The Finite Volume Method in Computational Fluid Dynamics: An Advanced Introduction with OpenFOAM® and Matlab (Fluid Mechanics and Its Applications Book 113)

This textbook explores both the theoretical foundation of the Finite Volume Method (FVM) and its applications in Computational Fluid Dynamics (CFD). Readers will discover a thorough explanation of the FVM numerics and algorithms used for the simulation of incompressible and compressible fluid flows, along with a detailed examination of the components needed for the development of a collocated unstructured pressure-based CFD solver. Two particular CFD codes are explored. The first is uFVM, a three-dimensional unstructured pressure-based finite volume academic CFD code, implemented within Matlab. The second is OpenFOAM®, an open source framework used in the development of a range of CFD programs for the simulation of industrial scale flow problems. With over 220 figures, numerous examples and more than one hundred exercise on FVM numerics, programming, and applications, this textbook is suitable for use in an introductory course on the FVM, in an advanced course on numerics, and as a reference for CFD programmers and researchers. The Finite Volume Method in Computational Fluid Dynamics: An Advanced Introduction with Open FOAM庐 and Matlab庐 4 Preface 6 Acknowledgments 8 Contents 10 About the Authors 24 Part I: Foundation 26 1 Introduction 28 Abstract 28 1.1 What Is Computational Fluid Dynamics (CFD) 28 1.2 What Is the Finite Volume Method 29 1.3 This Book 30 1.3.1 Foundation 30 1.3.2 Numerics 31 1.3.3 Algorithms 32 1.3.4 Applications 33 1.4 Closure 33 2 Review of Vector Calculus 34 Abstract 34 2.1 Introduction 34 2.2 Vectors and Vector Operations 35 2.2.1 The Dot Product of Two Vectors 36 2.2.2 Vector Magnitude 36 2.2.3 The Unit Direction Vector 37 2.2.4 The Cross Product of Two Vectors 37 2.2.5 The Scalar Triple Product 39 2.2.6 Gradient of a Scalar and Directional Derivatives 40 2.2.7 Operations on the Nabla Operator 42 2.2.8 Additional Vector Operations 44 2.3 Matrices and Matrix Operations 45 2.3.1 Square Matrices 46 2.3.2 Using Matrices to Describe Systems of Equations 48 2.3.3 The Determinant of a Square Matrix 48 2.3.4 Eigenvectors and Eigenvalues 51 2.3.5 A Symmetric Positive-Definite Matrix 52 2.3.6 Additional Matrix Operations 53 2.4 Tensors and Tensor Operations 54 2.5 Fundamental Theorems of Vector Calculus 57 2.5.1 Gradient Theorem for Line Integrals 57 2.5.2 Green's Theorem 58 2.5.3 Stokes' Theorem 59 2.5.4 Divergence Theorem 60 2.5.5 Leibniz Integral Rule 62 2.6 Closure 63 2.7 Exercises 64 References 66 3 Mathematical Description of Physical Phenomena 68 Abstract 68 3.1 Introduction 68 3.2 Classification of Fluid Flows 69 3.3 Eulerian and Lagrangian Description of Conservation Laws 70 3.3.1 Substantial Versus Local Derivative 71 3.3.2 Reynolds Transport Theorem 72 3.4 Conservation of Mass (Continuity Equation) 73 3.5 Conservation of Linear Momentum 75 3.5.1 Non-Conservative Form 76 3.5.2 Conservative Form 77 3.5.3 Surface Forces 77 3.5.4 Body Forces 79 3.5.4.1 Gravitational Forces 79 