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MULTI-BLOCKCOMPUTATIONSANDTURBULENCEMODELING FORTURBOMACHINERYFLOWS By JIANLIU ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOF THEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA ACKNOWLEDGEMENTS Iwouldliketoexpressmydeepestappreciationtomyadvisor,ProfessorWeiShyy, for his advice, encouragement, friendship and constant support. I have benefitted substantially fromhisexperience,knowledgeandphilosophy,bothprofessionallyand personally. Without his enthusiasm and commitment to this challenging project, my explorationinthisworkcouldnothavebeenaccomplished. Iwouldalsoliketosincerelythankmycommitteemembers,ProfessorsChen-Chi Hsu,UlrichH.Kurzweg,BernardM.LeadonandKermitN.Sigmonfortheirteaching, supportandencouragementthroughoutmytimeattheUniversityofFlorida. VeryspecialthanksaregiventoallmycolleaguesintheCFDgroup,especially, JeffreyWright,MadhukarRao,S.S.Thakur,RichardSmith,H.S.Udaykumar,Venkata Krishnamurty,Shin-JyeLiang,andEdwinBlosch,fortheirdirectandindirectcontributions IamgratefulandfortheirgenuinefriendshipIenjoy. Iwouldalsoliketothankmy colleaguesGuobaoGuoandGuang-WuChenfortheirhelpinmanyaspects. Lastbutnotleast,Iamdeeplyindebtedtomywife,Jie,forherunderstanding, encouragementandsupportinbothmyacademicpursuitsandlifeingeneral. ii TABLEOFCONTENTS pages ACKNOWLEDGEMENTS ii ABSTRACT vi CHAPTERS 1 INTRODUCTION 1 2 PRESSURE-BASEDMETHODFORTHENAVTER-STOKES EQUATIONS 5 2.1GoverningEquations 5 2.2DiscretizationoftheMomentumEquations 6 2.2.1First-OrderUpwindScheme 9 2.2.2CentralDifferenceScheme 10 2.2.3Second-OrderUpwindScheme 11 2.3DiscretizationofthePressureCorrectionEquation 12 2.4PhysicalBoundaryConditions 14 2.4.1BoundaryConditionsfortheMomentumEquations 14 2.4.2BoundaryConditionsforthePressureCorrectionEquation ... 15 2.5TheSolutionProcedure 16 2.6ConcludingRemarks 17 3 MULTI-BLOCKMETHODFORTHENAVTER-STOKESEQUATIONS 18 3.1Background 18 3.2NotationofMulti-BlockGrids 21 3.3AnalysisofModelEquations 21 3.3.1 1-DPoissonEquation 21 3.3.22-DPoissonEquation 25 3.3.31-DConvection-DiffusionEquation 27 3.4AnalysisoftheNaiver-StokesEquations 28 3.5Multi-BlockInterfaceTreatmentsfortheNavier-StokesEquations.. 30 3.5.1Multi-BlockGridArrangement 30 3.5.2InterfaceTreatmentfortheMomentumEquations 31 iii 3.5.3InterfaceTreatmentforthePressureCorrectionEquation 34 3.5.4InterfaceTreatmentforMassFlux 39 3.6SolutionMethods 42 3.7SolutionStrategyfortheNavier-StokesEquations onMulti-BlockGrids 43 3.7.1SolutionProcedureforSingleBlockComputations 43 3.7.2SolutionProcedureforMulti-BlockComputations 44 3.7.3ConvergenceCriterion 45 3.8ConcludingRemarks 46 4 DATASTRUCTURES 48 4.1GeneralInterfaceOrganization 48 4.2Interpolation 48 4.2.1GridVertexBasedInterpolation 49 4.2.2GridFaceCenterBasedInterpolation 52 4.3InterpolationwithLocalConservativeCorrection 54 4.3.1GridCellSplit 55 4.3.2FluxInterpolation 58 4.3.3LocalFluxCorrection 59 4.4ConcludingRemarks 61 5 ASSESSMENTOFTHEINTERFACETREATMENTS 63 5.1AssessmentsofConservativeTreatmentsintheN-NBoundary ConditionforthePressureCorrectionEquation 63 5.2ComparisonofTwoDifferentInterfaceConditions forthePressureCorrectionEquation 75 5.3SimulationsoftheHydraulicTurbineDistributorFlows 79 5.3.1CaseA:5Blades 79 5.3.2CaseB:4Blades 81 5.4ConcludingRemarks 85 6 TWO-EQUATIONTURBULENCEMODELINGWITH NON-EQUILIBRIUMEFFECT 88 6.1Background 88 6.2TurbulentTransportEquations 92 6.2.1Reynolds-StressTransportEquation 92 6.2.2K-eTransportEquations 93 6.3ImplementationoftheK-eModelfor2-DFlow 95 6.3.1BoundaryConditionsandWallTreatment 96 iv 6.3.2ImplementationoftheWallShearStressin MomentumEquations 99 6.3.3ImplementationoftheK-eEquationsnearWallBoundaries ... 