Cataloging-inPublicationDataDivisionofInformationandDocumentationTomita, Jesuino TakachiThree-Dimensional Flow Calculations of Axial Compressors and Turbines Using CFDTechniques / Jesuino Takachi Tomita.Sao Jose dos Campos, 2009.229f.Thesis of Doctor of Science Course of MechanicalAeronautical Engineering. Area ofAerodynamics, Propulsion and Energy. Technological Institute of Aeronautics, 2009. Advisor:Prof. Dr. Joao Roberto Barbosa.1. CFD. 2. Gas Turbine. 3. Axial Compressor. 4. Numerical Methods. 5. Compressible Flows.I. Aerospace Technical Center. Technological Institute of Aeronautics. Division ofMechanicalAeronautical. II. Title.BIBLIOGRAPHICREFERENCETOMITA,JesuinoTakachi. Three-DimensionalFlowCalculationsofAxialCompressorsandTurbinesUsingCFDTechniques. 2009. 229f. ThesisofDoctorofScienceTechnologicalInstituteofAeronautics,SaoJosedosCampos.SESSIONOFRIGHTSAUTHOR NAME:JesuinoTakachiTomitaPUBLICATION TITLE:Three-DimensionalFlowCalculationsofAxialCompressorsandTurbinesUsingCFDTechniques.TYPE OF PUBLICATION/YEAR:Thesisofdoctoral /2009It isgrantedtoAeronautics Instituteof Technologypermissiontoreproducecopies ofthisthesisandtoonlyloanortosell copiesforacademicandscienticpurposes. Theauthor reserves other publicationrights andnopart of this thesis canbe reproducedwithouttheauthorizationoftheauthor.JesuinoTakachiTomitaRuaCarneirodaCunha,1228CEP04144-001Apto144S aoPauloSPBrasilTHREE-DIMENSIONALFLOWCALCULATIONSOFAXIALCOMPRESSORSANDTURBINESUSINGCFDTECHNIQUESJesuinoTakachiTomitaThesisCommitteeComposition:Prof. Dr. NideGeraldoC.R.FicoJ unior Presidente - ITAProf.Dr. Jo aoRobertoBarbosa Advisor - ITAProf. Dr. EdsonLuizZaparoli MembroInterno - ITAProf. Dr. NelsonManzanaresFilho MembroExterno - UNIFEIProf. Dr. Jo aoBatistaPessoaFalc aoFilho MembroExterno - IAEITAA minha mae Izabel TiyokaTomita, a minha linda es-posa Thaisa Talarico HyppolitoTomita e ao meu orientadorProfessor Jo ao Roberto Bar-bosa, pelo apoio e conanca du-ranteessesanosdeestudosemturbinasagas.AgradecimentosDuranteosanosemqueestudeinoITA,pudeaprenderefazermuitomaisdoqueeuesperava,quandonesseInstitutoentreipelaprimeiravez. TambemaprendibastantecomosamigosqueznoCTA.Asomadetudoqueganheiduranteessetempo emuitomaisdoqueconhecimentotecnico-cientco.Econhecimentodevida.AgradecoaDeus,porpermitiraconclus aodemaisumaetapadaminhavida.AgradecooapoiodeminhamaeIzabelTiyokaTomita,quedesdeoprimeirodiadeauladaminhavida,meincentivouemedeusuporteparaqueeupudesseconcluirmaisumtrabalho.AgradecoaminhalindaesposaThaisaTalaricoHyppolitoTomita,queduranteessesanos,temcompreendidotodoomeuesforco,durantein umerasnoites,nsdesemanaeferiados,naqualqueigerandogeometriasdecompressoreseturbinas,malhasecompilandoprogramas.AgradecoatodososturbineirosdoGrupodeTurbinasdoITA(Cleverson,Franco,Santin,Luciano,Gustavo,Dulceneia,Renato,Dora,DiFiori,Helder,M arcioMendonca,e Daniel) e n ao poderia deixar de agradecer o Demerval, pela amizade e companheirismo.ReconhecotodaaajudaetodooconhecimentoquerecebidoProfessorJo aoRobertoBarbosa,que,semd uvida, eumareferenciainternacionalemturbinasagas. Agrade coportermedadoaoportunidadedemergulharnomundodasturbom aquinas. Esperopoderfazeromesmocomasproximasgerac oes.Obrigado!Anatureza,eaartedeDeus.DanteAlighieri).ResumoComoadventodepotentescomputadores, aDin amicadosFluidosComputacional(DFC) temsidovastamente utilizadapor pesquisadores e cientistas parainvestigar ocomportamentodeescoamentoseavariacaodaspropriedadesdosmesmos. Ocustodesimulac aodeDFC emuitopequenocomparadocomoarsenalexperimentalcomobancosdeensaioet uneisdevento. Nos ultimosanos,muitospacotescomerciaisdeDFCforamdesenvolvidos, algunsdelespossuemproeminencianaind ustriaenaacademia. Porem,algunscalculosespeccosdeDFCsaocasosmuitoparticularese` asvezesnecessitamdeatenc ao especial devido a complexidade do escoamento. Nesses casos, uma pesquisa metic-ulosatorna-senecess aria. Esteeocasodocalculodeescoamentosemturbom aquinas.Odesenvolvimentodec odigosdeDFCaplicadosemsimula coesdeescoamentosemtur-bomaquinas e os detalhes das implementa coes s ao assuntos muito reservados. Um pequenon umero de instituic oes possui esse tipo de conhecimento. Cada c odigo de DFC possue suaparticularidade. Desenvolverumc odigofonteparticular eumassuntomuitointeressantenosensoacademico.Nesse trabalho, umc odigocomputacional escritoemFORTRAN, foi desenvolvidoparacalcularescoamentosinternosemturbomaquinasusandotecnicasdaDFC. Opro-gramaecapazdecalcularescoamentostridimensionaisn aosomenteemturbomaquinas.Porexemplo, escoamentosinternoseexternoscomobocaiseaerofoliospodemsercal-viiiculados. Otratamentodadonoc odigopermiteousodemalhasn ao-estruturadascomelementoshexaedricos. Escoamentosenvolvendoasequac oesdeEuler, Navier-Stokeseescoamentosturbulentospodemsercalculados, dependendodanecessidadedousuario.Diferentesesquemasnumericosforamimplementadosparaaintegra caonotempoenoespaco. Metodosnumericosparamelhoraraestabilidadeeaumentaropassonotempo(passo no tempo variavel, suavizacao implcita do resduo) foram tambem implementadosetodososdetalhesest aodescritosnessetrabalho.Aorigemdoc odigocomputacional eparasimular escoamentos emcompressores eturbinas. Dessaforma, osistemadereferenciarotacional enao-rotacional ecalculadosimultaneamente. Dessa forma, o proceso de vericac ao e valida cao do codigo foi realizadoparaambosossistemas.Umprocedimentodeprojeto, passo-a-passo, eapresentadonessetrabalho.Emuitoimportantemencionar queparaoentendimentocompletodafsicadoescoamentoemcompressoreseturbinasoprojetistadevepossuirumsolidoconhecimentodeoperac aodoscomponentesdeumaturbinaag as.AbstractWith the advent of powerful computer hardware, Computational Fluid Dynamics (CFD)hasbeenvastlyusedbyresearchesandscientiststoinvestigateowbehavioranditsprop-erties. ThecostofCFDsimulationisverysmallcomparedtotheexperimentalarsenalastest facilitiesandwind-tunnels. Inthelast yearsmanyCFDcommercial packagesweredevelopedandsome of thempossess prominence inindustryandacademia. However,somespecicCFDcalculationsareparticularcasesandsometimesneedspecialattentionduetothecomplexityoftheow. Inthesecases,meticulousresearchbecomesnecessary.Thisisthecaseofturbomachineryowcalculations. ThedevelopmentofCFDcodesap-pliedtoturbomachineryowsimulationsanditsimplementationissuesarenotavailable.Afewinstitutionshavethistypeof knowledge. EachCFDcodehasitsparticularities.DevelopingaCFDcodeisveryinterestsubjectinacademia.Inthiswork,acomputationalcode,writteninFORTRAN,wasdevelopedtocalculateinternal owsinturbomachinesusingCFDtechniques. Thesolveriscapableofcalculat-ingthethree-dimensional owsnot onlyforturbomachines. Forinstance, internal andexternal ows of nozzles andairfoils canbecalculated. Theapproachusedallows theuseof unstructuredmeshesof hexahedral elements. Euler, Navier-Stokesandturbulentequationscanbecalculateddependingontheusersettings. Dierentnumerical schemeswereimplementedfortimeandspaceintegration. Numericaltoolstoimprovethestabilityxandtoincreasethetime-step(local time-stepandimplicitresidual smoothing)werealsoimplementedandall detailsaredescribedinthiswork.Theoriginofthissolveristosimulateowsincompressorsandturbines. Therefore,bothrotatingandnon-rotatingframesof referencearecalculatedsimultaneously. Hence,thevericationandvalidationprocesseswererunforbothinertial andnon-inertial sys-tems.Astep-by-stepdesignprocedureispresentedinthiswork. Itisveryimportanttomen-tion that to have a complete understanding of the ow physics in compressors and turbinesthedesignermusthaveasolidknowledgeoftheoperationofgasturbinecomponents.ContentsListofFigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvListofTables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiListofAbbreviationsandAcronyms. . . . . . . . . . . . . . . . xxiiiListofSymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.3 PreviousDevelopments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.3.1 CFDinAcademicResearchandIndustry . . . . . . . . . . . . . . . . . . . 341.3.2 CFDonGasTurbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.3.3 CFDonAxialCompressorsandTurbines . . . . . . . . . . . . . . . . . . . 381.4 AuthorContribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441.5 WorkOrganization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 MathematicalFormulation . . . . . . . . . . . . . . . . . . . . 512.1 Theuid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.2 Theow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.2.1 Turbulence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56CONTENTS xii2.2.2 TurbulenceModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 NumericalFormulation. . . . . . . . . . . . . . . . . . . . . . . 683.1 DiscretizationModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.1.1 Finite-VolumeMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.2 SpatialIntegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.2.1 TheCenteredSchemeofJameson . . . . . . . . . . . . . . . . . . . . . . . 723.2.2 TheUpwindSchemeofVanLeer: FluxVectorSplitting . . . . . . . . . . . 733.2.3 TheUpwindSchemeofRoe: FluxDierenceSplitting . . . . . . . . . . . 763.2.4 Reconstruction Based on Approximate Monotone Upstream-Centered SchemesforConservationLaws(MUSCL) . . . . . . . . . . . . . . . . . . . . . . . 803.2.5 VenkatakrishnansLimiter . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.2.6 DiscretizationofViscousFluxes . . . . . . . . . . . . . . . . . . . . . . . . 833.3 TimeIntegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.3.1 TheSchemeofMacCormack(1969) . . . . . . . . . . . . . . . . . . . . . . 843.3.2 TheSchemeofRunge-Kutta . . . . . . . . . . . . . . . . . . . . . . . . . . 853.4 NumericalStabilityandConvergenceAcceleration. . . . . . . . . . . 863.4.1 ArticialDissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.4.2 ImplicitResidualSmoothing . . . . . . . . . . . . . . . . . . . . . . . . . . 923.5 Spatial andTimeIntegrationof theSpalartAllmarasTurbulenceModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.6 UnstructuredMeshTreatment . . . . . . . . . . . . . . . . . . . . . . . 963.7 InitialConditions,BoundaryConditionsandRowsInterface . . . . 973.7.1 InitialConditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983.7.2 BoundaryConditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.7.3 NumericalStopCriteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111CONTENTS xiii4 ComputationalImplementation . . . . . . . . . . . . . . . . . 1134.1 CodeStructureandImplementationIssues . . . . . . . . . . . . . . . 1135 CodeVerificationandValidation . . . . . . . . . . . . . . . 1245.1 InviscidCase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.2 LaminarCase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.3 TurbulentFlowCase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1466.1 FlowSimulationinTurbomachines. . . . . . . . . . . . . . . . . . . . . 1466.1.1 Single-StageAxial-FlowTurbineSimulation . . . . . . . . . . . . . . . . . 1476.1.2 SingleRotorwithLowAspectRatio . . . . . . . . . . . . . . . . . . . . . . 1656.2 MultistageAxial-FlowCompressorSimulation . . . . . . . . . . . . . 1706.2.1 SpecicationofDesignParameters . . . . . . . . . . . . . . . . . . . . . . . 1726.2.2 PreliminaryDesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1726.2.3 StreamlineCurvatureMethod . . . . . . . . . . . . . . . . . . . . . . . . . 1846.2.4 3-DFlowCalculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1967 CommentsandConclusions . . . . . . . . . . . . . . . . . . . . 2077.1 TheCFDsolverasaResearchandTeachingTool . . . . . . . . . . . 2098 FutureImplementations. . . . . . . . . . . . . . . . . . . . . . . 2108.1 Pre-Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2118.2 Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2128.3 Post-Processing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215AppendixA ArtificialDissipationIssues. . . . . . . . . . . 225CONTENTS xivA.1 StencilAppliedonArticialDissipationImplementation. . . . . . . 225AnnexA RotatingFrameofReference . . . . . . . . . . . 228A.1 CoriolisandCentrifugalForce. . . . . . . . . . . . . . . . . . . . . . . . 228Glossary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230ListofFiguresFIGURE1.1Gasturbineengineanditscomponents . . . . . . . . . . . . . . . . 37FIGURE1.2Multidisciplinaryteamonturbomachinerydesign . . . . . . . . . . 38FIGURE2.1Schematicvelocityvectordiagram. . . . . . . . . . . . . . . . . . . 53FIGURE3.1Schemebasedonpiecewiselinearreconstruction . . . . . . . . . . . 81FIGURE3.2Scheme implemented to set the boundary condition in a ghost element 98FIGURE3.3Representationofamixing-planeoutletoncompressors . . . . . . . 110FIGURE3.4Representationofamixing-planeinletoncompressors . . . . . . . . 110FIGURE4.1Schemecreatedtoidentifythenumberofblades . . . . . . . . . . . 117FIGURE4.2Schemecreatedtoidentifythemixing-planes . . . . . . . . . . . . . 118FIGURE5.1Nozzlegeometryandmesh(owisfromlefttoright) . . . . . . . . 126FIGURE5.2Experimentalandnumericalresultsforstaticpressureratio2-in-viscidow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127FIGURE5.3Experimental andnumerical results for staticpressureratio5forcenteredandupwindschemes-inviscidow . . . . . . . . . . . . . 