Computational Fluid Dynamics

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface to the First Edition......Page 21
Preface to the Revised Second Edition......Page 24
PART ONE: PRELIMINARIES......Page 25
1.1.1 HISTORICAL BACKGROUND......Page 27
1.1.2 ORGANIZATION OF TEXT......Page 28
1.2 ONE-DIMENSIONAL COMPUTATIONS BY FINITE DIFFERENCE METHODS......Page 30
1.3 ONE-DIMENSIONAL COMPUTATIONS BY FINITE ELEMENT METHODS......Page 31
Stiffness (Diffusion or Viscosity) Matrix......Page 33
Neumann Boundary Vector......Page 34
1.4.1 FVM VIA FDM......Page 35
1.5 NEUMANN BOUNDARY CONDITIONS......Page 37
1.5.1 FDM......Page 38
1.5.3 FVM VIA FDM......Page 39
1.5.4 FVM VIA FEM......Page 40
FDM......Page 41
FEM......Page 42
FVM via FDM......Page 43
1.6.2 NEUMANN BOUNDARY CONDITIONS......Page 44
FDM......Page 45
FEM......Page 46
FEM......Page 47
1.7 SUMMARY......Page 48
REFERENCES......Page 50
2.1 CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS......Page 53
2.2 NAVIER-STOKES SYSTEM OF EQUATIONS......Page 57
2.3 BOUNDARY CONDITIONS......Page 62
2.4 SUMMARY......Page 65
REFERENCES......Page 66
PART TWO: FINITE DIFFERENCE METHODS......Page 67
3.1 SIMPLE METHODS......Page 69
3.2 GENERAL METHODS......Page 70
Backward Difference Formulas......Page 72
Central Difference Formulas......Page 73
3.3 HIGHER ORDER DERIVATIVES......Page 74
3.4 MULTIDIMENSIONAL FINITE DIFFERENCE FORMULAS......Page 77
3.5 MIXED DERIVATIVES......Page 81
3.6 NONUNIFORM MESH......Page 83
Higher Order Accuracy for the First Order Derivatives......Page 84
3.8 ACCURACY OF FINITE DIFFERENCE SOLUTIONS......Page 85
REFERENCES......Page 86
4.1.1 FINITE DIFFERENCE FORMULATIONS......Page 87
Line Gauss-Seidel Iteration Method......Page 89
Alternating Direction Implicit (ADI) Method......Page 90
4.2 PARABOLIC EQUATIONS......Page 91
Forward-Time/Central-Space (FTCS) Method......Page 92
Crank-Nicolson Method......Page 95
4.2.3 ALTERNATING DIRECTION IMPLICIT (ADI) SCHEMES......Page 96
4.2.4 APPROXIMATE FACTORIZATION......Page 97
4.2.6 THREE DIMENSIONS......Page 99
4.2.7 DIRECT METHOD WITH TRIDIAGONAL MATRIX ALGORITHM......Page 100
Euler’s Forward Time and Forward Space (FTFS) Approximations......Page 101
Euler’s Forward Time and Backward Space (FTBS) Approximations – First Order Upwind Scheme......Page 102
Lax- MethodWendroff......Page 104
Richtmyer Multistep Scheme......Page 105
MacCormack Multistep Scheme......Page 106
Lax-Wendroff Method......Page 107
Beam-Warming Implicit Method......Page 109
4.4 BURGERS’ EQUATION......Page 111
DuFort-Frankel Explicit Scheme......Page 112
MacCormack Implicit Scheme......Page 113
4.4.2 RUNGE-KUTTA METHOD......Page 114
4.5.2 EVALUATION OF SOURCES OF ERRORS......Page 115
Diffusive Transport......Page 116
Convective Transport......Page 117
4.6.1 DETERMINATION OF JACOBIANS AND TRANSFORMED EQUATIONS......Page 118
4.6.2 APPLICATION OF NEUMANN BOUNDARY CONDITIONS......Page 121
4.7.1 ELLIPTIC EQUATION (HEAT CONDUCTION)......Page 122
4.7.2 PARABOLIC EQUATION (COUETTE FLOW)......Page 124
Implicit Schemes......Page 125
4.7.4 HYPERBOLIC EQUATION (SECOND ORDER WAVE EQUATION)......Page 127
4.7.5 NONLINEAR WAVE EQUATION......Page 128
REFERENCES......Page 129
5.1 GENERAL......Page 130
5.2 ARTIFICIAL COMPRESSIBILITY METHOD......Page 131
