Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 2
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. . . . . . . . . . . . .3.6.1Conservation of Energy in Terms of SpecificInternal Energy . . . . . . . . . . . . . . . . . . . . .3.6.2Conservation of Energy in Terms of SpecificEnthalpy. . . . . . . . . . . . . . . . . . . . . . . . . .3.6.3Conservation of Energy in Terms of SpecificTotal Enthalpy . . .
. . . . . . . . . . . . . . . . . .3.6.4Conservation of Energy in Termsof Temperature . . . . . . . . . . . . . . . . . . . . .3.7General Conservation Equation . . . . . . . . . . . . . . . . .3.8Non-dimensionalization Procedure . . . .
. . . . . . . . . . .3.9Dimensionless Numbers . . . . . . . . . . . . . . . . . . . . . .3.9.1Reynolds Number . . . . . . . . . . . . . . . . . . .3.9.2Grashof Number . . . . . . . . . . . . . . . . . . . .3.9.3Prandtl Number. . . . . . . . . . . . . . . . . . . . .3.9.4Péclet Number . . . . . . . . . . . . . .
. . . . . . .3.9.5Schmidt Number . . . . . . . . . . . . . . . . . . . ..................................2829323233343537383941...............434344..................454647485051525254..........5557.....60.....61.....61.........626567727273737575.....................................................................................Contents3.9.6Nusselt Number3.9.7Mach Number. .3.9.8Eckert Number .3.9.9Froude Number .3.9.10Weber Number .3.10 Closure . . .
. . . . . . . . . .3.11 Exercises . . . . . . . . . . . .References. . . . . . . . . . . . . . . . .xi........................................................................................................................................................................................................77777879798080824The Discretization Process. . . . . . .
. . . . . . . . . . . . . . . . . . .4.1The Discretization Process . . . . . . . . . . . . . . . . . . . . .4.1.1Step I: Geometric and Physical Modeling . . . .4.1.2Step II: Domain Discretization . . . . . . . . . . .4.1.3Mesh Topology. . . . . . . . . . . .
. . . . . . . . . .4.1.4Step III: Equation Discretization . . . . . . . . . .4.1.5Step IV: Solution of the Discretized Equations4.1.6Other Types of Fields . . . . . . . . . . . . . . . . .4.2Closure . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . .....................................858587889093981001015The Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . . . .5.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.2The Semi-Discretized Equation . . . . . . . . . . . . . . .
. . . .5.2.1Flux Integration Over Element Faces . . . . . . . .5.2.2Source Term Volume Integration. . . . . . . . . . .5.2.3The Discrete Conservation Equationfor One Integration Point . . . . . . . . . . . . . . . .5.2.4Flux Linearization . . . . . . .
. . . . . . . . . . . . . .5.3Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . .5.3.1Value Specified (Dirichlet Boundary Condition)5.3.2Flux Specified (Neumann Boundary Condition).5.4Order of Accuracy. . . . . . . . . . . . . . . . . . . . . . . . . . . .5.4.1Spatial Variation Approximation . . . . . . . .
. . .5.4.2Mean Value Approximation . . . . . . . . . . . . . .5.5Transient Semi-Discretized Equation . . . . . . . . . . . . . . .5.6Properties of the Discretized Equations . . . . . . . . . . . . .5.6.1Conservation. . . . . . . . . . . . .
. . . . . . . . . . . .5.6.2Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . .5.6.3Convergence . . . . . . . . . . . . . . . . . . . . . . . . .5.6.4Consistency . . . . . . . . . . . . . . . . . . . . . . . . .5.6.5Stability . . . . . . . . . . . . . . . . . . .
. . . . . . . . .5.6.6Economy . . . . . . . . . . . . . . . . . . . . . . . . . . .5.6.7Transportiveness . . . . . . . . . . . . . . . . . . . . . .5.6.8Boundedness of the Interpolation Profile . . . . ................103103104105107......................................................108109111111112113113114117118118119119120120120120121xiiContents5.7Variable Arrangement . . . . . . . . . . . .