128FIGURE5.4Continuity residue histories for dierent spatial discretization schemes-inviscidow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129FIGURE5.5Machnumbercontoursforcenteredscheme . . . . . . . . . . . . . . 129FIGURE5.6Machnumbercontoursforupwindscheme-inviscidow . . . . . . 129LISTOFFIGURES xviFIGURE5.7MeshgeneratedforNACA0012airfoilowcalculation . . . . . . . . 130FIGURE5.8PressurecoecientdistributionontheNACA0012airfoilwithzeroangle-of-attackandMn= 0.8: centered-dierence . . . . . . . . . . 131FIGURE5.9PressurecoecientdistributionontheNACA0012airfoilwithzeroangle-of-attackandMn= 0.8: upwind. . . . . . . . . . . . . . . . . 131FIGURE 5.10 Machnumbercontoursforcenteredscheme . . . . . . . . . . . . . . 132FIGURE 5.11 Machnumbercontoursforupwindscheme . . . . . . . . . . . . . . 132FIGURE 5.12 Staticpressurecontoursforcenteredscheme . . . . . . . . . . . . . 133FIGURE 5.13 Staticpressurecontoursforupwindscheme . . . . . . . . . . . . . . 133FIGURE 5.14 Meshusedtocalculatetheowonaat-plate . . . . . . . . . . . . 134FIGURE 5.15 Continuityresiduehistory: rstcase. . . . . . . . . . . . . . . . . . 136FIGURE 5.16 Analytical andnumerical solutions of owwithM=0.3onaat-platewith10nodesinsideoftheboundary-layer: rstcase . . . 136FIGURE 5.17 Continuityresiduehistory: secondcase . . . . . . . . . . . . . . . . 137FIGURE 5.18 AnalyticalandnumericalsolutionsofanowwithM=0.3onaat-platewith15nodesinsideoftheboundary-layer: secondcase . 137FIGURE 5.19 Continuityresiduehistory: thirdcase . . . . . . . . . . . . . . . . . 138FIGURE 5.20 AnalyticalandnumericalsolutionsofanowwithM=0.3onaat-platewith15nodesinsideoftheboundary-layer: thirdcase . . 138FIGURE 5.21 Continuityresiduehistory: fourthcase . . . . . . . . . . . . . . . . 139FIGURE 5.22 AnalyticalandnumericalsolutionsofanowwithM=0.3onaat-platewith10nodesinsideoftheboundary-layer: fourthcase. . 139FIGURE 5.23 Continuityresiduehistory: fthcase . . . . . . . . . . . . . . . . . . 140FIGURE 5.24 AnalyticalandnumericalsolutionsofanowwithM=0.3onaat-platewith12nodesinsideoftheboundary-layer: fthcase . . . 140FIGURE 5.25 Analytical andnumerical solutionsof aowwithM=0.3onaat-platewith15nodesinsideoftheboundary-layer: sixthcase . . 141LISTOFFIGURES xviiFIGURE 5.26 Detailsof theMachnumbercontoursinsideof theboundary-layerontheat-plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142FIGURE 5.27 Detailsofthevelocityvectorsprolesinsideoftheboundary-layerontheat-plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142FIGURE 5.28 Meshgeneratedfornozzleowcalculation-turbulentow . . . . . 143FIGURE 5.29 Comparison of experimental and numerical results for static pressureratio5forcenteredandupwindschemes-turbulentow . . . . . . 144FIGURE 5.30 Continuityresiduehistoriesfortwospatialdiscretizationschemes . . 144FIGURE 5.31 Machnumbercontours-turbulentow . . . . . . . . . . . . . . . . 145FIGURE 5.32 Velocityvectors-turbulentow . . . . . . . . . . . . . . . . . . . . 145FIGURE6.1Auxiliarycurvestodrawingasingle-stageaxial-owturbine . . . . 148FIGURE6.23-DsoliddrawingoftheNGVandrotorofasingle-stageaxial-owturbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148FIGURE6.3Full3-Dviewofansingle-stageaxial-owturbine . . . . . . . . . . 149FIGURE6.4Single-stageaxial-owturbineH-griddomain. . . . . . . . . . . . . 149FIGURE6.5CloseupoftheleadingedgeofanH-grid . . . . . . . . . . . . . . . 150FIGURE6.6Single-stageaxialturbineH-O-H-griddomain . . . . . . . . . . . . . 150FIGURE6.7Closeup of the stator trailing edge and rotor leading edge of a single-stageaxial-owturbine . . . . . . . . . . . . . . . . . . . . . . . . . 151FIGURE6.8Single-stageaxial-owturbineO-griddomain. . . . . . . . . . . . . 151FIGURE6.9Closeup of the O-grid around the stator and rotor blades of a single-stageaxial-owturbine . . . . . . . . . . . . . . . . . . . . . . . . . 152FIGURE 6.10 Single-stageaxial-owturbinemass-owoutletconvergencehistory-inviscidcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153FIGURE 6.11 Static pressure contour of a single-stage axial-ow turbine - inviscidcase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153LISTOFFIGURES xviiiFIGURE 6.12 Closeupoftheowacrossthemixing-planeofasingle-stageaxial-owturbine-inviscidcase . . . . . . . . . . . . . . . . . . . . . . . 154FIGURE 6.13 Closeupoftheowatrotorsuctionandpressuresurfaces-inviscidcase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155FIGURE 6.14 H-gridusedforasingle-stageaxial-owturbine . . . . . . . . . . . . 155FIGURE 6.15 Mixing-planeoutlet (MPO) andmixing-planeinlet (MPI) ratio-turbulentcasewithH-grid . . . . . . . . . . . . . . . . . . . . . . . 156FIGURE 6.16 Single-stageaxial-owturbinemass-owoutletconvergencehistoryofthe-turbulentcasewithH-grid. . . . . . . . . . . . . . . . . . . 156FIGURE 6.17 Single-stage axial-ow turbine velocity vectors distribution along thestatorbladerow-turbulentcasewithH-grid. . . . . . . . . . . . . 157FIGURE 6.18 Closeup of the reverse ow at the rotor suction side - turbulent casewithH-grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157FIGURE 6.19 SchemeoftheblockscreatedtogeneratetheO-grid . . . . . . . . . 158FIGURE 6.20 DetailoftheO-gridontheturbinecasing. . . . . . . . . . . . . . . 158FIGURE 6.21 Detailofthegapbetweentheturbinerotorandcasing . . . . . . . . 159FIGURE 6.22 Meshelementsdistributiononthetipclearanceregion. . . . . . . . 159FIGURE 6.23 DetailoftheO-gridaroundtherotorbladetipandtherenementoftheclearanceregion . . . . . . . . . . . . . . . . . . . . . . . . . 160FIGURE 6.24 Single-stage axial-ow turbine mass-ow outlet convergency history-turbulentcasewithO-grid . . . . . . . . . . . . . . . . . . . . . . 160FIGURE 6.25 Mixing-plane outlet to mixing-plane inlet ratio - turbulent case withO-grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161FIGURE 6.26 Single-stage axial-ow turbine pressure ratio monitoring - turbulentcasewithO-grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161FIGURE 6.27 Distributionofthevelocityvectorsintheturbinerotor-turbulentcasewithO-grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162LISTOFFIGURES xixFIGURE 6.28 Eect of Coriolis force in the velocity eld close to the turbine rotorwall-turbulentcasewithO-grid. . . . . . . . . . . . . . . . . . . . 162FIGURE 6.29 Global view of the gas expansion along the turbine stage - turbulentcasewithO-grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163FIGURE 6.30 Detailofthestaticpressurecontoursalongtheturbinestage-tur-bulentcasewithO-grid. . . . . . . . . . . . . . . . . . . . . . . . . 163FIGURE 6.31 Detailofthestaticpressurecontoursalongtheturbinestage-tur-bulentcasewithO-grid. . . . . . . . . . . . . . . . . . . . . . . . . 164FIGURE 6.32 Detail of the leakage ow from pressure surface to suction surface oftheturbinerotor-turbulentcasewithO-grid . . . . . . . . . . . . 165FIGURE 6.33 3-Dviewoftherotor . . . . . . . . . . . . . . . . . . . . . . . . . . 166FIGURE 6.34 Detailoftherotorblade . . . . . . . . . . . . . . . . . . . . . . . . 166FIGURE 6.35 3-DO-gridaroundtherotorblade . . . . . . . . . . . . . . . . . . . 166FIGURE 6.36 3-Dviewoftherotor . . . . . . . . . . . . . . . . . . . . . . . . . . 167FIGURE 6.37 Detailoftherotorblade . . . . . . . . . . . . . . . . . . . . . . . . 167FIGURE 6.38 3-DO-gridaroundtherotorblade . . . . . . . . . . . . . . . . . . . 167FIGURE 6.39 Rotor outlet mass-owconvergence history- turbulent case withO-grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168FIGURE 6.40 Rotorpressureratio-turbulentcasewithO-grid. . . . . . . . . . . 168FIGURE 6.41 Increase of static pressure along the rotor blade - turbulent case withO-grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169FIGURE 6.42 Staticpressureintwodierentplanes(nearofhubandnearoftip)-turbulentcasewithO-grid . . . . . . . . . . . . . . . . . . . . . . 169FIGURE 6.43 Velocity vector distribution near of the rotor leading edge - turbulentcasewithO-grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170FIGURE 6.44 Deexionandproleloss . . . . . . . . . . . . . . . . . . . . . . . . 173FIGURE 6.45 5-stageaxial-owcompressormap . . . . . . . . . . . . . . . . . . . 177LISTOFFIGURES xxFIGURE 6.46 Compressorbleedschedule . . . . . . . . . . . . . . . . . . . . . . . 177FIGURE 6.47 5-stageaxial-owcompressormapusingBOV . . . . . . . . . . . . 178FIGURE 6.48 5-stageaxial-owcompressoreciencyusingBOV. . . . . . . . . . 178FIGURE 6.49 CompressorVIGVschedule . . . . . . . . . . . . . . . . . . . . . . . 179FIGURE 6.50 5-stageaxial-owcompressormapusingVIGV. . . . . . . . . . . . 179FIGURE 6.51 5-stageaxial-oweciencyusingVIGV. . . . . . . . . . . . . . . . 180FIGURE 6.52 VIGVandbleedschedule. . . . . . . . . . . . . . . . . . . . . . . . 180FIGURE 6.53 Sketchofasingleshaftfreepowerturbineunit . . . . . . . . . . . . 182FIGURE 6.54 Compressorcharacteristics: pressureratio. . . . . . . . . . . . . . . 183FIGURE 6.55 CompressorCharacteristics: eciency. . . . . . . . . . . . . . . . . 184FIGURE 6.56 Designed5-stageaxial-owcompressor . . . . . . . . . . . . . . . . 197FIGURE 6.57 3-Dviewofthe5-stageaxial-owcompressorbladeproles . . . . . 198FIGURE 6.58 Computationaldomainofthe5-stageaxial-owcompressordomain 198FIGURE 6.59 Axialviewofthe5-stageaxial-owcompressordomain . . . . . . . 198FIGURE 6.60 GeneralviewoftheO-gridgeneratedtothe5-stageaxial-owcom-pressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199FIGURE 6.61 Meshonthebladesurfaces . . . . . . . . . . . . . . . . . . . . . . . 199FIGURE 6.62 Computational domain and mesh structure on the hub of the rst-stage200FIGURE 6.63 DetailoftheO-gridmeshtypearoundtheblades . . . . . . . . . . 200FIGURE 6.64 5-stageaxial-owcompressoroutletmass-owconvergencehistory . 202FIGURE 6.65 5-stageaxial-owcompressor eciency variationduring thenumer-icaliteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202FIGURE 6.66 Staticpressurecontoursinthe5-stageaxial-owcompressor . . . . 203FIGURE 6.67 Detailofthevelocityvectorsatthecompressoroutlet . . . . . . . . 203FIGURE 6.68 Detailofthemixing-planevelocitydistribution. . . . . . . . . . . . 204LISTOFFIGURES xxiFIGURE 6.69 Detail of themixing-planevelocitydistributionatthecompressorthird-stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204FIGURE 6.70 Totalpressuredistributionalongthe5-stageaxial-owcompressor . 205FIGUREA.1Stencilusedtocalculatethearticialdissipationterms . . . . . . . 225FIGUREA.2Firstschemeto calculatethesecondandfourthorder termsofarti-cialdissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226FIGUREA.3Secondschemetocalculatethesecondandfourthorder terms ofarticialdissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . 226FIGUREA.4Thirdschemetocalculatethesecondandfourthordertermsofar-ticialdissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227ListofTablesTABLE6.1 Distributionofrotorbladeinletanglealongtherotorheight . . . . 193TABLE6.2 Distributionofrotorbladeoutletanglealongtherotorheight. . . . 193TABLE6.3 Distributionofstatorbladeinletanglealongthestatorheight . . . 193TABLE6.4 Distributionofstatorbladeoutletanglealongthestatorheight. . . 193TABLE6.5 Distributionofthebladespace-chordratioalongthebladeheight . 194TABLE6.6 Numberofbladesforeachrow. . . . . . . . . . . . . . . . . . . . . 194TABLE6.7 Rotorloadingfactordistribution. . . . . . . . . . . . . . . . . . . . 194TABLE6.8 Distributionofrotorincidenceowanglealongtherotorheight . . 194TABLE6.9 Distributionofstatorincidenceowanglealongthestatorheight . 195TABLE6.10Distributionofrotordeviationowanglealongtherotorheight . . 195TABLE6.11Distributionofstatordeviationowanglealongthestatorheight . 195TABLE6.12RotordeHallernumberdistribution. . . . . . . . . . . . . . . . . . 195TABLE6.13StatordeHallernumberdistribution . . . . . . . . . . . . . . . . . 196TABLE6.14AxialMachnumberforeachbladerow . . . . . . . . . . . . . . . . 196TABLE6.15ComparisonbetweenstreamlinecurvatureandCFDresults . . . . . 205ListofAbbreviationsandAcronyms1-D Uni-dimensional2-D Bi-dimensional3-D Three-dimensionalAFCC AxialFlowCompressorCodeAIAA AmericanInstituteofAeronauticsandAstronauticsBOV Bleed-ovalveCONV ConvectiveuxvectorCFD ComputationalFluidDynamicCFL CourantFriedrichLewynumberDISS ArticialdissipationvectorDP Design-pointFDS FluxDierenceSplittingFVM FiniteVolumeMethodFVS FluxVectorSplittingGTAnalysis GasTurbineAnalysisIRS ImplicitResidualSmoothingMP Mixing-PlaneMPO Mixing-PlaneOutletLISTOFABBREVIATIONSANDACRONYMS xxivMPI Mixing-PlaneInletMUSCL MonotoneUpstream-CenteredSchemesforConservationLawsNGV NozzleGuideVaneODP O-design-pointRHS Right-HandSidesm SurgemarginSA Spalart-AllmarasSLC StreamlineCurvatureMethodVIGV VariableInletGuideVaneVISC ViscousuxvectorVSV VariablestatorvaneListofSymbolsLatinCharactersa SpeedofsoundAiPArameterforthearticialdissipationtermb Functionofinletair-angleARoeRoe-matrixC Convectiontermc BladechordCdDragcoecientCPPressurecoecientcpGasspecicheatatconstantpressurecvGasspecicheatatconstantvolumeD DiusiontermD2SeconddiusiontermD(Qi) ArticialdissipationtermsDest DestructiontermDISS ArticialdissipationvectorLISTOFSYMBOLS xxvie Totalenergyperunitofvolume,EulerE, F, G Fluxvectors

