Navier-Stokes Turbulence: Theory and Analysis

Preface......Page 5
 References......Page 7
Acknowledgements......Page 8
Contents......Page 9
Abbreviation......Page 18
List of Figures......Page 20
1 Introduction......Page 38
 1.1 Notation......Page 47
 References......Page 51
2 Navier–Stokes Equations......Page 54
 2.1 Spatial/Eulerian Description......Page 55
  2.1.1 Incompressible Fluids......Page 56
  2.1.2 Compressible Fluids......Page 58
  2.1.3 Symmetries of the Euler and Navier–Stokes pdes......Page 59
 2.2 Fundamental Properties of the Solutions for a Single Incompressible Fluid......Page 65
 2.3 Rotation and Vorticity in the Spatial Description......Page 70
  2.3.2 Vorticity pde for Compressible Fluids......Page 72
 2.4 Lamb Vector Dynamics for Incompressible Fluids......Page 74
 2.5 Material/Lagrangean Description......Page 77
  2.5.2 Incompressible Fluids......Page 78
  2.5.3 Compressible Fluids......Page 79
 2.6 Rotation and Vorticity in the Material Description......Page 81
  2.6.1 Divergence of the Material Vorticity......Page 82
  2.6.2 Material Vorticity in Plane Flows......Page 83
  2.6.3 Frozen Vector Fields......Page 84
 2.7 Velocity Gradient Tensor in the Spatial Description......Page 86
 2.8 Problems for this Chapter......Page 87
 References......Page 89
3 Basic Properties of Turbulent Flows......Page 91
 3.1 Working Definition of Turbulence......Page 99
 3.2 Asymptotic Properties of Turbulent Flows......Page 102
 3.3 Number of Degrees of Freedom of the Turbulence Attractor for Maintained Turbulence......Page 103
 3.4 Problems for this Chapter......Page 105
 References......Page 107
4 Flow Domains and Bases......Page 109
 4.1 Homogeneous Turbulence Domain calDht......Page 110
 4.2 Periodic Pipe Flow Domain calD......Page 111
  4.2.1 Schauder Basis for Scalar Fields......Page 112
  4.2.2 Schauder Basis for Vector Fields with Non-zero and Zero Dilatation......Page 113
  4.3.1 Semi-infinite Pipe: z0=0 and z1=infty......Page 116
  4.3.2 Boundary Layer Domain calDbl......Page 118
  4.3.3 Free Shear Layer Domain calDfs......Page 121
 4.4 Problems for this Chapter: Flow Domains and Bases......Page 124
 References......Page 127
 5.1 Solution Sets......Page 128
 5.2 Phase Space for the Turbulence Measure: Incompressible Fluids and Homogeneous Boundary Conditions......Page 129
 5.3 Phase Space: Noncompact Domains......Page 131
 5.4 Argument/Test Function Space for Characteristic Functionals......Page 132
 5.5 Example for a Nuclear Space......Page 133
 References......Page 134
6 Probability Measure and Characteristic Functional......Page 135
 6.1 Elementary Properties of the Turbulence Measure......Page 136
 6.2 Turbulence Measure for Incompressible Fluids......Page 138
  6.3.1 Cylinder Measures......Page 139
  6.3.2 Kolmogorov Extension Theorem......Page 140
  6.3.4 Wiener Measure......Page 142
 References......Page 146
 7.1 Elliptic fdes......Page 148
 7.2 Parabolic fdes......Page 149
 Reference......Page 150
8 Characteristic Functionals for Incompressible Turbulent Flows......Page 151
 References......Page 153
9 Fdes for the Characteristic Functionals......Page 154
 9.2 The Hopf fde for the Spatial Characteristic Functional......Page 155
  9.2.1 The Pressure Gradient in the Spatial Description......Page 158
  9.2.2 Green\'s Function for Periodic Pipe Flow......Page 162
  9.2.4 The Hopf Equation Version Ix: Stochastic External Force......Page 168
