A compressible three-dimensional design method for

INTERNATIONALJOURNALOFNUMERICALMETHODSINFLUIDS,VOL.13,599-624(1991)ACOMPRESSIBLETHREE-DIMENSIONALDESIGNMETHODFORRADIALANDMIXEDFLOWTURBOMACHINERYBLADESM.ZANGENEHMechanicalEngineeringDepartment,UniversityCollegeLondon,TorringtonPlace,LondonWClE7JE,U.K.SUMMARYAfullythree-dimensionalcompressibleinversedesignmethodforthedesignofradialandmixedflowturbomachinesisdescribed.Inthismethodthedistributionofthecircumferentiallyaveragedswirlvelocityrvoonthemeridionalgeometryoftheimpellerisprescribedandthecorrespondingbladeshapeiscomputediteratively.Twoapproachesarepresentedforsolvingthecompressibleflowproblem.Intheapproximateapproachthepitchwisevariationindensityisneglectedandasaresultthealgorithmissimpleandefficient.Intheexactapproachthevelocitiesanddensityarecomputedthroughoutthethree-dimensionalflowfieldbyemployingafastfouriertransforminthetangentialdirection.Theresultsoftheapproximateandexactapproacharecomparedforthecaseofahigh-speed(subsonic)radial-inflowturbineanditisshownthatthedifferencebetweenthebladeshapescomputedbythetwomethodsiswellwithinthemanufacturingtolerances.Themethodwasvalidatedbycalculatingtheflowthroughadesignedhigh-speedradial-inflowturbinebyusingathree-dimensionalinviscidEulersolver.VerygoodcorrelationwasobtainedbetweenthespecifiedandcomputedTV&distributions.KEYWORDSRadialturbomachineryInversedesignPotentialflow1.INTRODUCTIONTherearetwomainapproachestotheproblemofaerodynamicdesignofturbomachineryblades,thedirectandtheinverseapproach.Inthedirectapproachtheflowiscomputedforagivenbladegeometry,whileintheinverseapproachtherequiredflowdistributionisspecifiedandthecorrespondingbladegeometryiscomputed.Inrecentyears,asaresultofdevelopmentincomputationalfluiddynamics,considerableprogresshasbeenmadeinthenumericalsolutionofthedirectproblemofturbomachinerydesign.Forexample,Denton'hasnumericallysolvedthethree-dimensionalEulerequationsofmotionandDawes'andHahetaL3havesolvedthe3DNavier-Stokesequations.Suchmethodsareofsubstantialvaluetothedesigner,whocanusethemtoanalysetheflowconditionsalongvanesandblades.Ideally,itshouldthenbepossibletomodifythebladeshapeiftheflowconditionsarenotthoserequired.Inpractice,however,therearedifficultiesindeterminingthedegreeanddirectionofanymodifications,whichdifficultiesarecompoundedbythefactthatachangeofbladeshapeatanylocationaffectstheflowatotherpartsoftheblade.Thisisparticularlysointhecaseofradialturbomachinery,wherethebladegeometryandtheflowfieldarecomplicatedandthree-dimensional.Asaresult,themostrationalapproachfordesigningradialturbo-machinerybladesistouseathree-dimensionalinversedesignmethod.0271-2091/91/150599-26$13.0001991byJohnWiley&Sons,Ltd.ReceivedAugust1990RevisedNovember1990600M.ZANGENEHAlargenumberofinversedesignmethodsareavailableintwodimensionsandarewidelyusedinthedesignofaxialturbomachineryblades(e.g.References4-7).Ideallyonewouldliketoprescribethepressureorvelocitydistributiononthepressureandsuctionsurfacessothatthebladesaredesignedwithoptimizedboundarylayers(e.g.References4and7).However,thistypeofdesignspecificationhasnocontroloverbladethicknessandcouldresultinanill-posedproblem.Forexample,Lighthil14foundthatitisimpossibletospecifythevelocitydistributiononthepressureandsuctionsurfacestogetherwiththeupstreamanddownstreamboundarycondi-tions.Asaresultsomeauthorsprefertoprescribethevelocity(orpressure)distributiononthesuctionsurfaceandthethicknessdistribution(e.g.Reference9,whereasothershaveproposedmethodsinwhichthebladepressureloadingisprescribedtogetherwithathicknessdistribution(e.g.Reference8).Inthreedimensions,additionalconstraintsonthechoiceofthedesignspecificationarerequiredinordertoavoidill-posedproblems.Forinstance,thevalueofthepressurealongthehubofaradialturbomachineryvaneisaffectedbythepressurealongtheshroudandasaresultitisnotpossibletospecifythepressureonthehubandshroudindependently.Themainimplicationofthisfactisthatveryfewthree-dimensionalinversedesignmethodsareavailableatpresent.Totheauthor'sknowledgeonlyfivethree-dimensionalinversedesignmethodshavebeenreportedintheliterat~re.~-'~Inallthesemethodseitherthebladeloadingorthecirculationdistributionisprescribedtogetherwithathicknessdistribution.However,allthesemethodsareaffectedbyshortcomingswhichlimittheirapplicationtoproblemsofpracticalinterest.ThemethodofZhaoetaL9islimitedtohigh-solidityblades,whilethemethodsofOckuroumuandMcCuneandFalcaoarelimitedtothedesignofannularcascadesofinfinitelythinbladeswithconstanthubandtipradiusandsmallcamberinincompressibleflow.AlthoughthemethodsofTanetal.,Borges13andGhalyandTan'4cancopewithhighlyloadedblades,theyarealsolimitedtoincompressibleflow.Thispaperisconcernedwiththedevelopmentofathree-dimensionalinversedesignmethodapplicabletoradialandmixedflowmachinesinsubsoniccompressibleflow.ThemethodisanextensionofHawthorneetal.'s15approachtothedesignprobleminwhichthebladesarerepresentedbysheetsofvorticitywhosestrengthisdeterminedbyaspecifieddistributionofcircumferentiallyaveragedswirlvelocityYV,,(directlyrelatedtotheboundcirculation27rrV0)definedaswhereBisthenumberofblades.Inthisway