F ForceeiInternalenergy

F BodyforcesH Specictotalenthalpyi Incidenceanglei, j, k Directionsofthecoordinatesytem(i0)10Zero-camberincidenceangle(iDi2D) Correctiontoaccountfortwodimensionaleectsk ConstantusedintheVenkatakrishnanslimiterfunctionkblBlockagecoecientkThicknesscorrectionforzerocamberdeviationanglekiCorrectionfactorl0CharacteristiclengthL Stateofthecontrol-volumeontheleftm Slopefactorforminimumlossdeviation,MeridionalM MachnumberMn0InletabsoluteMachnumberatthebladeMniInletrelativeMachnumberatthebladeMncoptOptimumMachnumberatthebladeinletml Minimumlossconditionn Normalvector,timeintegration,slopefactorN RotationalspeedLISTOFSYMBOLS xxviiP StaticpressurePeSumoftheconvectiveuxesPtTotalorstagnationpressurePr Prandtlnumber,pressureratioPrtTurbulentPrandtlnumberProd ProductiontermqjHeattransfervectorQ ConservedvariablevectorR Gasconstant,Riemanninvariantsterms,Rotor,Stateofthecontrol-volumeontheright,RadiiRcRadiusofcurvaturer distancefromthecell/element-centroidtothefacemidpointofthecell/elementRe0ReynoldsnumberRe0Reynoldsnumbercalculatedbydimensionlessvariabless BladespacingS ConstantfortheSutherlandlawequation,Areavector,Magnetudeofvorticity,Distancealongbladeedges,Statort Time,Tangentialvector,ThicknessT StaticTemperature,TransitiontermTtTotalorstagnationtemperaturet/c Bladethickness-chordratioU PeriphericalvelocityuFrictionvelocityV Volume,AbsolutevelocityLISTOFSYMBOLS xxviiiVmMeridionalvelocityx, y, z coordinatesystemz Isthecalculationlocationalongthecompressoraxialdistancez0Isthereferenceaxiallocationy+TurbulentdimensionlessdistanceypDistancebetweentherstnodeonthewallandthewallsurfaceW RelativevelocityGreekCharacters Flowangle Bladeanglet Time-step parameterusedintheHartenentropycorrection,Parameteroftheimplicitresidualsmoothing,DeviationangleBoundary-layerdisplacementthickness(d/di)2DSlopeatreferenceincidence(0)10Zero-camberdeviationangleatreferenceminimumlossincidenceanglededucedfromlowspeedcascadedatafor10percentthick2ParameterusedintheVenkatakrishnanslimiterfunction,Bladedeectionangle1Meridionalstreamlineinclinationanglerelativetotheaxialdirection Freestreamcondition Streamlineslope,Deectionangle2Parameterofthesecond-orderarticialdissipationLISTOFSYMBOLS xxix4Parameterofthefourth-orderarticialdissipation Limiter Specicheatratio Bladesweep SeconddynamicviscositycoecientcConvectiveuxJacobian vonKarmanconstant Dynamicviscositycoecientn Normalvector kinematicviscositycoecient Gradientoperator Divergentoperator2Laplacianoperator Bladecamberangle,Flowanglerelativetotheboundary Modiededdyviscositycoecient Pressuregradientsensor SourcetermtTurbulentkinematicviscositycoecient AngularvelocitypTotalpressurelosscoecient Density BladeSolidity Angleofperiodicity,boundary-layermomentumthicknessijShearstresstensorLISTOFSYMBOLS xxx PressuregradientsensorSubscripts0 Referencevalue1 Inlet2 OutletB Blockagec CentrifugalCO Coriolise Eulerequationtermf Fluctuationtermghost GhostelementH Hublocationin Internal(adjacentneighbor)i, j, k Vectorcomponents,Directionsin internalk faceL Stateofthecontrol-volumeontheleftm Meridionalmin Minimummn Normaldirection,Timeintegrationstepneig Neighbornfaces NumberoffacesLISTOFSYMBOLS xxxiout Outletx, y, z Cartesiancoordinates,ComponentsofvelocitiesR Stateofthecontrol-volumeontheright,Radials Stallr Radiiv Viscousequationtermt FluctuationtermT Tiplocation(casinglocation)w WallSuperscripts1 Inlet2 Outlet+ Positiveeigenvalues Negativeeigenvalues Nominal Average

