Modern experiments and numerical simulations show that the long-known coherent structures in turbulence take the form of elongated vortex tubes and vortex sheets. The evolution of vortex tubes may result in spiral structures which can be associated with the spectral power laws of turbulence. The mutual stretching of skewed vortex tubes, when they are close to each other, causes rapid growth of vorticity. Whether this process may or may not lead to a finite-time singularity is one of the famous open problems of fluid dynamics. This book contains the proceedings of the NATO ARW and IUTAM Symposium held in Zakopane, Poland, 2-7 September 2001. The papers presented, carefully reviewed by the International Scientific Committee, cover various aspects of the dynamics of vortex tubes and sheets and of their analogues in magnetohydrodynamics and in quantum turbulence. The book should be a useful reference for all researchers and students of modern fluid dynamics.
Author(s): K. Bajer, H.K. Moffatt
Series: Fluid Mechanics and Its Applications
Edition: 1
Publisher: Springer
Year: 2003
Language: English
Pages: 382
Tubes, Sheets and Singularitiesin Fluid Dynamics......Page 4
PROFESSOR ANTHONY E. PERRY - 19.2.1937 TO 3.1.2001......Page 8
Contents......Page 12
Preface......Page 16
Part I: VORTEX STRUCTURE, STABILITY ANDEVOLUTION......Page 18
1. Introduction......Page 20
2. The onset of chaos in vortex sheet flow......Page 21
3. Azimuthal waves, vortex ring merger......Page 23
Acknowledgments......Page 28
References......Page 29
1. Introduction......Page 30
3. Results......Page 32
References......Page 35
1. Introduction......Page 36
2. The basic vortex flow.......Page 37
3. Three-dimensional stability equations.......Page 38
4. Numerical simulations.......Page 40
References......Page 41
1. Introduction......Page 42
2. Perturbations on a stretched shear layer......Page 43
3. Optimal perturbations......Page 44
References......Page 47
1. Introduction......Page 48
2. Similarity solution......Page 49
3. Conclusions......Page 50
References......Page 53
1. Introduction......Page 54
2. Setting of linear stability problem......Page 56
3. Kelvin waves......Page 57
4. Linear instability of an elliptic vortex......Page 58
5. Linear instability of a vortex ring......Page 61
References......Page 65
1. Introduction......Page 66
2. Equation of motion and description of the computational algorithm......Page 67
3. Examples of the numerical results......Page 68
4. Closing remarks......Page 69
References......Page 71
1. Introduction......Page 72
2. Formulation of the problem and results......Page 73
References......Page 77
Introduction......Page 78
2. Experimental Results......Page 81
References......Page 83
Part II: SINGULAR VORTEX FILAMENTS......Page 84
2. Vortex dynamics and superfluid turbulence......Page 86
3. Complexity measures......Page 88
4. Results......Page 90
References......Page 91
1. Introduction......Page 92
2. Square-shaped configuration......Page 93
3. Diamond-shaped configuration......Page 95
References......Page 97
1. Introduction......Page 98
2. Statement of the problem......Page 100
3. Solution and analysis......Page 102
References......Page 103
1. Introduction......Page 104
2. Formulation......Page 105
3. Advection problem analysis......Page 106
References......Page 109
1. Introduction......Page 110
2. Time evolution of anisotropy of the binormal to vortex lines in a tangle......Page 112
References......Page 115
1. Introduction......Page 116
2. Basic definitions and current conservation in quantum mechanics......Page 117
5. A general method of generating vortex solutions......Page 118
References......Page 121
Part III: MAGNETIC STRUCTURE, TOPOLOGYAND RECONNECTION......Page 122
Magnetic dissipation: spatial and temporal structure......Page 124
2. Boundary driven magnetic dissipation......Page 125
3. Suspended / resumed boundary driving......Page 127
4. Coronal heating experiments......Page 128
5. Discussion and concluding remarks......Page 130
References......Page 131
1. Introduction......Page 132
2. Formation of current sheets in two dimensions......Page 133
3. Dissipation of current sheets in two dimensions......Page 135
4. Three-dimensional reconnection......Page 137
5. Can reconnect ion heat the solar corona?......Page 138
6. How do solar flares and coronal mass ejections occur?......Page 139
References......Page 141
2. The initial configuration......Page 142
3. The evolution equations......Page 144
4. Boundary-layer approximation......Page 145
5. Weak field limit......Page 146
6. Saddle point reconnection......Page 147
References......Page 149
1. Introduction......Page 150
2. Three-dimensional reconnection......Page 152
3. Reconnection solutions in HD......Page 153
References......Page 155
1. Magnetic and vortex knots as standard embeddings......Page 156
2. Helicity, linking and average crossing numbers......Page 157
3. Topological bounds on magnetic energy......Page 159
4. Vortex tangles and complexity......Page 160
References......Page 161
1. Introduction......Page 162
