Closure strategies for turbulent and transitional flows

Cover......Page 1
Half-title......Page 4
Title......Page 5
Copyright......Page 6
CONTENTS......Page 7
CONTRIBUTORS......Page 10
PREFACE......Page 13
ACRONYMS......Page 14
Introduction......Page 17
1 Introduction......Page 25
Nomenclature......Page 26
2 Reynolds-averaged Navier–Stokes formulation......Page 27
3 Linear eddy viscosity models......Page 28
3.1 Zero-equation models......Page 29
3.2 ‘Half-equation’ models......Page 31
3.3 One-equation models......Page 33
3.4 Two-equation models......Page 34
3.5 Near-wall integration......Page 39
4 Nonlinear eddy viscosity models......Page 41
4.1 Quadratic and cubic tensor representations......Page 44
4.2.1 Implicit algebraic stress model......Page 46
4.2.2 Explicit solution......Page 48
4.2.3 Curvature effects and the equilibrium assumption......Page 51
4.2.4 Turbulent transport and viscous diffusion assumption......Page 54
References......Page 59
1 Introduction......Page 63
2.1 The model equation for…......Page 64
2.2 The model equation for Epsilon......Page 70
2.3 Second-moment closure for scalar fields for high-Peclet-number flows......Page 71
2.3.1 The Model Equation for Scalar Flux…......Page 72
2.4 The algebraic stress/flux models (ASM/AFM)......Page 73
3 Advanced Differential Second-Moment Closures......Page 74
3.1 Some improvements to the modelled…equation......Page 75
3.2 Some modifications to the Epsilon equation......Page 81
4 Potential of Differential Second-Moment Closures for Modelling Complex Flow Phenomena......Page 84
5.1 The wall-function approach and its deficiency......Page 94
5.2 Models with near-wall and low-Re-number modifications......Page 96
5.2.1 A low-Re-number DSM......Page 98
6 Some Illustration of DSM Performance......Page 101
6.1 Some comments on low-Re-number DSMs......Page 110
7 Concluding Remarks......Page 112
References......Page 113
1 Introduction......Page 118
2 A TCL closure of the Reynolds stress transport equations......Page 119
2.1 Pressure-Strain Processes......Page 120
2.1.1 Mean-Strain (or ‘Rapid’) Part of Phi......Page 121
2.1.2 Turbulence (or ‘Slow’) Part of Pressure-Strain......Page 124
2.2 Dissipation......Page 125
3.1 Free Shear Flows......Page 126
3.2 Flows Near Plane Surfaces......Page 128
3.3 Flow Over Complex Surfaces......Page 130
4 Scalar flux modelling......Page 132
4.1 Mean-Strain (or ‘Rapid’) Part of Pressure-Scalar Gradient Correlation…......Page 133
4.2 Turbulence (or ‘Slow’) Part of Pressure-Scalar Gradient Correlation…......Page 136
5 Applications to the computation of the scalar field in free shear flows......Page 137
References......Page 139
1 Non-local wall effects......Page 143
2 A justification......Page 146
3 Its use with Reynolds Stress Transport equations......Page 149
3.1 Variants......Page 152
4.1 Insensitivity to the homogeneous model......Page 153
4.2 Rotating cylinder......Page 155
4.3 Rotating Channel Flows......Page 157
4.4 Axially Rotating Pipe......Page 163
4.5 Square Duct......Page 165
References......Page 166
1 Introduction......Page 169
2 Second-moment Closure......Page 171
3.1 General Considerations......Page 173
3.2 Collocated-storage, Pressure-based Algorithms......Page 175
3.3 Analysis of Fourth-Order Smoothing......Page 181
3.4 Source-Term Linearization......Page 182
3.5 Wall Conditions......Page 183
3.6 Approximation of Turbulence Convection......Page 186
3.7 Multigrid Acceleration......Page 187
3.8 Density-based Scheme......Page 189
4.1 Overview......Page 193
4.2 Prolate Spheroid......Page 194
4.3 Fin-Plate Junction......Page 195
5 Concluding Remarks......Page 200
References......Page 201
1 Introduction......Page 204
2 Two-equation model of turbulence for velocity field......Page 207