ReynoldsaverageuctuationsFavreuctuations Mass-averagemean Dimensionless1 Introduction1.1 ObjectivesThegoal ofthisworkistodevelopatool tocalculatetheoweldwithinaxial-owturbomachines. Alongwayis necessarytoreachanaccurate andreliable result onturbomachinerynumerical simulations, hencemanyotherobjectivesshouldbereachedbeforeunderstandingthecomplexprocessthatinvolvescompressorandturbinedesignandtheirperformanceanalysis. Withtheoweldcalculatedalongtheturbomachinerystreamwise, blade-to-bladeandspanwise, it is possibletoanalyzethephysical aspectsof theowatnearwall regions, theboundary-layerbehaviorandtheregionswithhighow losses. With these results, the compressor designer can enhance the compressor bladegeometry and rows matching aiming eciency improvement. In this work, ComputationalFluidDynamics (CFD) techniques areappliedtocalculatepressure, temperatureandvelocitydistributions of axial-owcompressor andturbinechannels. AcomputationalcodewritteninFORTRAN,wasdevelopedtosolvethethree-dimensionalNavier-Stokesequationsinsteady-statecondition.CHAPTER1. INTRODUCTION 331.2 MotivationGas turbines are used in various applications as: aerospace, marine and land vehicles.Industrial gasturbinestoenergygenerationarealsovastlyapplied. Theuseof turbo-machines is enormous. Anysmall gaininturbomachineryeciencyandperformancetranslatesintoamajoreconomicworldwide.Thethreemostimportantcomponentsof gasturbineengineare: compressor, com-bustionchamber andturbine. Thedesigner shoulddesignandtest eachoneof thesecomponentsusingexperimental andnumerical tools. Experimental processesgenerallyareveryexpensive. Computationalsimulationsareapromisingmeansforalleviatingthecostof thetime-consumingandexpensiveexperimental process. If thedesignerstartstheexperimentationuponanaccuratepreliminarydesign, thecostontestfacilitiesandthecomponentsmanufacturingcanbesignicantlyreduced. Thedevelopmentofnumer-ical tools increased dramatically as computers became more powerful in the last decades.TheGasTurbineGroupatITAhasdevelopedseveralcomputationalcodes. Acomputa-tional program, namedAxial FlowCompressorCode(AFCC), developedbyTomita[1]calculatesthepreliminarydesignofaxial-owcompressorsanditsperformance, becom-ingpossibletoobtainthecompressormap. Techniquestoimproveo-designoperation,suchasVariableInletGuideVanes(VIGV),VariableStatorVanes(VSV)andBleed-of-Valve(BOV)areimplementedinAFCC. Theseresultscanbeusedasinputdataforastreamline curvature program developed by Barbosa [2]. By using a streamline curvatureprogramitispossibletocalculateallcompressorgeometrydimensions(annulus,blades,diameters, length) using streamlines along the blade spanwise. A study of water injectiononthecompressorinletandofthecompressorpost-stallbehaviorhasbeendeveloped. AcomputationalcodedevelopedbyJesus[3]canbeusedtodesignandperformanceanal-CHAPTER1. INTRODUCTION 34ysisof axial turbines. TheGasTurbineGroupatITAalsohasanenginedeck, namedGTAnalysis, developedbyBringhenti[4]. Thisenginedeckiscapableofcalculatingtheengineperformanceof several gasturbinecongurationssuchasturbo-shaft, turbofan,turbojet,gasgeneratorwithfreeturbine,twinortriplespoolinstallationsforbothaero-nautical and industrial interests considering steady-state operation. Silva [5] implementedthetransientmoduleintheGTAnalysiscode. Arobustobject-orientedmodellingoftheenginedeckframeworkispresentedinreferences[6]and[7].Thegoalofthisworkistostudyaxial-owturbomachinesoweldusingCFDtech-niques. Twooptionsarepossible: eithertouseaCFDcommercialpackageortodevelopan in-house computational code. Considering that the historical of the Gas Turbine Groupat ITA is to develop their own computational tools for academic research, the last optionwaschosen. Withanin-houseCFDcodeotherresearcherswouldbeabletoimplementnewtechniquestoimprovethedesignprocess,toreducemachinetimeandtoallowopti-mizations.1.3 PreviousDevelopments1.3.1 CFDinAcademicResearchandIndustryCurrently, therearemanyCFDpackagesavailabletocalculatetheuidmotionforboth internal and external ows, with several congurations and installations. In particu-lar, for industry this is a very good option to obtain a fast problem solution with low cost.Overthelastdecades, majorprogresshasbeenmadeinareassuchasgridgeneration,turbulencemodelling,boundaryconditions,pre-andpost-processingcomputerarchitec-CHAPTER1. INTRODUCTION 35ture,amongothers. TheboundaryconditionsencounteredinturbomachineryareamongthemostcomplexinCFD[8].CFD provides a complementary tool for simulation, design optimization and, moreover,analysis of complex three-dimensional ows experimentaly inaccessible. For academy andindustrialpurposesitisapowerfultooltostudyaspecicowbehaviorasmulti-phase,ablationof re-entryvehicles, jets, ice formationat leadingedge of wings, componentcooling, porous-media, gas and oil extraction and process, thermal comfort, high-lift con-gurations, chemical reaction, weather forecast, amongothers. Most improvements ofCFDtechniquesandcodesareduetoacademiceortsorpartnershipbetweenacademiaandindustry.CFDhasbeenvastlyappliedtotheautomotiveindustry, mainlytoracingcars, toincreasethethermaleciencyofenginesandtoimprovethecaraerodynamics[9].Inaeronauticalandaerospaceareas,CFDhasbeenusedmoreandmore,forexampletosimulateaerodynamicimprovements. Withtheadvanceof computer hardwareandsoftware, numerical simulationsbecamewidelyusedaswell asexperimental tests. Thelast generation of CFD codes applied in aeronautical engineering are powered by an opti-mization technique [10] to improve the geometrical design of airfoils, fuselage and nacelles,decreasingdragandincreasingaerodynamiceciency. Acourseof PROPESA[11] ex-pressedthisdevelopmentas: Anelectronicversionof awindtunnel isemergingasausefultoolforaerodynamistis. CalledComputationalFluidDynamicsituseshigh-speedcomputerstogeneratemathematicallytheowofuidoveracomputer-designedmodel.Thecomputercananalysetheaerodynamicforcesontheaircraftssurface, quicklysortthroughalargenumberof possibledesignmodicationsandpresentthebestsolution.Energyconservationisoneof themaingoalsof wind-tunnel testingandCFDresearch.CHAPTER1. INTRODUCTION 36Commercial airlines in the United States consume about 40 billion liters of fuel every year;evenaoneper-centimprovementinfueleciencywouldsavemillionsoflitersoffuel.One example of the use of CFD as a design tool is the design of the A330/A340 Airbus[11]:Stringent Design Target: 10% reduction in cruise drag compared with the A310 and33%reductionincruisedragcomparedwiththeMD11.CFDDesign: 800dierentwinggeometriesanalysed. Time: 2years; Cost: 0.5million.WindTunnelDesign: 800dierentwinggeometriestested. Time: 150years;Cost:65million.1.3.2 CFDonGasTurbinesAhighperformance gas turbine depends oncompressor, combustionchamber andturbine eciencies. A typical turbo-fan engine with all components is show in Figure 1.1.Tosimulateall gasturbinecomponentssimultaneously, alargememorystorageandnumberofprocessorsarenecessary[12]. Generally,eachcomponentissimulatedindivid-ually[12, 13] includinginletandoutletducts, suchasthegasturbinenacelle[14, 15],[16, 17, 18]. Problems ingeometrical nacelledesignwill causeanengineperformancedegradation because the compressor eciency decreases due to the ow conditions at faninlet.Another point is that some ight conditions are very dicult to reproduce during thegas turbine tests in a test facility. With the advance of CFD many situations such as take-CHAPTER1. INTRODUCTION 37FIGURE1.1Gasturbineengineanditscomponentso, landingandcruisearecalculatedwithhighaccuracyandreliability. Testsinvolvinghighaltitudeandiceformationatnacellelipsurfaceoratwingleadingedgeareanotherexamplesofcomplexandexpensivetaskstobedoneexperimentally.CFD has been used to improve gas turbine components design, decreasing the internalow losses by components geometry optimizations. The position of the engine installationonanairplanecanbecarefullystudiedusingCFDmainlytounderstandtheproblemswithundesirabledragforcesgenerationcombinedwithparasitedragforcesgeneratedbythefuselage.Themixingowinaturbofanengineat theexhaust duct or noisesuppressor canbecalculateandoptimizedtodecreasetheowlosses. ThenoisefromfanbladeandfromexhaustsystemscanbecalculatedusingtheCFDtoolscoupledwiththeempiricalcorrelations. SomestudiesatRolls-Royce[19]involvingCFDongasturbineenginesusetheoptimizationoftheguidevanesinthebypassducttominimizeexcitationofthefanrotorusingsensitivitygradients.CHAPTER1. INTRODUCTION 381.3.3 CFDonAxialCompressorsandTurbinesModernmultistageturbomachinesarealreadytheresultof ahighlycomplexdesignprocess based on the designer expertise, supported by numerous computational resources(hardwareandsoftware)andstill nallydevelopedandimprovedbyvarioustests. Allturbomachinery components are aected by aerodynamic requirements, aerodynamic andstructural loads (pressure and temperature), heat transfer mechanisms and material prop-erties. Thestructuralloadsshouldbesatisedbythelifetargets. ModerntechniquestodesignturbomachineryneedadetailedworkdivisionasshowinFigure1.2. Theturbo-machinery eld has multidisciplinary nature [20] and a multidisciplinary engineering teambecomes indispensable. CFD has fostered an unied approach to turbomachinery analysisandisusedasdesigntool.Numerical MethodsApplied and Pure MathematicsManufacturing EngineeringControl SystemsAcousticFluid Mechanics:Inviscid FlowViscous FlowTurbulent FlowGas DynamicsCFDAero -Thermodinamic and Heat TransferSolid Mechanical and VibrationsMaterial ScienceTurbomachinery:ResearchDevelopmentDesignManufacturingMaintenceExperimentalFIGURE1.2MultidisciplinaryteamonturbomachinerydesignCHAPTER1. INTRODUCTION 39Gasturbinedesigncomponentsarebasedondesignbyanalysis,wherebythebladingisassessedusingCFDcodes. Duringthedesignstagethedesigner repeatedlyadjuststhebladegeometryandchanneluntilndingasuitablegeometrythatcombinesaccept-ableaerodynamicperformancewithlowstress levels, andit is practical andeconomictomanufacture. Thisprocessofcontinual renementofthematchgeometrycanbete-dious, expensiveandtimeconsuming, sinceateachstagethegeometrical dataforthenecessaryaerodynamicorstressanalysismustbedetermined. Optimizationtechniquessuch as genetic algorithms, neural network and inverse design can be used to enhance thepreliminary design. Eciency is probably the most important performance parameter formost turbomachines, speciallyfor gas turbines whether usedfor aircraft propulsionorforland-basedpowerplants,becausetheirnetpoweroutputisthereferencebetweentheturbineworkandthecompressorwork.Advances are still possible, not only in the eciency itself but also in the amount, andhencecost. Factorsinuencing eciency areextremely complexmainly forhighpressurecompressor due to the diculties during the rotor and stator rows matching. The adventofmodernnumericalmethodsappliedbothwholeenginesandtoindividualcomponents.Theowthroughturbomachinesisoneof themostcomplicatedintheeldof uiddynamics practice. Athree-dimensional oweldcalculationinsideturbomachineryisalongandcomplextask. Firstly, becauseitisnecessarytopreviouslyknowsomeim-portantpointsofthecompressoroperationtosetupcorrectparametersintheboundaryconditions. Toobtaintheseparameters it is necessarytoknowthemachinedesignedanditsoperational characteristics. Remindingthat, nowadays, withCFDtechniquesitisimpossibletodesignanaxialcompressor,howeveritispossibletoimprovethedesign.But, inmanyinstances, CFDistheonlytool availablethatprovidesdetailedoweldCHAPTER1. INTRODUCTION 40information. Actual testingof turbomachinerywithdetailedmeasurementsinrotatingbladerowspassagesiscumbersome, expensiveand, inmanycases, impossible. Athe-ory of three-dimensioanal ow of turbomachines using streamline, stream surface, streamfunctionispresentedbyXuet. al. [21]. Xuet. al., alsopresentthemethodsofsolvinginversedesignproblemonthestreamsurfaces[22].Theoweldwithinadvancedaxialcompressorsforbothaircraftpropulsionsystemsand industrial gas turbines, specically high bypass ratio turbofan engines, is characterizedbythepresenceofmixedsubsonic,transonicandsupersonicregions,shockwaves,shockboundary-layerinteractionsandeectsof three-dimensionalityandunsteadinessof theow. Theowwithinanaxial compressorhassomeimportantpointstobediscussedrelatedtoitsphysical aspects. Innature, thevelocityeldmovesfromahighpressureregiontoalowpressureregion. Thepowerrequiredtoincreasethetotal pressurefromthecompressorinlettothecompressoroutletisprovidedbytheturbine, inwhichthecompressor is attachedbyashaft. Certainly, duringthe energytransfer process theowlossesareinevitable[23, 24]. Isdiculttoquantifyseparatelyall owlosssources(secondaryow, prole losses, frictionlosses, owseparationmainlyat blade suctionsurface, prolelosses, shockwavelosses, owleakageatbladetipregion, endwall lossesandothers). Decreasingtheselossesmeanstoincreasethecompressoreciency.Current eorts from the CFD community have been very important to understand theow for both external and internal cases and, if possible, to change the compressor oweldbygeometrical congurations, decreasingtheentropygeneration. Flowpropertiesmaybecalculatedfor aprescribedcompressor geometryandaset of boundaryconditions,solvingthesystemofdierentialequationsusuaalycalledconservationequations(mass,momentumandenergy). However, thisworkisnottrivial. ThetoolsusedtodetermineCHAPTER1. INTRODUCTION 41the preliminarydesignshouldbe veryrobust [25, 26], accurate andreliable, becauseif thepressureratiocalculatedduringthecompressordesignisnotclosetotheactualpressureratio(obtainedinthetestfacility),theboundaryconditionswillbeunrealistic.An incorrect pressure ratio can take the compressor to operate in an instability range [27].Theowbehaviorwhenthecompressoroperateinaninstabilitypointiscompletelyunsteady. Dawes[28]presentsaworkbasedonunsteadyowsassociatedwiththeinter-bladerowinteractionsincompressorstages. Thecentrifugalandaxialcompressorswerestudied. Hence,allnumericaltreatmentgivenbetweentherowsinterfacemustbedier-entcomparedtothenumerical implementationforsteady-statecondition, includingthenumerical treatment at inow and outow boundaries due to the high ow perturbations.Compressorinstableoperationpushthemachinetoworkinsurgeorchokepoints.Eachcomponent,suchascompressororturbine,haveitsparticularities. Denton[29],describe the diculties to simulate high pressure compressor due to the numerical surgecausedbythe complexityinstart the CFDcalculations for compressors because thebladescanbeoperatingnearstall point. Hence, thecalculationmayfail asaresultofthetransient inducedbytheinitial guess rather thanbecauseof agenuinelyunstableoperatingpoint.IntheworkdevelopedbyDawes[30], ispossibletoobservethedicultiesinobtainboundaryconditionatcompressoroutletduetothecomplexphysicalaspects. Theow-eldinsideasingle-stageaxial compressorwascalculatedwithandwithoutthestator.Thevariationofstaticpressurealongtheoutletrotorspanwiseisclearlyandsometimestheimpositionof aconstant staticpressureor simpleradial equilibriumnot representcorrectlytheactualvaluesduetothetip-clearanceandstaticpressuredropatcasing.For unsteadyows inturbomachineryapplications Chen, CelestinaandAdamczykCHAPTER1. INTRODUCTION 42[31] presents twounsteadywake-bladerowinteractionmodels andarotor- stator un-steadyinteractionmodels. Thesewake-bladerowmodelshadproducedresultsthatwerequalitativeagreementwiththeresultsfromtherotor-statorinteractionmodelandquan-titativeagreement was reasonableandcouldbeimproveduponbyfurther developingimprovementsintheinletboundaryconditionsemployedinthetime-shiftmodel.Several researchers on turbomachine area participated in the AGARD Propulsion andEnergetics Panel set up Working Group 26 as reported in [32] to help to clarify the issuesinvolvingCFDappliedtoaxialcompressorsusingaswidearangeofcodesaspossibleoftwo representative single blade row cases: NASA Rotor 37 and an annular turbine cascadetested by DLR. This report presented a large amount of information about the