  9.2.5 The Hopf Equation II: Homogeneous Dirichlet Conditions......Page 169
  9.2.6 Spectral Version of the Hopf fde I......Page 170
  9.2.7 Convection as Diffusion in Hilbert Space......Page 171
 9.3 The Lewis–Kraichnan Equation for the Space–Time Functional......Page 172
  9.3.1 Lewis–Kraichnan fde Version I......Page 173
  9.3.2 Lewis–Kraichnan fde Version II......Page 174
  9.3.3 Variational Derivatives of a Gaussian Characteristic Functional......Page 177
  9.4.1 Variational and Standard Laplacians......Page 179
  9.4.2 Lévy-Laplacian......Page 181
  9.4.3 The Laplacians Applied to a Gaussian Characteristic Functional......Page 184
 9.5 Fde for the Characteristic Functional in the Material Description......Page 187
  9.5.1 Gateaux Derivative of Velocity with Respect  to the External Force......Page 190
 9.6 Problems for this Chapter......Page 191
 References......Page 192
10 Solution of Hopf-Type Equations in the Spatial Description......Page 194
 10.1 Steady-State Solutions......Page 195
 10.2 fde in Cartesian Coordinates......Page 196
 10.3 fde in Cylindrical Coordinates......Page 197
 10.4 Steady-State Solutions for Periodic Pipe Flow......Page 198
  10.4.1 Steady-State fde in a Solenoidal ONS Basis calBe for calN......Page 200
 10.5 Variational Hessian and the pde for θN......Page 203
 10.6 Boundary Conditions......Page 206
 10.7 Problems for this Chapter......Page 207
 References......Page 208
11 Finite-Dimensional Characteristic Functions, Pdfs and Cdfs Based on the Dirac Distribution......Page 209
 11.1 The Role of the Pressure......Page 211
 11.2 Transport Equation for the Multipoint Cdf F4N......Page 213
 11.4 Pdf Equation in Terms of Conditional Expectations......Page 226
 11.5 Problems for this Chapter......Page 229
 References......Page 231
12 Properties and Construction of Mappings......Page 232
 12.1 Elementary Properties of Mappings......Page 233
 12.2 Bijective Mappings for Probability Measures......Page 234
  12.2.1 Bijective Mappings for Characteristic Functionals......Page 235
 12.3 Construction of Mappings......Page 236
 12.4 Local and Global Mappings......Page 238
  12.4.1 Construction of Global Maps......Page 240
  12.4.2 Construction of Local Maps......Page 243
 References......Page 244
 13.1 The CCK Method......Page 246
  13.1.1 Single, Bounded Scalar in Homogeneous Turbulence......Page 247
  13.1.2 Mapping Equation for a Single Scalar......Page 250
  13.1.3 Solution of the Mapping pde for a Single Scalar......Page 255
  13.1.4 Initial Values for Mapping calX(η;0)......Page 258
 13.2 Mapping Method for the Characteristic Function......Page 263
  13.2.1 Transport pde for the Image θ[θG]......Page 267
 13.3 Problems for this Chapter......Page 274
 References......Page 275
 14.1 Single-Point Mapping Equation for Homogeneous Turbulence......Page 277
  14.1.1 Time-Dependent Reference Measures......Page 282
 14.2 Multipoint Mapping Method in the Spatial Description......Page 283
 14.3 Multipoint Mapping Method in the Material Description......Page 287
 14.4 Mapping Method for Two-Point Pdfs in the Material Description......Page 291
 14.5 A Pdf Reduction Property for Multidimensional Mappings......Page 293
 14.6 Summary......Page 294
 References......Page 295
 15.1 Fourier Transform (FT)......Page 296
 15.2 Fourier Transformed Navier–Stokes Equations......Page 297