3.1 Governing Equations......Page 209
3.2 Wall-Limiting Behavior of Velocity and Temperature......Page 210
3.3.1 Modelled Epsilon equations......Page 211
3.3.2 Assessment procedure......Page 212
3.3.3 Models for assessment......Page 213
3.3.4 Assessment results......Page 214
3.4.1 Modelling the eddy diffusivity for heat Alpha......Page 217
(a) Modelling of…......Page 219
(d) Modelled Epsilon-equation......Page 221
(e) Model functions and constants......Page 222
(f ) Second-order closure modelling......Page 224
3.4.3 Modelling the k equation......Page 226
3.5 Discussion of Predictions with Proposed Models......Page 227
3.5.2 Channel flow with heat transfer (constant-heat-flux wall and constant-temperature wall)......Page 228
3.5.3 Boundary-layer flows with uniform-temperature or uniform-heat-flux wall......Page 230
3.5.4 Constant wall temperature followed by adiabatic wall......Page 231
3.5.5 Constant heat flux followed by adiabatic wall......Page 233
3.5.6 Double-pulse heat input......Page 234
4.1.1 Eddy diffusivity…......Page 237
4.1.2 Modelling the k-equation......Page 239
4.1.3 Modelling the…......Page 240
4.2 Model Performance in Thermal Fields......Page 243
4.2.2 Channel flow with a uniform wall heat flux......Page 245
4.2.3 Heat transfer in channel flow with injection and suction......Page 247
(a) Reynolds number dependence......Page 249
(b) Prandtl number dependence......Page 251
(a) Comparison of turbulence quantities in air flow......Page 255
(b) Turbulent heat transfer for various Prandtl number fluids......Page 256
5 Conclusions......Page 259
References......Page 260
1 Introduction......Page 264
2 Turbulence scales and resolution requirements......Page 266
3 Numerical methods......Page 268
4 Validation Issues......Page 271
5 Applications......Page 272
5.1 Flow visualisation......Page 274
5.2 A posteriori model testing for LES and RANS......Page 276
5.3 A priori model testing for LES and RANS......Page 277
5.4 Differential a priori for RANS......Page 279
References......Page 280
1.1 Resolution requirements of DNS......Page 283
2 Governing Equations and Filtering......Page 284
2.1 Schumann’s approach......Page 285
2.2 Filtering......Page 286
2.3 Variable filter size......Page 289
2.4 Implicit versus explicit filtering......Page 290
3.1 Introduction......Page 291
3.2 Smagorinsky model......Page 292
3.3 Dynamic procedure......Page 293
3.4 Scale similarity models......Page 295
3.5 Further models and comparative discussion......Page 296
4.1 Discretization schemes in space and time......Page 297
4.2 Analysis of numerical schemes for LES......Page 298
5 Boundary conditions......Page 301
5.1 Resolution of the near-wall region......Page 302
5.2 Wall functions......Page 303
5.3 Other approaches......Page 304
5.4 Inflow and outflow conditions......Page 305
5.5 Sample computations......Page 306
6 Concluding Remarks......Page 310
References......Page 311
1 Introduction......Page 315
2 Nonlinearity and non-locality......Page 316
3.1 Second-order, one-point [2,1] models......Page 321
3.2 Third-order, two-point [3,2] models......Page 324
4 From RDT to anisotropic two-point closures......Page 325
5.1 General features......Page 330
5.2 Pure rotation......Page 333
5.3 Pure stratified homogeneous turbulence......Page 334
6 Concluding remarks......Page 338
References......Page 340
1.1 Instantaneous conservation equations......Page 344
2.1 One-point statistics......Page 345
2.2 Joint probabilities......Page 346
2.4 Favre averaging......Page 347
3 Averaged equations......Page 348
3.1 Reynolds-stress equations......Page 349
3.2 Reynolds-flux and scalar-variance equations......Page 350
4 One-point scalar PDF closure......Page 351
5 One point joint velocity-scalar PDF closure......Page 352
References......Page 353
