Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 68
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Any high order base scheme can be bounded using an ad-hoc setof curves.12.3High Resolution (HR) Schemes439~ 1 shouldTo construct a HR scheme, the monotonic profile in the range 0 /Cpass through the points (0, 0) and (1, 1), while remaining within the upper triangularshaded region on the NVD (Fig.
12.4). On the other hand, in the non-monotonic range,~ \0 and/or /~ [ 1; the profile should follow the upwind profile. A number ofi.e., /CCwell-known High-Resolution schemes built in this manner are illustrated in Fig. 12.6.For improved convergence behavior, any composite HR scheme should avoidhard angles at its profile connection points as well as at its horizontal and verticalprofiles. For example with the SMART scheme, which is constructed using theQUICK scheme, the convergence can be substantially improved by a minor~ ¼ 3/~ in themodification to the vertical portion of its composite profile using /fC~ 1=6: For the STOIC schemes the modification is applied in theregion 0 /C~ 1=5 region.
Also the horizontal portion of the composite profile for both0/CSMART and STOIC may be slightly modified to further improve convergence. For~ 1;example, one such modification is to impose a linear profile for 9=10 /f~ 1 altering the last portion of the profile to bewhich corresponds to 7=10 /C~ =3 þ 2=3: Other modifications are also possible (e.g., one may~ ¼/given by /fCdecide to modify the horizontal portion of the composite profile in the~ 1 region).
A similar modification can also be made for the bounded CD0:95 /fscheme to improve its convergence characteristics. The modified NVDs ofSMART, STOIC, and SUPERBEE schemes are shown in Fig. 12.7.Mathematically, the functional relationships of the composite HR schemesdisplayed in Figs. 12.6 and 12.7 are given by83~>~ 1>/C0/>C>2><2~ ¼ 111ð12:14ÞMINMOD /f~ þ~ 1//>CC>>222>>:~elsewhere/CBounded CD8>~ þ1<1/C~22/f ¼>:~/COSHERSMART~ ¼/f~ ¼/f83~>>>< 2 /C1>>>:~/C~ 10/Cð12:15Þelsewhere~ 20/C32 ~ /C 13elsewhereð12:16Þ83~3>>/ þ>>4 C 8<~ 0/C1>>>>:~/C5 ~ /C 16elsewhere56ð12:17Þ44012 High Resolution SchemesfffBOUNDED CDMINMODOSHER1113/ 43/ 43/ 41/ 20001/ 2f11/ 2C1CCffMUSCL1/ 21SMARTSTOIC1113/ 43/ 43/ 41/ 23/ 81/ 4001/ 2101/ 2Cf11/ 2C1CSUPERBEE13/ 41/ 201/ 2 2 / 31CFig.
12.6 NVD of several HR schemesff3/ 4fSTOICSMART19 / 1019 / 1013/ 43/ 42/33/ 5SUPERBEE1/ 2001/ 61/ 2 7 / 10101/ 5C1/ 2 7 / 1011/ 3 1/ 2 2 / 3CFig. 12.7 NVD of the modified SMART, STOIC, and SUPERBEE schemes1C12.3High Resolution (HR) SchemesModified SMART4418>~>3/>C>>>>>>>~ þ3<3/C~48/f ¼>>>1>~ þ2>/>C>>33>>:~/161 ~7 /C 610~ 0/CC81~1>>> /C þ>22>>>>>33< ~/C þ~STOIC /f ¼ 48>>>>>1>>>>:~/C8>~>3/>C>>>>>1~1>>/C þ>>>22><3~ ¼ 3~Modified STOIC /f/ þ>>4 C 8>>>>1~2>>>/C þ>>33>>>:~/121 ~5 /C 265 ~ /C 16elsewhere~ 0/C15~ 1/C2~ 7/C10MUSCLSUPERBEE15127~ 1/C10ð12:20Þelsewhere~ 10/C41 ~3 /C 443 ~ /C 14elsewhere81 1~>>> 2 þ 2 /C>>>>><3 ~~/f ¼ 2 /C>>>>1>>>>:~/Cð12:19Þ~ 0/CC8>~>2/>C>>>>>></~ þ1C~4/f ¼>>>>1>>>>>:~/Cð12:18Þ7~ 1/C10elsewhere~ 10/C21 ~2 /C 232 ~ /C 13elsewhereð12:21Þð12:22Þ44212 High Resolution SchemesModified SUPERBEE8>~>> 2/C>>>>>1>~ þ1>/>C>>22<~3/f ¼~/>>>2 C>>>>>>1>>>>:~/131~ /C 2~ 2/C3~ 0/C13122 ~ /C 13elsewhereCð12:23ÞMany other schemes were developed following this methodology such as CLAM[22], UTOPIA [23], SHARP [8], and ULTRA-SHARP [24, 25], to cite a few.Example 3Derive the NVF form of the SMART scheme.SolutionStarting with the QUICK scheme, we have3~3~/f ;QUICK ¼ /C þ48On the NVF diagram this looks like the black line.
In order to enforce theCBC, the interpolation profile of the QUICK scheme is modified to follow theblue lines as shown in the figure below32+ 3C3+ 8411CC33fCCKIQU3/ 43/ 801/ 21C12.3High Resolution (HR) Schemes443Thus the bounded QUICK scheme, now called the SMART scheme, iswritten as8>~>3/>C>>>>>3> /~ þ3<C48~/f ¼>2>1 ~>>/C þ>>33>>>:~/C12.4~ 10/C61 ~7 /C 6107~ 1/C10elsewhereThe TVD FrameworkAnother popular approach for developing HR convective schemes is the TotalVariation Diminishing (TVD) framework.