2. The Euler equations......Page 164
3. MHD equations......Page 165
4. Helicity conservation laws......Page 166
References......Page 167
1. Introduction......Page 168
2. Higher order invariants......Page 169
3. Generalisation of the third-order invariant......Page 170
4. Example magnetic field......Page 172
References......Page 173
1. Introduction......Page 174
2. Kinematic dynamo basics......Page 175
3. Steady spatially-periodic velocity field......Page 176
4. Slowly varying Fourier amplitudes......Page 177
5. Diffusionless asymptotics......Page 179
6. Diffusive asymptotics......Page 182
7. Summary......Page 184
References......Page 185
Part IV: VORTEX STRUCTURE INTURBULENT FLOW......Page 186
1. Introduction......Page 188
2. Small-scale model of passive-scalar mixing in turbulence......Page 189
3. Large-eddy simulation of turbulence......Page 191
4. Large-eddy simulation of channel flow......Page 192
5. Large-eddy simulation of passive-scalar mixing......Page 194
References......Page 196
1. Introduction......Page 198
2. Low-pressure vortex......Page 199
3. Tracking of individual vortices......Page 200
4. Structure of Tubular Vortices......Page 201
5. Reynolds Number Dependence......Page 203
6. Role of Vortices......Page 205
7. Concluding Remarks......Page 206
References......Page 207
1. Introduction and Overview......Page 208
2. Geometry, parameters and numerical methods......Page 210
3. Phenomena and time epochs......Page 211
4. Discussion, conclusion and future directions......Page 215
References......Page 216
1. Introduction......Page 218
2. Kinematics of passive vorticity......Page 219
3. Properties of the deformed blob......Page 221
4. Spectral analysis of a single blob......Page 222
5. Time-averaged spectra......Page 224
6. Summary......Page 226
References......Page 227
1. Introduction......Page 228
2. Wavelet algorithm for vortex extraction......Page 229
3. Application to a turbulent mixing layer......Page 231
4. Perspectives......Page 232
References......Page 233
1. Shear-stratified turbulent flows......Page 234
2. Coherent vortex extraction using wavelets......Page 236
4. Application to the potential vorticity field......Page 238
References......Page 244
1. Introduction......Page 246
2. The energy balance and the near-wall region......Page 247
3. The energy generation mechanism......Page 249
4. The inactive motions near the wall......Page 252
References......Page 256
1. Introduction......Page 258
2. Experimental facilities and test......Page 259
3. Clustering and conditional sampling analysis......Page 260
References......Page 263
1. Introduction......Page 264
2. One-parameter flows......Page 265
3. Fine scale motions......Page 266
4. The inertial subrange......Page 267
5. The geometry of dissipating fine scale motion......Page 268
References......Page 277
1. Introduction......Page 278
2. The compressible Lundgren transformation......Page 279
4. Numerical experiments......Page 280
5. Conclusion......Page 282
References......Page 283
Part V: FINITE-TIME SINGULARITY PROBLEMS......Page 284
1. Introduction......Page 286
2. Representation theory of 3-D discrete groups......Page 288
3. Blowup and Symmetry......Page 289
4. Anti-parallel vortex collapse and the group......Page 291
5. Orthogonal Dipoles and the Group......Page 292
6. Taylor-Green Vortex and the Group......Page 294
7. Octahedral Flows and the Group......Page 295
References......Page 299
1. Introduction......Page 302
2. Ideal Fluids......Page 304
3. Viscous Fluids......Page 306
4. Summary......Page 309
References......Page 310
1. Introduction......Page 312
2. The ideal MHD problem......Page 314
3. Concluding remarks......Page 320
References......Page 321
1. Introduction......Page 322
2. Basic equations......Page 323
3. Breaking of vortex lines......Page 325
4. Super-weak collapse......Page 327
5. Numerical results......Page 328
Acknowledgments......Page 332
References......Page 333
1. Introduction......Page 334
2. The Symmetries of the Kida flow......Page 336
3. Sufficient condition for finite time singularity......Page 337
4. Positivity of......Page 340
5. Quasi self-similarity......Page 342
References......Page 345
1. Introduction......Page 346
2. Hamiltonian dynamics of vortex filaments......Page 348
3. Singularity in long-scale nonlinear dynamics......Page 349
References......Page 351
1. Introduction......Page 352
2. Preliminaries......Page 353
3. Unique solvability......Page 354
4. Stabilisation......Page 356
References......Page 357
Part VI: STOKES FLOW AND SINGULAR BEHAVIOURNEAR BOUNDARIES......Page 358
1. Introduction......Page 360
3. Lubrication......Page 361
5. Experiment......Page 362
6. Model of the contact......Page 363
8. Summary......Page 364
References......Page 365
1. Introduction......Page 366
2. Earlier results......Page 367
3. The expansion method......Page 368
4. Summary......Page 370
References......Page 371
1. Introduction......Page 372
2. Statement of the problem and method of solution......Page 373
3. Numerical results......Page 375
References......Page 377
