Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 23
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(5.2)], which5.6 Properties of the Discretized EquationsFig. 5.12 Illustration of thetransportive property of fluidflow121Pe = 0Pe > 0WCEmay become hyperbolic under certain conditions. The implications on the finitevolume equation, visualized in Fig. 5.12, can be explained as follows. If there is aconstant source of / within an element C in a flow field with uniform velocity anddiffusivity, then the shapes of the contours of constant scalar / will be influencedby the ratio of convection to diffusion strengths, i.e., the Péclet number (Pe) definedasPe ¼Convection strengthqu¼Diffusion strengthC=Dxð5:39ÞBased on Eq.
(5.39), the case Pe = 0 indicates that the transport of / is governedby diffusion, which has an elliptic behavior [8]. In this case, isolines of / arecircular and the value of / at C is influenced by the surrounding nodes W andE (Fig. 5.12). Increasing convection effects (i.e., increasing Pe), the circular contours become elliptic in shape and the region influencing the value of / at C shiftsin the direction of the flow. Therefore for high Pe flows, events at node C will havea weak or no influence on upstream nodes, while downstream nodes will bestrongly affected.Failure to observe this requirement in the selected discretization schemes cangive rise to unstable solutions (i..e., unphysical oscillations).5.6.8 Boundedness of the Interpolation ProfileEnsuring conservation does not guarantee that other important properties of theoriginal partial differential equation will be maintained by the discretized equation.For example physical considerations lead to the conclusion that in the absence of1225 The Finite Volume Methodsources, the value of the conserved variable / within the domain should bebounded by the values at the domain boundaries [14].
The discretized equationXaC /C þaF /F ¼ bCð5:40ÞFNBðCÞwould fulfill this requirement when the nodal /C values satisfy the followingconstraint:N N min /Fi /C max /Fii¼1i¼1ð5:41Þwhere Fi represents the ith neighbor of C and N their number. This can be achievedby a judicial control of the discretization schemes and their linearization, as will bedetailed in later chapters.5.7 Variable ArrangementWhile the cell-centered variable arrangement is the one selected in OpenFOAM®and generally preferred with the FVM (Fig. 5.13a), vertex-centered [15](Fig.
5.13b) variable arrangement methods have also been used. Twovertex-centered arrangements have been adopted resulting in either overlappingelements or a dual mesh, respectively. Because the dual mesh method is morepopular, it is described next as a representative of vertex-centered methods.(a)(b)Cell-centeredVertex-centeredFig. 5.13 a Cell-centered arrangement; b vertex-centered arrangement5.7 Variable Arrangement1235.7.1 Vertex-Centered FVMIn a vertex-centered arrangement the flow variables are stored at the vertices (orgrid points), with elements constructed around the variable locations by using theconcept of a dual mesh and dual cells [16] (Fig.
5.14). Adopting this concept, a cellcan be created around a grid point in several ways. In two dimensions, oneapproach (Fig. 5.14a) is by connecting the centroids of the cells having the gridpoint in common. As displayed in Fig. 5.14b a second possibility is to join thecentroids of the surrounding elements to the centroids of their faces. The sameconstruction procedure can be used in three dimensional situations where an element around a grid point is created by properly connecting surrounding cells’centroids, faces’ centroids, and edges’ centroids resulting in non-overlappingelements.The use of a vertex-centered arrangement of variables allows for an explicitprofile to be defined over the elements in terms of the vertex variables.
In this casethe variables represent point values and variation through the element can becomputed using shape functions or interpolation profiles. This approach permits anaccurate resolution of face fluxes for all mesh topologies, but yields a lower orderaccuracy of element-based integrations since the vertex is not necessarily at theelement centroid. Moreover, it increases the storage requirements due to the creation of larger matrix. Furthermore handling of boundary conditions in cell-vertexschemes, as shown in Fig.
5.14b, requires additional treatment in order to ensure aconsistent solution at boundary points shared by multiple grid blocks. Still a majordisadvantage is the need to base the mesh on a set of element types for which ashape function can be defined.Another shortcoming of the vertex-centered scheme with dual elements appearsat solid boundaries where only a portion of an element is formed and the nodewhere values are stored is at the wall (Fig. 5.14b). In a regular cell, the integrationof face fluxes results in a residual located at the main point inside the element wherevalues are stored. In this case however, the residuals will be stored at the wall(a)(b)Fig.
