ORGANISING COMMITTEE

J. Jimenez (Director) Departamento de Mecanica de Fluidos Esc. Teen. Sup. Ing. Aeronauticos PI. Cardenal Cisneros 3 28040 -Madrid SPAIN

P. Huerre Dept. Mecanique Ecole Polytecnique, Departamentale 36 91128 Palaiseau cedex FRANCE

A. Lifian Departamento de Mecanica de Fluidos Esc. Teen. Sup. Ing. Aeronauticos PI. Cardenal Cisneros 3 28040 -Madrid SPAIN

Y. Pomeau Groupe de Physique Statistique de l'ENS 24, Rue Lhomond 75231 Paris Cedex 05 FRANCE

P. Saffman Applied Mathematics (217-50) California Institute of Technology Pasadena, CA 91125 USA

357

LECTURERS

G. Ahlers Lehrstuhl fur Theoretische Physik IV Universitat Bayreuth Postfach 101251 8580 Bayreuth GERMANY

J .C. Antoranz Dpto. Fisica Fundamental Univ. Educacion a Distancia Apto. 50487 28080 Madrid SPAIN

Roberto Benzi Universita di Roma Dip. di Fisica Via Orazio Raimondo 1-00173 Roma ITALY

T. Bohr Niels Bohr Institute Blegdamsvej, 17 2100 Copenhagen DENMARK

H.R. Brand FB 7 Department of Physics University of Essen D 4300 Essen 1 GERMANY

H. Chate Institut de Recherche Fondamentale DPh-G/PSRM, CEN Saclay F 91191 Gif-sur-Yvette cedex FRANCE

J-M. Chomaz CNMR 42 Av. G. Coriolis 31057 Toulouse FRANCE

P. Clavin Laboratoire de Recherche en Combustion Universite de Provence Centre Saint Jerome 13397 Marseille Cedex 13 FRANCE

R.J. Deissler Center for Nonlinear Studies Los Alamos National Laboratory Los Alamos, N .M. 87545 USA

Stephan Fauve Ecole Normale Superieur Lyon 46 allee d'ltalie 69364 Lyon Cedex 07 FRANCE

359

H. Fiedler Technische Universitat Berlin Hermann-Fottinger Institut Strasse des 17 Juni D-1000 Berlin 12 GERMANY

M. Gharib Dept. of AMES, R-011 Univ. California San Diego La Jolla, CA. 92093 USA

J.D. Gibbon Mathematics Department Imperial College London, SW7 2AZ UNITED KINGDOM

J .A. Hernandez Ramos Departamento de Mecanica de Flufdos Esc. Teen. Sup. Ing. Aeronauticos PI. Cardenal Cisneros 3 28040 -Madrid SPAIN

E. Hopfinger Institut de Mechanique de Grenoble B.P.68 38402 S.-Martin d'Heres Cedex FRANCE

M. Jensen Nordita Blegdamsvej, 17 2100 Copenhagen DENMARK

J.e. Lasheras Department of Mechanical Engineering University of Southern California Los Angeles, CA 90089-1191 USA

360

M. Lesieur Institut de Mechanique de Grenoble B.P.53X 38041 Grenoble Cedex FRANCE

J.L. Lumley Sibley School of Mechanical and Aerospace Engineering Upson and Grumman Hall Cornell Bniversity Ithaca, NY 14853-7501 USA

L. Kleiser DFLVR Institute for Theoretical Fluid Mechanics Bunsenstrasse 10 D-3400 Gottingen GERMANY

R. MacKay Mathematics Institute University of Warwick Coventry CV 4 7 AL UNITED KINGDOM

J.M. Massaguer E.T.S. Ingenieros Telecomunicaci6n Universidad Politecnica de Cataluiia Jordi Girona Salgado 31 08034 Barcelona SPAIN

O. Metais Institut de Mechanique de Grenoble B.P.53X 38041 Grenoble Cedex FRANCE

P. Moin Mechanical Engineering Department Stanford University Stanford, CA 94305 USA

D. Papailiou Dept. Mechanical Engineering University of Patras llio 26001 Patras GREECE

C. Perez Garda Dpto. Ffsica Facultad de Ciencias U niversidad de Navarra E-31080 Pamplona SPAIN

A. Pumir Laboratoire de Physique Statistique Ecole Normale Superieure 24, Rue Lhomond F-75231 Paris Cedex 05 FRANCE

W. Reynolds Mechanical Engineering Department Stanford University Stanford, CA 94305 USA

Renzo llicca Dept. AMTP Silver Street Cambridge, CB3 9EW UNITED KINGDOM

Jean-Pierre llivet Observatoire de Nice, BP139 06003 Nice Cedex FRANCE

Erika Roesch Max-Planck Inst. fUr Stromungsforschung Bunsenstr. 10 D-3400 Gottingen GERMANY