. . . .5.7.1Vertex-Centered FVM . . . . . . . . .5.7.2Cell-Centered FVM . . . . . . . . . . .5.8Implicit Versus Explicit Numerical Methods .5.9The Mesh Support. . . . . . . . . . . . . . . . . . .5.10 Computational Pointers . . . . . . . . . .
. . . . .5.10.1uFVM . . . . . . . . . . . . . . . . . . . .5.10.2OpenFOAM® . . . . . . . . . . . . . . .5.11 Closure . . . . . . . . . . . . . . . . . . . . . . . . . .5.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . .References. . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .67....................................................................................................................................122123124126127128128129133133134The Finite Volume Mesh . . . . . . . . . . . . . . . . . . . . . . .6.1Domain Discretization . . . . . .
. . . . . . . . . . . . . .6.2The Finite Volume Mesh . . . . . . . . . . . . . . . . . .6.2.1Mesh Support for Gradient Computation6.3Structured Grids . . . . . . . . . . . . . . . . . . . . . . . .6.3.1Topological Information . . . . . . . . . . . .6.3.2Geometric Information . . . .
. . . . . . . . .6.3.3Accessing the Element Field . . . . . . . . .6.4Unstructured Grids . . . . . . . . . . . . . . . . . . . . . .6.4.1Topological Information (Connectivities)6.5Geometric Quantities . . . . . . . . . . . . . . . .
. . . . .6.5.1Element Types . . . . . . . . . . . . . . . . . .6.5.2Computing Surface Area and Centroidof Faces . . . . . . . . . . . . . . . . . . . . . . .6.6Computational Pointers . . . . . . . . . . . . . . . . . . .6.6.1uFVM . . . . . . . . . . . . . . . . . . . . . . . .6.6.2OpenFOAM® . . . . . . .
. . . . . . . . . . . .6.7Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.8Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................................................................................................137137138139142142144145146147152153........................................................154162162164170170170The Finite Volume Mesh in OpenFOAM® and uFVM7.1uFVM . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .7.1.1An OpenFOAM® Test Case . . . . . . . .7.1.2The polyMesh Folder. . . . . . . . . . . . .7.1.3The uFVM Mesh. . . . . . . . . . . . . . . .7.1.4uFVM Geometric Fields. . . . . . .
. . . .7.1.5Working with the uFVM Mesh . . . . . .7.1.6Computing the Gauss Gradient . . . . . .7.2OpenFOAM® . . . . . . . . . . . . . . . . . . . . . . . . .7.2.1Fields and Memory . . . . . . . . . . . . . .7.2.2InternalField Data . . . . . . . . .
. . . . . .........................................................................................173173173175178183187188191197199...........Contentsxiii7.2.3BoundaryField Data . . . .7.2.4lduAddressing . . . . . . . .7.2.5Computing the Gradient .7.3Mesh Conversion Tools . . . . . . . .7.4Closure . . . . . . . . . . . . . . . .
. . .7.5Exercises . . . . . . . . . . . . . . . . . .References. . . . . . . . . . . . . . . . . . . . . . .............................200200202204205205207Spatial Discretization: The Diffusion Term. . . . . . . . . . . . . . .8.1Two-Dimensional Diffusion in a Rectangular Domain . . .8.2Comments on the Discretized Equation . . . . . . . . . . . . .8.2.1The Zero Sum Rule . .
. . . . . . . . . . . . . . . . . .8.2.2The Opposite Signs Rule . . . . . . . . . . . . . . . .8.3Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . .8.3.1Dirichlet Boundary Condition . . . . . . . . . . . . .8.3.2Von Neumann Boundary Condition . . . . . . . . .8.3.3Mixed Boundary Condition . . . . .
. . . . . . . . .8.3.4Symmetry Boundary Condition . . . . . . . . . . . .8.4The Interface Diffusivity . . . . . . . . . . . . . . . . . . . . . . .8.5Non-Cartesian Orthogonal Grids . . . . . . . . . . . . . . . . . .8.6Non-orthogonal Unstructured Grid. . . . . . . . . . . . . . . . .8.6.1Non-orthogonality . . . . . . . . .
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