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Approximation Schemes for Convective Term - Structured Grids - Common -- CFD-Wiki, The Free CFD Reference
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4/18/2015 ApproximationSchemesforconvectivetermstructuredgridsCommonCFDWiki,thefreeCFDreference
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ApproximationSchemesforconvectivetermstructuredgridsCommonFromCFDWiki
Contents1DiscretisationSchemesforconvectivetermsinGeneralTransportEquation.FiniteVolumeFormulation,structuredgrids2Introduction3BasicEquationsofCFD4ConvectionSchemes5BasicDiscretisationschemes
5.1CentralDifferencingScheme(CDS)5.2UpwindDifferencingScheme(UDS)also(FirstOrderUpwindFOU)5.3HybridDifferencingScheme(HDSalsoHYBRID)5.4PowerLawScheme(alsoExponentialschemeorPLDS)
6HighResolutionSchemes(HRS)6.1ClassificationofHighResolutionSchemes
6.1.1Linearschemes6.1.2Kappaformulation,KappaSchemesandOtherschemes6.1.3NonLinearschemes
6.2NumericalImplementationofHRS(Defferedcorrectionprocedure)7Diagonaldominancecriterion8NormalisedVariablesFormulation(NVF)9NormalisedVariablesDiagram(NVD)10NormalisedVariableandSpaceFormulation(NVSF)11ConvectionBoundednessCriterion(CBC)12S.K.Godunovtheorem13MonotonicityCriterion14TotalVariationDiminishing(TVD)SimplifiedDescription
14.1Generalissues15TotalVariationDiminishingDiagram(Swebydiagram)
15.1Fluxlimitingformulation16DiscretizationschemesQualityCriterions
DiscretisationSchemesforconvectivetermsinGeneralTransportEquation.
4/18/2015 ApproximationSchemesforconvectivetermstructuredgridsCommonCFDWiki,thefreeCFDreference
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FiniteVolumeFormulation,structuredgrids
Introduction
Thissectiondescribesthediscretizationschemesofconvectivetermsinfinitevolumeequations.Theaccuracy,numericalstability,andboundnessofthesolutiondependonthenumericalschemeusedfortheseterms.Thecentralissueisthespecificationofanappropriaterelationshipbetweentheconvectedvariable,storedatthecellcenter,anditsvalueateachofthecellfaces.
BasicEquationsofCFD
Alltheconservationequationscanbewritteninthesamegenericdifferentialform:
(1)
Equation(1)isintegratedoveracontrolvolumeandthefollowingdiscretizedequationfor isproduced:
(2)
where isthesourcetermforthecontrolvolume ,and and represent,respectively,theconvectiveanddiffusivefluxesof acrossthecontrolvolumeface
Theconvectivefluxesthroughthecellfacesarecalculatedas:
(1)
where isthemassflowrateacrossthecellface .Theconvectedvariable associatedwiththismassflowrateisusuallystoredatthecellcenters,andthussomeformofinterpolationassumptionmustbemadeinordertodetermineitsvalueateachcellface.Theinterpolationprocedureemployedforthisoperationisthesubjectofthevariousschemesproposedintheliterature,andtheaccuracy,stability,andboundednessofthesolutiondependontheprocedureused.
Ingeneral,thevalueof canbeexplicityformulatedintermsofitsneighbouringnodalvaluesbyafunctionalrelationshipoftheform:
4/18/2015 ApproximationSchemesforconvectivetermstructuredgridsCommonCFDWiki,thefreeCFDreference
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(1)
where denotestheneighbouringnode values.Combiningequations(\ref{eq3})through(\ref{eq4a}),thediscretizedequationbecomes:
(1)
ConvectionSchemes
Alltheconvectionschemesinvolveastencilofcellsinwhichthevaluesof willbeusedtoconstructthefacevalue
Whereflowisfromlefttoright,and isthefaceinquestion.
meanUpstreamnode
meanCentralnode
meanDownstreamnode
Inthefirstplot,itisnotsonatraltothinkthecentralnode"C"notasthepresentnode"P".Itmaybethoughtasthefirstnodetotheupstreamdirectionofthesurfaceinquestion"f".