3.5.4.2 System Rotation 80 3.5.5 Stress Tensor and the Momentum Equation for Newtonian Fluids 80 3.6 Conservation of Energy 82 3.6.1 Conservation of Energy in Terms of Specific Internal Energy 85 3.6.2 Conservation of Energy in Terms of Specific Enthalpy 86 3.6.3 Conservation of Energy in Terms of Specific Total Enthalpy 86 3.6.4 Conservation of Energy in Terms of Temperature 87 3.7 General Conservation Equation 90 3.8 Non-dimensionalization Procedure 92 3.9 Dimensionless Numbers 97 3.9.1 Reynolds Number 97 3.9.2 Grashof Number 98 3.9.3 Prandtl Number 98 3.9.4 P茅clet Number 100 3.9.5 Schmidt Number 100 3.9.6 Nusselt Number 102 3.9.7 Mach Number 102 3.9.8 Eckert Number 103 3.9.9 Froude Number 104 3.9.10 Weber Number 104 3.10 Closure 105 3.11 Exercises 105 References 107 4 The Discretization Process 110 Abstract 110 4.1 The Discretization Process 110 4.1.1 Step I: Geometric and Physical Modeling 112 4.1.2 Step II: Domain Discretization 113 4.1.3 Mesh Topology 115 4.1.4 Step III: Equation Discretization 118 4.1.5 Step IV: Solution of the Discretized Equations 123 4.1.5.1 Direct Methods 124 4.1.5.2 Iterative Methods 124 4.1.6 Other Types of Fields 125 4.2 Closure 126 5 The Finite Volume Method 128 Abstract 128 5.1 Introduction 128 5.2 The Semi-Discretized Equation 129 5.2.1 Flux Integration Over Element Faces 130 5.2.2 Source Term Volume Integration 132 5.2.3 The Discrete Conservation Equation for One Integration Point 133 5.2.4 Flux Linearization 134 5.3 Boundary Conditions 136 5.3.1 Value Specified (Dirichlet Boundary Condition) 136 5.3.2 Flux Specified (Neumann Boundary Condition) 137 5.4 Order of Accuracy 138 5.4.1 Spatial Variation Approximation 138 5.4.2 Mean Value Approximation 139 5.5 Transient Semi-Discretized Equation 142 5.6 Properties of the Discretized Equations 143 5.6.1 Conservation 143 5.6.2 Accuracy 144 5.6.3 Convergence 144 5.6.4 Consistency 145 5.6.5 Stability 145 5.6.6 Economy 145 5.6.7 Transportiveness 145 5.6.8 Boundedness of the Interpolation Profile 146 5.7 Variable Arrangement 147 5.7.1 Vertex-Centered FVM 148 5.7.2 Cell-Centered FVM 149 5.8 Implicit Versus Explicit Numerical Methods 151 5.9 The Mesh Support 152 5.10 Computational Pointers 153 5.10.1 uFVM 153 5.10.2 OpenFOAM庐 154 5.11 Closure 158 5.12 Exercises 158 References 159 6 The Finite Volume Mesh 162 Abstract 162 6.1 Domain Discretization 162 6.2 The Finite Volume Mesh 163 6.2.1 Mesh Support for Gradient Computation 164 6.3 Structured Grids 167 6.3.1 Topological Information 167 6.3.2 Geometric Information 169 6.3.3 Accessing the Element Field 170 6.3.3.1 Discretization Indexing 171 6.4 Unstructured Grids 171 6.4.1 Topological Information (Connectivities) 172 6.5 Geometric Quantities 177 6.5.1 Element Types 178 6.5.2 Computing Surface Area and Centroid of Faces 179 6.5.2.1 Surface of a Triangle 180 6.5.2.2 Volume and Centroid of Elements 183 6.5.2.3 Face Weighting Factor 184 6.6 Computational Pointers 187 6.6.1 uFVM 187 6.6.2 OpenFOAM庐 189 6.6.2.1 Area and Centroid of Faces 190 6.6.2.2 Volume and Centroid of Elements 192 6.7 Closure 195 6.8 Exercises 195 References 195 7 