5.3.1 SEMI-IMPLICIT METHOD FOR PRESSURE-LINKED EQUATIONS (SIMPLE)......Page 132
5.3.2 PRESSURE IMPLICIT WITH SPLITTING OF OPERATORS......Page 136
Two-Dimensional Vorticity Transport Equation......Page 139
Three-Dimensional Vorticity Transport Equations......Page 141
5.5 SUMMARY......Page 142
REFERENCES......Page 143
CHAPTER SIX: Compressible Flows via Finite Difference Methods......Page 144
6.1.1 GOVERNING EQUATIONS......Page 145
(a) Artificial Viscosity with Nonconservative Equation......Page 147
(b) Artificial Viscosity with Conservative Equation......Page 149
(c) Artificial Compressibility......Page 150
(d) Artificial Flux or Flux Upwinding......Page 151
(e) Over-Relaxation Scheme......Page 152
6.2 EULER EQUATIONS......Page 153
6.2.1.1 Quasilinearization of Euler Equations......Page 154
6.2.1.2 Eigenvalues and Compatibility Relations......Page 156
6.2.1.3 Characteristic Variables......Page 158
6.2.2 CENTRAL SCHEMES WITH COMBINED SPACE-TIME DISCRETIZATION......Page 160
6.2.2.2 Lax-Wendroff Second Order Scheme......Page 162
6.2.2.3 Lax-Wendroff Method with Artificial Viscosity......Page 163
6.2.2.4 Explicit MacCormack Method......Page 164
6.2.3 CENTRAL SCHEMES WITH INDEPENDENT SPACE-TIME DISCRETIZATION......Page 165
6.2.4.1 Flux Vector Splitting Method......Page 166
6.2.4.2 Godunov Methods......Page 169
Roe’s Approximate Riemann Solver......Page 171
(1) Variable Extrapolation – MUSCL Approach......Page 172
(2) Flux Extrapolation Approach......Page 173
6.2.6 SECOND ORDER UPWIND SCHEMES WITH HIGH RESOLUTION (TVD SCHEMES)......Page 174
(1) Definition of High Resolution Schemes......Page 175
(2) TVD Schemes with Limiters......Page 178
(3) Time Integration Methods for TVD Schemes......Page 181
6.2.7 ESSENTIALLY NONOSCILLATORY SCHEME......Page 187
6.2.8 FLUX-CORRECTED TRANSPORT SCHEMES......Page 189
6.3 NAVIER-STOKES SYSTEM OF EQUATIONS......Page 190
6.3.1 EXPLICIT SCHEMES......Page 191
6.3.2 IMPLICIT SCHEMES......Page 193
6.3.3 PISO SCHEME FOR COMPRESSIBLE FLOWS......Page 199
6.4.1 GENERAL......Page 202
6.4.2 PRECONDITIONING MATRIX......Page 203
6.5.1 BASIC THEORY......Page 204
6.5.2 FLOWFIELD-DEPENDENT VARIATION PARAMETERS......Page 207
6.5.3 FDV EQUATIONS......Page 209
6.5.4 INTERPRETATION OF FLOWFIELD-DEPENDENT VARIATION PARAMETERS......Page 211
6.5.5 SHOCK-CAPTURING MECHANISM......Page 212
6.5.6 TRANSITIONS AND INTERACTIONS BETWEEN COMPRESSIBLE AND INCOMPRESSIBLE FLOWS......Page 215
6.5.7 TRANSITIONS AND INTERACTIONS BETWEEN LAMINAR AND TURBULENT FLOWS......Page 217
6.6.1 ARTIFICIAL VISCOSITY FLUX LIMITERS......Page 219
6.6.2 FULLY IMPLICIT HIGH ORDER ACCURATE SCHEMES......Page 220
6.7.1.1 One-Dimensional Boundary Conditions......Page 221
Characteristic Boundary Conditions......Page 222
Extrapolation Methods......Page 225
Characteristic Extrapolation Methods......Page 227
6.7.1.3 Nonreflecting Boundary Conditions......Page 228
6.7.2 NAVIER-STOKES SYSTEM OF EQUATIONS......Page 229
Case 1......Page 231
6.8.2 TRIPLE SHOCK WAVE BOUNDARY LAYER INTERACTIONS USING FDV THEORY......Page 232
6.9 SUMMARY......Page 237
REFERENCES......Page 238
7.1 GENERAL......Page 242
7.2.1 NODE-CENTERED CONTROL VOLUME......Page 243
7.2.2 CELL-CENTERED CONTROL VOLUME......Page 247
7.2.3 CELL-CENTERED AVERAGE SCHEME......Page 249
