Acknowledgments .............................................. xiii
1 Introduction ................................................. 1
1.1 Natural and Technological Counterflows .................. 1
1.2 Physical Mechanisms of Counterflows ..................... 2
1.2.1 Accumulation ..................................... 2
1.2.2 Swirl Effect ..................................... 4
1.2.3 Separation ....................................... 7
1.2.4 Thermal Convection ............................... 8
1.3 Counterflow Applications, Control, and Stability ........ 9
1.4 Approach ............................................... 10
2 Accumulation Counterflows ................................... 11
2.1 Conical Similarity Flows ............................... 11
2.2 Conical Jets ........................................... 13
2.2.1 Equation Reduction .............................. 13
2.2.2 Jet in the Free Space ........................... 14
2.2.3 Jet Above a Plane ............................... 15
2.3 Super-Collimated Jet ................................... 16
2.4 Capillary Jet .......................................... 20
2.4.1 Features of Capillary Jets ...................... 20
2.4.2 Conical Similarity Model of the Meniscus Flow ... 22
2.4.3 Numerical Simulations of the Cone-Jet Flow ...... 24
3 Bifurcation of Swirl in Conical Counterflows ................ 28
3.1 Observations of Spontaneous Swirl Appearance ........... 28
3.2 Bifurcation of Swirl as Symmetry Breaking .............. 30
3.2.1 Reduction to a Boundary-Value ODE Problem ....... 30
3.2.2 Necessary Conditions for Swirl Bifurcation ...... 32
3.3 Swirl Appearance in Capillary Flows .................... 33
3.3.1 Two-Medium Flows ................................ 33
3.3.2 Swirl Origination ............................... 34
3.3.1 Swirl Development ............................... 35
3.3.4 Two-Cell Circulation ............................ 37
3.3.5 Collimated Annular Jets ......................... 38
3.3.6 Swirl Bifurcation in the Meniscus Flow .......... 40
3.4 Swirl Appearance in Electro-Vortex Flows ............... 43
3.4.1 Problem Formulation ............................. 43
3.4.2 Forced Swirl .................................... 44
3.4.3 Multi-Cell Counterflows ......................... 46
3.4.4 Self-Swirling ................................... 51
3.5 Mechanism of Swirl Appearance in Conical Flows ......... 56
3.5.1 Comparison of Self-Swirling Capillary and
Electro-Vortex Flows ............................ 56
3.5.2 Mechanism of Swirl Accumulation ................. 56
3.5.3. Destroyed Bifurcation ................................ 58
4 Bifurcation of Counter-Swirl ................................ 60
4.1 Outline of Stability and Bifurcation Features .......... 60
4.2 Parallel Jetlike Flows ................................. 62
4.3 Secondary Flows ........................................ 62
4.4 The Lyapunov-Schmidt Method ............................ 63
4.5 Bifurcations in the Jetlike Flows ...................... 66
4.6 MHD Flow in an Annular Pipe ............................ 68
4.7 Solving Stability Problems for Large Re ................ 68
4.8 Bifurcations in the Annular-Pipe Flows ................. 70
5 Conical Counterflows Driven by Swirl ........................ 73
5.1 Swirling Jet Above a Plane ............................. 73
5.1.1 Reduction to a Boundary-Value ODE Problem ....... 73
5.1.2 Asymptotic Analysis of Two-Cell Flow ............ 75
5.1.3 Hysteresis ...................................... 79
5.1.4 Vortex Breakdown ................................ 83
5.1.5 Vortex Consolidation ............................ 85
5.1.6 Cusp Catastrophe ................................ 87
5.1.7 Near-Plane Outflow .............................. 90
5.2 A Half-Line Vortex in a Free Space ..................... 92
5.2.1 Tornado and Delta-Wing Vortices ................. 92
5.2.2 Multiple Solutions .............................. 94
5.2.3 Modeling Turbulent Vortex Breakdown ............. 98
5.3 Swirling Jets in Conical Regions ....................... 98
5.3.1 Suction Devices and Their Modeling .............. 98
5.3.2 Asymptotic Analysis ............................ 100
5.3.3 Decomposition of the Flow Force ................ 102
5.3.4 Descending One-Cell Flow ....................... 103
5.3.5 Ascending One-Cell Flow ........................ 106
5.3.6 Flow Inside the θC = 45° Cone .................. 107
5.3.7 Flow Outside the θC = 45° Cone ................. 109
5.3.8 Pressure Peak in Swirling Annular Jets ......... 112
5.4 Super-Collimation in Swirling Counterflows ............ 114
5.4.1 Bipolar Jet Induced by Vortex-Sink Accretion ... 114
5.4.2 Analysis of Super-Collimation .................. 115
5.4.3 Vortex-Wall Interaction as a Model Tornado ..... 118
6 Jetlike Swirling Counterflows .............................. 122
6.1 Power-Law Jets ........................................ 122
6.1.1 Introduction ................................... 122
