Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 20
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In Chap. 10,an algebraic multigrid method will be used to accelerate the rate of convergence ofiterative schemes and improve their performance.4.1.6Other Types of FieldsIn addition to the element field introduced above, other fields are defined fordifferent purposes. Two such fields include the face field and the vertex field thatare briefly described below.The face field consists of the array of values defined at the centre of the faces. Asshown in Fig.
4.15, it defines a number of arrays for the interior faces and thevarious patch faces. The face field is used, for example, to define the face massfluxes for use when solving advective and flow problems.The vertex field schematically depicted in Fig. 4.16 stores variables at the vertices; these again are grouped into interior vertices and patch vertices. Vertex fieldsare usually used for post processing, and in some cases for gradient computation.4.2 ClosurePatch#2101Patch#1Face FieldPatch#3interior12 3patch#112 3patch#212 34patch#312 34......3325Fig. 4.15 Face fieldPatch#2Patch#1Vertex FieldPatch#3interior12 3patch#112 3patch#212 34patch#312 348...25Fig.
4.16 Vertex field4.2ClosureThis chapter overviewed the discretization process, underlining on the way thebasic ingredients needed for the development of a CFD code. The coming chapterswill take each of these ingredients and dissect it, while developing the “uFVM” andlearning about the industrial open source CFD library, denoted by OpenFOAM®.The next two chapters will further expand on the idea of finite volume mesh andfinite volume discretization.Chapter 5The Finite Volume MethodAbstract Similar to other numerical methods developed for the simulation of fluidflow, the finite volume method transforms the set of partial differential equationsinto a system of linear algebraic equations.
Nevertheless, the discretization procedure used in the finite volume method is distinctive and involves two basic steps. Inthe first step, the partial differential equations are integrated and transformed intobalance equations over an element. This involves changing the surface and volumeintegrals into discrete algebraic relations over elements and their surfaces using anintegration quadrature of a specified order of accuracy. The result is a set ofsemi-discretized equations. In the second step, interpolation profiles are chosen toapproximate the variation of the variables within the element and relate the surfacevalues of the variables to their cell values and thus transform the algebraic relationsinto algebraic equations.
The current chapter details the first discretization step andpresents a broad review of numerical issues pertaining to the finite volume method.This provides a solid foundation on which to expand in the coming chapters wherethe focus will be on the discretization of the various parts of the general conservation equation. In both steps, the selected approximations affect the accuracy androbustness of the resulting numerics. It is therefore important to define someguiding principles for informing the selection process.5.1 IntroductionThe popularity of the Finite Volume Method (FVM) [1–3] in Computational FluidDynamics (CFD) stems from the high flexibility it offers as a discretization method.Though it was preceded for many years by the finite difference [4, 5] and finiteelement methods [6], the FVM assumed a particularly prominent role in the simulation of fluid flow problems and related transport phenomena as a result of thework done by the CFD group at Imperial College in the early 70 s under thedirection of Professor Spalding [7], with such contributors as Patankar [8], Gosman[9], and Runchal [10, 11] to cite a few.
The FVM owes much of its flexibility andpopularity to the fact that discretization is carried out directly in the physical spacewith no need for any transformation between the physical and the computational© 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_51031045 The Finite Volume Methodcoordinate system.
Furthermore its adoption of a collocated arrangement [12] madeit suitable for solving flows in complex geometries. These developments haveexpanded the applicability of the FVM to encompass a wide range of applicationswhile retaining the simplicity of its mathematical formulation. Another importantaspect of the FVM is that its numerics mirrors the physics and the conservationprinciples it models, such as the integral property of the governing equations, andthe characteristics of the terms it discretizes.
In what follows the semi-discretizedform of a general scalar equation is derived. Then the properties required from thediscretization method are discussed along with some guiding principles. Thechapter ends with a discussion of a number of issues pertinent to the FVM. Thetransformation of the semi-discretized equation into algebraic equations will be thesubject of a number of chapters to follow.5.2 The Semi-Discretized EquationIn step 1 of the finite volume discretization process, the governing equations areintegrated over the elements (or finite volumes) into which the domain has beensubdivided, then the Gauss theorem is applied to transform the volume integrals ofthe convection and diffusion terms into surface integrals. Following this step, thesurface and volume integrals are transformed into discrete ones and integratednumerically through the use of integration points (ip).