7.3.1 3-D GEOMETRY DATA STRUCTURE......Page 251
7.3.2 THREE-DIMENSIONAL FVM EQUATIONS......Page 256
7.4 FVM-FDV FORMULATION......Page 258
REFERENCES......Page 263
PART THREE: FINITE ELEMENT METHODS......Page 265
8.1 GENERAL......Page 267
8.2 FINITE ELEMENT FORMULATIONS......Page 269
8.3 DEFINITIONS OF ERRORS......Page 278
Energy Norm Error......Page 279
Matrix Norms......Page 280
8.4 SUMMARY......Page 283
REFERENCES......Page 284
9.1 GENERAL......Page 286
9.2.1 CONVENTIONAL ELEMENTS......Page 288
9.2.2 LAGRANGE POLYNOMIAL ELEMENTS......Page 293
9.2.3 HERMITE POLYNOMIAL ELEMENTS......Page 295
Cartesian Coordinate Triangular Elements......Page 297
Triangular Element with Origin on One Side......Page 300
Natural Coordinate Triangular Element......Page 302
Lagrange and Hermite Elements......Page 308
9.3.3 QUADRILATERAL ISOPARAMETRIC ELEMENTS......Page 310
9.4.1 TETRAHEDRAL ELEMENTS......Page 322
9.4.2 TRIANGULAR PRISM ELEMENTS......Page 326
9.4.3 HEXAHEDRAL ISOPARAMETRIC ELEMENTS......Page 327
9.5 AXISYMMETRIC RING ELEMENTS......Page 329
9.6 LAGRANGE AND HERMITE FAMILIES AND CONVERGENCE CRITERIA......Page 330
REFERENCES......Page 332
10.1.1 TWO-DIMENSIONAL ELLIPTIC EQUATIONS......Page 333
(a) Standard Approach......Page 339
(b) Lagrange Multipliers Approach......Page 342
10.1.3 SOLUTION PROCEDURE......Page 344
10.1.4 STOKES FLOW PROBLEMS......Page 348
Mixed Methods......Page 349
Penalty Methods......Page 350
10.2.1 PARABOLIC EQUATIONS......Page 351
Temporal Parameter......Page 353
Explicit Scheme......Page 354
Implicit Scheme......Page 355
10.2.2 HYPERBOLIC EQUATIONS......Page 356
10.2.3 MULTIVARIABLE PROBLEMS......Page 358
10.2.4 AXISYMMETRIC TRANSIENT HEAT CONDUCTION......Page 359
10.3.1 CONJUGATE GRADIENT METHODS (CGM)......Page 361
10.3.2 ELEMENT-BY-ELEMENT (EBE) SOLUTIONS OF FEM EQUATIONS......Page 364
10.4.1 SOLUTION OF POISSON EQUATION WITH ISOPARAMETRIC ELEMENTS......Page 366
10.4.2 PARABOLIC PARTIAL DIFFERENTIAL EQUATION IN TWO DIMENSIONS......Page 367
REFERENCES......Page 370
11.1 BOUNDARY AND INITIAL CONDITIONS......Page 371
11.1.1 INCOMPRESSIBLE FLOWS......Page 372
11.1.2 COMPRESSIBLE FLOWS......Page 377
11.2.1 LINEARIZED BURGERS’ EQUATIONS......Page 379
Numerical Diffusion......Page 381
11.2.2 TWO-STEP EXPLICIT SCHEME......Page 382
Euler Method......Page 386
Leapfrog Method......Page 387
Crank-Nicolson Method......Page 388
11.2.4 CONVERSION OF IMPLICIT SCHEME INTO EXPLICIT SCHEME......Page 389
11.2.5 TAYLOR-GALERKIN METHODS FOR NONLINEAR BURGERS’ EQUATIONS......Page 390
11.3 NUMERICAL DIFFUSION TEST FUNCTIONS......Page 391
11.3.1 DERIVATION OF NUMERICAL DIFFUSION TEST FUNCTIONS......Page 392
11.3.2 STABILITY AND ACCURACY OF NUMERICAL DIFFUSION TEST FUNCTIONS......Page 393
11.3.3 DISCONTINUITY-CAPTURING SCHEME......Page 400
11.4.1 GENERALIZED PETROV-GALERKIN METHODS FOR UNSTEADY PROBLEMS......Page 401
11.4.2 SPACE-TIME GALERKIN/LEAST SQUARES METHODS......Page 402
11.5.1 NEWTON-RAPHSON METHODS......Page 404
11.5.2 ELEMENT-BY-ELEMENT SOLUTION SCHEME FOR NONLINEAR TIME......Page 405
11.5.3 GENERALIZED MINIMAL RESIDUAL ALGORITHM......Page 408
11.5.4 COMBINED GPG-EBE-GMRES PROCESS......Page 415
11.5.5 PRECONDITIONING FOR EBE-GMRES......Page 420
11.6.2 PURE CONVECTION IN TWO DIMENSIONS......Page 423
