Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 69
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(12.28)]: 1 w reþ1 01 þ w rw 12 reþ2ð12:36Þwhich can be expanded to1 w ðr Þ 11 wðr Þ 1wðr Þ wðr Þ 0 ) wðr Þ 1 ) wðr Þ 22 r22 r2rð12:37Þ1 w ðr Þ 11 wðr Þ 1w ðr Þ1þ wðr Þ 1 ) wðr Þ 0 ) wðr Þ 02 r22 r2r1þor simply0 wðr Þ w ðr Þ2rð12:38ÞIf in addition to having wðr Þ 0; a condition is imposed whereby wðr Þ ¼ 0 fornegative values of r, then the above conditions will be satisfied ifw ðr Þ 2andwðr Þ 2rð12:39ÞCombining all conditions that the limiter has to satisfy to produce a TVDscheme, a criterion similar to the CBC can be developed and is given byminð2r; 2Þ r [ 0w ðr Þ ¼ð12:40Þ0r0(r )22rTVD Monotonicity Region0rFig.
12.9 TVD monotonicity region on a r w diagram12.4The TVD Framework447(r )r2rDOWNWIND2USO1CD0UPWIND1rFig. 12.10 Limiters of SOU and CD schemes on a r w diagramAs depicted in Fig. 12.9, these conditions can be drawn on a r w diagram,which is also denoted by Sweby’s diagram, to show the TVD monotonicity region(blue region in the plot). Using this diagram, it is simple to grasp the formulation ofTVD schemes. Any flux limiter wðr Þ formulated to lie within the TVD monotonicity region yields a TVD scheme. Sweby’s diagram is very similar to the NVDpresented above.The limiters for all schemes presented so far can be derived and their functionalrelationships drawn on Sweby’s diagram. In specific the limiter of the CD is easily obtained from Eq.
(12.29) as wCD rf ¼ 1 while that of the SOU scheme can becomputed as follows: 131/f ¼ /C þ wSOU rf ð/D /C Þ ¼ /C /U222//U) wSOU rf ¼ C¼ rf/D /Cð12:41ÞThe limiters for both schemes are displayed in Fig. 12.10.Sweby [27] also noted that because w rf ¼ 0 for rf \0; second order accuracyis lost at extrema of the solution.
The SOU and CD schemes are second orderschemes and by inspecting Fig. 12.10 it is clearly seen that both of them passthrough the point (1, 1). In addition, as demonstrated in the work of Van Leer [28],any second order scheme can be written as a weighted average of the CD and SOUschemes. Thus for a scheme to be second order its limiter has to pass through thepoint (1, 1) and, as shown in Fig. 12.10, its limiter should lie in the region borderedby the CD and SOU limiters (blue region in the plot). The corresponding region onan NVD is shown in Fig. 12.11.Fig. 12.11 Region on anNVD equivalent to the TVDmonotonicity region on aSweby’s diagram for secondorder schemes12 High Resolution SchemesSOU448f13/4CD1/201/ 21CAdopting this approach and following the procedure used with the SOU scheme,the functional relationships of the limiters for many of the HO schemes presentedabove can be easily computed and are given by 8Upwindwrf ¼ 0>>>>Downwind w rf ¼ 2>>> 1 þ rf>>< FROMMw rf ¼2 ð12:42ÞSOUwrf ¼ rf>>>>>CDw rf ¼ 1>>> 3 þ rf>: QUICKw rf ¼4The FROMM scheme is the average of the CD and SOU scheme.
Its functionalrelationship is mathematically written as91 /C þ /D 31/ /U >=/f ¼þ /C /U ¼ /C þ D 1 þ rf22224) w rf ¼>12;/f ¼ /C þ w rf ð/D /C Þ2ð12:43ÞThe functional relationships of these limiters are displayed in Fig. 12.12. Withthe exception of the upwind scheme limiter all others are seen not to be totally lyingwithin the monotonicity region. As such these schemes are unbounded.By limiting the w rf functions of the various schemes given above to lie withinthe monotonicity region displayed in Fig. 12.9, these HO schemes are transformedinto HR TVD schemes.