 References......Page 301
 16.1 External Intermittency......Page 303
  16.2.1 The Intermittency Generating Function  of Wilczek et al. Wilczek13......Page 306
 References......Page 308
 17.1 Energy Cascade......Page 310
 17.2 Basic Assumptions of Kolmogorov–Onsager Theory......Page 313
 17.3 Remarks......Page 317
 References......Page 318
 18.1 Homogeneous and Isotropic Turbulence......Page 319
 Reference......Page 321
19 Length and Time Scales......Page 322
 19.2 Scale Relations......Page 326
 19.3 Taylor\'s Length Scale......Page 327
 19.4 Macroscales......Page 328
 References......Page 329
20 The Structure of Turbulent Flows......Page 330
 20.3 Characteristic Scales......Page 331
  20.4.1 Line-Type Structures......Page 332
  20.4.2 Surface-Type Structures......Page 333
  20.4.3 Volume-Type Structures......Page 335
 20.5 Material/Lagrangean Structures......Page 336
  20.5.1 Vortex Rings......Page 337
  20.5.2 Chaplygin–Lamb Dipole......Page 342
  20.5.3 Hill\'s Vortex as Solution of the Bragg–Hawthorne pde......Page 345
 20.6 Large-Scale Motion in Jupiter\'s Atmosphere......Page 348
 20.7 Problems for this Chapter......Page 350
 References......Page 352
21 Wall-Bounded Turbulent Flows......Page 356
 21.1 Statistical Description of Turbulent Flows......Page 357
  21.1.1 Moment Equations Derived from the Navier–Stokes pdes......Page 359
  21.1.2 Moment Equations Derived from the Leray Version  of the Navier–Stokes pdes......Page 361
 21.2 Pipe Flow......Page 366
  21.2.1 (iii.2) Numerical Aspects......Page 367
  21.2.2 (iii.3), (iv.1) Theoretical Aspects......Page 369
  21.2.3 Fully Developed Turbulent Flow Through Circular Pipes......Page 370
 21.3 DNS Examples......Page 371
  21.3.1 Turbulent Pipe Flow......Page 372
  21.3.2 Compressible Boundary Layer Flows......Page 376
 21.4 Problems for this Chapter......Page 378
 References......Page 379
22 The Limit of Infinite Reynolds Number for Incompressible Fluids......Page 381
 22.1 A Bounded Vector Field Derived From Vorticity......Page 384
 22.2 The Kolmogorov–Onsager Conjecture......Page 386
  22.2.1 Experimental and Numerical Evidence......Page 387
  22.2.2 Analysis......Page 388
  22.2.3 Weierstrass Function......Page 391
 22.3 Duchon–Robert Distribution......Page 393
 22.4 The Physical Limit: Extrapolation of Smits\' Formula for Pipe Flow......Page 394
 22.5 The Mathematical Limit......Page 397
  22.5.1 Hopf fde for the Limit (ii) of Infinite Reynolds Number......Page 398
 22.6 Problems for this Chapter......Page 400
 References......Page 401
 23.2 Limits......Page 403
 23.3 Caratheodory, Hausdorff and Box Dimensions......Page 404
 23.5 Levi-Civita Symbol......Page 406
 23.6 Multi-index Notation......Page 407
 23.7 Function Spaces......Page 408
 23.9 Ergodicity......Page 412
 23.10 Leray–Stokes Operator......Page 414
 23.11 n-Sphere and n-Ball......Page 416
 23.12 Differentiation of Functionals and Measures......Page 417
  23.12.1 Lebesgue Integral......Page 427
 23.13 Examples......Page 439
 23.14 Problems for Sect. 23.14......Page 440
 23.15 Spherical Coordinates in Rn......Page 441
  23.16.1 Integrand xk......Page 443
  23.16.2 Integrand x2k......Page 444
  23.16.3 Integrand xkxl for kneql......Page 445
  23.16.4 Surface Average of an Analytic Function Over the n-Sphere......Page 446
  23.17.1 Properties of n-Sphere and n-Ball as nrightarrowinfty......Page 450