A. Roshko Aeronautics Department California Institute of Technology Pasadena, CA 91125 USA

Miguel A. Rubio Dpto. Ffsica Fundamental Univ. Educaci6n a Distancia Apto. 50487 28080 Madrid SPAIN

C. Simo Dept. Matematicas Aplicadas y AnaIisis Facultad de Matematicas Universidad de Barcelona Plaza de la Universidad Barcelona SPAIN

C. Van Atta Department of Applied Mechanics and Engineering Sciences (R-013) University of California, San Diego La Jolla, Ca. 92093-0413 USA

M.G. Velarde Facultad de Ciencias Univ. Educaci6n a Distancia Apto. 60141 28071 Madrid SPAIN

D. Walgraef Service de Chimie-Physique Universite Libre de Bruxelles Campus Plaine, C.P. 231 B-1050 Bruxelles BELGIUM

I. Wygnanski Faculty of Engineering Tel-Aviv University Ramat-Aviv Tel-Aviv, 69978 Israel ISRAEL

361

N. Zabusky Dept. Mechanical and Aerospace Eng. College of Engineering Rutgers University, P.O. Box 909 Piscataway, N J 08855-0909 USA

Stephane Zaleski Lab. Physique Statistique Ecole Normale Superieure 24 Rue Lhomond 75231 Paris cedex 05 FRANCE

362

W. Zimmerman Lehrstuhl fur Theorische Physik U niversitat Bayreuth Postfach 101251 8580 Bayreuth GERMANY

PARTICIPANTS

J.C. Agiif IBM Scientific Centre Paseo Castellana 4 28046 -Madrid SPAIN

Roque Corral Departamento de Mecanica de Flufdos Esc. Teen. Sup. Ing. Aeromiuticos PI. Cardenal Cisneros 3 28040 -Madrid SPAIN

Owen E. Cote Air Force Office of Scientific Research European Off. Aerospace Res. & Dev. 233 Old Marylebone Rd. London NW1 5TH UNITED KINGDOM

M. de la Torre Dpto. Ffsica Fundamental Univ. Educacion a Distancia Apto.50487 28080 Madrid SPAIN

C. Dopazo Dept. Mecanica de Fluidos E. T .S.1. Industriales Marfa Zambrano 50 Polfgono Actur 50015, Zaragoza SPAIN

Miguel A. Fernandez Sanjuan Dpto. de Ffsica ETS Arquitectura Universidad Politecnica de Madrid 28040 Madrid SPAIN

M. Gaster Engineering Department Cambridge University Cambridge, CB3 9EW UNITED KINGDOM

F. Higuera Departamento de Mecanica de Flufdos Esc. Teen. Sup. Ing. Aeronauticos PI. Cardenal Cisneros 3 28040 -Madrid SPAIN

P. Juvet Mechanical Engineering Department Stanford University Stanford, CA 94305 USA

Carlos Martel Departamento de Mecanica de Flufdos Esc. Teen. Sup. Ing. Aeronauticos PI. Cardenal Cisneros 3 28040 -Madrid SPAIN

363

Roy E. Reichenbach Aeronautics and Mechanics Branch US Army European Res. Off. 233 Old Marylebone Rd. London NWI 5TH UNITED KINGDOM

Ezequiel del Rio-Fernandez Dpto. Fisica Fundamental Univ. Educacion a Distancia Apto. 50487 28080 Madrid SPAIN

Maurice Rossi Mecanique Theorique Universite de Paris VI Tour 66- 4, Place Jussieu 75230 Cedex 05, Paris FRANCE

J aume Timoneda Matematicas Aplicadas i AmUisis Univ. de Barcelona Gran VIa 585 08091 Barcelona SPAIN

364

Douglas A. Varela 15 Flag St. Massachusetts Institute of Technology Cambridge, MA 02139 USA

P.D. Weidman Department of Mech. Engineering University of Colorado Boulder, Co. USA

J. Zufiria IBM Scientific Centre Paseo Castellana 4 28046 -Madrid SPAIN

INDEX

Absolute instabilities, 23, 24-25, 58, 63 Acoustic excitation, 89, 90-91, 93 Active control of shear flow, 58 Aliasing errors, 124, 125 Analytic function theory, 135 Anisotropic diffusion terms, 321, 322, 326 Annular chaos, 242 Anti-Fourier transforms, 195 Argon laser technique, 16 Artificial compressibility, 177 Axial forcing

in laminar, co-flowing forced jets, 95, 96, 99,101,102,107

in round jets, 89, 90 Axisymmetric forcing, 97, 100, 343 Azimuthal forcing

in laminar, co-flowing forced jets, 95, 96, 97,99,100,102,105,107

puffs and, 343

Barber-pole instabilities, 251 Benard problem, 247-256, see also Parallel

flows Benney-Lin equations, 278 Betchov-Da Rios equations, 257-260 Bifurcations, 89, 350

in dynamical systems theory, 211, 218-219 Hopf, 320, 322-323, 327 noise effect on, 183-188 nonlinear oscillators and, 289-292,