The Finite Volume Mesh in OpenFOAM and uFVM 198 Abstract 198 7.1 uFVM 198 7.1.1 An OpenFOAM Test Case 198 7.1.2 The polyMesh Folder 200 7.1.3 The uFVM Mesh 203 7.1.4 uFVM Geometric Fields 208 7.1.4.1 The Element Fields 208 7.1.4.2 The Face Fields 210 7.1.4.3 The Node Field 211 7.1.5 Working with the uFVM Mesh 212 7.1.5.1 Looping Over Elements 212 7.1.5.2 Looping Over Faces 213 7.1.6 Computing the Gauss Gradient 213 7.2 OpenFOAM庐 216 7.2.1 Fields and Memory 222 7.2.2 InternalField Data 224 7.2.3 BoundaryField Data 225 7.2.4 lduAddressing 225 7.2.5 Computing the Gradient 227 7.3 Mesh Conversion Tools 229 7.4 Closure 230 7.5 Exercises 230 References 232 Part II: Discretization 234 8 Spatial Discretization: The Diffusion Term 236 Abstract 236 8.1 Two-Dimensional Diffusion in a Rectangular Domain 236 8.2 Comments on the Discretized Equation 241 8.2.1 The Zero Sum Rule 241 8.2.2 The Opposite Signs Rule 242 8.3 Boundary Conditions 242 8.3.1 Dirichlet Boundary Condition 243 8.3.2 Von Neumann Boundary Condition 245 8.3.3 Mixed Boundary Condition 247 8.3.4 Symmetry Boundary Condition 248 8.4 The Interface Diffusivity 249 8.5 Non-Cartesian Orthogonal Grids 264 8.6 Non-orthogonal Unstructured Grid 266 8.6.1 Non-orthogonality 266 8.6.2 Minimum Correction Approach 267 8.6.3 Orthogonal Correction Approach 268 8.6.4 Over-Relaxed Approach 268 8.6.5 Treatment of the Cross-Diffusion Term 269 8.6.6 Gradient Computation 269 8.6.7 Algebraic Equation for Non-orthogonal Meshes 270 8.6.8 Boundary Conditions for Non-orthogonal Grids 277 8.6.8.1 Dirichlet Boundary Condition 277 8.6.8.2 Neumann Boundary Condition 278 8.6.8.3 Mixed Boundary Condition 278 8.7 Skewness 279 8.8 Anisotropic Diffusion 280 8.9 Under-Relaxation of the Iterative Solution Process 281 8.10 Computational Pointers 283 8.10.1 uFVM 283 8.10.2 OpenFOAM庐 285 8.11 Closure 290 8.12 Exercises 290 References 295 9 Gradient Computation 298 Abstract 298 9.1 Computing Gradients in Cartesian Grids 298 9.2 Green-Gauss Gradient 300 9.3 Least-Square Gradient 310 9.4 Interpolating Gradients to Faces 314 9.5 Computational Pointers 315 9.5.1 uFVM 315 9.5.2 OpenFOAM庐 320 9.6 Closure 323 9.7 Exercises 323 References 327 10 Solving the System of Algebraic Equations 328 Abstract 328 10.1 Introduction 328 10.2 Direct or Gauss Elimination Method 330 10.2.1 Gauss Elimination 330 10.2.2 Forward Elimination 331 10.2.3 Forward Elimination Algorithm 332 10.2.4 Backward Substitution 332 10.2.5 Back Substitution Algorithm 333 10.2.6 LU Decomposition 333 10.2.7 The Decomposition Step 335 10.2.8 LU Decomposition Algorithm 336 10.2.9 The Substitution Step 337 10.2.10 LU Decomposition and Gauss Elimination 337 10.2.11 LU Decomposition Algorithm by Gauss Elimination 338 10.2.12 Direct Methods for Banded Sparse Matrices 340 10.2.13 TriDiagonal Matrix Algorithm (TDMA) 341 10.2.14 PentaDiagonal Matrix Algorithm (PDMA) 342 10.3 Iterative Methods 344 10.3.1 Jacobi Method 348 10.3.2 Gauss-Seidel Method 350 10.3.3 Preconditioning and Iterative Methods 352 10.3.4 Matrix Decomposition Techniques 354 10.3.5 Incomplete LU (ILU) Decomposition 354 10.3.6 