1 Introduction......Page 357
2 Turbulence Lengthscales in Impingement and Reattachment Regions......Page 358
2.1 Lengthscale Correction Terms......Page 359
2.2 Alternative Lengthscale Equations......Page 361
3 Turbulence Energy Production in Impingement Regions......Page 362
3.1 Weaknesses of the Eddy-Viscosity Formulation......Page 363
3.2 Kato–Launder Modification......Page 366
3.3 Durbin’s Modification......Page 367
4.2 Reynolds Stress Transport Models......Page 368
References......Page 374
1 Introduction......Page 377
2 LES Methods Used......Page 378
3.1 Square Cylinder......Page 380
3.2 Circular Cylinder......Page 388
4.1 Single Cube......Page 395
4.2 Matrix of Cubes......Page 399
5 Conclusions......Page 401
References......Page 404
1 Introduction......Page 408
2 Numerical schemes for LES in complex geometry......Page 409
3 Application to a Tube Bundle and Cylinder......Page 412
4 Fluid-Structure Coupling......Page 414
5 Acoustic Source Terms......Page 415
6 Toward Industrial LES......Page 417
Conclusions......Page 420
References......Page 421
2.1 The Second-Moment Equations......Page 423
2.2 Modelling the Dissipation and Diffusion Processes......Page 425
2.3 TCL Modelling of Pressure Interaction in Buoyancy Affected Flows......Page 426
2.4 Some Applications of the Model to Buoyant Flows......Page 428
3.1 Modelling......Page 431
3.2 Application......Page 435
References......Page 438
1 Introduction......Page 440
2 The closure strategy......Page 443
3 Algebraic models for triple correlations......Page 445
4 Third-order closure models......Page 451
5 Assessment of cumulant model......Page 455
6 Conclusions......Page 458
Appendix......Page 459
References......Page 463
1 The role of bypass transition......Page 465
2 Transition induced by free-stream turbulence......Page 468
2.1 Streaks, spots and shear filtering......Page 470
3 Transition induced by periodic passing wakes......Page 473
References......Page 478
1 Introduction......Page 480
2 Applying turbulence models to prediction of transitional flows......Page 481
2.1 Low-Reynolds Number Transport Models......Page 483
2.2 Alternative Correlation and Intermittency-Scaling Models......Page 484
2.3 Typical Results For Simple Test Cases......Page 487
3 Some simple model refinements for improving predictions......Page 490
3.1 Integral Methods......Page 491
3.3 Two-Equation Models......Page 492
3.4 Non-Linear Viscosity Models......Page 495
3.5 Reynolds Stress Transport Models......Page 497
4 Towards practical computations for engineering flows......Page 500
Acknowledgements......Page 503
References......Page 504
2 Results from by-pass transition simulations......Page 509
2.2 Reynolds Stress Balances......Page 510
2.3 Deduced Transition Mechanisms......Page 513
2.4 Results from Numerical ‘Experiments'......Page 515
2.5 Implications for Models in General......Page 516
3.1 Skin Friction Development......Page 518
3.2 Predicted Reynolds Stress and Turbulence Energy Balances......Page 520
3.3 Non-Local Pressure-Velocity Modelling Refinements......Page 521
4 Additional allowance for intermittency transport......Page 522
5 Other possible approaches......Page 528
6 Concluding remarks concerning best choice current models......Page 531
References......Page 534
1 Introduction......Page 538
2 Background: Experimental and Simulation Data......Page 539
3 Compressible Navier–Stokes Equations......Page 540
2.1 Reynolds-averaged equations......Page 541
2.2 Morkovin’s Hypothesis and the Strong Reynolds Analogy......Page 544
2.3 Turbulent transport equations......Page 548
3 Turbulent Shear Layers......Page 550
3.1.1 Law of the wall......Page 551
3.2.1 Mean density gradient......Page 553
3.2.2 Turbulent heat flux......Page 554
3.3 Mixing Layers......Page 555