BasicDiscretisationschemes
CentralDifferencingScheme(CDS)
Italsocanbeconsideredaslinearinterpolation.
Themostnaturalassumptionforthecellfacevalueoftheconvectedvariable wouldappeartobetheCDS,whichcalculatesthecellfacevaluefrom:
(1)
orformorecommoncase:
4/18/2015 ApproximationSchemesforconvectivetermstructuredgridsCommonCFDWiki,thefreeCFDreference
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(1)
wherethelinearinterpolationfactorisdefiniedas:
(1)
normalizedvariablesuniformgrids
(1)
normalizedvariablesnonuniformgrids
(1)
Thisschemeis2ndorderaccurate,butisunboundedsothatnonphysicaloscillationsappearinregionsofstrongconvection,andalsointhepresenceofdiscontinuitiessuchasshocks.TheCDSmaybeuseddirectlyinverylowReynoldsnumberflowswherediffusiveeffectsdominateoverconvection.
UpwindDifferencingScheme(UDS)also(FirstOrderUpwindFOU)
TheUDSassumesthattheconvectedvariableatthecellface isthesameastheupwindcellcentrevalue:
4/18/2015 ApproximationSchemesforconvectivetermstructuredgridsCommonCFDWiki,thefreeCFDreference
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(1)
normalisedvariables
(1)
TheUDSisunconditionallyboundedandhighlystable,butasnotedearlieritisonly1storderaccurateintermsoftruncationerrorandmayproduceseverenumericaldiffusion.Theschemeisthereforehighlydiffusivewhentheflowdirectionisskewedrelativetothegridlines.
(1)
(1)
UDSmaybewrittenas
(1)
orinmoregeneralform
(1)
where
(1)
(1)
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HybridDifferencingScheme(HDSalsoHYBRID)
TheHDSofSpalding[1972]switchesthediscretizationoftheconvectiontermsbetweenCDSandUDSaccordingtothelocalcellPecletnumberasfollows:
(1)
(1)
ThecellPecletnumberisdefinedas:
(1)
inwhich and arerespectively,thecellfaceareaandphysicaldiffusioncoefficient.When ,CDScalculationstendstobecomeunstablesothattheHDSrevertstotheUDS.Physicaldiffusionisignoredwhen .
TheHDSschemeismarginallymoreaccuratethantheUDS,becausethe2ndorderCDSwillbeusedinregionsoflowPecletnumber.
D.B.Spalding(1972),"Anovelfinitedifferenceformulationfordifferentexpressionsinvolvingbothfirstandsecondderivatives",Int.J.Numer.Meth.Engng.,4:551559,1972.
PowerLawScheme(alsoExponentialschemeorPLDS)
Patankar,S.V.(1980),NumericalHeatTransferandFluidFlow,ISBN0070487405,McGrawHill,NewYork.
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HighResolutionSchemes(HRS)
ClassificationofHighResolutionSchemes
HRScanbeclassifiedaslinearornonlinear,wherelinearmeanstheircoefficientsarenotdirectfunctionsoftheconvectedvariablewhenappliedtoalinearconvectionequation.Itisimportanttorecognisethatlinearconvectionschemesof2ndorderaccuracyorhighermaysufferfromunboundedness,andarenotunconditionallystable.
Nonlinearschemesanalysethesolutionwithinthestencilandadaptthediscretisationtoavoidanyunwantedbehavior,suchasunboundedness(seeWaterson[1994]).ThesetwotypesofschemesmaybepresentedinaunifiedwaybyuseoftheFluxLimiterformulation(WatersonandDeconinck[1995]),whichcalculatesthecellfacevalueoftheconvectedvariablefrom:
(1)
where istermedalimiterfunctionandthegradientration isdefinedas:
(1)
ThegeneralisationofthisapproachtohandlenonuniformmesheshasbeengivenbyWaterson[1994]
Fromtheequation(\ref{eq9})itcanbeseenthat givestheUDSand givestheCDS.