11.7 SUMMARY......Page 426
REFERENCES......Page 428
12.1.1 MIXED METHODS......Page 431
12.1.2 PENALTY METHODS......Page 432
12.1.3 PRESSURE CORRECTION METHODS......Page 433
12.1.4 GENERALIZED PETROV-GALERKIN METHODS......Page 434
12.1.5 OPERATOR SPLITTING METHODS......Page 435
12.1.6 SEMI-IMPLICIT PRESSURE CORRECTION......Page 437
12.2 VORTEX METHODS......Page 438
Three-Dimensional Biharmonic Equation with Stream Function......Page 439
The Curl of Three-Dimensional Vorticity Transport Equations......Page 441
12.2.2 TWO-DIMENSIONAL ANALYSIS......Page 442
12.2.3 PHYSICAL INSTABILITY IN TWO-DIMENSIONAL INCOMPRESSIBLE FLOWS......Page 443
Three-Dimensional Vorticity Transport Equations......Page 445
REFERENCES......Page 448
13.1 GOVERNING EQUATIONS......Page 450
Nondimensional Form of Navier-Stokes System of Equations......Page 452
13.2.1 TAYLOR-GALERKIN METHODS......Page 454
13.2.2 TAYLOR-GALERKIN METHODS WITH OPERATOR SPLITTING......Page 457
13.2.3 GENERALIZED GALERKIN METHODS......Page 459
13.3.1 NAVIER-STOKES SYSTEM OF EQUATIONS IN VARIOUS VARIABLE FORMS......Page 460
13.3.2 THE GPG WITH CONSERVATION VARIABLES......Page 463
13.3.3 THE GPG WITH ENTROPY VARIABLES......Page 465
13.3.4 THE GPG WITH PRIMITIVE VARIABLES......Page 466
13.4 CHARACTERISTIC GALERKIN METHODS......Page 467
13.5 DISCONTINUOUS GALERKIN METHODS OR COMBINED FEM/FDM/FVM METHODS......Page 470
13.6.1 BASIC FORMULATION......Page 472
13.6.2 INTERPRETATION OF FDV PARAMETERS ASSOCIATED WITH JACOBIANS......Page 475
13.6.3 NUMERICAL DIFFUSION......Page 477
13.6.4 TRANSITIONS AND INTERACTIONS BETWEEN COMPRESSIBLE AND......Page 478
13.6.5 FINITE ELEMENT FORMULATION OF FDV EQUATIONS......Page 479
The Neumann boundary conditions......Page 482
(2) Two-dimensional Supersonic Flows (Euler Equations) with Two-Step TGM......Page 484
(3) Examples for FDV Methods......Page 486
(4) Driven Cavity Flow Problems to Test Compressibility/Incompressibility Characteristics......Page 489
(5) Hypersonic Flow Solutions by the FDV Method, M = 20, Re = 300,000, with Impinging Shock Wave on a Flat Inlet Combustion Chamber......Page 491
REFERENCES......Page 493
14.1 SPECTRAL ELEMENT METHODS......Page 496
Chebyshev Polynomials......Page 497
Legendre Polynomials......Page 500
14.1.2 SPECTRAL ELEMENT FORMULATIONS BY LEGENDRE POLYNOMIALS......Page 501
14.1.3 TWO-DIMENSIONAL PROBLEMS......Page 505
14.1.4 THREE-DIMENSIONAL PROBLEMS......Page 509
14.2.1 LSM FORMULATION FOR THE NAVIER-STOKES SYSTEM OF EQUATIONS......Page 512
14.2.2 FDV-LSM FORMULATION......Page 513
14.2.3 OPTIMAL CONTROL METHOD......Page 514
14.3 FINITE POINT METHOD (FPM)......Page 515
14.4.1 SHARP FIN INDUCED SHOCK WAVE BOUNDARY LAYER INTERACTIONS......Page 517
14.4.2 ASYMMETRIC DOUBLE FIN INDUCED SHOCK WAVE BOUNDARY......Page 520
REFERENCES......Page 523
15.1 GENERAL......Page 525
15.2.1 BURGERS’ EQUATIONS......Page 526
Generalized Petrov-Galerkin with PISO......Page 532
(2) FVM with PISO Approach......Page 534
(3) FVM with Upwind Finite Elements......Page 535
15.2.3 THREE-DIMENSIONAL PROBLEMS......Page 536
(1) Two-Dimensional Euler Equations, Scramjet Flame Holder Problem......Page 537
(3) Backward Facing Step with Forced Convection......Page 540
15.4 SUMMARY......Page 541
REFERENCES......Page 542