Incomplete LU Factorization with no Fill-in ILU(0) 355 10.3.7 ILU(0) Factorization Algorithm 356 10.3.8 ILU Factorization Preconditioners 356 10.3.9 Algorithm for the Calculation of {{\bf D}}^{*} in the DILU Method 357 10.3.10 Forward and Backward Solution Algorithm with the DILU Method 358 10.3.11 Gradient Methods for Solving Algebraic Systems 358 10.3.12 The Method of Steepest Descent 360 10.3.13 The Conjugate Gradient Method 362 10.3.14 The Bi-conjugate Gradient Method (BiCG) and Preconditioned BICG 365 10.4 The Multigrid Approach 368 10.4.1 Element Agglomeration/Coarsening 370 10.4.2 The Restriction Step and Coarse Level Coefficients 371 10.4.3 The Prolongation Step and Fine Grid Level Corrections 374 10.4.4 Traversal Strategies and Algebraic Multigrid Cycles 374 10.5 Computational Pointers 375 10.5.1 uFVM 375 10.5.2 OpenFOAM庐 376 10.6 Closure 383 10.7 Exercises 383 References 387 11 Discretization of the Convection Term 390 Abstract 390 11.1 Introduction 390 11.2 Steady One Dimensional Convection and Diffusion 391 11.2.1 Analytical Solution 391 11.2.2 Numerical Solution 393 11.2.3 A Preliminary Derivation: The Central Difference (CD) Scheme 394 11.2.4 The Upwind Scheme 400 11.2.5 The Downwind Scheme 404 11.3 Truncation Error: Numerical Diffusion and Anti-Diffusion 405 11.3.1 The Upwind Scheme 406 11.3.2 The Downwind Scheme 407 11.3.3 The Central Difference (CD) Scheme 408 11.4 Numerical Stability 410 11.5 Higher Order Upwind Schemes 413 11.5.1 Second Order Upwind Scheme 414 11.5.2 The Interpolation Profile 415 11.5.3 The Discretized Equation 415 11.5.4 Truncation Error 416 11.5.5 Stability Analysis 417 11.5.6 The QUICK Scheme 417 11.5.7 The Interpolation Profile 418 11.5.8 Truncation Error 419 11.5.9 Stability Analysis 419 11.5.10 The FROMM Scheme 420 11.5.11 The Interpolation Profile 420 11.5.12 The Discretized Equation 421 11.5.13 Truncation Error 422 11.5.14 Stability Analysis 422 11.5.15 Comparison of the Various Schemes 423 11.5.16 Functional Relationships for Uniform and Non-uniform Grids 424 11.6 Steady Two Dimensional Advection 425 11.6.1 Error Sources 429 11.7 High Order Schemes on Unstructured Grids 431 11.7.1 Reformulating HO Schemes in Terms of Gradients 432 11.8 The Deferred Correction Approach 434 11.9 Computational Pointers 436 11.9.1 uFVM 436 11.9.2 OpenFOAM庐 438 11.10 Closure 446 11.11 Exercises 447 References 451 12 High Resolution Schemes 454 Abstract 454 12.1 The Normalized Variable Formulation (NVF) 454 12.2 The Convection Boundedness Criterion (CBC) 461 12.3 High Resolution (HR) Schemes 463 12.4 The TVD Framework 468 12.5 The NVF-TVD Relation 475 12.6 HR Schemes in Unstructured Grid Systems 481 12.7 Deferred Correction for HR Schemes 481 12.7.1 The Difficulty with the Direct Use of Nodal Values 483 12.8 The DWF and NWF Methods 484 12.8.1 The Downwind Weighing Factor (DWF) Method 485 12.8.2 The Normalized Weighing Factor (NWF) Method 488 12.8.2.1 The NWF Method in the Context of the TVD 491 12.9 Boundary Conditions 492 12.9.1 Inlet Boundary Condition 493 12.9.2 Outlet Boundary Condition 495 12.9.3 Wall Boundary Condition 496 