dicultiesmainlytopredictthepressureloss(wasupto40%inerror). Somecodesshownin[32]wereunabletopredictcorrectlythehighlythree-dimensionalsecondaryow.CFDcodesaredevelopedspeciallytosolvespecicproblems[33, 34]. Forexample,internal or external ows, incompressible or compressible, low or high Reynolds numbers,withorwithoutchemical reactionsasdissociationsthatoccurinthehypersonicows,porous-media, unsteadyor steady-state, amongothers. There are specic theoreticalandnumerical formulationsforeachcase. Forcomplexgeometriesandowsthemeshgeneration and numerical implementation become more dicult. Some authors [35, 36, 37]studiedspecicmethodologies tocalculateregions as shockwaves withhighaccuracy.Segunpta [38] develop a new compact nite-volume scheme based on a ux-vector splittingdiscretization. Thisschemehasbeenanalyzedbymatrix-spectralanalysisdevelopedbyauthor.Yee [39] present amathematical formulationfor high-order schemes usingexplicitand implicit multidimensional compact high-resolution shock-capturing methods for EulerCHAPTER1. INTRODUCTION 43equations. For acompressor bladerow, athird-order accuratehigh-resolutionschemeispresentedby[40]. Inthiswork, athree-dimensional transonicowinsideanisolatecompressorbladerowispresented. TheENO3schemewiththelower-uppersymmetricGauss-Seidel (LU-SGS) algorithm is adopted to improve the computational eciency. Theresultsprovidesgoodresolutionof theshocksystemandthewakesinow. Yang[41],Lacor and [42],Chima and Liou [43] present other spatial dicretization schemes based onhigh-ordermethods.The progress towards three-dimensional ow solvers is reported by Frink [44] in detail,including turbulent viscous ows. In this work, Frink presents an unstructured grid solverthatupgradeitsformal accuracytosecondordertosolveNAvier-Stokesontetrahedralcells. Theaccuracyis increasedbyapseudoLaplacianweightedaveragingalgorithmwhichproducesrobustconvergenceandpermitshigh-orderboundaryconditions.Amethodtosolveturbulentowproblemsonthree-dimensional unstructuredgridswith cell-centered spatial discretization and an implicity backward-Euler time-step schemeis presented by Frink [45]. For turbomachinery ow simulations some modications in theNavier-Stokes equations are necessary as discussed in forthcoming chapters [46, 8, 47, 48,49,50]duetotheadditionaltermstorepresentthecentrifugalandCoriolisforces.CFD techniques applied to turbomachines are vastly used to study the oweld withincomponents such as the compressor and the turbine aiming optimization. For compressorandturbinecasessomespecicnumerical treatmentisnecessarybecausetwoframesofreferencemust beused: rotatingandnon-rotating. Arotor rowis treatedwithintherotating frame of reference and a stator row within the non-rotating (or stationary) frameof reference. Acomparisonabouttheuseof theCFDcodesdevelopedtocalculatetheconservation laws using these two dierent reference frames based on absolute and relativeCHAPTER1. INTRODUCTION 44velocitiescanbefoundin[51].In[52] itisreportedatwo-dimensional owthroughabladerow(withrotatingandstationarycascades) of acompressor fanusingaformulationbasedonblade-to-bladestreamsurfaceforonepassageofabladerow. In[53]itispresentedathree-dimensionalanalysisof theoweldwithinatwo-stagefuel turbineusedonthespaceshuttlemainengines.AbignetworkorganizedbytheEuropeanCommunitycalledQNET-CFD[54] pre-sentedseveralpublicationsofturbomachineryoweldsimulations[55]. Turbulentmod-ellingforturbomachinery[56,57]wasalsodiscussed[58].1.4 AuthorContributionThethreemainrequirementsofthesupervisorare:1. ThethesismustcontributewiththeimprovementoftheaxialdesignproceduresinuseintheGasTurbineGroupatITA;2. Thethesismightproducedatafortheestablishmentoflossmodelstobeincorpo-ratedtothestreamlinecurvaturecode,allowingadditionalbladeproles;3. DeveloptheCFDcodeinamodularbasistoserveasaplatformforfuturestudies.In the Master of Science program,the author worked with compressor design and o-designperformancecalculations. Hence,themeanlineandstreamlinecurvaturemethodswerevastlystudiedandapplied. Someresults of compressor designandperformanceanalysis calculatedusingAFCCprogramcanbe foundinreferences [1, 59, 60]. Theuse of the AFCC (Axial Flow Compressor Code) and GTAnalysis (Gas Turbine Analysis)CHAPTER1. INTRODUCTION 45programs to predict the gas turbine engine performance using VIGV (Variable Inlet GuideVanes) andBOV(Bleed-of-Valve) canbefoundin[61, 62]. TheAFCCprogramwasusedintheDoctoral ProgramintheFaculdadedeEngenhariadeITAJUBA-EFEItodevelop a module to optimize the design of axial compressors using Sequential QuadraticProgramming(SQP). Theresults arepresentedinreference[63] andtheenginecycleimprovement using this optimization technique in the axial compressor when it is installedin a gas turbine engine is presented in [64]. The most eective design charts for use in thepreliminarydesignprocessarethosethatallowtheengineertorapidlyidentifydesignsthatareunacceptableorprovideguidanceontheroutetowardsthedesignoptimization.Thepurposehereistodevelopatool tocalculatethree-dimensional oweldwithinaxial turbomachines. Thedevelopingprocessofthiskindofcomputational codeisverycumbersome. Anotheroptionwouldbetouseacommercial package. Thequestionis:whynottouseacommercialpackage?DesigntoolsthatcombineCFDandoptimizationtechniques,forexample,dragmin-imization, fuselageoptimizationandinverseproblemappliedtothebladedesign, needsomespecialattentionduringthenumericalimplementationprocessandsometimesonlywiththeCFDsourcecodeitispossibletoimplementthesetechniques. Inmostcases,turbomachinerydesignersmustunderstandspecicbehaviorsof themachinetodesignimprovementsinordertodecreaselosssources.Commercialpackagesarewidelyusedinindustry,alreadyvalidatedandveriedtheyare user friendly due to the Graphic User Interface (GUI). Several problems in the appliedengineercanbesolvedusingthesepackages. However, inacademia, foranautonomousresearchlaboratorythatworkswiththestate-of-artongasturbineenginedesign, itisof greatimportancetoownsourcecodesinordertoformhumanresourcescapableofCHAPTER1. INTRODUCTION 46continuingthescienticdevelopmentofthecomputationalcodes.DuringmanyyearstheGasTurbineGroupatITA,nowadaysCenterofReferenceonGas Turbines, has developed numerical tools to design and performance calculations of gasturbines components as compressors and turbines. Besides, an engine deck was developedtobecomepossiblethestudiesinvolvinggasturbineperformancewithcongurationsas: gas generator withfree turbine, turbo-shaft, turbo-fanandturbo-jet withsingle ormultiplespools. Thedesignof compressor or turbineis astressedinteractiveprocessbetweenthedesignerandthecomputationalcodeduetothehighnumberofparameterstobeanalyzedandthatitshouldbeconsideredtoobtainareliabledesign.InBrazil,gasturbinesisastrategicissuemainlyduetothestronggrowthinenergyconsumption. TheGasTurbineGroupatITAworriesabouttoolsdevelopingcapabletoassure success in the gas turbine components design process. This is the time to create thehumanresourcestoworkintheprojectsinvolvingnewfamiliesofenginesfortheenergyareaaiminghigheciency,reliability,lowcostandlowpollutantemissions.However, alongwayisnecessarytostarttheCFDcalculation. Firstly, theturbo-machinerypreliminarydesignwascarriedoutandlaterthedesignrenement. Obtainedthe turbomachinery geometrical dimensions from the design tools a Computer Aided De-sign(CAD)softwaresisusedtogeneratethegeometry. Generally, turbomachineshasacomplexgeometrymainlyonthecompressorbladeproles. Thiscausesmanydicultiesspendingalongtimeduringthemeshgenerationprocess.Astep-by-stepdesignprocedureis presentedinthis work. It is veryimportant tomention that to a complete understanding of the ow physics in compressors and turbinesthedesignershouldhaveasolidknowledgeongasturbinecomponentsoperation.CHAPTER1. INTRODUCTION 471.5 WorkOrganizationThisworkisdividedineightchapters.Chapteronedescribes the importance and motivation of applying CFD to simulateturbomachine ows. The researchareas of the Gas Turbine Groupat ITAhas beenpreviouslydescribed,whoseauthorsandworksarementionedinthisthesis.Chaptertwodescribesthetheoreticalformulationofthegoverningequations. Thischapter starts withtheideal gas formulation. Thegeneral equations of uidmechan-icsareobtainedusingbothReynolds-AverageNavier-Stokes(RANS)andFavreaveragedenitions. Allvariables,bothdependentandindependent,aretreatedindimensionlessform. A rotating-frame of reference is added to the momentum equations to quantify theadditional forces that occur intherotatingregion. Anone-equationSpalart-Allmarasturbulencemodelisalsoimplementedanddescribed.Chapter threedescribes the numerical formulation of the governing equations. Threediscretizationmethodsofthespatial integration(central-dierence, ux-vectorsplittingandux-dierencesplitting)andtwotimeintegration(Runge-KuttaandMacCormack)are presented. Damping functions as articial dissipation models to avoid numerical insta-bilities for centered schemes and numerical methods such as Implicit Residual Smoothing(IRS) and variable time-step, to accelerate the time-marching process are discussed. Theturbulence equation discretization is presented for both time and spatial integration. Theunstructured mesh treatment is briey commented. The implementation of the initial andboundaryconditionsandthetreatmentoftherotor-statorinterfacearealsodescribed.Chapterfourdescribesthestructureofthecomputationalcodeanditsimplemen-tationissues. Thestep-by-stepcalculationandtheimplementationprocessinvolvedinCHAPTER1. INTRODUCTION 48thesolverdevelopmentandsomecommentsandalsorecommendationsoftheauthorarepresented.Chaptervedescribes the code verication and validation. A step-by-step procedureis described to verify the convective and diusive momentum equations terms. The Eulerequationswerecalculatedforasupersonicnozzle, usingtwospatial integrationschemes(centered and upwind) and dierent pressure ratio values. A NACA0012 airfoil with zeroangle-of-attackwasalsosimulated, consideringthetransonicregimewithMachnumberequalto0.8. Thecenteredandupwindspatialintegrationschemeswerevastlytestedtoanalyzethecodenumericalstability. Theimplementationofdiusivetermsweretestedwithlaminar owonaat-platefor dierent articial dissipationmodels for centeredspatial integrationscheme. Theupwindschemewasalsocalculated. Theat-platenu-merical resultswerecomparedwiththeanalytical solutionof Blasius. ThenozzleusedintheEulerequationsvalidationwasalsosimulatedusingturbulentow. TheresultsofturbulentowsimulationinsidethenozzlewereimprovedontheregionafterthenozzlethroatwhencomparedtotheEulerequationsresults.Chaptersixdescribestheresultsforturbomachineryowsimulations. ThemeshgenerationprocessandthemeshtypesasH-grid, H-O-H-gridandO-gridarepresented.Casesinvolvingturbinesandcompressorsweresimulated. Aesingle-stageaxial-owtur-bine, asingle compressor rotor withlowaspect-ratioandafull multistage axial-owcompressordesignprocedureandCFDcalculationswerecarriedout.Thestudycaseinvolvingasingle-stageaxial-owturbinepresentstheinteractionofturbine design methodology and CFD techniques to improve the ow through the turbinerotor. Eulerandturbulentowwerecalculatedusingdierentmeshtypes(H-gridandO-grid). The ow on the stator outlet and rotor inlet is evaluated with the use of mixing-CHAPTER1. INTRODUCTION 49plane. Therotorbladegeometricalchangesaredescribedtosolvetheproblemofreverseow on the rotor suction side. After the geometrical modications an O-grid was generatedwithtip-clearancebetweenrotortipsurfaceandturbinecasingtoverifythetipleakagephenomena. DierenttypesofnumericalinitializationtostarttheCFDcalculationsarediscussed.Based on a 1MW gas turbine engine, a multistage axial-ow compressor was designedandanalyzedinfoursteps:Preliminarydesign: basedonthemeanlinetechnique. Thecompressorisdesignedanditsperformanceiscalculatedfordierentrotational speeds. Thegasturbineengine deck is used to calculate the engine running line on the compressor map. Toolsas VIGV(VariableInlet GuideVanes) andBOV(Bleed-of-Valve) areconsideredtoimprove the operationrange inthe o-designcondition. These analyzes arenecessarytostartthecompressordesignrenementusingthestreamlinecurvaturemethod.Streamlinecurvaturemethod: usedtocalculateallcompressorgeometricaldimen-sionsbasedonthestreamlinesdistributiononthebladeleadingandtrailingedgesfromhub-to-tip. Atthispoint, therotorandstatorblades, suchasinletandout-letbladeanglesincludingthecompressorannulusdimensionsaredetermined. Lossmodels(secondarylosses,frictionlosses,prolelosses,lossesduetoReynoldsnum-bereects)toaccounttheowviscouseectsinsidethecompressorarecalculatedand calibrated based on the designer expertise. A CAD software is used to generatethe3 Dviewofthecompressor.Meshgeneration: Once obtainedthe compressor geometry, the meshgenerationCHAPTER1. INTRODUCTION 50process is started and the computational domain is presented considering the use ofperiodicitycondition.CFD calculation: Based on the streamline curvature results the boundary conditionand numerical initialization process were carried out to start the CFD calculations.General aspectsbetweendesignmethodologyandCFDresultsarediscussedforallstudycasesinvolvingturbomachines.Chaptersevendescribes important comments and conclusions about the cases sim-ulated, their diculties andnumerical aspects. Abrief discussionis done about theimportanceoftheCFDsolverasresearchandteachingtool.Chaptereightdescribes the future implementations to improve the code robustnessfrompre-processingdatauntil post-processingresults. Thetechniques toimprovetheCFD solver robustness based on numerical aspects including the mesh generation processaredescribedbasedonturbomachineryowsimulations.2 MathematicalFormulation2.1 TheuidAimingattheowcalculationingasturbinespartslikecompressorsandturbines,theuidisusuallytheatmosphericairortheproductsofcombustionofoil-derivedfuelsinair. Itisobservedthattheamountof fuel burnedingasturbinesisaround2%oftheairmassowenteringthecombustionchamber,sothattheuidpropertiesinagasturbine can be considered similar to those of the air. At the usual ranges of pressures andtemperatures encountered in todays gas turbines, air behaves as a newtonian perfect gaswithtemperature-dependentspecicheatsandviscosity.Bearinginmindthatthetotalspecicenergy(byvolume)isgivenbye = _ei +12|W|2_, (2.1)anditfollowsthattheinternalenergy(ei)canbecalculatedbyei=e 12WiWi. (2.2)Fromtheperfectgasequationanduseful denitions, p=RT, cp cv=R, =cp/cvCHAPTER2. MATHEMATICALFORMULATION 52andei= cvT,itfollowsthatp = ( 1)ei; (2.3)assumedthatthespecicheatratioisknown,sothat =pRT . (2.4)Aniterativeprocessisneedtogettheapproximatevalueof . FromthecalculatedT,theuidpropertiescanbecalculated[65]:cp= cp(T) =8.9883655 104T4+ 1.0126885 1010T3++ 4.739514 108T23.2961648 105T, (2.5)cv= cv(T) = R cp(T). (2.6)ThemoleculardynamicviscosityiscalculatedbySutherlandlaw: = bT32(s +T), (2.7)where,b = 1.458 106ands = 110.4. ThetotalenthalpyisgivenbyH= ei +p+12|W|2= h +12|W|2. (2.8)TheheatconductioniscalculatedbyFourierlawandisrepresentedbytheheatux q : q = T= Prhxi, (2.9)CHAPTER2. MATHEMATICALFORMULATION 53wherePristhePrandtlnumberandisthethermalconductivitycoecientgivenby: =9.0534 1016T4+ 5.1523 1012T3+ 2.9869 108T2++ 8.5405 105T+ 2.6321 103(2.10)2.2 TheowHigh performance gas turbines require very fast uid ows inside the compressors andturbinesbladepassages. Bladepassagearechannelsthatdeecttheowusuallyaxedfollowedbyarotating(inturbines)orarotatingfollowedbyaxed(incompressors).Thisconstructioncharacteristiccausestheow, enteringthexedcascadethatfollowsarotatingone, tobelocallytransient, evensoitmaybeconsideredsteadyglobally, aswhen the turbomachine is run at constant speed. Gaps existing at the top of the rotatingpassagesallowuidowfromonepassagetotheother. Experiencehasshownthattheowishighly3 D, viscousandturbulent, sothat, inthiswork, itismodelledusingthefullequationsosconservationofmass, momentumandenergy, writteninarotatingframeof reference. Hence, therelationof absoluteandrelativevelocitiesisrepresentedbyFigure2.1.FIGURE2.1Schematicvelocityvectordiagramwhere