3.3.1 Characteristic features......Page 556
3.3.2 Experimental and DNS results for RANS model validation......Page 561
4 Shock Wave/Turbulence Interactions......Page 567
4.1 Shock wave/homogeneous turbulence interaction......Page 569
4.1.1 Linear interaction analysis (LIA)......Page 570
4.1.2 Comparison with experimental results......Page 580
4.2 Shock/boundary-layer interaction......Page 585
5 Concluding Remarks......Page 589
Acknowledgements......Page 591
References......Page 592
1 Introduction......Page 598
2 Derivation of the Scalar PDF Equation......Page 600
2.1 Terms Appearing in the PDF Equation......Page 602
3 Solution Aspects of PDF Evolution Equations......Page 603
4 Basic Modelling Requirements of the Molecular Mixing Term......Page 604
4.1 Linear Mean Square Estimation (LMSE) closure......Page 606
4.2.1 Curl’s Mixing Model......Page 608
4.2.2 Modified Curl......Page 610
4.4 Binomial Models......Page 612
4.5 Mapping Closures......Page 615
4.6 EMST Mixing Model......Page 618
4.7 Models for the Turbulent Transport of the PDF......Page 619
5.1 Eulerian Particle Method......Page 620
5.2.1 Transport......Page 622
5.3 Stochastic Field Methods......Page 624
5.4 Chemical Reaction......Page 625
6 Applications......Page 626
6.2 Counterflow Flames......Page 627
7 Concluding Remarks......Page 634
References......Page 637
1 PDF transport equation......Page 642
2 Monte Carlo solution method......Page 644
3 Hybrid flow field model......Page 646
4.1 Langevin models......Page 648
4.2 Third-moment and turbulent scalar-flux equations......Page 650
Modeled turbulent scalar-flux equations......Page 651
4.3 Consistency......Page 652
5 Developments in Lagrangian turbulence modeling......Page 653
6 Chemical reaction......Page 654
Constrained-equilibrium (CE) model......Page 655
Intrinsic Low-Dimensional Manifold method......Page 656
7.1 Introduction......Page 657
7.2 Test case specification......Page 658
Hybrid turbulence model......Page 659
Scalar micro-mixing......Page 660
Constrained-Equilibrium results......Page 661
ILDM results......Page 663
Results: micro-and macro-mixing......Page 665
7.5 Discussion......Page 667
References......Page 668
1 Introduction......Page 675
2 VLES and Hybrid RANS/LES methods......Page 677
3 The Time-Dependent RANS (T-RANS)......Page 678
3.1 Equations......Page 680
3.2 The subscale model......Page 681
3.3 Evaluation of second moments......Page 682
4 T-RANS Simulation of Rayleigh–Bénard Convection......Page 683
4.1 Rayleigh–Bénard convection over a flat wall......Page 684
4.2 Effects of Wall Topology: Rayleigh–Bénard Convection over Wavy Walls......Page 692
4.3 The Nusselt–Rayleigh Number Correlation......Page 696
5 Conclusions......Page 697
References......Page 698
1 Introduction......Page 701
2 Use of higher moments to construct PDFs in stratified flows......Page 702
3 Constructing PDFs in the convective PBL......Page 703
4 Modeling the turbulent transport of substance......Page 713
5 Conclusions......Page 714
References......Page 716
1 Introduction......Page 718
2 Backward-facing step......Page 719
3 Flat-plate separation bubble with turbulent detachment......Page 722
4 Flat-plate separation bubble with laminar detachment......Page 726
References......Page 733
1 Introduction......Page 736
2 The filtering and inversion approach to LES......Page 739
3 Non-uniform filters and LES of complex flows......Page 744
4 Interaction between numerical and modelling errors......Page 747
5 Some guidelines for predictable LES......Page 750
References......Page 753
1.1 Linear theories......Page 756
1.2 Transport models for statistical quantities......Page 760
2 Anisotropic turbulence with dispersive waves......Page 763
3 Concluding remarks......Page 765
References......Page 768