12.9.4 Symmetry Boundary Condition 497 12.10 Computational Pointers 497 12.10.1 uFVM 497 12.10.2 OpenFOAM庐 500 12.11 Closure 508 12.12 Exercises 508 References 512 13 Temporal Discretization: The Transient Term 514 Abstract 514 13.1 Introduction 514 13.2 The Finite Difference Approach 517 13.2.1 Forward Euler Scheme 517 13.2.2 Stability of the Forward Euler Scheme 519 13.2.2.1 Stability of a Transient-Advection Case 520 13.2.2.2 Stability of a Transient-Diffusion Case 521 13.2.2.3 Stability of a Transient-Convection-Diffusion Case 522 13.2.3 Backward Euler Scheme 523 13.2.4 Crank-Nicolson Scheme 525 13.2.5 Implementation Details 527 13.2.6 Adams-Moulton Scheme 528 13.3 The Finite Volume Approach 532 13.3.1 First Order Transient Schemes 533 13.3.2 First Order Implicit Euler Scheme 533 13.3.2.1 Numerical Diffusion 534 13.3.3 First Order Explicit Euler Scheme 535 13.3.3.1 Numerical Anti-Diffusion 536 13.3.4 Second Order Transient Euler Schemes 537 13.3.5 Crank-Nicholson (Central Difference Profile) 537 13.3.5.1 Numerical Accuracy 538 13.3.6 Second Order Upwind Euler (SOUE) Scheme 539 13.3.6.1 Numerical Accuracy 540 13.3.7 Initial Condition for the FV Approach 540 13.4 Non-Uniform Time Steps 544 13.4.1 Non-Uniform Time Steps with the Finite Difference Approach 544 13.4.1.1 Crank-Nicolson Scheme 544 13.4.2 Adams-Moulton (or SOUE) Scheme 546 13.4.3 Non-Uniform Time Steps with the Finite Volume Approach 547 13.4.4 Crank-Nicolson Scheme 548 13.4.5 Adams-Moulton (or SOUE) Scheme 549 13.5 Computational Pointers 550 13.5.1 uFVM 550 13.5.2 OpenFOAM庐 551 13.6 Closure 554 13.7 Exercises 554 References 558 14 Discretization of the Source Term, Relaxation, and Other Details 560 Abstract 560 14.1 Source Term Discretization 560 14.2 Under-Relaxation of the Algebraic Equations 563 14.2.1 Under-Relaxation Methods 564 14.2.2 Explicit Under-Relaxation 565 14.2.3 Implicit Under-Relaxation Methods 565 14.2.3.1 Patankar's Under-Relaxation 565 14.2.3.2 E-Factor Relaxation 566 14.2.3.3 False Transient Relaxation 569 14.3 Residual Form of the Equation 569 14.3.1 Residual Form of Patankar's Under-Relaxation 570 14.4 Residuals and Solution Convergence 571 14.4.1 Residuals 571 14.4.2 Absolute Residual 572 14.4.3 Maximum Residual 572 14.4.4 Root-Mean Square Residual 572 14.4.5 Normalization of the Residual 573 14.5 Computational Pointers 574 14.5.1 uFVM 574 14.5.1.1 Source Term Linearization 574 14.5.1.2 Under-Relaxation 574 14.5.2 OpenFOAM庐 575 14.5.2.1 Source Term Linearization 575 14.5.2.2 Under-Relaxation 577 14.6 Closure 580 14.7 Exercises 580 References 582 Part III: Algorithms 584 15 Fluid Flow Computation: Incompressible Flows 586 Abstract 586 15.1 The Main Difficulty 586 15.2 A Preliminary Derivation 588 15.2.1 Discretization of the Momentum Equation 589 15.2.2 Discretization of the Continuity Equation 590 15.2.3 The Checkerboard Problem 590 15.2.4 The Staggered Grid 592 15.2.5 The Pressure Correction Equation 594 15.2.6 The SIMPLE Algorithm on Staggered Grid 597 15.2.7 Pressure Correction Equation in Two Dimensional Staggered Cartesian Grids 603 15.2.8 Pressure Correction