Wisthevectorofrelativevelocity,

V isthevectorofabsolutevelocityand

UCHAPTER2. MATHEMATICALFORMULATION 54isthevectoroftangentialvelocity,givenby

U= r. Hence,from2.1onecanobtain

V=

W+ r, (2.11)with r =________yx0________. (2.12)UsingtheEinsteinnotation,theconservationequationsaredenedbyMasst+xj(Wj) = 0, (2.13)Momentumt(Wi) +xi(WiWj) +pxiijxj+ = 0, (2.14)Energyet+xj[(e +p)WjijWi +qj] = 0. (2.15)Theviscousstressesareconsideredlinearandproportionaltotherateofstrain,andarecalculatedusingthenewtonianuidtheory,hence:ij= _Wixj+Wjxi_+i,j_Wkxk_, (2.16)where is the molecular dynamic viscosity and is the second coecient of viscosity. InthisworktheStokeshypothesisisused, hence= 2/3l. ThevariableinequationCHAPTER2. MATHEMATICALFORMULATION 552.14representssometermsrelatedtotherotationoftheuidowinrotatingchannels.DetailsofthesourcetermarepresentedintheAnnexA.Thecompleteformofthemomentumequationsare,therefore:t(Wx)+x(WxWx+pxx)+y(WyWxyx)+z(WzWxzx) = 2x+2Wy,(2.17)t(Wy)+x(WxWyyx)+y(WyWy+pyy)+z(WzWyzy) = 2y2Wx,(2.18)t(Wz) +x(WxWzzx) +y(WzWyyz) +z(WzWz +p zz) = 0, (2.19)whereistheangularvelocity.Analyzingthemagnitudesofthevariablesinvolvedintheaboveequationsitisseenthat they dier by several order of magnitudes. Bearing in mind their numerical solution,it is possible to avoid the numerical errors related to those dierences in numerical valuesbynormalizingthevariables. Thisisdonethroughthenon-dimensionalizationequations =0,Wi=Wia0, =0, x =xl0,T=TT0,t = ta0l0, y=yl0, e =ea20, =0, z=zl0, =0,P=PP0,and the Reynolds number is calculated by Re0= 0a0l0/0for internal ow computationsandRe0=0|

W|l0/0for external owcomputations withtheMachnumber M=|

W|/a0, wherethesubscripts0and meansreferential propertiesfromthestagnationconditions andfreestreamconditions, respectively. Thesenewdenedparameters areCHAPTER2. MATHEMATICALFORMULATION 56called =dimensionlessdensity, x=dimensionlesscoordinateofaspecicpoint,thesameruleisvalidforyandzcoordinates,Wi=dimensionlessvelocity,T=dimensionlesstemperature, e=dimensionlesstotalenergy, =dimensionlessmoleculardynamicviscosity, =dimensionlessthermalconductivity,t=dimensionlesstime,=dimensionlesssecondviscositycoecient,P=dimensionlesspressure,Re0=Reynoldsnumbercalculatedusingdimensionlessvariables.After the transformation of variables indicated above and collection of terms, a similarsetofequationsofconservationisobtained. Detailsarenotsuppliedhere.2.2.1 TurbulenceTurbulence as understood today states that the ow properties may be represented asthesumofmeanpropertyandauctuationoftheproperty:A = A +A

, (2.20)CHAPTER2. MATHEMATICALFORMULATION 57A =A +A (2.21)wheretheoverbarandtildestandsforthemeanandtheprimeanddoubleprimestandsfortheuctuationcomponentsofthepropertyA.Therearedierentwaystoextracttheaveragevalues. Inthiswork, theReynoldsAveraged Navier-Stokes (RANS) and Favre average methods are used, with the denitions,respectively,AReynolds=limT1T_t+TtAdT, (2.22)AFavre=1limT1T_t+TtAdT, (2.23)whereistheReynoldsaverageof(density)andTisatimeinterval.Following successful applications reported in the literature [66], in this work the aver-agingproceduresusethedenitionsWi= Wi +Wi,H=H +H,e = e +e,T=T+T, (2.24)qi= qi +q

i, = +

,p = P+p

,Applyingtheaveragesindicatedbyequations2.24tothenon-dimensionalizedequa-CHAPTER2. MATHEMATICALFORMULATION 58tions,thefollowingequationswillresult:t+xj(

Wj) = 0 (2.25)t(

Wi) +xi_

Wi

Wj_+Pxixj(ijWiWj) + = 0 (2.26)et+xj_(e +P)

Wj Wi(ijWiWj) Wi(ijWiWj2) +qj +Wjh_= 0(2.27)Equations2.25,2.26and2.27aresimilartoequations2.13,2.14and2.15exceptforthefollowingtermsassociatedwiththeturbulence:ij= WiWj=usuallynamedtheReynoldsStressTensor(RST),12WiWi=usuallynamesthekineticenergyofthe(turbulent)uctuations,qTi= Wih=usuallynamedthemoleculartransportofheat.After the non-dimensionalization and collection of the appropriate terms, it is possibletowritethisturbulent systemofpartialdierentialequationsinmatricialform Qt+ E x+ F y+ G z+ = 0, (2.28)whereQiscalledthevectorofconservedvariables, E, FandGtheuxvectorsandthesourcetermassociatedtotherotationoftheframeofreference.In equation 2.28 the over hat stands for non-dimensional variables and the overbar fortheaverages(RANSorFavre)asindicatedbyequations2.22and2.23.Theuxvectorsaremadeof non-viscous, viscous andturbulent parts, asindi-CHAPTER2. MATHEMATICALFORMULATION 59catedbyequations2.29,2.30and2.31E= Ee +Ev +Et, (2.29)F= Fe +Fv +Ft, (2.30)G = Ge +Gv +Gt, (2.31)The convective components are represented by the subscript e. The viscous componentscontainonlythetermswiththeviscosityandarerepresentedbythesubscriptv. Theturbulent components only the terms containing the uctuations and are represented bysubscriptt. Thematricialrepresentationsofthetermsare,thereforeQ =___

Wx

Wy

Wz e___, (2.32)Ee=___

Wx

Wx

Wx +P

Wx

Wx

Wx

Wx( e +P)

Wx___, (2.33)CHAPTER2. MATHEMATICALFORMULATION 60Fe=___

Wy

Wy

Wx

Wy

Wy +P

Wz

Wy( e +P)

Wy___, (2.34)Ge=___

Wz

Wz

Wx

Wz

Wy

Wz

Wz +P( e +P)

Wz___, (2.35)Et=___0 Wx Wx xx Wx Wy yx Wx Wz zx( e +P) Wx xf___, (2.36)Ft=___0 Wy Wx yx Wy Wy yy Wz Wy zy( e +P) Wy yf___, (2.37)CHAPTER2. MATHEMATICALFORMULATION 61Gt=___0 Wz Wx zx Wz Wy yz Wz Wz zz( e +P) Wz zf___, (2.38)Ev=1Re0___0 xx yx zx xx

Wx + yx

Wy + zx

Wz +_ lPrl+ tPrt_ ei x___, (2.39)Fv=1Re0___0 yx yy zy yy

Wy + yx

Wx + yz

Wz +_ lPrl+ tPrt_ ei y___, (2.40)Gv=1Re0___0 zx zy zz zz

Wz + zx

Wx + zy

Wy +_ lPrl+ tPrt_ ei z___, (2.41)CHAPTER2. MATHEMATICALFORMULATION 62 =___0 2 x + 2

Wy 2 y 2

Wx00___. (2.42)The variables that have already appeared have the same meaning (but non-dimensionalized).Duringthenon-dimensionalizationandaveragingof thevariablesotherdenitioncameout: = l +t, (2.43)whereistheactual coecientof viscosity, listhemolecularcoecientof viscosity,tistheturbulentcoecientofviscosity,PrlisthelaminarPrandtlnumber,PrtistheturbulentPrandtlnumber,assumedhereequalto0.9,istheratioofspecicheatsand xf= xxWx+ xyWy+ xzWz+ xx

