Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 59
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11.6. The upwind scheme basicallymimics the basic physics of advection in that the cell face value is made dependenton the upwind nodal value, i.e., dependent on the flow direction. In this case the cellface values for the configuration displayed in Fig. 11.6 are given by37611 Discretization of the Convection TermeCwWEuwwWWuwWwueeCueeEEEFig. 11.6 The upwind scheme profile/e ¼/C/Eif m_ e [ 0if m_ e \ 0and/w ¼/C if m_ w [ 0/W if m_ w \ 0ð11:33Þwhere m_ e and m_ w are the mass flow rates at faces e and w given bym_ e ¼ ðqv SÞe ¼ ðquSÞe ¼ ðquDyÞem_ w ¼ ðqv SÞw ¼ ðquSÞw ¼ ðquDyÞwð11:34ÞThus, the advection flux at face e can be written asm_ e /e ¼ km_ e ; 0k/C km_ e ; 0k/E¼ FluxCeConv /C þ FluxFeConv /E þ FluxVeConvð11:35ÞwhereFluxCeConv ¼ km_ e ; 0kFluxFeConv ¼ km_ e ; 0kFluxVeConvð11:36Þ¼0In Eqs.
(11.35) and (11.36), the term ka; bk represents the maximum of a and b.Moreover, a similar relation can be derived for the advection flux at face w and isgiven bym_ w /w ¼ km_ w ; 0k/C km_ w ; 0k/W¼ FluxCwConv /C þ FluxFwConv /W þ FluxVwConvð11:37Þwhere nowFluxCwConv ¼ km_ w ; 0kFluxFwConv ¼ km_ w ; 0kFluxVwConv¼0ð11:38Þ11.2Steady One Dimensional Convection and Diffusion377Denoting the contribution of the diffusion flux with a superscript Diff, then substitution into Eq. (11.15) yieldsFluxCeConv þ FluxCeDiff þ FluxCwConv þ FluxCwDiff /Cþ FluxFeConv þ FluxFeDiff /E þ FluxFwConv þ FluxFwDiff /W ¼ 0ð11:39Þwhich can be modified into the formaC /C þ aE /E þ aW /W ¼ 0ð11:40ÞwithaE ¼ FluxFeConv þ FluxFeDiffSe¼ km_ e ; 0k C/edxeaW ¼ FluxFwConv þ FluxFwDiffSw¼ km_ w ; 0k C/wdxwXConvaC ¼FluxCfþ FluxCfDifffð11:41ÞSeSwþ C/w¼ km_ e ; 0k þ km_ w ; 0k þ C/edxedxw¼ ðaE þ aW Þ þ ðm_ e þ m_ w Þ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}¼0XConvbC ¼ FluxVfþ FluxVfDifff¼0It is easily seen that the upwind scheme yields negative neighbor coefficients, andprovided continuity is satisfied, the coefficient at the main node is given by,aC ¼ ð aW þ aE Þð11:42Þwhich guarantees the boundedness property.Invoking the continuity constraint in Eq.
(11.41) and assuming uniform grid andconstant diffusion coefficient, the value for /C in terms of /E and /W is obtainedfrom Eq. (11.40) and Eq. (11.41) as/C /W2 þ jj PeL ; 0jj2 þ jj PeL ; 0jj¼¼4 þ jPeL j/E /W 4 þ jj PeL ; 0jj þ jjPeL ; 0jjð11:43Þ37811 Discretization of the Convection TermFig. 11.7 Comparison of solutions obtained analytically and numerically using the upwind andcentral difference schemes for the one dimensional convection and diffusion problemThe /C profile generated using the upwind scheme (Eq.
11.43) is compared inFig. 11.7 with similar ones obtained analytically (Eq. 11.29) and numerically viathe central difference scheme (Eq. 11.28). At low PeL values, profiles indicatethat the upwind scheme is not as accurate as central differencing. This isexpected since the upwind profile is first order accurate while the linear profileis second order accurate. At high PeL values, the central difference scheme isunstable as its solution is unbounded and physically incorrect. On the other hand,even though the upwind scheme is not particularly accurate, it is physicallycorrect.Thus there appears to be a tradeoff between accuracy and stability.