6.1.2 Problem Formulation ............................ 123
6.1.3 Features of Power-Law Jets ..................... 125
6.2 Analytical Modeling of Multiple Counterflows .......... 132
6.2.1 Motivation ..................................... 132
6.2.2 Generalized Vortex Sink ........................ 134
6.2.3 Shape of the Surface of Revolution ............. 136
6.2.4 Inner Solutions ................................ 139
6.2.5 Composite Vortex Sink .......................... 142
6.2.6 Applications of the Generalized Vortex Sink .... 148
6.2.7 Applications of the Composite Solutions ........ 152
6.3 Swirling Counterflows in a Capillary Meniscus ......... 160
6.3.1. Effects of Swirling Gas Jet .................... 160
6.3.2. Analysis of Changing Flow Topology ............. 164
7 Swirling Counterflows in Cylindrical Devices ............... 168
7.1 Swirl-Decay Mechanism ................................. 168
7.1.1 Elongated Counterflows ......................... 168
7.1.2 Problem Formulation ............................ 168
7.1.3 Modeling Swirl Decay ........................... 170
7.1.4 Velocity Profiles .............................. 172
7.1.5 Pressure Distribution .......................... 173
7.1.6 End-Wall Effects ............................... 175
7.2 Modeling Counterflows in Vortex Separators ............ 177
7.2.1 Introduction ................................... 177
7.2.2 Two Flow Components ............................ 178
7.2.3 Core Flow Features ............................. 181
7.2.4 Flow Approximation Near End Walls .............. 184
7.2.5 Particle Trajectories .......................... 185
7.2.6 Pressure Distribution .......................... 187
7.2.7 Centrifugal Stratification ..................... 188
7.2.8 Summary of the Asymptotic Analysis ............. 189
7.3 Numerical Study of Vortex Breakdown and Double
Counterflow ........................................... 190
7.3.1 Technological Importance of Local and Global
Circulations ................................... 190
7.3.2 Formulation of the Numerical Problem ........... 190
7.3.3 Development of Global Counterflow as Swirl
Number Increases ............................... 192
7.3.4 Development of Global Counterflow as Re
Increases ...................................... 193
7.3.5 Comparison with the Asymptotic Theory .......... 194
7.3.6 Vortex Breakdown Development ................... 196
7.3.7 Double Counterflow Development ................. 199
7.3.8 Summary of Double Counterflow Features ......... 205
7.4 Double Counterflow in a Vortex Trap ................... 206
7.4.1 Technological Importance of Vortex Traps ....... 206
7.4.2 Development of a Global Counterflow ............ 207
7.4.3 Analytical Approximation of the Global
Counterflow in the Vortex Trap ................. 209
7.4.4 Solid Particle Trajectory in the Single
Counterflow .................................... 210
7.4.5 Double Counterflow in the Vortex Trap .......... 212
7.4.6 Solid Particle Trajectories in the Double
Counterflow .................................... 213
7.4.7 Development of Karman Vortex Street ............ 216
7.4.8 Summary of the Vortex Trap Features ............ 217
8 Separation Counterflows .................................... 219
8.1 Counterflows in a Plane Diverging Channel ............. 219
8.1.1 Brief Literature Review ........................ 219
8.1.2 Problem Formulation ............................ 220
8.1.3 Patterns of Jeffery-Hamel Counterflows ......... 221
8.1.4 Scaling ........................................ 223
8.1.5 Counting ....................................... 223
8.2 Counterflows Due to Bifurcations of Vortex Source
Flow .................................................. 225
8.2.1 Equations for Disturbances ..................... 225
8.2.2 Bifurcation Character .......................... 227
8.2.3 Phase Pattern and Asymptotic Features .......... 228
8.2.4 Spiral Vortices ................................ 229
8.3 Stability of Plane Counterflows ....................... 233
8.3.1 Approach ....................................... 233
8.3.2 Stability of Vortex-Source Flow ................ 235
8.3.3 Spatial Stability of the Jeffery-Hamel Flow .... 236
8.4 Transition Flows ...................................... 238
8.4.1 Jet in the Sink Flow ........................... 238
8.4.2 Tripolar Jet ................................... 240
8.4.3 Attachment Flow in the Diverging Channel ....... 242
8.4.4 Jet Emerging from a Slit in a Wall ............. 243
8.4.5 Jet Emerging from a Thin Plane Channel ......... 245
8.5 Summary of Plane Counterflow Features ................. 247
8.5.1 Spatial Instability ............................ 247
8.5.2 Further Applications ........................... 248
8.5.3 Limitations .................................... 250
8.6 Counterflows Due to Internal Separation in Spatial
Conical Flows ......................................... 250
8.6.1 Introduction ................................... 250
