Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 22
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. ffl::} ” is the inner product of two|fflfflffl.{zfflfflnðn1Þtimesnth rank tensors, all yielding a scalar.1145 The Finite Volume MethodFig. 5.7 Variation within anelement(x ) xxCCCCComparison between the assumed variation given by Eq. (5.27) and the Taylorseries expansion [Eq. (5.28)] indicates an error proportional to |(x − xC)2| andimplying a second order spatial accuracy.5.4.2 Mean Value ApproximationThe accuracy of the mean value approximation can be derived by integrating thevariable /ðxÞ over the cell of centroid C depicted in Fig. 5.8 and leading to/C ¼1VCZ/ dVVC1¼VCZ hi/C þ ðx xC Þ ðr/ÞC þ O jx xC j2 dVVC¼/CVCZdV þVC1VCZðx xC Þ ðr/ÞC dV þVC¼ /C þ O jx xj21VCZO jx xC j2 dVð5:29ÞVCwhere VC is the volume of the element.
The second term is equal to zero becausepoint C is the centroid of the element. Neglecting the last term in the equationintroduces an error of order 2, demonstrating that the mean value approximation issecond order accurate.5.4 Order of Accuracy115Fig. 5.8 Mean-value theoremCCThe convective flux at face f of element C shown in Fig. 5.9 is computed asqf v f Sf /f ¼Zðqv/Þ dSfZ¼h2 iqf vf /f þ x xf ðr/Þf þ O x xf dSfZ¼ qf vf /f ZdS þf¼ qf vf SfhZx xf ðr/Þf qf vf dS þf2 O x xf qf vf dSð5:30Þf2 i/f þ O x xf where again the second term is equal to zero as xf is the centroid of the respectiveelement surface.
Here subscript f denotes the value of the variable, in this case /, atthe face centroid and S is the outward pointing face surface vector.FJf,DSf = ()f SffDiffusion fluxJ f ,C S f = ( v ) f S fConvection fluxCFig. 5.9 Convection and diffusion fluxes at element faces1165 The Finite Volume MethodFor the diffusive flux, truncating second order terms and higher, the followingcan be written:ZfC/ r/ dS ¼Z h2 i dSC/ r/ f þ x xf r C/ r/ f þ O x xf f¼ C/ r/Zff32Z2 7 /6 dS þ 4x xf dS5 : r C r/ f þ O x xf ð5:31Þf2 ¼ C r/ f Sf þ O x xf /Hence for a second order method, the surface integral and the variation of / withinthe element should be second order accurate.Higher order accuracy can be achieved by increasing the order of accuracy of thesurface integral or of the assumed profile for /. One method developed by Lilekand Peric [13] in two dimensions used a fourth order accurate surface integral,evaluated via Simpson’s rule asZ/ip1 þ 4/ip2 þ /ip3/dS ¼Sf6ð5:32ÞfAs shown in Fig.
5.10, for face f the value of / is needed at three integrationpoints, i.e., the surface centroid ip2, and the vertices of the face ip1 and ip3.To obtain a formally fourth order accurate discretization, the assumed profile of/ within the element should also be fourth order accurate such that1/ðxÞ ¼ /C þ ðx xC Þ ðr/ÞC þ ðx xC Þ2 :ðrr/ÞC þ 2ð5:33ÞIn this case the accuracy of the gradient calculation has to be second order orhigher and that of the Hermitian first order or higher.Fig. 5.10 A element facef showing the integrationpoints where values areneeded for a higher orderapproximation of the surfaceintegralFip1fip2ip35.5 Transient Semi-Discretized Equation1175.5 Transient Semi-Discretized EquationFor unsteady state the temporal term in Eq. (5.1) is retained and in addition tointegrating over the volume of the element, integration over time is also needed.
Inthis case the integrated equation becomes0132tþDt ZtþDtZZZX@ ðq/ÞBC76dVdt þ@ ðqv/Þf dSA5dt4@tf nbðCÞttVCtþDtZf013ZXBC76@ ðCr/Þf dSA5dt42f nbðCÞttþDtZ¼264tZfð5:34Þ37Q/ dV5dtVCFurther simplification of this equation requires a choice with regard to the timeintegration accuracy required. This will be dealt with in Chap. 13.