293-294, 295 pattern formation and, 319, 320, 322-323,

325,326,327 puffs and, 275, 345 subcritical instabilities in, see Subcritical

bifurcations Billows, 143, 145, 146, 147-148, 149, 150,

151, 152, 337, see also Waves Biot-Savart law, 203, 266 Blasius theory, 27, 277 Blooming jets, 89, 94 Bluffbodies, 43, 49, 51-53 Boltzmann equations, 339 Boundary layers

DNS of, 123, 128-130, 167, 170-172,332 dynamical systems theory and, 211-219

see also Dynamical systems theory

Boundary layers (cont'd) in laminar, co-flowing forced jets, 101 in thermal turbulence, 167-170, 175 vortex shedding and, 17 in wall jets, 68, 73

Boussinesq equations, 155, 160, 162, 170, 177, see also Non-Boussinesq conditions,

Braid regions DNS of, 145, 146 in laminar, co-flowing forced jets, 100,

102, 104 Brown-Rebollo concentration probe, 4-5 Brunt-Vaisala frequencies, 14, 161 Bubble formation, 133-141

core bursting and, 207 equations in, 134 evolution of disturbances in, 137-138 one-dimension solutions to, 138-139 two-dimensional evolution and, 139-141 uniform state instabilities in, 135-137

Buoyancy, 19, 162, 168,247,250 Burgers vortices, 201, 202, 203

Cantor sets, 230, 299 Cartesian coordinates, 159, 249, 276 Cauchy transformations, 262 Central manifold theory, 191 CGL equations, see Complex

Ginzburg-Landau equations Channel flows, 135-137, 140,141,334 Chaotic advection, 3 Chaotic attractors, 290, 297 Chaotic dynamics, 111, 112 Chaotic repellors, 299 Closed flows, 251-255 Closure model, of bubble formation, 134 Coanda effect, 59 Codimension-two bifurcation theory, 287 Coherent chaos, 297-302 Coherent motion, 81, 82 Coherent structures, 58, 68, 203, 211, 354

DNS of, 143-152,332,333 Complex Ginzburg-Landau (CGL) equations,

309-316 lattice theorem for, 311-312,315

Contractible chaos, 242 Convective chaos, 302-304

365

Convective instabilities, 58, 136, see also Nonlinear convective

instabilities Cooperative instabilities, 203 Core bursting, 202, 207 Corrsin-Oboukov theory, 157,158,159 Couette flow, 169, 170, 176,250-251,277,

345, see also Taylor-Couette flow Coupled map lattices, 297-307

coherent to incoherent chaos in, 297-302 convective chaos and, 302-304 power laws in, 304

Cylinders, 51-53, 115, 117 Dense fluidized beds, 133-141, see also

Bubble formation Desingularization, 204 DiffusiW, 155, 156, 157, 158, 162 Direct numerical simulation (DNS), 123-130,

352 of coherent structures, 143-152, 332, 333 as experiments, 331-339 of intermittency, 221,222 minimal flow unit in, 123, 128-130 outflow boundary conditions and, 125-128 of puffs, 341 ofRBC, 170-172, 176 spatial discretization in, 124-125

Dissipation, 222, 230, 310, 353, 354 noise effect on, 183-188

Double helices, 17 Doubly periodic channel flows, 247, 248, 249,

250,252,254 Drag

in dynamical systems theory, 211, 218-219 external excitation and, 79, 82, 83, 86

Duffing oscillator, 287 Dynamical systems theory, 211-219, 234

energy transfer model in, 213 equations in, 212-213 physical interpretation of, 214-217 Proper Orthogonal Decomposition in,

211-212, 218 wall region flow implications in, 213-214

Eckhaus instabilities, 321 Eddies, 349

in dynamical systems theory, 214, 217 in Kolmogorov spectra, 221 in large scale vortices, 33 in mixing transitions, 4, 9, 10 in thermal convection, 173

E.D.Q.N.M. calculations, 158, 159 Einstein convention, 276 Electro-convection, 183, 184 Energy transfer model, 213 Enstrophy, 261-263,353 Equilibrium hypothesis, 33, 38 Equivalent iterative map, 292-294 Euler equations, 243, 309, 351

in fIlamentation, 207 invariants of, 257, 259, 260-263 puffs and, 344

366

Euler equations (cont'd) in reconnection, 206 vortex line stretching and, 265, 266, 269

