Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab.pdf), страница 3
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. . . . . . . . . . . .8.6.2Minimum Correction Approach . . . . . . . . . . . .8.6.3Orthogonal Correction Approach . . . . . . . . . . .8.6.4Over-Relaxed Approach . . . . . . . . . . . . . . . . .8.6.5Treatment of the Cross-Diffusion Term . . . . . .8.6.6Gradient Computation . . . . . . . . . . . . . . . . . .8.6.7Algebraic Equation for Non-orthogonal Meshes8.6.8Boundary Conditions for Non-orthogonal Grids.8.7Skewness . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.8Anisotropic Diffusion . . . . . . . . . . . . . . . . . . . . . . . . .8.9Under-Relaxation of the Iterative Solution Process . . . . .8.10 Computational Pointers . . . . . . . . . . . . . . . . . . . . . . . .8.10.1uFVM . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .8.10.2OpenFOAM® . . . . . . . . . . . . . . . . . . . . . . . .8.11 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.12 Exercises . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . .References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...........................................................................................211211216216217217218220222223224239241241242243243244244245252254255256258258260265265270Part II8.........................................................................................................DiscretizationxivContents..................................................................27327327528528929029029529829830210 Solving the System of Algebraic Equations.
. . . . . . . . . . . .10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.2 Direct or Gauss Elimination Method . . . . . . . . . . . . .10.2.1Gauss Elimination . . . . . . . . . . . . . . . . . . .10.2.2Forward Elimination . . . . . . . . . .
. . . . . . .10.2.3Forward Elimination Algorithm . . . . . . . . . .10.2.4Backward Substitution . . . . . . . . . . . . . . . .10.2.5Back Substitution Algorithm . . . . . . . . . . . .10.2.6LU Decomposition . . . . . . . . . . . . . . . . . .10.2.7The Decomposition Step . . .
. . . . . . . . . . .10.2.8LU Decomposition Algorithm . . . . . . . . . . .10.2.9The Substitution Step. . . . . . . . . . . . . . . . .10.2.10 LU Decomposition and Gauss Elimination . .10.2.11 LU Decomposition Algorithm by GaussElimination. . . . . . . . . . . . . . . . . .
. . . . . .10.2.12 Direct Methods for Banded Sparse Matrices .10.2.13 TriDiagonal Matrix Algorithm (TDMA) . . . .10.2.14 PentaDiagonal Matrix Algorithm (PDMA) . .10.3 Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . .
. .10.3.1Jacobi Method . . . . . . . . . . . . . . . . . . . . .10.3.2Gauss-Seidel Method . . . . . . . . . . . . . . . . .10.3.3Preconditioning and Iterative Methods . . . . .10.3.4Matrix Decomposition Techniques. . . . . . . .10.3.5Incomplete LU (ILU) Decomposition . . . . . .10.3.6Incomplete LU Factorizationwith no Fill-in ILU(0) .
. . . . . . . . . . . . . . .10.3.7ILU(0) Factorization Algorithm . . . . . . . . . .10.3.8ILU Factorization Preconditioners . . . . . . . .10.3.9Algorithm for the Calculation of Din the DILU Method . . . . . . . . . . . . . . . . .10.3.10 Forward and Backward Solution Algorithmwith the DILU Method . . . . . .
. . . . . . . . ..................................................................303303305305306307307308308310311312312..................................................313315316317319323325327329329...............330331331.....332.....3339Gradient Computation . . . . . . . .
. . . . . . . . .9.1Computing Gradients in Cartesian Grids9.2Green-Gauss Gradient . . . . . . . . . . . . .9.3Least-Square Gradient . . . . . . . . . . . . .9.4Interpolating Gradients to Faces . . . . . .9.5Computational Pointers . . . . . . . . . . . .9.5.1uFVM . . . . . . . . . . . . . . . . .9.5.2OpenFOAM® . . . . .
. . . . . . .9.6Closure . . . . . . . . . . . . . . . . . . . . . . .9.7Exercises . . . . . . . . . . . . . . . . . . . . . .References. . . . . . . . . . . . . . . . . . . . . . . . . . ....................................................................................................Contentsxv10.3.11Gradient Methods for Solving AlgebraicSystems . . . .
. . . . . . . . . . . . . . . . . . . . . .10.3.12 The Method of Steepest Descent . . . . . . . . .10.3.13 The Conjugate Gradient Method . . . . . . . . .10.3.14 The Bi-conjugate Gradient Method (BiCG)and Preconditioned BICG . . . . . . . . . . . . . .10.4 The Multigrid Approach. . .