Equation in Three Dimensional Staggered Cartesian Grid 606 15.3 Disadvantages of the Staggered Grid 607 15.4 The Rhie-Chow Interpolation 610 15.5 General Derivation 613 15.5.1 The Discretized Momentum Equation 613 15.5.2 The Collocated Pressure Correction Equation 617 15.5.3 Calculation of the {{\cal D}}_{f} Term 621 15.5.3.1 Minimum Correction Approach 621 15.5.3.2 Orthogonal Correction Approach 621 15.5.3.3 Over-Relaxed Approach 622 15.5.4 The Collocated SIMPLE Algorithm 622 15.6 Boundary Conditions 627 15.6.1 Boundary Conditions for the Momentum Equation 628 15.6.1.1 Wall Boundary Conditions 629 15.6.1.2 Inlet Boundary Conditions 633 15.6.1.3 Outlet Boundary Conditions 636 15.6.1.4 Symmetry Boundary Condition 639 15.6.2 Boundary Conditions for the Pressure Correction Equation 642 15.6.2.1 Wall Boundary Condition 643 15.6.2.2 Inlet Boundary Conditions 643 15.6.2.3 Outlet Boundary Conditions 645 15.6.2.4 Symmetry Boundary Condition 646 15.6.2.5 The Relative Nature of Pressure 646 15.7 The SIMPLE Family of Algorithms 646 15.7.1 The SIMPLEC Algorithm 648 15.7.2 The PRIME Algorithm 649 15.7.3 The PISO Algorithm 650 15.8 Optimum Under-Relaxation Factor Values for v and p^{\prime} 653 15.9 Treatment of Various Terms with the Rhie-Chow Interpolation 655 15.9.1 Treatment of the Under-Relaxation Term 655 15.9.2 Treatment of the Transient Term 656 15.9.3 Treatment of the Body Force Term 657 15.9.4 Combined Treatment of Under-Relaxation, Transient, and Body Force Terms 661 15.10 Computational Pointers 661 15.10.1 uFVM 661 15.10.2 OpenFOAM庐 663 15.10.2.1 Pressure Correction SIMPLE Solvers 663 15.11 Closure 674 15.12 Exercises 674 References 678 16 Fluid Flow Computation: Compressible Flows 680 Abstract 680 16.1 Historical 680 16.2 Introduction 681 16.3 The Conservation Equations 682 16.4 Discretization of the Momentum Equation 683 16.5 The Pressure Correction Equation 684 16.6 Discretization of The Energy Equation 688 16.6.1 Discretization of the Extra Terms 688 16.6.1.1 The Specific Heat Term 688 16.6.1.2 The Substantial Derivative Term 689 16.6.1.3 The Dissipation Term 689 16.6.1.4 The Viscous Dissipation Term 689 16.6.1.5 The Source/Sink Term 690 16.6.2 The Algebraic Form of the Energy Equation 690 16.7 The Compressible SIMPLE Algorithm 691 16.8 Boundary Conditions 692 16.8.1 Inlet Boundary Conditions 694 16.8.1.1 Subsonic Flow at Inlet 694 16.8.1.2 Supersonic Flow at Inlet 696 16.8.2 Outlet Boundary Conditions 697 16.8.2.1 Subsonic Flow at Outlet 697 16.8.2.2 Supersonic Flow at Outlet 698 16.9 Computational Pointers 698 16.9.1 uFVM 698 16.9.2 OpenFOAM庐 699 16.10 Closure 712 16.11 Exercises 712 References 714 Part IV: Applications 716 17 Turbulence Modeling 718 Abstract 718 17.1 Turbulence Modeling 718 17.2 Reynolds Averaging 721 17.2.1 Time Averaging 721 17.2.2 Spatial Averaging 721 17.2.3 Ensemble Averaging 722 17.2.4 Averaging Rules 722 17.2.5 Incompressible RANS Equations 722 17.3 Boussinesq Hypothesis 724 17.4 Turbulence Models 725 17.5 Two-Equation Turbulence Models 725 17.5.1 Standard k - 蔚 Model 725 17.5.2 The k - 蠅 Model 727 17.5.3 The Baseline (BSL) k -蠅 Model 729 17.5.4 The Shear Stress Transport (SST) k - 蠅 Model 730 17.6 Summary of Incompressible Turbulent Flow Equations 732 17.7 Discretization of the Turbulent Flow Equations 732 17.7.1 The Discretized Form of the k Equation 733 17.7.2 The Discretized Form of the epsilon Equation 733 17.7.3 The Discretized Form of the omega Equation 734 17.8 Boundary Conditions 735 17.8.1 Modeling Flow Near the Wall 735 17.8.2 Standard Wall Functions 736 17.8.3 Improved Wall Functions 741 17.8.4 Scalable Wall Functions 743 17.8.5 Wall Boundary Conditions for Low Reynolds Number Models 744 17.8.6 Automatic Near-Wall Treatment 745 17.8.7 Near-Wall Heat Transfer 746 17.8.8 Other Boundary Conditions 747 17.9 Calculating Normal Distance to the Wall 748 17.10 Computational Pointers 750 17.10.1 The k - 蔚 Model 752 17.10.2 The SST k - 蠅 Model 759 17.10.3 simpleFoamTurbulent 763 17.11 Closure 765 17.12 Exercises 765 References 767 18 Boundary Conditions in OpenFOAM庐 and uFVM 770 Abstract 770 18.1 Boundary Conditions in OpenFOAM庐 770 18.2 Boundary Condition Customization 772 18.3 Development of a New BC: No Slip Wall Condition 777 18.4 The No-Slip Boundary Condition in uFVM 781 18.5 Closure 784 Reference 784 19 An OpenFOAM庐 Turbulent Flow Application 786 Abstract 786 19.1 Introduction 786 19.2 The Ahmed Bluff Body 786 19.3 Domain Discretization 788 19.3.1 Initial and Boundary Conditions 793 19.3.2 Systems Files 795 19.3.3 Running the Solver 798 19.4 Conclusion 801 References 801 20 Closing Remarks 802 Appendix: uFVM 804 A.1 Introduction 804 A.2 The Base Structure 804 A.3 Reading the Mesh 806 A.4 Setting-Up the Model 809 A.5 Setup the Computational Fields 811 A.6 Equation Discretization (Assembly) 811 A.6.1 Equation Assembly 811 A.6.2 Solving the Equations 812 A.6.3 Computing the Residuals 813 A.7 Plotting Utilities 813 A.8 Interpolation Schemes 815 A.9 Test Cases 816 A.10 Closing Remarks 816 Front Matter....Pages i-xxiii Front Matter....Pages 1-1 Introduction....Pages 3-8 Review of Vector Calculus....Pages 9-42 Mathematical Description of Physical Phenomena....Pages 43-83 The Discretization Process....Pages 85-101 The Finite Volume Method....Pages 103-135 The Finite Volume Mesh....Pages 137-171 The Finite Volume Mesh in OpenFOAM® and uFVM....Pages 173-207 Front Matter....Pages 209-209 Spatial Discretization: The Diffusion Term....Pages 211-271 Gradient Computation....Pages 273-302 Solving the System of Algebraic Equations....Pages 303-364 Discretization of the Convection Term....Pages 365-427 High Resolution Schemes....Pages 429-488 Temporal Discretization: The Transient Term....Pages 489-533 Discretization of the Source Term, Relaxation, and Other Details....Pages 535-557 Front Matter....Pages 559-559 Fluid Flow Computation: Incompressible Flows....Pages 561-654 Fluid Flow Computation: Compressible Flows....Pages 655-690 Front Matter....Pages 691-691 Turbulence Modeling....Pages 693-744 Boundary Conditions in OpenFOAM® and uFVM....Pages 745-759 An OpenFOAM® Turbulent Flow Application....Pages 761-776 Closing Remarks....Pages 777-777 Back Matter....Pages 779-791