Using theupwind scheme, solutions are better behaved and bounded even at high Pécletnumbers. This is however achieved at the cost of low accuracy. On the otherhand, the second order central difference scheme becomes unstable beyond acertain value of PeL resulting in physically erroneous solutions. Both schemesseem to be infected by errors, one affecting accuracy while the other affectingstability. What are these errors? This will be discussed after introducing thedownwind scheme.11.2Steady One Dimensional Convection and DiffusionCwWeuwwWWuwW379wEueeueCeEEEFig. 11.8 The downwind scheme profile11.2.5 The Downwind SchemeIt is interesting to see what happens if a scheme opposite to the upwind scheme isused, i.e., the downwind scheme [7, 8]. For this interpolation profile, displayedin Fig.
11.8, the value at the node on the downwind side of the interface is takento represent the value at the interface. Thus, the values at faces e and w arecalculated as/e ¼/E/Cif m_ e [ 0if m_ e \ 0and/w ¼/W if m_ w [ 0/C if m_ w \ 0ð11:44ÞUsing Eq. (11.44), the advection fluxes at the faces can be written asm_ e /e ¼ km_ e ; 0k/C þ km_ e ; 0k/E¼ FluxCeConv /C þ FluxFeConv /E þ FluxVeConvm_ w /w ¼ km_ w ; 0k/C þ km_ w ; 0k/Wð11:45Þ¼ FluxCwConv /C þ FluxFwConv /W þ FluxVwConvSubstitution of the above values in the discretization equation, Eq.
(11.15), yieldsFluxCeConv þ FluxCeDiff þ FluxCwConv þ FluxCwDiff /Cþ FluxFeConv þ FluxFeDiff /E þ FluxFwConv þ FluxFwDiff /W ¼ 0ð11:46Þwhich can be modified into the formaC /C þ aE /E þ aW /W ¼ 0ð11:47Þ38011 Discretization of the Convection TermwithaE ¼ FluxFeConv þ FluxFeDiffSe¼ km_ e ; 0k C/edxeaW ¼ FluxFwConv þ FluxFwDiffSw¼ km_ w ; 0k C/wdxwXConvaC ¼FluxCfþ FluxCfDifffð11:48ÞSeSw km_ w ; 0k þ C/wdxedxw¼ ðaE þ aW Þ þ ðm_ e þ m_ w Þ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}¼0XConvbC ¼ FluxVfþ FluxVfDiff¼ km_ e ; 0k þ C/ef¼0Invoking the continuity constraint in Eq. (11.48) and assuming uniform grid andconstant diffusion coefficient, the value for /C in terms of /E and /W is obtained as/C /W2 jjPeL ; 0jj2 jjPeL ; 0jj¼¼4 jPeL j/E /W 4 jj PeL ; 0jj jjPeL ; 0jjð11:49ÞWithout plotting Eq.
(11.49) it is clear that as jPeL j ! 4 the solution becomescompletely unbounded confirming once more the analysis presented above. Thedownwind scheme may be beneficial when blended with other schemes to predictsharp interfaces [7, 8]. Nevertheless its introduction here will be exploited in the nextsection to give better insight into stability.11.3Truncation Error: Numerical Diffusionand Anti-DiffusionTruncation error occurs due to the approximate nature of the discretization processand is more easily analyzed for one dimensional situations on Cartesian meshes.The diffusion and anti-diffusion of the upwind, downwind, and central differenceschemes are presented next.11.3Truncation Error: Numerical Diffusion and Anti-Diffusion38111.3.1 The Upwind SchemeConsidering the equation discretized via the upwind scheme, the intention is torecover the integral equation while accounting for the truncation error.