. . . . . . . . . . . . . . . . . . .10.4.1Element Agglomeration/Coarsening . . . . . . .10.4.2The Restriction Step and Coarse LevelCoefficients . . . . . . . . . . . . . . . . . . . . . . .10.4.3The Prolongation Step and Fine Grid LevelCorrections . . . . . . . . . . . . . .
. . . . . . . . . .10.4.4Traversal Strategies and Algebraic MultigridCycles . . . . . . . . . . . . . . . . . . . . . . . . . . .10.5 Computational Pointers . . . . . . . . . . . . . . . . . . . . . .10.5.1uFVM . . . . . . . . . . . . . . . . . . . . . . .
. . . .10.5.2OpenFOAM® . . . . . . . . . . . . . . . . . . . . . .10.6 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................333335337...............340343345.....346.....349...................................34935035035135835836211 Discretization of the Convection Term . .
. . . . . . . . . . . . . . . .11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.2 Steady One Dimensional Convection and Diffusion . . . . .11.2.1Analytical Solution . . . . . . . . . . . . . . . . . . . .11.2.2Numerical Solution . . . . . . . . . . . . . . . . . . . .11.2.3A Preliminary Derivation: The CentralDifference (CD) Scheme . . .
. . . . . . . . . . . . .11.2.4The Upwind Scheme . . . . . . . . . . . . . . . . . . .11.2.5The Downwind Scheme . . . . . . . . . . . . . . . . .11.3 Truncation Error: Numerical Diffusion and Anti-Diffusion11.3.1The Upwind Scheme . . . . . . . . . . . . . . . . . . .11.3.2The Downwind Scheme . . . . . . . . . . . . . . . . .11.3.3The Central Difference (CD) Scheme. .
. . . . . .11.4 Numerical Stability . . . . . . . . . . . . . . . . . . . . . . . . . . .11.5 Higher Order Upwind Schemes. . . . . . . . . . . . . . . . . . .11.5.1Second Order Upwind Scheme . . . . . . . . . . . .11.5.2The Interpolation Profile. . . . . . . . . . . . . . . . .11.5.3The Discretized Equation . . . .
. . . . . . . . . . . .11.5.4Truncation Error . . . . . . . . . . . . . . . . . . . . . .11.5.5Stability Analysis . . . . . . . . . . . . . . . . . . . . .11.5.6The QUICK Scheme . . . . . . . . . . . . . . . . . . .11.5.7The Interpolation Profile. . . . . .
. . . . . . . . . . .11.5.8Truncation Error . . . . . . . . . . . . . . . . . . . . . ................365365366366368...................................................369375379380381382383385388389390390391392392393394xviContents11.5.911.5.1011.5.1111.5.1211.5.1311.5.1411.5.1511.5.16Stability Analysis . . . . .
. . . . . . . . .The FROMM Scheme . . . . . . . . . . .The Interpolation Profile. . . . . . . . . .The Discretized Equation . . . . . . . . .Truncation Error . . . . . . . . . . . . . . .Stability Analysis . . . . . . . . . . . .
. .Comparison of the Various Schemes .Functional Relationships for Uniformand Non-uniform Grids . . . . . . . . . .11.6 Steady Two Dimensional Advection . . . . . . . .11.6.1Error Sources . . . . . . . . . . . . . . . . .11.7 High Order Schemes on Unstructured Grids . . .11.7.1Reformulating HO Schemes in Termsof Gradients . . . . .
. . . . . . . . . . . . .11.8 The Deferred Correction Approach . . . . . . . . .11.9 Computational Pointers . . . . . . . . . . . . . . . . .11.9.1uFVM . . . . . . . . . . . . . . . . . . . . . .11.9.2OpenFOAM® . . . . . . . . . . . . . . . . .11.10 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.11 Exercises . .
. . . . . . . . . . . . . . . . . . . . . . . . .References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12 High Resolution Schemes . . . . . . . . . . . . . . . . . . . .12.1 The Normalized Variable Formulation (NVF) . .12.2 The Convection Boundedness Criterion (CBC) .12.3 High Resolution (HR) Schemes. .
. . . . . . . . . .12.4 The TVD Framework . . . . . . . . . . . . . . . . . .12.5 The NVF-TVD Relation. . . . . . . . . . . . . . . . .12.6 HR Schemes in Unstructured Grid Systems . . .12.7 Deferred Correction for HR Schemes. . . . . . . .12.7.1The Difficulty with the Direct Useof Nodal Values . . . . . . . . .
