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Inviscid Transonic Flow around a Sphere

Permanent Link: http://ufdc.ufl.edu/UFE0024101/00001

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Title: Inviscid Transonic Flow around a Sphere
Physical Description: 1 online resource (45 p.)
Language: english
Creator: Karanjkar, Parag
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: 3d, behavior, cfd, euler, flow, inviscid, mach, modelling, rocflump, shocks, simulation, sphere, thesis, transonic, vortex, wake
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Inviscid transonic flows governed by the Euler equations are computed. Botta performed an exhaustive analysis on transonic flows over a cylinder after Pandolfi and Larocca paved way for the understanding of transonic flows. Using their work as a foundation, our study was extended to the case of a sphere. Spatial discretization was carried out using finite volume method, upwind schemes were used to compute inviscid fluxes and the computations were performed by a fourth order accurate Runge-Kutta method. Two grids have been used for the analysis of the flow. The work concentrates on the analysis of the behavior of the flow and also the effects experienced by the sphere. Four cases of freestream Mach numbers of 0.6, 0.7, 0.8 and 0.95 have been investigated in this work. It is observed that the drag on the sphere rises greatly with the increase in the Mach number. The shocks forming on the sphere are much sharper in the higher Mach number cases and further in the downstream section. The nature of the flow becomes more periodic as the freestream Mach number increases.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Parag Karanjkar.
Thesis: Thesis (M.S.)--University of Florida, 2008.
Local: Adviser: Haselbacher, Andreas.
Local: Co-adviser: Balachandar, Sivaramakrishnan.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-12-31

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0024101:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024101/00001

Material Information

Title: Inviscid Transonic Flow around a Sphere
Physical Description: 1 online resource (45 p.)
Language: english
Creator: Karanjkar, Parag
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: 3d, behavior, cfd, euler, flow, inviscid, mach, modelling, rocflump, shocks, simulation, sphere, thesis, transonic, vortex, wake
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Inviscid transonic flows governed by the Euler equations are computed. Botta performed an exhaustive analysis on transonic flows over a cylinder after Pandolfi and Larocca paved way for the understanding of transonic flows. Using their work as a foundation, our study was extended to the case of a sphere. Spatial discretization was carried out using finite volume method, upwind schemes were used to compute inviscid fluxes and the computations were performed by a fourth order accurate Runge-Kutta method. Two grids have been used for the analysis of the flow. The work concentrates on the analysis of the behavior of the flow and also the effects experienced by the sphere. Four cases of freestream Mach numbers of 0.6, 0.7, 0.8 and 0.95 have been investigated in this work. It is observed that the drag on the sphere rises greatly with the increase in the Mach number. The shocks forming on the sphere are much sharper in the higher Mach number cases and further in the downstream section. The nature of the flow becomes more periodic as the freestream Mach number increases.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Parag Karanjkar.
Thesis: Thesis (M.S.)--University of Florida, 2008.
Local: Adviser: Haselbacher, Andreas.
Local: Co-adviser: Balachandar, Sivaramakrishnan.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0024101:00001


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Igratefullyacknowledgemyadvisor,Dr.AndreasHaselbacher,AssistantProfessorofMechanicalandAerospaceEngineeringattheUniversityofFloridaforhisconstantsupport.Ithankhimforhishelpfulinsightsandguidance.IwouldalsoliketothankDr.S.Balachandar,chairofMechanicalandAerospaceEngineering,forhisinvaluableguidanceinthisresearch.IthankhimandDr.RenWeiMeiforservingonmysupervisorycommittee.IalsoacknowledgethesupportprovidedbytheHigh-PerformanceComputingCenteratUniversityofFlorida.Lastbutnotleast,IthankmycolleaguesintheComputationalMultiphysicsGroup,especiallyManojParmar,fortheircontinuoushelpandencouragement. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 6 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 10 2PREVIOUSWORK ................................. 11 3SOLUTIONMETHOD ................................ 17 3.1OverviewoftheCode .............................. 17 3.2GridGeneration ................................. 20 3.3InitialandBoundaryConditions ........................ 21 3.4Perturbation ................................... 21 4RESULTS ....................................... 25 4.1FreestreamMachnumberofM1=0:6 .................... 25 4.2FreestreamMachnumberofM1=0:7 .................... 26 4.3FreestreamMachnumberofM1=0:8 .................... 26 4.4FreestreamMachnumberofM1=0:95 ................... 27 5CONCLUSIONANDSCOPEOFFUTUREWORK ............... 42 REFERENCES ....................................... 44 BIOGRAPHICALSKETCH ................................ 45 5

