Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 26
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It continues with the computation of relevant geometric information for the various components of the computational mesh, and is completed bycapturing the topology of these components, i.e., how they are related and locatedone with respect to the other. Thus the result of the domain discretization step is notonly the set of non-overlapping elements and other related geometric entities andthe generated information about their geometric properties, but also the topologicalinformation about their arrangement and relations.
It is this combined informationthat defines the finite volume mesh. The objective of this chapter is to clarify thetopological and geometric requirements of the finite volume mesh.6.1Domain DiscretizationThe discretization of the physical domain, or mesh generation, produces a computational mesh (Fig. 6.1) on which the governing equations are subsequentlysolved. The methods and techniques used for domain discretization have changeddrastically over the last few decades [1, 2], and, nowadays, have become mostlyautomated [3–6].
Before reviewing the types of elements commonly used in acomputational mesh, the characteristics and attributes that the mesh system shouldpossess in order to be employed in the context of the finite volume method are firstdescribed. These attributes will be presented in the context of computing the gradient of a variable ϕ on both a structured and an unstructured triangular mesh.In general a geometric domain may be discretized using either a structured or anunstructured grid system. In a structured mesh, three dimensional elements aredefined by their local indices (i, j, k). A structured grid system has many coding and© Springer International Publishing Switzerland 2016F.
Moukalled et al., The Finite Volume Method in Computational Fluid Dynamics,Fluid Mechanics and Its Applications 113, DOI 10.1007/978-3-319-16874-6_61371386 The Finite Volume Mesh(a)(b)Tsink InsulatedTmicroprocessornodefaceelement(c)nodeelementfaceFig. 6.1 a Domain of interest, b domain discretized using a uniform grid system, and c domaindiscretized using an unstructured grid system with triangular elementsperformance advantages but suffers from a limited geometric flexibility.
Additionalflexibility in the generation of structured meshes can be achieved by employingmultiple blocks to define the geometry, with a structured mesh generated for eachblock either independently from other blocks or jointly.Another way to make the mesh generation more flexible is to avoid the use ofstructured grids with their implicit topological information, and to adopt anunstructured mesh with explicit topological information based on connectivitytables and geometric entity numbering.Structured grids remained the staple of numerical simulation for a long time andit is only in the past two decades that unstructured grids became more popular [7].Starting in early 1970s, interest in automatic mesh generation escalated as problemsize increased and manual mesh generation became too time consuming [8].
Thefirst methods were semi-automatic with an operator manually placing points in thecomputational domain and then, in a second step, using a computer to generate themesh. Nowadays the whole process is fully automated with both points and elements generated automatically.Most modern CFD codes have the ability to use unstructured grids in addition toa variety of hybrid multiblock grids [9].
OpenFOAM® [10] uses unstructured gridsbut can also use conforming and non-conforming multiblock grids [11]. The finitevolume method will be presented here in the context of an unstructured grid system.However as the unstructured finite volume mesh requirements are defined, itscharacteristics will be compared to those of a structured grid system.6.2The Finite Volume MeshThe discussion for the requirements of the finite volume mesh will be contextualized in terms of a simple problem, namely the computation of the gradient of anelement field. The gradient will first be computed on a structured grid and then overan unstructured grid; the differences will help clarifying a number of issues.6.2 The Finite Volume Mesh6.2.1139Mesh Support for Gradient ComputationMany techniques can be used to compute the gradient of an element field, and thesewill form the subject of Chap.
9. The method adopted in this chapter is based on theGreen-Gauss theorem [12, 13]. It is relatively straightforward and can be used for avariety of topologies and grids (structured/unstructured, orthogonal/nonorthogonal,etc.). The starting point is defining the average gradient over a finite volume element of centroid C and volume VC as1r/C ¼VCZr/ dVð6:1ÞVCThen, using the divergence theorem, the volume integral is transformed into asurface integral yieldingZ1r/C ¼/ dSð6:2ÞVC@VCwhere dS is the outward pointing surface vector. In the presence of discrete faces,Eq. (6.2) can be written asX Zr/C VC ¼/ dSð6:3Þ@VCfaceNext the integral over a cell face is approximated using the mid-point integrationrule to become equal to the interpolated value of the field at the face centroidmultiplied by the face area, resulting inr/C ¼1 X/ SfVC f ¼nbðCÞ fð6:4ÞBy reviewing Eq.