Many TVD schemes have been developed in that mannerand the limiters for a number of them are shown in Fig. 12.13a–d with the functional relationships of their limiters given by12.4The TVD Framework449( rf )2rfUSO2MOMKQUICFRDOWNWIND1CD3/41/20UPWINDrf1Fig. 12.12 High Order schemes and TVD monotonicity region on Sweby’s diagram(b)(a)( rf )( rf )MINMOD2rfOSHER2rfrf2211rf00rf1rf1(d)(c)( rf )( rf )2rfMUSCL2rfrfSUPERBEErf22Van Leer11001rf1rfFig. 12.13 Limiters of the a MINMOD, b OSHER, c MUSCL, and d SUPERBEE TVD schemeson a Sweby diagram45012 High Resolution Schemes8SUPERBEE>>>>>>>MINMOD>><OSHER>>>>Van Leer>>>>>:MUSCL12.5 w rf ¼ max 0; min 1; 2rf ; min 2; rf w rf ¼ max 0; min 1; rf w rf ¼ max 0; min 2; rf rf þ rf w rf ¼1 þ rf w rf ¼ max 0; min 2rf ; rf þ 1 =2; 2ð12:44ÞThe NVF-TVD RelationBoth NVF and TVD formulations enforce Boundedness following differentapproaches, which can be demonstrated to be somewhat related.
This is done by~ ; then comparing the NVF-CBCfirst deriving a relation between rf and /C(Eq. 12.13) with the TVD-CBC (Eq. 12.40), and finally presenting the generaltransformation that allows the functional relationship of any TVD scheme to bewritten in the NVF framework and vice versa.~ can be easily derived starting with the definitionThe relation between rf and /Cof rf and is obtained as/C /Uð/C /U Þ=ð/D /U Þ¼/D /C ð/D /U þ /U /C Þ=ð/D /U Þ~/C~ ¼ rf¼)/C~1 þ rf1/rf ¼ð12:45ÞCUsing Eq.
(12.45) a number of linear schemes can be compared in the twoframeworks. The limiter w rf ¼ 0; which represents the Upwind scheme in theTVD formulation is also equivalent to the upwind scheme in the NVF formulation ~ ¼/~ ). This follows from the fact that w rf ¼ 0 ) / ¼ / ) /~ ¼/~ :(i.e., /fCfUfCThe upwind scheme is imposed as a limit for the TVD-CBC when rf 0; theequivalent condition in the NVF-CBC is obtained as(~ 0~//CCrf 0 )0 )ð12:46Þ~~1// [1CCThese also represent the conditions for imposing the Upwind scheme in theNVF-CBC.Moreover, on the NVF-CBC, the functional relationship has to increase~ 1: On Sweby’s diagram the region extendsmonotonically in the region 0 /Cover the interval 0 rf \ þ1: Both regions represent the same interval as demonstrated by the following relation:12.5The NVF-TVD Relation451~ ! 1 ) rf ¼/C~/C! þ1~1/ð12:47ÞCFurther, for the TVD-CBC condition w rf 2ð12:48Þthe equivalent condition in the NVF-CBC can be obtained as follows:9 =w rf ¼ 2~ ¼1) /f ¼ /C þ ð/D /C Þ ¼ /D ) /1 f/f ¼ /C þ w rf ð/D /C Þ ;2ð12:49ÞThus, ~ 1w rf 2 ) /fð12:50Þwhich is the condition that should be satisfied by the NVF-CBC.