CHAPTER SIXTEEN: Relationships Between Finite Differences and Finite Elements and Other Methods......Page 543
(2) Parabolic Equations......Page 544
(3) Hyperbolic Equations......Page 546
(2) Lax-Wendroff Scheme......Page 548
(4) Explicit MacCormack Scheme......Page 549
(7) TVD Scheme......Page 550
(1) Taylor-Galerkin Methods (TGM) with Convection and Diffusion Jacobians......Page 552
(2) Taylor Galerkin Methods (TGM) with Convection Jacobians......Page 554
(3) Generalized Petrov-Galerkin......Page 555
Green’s Function and Boundary Integral Equation......Page 556
Helmholtz Equations......Page 557
Diffusion Equations......Page 558
Eulerian Difference Equations......Page 559
Lagrangian Difference Equations......Page 561
16.4.4 MONTE CARLO METHODS (MCM)......Page 562
REFERENCES......Page 564
PART FOUR: AUTOMATIC GRID GENERATION, ADAPTIVE METHODS, AND COMPUTING TECHNIQUES......Page 565
(a) Lagrange Polynomials......Page 567
(b) Hermite Polynomials......Page 568
(c) Cubic Spline Functions......Page 569
17.1.2.1 Domain Vertex Method......Page 571
17.1.2.2 Transfinite Interpolation Methods (TFI)......Page 579
17.2.1.1 Derivation of Governing Equations......Page 585
17.2.1.2 Control Functions......Page 591
17.2.2 HYPERBOLIC GRID GENERATOR......Page 592
17.2.2.1 Cell Area (Jacobian) Method......Page 594
17.2.2.2 Arc-Length Method......Page 595
17.3.1 ELLIPTIC PDE METHODS......Page 596
17.3.1.1 Differential Geometry......Page 597
17.3.1.2 Surface Grid Generation......Page 601
17.3.2.1 Points and Curves......Page 603
(a) Lagrange Polynomial......Page 604
(c) Bezier Curve......Page 605
17.3.2.2 Elementary and Global Surfaces......Page 607
17.3.2.3 Surface Mesh Generation......Page 608
(3) Creation of Mesh......Page 609
17.4 MULTIBLOCK STRUCTURED GRID GENERATION......Page 611
REFERENCES......Page 614
18.1 DELAUNAY-VORONOI METHODS......Page 615
18.1.1 WATSON ALGORITHM......Page 616
18.1.2 BOWYER ALGORITHM......Page 621
18.1.3 AUTOMATIC POINT GENERATION SCHEME......Page 624
18.2 ADVANCING FRONT METHODS......Page 625
Mesh Smoothing......Page 628
18.3 COMBINED DVM AND AFM......Page 630
18.4.1 DVM IN 3-D......Page 631
18.4.2 AFM IN 3-D......Page 632
18.4.4 EXAMPLE PROBLEMS......Page 633
18.5 OTHER APPROACHES......Page 634
18.5.1 AFM MODIFIED FOR QUADRILATERALS......Page 635
18.5.2 ITERATIVE PAVING METHOD......Page 637
18.5.3 QUADTREE AND OCTREE METHOD......Page 638
REFERENCES......Page 639
19.1.1.1 Basic Theory......Page 641
19.1.1.2 Weight Functions in One Dimension......Page 643
19.1.1.3 Weight Function in Multidimensions......Page 645
19.1.2.1 Variational Formulation......Page 646
19.1.2.2 Smoothness Orthogonality and Concentration......Page 647
19.2 UNSTRUCTURED ADAPTIVE METHODS......Page 651
19.2.1.1 Error Indicators......Page 652
19.2.1.2 Two-Dimensional Quadrilateral Element......Page 654
Three Irregular Nodes......Page 658
Six Hanging Nodes......Page 662
19.2.2 MESH MOVEMENT METHODS (r-METHODS)......Page 663
19.2.3 COMBINED MESH REFINEMENT AND MESH MOVEMENT METHODS......Page 664
Mesh Stretching......Page 665
Local Remeshing......Page 666
19.2.4 MESH ENRICHMENT METHODS (p-METHODS)......Page 668
19.2.5 COMBINED MESH REFINEMENT AND MESH ENRICHMENT METHODS......Page 669
19.2.6 UNSTRUCTURED FINITE DIFFERENCE MESH REFINEMENTS......Page 674