6.4 and Fig. 6.2, it is clear that to compute the average of thegradient over the control element C, information about the face area and direction(Sf) is required, as well as informationabout the neighboring elements and the ϕvalues at the element centroids /C ; /Fk . This information is needed to compute thevalue of ϕ at the interface (ϕf), which will have to be interpolated in some fashion.A profile for the variation of the dependent variable ϕ between nodal values isassumed, which basically introduces an approximation in the evaluation of thegradient. In all cases the value of ϕf will have to be computed at each face centroid.Assuming a linear profile for the variation of ϕ between the elements C andF straddling the interface f, an approximate value for ϕf, denoted by /f , can becomputed as1406 The Finite Volume MeshF1Sf1ffCFig.
6.2 Gradient computation/f ¼ gF /F þ gC /Cð6:5ÞOne way to calculate the weight factors gF and gC is given bygF ¼VCVC þ VFgC ¼VF¼ 1 gFVC þ VFð6:6ÞOther interpolation practices may be used some of which will be explained later inthe chapter.Example 1Compute the gradient for the two fields given in Table 6.1 over the twodimensional cell shown in Fig. 6.3 using the surface vector values given inthe table.Table 6.1 Geometric data for Example 1SfC123456(−2.4, −3.24)(2.4, −3.48)(4.1, −6.7)(2.2, 3.7)(−2.64, 2.9)(−3.66, 6.82)VField (1)Field (2)37.81111111610953486.2 The Finite Volume Mesh141F4F5f4f5F6f6Cf1f3F3f2F1F2Fig.
6.3 A two dimensional cell for Example 1SolutionFor case 1, the field is constant with a value of 1, so the value of the gradientis expected to be 0.r/C ¼1 X/ SfVC f ¼nbðCÞ f1 / Sf þ /f2 Sf2 þ /f3 Sf3 þ /f4 Sf4 þ /f5 Sf5 þ /f6 Sf6VC f1 11ð2:4i 3:24jÞ þ 1ð2:4i 3:48jÞ þ 1ð4:1i 6:7jÞ1r/C ¼37:8 þ 1ð2:2i þ 3:7jÞ þ 1ð2:64i þ 2:9jÞ þ 1ð3:66i þ 6:82jÞ¼¼ ð0i þ 0jÞThis is actually a property of the surfaces of closed elements, provided theyall are pointing outward (or inward) they always sum to zero.For case 2 the gradient is computed asr/C ¼1 X/ SfVC f ¼nbðCÞ f1 /f1 Sf1 þ /f2 Sf2 þ /f3 Sf3 þ /f4 Sf4 þ /f5 Sf5 þ /f6 Sf6VC!10ð2:4i 3:24jÞ þ 9ð2:4i 3:48jÞ þ 5ð4:1i 6:7jÞ1¼37:8 þ 3ð2:2i þ 3:7jÞ þ 4ð2:64i þ 2:9jÞ þ 8ð3:66i þ 6:82jÞ¼¼ 15:14i 19:96j1426.36 The Finite Volume MeshStructured GridsFor a regular structured grid, every interior cell in the domain is connected to thesame number of neighboring cells.
These neighboring cells (Fig. 6.4) can beidentified using the indices i, j, and (k) in the x, y, and (z) coordinate direction,respectively, and can be directly accessed by incrementing or decrementing therespective indices. This allows for lower memory usage since topological information is embedded in the mesh structure through the indexing system. This alsoleads to greater efficiency in coding, cache utilization, and vectorization. Structuredgrids were widely used in the development of the Finite Volume and FiniteDifference methods.In a structured grid, one can associate with each computational cell an orderedset of indices (i, j, k), where each index varies over a fixed range, independently ofthe values of the other indices, and where neighboring cells have associated indicesthat differ by plus or minus one.
Thus, if there are Ni, Nj, and Nk elements in the i, j,and k index direction, respectively, then the total number of elements in the domainis Ni Nj Nk. In three-dimensional spaces, elements are hexagons with 6 facesand 8 vertices, with each interior element having 6 neighbors. In two-dimensions,elements are quadrilaterals with 4 faces and 4 vertices, with each interior elementhaving 4 neighbors.Njj +1jj 111i 1ii +1NiFig.
6.4 Local indices and topology6.3.1Topological InformationGlobal indices are generally used for building the full system of equations over thecomputational domain, while local indices are employed to define the local stencilfor an element, information that is useful during the discretization process.
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