5.14 Vertex-centered arrangement: a dual cells connecting centroids of cells, and b dual cellsconnecting centroids of cells to centroids of faces1245 The Finite Volume Methodboundary creating a discrepancy and causing an increase in the discretization errorthere in comparison with cell-centered schemes.
Additional complications may alsoarise at sharp edges and branch cuts.5.7.2 Cell-Centered FVMThe cell-centered variable arrangement is currently the most popular type of variablearrangement used with the FVM. With this practice, the variables and their relatedquantities are stored at the centroids of grid cells or elements. Thus, the elements areidentical to the discretization elements and, in general, the method is second orderaccurate since all quantities are computed at element and face centroids, where thedifference between the value of the variable and its average is O(Δx2).
Variationswithin the cell can be re-constructed using a Taylor series expansion. Anotheradvantage of the cell-centered formulation is that it allows for the use of generalpolygonal elements with no need for pre-defined shape functions. This permits astraightforward implementation of a full multigrid strategy.However two important disadvantages of the method are its treatment ofnon-conjunctional elements and the manner the diffusion term is discretized onnon-orthogonal cells. The first issue influences the accuracy of the method, thesecond its robustness, while both being affected by the quality of the mesh.Consider the two-cell arrangement shown in Fig.
5.15. It is clear that anyaverage of a value defined at C and F will be defined at f 0 rather than f, the centroidof the face. Thus any discretization procedure using this interpolated value will nothave an O(Δx2) accuracy.For a cell-centered scheme the discretization error depends strongly on thesmoothness of the grid. Results for such a situation are displayed in Fig. 5.16. Thephysical situation displayed in Fig. 5.16a represents an annulus between two horizontal cylinders of rhombic cross-sections.
The inner cylinder is maintained at theuniform hot temperature Th while the outer cylinder is kept at the cold temperatureTc. The difference in temperature creates density variation within the enclosure andgives rise to buoyancy forces that establish a flow field. Using the grid systemSfCfFFig. 5.15 Two non-junctional cell-centered elements5.7 Variable Arrangement(a)125(b)(c)(d)Fig. 5.16 a Grid used in solving for natural convection in the annulus between horizontalcylinders of rhombic cross sections; b isotherms over the domain; c a magnified region showingthe grid around the horizontal centerline of the domain; d a magnified region around the horizontalcenterline showing the kinks in the generated isothermsdisplayed in Fig.
5.16a, the velocity and temperature distributions within theenclosure are obtained numerically using a finite volume method. Generated isotherms are displayed in Fig. 5.16b. By carefully inspecting Fig. 5.16b small kinkscan be seen in the region around the horizontal centerline of the domain.
The regionaround the kinks is magnified and the grid and isotherms in that region are displayed in Fig. 5.16c, d, respectively. The kinks in the isotherms are clear in themagnified plot and are due to the use of the grid with slope discontinuity causing adiscretization error that cannot be reduced independent of how much the grid isrefined. This zero order error does not arise with a cell-vertex scheme. Despite thisfact, for a sufficiently smooth grid, the cell-centered arrangement can attain accuracy of order two or higher.The other issue that affects cell centered FVM is the treatment ofnon-orthogonality in the discretization of the diffusion term. This will be covered indetail in Chap. 8, however a brief explanation is useful at this stage.In the discretization of the diffusion term (Fig.
15.17) the value of the angle θextending between the unit vector e (which is in the direction of the line joining thecentroids of the elements C and F) and the unit vector n (which is normal to the faceshared by the elements C and F) affects the degree of implicitness and hence therobustness of the method applied in the discretization of the diffusion term.1265 The Finite Volume MethodFig.
15.17 A non-orthogonalelementSft nefFd CFCFor discretization, the diffusion term is generally written asr/ S ¼r/ E|fflfflffl{zfflfflffl}Implicit orthogonallike contributionþr/ T|fflfflffl{zfflfflffl}ð5:42ÞExplicit nonorthogonal like contributionThe larger the angle θ is, the larger the explicit term will be, and the less robustthe discretization method becomes.To summarize, the vertex-centered scheme with dual elements and thecell-centered scheme are numerically very similar in the interior of a domain forsteady state calculations. The only situation where the performance of thevertex-centered scheme is superior to that of the cell-centered scheme is over adistorted grid.
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