Expansion theorem, 260 External excitation, 67-86

in forced flows, 67, 75-86 in unforced flows, 67, 68-75, 76, 78, 82, 84,

85,86 Extrusive filamentation, 206

Fary-Milnor theorem, 260 Fencheltheorem, 260 Filamentation, 202, 206-207, see also Vortex

filaments Finite amplitude instabilities, 274-278 Finite-difference simulations, 124-125, 126 Floquet multipliers, 235, 291, 295 FLOSIAN, 146 Flow behind a backwards-facing step, 143,

149-150,152 Flow visualizations, see also specific types

of {iQrtler instability, 29 oflaminar, co-flowing forced jets, 99 of large scale vortices, 34 of mixing transitions, 3, 9 of vortex shedding, 13

Fluctuations, 319,320, 324-327 Forced flows, 67, 75-86 Fourier modes

in bubble formation, 137 in coupled map lattices, 301, 307 DNS of, 124, 125, 146 in dynamical systems theory, 212, 217 external excitation effect on, 82 RCs and, 194, 195 L.E.S. of, 156, 157 in mixing transitions, 9 in parallel flows, 250, 256 pattern formation and, 320 in Ruelle-Takens route, 236 in shell model, 222 vortex line stretching and, 266

Fractals, 230, 348-349 Frechet derivatives, 276 Free-shear flows, 68, 75, 97

DNS of coherent structures in, 143-152

mixing transitions in, 3-10, see also Mixing transitions

Free-shear layers, 3,5,8,96,99 Froude numbers

in bubble formation, 134 L.E.S. of, 161, 162 vortex shedding and, 13-14, 19

Galerkin projections, 211, 212 Galerkin truncations, 255-256 Gaussian distributions, 176

L.E.S. of, 159-160, 162 Gaussian white noise, 325, 326 Gaussian window, 137 Gauss linking number, 261

Ginzburg-Landau equations, 25, 191, 278, 319, see also Complex Ginzburg-Landau equations

Gledzer model, 222 Global instabilities, 58 Gortler instabilities, 23-31

forcing effects on, 29-30 forcing means used in, 28 geometry with counter-profile, 27-28,29,

31 geometry without counter-profile, 27, 29,

31 Gravity,161 Green's function, 23

Hairpin vortex filaments, 145, 146, 149 Hamiltonian systems, 207, 225, 259-260, 348,

351,353,354 Hard turbulence, see Strong turbulence Hartree approximations, 325 Hasimoto transformations, 258, 260 Heisenberg parameters, 213, 214, 215 Helical-pairing instabiliw, 143, 147-149, 151,

152 Helicity

in Betchov-Da Rios equations, 259-260 in Euler equations, 260-262 in homogeneous fluids, 17 puffs and, 345 in round jets, 89, 90, 91, 93

Helmholtz oscillator, 97, 287-295 equivalent iterative map and, 292-294 intermittency in, 289-292, 294

Heteroclinic cycle, 219 Hexagonal convective cells (HCs), 191-197

under non-Boussinesq conditions, 191-192, 194-196, 197

stationary defects in, 196-197 High-order upwind-based schemes, 125 Hilbert-Schmidt theory, 211 Hill spherical vortices, 201 Homogeneous fluids, 13, 14-17 Homogeneous turbulence, 335, 349 Hopfbifurcations, 320, 322-323, 327 Horseshoe vortices, 28 Hot-wire anemometry, 335, 337

of Gortler instabilities, 26, 30 in mixing transitions, 3 in round jets, 90, 94 of vortex shedding, 18,21,52-53 in wall jets, 68,75

Hydrodynamical turbulence, 297 Hyperbolic attractors, 234, 244 Hyperbolicity, 233, 237 Hyperbolic-tangent velocity profile, 143, 144 Hypersonic re-entry vehicles, 334 Hysteresis

in Gortler instability, 24,29 in nonlinear oscillators, 288, 290, 291

Icosahedral lattices, 325 Incoherent chaos, 297-302

Individual instabilities, 203 Infinitesimal perturbations, 134, 135 Inflow boundary conditions, 125-128 Inhomogenuos chemical reactions, 297 Instabilities, see specific types Instantaneous Lyapunov exponents, 222 Intermittency, 221-230, 355

dynamical systems theory on, 202, 218 fractal structure of, 230 L.E.S. of, 155-164, see also Large-eddy

simulation Lyapunov exponents and, 221, 222,

225-230 in nonlinear oscillators, 287, 289-292, 294,

295 shell model of, 221, 222-225

Intrusive filamentation, 206 Inviscid theory, 309,310, 313

in laminar, co-flowing forced jets, 99, 100 in mixing transitions, 5 in thermal convection, 169, 170