102 6.4TurbulenceModelingforNon-Equilibrium Flows 104 6.5CaseStudies 107 6.5.1BackwardFacingStepFlow 107 6.5.2HillFlowInsideaChannel Ill 6.5.33-DDiffuserFlow 121 6.6ConcludingRemarks 122 7 TWO-EQUATIONTURBULENCEMODELINGWITH ROTATIONALEFFECT 129 7.1Background 129 7.2TheMomentumandTurbulentTransportEquations 131 7.3TheoreticalAnalyses 134 7.3.1DisplacedParticleAnalysis 134 7.3.2SimplifiedReynolds-StressAnalysis 137 7.4TheModelwithRotationalEffect 140 7.5AssessmentoftheModifiedModel 142 7.5.1RotatingChannelFlow 142 7.5.2RotatingBackwardFacingStepFlow 147 7.5.3FlowSimulationinATorqueConverter 162 7.6ConcludingRemarks 164 8 SUMMARYANDSUGGESTIONSFORFUTUREWORK 168 REFERENCE 171 BIOGRAPHICALSKETCH 178 AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulfillmentofthe RequirementsfortheDegreeofDoctorofPhilosophy MULTI-BLOCKCOMPUTATIONSANDTURBULENCEMODELING FORTURBOMACHINERYFLOWS By JianLiu May1996 Chairperson:WeiShyy MajorDepartment:AerospaceEngineering,MechanicsandEngineeringScience Turbomachineryflowsareencounteredinmanyindustrialdevices.Tosuccessfully analyzetheflowbehavior,bothnumericaltreatmentsregardinggriddingtechniquesin complexgeometries,andphysicalmodelingregardingrotatingturbulenceclosures,needto be developed. In this work, a pressure-based multi-block grid method for solving incompressibleviscousflowsincomplexgeometriesispresented.Themethodemploysa finitevolumeapproachandastaggeredgrid, andsolvesthegoverningequationsina sequentialmanner.Issuesregardingdiscontinuousgridinterfacetreatmentforsolvingthe Navier-Stokes equations in multi-block domain are investigated. For the momentum equations,thedependentvariablesbetweenblockscanbedirectlyinterpolated.Forthe pressurecorrectionequation,itisfoundthatwiththelocalconservativeinterfacemassflux treatment with certain degree of accuracy, both the Neumann-Neumann and Neumann-Dirichlet boundaryconditionscanyieldsatisfactorysolutions. vi The closure issues ofthe K-e two-equation turbulence model are discussed. Specifically, theoriginalandnon-equilibriumK-emodelsarecompared.Itisobservedthat theequilibriumconditionbetweenproduction anddissipationoftheturbulentkinetic energy, whichisthebasisoftheoriginalK-emodel,isnotsatisfiedingeneral.The non-equilibriumK-emodel,wherethe£-equationismodifiedtoaccountfortheimbalance betweentheproductionanddissipationterms,inessenceintroducesanextratimescalein themodel,andappearstoproducebetterpredictionsforcomplexflows. TheK-eturbulencemodelwithrotationaleffectisalsoinvestigated.Theanalysis showsthatinthecontextoftheK-and£-equations, therotationaffects theturbulent quantitiesonlyindirectlythroughtheReynolds-stressterms.Thisindirectrotationaleffect isnotaccountedfor,intheoriginalK-emodel.Theanalysisalsoclearlyidentifiestheeffect of the system rotation interacting with structure of turbulence and the primary nondimensionalparameter.Therotationaleffectismodeledthroughthemodificationofthe sourceterminthe£-equation.Thenumericalandmodelingtechniquesareusedtosolvetwo- andthree-dimensionalturbomachineryflowproblems. vii CHAPTER 1 INTRODUCTION With the rapid advance of numerical methods and computer technologies, computationalfluiddynamics(CFD)isbecominganimportanttoolinanalyzingfluid physicsandinengineeringdesign.However,formanyflowproblemsinvolvingcomplex physicsandgeometries,suchasturbomachineryflows,gridgenerationbecomesamajor issue.Basicallytwodifferentapproacheshavebeendeveloped:(1)unstructuredgrid,and (2)multi-blockstructuredgrid.Eachoftwomethodshasitsadvantagesanddisadvantages. Withunstructuredgrids,theflowdomainistypicallydiscretizedwithanarrayofgrid cellswhichhaveanirregularconnectivity.Usuallythegridiscomposedofasinglecelltype, typicallyeithertriangularorquadrilateral(intwodimensions)andtetrahedral(inthree dimensions).Unstructuredgridsareveryflexibleindiscretizingcomplexgeometries.With thedevelopmentofcomputationalgeometryanddatastructure,discretizationofeventhe mostcomplexgeometriescanbehandledinanefficientway(LohnerandParikh 1988, Morganetal. 1991,Braatenand Connell 1996, Kallinderiset al. 1996). In addition, unstructured grids are well suited for solution adaptive methods involving local grid refinementwithoutintroducinggriddiscontinuity.However,unstructuredgridshavesome disadvantages.Intermsofcomputermemoryandcomputationalefficiency,duetothelack ofinherentorderofgridnodes,theflowsolverswithunstructuredgridsmayoccupymore computermemorycomparedtothesolverswithstructuredgrids.Thesameproblemalso makesitdifficultfortheflowsolvertobeefficientlyparalleledtoenhancethecomputational efficiency.Regardingsolutionaccuracy,unstructuredgridmethodsposeanotherchallenge forflowproblemscontainingthinlayers,suchasboundarylayers.Theabilitytoaccurately