For fixed grids,where the volume and surface of each element are constant in time, the first termcan be integrated astþDt ZZtVC@ ðq/ÞdV dt ¼@ttþDtZtwhereq/C ¼1VC0@B@@tZ1ZCq/dVAdt ¼tþDt Z@ q/VC dt@tð5:35ÞtVC q/dV ¼ ðq/ÞC þ O D2ð5:36ÞVCSubstituting back, Eq. (5.34) reduces totþDtZt@ ðq/ÞVC dt þ@ttþDtZ130ZX6C7B4@ ðqv/Þf dSA5dt2f nbðCÞttþDtZf130ZX6C7B4@ ðCr/Þf dSA5dt2tf nbðCÞtVCf32tþDt ZZ76¼4 Q/ dV5dtð5:37Þ1185 The Finite Volume Methodand using the midpoint rule, Eq. (5.37) becomestþDtZt@ ðq/ÞVC dt þ@ttþDtZ2X43ðqv/Þf Sf 5dtf nbðCÞttþDtZ24ttþDtZ¼X3ðCr/Þf Sf 5dtð5:38Þf nbðCÞQ/C VC dttGoing forward beyond this point requires some assumptions as to how thevariable is changing in time.5.6 Properties of the Discretized EquationsAs the size of the element tends to zero, the numerical solution is expected to be theexact solution of the general conservation equation [e.g., Eq. (5.2)] irrespective ofthe interpolation profile used to evaluate the element / values.
However, sincefinite volumes are used, it is crucial for the discretized equations to possess someproperties in order to ensure a meaningful solution field. These properties arediscussed next.5.6.1 ConservationFrom a physical point of view it is very important for the transported variables,which are generally conservative quantities (e.g., mass, energy, etc.), to be conserved in the discretized solution domain too, otherwise results may be unrealistic.This property is inherent to the FVM because the fluxes integrated at an elementface are based on the values of the elements sharing the face [8]. Thus for anysurface common to two elements, the flux leaving the face of one element will beexactly equal to the flux entering the other element through that same face. Thusthese fluxes are of equal magnitudes but of opposite signs (Fig.
5.11). Any methodthat possesses this property is said to be conservative.5.6 Properties of the Discretized EquationsFluxC fC+ FluxFfF+ FluxV fFluxT f =F1119FluxFfF+ FluxC fFluxT fC+ FluxV fCSfCFluxC fCfF1SffFig. 5.11 Fluxes on neighboring elements5.6.2 AccuracyAccuracy refers to how close a numerical solution is to the exact solution. However,in most of the cases the exact solution for the problem to be solved is unknown.Therefore a direct comparison to check accuracy is not possible. An alternative is toconsider the truncation error as a measure of accuracy. The error associated with thefirst step discretization of the various terms presented above was O|(x − xf)|2, whichrepresents a second order accuracy.
This means that if the number of grid points isdoubled then the discretization error will be reduced by a factor of 4. The truncationerror of a discretization scheme is the largest truncation error of each of the individual terms in the discretized equation. The discretization error does not give thevalue of the error on a certain grid. It does tell, however, how fast the error willdecrease with grid refinement. The higher the order of the error the faster it willdecrease with mesh refinement.5.6.3 ConvergenceThe nonlinear nature of the conservation equations dealt with here necessitates aniterative approach.
Starting with an initial guess, solutions are obtained byrepeatedly applying a solution algorithm with the solution at the end of an iterationused as an initial guess for the following iteration. Ideally a solution is said to beconverged when it does not change any further as iterations progress. In practice,however, a solution is established converged when changes between two consecutive iterations fall below a vanishing quantity ε. In general convergence is used toindicate the obtainment of a solution with any method.
Sometimes the term“convergence” is used to indicate the attainment of a grid independent solution, i.e.,a solution that does not change with any further grid refinement.1205 The Finite Volume Method5.6.4 ConsistencyA solution to an algebraic equation approximating a given partial differentialequation is said to be consistent if, at each point in the solution domain, thenumerical solution approaches the exact solution of the partial differential equationas the time step and grid spacing tend to zero, i.e., as the discretization errorapproaches zero.
For this to be true, the discretization error should be some powerof Δt and/or Δx. If the discretization error is expressed in terms of Δx/Δt then, forconsistency, Δx should tend to zero at a faster rate than Δt.5.6.5 StabilityStability refers to the behavior of the discretized equations to be resolved by aniterative solver.
It indicates whether the resulting system of algebraic equations canbe solved under a variety of initial and boundary conditions. In this sense stability isnot really a property of the discretization process but rather a property of theresulting system of equations.
As mentioned in a previous chapter, a sufficientcondition for a system of linear equations to be stable and converge to a solution isfor it to satisfy the Scarborough criterion, i.e., for its matrix of coefficients to bediagonally dominant.For transient problems, a stable numerical scheme keeps the error in the solutionbounded as time marching proceeds. The use of explicit or implicit transientschemes has direct impact on the stability of the numerical method. The stability ofexplicit methods is ensured by limiting the size of the time step.
On the other hand,the stability of implicit methods can be enhanced by under-relaxing the discretizedset of algebraic equations either through the use of under relaxation factors or byapplying the false-transient approach, as described in a later chapter.5.6.6 EconomyEconomy is an important consideration in the development and application of CFDcodes. It is understood to mean that the time required to compute realistic flowproblems should not be prohibitive.5.6.7 TransportivenessThe directional properties exhibited by fluid transport are well known and aresignaled by the change in the type of the transport scalar equation [Eq.
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