Isotropic turbulence, 349 DNS of, 124, 125, 126, 127, 152,335 L.E.S. of, 155, 156-160, 161, 162, 163 pattern formation and, 321, 322

Jacobian matrices, 225, 252, 253

KAM theory, 238 Kapitza instabilities, 278 Kaplan-Yorke dimension, 225, 226 Karhunen-Loeve expansion, see Proper

Orthogonal Decomposition Karman vortex streets, 111-119, 350

control of, 115-119 differential equations in, 112, 114-115,

117,118,119 model for, 113-115

Kelvin-Helmholtz instabilities DNS of, 143, 144, 146, 147-148, 149, 151 in large scale vortices, 43 mixing transitions and, 3, 5, 7, 9 in vortex shedding, 14, 17, 19

Kinetic energy, 352, 353 DNS of, 130, 143 L.E.S. of, 157-158, 159, 161, 162, 163, 164

Knotted vortices, 260, 261, 262 Kolmogorov spectra, 349, 351, 352, 354-355

coupled map lattices and, 304 intermittency and, 221-230, see also

Intermittency L.E.S. of, 157, 159, 164 mixing transitions and, 5, 9 vortex dynamics and, 201, 202, 204

Korteweg-DeVries equations, 207 KPP problem, 345 Krutzsch instability, 203

Lagrangian derivatives, 260, 265 Laminar, co-flowing forced jets, 95-107

axial forcing in, 95, 96, 99, 101, 102, 107

367

Laminar co-flowing forced jets (cont'd) azimuthal forcing in, 95, 96, 97, 99, 100,

102,105,107 Laminar flows, 3, 58, 75, 207, 334, see also

Laminar, co-flowing forced jets Landau equations, 63 Langevin equations, 304 Laplace terms, 302, 313 Large coherent structures, 68 Large-eddy simulation (L.E.S.), 155-164

of isotropic turbulence, 155, 156-160,161, 162, 163

of stably stratified turbulence, 155, 156, 160-163

of viscosity, see under Viscosity Large scale vortices, 33-49

stationary boundary conditions in, 58-59 turbulence memory and, 33, 34, 38 vortex decay in, 33, 38 vortex dipoles in, 43, 44 vortex generation in, 33, 38, 43, 45, 46, 48,

49 vortex pairing in, 33, 38, 43

Laser-induced fluorescence technique, 29 Lattice theorem, 311-312, 315 Law of the wall, 73, 74, 354 Legendre transforms, 112, 225, 303 L.E.S., see Large-eddy simulation LIA, see Localized induction approximation Linear instabilities, 206 Linearly tapered cylinders, 51-53 Linear oscillations, 287 Linear stability theory, 302 Local instabilities, 58 Localized induction approximation (IJA),

257-258,259-260,263 Lorenz model, 255 Lyapunov eigenvectors, 222, 227, 229 Lyapunov exponents, 226-227, 234

bifurcations and, 277 in coupled map lattices, 297, 299, 300,

301,304 instantaneous, 222 intermittency and, 221, 222, 225-230 Karman vortex streets and, 111 in Ruelle-Takens route, 235 s-waves and, 279

Macho turbulence, 237, see also Strong turbulence

Magnetic energy, 262 Melting theory of two-dimensional solids, 326 Miller integrators, 288 Minimal flow unit, 123, 128-130 Mixing layers, 3, 202, 350

DNS of, 123, 128, 143, 144-145 external excitation effect on, 68 Ruelle-Takens route and, 237 stationary boundary conditions and, 57,

59-64 tearing and, 203

Mixing transitions, 3-10, 123

368

Mixing transitions (cont'd) DNSof,333 small scale structures and, 4, 7-8, 9 streamwise vortices and, 3, 6-7, 9 vortex pairing and, 3, 8

Mode-locking strips, 239, 240 Modulational stability, 314 Moses method, 347, 350 Moving equilibrium, 33 Moving-front solutions, 279 Mushroom-type vortices, 145

Navier-Stokes equations, 52, 243, 309-310, 316,347,350,352

bifurcations and, 183, 188, 273, 274, 275 DNS of, 222, 336, 338, 339 in dynamical systems theory, 212, 214, 216 intermittency and, 221, 222, 224-225 L.E.S. of, 156 in mixing transitions, 9 in spatially developing mixing layers, 144 in temporal mixing layers, 146, 149 in thermal convection, 250, 251 vortex dynamics and, 201, 202 vortex line stretching and, 265, 266,

268-269 Nematic liquid crystal, 183, 184,320 NLSE, see Nonlinear Schrodinger equation Noise, see also White noise

in coupled map lattices, 299 dissipation and, 183-188 DNSand,338 in dynamical systems theory, 218