8.6.2 Governing Equations ............................ 252
8.6.3 Basic Flows .................................... 253
8.6.4 Experiment ..................................... 254
8.6.5 Linear Stability Approach ...................... 256
8.6.6 Instability of the Squire-Wang Flow ............ 257
8.6.7 Instability of Divergent Flow in a Conical
Region ......................................... 259
8.6.8 Instability of Marangoni Flow .................. 260
8.6.9 Concluding Remarks ............................. 264
9 Temperature Distribution in Swirling Counterflows .......... 266
9.1 Temperature Distribution in Conical Similarity Jets ... 266
9.1.1 Reduction of the Heat Equation ................. 266
9.1.2 Point Source of Heat in the Landau Jet ......... 267
9.1.3 Point Source of Heat in the Half-Line Vortex ... 267
9.1.4 Point Source of Heat in Long's Jet ............. 270
9.1.5 Heat Transfer in a Near-Wall Jet ............... 273
9.1.6 Summary of the Heat Transfer Features in
Conical Swirling Counterflows .................. 279
9.2 Temperature Distribution in Generalized Vortex-Sink ... 280
9.2.1 Reduction of Energy Equation ................... 280
9.2.2 Axisymmetric Temperature Distribution .......... 281
9.2.3 Spiral Thermal Distribution .................... 281
9.2.4 Species Distribution ........................... 283
9.2.5 Three-Dimensional Temperature Distribution ..... 284
9.3 Temperature Distribution in a Cylindrical
Counterflow ........................................... 285
10 Onset of Buoyancy Similarity Counterflows .................. 288
10.1 Development of Conical Buoyancy Bipolar Jets .......... 288
10.1.1 Introduction ................................... 288
10.1.2 Problem Formulation ............................ 289
10.1.3 Instability of the Rest State .................. 290
10.1.4 Weakly Nonlinear Analysis of Convection Onset .. 292
10.1.5 Development of Bipolar Convection via
Hysteresis ..................................... 294
10.1.6 Development of Strong Jets ..................... 296
10.1.7 Effects of Swirl on the Jets ................... 300
10.1.8 Stability of Conical Buoyancy-Driven Flows ..... 303
10.1.9 Concluding Remarks ............................. 306
10.2 Onset of Keplerian Buoyancy Flows ..................... 308
10.2.1 Introduction ................................... 308
10.2.2 Similarity Family .............................. 309
10.2.3 Keplerian Convection ........................... 311
10.2.4 Infinitesimal Disturbances of the Equilibrium
State .......................................... 312
10.2.5 Critical Rayleigh Numbers for Convection
Onset .......................................... 313
10.2.6 Neutral Modes for a Few Small Values of Racr ... 313
10.2.7 Concluding Remarks ............................. 314
11 Thermal Convection Counterflows ............................ 316
11.1 Model of a Free Convection Near a Black Smoker ........ 317
11.1.1 Reduction of the Boussinesq Equations .......... 317
11.1.2 Flow Features at Pr = 0 ........................ 318
11.1.3 Super-Collimation .............................. 319
11.2 Model of a Free Convection Near a Volcano ............. 321
11.2.1 Reduction of the Boussinesq Equations .......... 321
11.2.2 Flow Features at Pr = 0 ........................ 321
11.2.3 Super-Collimation .............................. 324
11.2.4 Thermal Quadruple on the Horizontal Wall ....... 326
11.2.5 Convection Inside a Conical Crater ............. 329
11.3 Centrifugal Convection ................................ 330
11.3.1 Introduction ................................... 330
11.3.2 Problem Formulation ............................ 331
11.3.3 Parallel Flow .................................. 332
11.3.4 End-Wall Effect ................................ 336
11.3.5 Rapid Rotation ................................. 338
11.4 Centrifugal Convection of a Perfect Gas ............... 339
12 Control of Vortex Breakdown ................................ 342
12.1 Introduction .......................................... 342
12.2 Experimental Study of VB Control ...................... 344
12.2.1 Experimental Setup and Technique ............... 344
12.2.2 Co-rotation .................................... 346
12.2.3 Counter-rotation ............................... 351
12.2.4 Concluding Remarks ............................. 356
12.3 Numerical Study of VB Control by Temperature
Gradients ............................................. 357
12.3.1 Problem Formulation ............................ 357
12.3.2 Numerical Procedure ............................ 358
12.3.3 Centrifugal Convection in a Rotating
Container ...................................... 359
12.3.4 Control of VB by Thermal Convection ............ 360
12.3.5 Suppressing VB by Centrifugal Convection for
Other Flow Configurations ...................... 367
12.3.6 Effects of Gravitational Convection ............ 368
12.3.7 Conclusions .................................... 368
