Table Of ContentMULTI-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,