Nonaxisymetric oscillating instabilities, 14 Non-Boussinesq conditions, 191-192, 194-196,

197,319 Nonlinear absolute instabilities, 23, 24-25 Nonlinear convective instabilities, 23-31, see

also Gortler instabilities Nonlinear instabilities

absolute, 23, 24-25 convective, see Nonlinear convective

instabilities DNS of, 124, 125 in filamentation, 206-207

Nonlinear oscillators, 287-295 equivalent iterative map and, 292-294 intermittency in, 287, 289-292, 294,295 Karman vortex streets and, 112, 115

Nonlinear Schrodinger equation (NLSE), 310 invariants of, 257, 258,260 s-waves and, 278, 279, 280-281

Nonlinear wavetrains, 138 Nontrivial orthogonal coordinates, 253 Nonvariational systems, 278-285 Nozzles

of laminar, co-flowing forced jets, 95, 96-97,99,100,101,104,105,107

of round jets, 89-90, 90, 91,93 of wall jets, 67, 68, 69-70,76,79,86

Numerical simulation, see Direct numerical simulation

Nusseltnumbers,167,168,170,171,173, 175,195

Oberbeck-Boussinesq approximations, 192 Oceanic measurements, 164 Ocean surface wave turbulence, 347 One-dimensional turbulence, 311, 347 Open flows, 248-250, 274-278 Orbits,235,239,242,292,353 Orthogonal coordinates, 253 Oscillatory instability, 253, 254, 255 Outflow boundary conditions, 125-128

Pade approximations, 124 Parallel flows, 247-256

poloidal field in, 252, 253-254, 256 subcritical bifurcations in, 273, 274-278 symmetry breaking in, 248, 254-256 three-dimensional, 248, 252-255 toroidal fields in, 252-253, 256 two-dimensional, 248

Passive control of shear flow, 57-59 Pattern formation, 319-327

flow field effects on, 320-322 fluctuations and, 319, 320, 324-327 spatial forcing in, 322-324

Peak-valley-counting technique, 8 Peclet numbers, 255-256 Penrosetilings, 325 Periodic attractors, 297 Periodic orbits, 235, 239, 242, 292 Plasma turbulence, 347 Plume bursting, 173, 174, 175 Poincare invariants, 353 Poincare return maps, 291 Poiseuille flows, 249, 250, 277, 321, 345 Poloidal fields, 252, 253-254, 256 Polymers, 59, 217, 218-219 Polynomial conservation laws, 258 Pomeau-Manneville intermittencies, 287 Porous disks, 34, 36, 37, 43 Potential energy, 161, 162 Power laws, 76,304 Prandtl numbers, 353

HCsand,l94 L.E.S. of, 157, 158-159, 162 in thermal convection, 247-256, see also

Parallel flows Preston tubes, 73, 74 Primary instabilities, 58 Proper Orthogonal Decomposition, 211-212,

218 Pseudo-heliciw, 257, 259, 263 Pseudo-spectral methods, 143, 146, 156 Puffs, 275, 341-345 Pulsating instabilities, 14

Rayleigh-Benard convection (RBC), 167, 167-177,350, see also Rayleigh numbers

bifurcations and, 183, 187 boundary layer in, 167-170 DNSof,170-172,176

Rayleigh-Bernard convection (RBC) (cont'd) under non-Boussinesq conditions,

191-192, 194-196 pattern formation and, 322, 324 in shear absence, 173-175, 176 shear induced, 172-173

Rayleigh numbers, 187, 188,192, 194,248, 255, see also Rayleigh-Benard convection

RBC, see Rayleigh-Benard convection RD equations, see Reaction diffusion

equations Reaction diffusion (RD) equations, 273, 274,

275-276 Recirculation zones, 14, 15, 21 Reconnection, 202, 204-206 Resonance regions, 239 Reynolds numbers, 309, 349, 350, 351, 352,

353,354,355 bifurcationsand,188,273,274,275,285 bluff bodies in, 51-53 in bubble formation, 134 DNS of, 123, 128, 143, 147, 331, 332,

333-334,335,338 in dynamical systems theory, 217 in Karman vortex streets, Ill, 113, 115,

116,350 in laminar, co-flowing forced jets, 95, 96,

99,101 large scale vortices and, 38 L.E.S. of, 156, 157,159 in mixing transitions, 3, 4, 5-6, 7, 8, 9, 10,

333 in parallel flows, 247, 253, 254 puffs and, 341, 342, 343, 344, 345 in reconnection, 204 in round jets, 89-94 thermal convection and, 170, 173 vortexdynamicsand,202 vortex line stretching and, 265, 269 vortex shedding at, 13-21, 51-53, see also