Wx+ xx Wx+ xy

Wy xy Wy+ xz

Wz+ xz Wz,(2.44) yf= yyWy+ yxWx+ yzWz+ yy

Wy+ yy Wy+ yx

Wx yx Wx+ yz

Wz+ yz Wz,(2.45) zf= zzWz+ zxWx+ zyWy+ zz

Wz+ zz Wz+ zx

Wx zx Wx+ zy

Wy+ zy Wy.(2.46)2.2.2 TurbulenceModelThereis not enoughaordablecomputational power todayfor theowcalculationusingtheaboveequations, sincetocapturetheuctuations of theowproperties anCHAPTER2. MATHEMATICALFORMULATION 63extremelyrenedmeshisrequired. Therefore, toavoidthisshortcoming, itispracticetodaytomodel theturbulentpartsof theequations. Detailsof thephysicsandof theavailableturbulencemodels, despitenotbeingthescopeofthiswork, canbeformal in,forexample,references[66,67,68,69,70,8,71,72].Basedontheresultsofpublishedliteraturerelatedtothecalculationofowsintur-bomachines[25, 32], theone-equationturbulencemodel developedbySpalart-Allmaras[73]waschosenforthiswork.2.2.2.1 TheSpalart-Allmarasone-EquationTurbulenceModelReference[73]containsdetailsofthephysicsbehindthemodel,andofthemodel. Itisanequationdevelopedfortheeddyviscosity ,validforhighReynoldsnumber:D Dt= cb1[1 ft2]S + 1_ (( + ) ) +cb2( )2_cw1fwcb12ft2__ d_2+ft1U2.(2.47)TheturbulentkinematicviscosityTisgivenby:T= fv1, (2.48)wherefv1=33+c3v1, = ,S= S + 2d2fv2, (2.49)isthevonKarmanconstantandSismagnitudeofvorticity. ThereforeS=_2ijij, (2.50)CHAPTER2. MATHEMATICALFORMULATION 64whereij=12_Wixj+Wjxi_, (2.51)andfv2= 1 1 +fv1, fw= g_1 +c6w3g6+c6w3_16, g= r +cw2(r6r), (2.52)r = SK2d2, ft2= ct3exp(ct42), (2.53)ft1= ct1gtexp_ct2w2tU2(d2+g2t+d2t)_, gt= min_0.1,Uwtxt_, (2.54)where xtisthegridspacingalongthewallatthenominatedtransitionpoint,dtisthedistancebetweenthelocal pointandthetransitonpointandwtisthevorticityatthewallatthenominatedtransitionpoint. Theotherconstantsaregivenby:cb1= 0.1355 , =23, cb2= 0.622 , = 0.41,cw1=cb1+(1 +cb1), cw2= 0.3 , cw3= 2,cv1= 7.1 , ct1= 1 , ct3= 1.2 , ct4= 0.5.CHAPTER2. MATHEMATICALFORMULATION 65Thenon-dimensionalformofequation2.47is:D Dt=cb1[1 ft2]S +1_ (( + ) ) +cb2( )2_1Re0+_cw1fwcb12ft2__ d_21Re0+ft1U2Re0, (2.55)where =0/0,S=S + 2 d2fv21Re0, r = SK2 d21Re0. (2.56)All othervariablesaredimensionlessbydenition. Applyingthedenitionof materialderivativeD( )Dt=( )t+

W.( ), (2.57)and assuming that on the near wall region the velocity is small, the density variation canbeneglected. Thenon-conservativeformofequation2.47becomes t= cb1[1 ft2]S +1_ (( + ) ) +cb2( )2_1Re0+_cw1fwcb12ft2__ d_21Re0.(

W) +ft1U2Re0. (2.58)Theidentity

W.( ) = .(

W ) .

WwasusedtoobtainEquation2.58.In the equations above, subscript t stand for transition points, so that the correspond-ingtermsaresettozerooutsidethetransitionregion.At walls =0. For freestreaminitial condition is usuallytakenintheinterval[0, /10].The physical process of uid motion is governed by the Navier-Stokes equations whereCHAPTER2. MATHEMATICALFORMULATION 66unsteady, convectionanddiusiontermsareconsidered. Theturbulentuidmotionisrelated to production, diusion and dissipation of eddies terms including unsteadiness andconvection. Eachoweldproblemhasdierentphysicalaspects. Basically,intheinnerregion the small eddies are formed and predominant. Large eddies are formed in the outerregion. Thesmallesteddiesgrowandbecomelargeeddiesandtheinverseprocessoccurstoo. Thisphenomenoniscalledproductionofturbulence. Duetotheeddiesmotioninaturbulentow,thereisatransportofuidpropertiesthatincreasesthemixingprocess.This process is callededdydiusion. Duringthis mixingprocess the smallest eddiesdominatetheenergydissipation. Theseeddiesareformedduetohighvelocitygradients,sotheviscousforcesbecomeconsiderableandthesmallesteddiesaredestroyed.TheSpalart-AllmarasturbulencemodelpresentedinEquation2.58canbewritteninthefollowingform: t= C( ) +D( ) +Prod( ) +Dest( ) +T, (2.59)wheretheconvectiontermisC( ) = .(

W)2, (2.60)thediusiontermisD( ) =1_ (( + ) ) +cb2( )2_, (2.61)theproductiontermisProd( ) = cb1[1 ft2]S , (2.62)CHAPTER2. MATHEMATICALFORMULATION 67thedestructiontermisDest( ) =_cw1fwcb12ft2__ d_2, (2.63)andthetransitiontermisT= ft1U2. (2.64)Detailsofthedevelopmentofalltermsandthecalibrationofthemodel,suchasthechoiceofconstantvalues,arepresentedin[73].3 NumericalFormulationInthischaptertheequationsdevelopedinthepreviouschaptersarepreparedforthesake of numerical solution. Stability, consistencyandconvergenceproblems aredealtwith. Inaddition, methods for convergenceacceleration, meshtreatments, initial andboundaryconditionsfortherotor-statorinteractionarediscussed. Essential ingredientsforanaccurateandecientsolutionoftheoweldare:Governingequationsandturbulencetransportequations,Goodinitialconditions,Enforcementofboundaryconditions,Adequatemeshresolutionsandelementsquality,Numerical tools as articial dissipation, accurate discretization methods and properassessment,Ecientnumericalalgorithmdevelopment,Hardwarearchitecture.CHAPTER3. NUMERICALFORMULATION 693.1 DiscretizationModelsFinite volume has been reported as a method which gives good results for ow boundbycomplexgeometriesusingunstructuredmesh. Itisthemethodadoptedinthiswork.The reader may refer to [74, 55, 68, 75] if comparisons to other methods are of his interest.Dierentspaceandtimediscretizationsareusedinthiswork, aimingataplatformforthetest,comparisonandapplicationofknownandtobedevelopedmethods. Forthespatial discretization the methods of Jameson (centered), van Leer (Flux Vector-Sppliting)and Roe (Flux-Dierence Sppliting), are studied, while for the temporal discretization theexplicitMacCormack(1969)andtheRunge-Kutta(vesteps)arechosen. Itispossibletocombinespatialandtemporaloptions,resultinginseveraldierentmethods.Central schemescannotrecognizeandsuppresstheodd-evendecouplingofthesolu-tion. Hence, the so-called articial dissipation has to be added for numerical stabilization.It is based on a blend of 2ndand 4thdierences scaled by the maximum eigenvalue of theconvectiveuxJacobian. ThesecombinationofanundividedLaplacianandbiharmonicoperatorisemployedforunstructuredmeshes. Thedetailsaredescribedinsection3.4.1.Upwind schemes are constructed by considering the physical properties of the convec-tive terms (Euler equations). These treatment distinguish the upstream and downstreaminuencesbasedonwavepropagationdirections. Twoupwingschemesareimplementedinthiswork: Flux-VectorSpplitingandFlux-DierenceSppliting.TheSpalart-Allmaras methodis usedfor turbulent calculations. This methodwaschosenbasedonthe goodresults reportedinthe literature whenappliedtoows inturbomachines, theapplicationborninmindinthiswork. Thediusionandadvectiontermsarediscretizedaccordingthesuggestionsof[73];thetimediscretizationisimplicitCHAPTER3. NUMERICALFORMULATION 70andfollowingthesuggestionofthesamereference(Eulerimplicit).3.1.1 Finite-VolumeMethodNeglectingthesourceterm,thesuperscriptsofnon-dimensionalizationofRANSandFavreaverages,theequationsinconservative,matricialformhavetherepresentationQt+Ex+Fy+Gz= 0, (3.1)withE= EeEv, (3.2)F= FeFv, (3.3)G = GeGv, (3.4)(3.5)whereEe, FeandGeareEuler(convection)termsandEv, FvandGvaretheviscousterms.Integrating Equation 3.1 on a nite-volume and applying the Gauss theorem to trans-formthevolumeintegralintoasurfaceintegral:_VQt dV= _V(

.

P)dV= _S(

P.n)dS, (3.6)with

P= E

i +F

j +G

k, (3.7)CHAPTER3. NUMERICALFORMULATION 71Sisthecontrolsurfaceand nistheoutwardnormalvector. DeningthevalueofQbyQ =1V_VQdV, (3.8)itfollowsthatQV=_VQdV, (3.9)and_VQt dV=t( QV ) = V Qt . (3.10)SubstitutionofEquation3.10intoEquation3.6yields Qt= 1V_S(

P.n)dS. (3.11)Forthei thcontrolvolumetheEquation3.11isapproximatedbyQit= 1Vinfaces

k=1(Ek

i +Fk

j +Gk

k).

Sk, (3.12)where kindicates the face of i th volume, nfaces is the number of faces for each volumeand

Skistheoutwardorientednormal areavectorof kface. All uxesarecalculatedbasedonthefacekofani thvolume. Inthiswork, thevaluesofowpropertiesareconsideredtobelocatedatvolumecentroid.3.2 SpatialIntegrationThespatialintegrationofvectorsE,FandGforeachkfacegivenbyEquation3.12isdividedintwoparts: theterminvolvingconvectiveuxandtheterminvolvingtheCHAPTER3. NUMERICALFORMULATION 72diusiveorviscousux. Foreachi thvolumetheconvectiveandviscousterms, arerepresentedbyCONV (Qi)andV ISC(Qi),respectively.About the convective terms, to avoid numerical instabilities associated to the centeredmethodcausedbyhighgradients, articialdissipationorarticialviscosityisused. Ar-ticial dissipationactsasadampingfunctiontolimithighgradients[76] andisaddedtothemodelusedinthiswork. Upwindschemes[77,78,43,79]donotrequirearticialdissipationbecausetheuxiscalculatedfollowingaspecicdirection, hencenaturallydissipative. Thisnumerical treatmentdealswithproblemsinhighgradientregionsus-ingthevector-splittingmethodology[80,81,82,83,84,67]. Both,centeredandupwindmethodsareusedinthisworkandarepresentedinforthcomingsections.3.2.1 TheCenteredSchemeofJamesonThe centered discretization [76] of ux balance for each i th control volume is basedoninformationfromthecurrent volumeandits neighbors. Theuxis calculatedbysummationof uxesatall facesof i thcontrol volume. ThevectorsE, FandGaredividedintwoterms,oneforconvectivetermsandotherforviscousterms,CONV (Qi) =nfaces

k=1(Ee(Qk)

i +Fe(Qk)

j +Ge(Qk)

k).

Sk. (3.13)Thediscretizationisbasedonthearithmetical averagesforacurrenti thcontrolvolumeanditsneighborsneiatk thface. TheuxesforeachfacecanbewrittenasCONV (Qk) =12(Qk +Qnei).