Pleasenotethatlineardoesnotmeanfirstorder
Linearschemes
Linearschemesarethoseforwhich islinearfunctionof
isupwinddifferencing(firstorderaccurate)
iscentraldifferencing(secondorderaccurate)
Kappaformulation,KappaSchemesandOtherschemes
kappaformulation
B.vanLeer(1985),"UpwinddifferencemethodsforaerodynamicsproblemsgovernedbytheEulerequations",LecturesinAppl.Math.,22:327336.
Higherorderschemesareusuallymembersofthe class,forwhich
(1)
Usingthisequationfacevariablescanbeexpressed:
inusualvariables
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(1)
innormalisedvariables
(1)
Themainschemesare
CDS(centraldifferencingscheme)QUICK(quadraticupwindscheme)LUS(linearupwindscheme)FrommCUS(cubicupwindscheme)
NonLinearschemes
Nonlinearschemesarethoseforwhich isnotalinearfunctionof .Theyfallintothreecategories,dependingonthelinearschemesonwhichtheyarebased.
QUICKbased:
SMART(piecewiselinear,bounded)
(1)
HQUICK(smooth)
(1)
UMIST(piecewiselinear,bounded)
(1)
CHARM(smooth,bounded)
(1)
(1)
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Frommbased:
MUSCL(piecewiselinear)
(1)
vanLeer(smooth)
(1)
OSPRE(smooth)
(1)
vanAlbada(smooth)
(1)
other:
Superbee(piecewiselinear)
(1)
MinMod(piecewiselinear)
(1)
Waterson,N.PandDeconinck,H(1995),"Aunifiedapproachtothedesignandapplicationofboundedhighordercovectionschemes",9thInt.Conf.onNumericalMethodsinLaminarandTurbulentFlow,Atlanta,USA,July1995,TaylorandDurbetakieds.,PineridgePress.
Waterson,N.P.(1994),"Developmentofboundedhighorderconvectionschemeforgeneralindustrialapplications",ProjectReport199433,vonKarmanInstituteforFluidDynamics,SintGenesiusRode,Belgium.
NumericalImplementationofHRS(Defferedcorrectionprocedure)
TheHRSschemescanbeintroducedintoequation(\ref{eq4b})byusingthedeferredcorrectionprocedureofRubinandKhosla[1982].Thisprocedureexpressesthecellfacevalue by:
(1)
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where isahigherordercorrectionwhichrepresentsthedifferencebetweentheUDSfacevalue andthehigherorderschemevalue ,i.e.
(1)
Ifequation(\ref{eq10a})issubstitutedintoequation(\ref{eq4b}),theresultingdiscretisedequationis:
(1)
where isthedeferredcorrectionsourceterms,givenby:
(1)
ThistreatmentleadstoadiagonallydominantcoefficientmatrixsinceitisformedusingtheUDS.
Thefinalformofthediscretizedequation:
(1)
Subscrit representsthecurrentcomputationalcell representthesixneighbouringcellsandrepresentstheprevioustimestep(transistentcasesonly)
Thecoefficientscontaintheappropriatecontributionsfromthetransient,convectiveanddiffusivetermsin(\ref{eq1})
P.K.KhoslaandS.G.Rubin(1974),"Adiagonallydominantsecondorderaccurateimplicitscheme",Comput.Fluids,2207209.
S.G.RubinandP.K.Khoshla(1982),"Polynomialinterpolationmethodforviscousflowcalculations",J.Comp.Phys.,Vol.27,pp.153.
Diagonaldominancecriterion
NormalisedVariablesFormulation(NVF)
B.P.Leonard(1988),"Simplehighaccuracyresolutionprogramforconvectivemodellingofdiscontinuities",InternationalJ.NumericalMethodsFluids,8:12911318.
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NormalisedVariablesDiagram(NVD)
AccordingtoLeonard[1988],forany(ingeneralnonlinear)characteristicsinthenormalizedvariablediagram(seefigurebelow):
Passingthrough isnecessaryandsufficientforsecondorderaccuracyPassingthrough withaslopeof0.75(forauniformgrid)isnecessaryandsufficientforthirdorderaccuracy
Thehorizontalandverticalcoordinatesofpoint inthenormalizedvariablediagram,andtheslopeofthecharacteristicsatthepoint forpreservingthethirdorderaccuracyforanonuniformgrid,canbeobtainedbysimplealgebrausingeqs.[.....]