  23.17.2 Limit nrightarrowinfty of Lebesgue/Riemann Surface Area and Volume......Page 451
 23.18 Gaussian Measures......Page 455
  23.18.1 Example: Lebesgue/Riemann and Gaussian Volumes of an n-Ball for n>2......Page 456
  23.18.2 Gaussian Measure on the Space of Infinite Sequences......Page 460
  23.18.3 Characteristic Functional......Page 463
 23.19 Lagrangean and Eulerian Statistics......Page 464
 23.20 Problems for Sects. 23.17 to 23.20......Page 466
 References......Page 467
24 Appendix B: Example for a Measure  on a Ball in Hilbert Space......Page 469
 Reference......Page 471
25 Appendix C: Scalar and Vector Bases  for Periodic Pipe Flow......Page 472
 25.1 Modified Jacobi Polynomials calPna,b(r)......Page 473
 25.3 Test Function Space calNp: Scalar Fields......Page 475
 25.4 Projected Functionals......Page 476
 25.5 Bases for the Test Function Space calNp......Page 477
 25.6 Discrete Fourier Transform w.r.t. the Angular Coordinate......Page 478
 25.7 Scalar Parity Conditions for Azimuthal Wavenumbers k0......Page 479
  25.7.2 Construction of Scalar Basis Functions......Page 480
 25.8 Radial Modes for Homogeneous Dirichlet Conditions......Page 481
 25.9 Representation of Scalar Fields......Page 484
 25.11 Bases in Phase calVp and the Test Function calNp Spaces......Page 488
 25.12 Vector Parity Conditions for Azimuthal Wavenumbers k0......Page 489
 25.13 Design Requirements for the Vector Basis Functions......Page 490
 25.14 Representation of a Vector Field......Page 491
 25.15 Construction of the Vector Modes for Homogeneous Dirichlet Conditions......Page 493
 25.16 Radial Shape Function......Page 494
 25.17 (i) Construction of a General Vector Basis......Page 497
 25.18 Divergence of the Basis......Page 498
  25.18.1 Numerical Test Functions: Non-zero Divergence......Page 499
 25.20 Solenoidal Vector Basis for Doubly Periodic Flow Fields, First Method......Page 501
  25.20.1 Modes k=m=0......Page 502
  25.20.2 Modes k>0,m=0......Page 503
  25.20.3 Axial Modes mneq0......Page 504
  25.20.4 Radial Derivative......Page 506
  25.20.5 Incomplete Orthonormality of the Solenoidal Vector Components......Page 507
 25.21 Solenoidal Vector Basis for Doubly Periodic Flow Fields, Second Method......Page 511
  25.21.2 Modified Vector Potential Set Up in Jacobi–Fourier Form......Page 513
 25.22 Gram–Schmidt Orthonormalization......Page 515
 References......Page 527
 26.1 Domain of Definition......Page 528
 26.2 Function Spaces: Scalar Fields......Page 530
 26.3 Poisson pde for the Disturbance Pressure......Page 531
 26.4 Fourier Transform of the Poisson pde......Page 532
  26.4.1 Parity Conditions for the Modal Form of Scalar Fields......Page 536
 26.5 Representation of the Pressure......Page 537
 26.6 Solution of the BVP for the Complementary  Pressure ph......Page 538
 26.7 Green\'s Theorem and its Application......Page 545
 26.8 Solution of the Poisson pde for the Disturbance Pressure......Page 557
 26.9 Leray Version of the Navier–Stokes pdes......Page 558
 References......Page 560
 27.1 Exact Moment pdes for Fully Developed Turbulent Channel Flow......Page 561
  27.1.1 The Closure Problem......Page 568
  27.1.2 Empirical Mean Velocity Profiles: The Nickels Interpolant......Page 574
 References......Page 577
28 Appendix F: Solutions to Problems......Page 579
 References......Page 729
 BookmarkTitle:......Page 732
Index......Page 734