REFERENCES......Page 676
20.1 DOMAIN DECOMPOSITION METHODS......Page 678
20.1.1 MULTIPLICATIVE SCHWARZ PROCEDURE......Page 679
20.1.2 ADDITIVE SCHWARZ PROCEDURE......Page 684
Restriction Process......Page 685
Prolongation Process......Page 686
20.2.3 MULTIGRID SOLUTION PROCEDURE ON UNSTRUCTURED GRIDS......Page 689
20.3.1 GENERAL......Page 690
SIMD and MIMD Structures......Page 691
Matrix-by-Vector Products in Parallel Processing......Page 693
20.3.3 PARALLEL PROCESSING WITH DOMAIN DECOMPOSITION......Page 695
New Trends in Parallel Processing......Page 696
The Static Load Balancing......Page 698
Dynamic Load Balancing......Page 699
20.4.1 SOLUTION OF POISSON EQUATION WITH DOMAIN DECOMPOSITION......Page 700
20.4.2 SOLUTION OF NAVIER-STOKES SYSTEM OF EQUATIONS......Page 702
20.5 SUMMARY......Page 707
REFERENCES......Page 708
PART FIVE: APPLICATIONS......Page 711
21.1 GENERAL......Page 713
Time Averages......Page 714
Mass (Favre) Averages......Page 715
Prandtl’s Mixing Length Model......Page 717
Cebeci-Smith Model......Page 718
Turbulent Heat Flux Vector......Page 719
21.3.3 TWO-EQUATION MODELS......Page 720
Wall Functions......Page 722
Dissipation Tensor......Page 724
Pressure-Strain Correlation Tensor......Page 725
21.3.5 ALGEBRAIC REYNOLDS STRESS MODELS......Page 726
21.3.6 COMPRESSIBILITY EFFECTS......Page 727
Modifications From Incompressible Flows......Page 728
21.4.1 FILTERING, SUBGRID SCALE STRESSES, AND ENERGY SPECTRA......Page 730
21.4.3 SUBGRID SCALE MODELING......Page 733
SGS Eddy Viscosity Model for Stress Tensor with Favre Averages......Page 734
Dynamic SGS Eddy Viscosity Model with Time Averages......Page 735
SGS Turbulent Diffusion and Viscous Diffusion Closures......Page 736
21.5.1 GENERAL......Page 737
21.5.2 VARIOUS APPROACHES TO DNS......Page 738
21.6 SOLUTION METHODS AND INITIAL AND BOUNDARY CONDITIONS......Page 739
21.7.1 TURBULENCE MODELS FOR REYNOLDS AVERAGED NAVIER-STOKES (RANS)......Page 740
(1) Incompressible Flows......Page 742
(2) Compressible Flows......Page 746
21.7.3 DIRECT NUMERICAL SIMULATION (DNS) FOR COMPRESSIBLE FLOWS......Page 750
21.8 SUMMARY......Page 752
REFERENCES......Page 755
22.1 GENERAL......Page 758
Mass Concentration,......Page 759
The Law of Mass Action......Page 760
Mixture Conservation Equations......Page 761
22.2.2 CONSERVATION OF MOMENTUM......Page 763
22.2.3 CONSERVATION OF ENERGY......Page 764
22.2.4 CONSERVATION FORM OF NAVIER-STOKES SYSTEM OF EQUATIONS......Page 766
22.2.5 TWO-PHASE REACTIVE FLOWS (SPRAY COMBUSTION)......Page 770
22.2.6 BOUNDARY AND INITIAL CONDITIONS......Page 772
22.3.1 SOLUTION METHODS OF STIFF CHEMICAL EQUILIBRIUM EQUATIONS......Page 774
22.3.2 APPLICATIONS TO CHEMICAL KINETICS CALCULATIONS......Page 778
22.4.1 FAVRE-AVERAGED DIFFUSION FLAMES......Page 779
22.4.2 PROBABILITY DENSITY FUNCTIONS......Page 782
Boundary Treatments and Numerical Solutions......Page 785
Direct Stress Model......Page 787
22.4.4 SGS COMBUSTION MODELS FOR LES......Page 788
22.5.1 GENERAL......Page 790
22.5.2 VIBRATIONAL AND ELECTRONIC ENERGY IN NONEQUILIBRIUM......Page 792
Vibrational Model......Page 796
Electronic Excitation Model......Page 797
Chemical Reaction Model......Page 798