Vortex shedding vortex splitting at, 51-53 in wall jets, 67-86, see also External

excitation Reynolds stresses

bifurcations and, 275 DNS of, 123, 127, 130 in dynamical systems theory, 211, 212,

213,214,216-217 external excitation and, 67, 71, 75, 79, 86 mixing transitions and, 6 parallel flows and, 250 puffs and, 344, 345

Riblets, flow over, 123, 129, 130 RNG theory, 351 Round jets, 89-94 Ruelle-Takens route, 233-244

criticisms of, 237-238 experimental observations on, 235-

236 uniform flows in, 233, 238-243

Runge-Kutta method, 137, 289

369

St. Venant's principle, 213 Schmidt numbers, 146, 337 Schwartz functions, 260, 262, 266 Secondary corrugation, 133 Secondary instabilities, 58 Shape instabilities, 253 Shear flows, 33, 123, 321, 349, see also

Free-shear flows stationary boundary conditions and, see

Stationary boundary conditions Shear instabilities, 247-256, see also Parallel

flows Shear layers, see also Free-shear layers

in large scale vortices, 43, 44, 47, 49 RBC in, 167-177, see also

Rayleigh-Benard convection in round jets, 89, 90, 91, 92, 93, 94 vortex shedding and, 17

Shear stresses, 73, 75, 79, 84 Shell models, 221, 222-225 Short-wave instabilities, 96, 203 Shroudedjets, 90, 91, 92, 94 Single helices, 17 Singularity formation, 202, 203-204 Sinuous instabilities, 14, 17-18 Sinusoidal driving force, 117, 118 Skin friction

DNSof,336 external excitation effect on, 67, 68, 73,

74,75,81,86 Skin-friction drag, see Drag Slugs

bifurcations and, 276 bubble formation and, 133, 135, 138, 139,

140, 141 Gortler instabilities and, 25 puffs and, 341, 342, 343

Small boxes, 274 Small scale structures, 4, 7-8, 9 Smoke-wire flow visualization, 52, 60 Soft turbulence, see Weak turbulence Spanwise forcing, 96, 99, 100 Spatial discretization, 124-125 Spatial forcing, 322-324 Spatially developing mixing layers, 127, 143,

144, 149-151, 152 Spectral simulations, 124-125, 126 Spheres,34,38,39,40,41,42,43

vortex shedding from, 13-21, see also Vortex shedding

Spiky turbulence, 310, 313, 315 Stable localized structures, see S-waves Stably stratified turbulence, 155, 156, 160-163 Stanton tubes, 73 Static control of shear flow, 58-59 Stationary boundary conditions, 57-65

control characteristics in, 62-63 feedback in, 57, 58, 61, 62, 64

Stochastic attractors, 234 Stochastic fields, 183,184,186,187,188,212 Stokes flow, 250, 344, 351 Straining field, 203, 204, 205

370

Strange attractors, 235,236,237,242,243, 244,348

defined,234 Stratified decay calculations, 156 Stratified fluids, 14, 19-21 Streamwise channels, 133 Streamwise forcing, 96, 97, 99, 100 Streamwise velocity, 67, 68, 76, 125, 214 Streamwise vortices

DNS of, 129, 130 Gortler instabilities and, 28 in laminar, co-flowing forced jets, 100,

101,102,104 mixing transitions and, 3, 6-7, 9 stationary boundary conditions and, 58

Strong turbulence, 309-316, see also Complex Ginzburg-Landau equations

Strouhal numbers in laminar, co-flowing forced jets, 95 in round jets, 89, 91 at stationary boundary conditions, 60, 63 in vortex shedding, 13, 17, 18, 19 in wall jets, 76

Stuart vortices, 146 Subcritical bifurcations, 273-286

in Gortler instabilities, 23, 24 parallel flow and, 273, 274-278 s-waves and, see S-waves

S-waves, 278-285 moving-front solutions of, 279-280 NLSE and, 278, 279, 280-281

Swift-Hohenberg equations, 324, 325 Swirl instabilities, 253, 254 Symmetry breaking

in parallel flows, 248, 254-256 pattern formation and, 319, 321

Taylor-Couette flows, 111,275,345,350, see also Couette flows

Taylor expansions, 112, 282 Taylor hypothesis, 125, 230 Taylor vortices, 277 Tearing, 203 Temporal mixing layers, 127, 143,146-149 Test Field Model closure calculations, 159 Thermal convection, 167-177,247-256, see

also Prandtl numbers, in thermal convection; Rayleigh-Benard convection

Thermal turbulence boundary layers in, 167-170, 175 shear absent in, 173-175 shear induced, 172-173

Thermohaline convection, 255 Thomson's circulation theorem, 17 Three-dimensional boundary layers, 128-130 Three-dimensional parallel flows, 248,