To clarify this approach andto develop an adequate appreciation for the subtleties of the advanced discretizationschemes discussed in later sections, the following example illustrates the application of the technique for a two-dimensional transport problem.As presented in Chap. 3, the conservation equation for a general scalar variable/ can be expressed as@ ðq/Þþ r ðqv/Þ ¼ r C/ r/ þ Q/|{z}|fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}@t|fflffl{zfflffl}transient termconvective termdiffusion termð5:1Þsource termThe steady-state form of the above equation is obtained by dropping the transientterm and is given byr ðqv/Þ ¼ r C/ r/ þ Q/ð5:2ÞBy integrating the above equation over the element C shown in Fig.
5.1; Eq. (5.2) istransformed toZZZr ðqv/ÞdV ¼r C/ r/ dV þQ/ dVð5:3ÞVCVCVC5.2 The Semi-Discretized Equation105Fig. 5.1 Conservation in adiscrete elementF1F2Convectionf2Diffusionf1TransientCf6f3F3F6Source/SinkVCf4f5F5F4Replacing the volume integrals of the convection and diffusion terms by surfaceintegrals through the use of the divergence theorem, the above equation becomesIIðqv/Þ dS ¼@VCC/ r/ dS þ@VCZQ/ dVð5:4ÞVCwhere bold letters indicate vectors, (·) is the dot product operator, Q/ represents thesourceH term, S the surface vector, v the velocity vector, / the conserved quantity,andthe surface integral over the volume VC.@VC5.2.1 Flux Integration Over Element FacesDenoting the convection and diffusion flux terms by J/;C and J/;D , respectively,their expressions are given byJ/;C ¼ qv/ð5:5ÞJ/;D ¼ C/ r/ð5:6Þ1065 The Finite Volume MethodFurther, defining the total flux J/ as the sum of the convection and diffusionfluxes, it can be written asJ/ ¼ J/;C þ J/;Dð5:7ÞReplacing the surface integral over cell C by a summation of the flux terms overthe faces of element C, the surface integrals of the convection, diffusion, and totalfluxes become0IJ/;C dS ¼B@XB@f facesðVC Þ@VCIZXB@f facesðVC Þ@VCð5:8Þ1CC/ r/ dSAð5:9Þf0J/ dS ¼Cðqv/Þ dSAf0J/;D dS ¼1Zf facesðVC Þ@VCIXZ1CJ/f dSAð5:10ÞfIn Eqs.
(5.8)–(5.10) the surface fluxes are evaluated at the faces of the elementrather than integrated within it. This transformation has important consequences onthe properties of the FVM, one of which is that it renders the method conservative,as will be discussed later.To proceed further with the discretization, the surface integral at each face of theelement in addition to the volume integral of the source term have to be evaluated.Using a Gaussian quadrature the integral at the face f of the element becomesZZX / /J dS ¼J n dS ¼J/ n ip xip Sfð5:11Þffipipðf Þwhere ip refers to an integration point and ip(f) the number of integration pointsalong surface f.
As seen in Fig. 5.2, a number of options are available with theiraccuracy depending on the number of integration points used and the weighingfunction ωip. For a simple mean value integration (Fig. 5.2a), also known as thetrapezoidal rule, only one integration point located at the centroid of the face is usedwith a weighing function of value equal to 1 (i.e., ip = ωip = 1).
This approximationis second order accurate and is applicable in two and three dimensions. Anotheroption (Fig. 5.2b) in two dimensions, which is third order accurate, involves twopffiffiffipffiffiffiintegration points (ip = 2) positioned at n1 ¼ ð3 3Þ=6 and n2 ¼ ð3 þ 3Þ=6where ξ is distance along the face measured from one end and normalized by thetotal length, with weights x1 ¼ x2 ¼ 1=2. A third option depicted in Fig. 5.2c uses5.2 The Semi-Discretized Equation(a)(b)FluxT f=FluxC f C + FluxF f107F1+ FluxV f=FluxC f C + FluxF fF1F2f2F6F3Sff11dCNf4F4two integration pointsF6f6f3f5F51CF6f6F4one integration point+ FluxV fF1F3f4F1f2F5F4=+ FluxF fF2f3f5f4C1Cf6f3FluxC f1dCN1CF3+ FluxV fSff11f2dCNF1FluxT fF1F2Sff1(c)FluxT ff5F5three integration pointsFig.
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