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Table page 3-1Timestepsforthecomputationofallcases ..................... 24 4-1Meanvaluesforthecoecientofforces ....................... 37 4-2Secondmomentsofabsolutevaluesfortheforcecoecients ........... 37 4-3Secondmomentsofuctuationsfortheforcecoecients ............. 41 6

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Figure page 2-1ResultforliftcoecientobtainedbyPandolandLarocca[ 3 ]forfullcylinderinM1=0:5andM1=0:6 ............................. 14 2-2ResultfordragcoecientobtainedbyPandolandLarocca[ 3 ]forfullcylinderinM1=0:5andM1=0:6 ............................. 14 2-3Divisionintothreetimeintervals;dragcoecientforcircularcylinderforowsM1=0:5andM1=0:98 .............................. 15 2-4AnalysisofCdandClplotsforM1=0:5byBotta[ 4 ] .............. 16 3-1Coarsegridusedinthecomputation:GridG1with33,034cellsonsphere ... 22 3-2Finergridusedinthecomputation:GridG2with154,210cellsonsphere .... 23 3-3Symbolicrepresentationofperturbationinducedinthiswork ........... 23 4-1ForcecoecientsforM1=0:6;blackcirclerepresentst=1:7s,thetimeatwhichsolutioncontoursareshownasingure 4-3 and 4-4 ............ 28 4-2AutocorrelationfunctionoftheforcecoecientsforM1=0:6 .......... 29 4-3FreestreamMachnumberofM1=0:6,G1,Machcontoursatt=1:7s 29 4-4FreestreamMachnumberofM1=0:6,G1,entropycontoursatt=1:7s 30 4-5ForcecoecientsforM1=0:7;blackcirclerepresentst=1:7s,thetimeatwhichsolutioncontoursareshownasingure 4-7 and 4-8 ............ 30 4-6AutocorrelationfunctionoftheforcecoecientsforM1=0:7 .......... 31 4-7FreestreamMachnumberofM1=0:7,G1,Machcontoursatt=1:7s 31 4-8FreestreamMachnumberofM1=0:7,G1,entropycontoursatt=1:7s 32 4-9ForcecoecientsforM1=0:8computedusingG1;blackcirclerepresentst=1:7s,thetimeatwhichsolutioncontoursareshownasingure 4-13 and 4-15 32 4-10ForcecoecientsforM1=0:8computedusingG2;blackcirclerepresentst=1:7s,thetimeatwhichsolutioncontoursareshownasingure 4-14 and 4-16 33 4-11AutocorrelationfunctionoftheforcecoecientsforM1=0:8,computedusingG1 ........................................... 33 4-12AutocorrelationfunctionoftheforcecoecientsforM1=0:8,computedusingG2 ........................................... 34 4-13FreestreamMachnumberofM1=0:8,G1,Machcontoursatt=1:7s 34 7

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35 4-15FreestreamMachnumberofM1=0:8,G1,entropyContoursatt=1:7s 35 4-16FreestreamMachnumberofM1=0:8,G2,entropyContoursatt=1:7s 36 4-17ForcecoecientsforM1=0:95computedusingG1;blackcirclerepresentst=1:7s,thetimeatwhichsolutioncontoursareshownasingure 4-21 and 4-23 36 4-18ForcecoecientsforM1=0:95computedusingG2;blackcirclerepresentst=1:7s,thetimeatwhichsolutioncontoursareshownasingure 4-22 and 4-24 37 4-19AutocorrelationfunctionoftheforcecoecientsforM1=0:95,computedusingG1 ........................................ 38 4-20AutocorrelationfunctionoftheforcecoecientsforM1=0:95,computedusingG2 ........................................ 38 4-21FreestreamMachnumberofM1=0:95,G1,Machcontoursatt=1:7s 39 4-22FreestreamMachnumberofM1=0:95,G2,Machcontoursatt=1:7s 39 4-23FreestreamMachnumberofM1=0:95,G1,entropycontoursatt=1:7s 40 4-24FreestreamMachnumberofM1=0:95,G2,entropycontoursatt=1:7s 40 8

