Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 21
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5.2 Surface integration of fluxes using a one integration point, b two integration points, andc three integration pointspffiffiffiffiffithree integration points (ip = 3) positioned at n1 ¼ ð5 15Þ=10; n2 ¼ 1=2, andpffiffiffin3 ¼ ð5 þ 15Þ=10, with weights ω1 = 5/18, ω2 = 4/9, and ω3 = 5/18. It is clearthat the computational cost rises with the number of integration points used in theapproximation.
With ip(f) denoting the number of integration points along face f,the general discretized relations for the convection and diffusion terms becomeIX Xðqv/Þ dS ¼xip ðqv/Þip Sfð5:12Þf facesðVÞ ipipðf Þ@VCIC/ r/ dS ¼XX f facesðVÞ ipipðf Þ@VCxip C/ r/ ip Sfð5:13Þ5.2.2 Source Term Volume IntegrationVolume integration is used for the source term. Adopting a Gaussian quadratureintegration, the volume integral of the source term is computed asZX /Q/ dV ¼Qip xip Vð5:14ÞVipipðVÞAs with surface flux integration, Fig. 5.3 shows different options for volumeintegration with their accuracy depending on the number of integration points used(ip) and the weighing function ωip.For one point Gauss integration (Fig.
5.3a), ip = ωip = 1 with the integration pointlocated at the centroid of the element. This approximation is second order accurateand is applicable in two and three dimensions. In two dimensions, four point gauss1085 The Finite Volume Method(a)(b)FluxT=FluxC C + FluxVFluxT=FluxC C + FluxVF1F1F1F2F2F2f1f1f2f4f2CF6f6f3F4f1f2CF3(c)FluxT=FluxC C + FluxVF3f5f4F5F4one integration pointCF6f6f3F3f5f4F5four integration pointsF6f6f3F4f5F5nine integration pointsFig.
5.3 Volume integration of source terms using a one integration point, b four integrationpoints, and c nine integration pointsintegration (Fig. 5.3b) involves the use of four integration points. The weights arecomputed as the product of the one dimensional weights and the integration points(ξ, η) are obtained from the one dimensional profiles. Therefore the function is calpffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffifficulated at 3 3 =6; 3 þ 3 =6 , 3 þ 3 =6; 3 þ 3 =6 , 3 3 =6;pffiffiffipffiffiffi pffiffiffi 3 3 =6, and3 þ 3 =6; 3 3 =6 with the weights being equal(ωip = 1/4 for ip = 1 to 4). The nine point gauss integration method (Fig. 5.3c)involves the use of nine integration points.
The accuracy increases with increasingthe number of integration points but so does the computational cost.5.2.3 The Discrete Conservation Equation for OneIntegration PointWhile the above terms can be discretized with any specified number of integrationpoints, it is customary for the finite volume method to use one integration point,yielding second order accuracy. This was found to be a good compromise betweenaccuracy and flexibility while keeping the method simple and relatively of lowcomputational cost. Following the mid-point integration approximation, thesemi-discrete steady state finite volume equation for element C shown in Fig. 5.4can be finally simplified toX f nbðCÞqv/ C/ r/ f Sf ¼ Q/C VCð5:15ÞThe aim of the second stage of the discretization process is to transformEq.
(5.15) into an algebraic equation by expressing the face and volume fluxes interms of the values of the variable at the neighboring cell centers. This linearization of the fluxes is at the core of the second discretization step.5.2 The Semi-Discretized Equation109Fig. 5.4 Fluxes at elementsurfacesF1F2f1f2CF3F6f6f3f5f4F5F45.2.4 Flux LinearizationAs schematically depicted in Fig. 5.2a, the face flux can be split into a linear part,function of the / values at the nodes straddling the face (i.e., /C and /F ), and anon-linear part, which includes the remaining portion that cannot be expressed interms of /C and /F .
The resulting equation can be written asJ/f Sf ¼ FluxTf ¼|fflfflffl{zfflfflffl}total fluxfor face fFluxCf|fflfflffl{zfflfflffl}/C þflux linearizationcoefficient for CFluxFf|fflfflffl{zfflfflffl}flux linearizationcoefficient for F/F þFluxVf|fflfflffl{zfflfflffl}nonlinearized partð5:16Þwhere FluxTf represents the total flux through face f, and is decomposed into threeterms. The first two terms represent the contributions of the two elements sharingthe face and are written via the linearization coefficients FluxCf and FluxFf.
The lastterm describes the nonlinear contribution that cannot be expressed in terms of /Cand /F and is given by the non-linear term FluxVf. The values of FluxCf, FluxFf,and FluxVf obviously depend on the discretized term and the scheme used for itsdiscretization.The flux linearization is thus obtained by substituting Eq (5.16) into the left handside of Eq. (5.15). Repeating for all cell faces yieldsX J/f Sf¼f nbðCÞX FluxTff nbðCÞ¼X f nbðCÞFluxCf /C þ FluxFf /F þ FluxVfð5:17Þ1105 The Finite Volume MethodThe linearization of the volume flux is performed, as shown in Fig.