(1) Global Two-Step Model (Quasi-1-D and 2-D Analysis), Rapid Expansion Diffuser......Page 799
(2) Comparison of Global Two-Step Model with Eighteen-Step Model, Ramjet Combustion......Page 803
(1) Turbulent Premixed Combustion Analysis......Page 804
(2) Turbulent Scramjet Flame Holder Combustion Analysis......Page 805
(3) Transverse Hydrogen Jet Injection......Page 808
22.6.3 PDF MODELS FOR TURBULENT DIFFUSION COMBUSTION ANALYSIS......Page 809
22.6.5 SPRAY COMBUSTION ANALYSIS WITH EULERIAN-LAGRANGIAN FORMULATION......Page 812
(1) Comparison of LES and DNS for Non-premixed Reacting Jet......Page 816
(2) LES analysis for Bluff Body Flame Stabilizer......Page 818
(3) DNS Analysis for Interaction of Isotropic Turbulence and Chemical Reactions......Page 820
(1) Inviscid Hypersonic Reacting Flow with Vibrational Energy......Page 822
(3) Thermochemical Nonequilibrium Hypersonic Flows with Two-Temperature Model......Page 823
(4) Thermomechanical Nonequilibrium Hypersonic Flows with Multi-Temperature Model......Page 825
REFERENCES......Page 826
23.1 INTRODUCTION......Page 830
23.2.1 BASIC EQUATIONS......Page 832
23.2.2 KIRCHHOFF’S METHOD WITH STATIONARY SURFACES......Page 833
23.2.4 KIRCHHOFF’S METHOD WITH SUPERSONIC SURFACES......Page 834
23.3.1 LIGHTHILL’S ACOUSTIC ANALOGY......Page 835
23.3.2 FFOWCS WILLIAMS-HAWKINGS EQUATION......Page 836
23.4.1 ENTROPY ENERGY GOVERNING EQUATIONS......Page 837
23.4.2 ENTROPY CONTROLLED INSTABILITY (ECI) ANALYSIS......Page 838
23.4.3 UNSTABLE ENTROPY WAVES......Page 840
(1) Fluid-Structure-Acoustic Interaction......Page 842
(2) Blade-Vortex-Interaction Hover Noise......Page 844
(3) Noise Level in Rocket Nozzle Exhaust......Page 849
(4) Noise Control in Perforated Muzzle Brake......Page 851
(1) Isotropic Turbulence......Page 856
(3) Circular Cylinder......Page 860
(4) Vortex Line......Page 861
(5) Airfoil Trailing Edge......Page 862
(1) Entropy Controlled Instability (ECI) Analysis Rocket Motor Combustion......Page 863
(2) Unstable Waves of Flame Propagation in a Closed Tube......Page 869
23.6 SUMMARY......Page 871
REFERENCES......Page 872
Planck’s Law......Page 875
Intensity of Radiation......Page 876
Emission......Page 877
Surface Radiation......Page 878
24.2.1 DIFFUSE INTERCHANGE IN AN ENCLOSURE......Page 879
24.2.2 VIEW FACTORS......Page 882
Surface A......Page 883
Angles and......Page 884
Radiation Boundary Conditions with View Factors......Page 886
24.2.3 RADIATIVE HEAT FLUX AND RADIATIVE TRANSFER EQUATION......Page 889
Optically Thin Limit......Page 893
Optically Thick Limit......Page 895
Radiative Equilibrium......Page 896
24.2.4 SOLUTION METHODS FOR INTEGRODIFFERENTIAL RADIATIVE......Page 897
24.3.1 COMBINED CONDUCTION AND RADIATION......Page 898
Formulation of Finite Element Equations......Page 901
Emitting and Absorbing Media......Page 904
Flow through Ducts......Page 905
Nonviscous, Nonconducting Flow over a Flat Plate......Page 906
Optically Thin Boundary Layer......Page 907
Optically Thick Boundary Layer......Page 909
Flow through Ducts......Page 910
Scattering Medium......Page 914
24.3.3 THREE-DIMENSIONAL RADIATIVE HEAT FLUX INTEGRAL FORMULATION......Page 916
(1) View Factors......Page 920
24.4.2 SOLUTION OF RADIATIVE HEAT TRANSFER EQUATION......Page 922
(2) 3-D Enclosure and Reflective Walls......Page 924