In solving numerically an advectionpartial differential equation for a variable / of the form presented so far, TotalVariation (TV) is defined asX/iþ1 /i TV ¼ð12:24Þiwhere i represents the index of a node in the spatial solution domain. A numericalmethod is said to be Total Variation Diminishing (TVD) if the TV in the solutiondoes not increase with time. Mathematically this is equivalent toð12:25ÞTV /tþDt TV ð/t ÞIn his seminal paper, Harten [26] proved that a monotone scheme is TVD, and aTVD scheme is monotonicity preserving.
A monotonicity preserving scheme doesnot create any new local extrema within the solution domain, i.e., the value of a localminimum is non-decreasing, and the value of a local maximum is non-increasing.It is not intended here to give full mathematical derivations of the TVDapproach. Rather, the intention is simply to explain the methodology for constructing TVD schemes. The approach used is based on the work of Sweby [27].Consider the unsteady one-dimensional convection equation (11.73), which wasused in the previous chapter to study the stability of convection schemes. In theabsence of diffusion and sources this equation reduces to@ ðq/Þ@ ðqu/Þ¼@t@x ffl}|fflfflfflfflfflffl{zfflfflfflfflfflð12:26ÞRHSThe general discretized form of the RHS term based on a five point stencil can bewritten as44412 High Resolution SchemesRHS ¼ að/C /U Þ þ bð/D /C Þð12:27Þwhere U, C, and D represent the far upstream, upstream, and downstream nodesshown in Fig.
12.1a. Sweby and Harten proved that a sufficient condition for anumerical scheme presented by Eq. (12.27) to be TVD or monotone is for thecoefficients per unit mass flow rate to satisfy the inequalitiesa 0; b 0;and0a þ b1ð12:28Þwhere the expressions for the coefficients a and b depend on the adopted convectionscheme. Referring back to the convection schemes presented above, it was foundthat the first order upwind scheme is very diffusive while the second order centraldifference scheme is highly dispersive. The need is for a scheme that lies somewhere between the upwind and the central difference schemes, i.e., a scheme thathas the stability of the upwind scheme and the accuracy of the central differencescheme.
Such a scheme can be constructed starting from the central differencescheme written as11/f ¼ ð/D þ /C Þ ¼ /C þ ð/D /C Þ|{z} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}2|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}2upwindCDð12:29Þantidiffusive fluxwhere the notation used earlier is adopted with C denoting the upwind node, D thedownwind node, and f the value at the cell face straddling the C and D nodes. Asimplied by Eq. (12.29), the central difference scheme can be written as the sum ofthe upwind scheme and a flux which is supposed to be anti-diffusive since the CDscheme is dispersive.
This flux is desirable as it makes the scheme second orderaccurate. The side effect is the unphysical oscillation it creates due to the decreasein numerical diffusion. Therefore a better approach would be one in which a portionof this anti-diffusive flux is added to the upwind scheme in such a way that thesecond order accuracy is preserved without creating any unphysical oscillations.One way to do that is to multiply this flux by a limiter function (also called limiteror flux limiter) that will prevent its excessive use in regions where oscillationsmight occur (e.g., across large gradients) while maximizing its contribution insmooth areas. Denoting such a limiter by wðr Þ; where r is usually taken as the ratioof two consecutive gradients, /f is calculated asmwWWmewWCweEeFig. 12.8 Convective fluxes in a one dimensional domainEE12.4The TVD Framework4451 / /U/f ¼ /C þ w rf ð/D /C Þ with rf ¼ C2/D /Cð12:30Þwhere U is the node upwind to C and D the node downwind to C.
In order topreserve the sign of the anti-diffusive flux, w rf is taken to be nonnegative.Therefore developing a TVD scheme reduces to finding limiters that will make thenumerical scheme TVD or monotone. The conditions that these limiters have tosatisfy in order for the convection scheme to be monotonicity preserving arederived next by invoking the flux limiter in the discretization of the RHS ofEq. (12.27) via the interface values given by Eq. (12.30).Considering the one dimensional domain shown in Fig. 12.8, the convectivefluxes at the element faces are given by1 m_ e /e ¼ /C þ w reþ ð/E /C Þ km_ e ; 0k21 /E þ w re ð/C /E Þ km_ e ; 0k21 þm_ w /w ¼ /C þ w rw ð/W /C Þ km_ w ; 0k2ð12:31Þ1 /W þ w rw ð/C /W Þ km_ w ; 0k2/// /EEW; re ¼ E;reþ ¼ C/E /C/C /E/ /E /W /WW;r ¼rwþ ¼ C/W /C w/C /WTo simplify the derivations to follow, a positive velocity is assumed.
Under theseconditions, the discretized form of the RHS of Eq. (12.24) is obtained as1 þ1 RHS ¼ m_ e /C þ w re ð/E /C Þ m_ w /W þ w rw ð/C /W Þ22ð12:32Þwhile the continuity equation is given bym_ e þ m_ w ¼ 0 ) m_ w ¼ m_ eð12:33ÞInvoking the continuity constraint, the RHS equation can be rearranged into3261 ð/E /C Þ 1 776RHS ¼ m_ e 61 þ w reþ w rw 7ð/C /W Þ542ð/ C / W Þ 2|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}1=reþ 1 w reþ1 ¼ m_ e 1 þ w rw ð/C /W Þ2 reþ2ð12:34Þ44612 High Resolution SchemesComparing Eq. (12.34) with Eq. (12.27), the values of a and b are found to be 1 w reþ1 ð12:35Þ w rw b ¼ 0a¼1þþ2 re2For the scheme to be TVD, the following should hold [Eq.
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