To do so, /Cand /W are written as functions of /e and /w , respectively, for the case when the flowis assumed to be in the positive x direction. In this case the upwind scheme results in/e ¼ /Cand/w ¼ /W :ð11:50ÞThus the one dimensional convection diffusion equation discretized using theupwind scheme simplifies toðquDyÞe /C ðquDyÞw /W d/d/Dy C/DyC/¼ 0:dxdxewð11:51ÞThe one dimensional Taylor series expansion of /C with respect to its value at cellface e is given by d/1 d2 //C ¼ /e þð xC xe Þ þð xC xe Þ 2 þ dx e2 dx2 e d/¼ /e ð xe xC Þ þ dx eð11:52Þand for a uniform grid as/C ¼ /e ðdxÞe 2d/ ðdxÞe 1 d 2 /þdx e 22 dx2 e2ð11:53ÞA similar expression can be obtained for /W and is given by d/ ðdxÞw 1 d 2 /ðdxÞw 2þ/W ¼ /w dx w 22 dx2 w2ð11:54ÞTruncating second order terms and higher and substituting into the advection term,the left hand side of the discretized equation becomes / d/DyðquDyÞe /C ðquDyÞw /W Cdx wed/ dxed/ dxw¼ ðquDyÞe /e ðquDyÞw /w dx e 2dx w 2 d/d/Dy C/Dy C/dxdxewd/DyCdx/ð11:55Þ38211 Discretization of the Convection Termwhich can be rearranged intoðquDyÞe /C ðquDyÞw /W d/d/Dy C/DyC/dxdxew¼ ðquDyÞe /e ðquDyÞw /w dxd/dxd/Dy C/ þ quDy C/ þ qu2 e dx2 w dxewð11:56ÞIt is clear now that the equation being solved has an added component of diffusion,which is called truncation error.
The value of the numerical diffusion is equal todx2C/truncation ¼ quð11:57ÞThis truncation error, also known as stream wise diffusion, reduces the accuracy ofthe solution by altering the magnitude of the diffusion coefficient and consequentlythe equation to be solved. Thus the convection and diffusion equation has aneffective modified value of the diffusion effects. On the other hand, this additionalstream wise numerical diffusion is desirable as it stabilizes the solution by keepingit bounded and physically correct.It is obvious that to reduce stream wise numerical diffusion a higher orderapproximation of the convection term is needed. However, as will be explained in alater section, this should be done in such a way that the solution remains bounded.11.3.2 The Downwind SchemeAs with the upwind scheme, /C and /W are written as functions of /e and /w ,respectively, for the case when the flow is assumed to be in the positive x direction.In this case the downwind scheme results in/e ¼ /Eand /w ¼ /Cð11:58Þand the discretized equation becomesðquDyÞe /E ðquDyÞw /C C/d/Dydx d/Dy C/¼0dxewð11:59Þ11.3Truncation Error: Numerical Diffusion and Anti-Diffusion383For a uniform grid, the one dimensional Taylor series expansions of /E and /Cwith respect to their values at cell faces e and w are given byðdxÞe 2d/ ðdxÞe 1 d 2 /þþ/E ¼ /e þdx e 22 dx2 e2 d/ ðdxÞw 1 d 2 /ðdxÞw 2þþ/C ¼ /w þdx w 22 dx2 w2ð11:60ÞTruncating second order terms and higher and substituting into the advection term,the left hand side of the discretized equation becomesðquDyÞe /E ðquDyÞw /C d/d/Dy C/DyC/dxdxew¼ ðquDyÞe /e ðquDyÞw /w dxd/dxd/Dy C/ quDy C/ qu2 e dx2 w dxewð11:61ÞFor this profile, numerical diffusion has a negative sign and is equal toC/truncation ¼ qudx2ð11:62Þwhich acts at decreasing the diffusion coefficient and in effect is an anti-diffusionerror.
Predictions using the downwind scheme are found to cause clipping of theadvected profiles. In fact solutions to the one dimensional convection-diffusionproblem generated using this scheme are more oscillatory than the CD scheme.11.3.3 The Central Difference (CD) SchemeThe truncation error for the CD scheme is a little more involved to obtain with thefinite volume method as computation of the gradient requires interpolated values atthe element faces and not at nodes where they are available. Assuming the velocityis known everywhere and the grid is uniform with a size of Dx, the approximation issimply the one introduced in the calculation of ð/e /w Þ, which can be written as11ð/e /w Þ ¼ ð/E þ /C Þ ð/C þ /W Þ ¼ ð/e /w Þ þ TE|fflfflfflfflfflffl{zfflfflfflfflfflffl} 2|fflfflfflfflfflffl{zfflfflfflfflfflffl}2InterpolatedExactð11:63Þ38411 Discretization of the Convection Termwhere TE refers to truncation error.
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