The last conditionimposed by the TVD-CBC on w rf is given by w rf 2rfð12:51ÞThe equivalent condition using the NVF-CBC is obtained as9 =w rf ¼ 2rf/ /Uð/ /C Þ ¼ 2/C /U) /f ¼ /C þ C1 /D /C D/f ¼ /C þ w rf ð/D /C Þ ;2ð12:52Þwhich can be normalized to yield~ ¼ 2/~/f ¼ 2/C /U ) /f /U ¼ 2/C 2/U ) /fCð12:53ÞThis is more restrictive than the NVF-CBC and is the only difference betweenthe two formulations. Based on this condition, the TVD-CBC and the modifiedNVF-CBC would look as shown in Fig. 12.14a with the monotonicity regionreduced to the upwind line and the blue area. While the modified TVD-CBC and~ ¼ 0 on the NVF-CBC corresponds to rf ¼ 0the NVF-CBC (i.e., the condition /Con the TVD-CBC) would look as shown in Fig. 12.14b.
Regarding second orderaccuracy, it was stated that for a TVD scheme to be second order accurate it has topass through the point (1, 1), i.e., wð1Þ ¼ 1: The equivalent values using the NVFare found as45212 High Resolution Schemes(a)f1DOWNWINDf=2Cr2rSOU(r )23/ 4DOWNWINDDINPWUSOUCD1CD00UPWINDr1(b)1/ 21Cf1DOWNWINDSOU(r )SOUr23/ 4DOWNWINDDINPWUCD1CD00UPWINDr1(c)1/ 21C( rf3/ 4DOWNWIND2U)=2DOWNWINDSO1r2rrCf(r )DINPWUSOUCD1CD00UPWINDr11/ 21CFig. 12.14 a TVD-CBC on Sweby and Normalized Variable Diagrams.
b NVF-CBC on Swebyand Normalized Variable Diagrams. c TVD-CBC on Sweby and Normalized Variable Diagramsfor second order schemesrf ¼ 1 )~/C~ ¼1/~ )/~ ¼ 0:5¼1)/CCC~1/C111/f ¼ /C þ wð1Þð/D /C Þ ¼ /C þ ð/D /C Þ ¼ ð/D þ /C Þ2221~ ¼ 0:75~ ¼1 1þ/) /f /U ¼ ð/D /U þ /C /U Þ ) /fC22ð12:54Þ12.5The NVF-TVD Relation453which is exactly the point Q(0.5, 0.75) found in the NFV.
As stated earlier, VanLeer demonstrated that any second order scheme can be written as a weightedaverage of the CD and SOU schemes. Therefore its functional relationship shouldlie between the functional relationships of the CD and SOU schemes with theirTVD-CBC monotonicity regions reduced to the upwind line and the blue areashown on a Sweby diagram and an NVD in Fig.
12.14c.The above procedure can be generalized to transform any TVD scheme into anequivalent NVF scheme and vice versa. Starting with a scheme in the NVFframework, the value at the face /f is expressed as ~ ð/ / Þ þ / with /~ ¼ /C /U/f ¼ f /CDUUC/D /Uð12:55Þwhereas for a TVD scheme /f is given by1 / /U/f ¼ /C þ w rf ð/D /C Þ with rf ¼ C2/D /Cð12:56ÞEquating the above two /f equations, yields 1 ~ ð/ / Þ þ //f ¼ /C þ w rf ð/D /C Þ ¼ f /CDUU2ð12:57ÞThus ~ ð/ / Þ f / ð/D /C ÞCDUð/ /U Þ~~ /¼22 C¼2 f /w rfCCð/D /U Þð/D /U Þð/D /U Þð12:58ÞThe term on the left hand side of the above equation can be modified to ð/D /C Þ ð/D /U /C þ /U Þ ~w rf¼ w rf¼ w rf 1 /Cð/D /U Þð/D /U Þð12:59Þleading to ~ ð/ / Þ ð/ / Þ~ /~f /f / CDUCUCC~ ¼2¼2¼wrw rf 1 /fC~ð/D /U Þ1/Cð12:60ÞEquation (12.60) may also be written as wr þ 2rff~f /C ¼ 2 1 þ rfð12:61Þ45412 High Resolution SchemesAs