Computational Fluid Dynamics

The second edition of Computational Fluid Dynamics represents a significant improvement from the first edition. However, the original idea of including all computational fluid dynamics methods (FDM, FEM, FVM); all mesh generation schemes; and physical applications to turbulence, combustion, acoustics, radiative heat transfer, multiphase flow, electromagnetic flow, and general relativity is still maintained. This unique approach sets this book apart from its competitors and allows the instructor to adopt this book as a text and choose only those subject areas of his or her interest. The second edition includes a new section on preconditioning for EBE-GMRES and a complete revision of the section on flowfield-dependent variation methods, which demonstrates more detailed computational processes and includes additional example problems. For those instructors desiring a textbook that contains homework assignments, a variety of problems for FDM, FEM, and FVM are included in an appendix. To facilitate students and practitioners intending to develop a large-scale computer code, an example of FORTRAN code capable of solving compressible, incompressible, viscous, inviscid, 1D, 2D, and 3D for all speed regimes using the flowfield-dependent variation method is made available. 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 "In this second edition of Computational Fluid Dynamics, the author presents up-to-date treatments of all computational methods of fluid dynamics, while maintaining the original idea of including all computational fluid dynamics methods. The breadth of information sets this book apart from its competitors and allows the instructor to adopt this book, selecting only those subject areas of his or her interest. The second edition includes a new section on preconditioning for EBE-GMRES and a complete revision of the section on flow field-dependent variation methods, which demonstrates more detailed computational processes and includes additional example problems. Homework examples facilitate students and practitioners intending to develop a large-scale computer code"-- "The development of modern computational fluid dynamics (CFD) began with the advent of the digital computer in the early 1950s. Finite difference methods (FDM) and finite element methods (FEM), which are the basic tools used in the solution of partial differential equations in general and CFD in particular, have different origins. In 1910, at the Royal Society of London, Richardson presented a paper on the first FDM solution for the stress analysis of a masonry dam. In contrast, the first FEM work was published in the Aeronautical Science Journal by Turner, Clough, Martin, and Topp for applications to aircraft stress analysis in 1956. Since then, both methods have been developed extensively in fluid dynamics"-- Machine generated contents note: Part I. Preliminaries: 1. Introduction; 2. Governing equations; Part II. Finite Difference Methods: 3. Derivation of finite difference equations; 4. Solution methods of finite difference equations; 5. Incompressible viscous flows via finite difference methods; 6. Compressible flows via finite difference methods; 7. Finite volume methods via finite difference methods; Part III. Finite Element Methods: 8. Introduction to finite element methods; 9. Finite element interpolation functions; 10. Linear problems; 11. Nonlinear problems/convection-dominated flows; 12. Incompressible viscous flows via finite element methods; 13. Compressible flows via finite element methods; 14. Miscellaneous weighted residual methods; 15. Finite volume methods via finite element methods; 16. Relationships between finite differences and finite elements and other methods; Part IV. Automatic Grid Generation, Adaptive Methods and Computing Techniques: 17. Structured grid generation; 18. Unstructured grid generation; 19. Adaptive methods; 20. Computing techniques; Part V. Applications: 21. Applications to turbulence; 22. Applications to chemically reactive flows and combustion; 23. Applications to acoustics; 24. Applications to combined mode radiative heat transfer; 25. Applications to multiphase flows; 26. Applications to electromagnetic flows; 27. Applications to relativistic astrophysical flows; Appendices.