in a flow modulated by the
Tollmien-Schlichting waves ............................ 207
11.2 Secondary instability in flows modulated by
streamwise vortices and streaks ....................... 209
11.3 Local high-frequency secondary instability ............ 215
References ................................................. 217
Further Reading ............................................ 219
Part IV Onset of turbulence ................................... 221
12 Nonlinear effects during the laminar-turbulent
transition ................................................. 223
12.1 Onset of nonlinearity ................................. 223
12.2 Basic nonlinear scenarios ............................. 225
12.2.1 K-and N-regimes of the laminar-turbulent
transition ..................................... 225
12.2.2 Oblique breakdown .............................. 228
12.2.3 Nonlinearity of locally separating boundary
layers ......................................... 229
12.3 Transition to turbulence in boundary layers at
a high free-stream disturbance level .................. 231
12.3.1 Linear Tollmien-Schlichting waves in the
presence of streaks ............................ 233
12.3.2 Interaction of the streaks with the Tollmien-
Schlichting waves .............................. 235
References ................................................. 237
Further Reading ............................................ 239
13 Generation of turbulence ................................... 243
13.1 Wave combinations and interimttency ................... 243
13.2 Turbulent spots ....................................... 247
13.2.1 Isolated turbulent spots ....................... 247
13.2.2 Interaction of turbulent spots ................. 250
13.2.3 Turbulent spots in the presence of other
disturbances ................................... 252
13.3 Super-late stage of the transition:'Deterministic
turbulence' ........................................... 253
References ................................................. 255
Further Reading ............................................ 257
A Basic engineering aspects of the laminar-turbulent
transition ................................................. 259
A.1 Transition prediction ................................. 259
A.2 Outline of the linear control theory .................. 262
References ................................................. 263
Further Reading ............................................ 264
Index ......................................................... 267
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