an example, the functional relationship of the UPWIND Scheme in the NVF~ ; its TVD limiter is found as~ ¼/framework is /fC ~ /~f /~ /~ CC/C~ ¼/~ ) w rf ¼ 2/¼2 C¼0ð12:62ÞfC~~1/1/CC The TVD limiter for the DOWNWIND Scheme is w rf ¼ 2; its NVF functionalrelationship can be obtained as wr þ 2r2 þ 2rff~ ¼f /~ ¼ f ¼ ¼1/ð12:63ÞfC2 1 þ rf2 1 þ rfKnowing the NVF form of the SOU scheme, its TVD limiter is computed as3~~/C /C~0:5/~ C/32C~ ) w rf ¼ 2 ~ ¼ / ¼ 2¼ ¼ rf ð12:64Þ/f2 C~~~1/1/1/CCCThe same is applicable to other schemes.Example 4Starting with the TVD-Van Leer formulation, derive the NVF-Van Leerscheme.SolutionThe TVD-Van Leer limiter is given by rf þ rf :w rf ¼1 þ rf Noting thatrf ¼~/C~1/Cits TVD functional relationship is transformed to /~/ ~C Cþ~~ 1 / 1/CCw rf ¼ / ~C 1þ~ 1 /C~ 1 / C ~~ /C þ /C~1 /C ¼ ~ ~ þ /1 /CC12.5The NVF-TVD Relation455Combining the above equation with the TVD relationship in normalizedform, which is given by, ~ ¼/~~ þ 1 w rf 1 //fCC2yields ~ ¼/~~ þ 1 w rf 1 //fCC2~ 1 / C ~~ /C þ /C ~11/C~ þ~¼/1/CC~ þ /~ 2 1 /CCThe following three cases are identified:~ \1a.
Case 1: 0\/C ~ ¼1/~ ¼/~ and 1 /~ ; thusIn this case /CCCC9 ~= 2w r f ¼ 2/C /~ ¼/~ þ/~ 1/~ ¼ 2/~~ / 1CCCCC~ ¼/~ þ w rf 1 /~ ; f/fCC2~ [1b. Case 2: /C ~ ¼/~ ¼/~ and 1 /~ 1; thusIn this case /CCCC9= w rf ¼ 0 ~~ ~ ;/f ¼ /C~ þ 1 w rf 1 /~ ¼//fCC2~ \0c. Case 3: /C ~ ¼1/~ ¼ /~ and 1 /~ ; thusIn this case /CCCC9= w rf ¼ 0 ~~ ~ ;/f ¼ /C~ þ 1 w rf 1 /~ ¼//fCC2Combining the results of the three cases into one NVF formulation, thefunctional relationship of the Van Leer Scheme becomes8 2< ~~~ \12/C /0\/CC~/f ;VanLeer ¼:/~otherwiseC45612.612 High Resolution SchemesHR Schemes in Unstructured Grid SystemsAs mentioned in Chap.
11, another alternative that can be followed to overcome thehurdle of not having a clear upwind location U in unstructured grids, which is~ or rf ; is to create a virtual one. As depicted inneeded in the calculation of /CFig. 12.15, the easiest way is to assume U to lie on the line joining the nodes C andD such that C is the midpoint of the segment joining the points U and D. With thisassumption and based on the analysis done earlier, the following can be written:/D /U ¼ r/C dUD ¼ 2r/C dCDð12:65Þfrom which the value of /U is computed as/U ¼ /D 2r/C dCDð12:66Þwhere dCD is the vector between the nodes C and D, and dUD is the vector betweennodes D and the virtual node U.
As mentioned above U is constructed such that C istaken to be the centre of the UD segment. With the value of /U computed, the useof either the NVF or the TVD approach proceeds as described above.Fig. 12.15 Virtual upwindnode in unstructured grids?UCDfDCf12.7?UdCDdCDDeferred Correction for HR SchemesThe numerical implementation of HR schemes is best understood through anexample.
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