(1) Combined Mode Heat Transfer Without and With Scattering and Viscous Dissipation......Page 926
(2) Effect of Radiation on Natural Convection......Page 929
24.4.5 THREE-DIMENSIONAL RADIATIVE HEAT FLUX INTEGRATION FORMULATION......Page 930
REFERENCES......Page 934
25.1 GENERAL......Page 936
25.2.1 NAVIER-STOKES SYSTEM OF EQUATIONS......Page 938
25.2.2 SURFACE TENSION......Page 940
25.2.3 SURFACE AND VOLUME FORCES......Page 942
25.2.4 IMPLEMENTATION OF VOLUME FORCE......Page 944
25.2.5 COMPUTATIONAL STRATEGIES......Page 945
25.3.1 LAMINAR FLOWS IN FLUID-PARTICLE MIXTURE WITH RIGID BODY......Page 947
25.3.2 TURBULENT FLOWS IN FLUID-PARTICLE MIXTURE......Page 950
25.3.3 REACTIVE TURBULENT FLOWS IN FLUID-PARTICLE MIXTURE......Page 951
Governing Equations for the Gas Phase......Page 952
Mathematical Modeling of Phase Interactions......Page 953
25.4.1 LAMINAR FLOWS IN FLUID-PARTICLE MIXTURE......Page 954
25.4.2 TURBULENT FLOWS IN FLUID-PARTICLE MIXTURE......Page 955
25.4.3 REACTIVE TURBULENT FLOWS IN FLUID-PARTICLE MIXTURE......Page 956
REFERENCES......Page 958
26.1 MAGNETOHYDRODYNAMICS......Page 961
26.2.1 BASIC EQUATIONS......Page 965
26.2.2 FINITE ELEMENT SOLUTION OF BOLTZMANN EQUATION......Page 967
26.3.1 INTRODUCTION......Page 970
26.3.2 CHARGED PARTICLE KINETICS IN PLASMA DISCHARGE......Page 973
(1) Strongly Anisotropic VDF......Page 975
(3) Weakly Collisional Discharges with Hot Plasma Effects......Page 976
26.3.3 DISCHARGE MODELING WITH MOMENT EQUATIONS......Page 977
26.3.4 REACTOR MODEL FOR CHEMICAL VAPOR DEPOSITION (CVD) GAS FLOW......Page 979
26.4.1 APPLICATIONS TO MAGNETOHYDRODYNAMIC FLOWS......Page 980
(1) Capacitively Coupled RF Glow Discharge......Page 981
(2) Two-dimensional Capacitively Coupled RF Glow Discharge......Page 982
(5) PECVD of SiO2 in a 3D ICP Reactor......Page 984
26.5 SUMMARY......Page 986
REFERENCES......Page 987
27.1 GENERAL......Page 989
27.2.1 RELATIVISTIC HYDRODYNAMICS EQUATIONS IN IDEAL FLOWS......Page 990
27.2.2 RELATIVISTIC HYDRODYNAMICS EQUATIONS IN NONIDEAL FLOWS......Page 992
Simplification with Minkowski Coordinate Transformation......Page 996
27.2.3 PSEUDO-NEWTONIAN APPROXIMATIONS WITH GRAVITATIONAL EFFECTS......Page 997
27.3.1 RELATIVISTIC SHOCK TUBE......Page 998
27.3.2 BLACK HOLE ACCRETION......Page 999
27.3.3 THREE-DIMENSIONAL RELATIVISTIC HYDRODYNAMICS......Page 1000
27.3.4 FLOWFIELD DEPENDENT VARIATION (FDV) METHOD......Page 1001
Special Relativistic Shock Tube......Page 1003
Dust Infall......Page 1004
27.4 SUMMARY......Page 1007
REFERENCES......Page 1008
APPENDIXES......Page 1011
APPENDIX A: Three-Dimensional Flux Jacobians......Page 1013
APPENDIX B: Gaussian Quadrature......Page 1019
Gaussian Quadrature by Legendre Polynomials......Page 1021
APPENDIX C: Two Phase Flow – Source Term Jacobians for Surface Tension......Page 1027
D.1 METRICS AND CHRISTOFFEL SYMBOLS......Page 1033
D.2 FDV Flux and Source Term Jacobians......Page 1035
Homework problems for CFD I......Page 1041
Homework Problems for CFD II......Page 1047
Homework Problems for CFD III......Page 1049
A Computer Program (Fortran 90) for the Solution of Navier-Stokes System of Equations Using the Flowfield-Dependent Variation (FDV) Method with Finite Elements......Page 1050
Index......Page 1053