Sk, (3.14)CHAPTER3. NUMERICALFORMULATION 73where nei stands for the neighbor volume. The vectors E, Fand G are written as functionsofthevectorofconservedvariablesQ.3.2.2 TheUpwindSchemeofVanLeer: FluxVectorSplittingTheuseof dampingfunctionslikearticial dissipationisavoidedwiththeFV Sfortheconvectiveterms. Thismethodisanupwindmethodthatisnaturallydissipative.StegerandWarming[85]developedthemethodfortheEulerequationssplittingtheeigenvalues of the Jacobian matrix. A disadvantage of this scheme is that the split uxesarenotcontinuouslydierentiableatthesonicregion,thusanon-smoothsolutionatthesonicregionmayappear withinthedomain. Toovercomethis problemvanLeer [83]proposedsomemodicationstotheStegerandWarmingscheme,detailsofwhichcanbefound in [80]. Basically the method uses a separation of the vectors Ee, Feand Gein twocontributionseach:Ee= E+e+Ee, (3.15)Fe= F+e+Fe, (3.16)Ge= G+e+Ge . (3.17)Contributions + and are associated with positive and negative eigenvalues, respectively.ForM 1andM 1thecalculationsarebasedontheuxforalldirections. Theschemewill bepresentedonlyforvectorEe. ForvectorsFeandGethedevelopmentoftermsissimilar.CHAPTER3. NUMERICALFORMULATION 74ForM 1:E+e= f, (3.18)Ee= {0 0 0 0 0}T, (3.19)andforM 1:E+e= {0 0 0 0 0}T, (3.20)Ee= f, (3.21)where,f=___Wx(W2x+a2/)WxWyWxWzWx[0.5(W2x+W2y+W2z) +a2/( 1)]___. (3.22)For 1 < M< 1thecalculationsarebasedontheforwarduxfforalldirectionsE+e= f+, (3.23)Ee= f, (3.24)CHAPTER3. NUMERICALFORMULATION 75where,f=___fmassfmass[( 1)Wx2a]/fmassWyfmassWzfmass{[( 1)Wx + 2a]2/[2(21)] + 0.5(W2y+W2z)}___, (3.25)withfmass= a[0.5(M 1)]2, (3.26)andthespatialintegrationfollowsthefollowingprocedure:Forpositiveoutwardnormalvectoratfacek:(Ee)k= E+e(Qi) +Ee(Qnei), (3.27)andfornegativeoutwardnormalvectoratfacek:(Ee)k= Ee(Qi) +E+e(Qnei). (3.28)Theconvectivetermiscalculatedby:CONV (Qi) =nfaces

k=1[(Ee)k

i + (Fe)k

j + (Ge)k

k].

Sk. (3.29)CHAPTER3. NUMERICALFORMULATION 763.2.3 TheUpwindSchemeofRoe: FluxDierenceSplittingTheFDSschemes evaluatetheconvectiveuxes at afaceof control volumefromleftandrightstatesbasedonideaof Godunov[86]. Several numerical schemesforthesolution of hyperbolic conservation laws are based on exploiting the information obtainedbyconsideringasequence of Riemannproblems. The exact solutionof the Riemannproblem, approximate Riemann solvers were developed by Osher et al. [77] and Roe [78].The methodology of Roe is applied quite often because of its high accuracy for boundarylayersandgoodresolutionofshocksregions.Inthiswork, RoesapproximateRiemannsolverwasimplementedintheframeworkof thecell-centeredscheme. It is basedonthedecompositionof theuxdierenceofconvective terms over a face of the control volume into an addition of wave contributions.At ak-face of the left (L) andright (R) states of acontrol volume the dierence isexpressedas(

Pe)R(

Pe)L= ( ARoe)k(

QR

QL), (3.30)whereARoedenotestheRoe-matrix. TheRoe-matrixisidentical totheconvectiveuxJacobian,c, wheretheowvariablesarereplacedbyRoe-averagedvariablespresentedbellowCHAPTER3. NUMERICALFORMULATION 77 =_ L R,Wx=WxLL +WxRRL +R,Wy=WyLL +WyRRL +R,Wz=WzLL +WzRRL +R, (3.31)H =HLL +HRRL +R, a =_( 1)( H q2),W =Wxnx +Wyny +Wznz, q2=W2x+W2y+W2z.With some algebra, inserting the diagonalisation of the Roe-matrix asARoe=T c T1decomposingtheRoesschemegivenbyequation3.30intowaveswhere(

Pe)R(

Pe)L=T c(

CR

CL), (3.32)where|c| = diag(|Wn|, |Wn|, |Wn|, |Wn| +a, |Wn| a)T. (3.33)ThematrixofeigenvectorsTandT1, aswell asthediagonal matrixofeigenvalues(c) are evaluated using equations 3.31. The characteristic variables

Crepresent the waveamplitudes, the eigenvaluesc are the associated wave speeds of the approximate Riemannproblem and theTeigenvectors are the waves themselves. The uxes of convective termsCHAPTER3. NUMERICALFORMULATION 78areevaluatedatK-facesofacontrolvolumeas(

Pe)k=12[

Pc(

QR) +

Pc(

QL) |ARoe|K(

QR

QL)]. (3.34)Theproductof |ARoe|andthedierenceoftheLandRstatesisgivenby|ARoe|(

QR

QL) = |

P1| +|

P2,3,4| +|

P5|, (3.35)where|

P1| = |W a|_p aW2 a___1Wx anxWy anyWz anzH a W__, (3.36)|

P2,3,4| = |W|____ p a___1WxWyWz q22__+ __0 WxWnx WyWny WzWnzWxWx +WyWy +WzWzWW_____,(3.37)CHAPTER3. NUMERICALFORMULATION 79|

P5| = |W+ a|_p + aW2 a___1Wx + anxWy + anyWz + anzH + a W__. (3.38)DiscretizationsbasedoncharacteristicsoftheEulerequationsseparatelyinterpolateow variables from the L and R states at face using non-symmetric formulae. The valuesof L and R state are used to calculate the convective ux through the face. Roes scheme isconsiderably more accuratethan centered scheme dueto the high resolution ofboundarylayers andthelower sensitivitytogriddistortions of theformer oneincomparisontocentral scheme. Thediscretizationmethodof theEulerequationsdevelopedbyRoeisoriginally of rst order. To increase the order of the above equations a reconstruction basedonMonotoneUpstream-CenteredSchemesforConservationLaws(MUSCL)approachis usedtoachievesecond-order accuracyas presentedinthesection3.2.4. Details ofMUSCLschemescanbefoundinreference[87].Equation 3.30 will produce an unphysical expansion shock in the case of stationary ex-pansion, in which

PcL=

PcR, but

QL =

QR causing the so-called carbuncle phenomenon,where the perturbation grows ahead of a strong bow shock along the stagnation line. Fur-thermore,theoriginalschemedoesnotrecognizethesonicpoint. Tosolvethisproblem,themodulusof theeigenvalues |c| = |W a| ismodiedusingtheentropycorrectionproposedbyHartenCHAPTER3. NUMERICALFORMULATION 80|c| =___|c|, if |c| > ;|c|2+22, if |c| ,(3.39)whereisasmall valuethatrepresentssomefractionof thelocal speedof sound. Inthis worktherangeusedis 0.05 0.15. Thesameentropycorrectionwas alsoimplementedtoW.3.2.4 Reconstruction Based on Approximate Monotone Upstream-CenteredSchemesforConservationLaws(MUSCL)Toreachthesecond-orderaccuracy, onepossibilityistousetheMUSCLapproachassuming that the solution to vary over the control volumes in a linear fashion. In order tocalculatetheLandRstates,a reconstruction oftheassumedsolutionvariation becomesnecessary. Inthisworkapiecewiselinearreconstructionwasimplemented. Hence, itisassumedthatthesolutionislinearlydistributedovercontrolvolume(Figure3.1). Withthis idea, the L and R states are calculated for the cell-centered scheme with the followingrelationsQL= QI+I(QI rL)QR= QJ+J(QJ rR) (3.40)where the QIand QJare the gradients at the cell-center I and Jrespectively, denotesa limiter function, riand rjare the distances from the cell-centroid to the face-midpointsofthecell.CHAPTER3. NUMERICALFORMULATION 81FIGURE3.1SchemebasedonpiecewiselinearreconstructionThis reconstructionmethodwas presentedbyBarthandJespersen[88] andcorre-sponds toaTaylor-series expansionaroundthe neighbouringcenters of the face of acontrol-volume. Theaboveschemerequiresthecomputationofgradientsatcell-centers.Thisisthepricetobepaidtouseahighorderupwindschemes. Limiterfunctionmustbeemployedinordertopreventthegenerationof spuriousoscillationsclosetostrongdiscontinuities. Limiters, in general, need the calculation of gradients also, requiring highcomputational eort. Inthis work, theVenkatakrishnanlimiter function[89, 90] wasimplementedandispresentedinsection3.2.5.3.2.5 VenkatakrishnansLimiterAs aforementioned, second and higher-order upwind spatial discretizations require theuseoflimitersinordertopreventthegenerationofoscillationsandspurioussolutionsinregions with high gradients as shock waves. The purpose of a limiter is to reduce the slopesusedtointerpolateaowvariableatthefaceofacontrolvolume, inordertoconstrainthesolutionvariations. Itisnotthescopeofthisworktodiscussandcomparelimiters.Reference[91]presentstheEulercomputationsusingexplicittimemarchingfordierentlimiterfunctionsasminmodandsuperbee. OtherlimiterfunctionwidelyusedbyCFDcode developers is the Barth and Jespersen limiter presented in reference [88]. Barth andJespersenslimiterenforcesamonotonesolution,butitisratherdissipativeandittendsCHAPTER3. NUMERICALFORMULATION 82to smear discontinuities. Furthermore, this formulation demonstrated the eectiveness oftheirmulti-dimensional limiterbycomputingoscillation-freetransonicowsolutionsonhighlyirregulartriangularmeshes.Inthis work, theVenkatakrisnans limiter functionwas implemented. This limiterpresents superior convergence properties. All details of this limiter function are describedinreference[89,90]. Theformulationofthislimiterfunctionimplementedinthiscodeispresentedbellowi= minj___12_(21,max +2)2 + 2221,max21,max + 222 + 1,max2 +2_, if2> 0;12_(21,min +2)2 + 2221,min21,min + 222 + 1,min2 +2_, if2< 0;1, if2= 0,(3.41)where1,max= QmaxQi,1,min= QminQi, (3.42)and2=12(Qi ri),Qmax= max(Qi, maxjQj),Qmin= min(Qi, minjQj), (3.43)whereQmaxandQminstandfortheminimumandmaximumvaluesonall neighborsofcontrol volume i including i itself; parameter 2is used to control the amount of limiting.CHAPTER3. NUMERICALFORMULATION 83Hence, setting 2equal to zero results in full limiting, but this may stall the convergence.Otherwise, if 2issettoalargevalue, thelimiterfunctionwill returnavaluearoundunity. In practice, it was found that 2should be proportional to a local length scale, i.e.,

2= (Kh)3, (3.44)whereKis aconstant set bytheuser andit is (1) andhis thecube-root of thevolumeof control volume. Inreference[90] thesolutiondependenceof several kvaluesarediscussed.3.2.6 DiscretizationofViscousFluxesTheviscoustermsareevaluatedusingcell-centeredschemeduetotheellipticnatureof thediusiveuxes, dynamicviscosity, heatconductioncoecientandstressestermsresulting:V ISC(Qi) =nfaces

k=1(Ev(Qk)

i +Fv(Qk)

j +Gv(Qk)

k).

Sk. (3.45)The discretization is based on the arithmetical averages for a current ith control volumeanditsneighborsneiatk thface. TheuxesforeachfacecanbewrittenasV ISC(Qk) =12(Qk +Qnei).

Sk, (3.46)where nei stands for a determined neighbor volume. The vectors E, Fand G are writtenasfunctionofthevectorofconservedvariablesQ.CHAPTER3. NUMERICALFORMULATION 843.3 TimeIntegrationThe used time integration scheme is explicit. The MacCormack (1969) and the Runge-Kutta(vesteps)schemeswereimplementedinthisworkandarepresentedbellow.3.3.1 TheSchemeofMacCormack(1969)The MacCormack scheme for time integration is based on a special version of the Lax-Wendromethod. Instabilityof centeredschemesisavoidedif predictor(forward)andcorrector(backward)stepsliketheusedbyVeuillot[34]forturbomachinerysimulations.Forpredictorstep,Q(n+1)i= Q(n)itiVi[CONV (Q(n)i)forwardV ISC(Q(n)i)forwardDISS(Q(n)i)forward].(3.47)Forthecorrectorstep:Q(n+1)i= Q(n)itiVi[CONV (Q(n+1)i)backwardV ISC(Q(n+1)i)backwardDISS(Q(n+1)i)backward].(3.48)Fortheupdate:Q(n+1)i=12[Q(n+1)i+Q(n+1)i]. (3.49)Constant Courant-Friedrichs-Lewy (CFL) number is used for the calculations of the localtime-stepconsidered,fromnton + 1timeindexes:ti= CFLlengthi(|

W| +ai). (3.50)CHAPTER3. NUMERICALFORMULATION 85The variable lenghticorresponds to the characteristic length of the volume, calculatedbased on the distance between the ith volume centroid and the closest face. The variableaiisthespeedofsoundforthei thcontrolvolume.The MacCormack time integration method presents a good convergence for 0 1 there is supersonic propagation of the information at the face, so thatCHAPTER3. NUMERICALFORMULATION 104ifqface n > 0thequantitiesatfaceareextrapolatedfrominteriorofcontrolvolume:face= in,(Wx)face= (Wx)in,(Wy)face= (Wy)in,(Wz)face= (Wz)in.Otherwise,ifqface