12.4 VB Control by Adding Near-Axis Swirl and Temperature
Gradients ............................................. 369
12.4.1 Vortex Breakdown Control by Adding Near-Axis
Rotation ....................................... 369
12.4.2 Near-Axis Rotation and Axial Temperature
Gradient ....................................... 374
12.5 Concluding Remarks .................................... 377
13 Magnetic Counterflows ...................................... 379
13.1 Problem Formulation ................................... 379
13.1.1 Governing Equations ............................ 379
13.1.2 Bifurcation in a Planar Sink Flow .............. 380
13.1.3 Reduction of the MHD Equations ................. 381
13.1.4 Linear Problem for a Swirl-Free Flow ........... 382
13.2 Magnetic Field Bifurcation in the Bipolar Accretion
Flow .................................................. 383
13.2.1 Flow Map ....................................... 383
13.2.2 Nonlinear MHD Problem .......................... 383
13.2.3 Asymptotic MHD Flow as Re → ∞ .................. 384
13.2.4 Bifurcation of Magnetic Field in a Super-
Collimated Flow ................................ 385
13.3 Magnetic Field Bifurcation in the Bipolar Vortex-Sink
Accretion Flow ........................................ 386
13.3.1 Flow Map ....................................... 386
13.3.2 Analytical Solution ............................ 388
13.3.3 Development of Hysteresis ...................... 390
13.4 Magnetic Field Bifurcation Near a Point Source
of Heat and Gravity ................................... 390
13.4.1 Linear Problem ................................. 390
13.4.2 Super-Collimated Convection .................... 395
13.4.3 MHD Bifurcation in the Super-Collimated
Convection ..................................... 396
13.4.4 Nonlinear MHD Problem .......................... 397
13.4.5 Swirling MHD Flows ............................. 398
13.4.6 Separated Branches of MHD Convection ........... 399
13.4.7 Features of MHD Flows .......................... 401
13.5 Instability Nature of MHD Bifurcation ................. 403
13.5.1 Formulation of the Stability Problem ........... 403
13.5.2 Linear Stability ............................... 404
13.5.3 Nonlinear Stability ............................ 405
13.5.4 Physical Interpretation ........................ 406
13.6 Bifurcation of Magnetic Field in an Electro-Vortex
Flow .................................................. 407
13.6.1 Problem Formulation ............................ 407
13.6.2 Bifurcation of the Meridional Induction ........ 408
13.6.3 Bifurcation in the Super-Collimated Flow ....... 409
14 Stability of Conical Flows ................................. 411
14.1 Formulation of the Stability Problem ................. 411
14.1.1 Transformation of Governing Equations .......... 411
14.1.2 Equations for Infinitesimal Disturbances ....... 413
14.1.3 Boundary Conditions ............................ 414
14.1.4 Eigenvalue Problem ............................. 415
14.2 Stability of the Fluid at Rest ........................ 416
14.2.1 Modified Equations for Disturbances ............ 416
14.2.2 Spectrum for the Unbounded Still Fluid ......... 416
14.2.3 Spectrum for a Conical Region .................. 419
14.3 Instability Nature of Folds and Hysteresis in
Swirl-Free Jets ....................................... 421
14.3.1 Multiple Flow States in Swirl-Free Jets ........ 421
14.3.2 Fold-Catastrophe Instability ................... 423
14.3.3 Space-Oscillatory Instability .................. 426
14.4 Deceleration Instability of Jets ...................... 430
14.4.1 Review of Stability Studies .................... 430
14.4.2 Stability of Swirl-Free Jets ................... 431
14.5 Instability of Swirling Jets .......................... 439
14.5.1 Stability of One-Cell Flows .................... 439
14.5.2 Stability of Two-Cell Flows .................... 442
14.6 Instability Nature of Swirl Bifurcation ............... 445
14.6.1 One-Phase Flow in a Capillary Meniscus ......... 445
14.6.2 Two-Phase Flow ................................. 446
14.6.3 Instability of the Flow Driven by Electric
Current ........................................ 447
14.7 Instability of Flows Diverging Near a Surface ......... 447
14.7.1 Azimuthal Instability of the Squire-Wang
Flow ........................................... 447
14.7.2 Diverging Electro-Vortex Flow .................. 448
14.7.3 Flow Near a Glacier ............................ 449
14.8 Concluding Remarks .................................... 451
14.8.1 Inner and Outer Modes .......................... 451
14.8.2 The Role of Similarity ......................... 452
14.8.3 Unsteadiness ................................... 453
14.8.4 Deceleration Instability ....................... 453
References .................................................... 457
Index ......................................................... 467
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