controltheaspectratioanddirectionalityofthegridcells,withoutincreasingtoomanygrid 1 2 cells,needtoberesolvedbeforeunstructuredgridscanbeefficientlyusedtosolvepractical viscousflowproblems. Ontheotherhand,structuredgrids,whichhavetheusualorderingofthegridcells, generallyrequirelesscomputermemoryandcanemploylinearequationsolverefficiently. Structuredgridscanalsobetteraccommodatedirectionalpreferencesuchasthinlayers.For complexflows,itisoftendifficulttogenerateasatisfactory,singlestructuredgridtocover theentire flow domain. In addition, forproblems ofmultiple length scales, a single structuredgridhasdifficultiestoresolveallflowfeatureswithareasonablenumberofgrid points.Allthesemakethemulti-blockstructuredgridveryattractive.Themulti-block structuredgridcanalleviatethosedifficultiesbecause(1)itcanreducethetopological complexity ofa single structured grid system by employing several blocks ofgrid, permittingeachindividualgridblocktobegeneratedindependentlysothatbothgeometry andresolutionintheboundaryregioncanbetreatedmoresatisfactorily;(2)gridlinesneed not to be continuous across grid interfaces, and local grid refinement and adaptive redistributioncanbeconductedmoreeasilytoaccommodatedifferentphysicallengthscales presentindifferentregions(Rai1984,ChesshireandHenshaw1990);(3)themulti-block method, in the spirit of divide-conquer, also provides a natural route for parallel computations(GroppandKeyes1992a,1992b). Inthemulti-blockmethod,becausethegridlinesmaynotbecontinuousacrossgrid blockinterfaces,informationbetweenblockshastobetransferredwithduecare.Suchan informationtransfermethodispreferablyeasytoimplement,whilemaintainingnecessary accuracy and computational efficiency. A majorrelated issue is that, formany flow problems,itisoftenimportanttouseconservativeinterfaceproceduretoensurethatphysical conservationlawsaresatisfied(Berger1987,Kallinderis1992,ChesshireandHenshaw. 1994). Suchaconsiderationcanimposestringentconstraintsontheconstructionofan effectiveinterfacescheme;furthermore,insomecases,theaboveneedscanconflictone another.Simultaneousachievementofbothconservationandaccuracycanbedifficult. 3 Inthepresentstudy,amethodologyforsolvingincompressibleviscousflowswith multi-blockstructuredgridintheframeworkofapressure-basedmethodisexplored.In Chapter 2, a pressure-based method for solving two-dimensional incompressible Navier-Stokesequationsinnon-orthogonalcurvilineargridissummarized. Theissuesof theinterfacetreatmentinthemulti-blockmethodareinvestigatedinChapter3.Specifically, theeffects ofnon-conservativeandconservativetreatments formassfluxonsolution accuracyarestudied.Differentwaystocouplepressurefieldsbetweenadjacentblocksand differentiterativestrategiesbetweenblocksarediscussed.InChapter4,datastructures associatedwithgridinterfacetreatmentaredetailed.Assessmentsaremadeoftheaccuracy andeffectivenessoftheseinterfaceschemes.Theapplicationsofthemulti-blockmethodare demonstratedthroughtheturbulentflowsaroundhydraulicturbinedistributorsinvolving cascadesofmultipleairfoils.ThesenumericalresultsarepresentedinChapter5. Besidesnumericaldiscretizationschemesandcomplexgeometries,thetreatmentof turbulentflowsisanotherissueofmuchpracticalimportancetothesuccessofCFD.Over thelastseveraldecades,variousturbulentmodelshavebeendevelopedandappliedtomany flowproblemswithvarieddegreesofsuccess.Thesemodelsrangefromthesimplestand leastexpensivealgebraicmodelstotheseven-equationReynolds-stressmodels,which include six equations for the Reynolds-stress components and one equation for the dissipationrateoftheturbulencekineticenergy.Amongthem,thetwo-equationmodelshave been most popular in complex engineering flow computations. The most popular two-equationmodelsistheK-emodel(LaunderandSpalding1974). However,thetwo-equationmodelsarederivedbasedoncertainassumptionsunder some specific flowconditions. One ofthemis assumingthattheproduction andthe dissipationrateoftheturbulentkineticenergyisunderequilibriumcondition.Whilethisis satisfactory for some simple flows, for complex flows including flow recirculation, streamlinecurvature,adversepressuregradient,etc.,theequilibriumconditiondoesnot exist,andtheoriginaltwo-equationmodelscannotgivesatisfactoryresults(ChenandKim,