252-255 Three-dimensional small scale structures, 4,

7-8,9 Three-dimensional turbulence, 309, 311, 347,

348,352,354 Thual-Fauve solitions, 345

1broidal chaos, 242 1broidal fields, 252-253, 256, 343 n-Torus, 235, 237 ~Torus,235,237,239,240,241 3-Torus, 235, 236

uniform flows on, 233, 238-243 Translative instabilities, 149 Truly incompressible method, 177 Thrbulence memory, 33, 34, 38 Thrbulent coupled map lattices, see Coupled

map lattices Thrbulent patches, see Vortex patches Thrbulent puffs, see Puffs Thrbulent scalar, 155-164, see also

Large-eddy simulation Thrbulent wall jets, see Wall jets '!\vo-dimensional boundary layers, 128-130 '!\vo-dimensional parallel flows, 248 '!\vo-dimensional shear layers, 167-177, see

also Rayleigh-Benard convection '!\vo-dimensional turbulence, 309, 347, 353 '!\vo-dimensional wakes, 33-49, see also Large

scale vortices

Unforced flows, 67, 68-75, 76, 78, 82, 84, 85, 86, 143

SUB = behind a backwards-facing step, 143, 149-150,152

Uniform flows, 233, 238-243 Uniform state instabilities, 135-137 Uniform wavetrains, 138

van der Pol oscillators, 111, 287 Velocity

bubble formation and, 133, 135, 136 DNS of, 124, 125, 126, 127, 129, 147,336 external excitation and, 81, 82 intermittency and, 222 in Karman vortex streets, 113-114 in laminar, co-flowing forced jets, 97 L.E.S. of, 155, 156-157, 158, 159, 160-163 in round jets, 89, 90, 91, 92, 93 as vorticity substitute, 353 in wall jets, 68-75, 76-79, 84, 86

Vertical channels, 139 ViscosiW, 354

in dynamical systems theory, 219 intermittency and, 222 in Kolmogorov theory, 352, 355 large scale vortices and, 43, 49 L.E.S. of, 155, 156, 157, 158,162,163-164 mixing transitions and, 9 RBC and, 167, 169 stationary boundary conditions and, 59 vortex line stretching and, 266, 269 in wall jets, 67, 68, 70, 73, 84, 86, 169

Vortex decay, 33, 38 Vortex dipoles, 43, 44 Vortex dynamics, 201-207

defined,201 Vortex filaments, 203, 205, see also

Filamentation

Vortex filaments (cont'd) Betchov-Da Rios equations and, 257-260 DNS of, 145-146, 149

Vortex generation, 33, 38, 43, 45, 46, 48, 49 Vortex lines, 337

stretching of, 265-269, 354 Vortex loops, 101, 104,105, 106, 107 Vortex pairing

DNS of, 143, 144 Gortler instability and, 28 in large scale vortices, 33, 38, 43 mixing transitions and, 3, 8

Vortex patches, 207, 273, 274-275, 277, 285 filamentation of, 202, 206-207 fusion and fission of, 202-203

Vortex rings, 99-100, 203, 343 Euler equations and, 262 in laminar, co-flowing forced jets, 95, 96,

101,102,103,104 puffs and, 341, 345 reconnection and, 204 in round jets, 89

Vortex shedding, 13-21 in bluff body wakes, 51-53 in homogeneous fluids, 13, 14-17 instability mode frequencies in, 17-18 in large scale vortices, 34, 38 in stratified fluids, 14, 19-21

Vortex sheets, 204 Vortex splitting, 51-53 Vortex streets, 38, see also Karman vortex

streets Vortex tubes, 99-100

core bursting in, 202, 207 DNSof,337 Euler equations and, 261, 262-263 in laminar, co-flowing forced jets, 96, 101 reconnection in, 202, 204-206 singularity formation and, 204

Vorticity DNSof,150 in dynamical systems theory, 219 in laminar, co-flowing forced jets, 95-107,

see also Laminar, co-flowing forced jets

in round jets, 91 velocity as substitute for, 353

Vorticity links, 260-263

Wakes, 68,117,203, see also '!\vo-dimensional wakes

in homogeneous fluids, 13, 14-17 Karman vortex streets and, 115 in stratified fluids, 19-21

Wall jets, 67-86, 336, see also External excitation

Wall stresses, 80, 81, 86 Wave breaking, 206,207 Wavelengths, 138, 139 Wave patterns, 320, 322-324 Waves, 202, 203, see also Billows Wavetrains, 138

371

Weak turbulence, 309-316, see also Complex Ginzburg-Landau equations

Whitehead links, 261, 262, 263 White noise, 144

372

Wimpy turbulence, 237, 273, see also Weak turbulence

Wmding ratios, 238, 239

Zeldovich themy, 345