(1)
(1)
(1)
where
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(1)
(1)
(1)
Forauniformqrid, and
Normalisedvariablediagramforvariouswellknownschemes
NormalisedVariableandSpaceFormulation(NVSF)
DarwishM.S.andMoukalledF.(1994),"NormalizedVariableandSpaceFormulationMethodologyforHighResolutionSchemes",Num.HeatTrans.,partB,vol.26,pp.7996.
AlvesM.A.,CruzP.MendesA.MagahaesF.D.PinhoF.T.,OliveiraP.J.(2002),"AdaptivemultiresolutionapproachforsolutionofhyperbolicPDEs",ComputationalMethodsinAppliedMechanicsandEngineering,191,39093928.
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ConvectionBoundednessCriterion(CBC)
ChoiS.K.,NamH.Y.andChoM.(1995),"Acomparisonofhighorderboundedconvectionschemes",ComputationalMethodsinAppliedMechanicsandengineering,Vol.121,pp.281301.
GaskellP.H.andLauA.K.C.(1988),"Curvativecompensatedconvectivetransport:SMART,anewboundednesspreservingtrasportalgorithm",InternationalJournalforNumericalMethodsinFluids,Vol.8,No.6,pp.617641.
GaskelandLauhaveformulatedtheCBCasfollows.Anumericalapproximationto isboundedif:
for , isboundedbelowbythefunction andabovebyunityandpassesthroughthepoints(0,0)and(1,1)
for or , isequalto
TheCBCisclearlyillustratedinfigurebelow,wheretheline andtheshadedareaaretheregionoverwhichtheCBCisvalid.TheimportanceoftheCBCistoprovideasufficientandnecessaryconditionforguaranteeingtheboundedsolutionifatmostthreeneighbouringnodalvaluesareusedtoapproximatefacevalues.Itiswellknownthatthepositivityoffinitedifferencecoefficientsisalsoasufficientconditionforboundedness,butthisisoverlystringent,fortheexistenseofnegativecoefficientsdoesnotneccesarilyleadtooverorundershoots.
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S.K.Godunovtheorem
MonotonicityCriterion
TotalVariationDiminishing(TVD)SimplifiedDescription
Generalissues
A.Harten(1984),"Onaclassofhighresolutiontotalvariationstablefinitedifferenceschemes",SIAMJ.Num.Analysis,21,p1.
A.Harten(1983),"Highresolutionschemesforhyperbolicconservationlaws",J.Comput.Phys.,49:357393,1983.
P.K.Sweby(1984),"Highresolutionschemesusingfluxlimitersforhyperbolicconservationlaws",SIAMJ.Num.Analysis,21,p995.
TVDcriterion
nonewlocalextremamustbecreatedthevalueofanexistinglocalminimummustbenondecreasingandthatofthelocalmaximummustbenonincreasing
TotalVariation(TV)ofafunction isdefinedby
(1)
TotalVariation(TV)ofanumericalsolutionisdefinedby
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(1)
where gridpointindex
forasetofdiscretedata
theTVisdefinedby
(1)
(1)
Formonotonicitytobesatisfied,thisTVmustnotbeincreased!
FinallyanumericalschemeissaidtobeTVDif:
(1)
where timesteporiterationindex
Usingnormalisedvaribles,TVDconditioncabbewritten:
(1)
(1)
Toobtaindifferencingscheme,satisfyingTVDcondition,fluxlimiter isincluded,whichdependsuponfunction'sgradients.
Inordertoprovidemonotonicityofthesolution,itisnecessarytoimplementcondition[K.Fletcher]
(1)
where
(1)
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TotalVariationDiminishingDiagram(Swebydiagram)
Fluxlimitingformulation
DiscretizationschemesQualityCriterions
ReturntoNumericalMethods
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