HighRe solutions for incompressible flow using the

JOURNALOFCOMPUTATIONALPHYSICS48,387-411(1982)High-ReSolutionsforIncompressibleFlowUsingtheNavier-StokesEquationsandaMultigridMethod*U.GHIA,K.N.GHIA,ANDC.T.SHINUniversityofCincinnati,Cincinnati,Ohio45221ReceivedJanuary15,1982Thevorticity-streamfunctionformulationofthetwo-dimensionalincompressibleNavier-Stokesequationsisusedtostudytheeffectivenessofthecoupledstronglyimplicitmultigrid(CSI-MG)methodinthedeterminationofhigh-Refine-meshflowsolutions.Thedrivenflowinasquarecavityisusedasthemodelproblem.SolutionsareobtainedforconfigurationswithReynoldsnumberashighas10.000andmeshesconsistingofasmanyas257x257points.ForRe=1000,the(129x129)gridsolutionrequired1.5minutesofCPUtimeontheAMDAHL470V/6computer.Becauseoftheappearanceofoneormoresecondaryvorticesintheflowfield,uniformmeshrefinementwaspreferredtotheuseofone-dimensionalgrid-clusteringcoordinatetransformations.1.INTRODUCTIONThepastdecadehaswitnessedagreatdealofprogressintheareaofcomputationalfluiddynamics.Developmentsincomputertechnologyhardwareaswellasinadvancednumericalalgorithmshaveenabledattemptstobemadetowardsanalysisandnumericalsolutionofhighlycomplexflowproblems.Forsomeoftheseapplications,theuseofsimpleiterativetechniquestosolvetheNavier-Stokesequationsleadstoaratherslowconvergencerateforthesolutions.Thesolutionconvergenceratecanbeseriouslyaffectedifthecouplingamongthevariousgoverningdifferentialequationsisnotproperlyhonoredeitherintheinteriorofthesolutiondomainoratitsboundaries.TherateofconvergenceisalsogenerallystronglydependentonsuchproblemparametersastheReynoldsnumber,themeshsize,andthetotalnumberofcomputationalpoints.Thishasledseveralresearcherstoexaminecarefullytherecentlyemergingmultigrid(MG)techniqueasausefulmeansforenhancingtheconvergencerateofiterativenumericalmethodsforsolvingdiscretizedequationsatanumberofcomputationalgridpointssolargeastobeconsideredimpracticalpreviously.*ThisresearchwassupportedinpartbyAFOSRGrant80-0160,withDr.JamesD.WilsonasTechnicalMonitor.3870021.9991/82/120387-25$02,00/OCopyrightC1982byAcademicPress,Inc.Allrightsofreproductioninanyformreserved.388GHIA,GHIA,ANDSHENThetheoreticalpotentialofthemultigridmethodhasbeenadequatelyexposedforthesystemofdiscretizedequationsarisingfromasingledifferentialequation(e.g.,[15,221).Infact,for1-Dproblems,Merriam[151hasshownthelikenessofthemultigridmethodtothedirectsolutionprocedureofcyclicreduction.Thispotentialhasbeenrealizedanddemonstratedinactualsolutionsofsampleproblems[3,5,8,131.Applicationofthemultigridtechniquetothesolutionofasystemofcouplednonlineardifferentialequationsstillposesseveralquestions,however,thatarecurrentlybeingstudiedbyvariousinvestigators[7,21,221.ThepresentstudyrepresentsanefforttoemploythemultigridmethodinthesolutionoftheNavier-StokesequationsforamodelflowproblemwithagoalofobtainingsolutionsforReynoldsnumbersandmeshrefinementsashighaspossible.Thefundamentalprincipleofthemultigridprocedureisfirstdescribedbriefly,thenitsapplicationtothegoverningequationsisdiscussedindetail.Finally,theresultsobtainedfortheshear-drivenflowinasquarecavityatReynoldsnumberashighas5000and10,000arepresented,togetherwiththeparticulardetailsthatneededtobeobservedinobtainingthesesolutions.2.BASICPRINCIPLEOFMULTIGRIDTECHNIQUEFollowingBrandtandDinar[7],thecontinuousdifferentialproblemconsideredisasystemofIpartialdifferentialequationsrepresentedsymbolicallyasLjCT(X)=Fj(~),j=1,2,...,1,YED,withthemboundaryconditionsBio(2)=G,(Z),i=1,2,...,m,XEaD,(2.1)(2.2)where0=(U,,U,,...,U,)aretheunknownvariables,X=(x,,x2,...,xd)arethedindependentvariablesofthed-dimensionalproblem,FjandGiareknownfunctionsondomainDanditsboundaryc?D,respectively,andLjandBiaregeneraldifferentialoperators.Afinite-differencesolutiontotheproblemdescribedbyEqs.(2.1)and(2.2)isdesiredinacomputationaldomainwithgridspacingh.Withasuperscripthtodenotethefinite-differenceapproximation,thelinearsystemofalgebraicequationsresultingfromaselecteddifferenceschemecanberepresentedas(2.3)AconventionaliterativetechniqueforsolvingEq.(2.3)consistsofrepeatedsweepsofsomerelaxationscheme,thesimplestbeingtheGauss-Seidelscheme,untilconvergenceisachieved.Itisoftenexperiencedthattheconvergenceofthemethodisfastonlyforthefirstfewiterations.ThisphenomenoncanbeexplainedifoneconsidersaFourieranalysisoftheerror.Brandt(51hasthusestimatedtheHIGHREINCOMPRESSIBLEFLOW389magnitudeofthesmoothingratepdefinedasthefactorbywhicheacherrorcomponentisdecreasedduringonerelaxationsweepoftheGauss-Seidelprocedure.ItisobservedthatGauss-Seidelrelaxationproducesagoodsmoothingrateforthoseerrorcomponentswhosewavelengthiscomparabletothesizeofthemesh;thesmoothingrateofmoreslowlyvaryingFouriercomponentsoftheerrorisrelativelypoor.Themultigridmethodisbasedprimarilyonthisfeature.Itrecognizesthatawavelengthwhichislongrelativetoatinemeshisshorterrelativetoacoarsermesh.Hence,afterthefirsttwoorthreeiterationsonagivenfinemesh,themultigridmethodswitchestoacoarsermeshwithstepsize2h,wheretheerrorcomponentswithwavelengthcomparableto2hcanberapidlyannihilated.Thefine-gridsolutiondeterminedinthefirststepthenneeds