5.3a, byexpressing it as a linear function of the element node value /C and is given byQ/C VC ¼ FluxT¼ FluxC /C þ FluxVð5:18ÞIn the case of a constant source term, the volume flux, which represents the righthand side of Eq. (5.15), reduces toFluxC ¼ 0FluxV ¼ Q/C VCð5:19ÞSubstitution of Eqs. (5.17) and (5.18) in Eq. (5.15), yields the sought afteralgebraic relation asXaC / C þðaF /F Þ ¼ bCð5:20ÞFNBðCÞwhere the relations between equation coefficients and flux linearization coefficientsare expressed asXaC ¼FluxCf FluxCf nbðCÞaF ¼ FluxFfXbC ¼ FluxVf þ FluxVð5:21Þf nbðCÞExample 1Find the linearization coefficients for the discretization of the convection termwhen the velocity field is in the positive direction using the approximation/f ¼ /C (this is known as the upwind scheme).SolutionJ/f ¼ ðqv/ÞfthusJ/f Sf ¼ ðqv/Þf Sf ¼ qf vf Sf /f ¼ m_ f /f5.2 The Semi-Discretized Equation111With /f ¼ /C , the coefficients in the total flux equation are obtained asFluxCf ¼ m_ fFluxFf ¼ 0FluxVf ¼ 0and the total flux is expressed asFluxTf ¼ m_ f /C þ 0/F þ 05.3 Boundary ConditionsThe evaluation of the fluxes at the faces of a domain boundary does not require, ingeneral, a profile assumption.
Rather a direct substitution is usually performed. Thetype of boundary conditions are numerous. However, two of the most widely usedones for general scalars are the Dirichlet and the Neumann boundary conditions. Inmathematical terms these are respectively a value specified (or a first type) and aflux specified (or a second type) boundary condition.5.3.1 Value Specified (Dirichlet Boundary Condition)Consider the case where some scalar / is being convected through an inlet.Assuming the diffusion of / to be negligible, the boundary condition can beexpressed as/b ¼ /b;specifiedð5:22ÞFor the boundary face shown in Fig. 5.5, the boundary flux is evaluated using theknown value of /b given by Eq.
(5.22). Therefore the value of the boundary flux isnot unknown, rather it can be directly evaluated asJ/b Sb ¼ J/;C Sbb¼ ðqv/Þb Sb¼ FluxCb /C þ FluxVb¼ ðqb vb Sb Þ/b ¼ m_ f /b;specifiedð5:23ÞThusFluxCb ¼ 0FluxVb ¼ m_ f /b;specifiedð5:24Þ1125 The Finite Volume MethodFig. 5.5 Dirichlet boundaryconditionSbntb,specifiedbebC5.3.2 Flux Specified (Neumann Boundary Condition)Considering the case shown in Fig. 5.6 where the boundary face of the elementC represents physically a wall where a flux for the quantity / is specified.Mathematically this is equivalent to writingJ/b Sb ¼ J/b nb Sb|fflfflffl{zfflfflffl}ð5:25Þspecified flux¼ qb;specified SbIn the above equation qb,specified is a known quantity specified by the user,representing the flux per unit area.Fig. 5.6 Neumann boundaryconditionSbntbebCqb,specified5.3 Boundary Conditions113ThusFluxCb ¼ 0FluxVb ¼ qb;specified Sbð5:26ÞThe treatment of these two boundary conditions and others will be detailed in thefollowing chapters along with the treatment of the terms related to step twodiscretization.5.4 Order of AccuracyAs discussed earlier, fluxes at the faces and sources over the element are evaluatedfollowing the mean value approach, i.e., using the value at the centroid of thesurface (midpoint rule) and cell, respectively.
This treatment, in addition to theassumed variation of / in space around point C, i.e., / ¼ /ðxÞ, determine theaccuracy of the discretization procedure. In the adopted method, / is assumed tovary linearly in space, i.e.,/ðxÞ ¼ /C þ ðx xC Þ ðr/ÞC where /C ¼ /ðxC Þð5:27Þ5.4.1 Spatial Variation ApproximationThe spatial variation of the variable / ¼ /ðxÞ within the element shown in Fig. 5.7can be described via a Taylor series expansion around point xC as1/ðxÞ ¼ /C þ ðx xC Þ ðr/ÞC þ ðx xC Þ2 : ðrr/ÞC21þ ðx xC Þ3 :: ðrrr/ÞC þ.
. . . . . . . . . . . . . .3!01þð5:28Þ1ðx xC Þn ::. . ffl::} @rr. . .r/ A þ. . . . . . . . . . . . . . .|fflfflffl.{zfflffl|fflfflfflfflfflffl{zfflfflfflfflfflffl}n!ðn1Þtimesn timesCThe expression (x − xC) in the equation and consequent ones represents the nthtensorial product of the vector (x − xC) with itself, producing an nth rank tensor.The operator (:) is the inner product of two 2nd rank tensors, (::) is the inner productof two 3rd rank tensors, and more generally “ ::.
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