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InviscidtransonicowsgovernedbytheEulerequationsarecomputed.BottaperformedanexhaustiveanalysisontransonicowsoveracylinderafterPandolandLaroccapavedwayfortheunderstandingoftransonicows.Usingtheirworkasafoundation,ourstudywasextendedtothecaseofasphere. Spatialdiscretizationwascarriedoutusingnitevolumemethod,upwindschemeswereusedtocomputeinvisciduxesandthecomputationswereperformedbyafourthorderaccurateRunge-Kuttamethod.Twogridshavebeenusedfortheanalysisoftheow. Theworkconcentratesontheanalysisofthebehavioroftheowandalsotheeectsexperiencedbythesphere.FourcasesoffreestreamMachnumbersof0.6,0.7,0.8and0.95havebeeninvestigatedinthiswork.ItisobservedthatthedragonthesphererisesgreatlywiththeincreaseintheMachnumber.TheshocksformingonthespherearemuchsharperinthehigherMachnumbercasesandfurtherinthedownstreamsection.ThenatureoftheowbecomesmoreperiodicasthefreestreamMachnumberincreases. 9

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Theowofacompressibleuidaroundabodycanbeclassiedintothreecategoriessubsonic,transonicandsupersonic.IfthefreestreamMachnumberisincreasedcontinuouslyfromzero,thetransonicrangebeginswhenthehighestlocalMachnumberreachesunity,andendswhenthelowestlocalMachnumberreachesunity[ 1 ].TheslowestMachnumberatwhichtheowoveranylocalregionreachessonicspeeds,isknownasthecriticalMachnumber.Thetransonicregimeischaracterizedbytheformationofpocketsofsupersonicowandrelativelyweakshocks.AppearanceoftheshockwavesonabodyleadstorapidincreaseinthedragcoecientwithincreasingMachnumber. Incontrasttothesubsonicandsupersonicregimeswheretheowmaybedescribedbylinearpartialdierentialequations,thetransonicregimeisalwaysnonlinear.Thismeansthatthestudyoftransonicowsalmostinvariablyrequirestheuseofacomputerandnumericalsolutions[ 2 ]. Acanonicalcaseofbasicinterestistheowovercylinderandspheres.Relativelylittleisknownaboutthetransonicinviscidowaboutcylindersandspheres.PandolandLarocca[ 3 ]pavedwaytotheunderstandingoftheinstabilitieswhichcharacterizetheowinthetransonicrange.Botta[ 4 ],furtheringonthisgroundwork,providedananalysisoftheentiretransonicrangeandofthebehavioroftheowoveracylinder.Continuingtothisknowledgebyconsideringthetransonicinviscidowoverthesphereisthecentralgoalofthisthesis.OurstudyconsideredowswithMachnumbersM1=0:6,M1=0:7,M1=0:8,M1=0:95areconsidered. 10

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Inthecontextofourstudy,twopriorcontributionsareofparticularimportance.PandolandLarocca[ 3 ]numericallyinvestigatedtransonicowsgovernedbyEulerequationsabouthalfandfulltwo-dimensionalcylindersforMachnumbersM1=0:5andM1=0:6.Thetoolusedintheirstudywasasecond-ordernite-dierencetime-dependentmethodbasedonablendof\lambda-scheme"and\ux-dierencesplitting"formulations[ 3 ].Theintegrationschemeadoptedisatwolevelalgorithmwhichachievessecondorderaccuracy.Twopolargridsusedforcomputationhad24and32intervalsintheradialdirection.Thehalfcylinderhad64intervalsinthecircumferentialdirectionwhilethefullcylinderhad128.Therewasnosteadycongurationobtained,insteadtheyobservedthedevelopmentofaperfectlyperiodicunsteadinesswhichwasmoreevidentatthehigherMachnumber.Theyalsoconclusivelyfoundthateventhesymmetric,fullcylindercongurationisunstable.Asymmetric,unsteadybutperiodicowswerefoundalongwiththeformationofeddiesbehindthecylinderwhichtrapthevorticity.Vorticitycanbecreatedininviscidowbycurvedshockwaves[ 4 ]ascanbearguedfromCrocco'stheorem.Perturbationwasinducedinthesolutionbythealgorithm.Eventhoughthealgorithmwassymmetric,thecomputer'sniteprecisioninducedslightasymmetry.Governingequationscankeepasymmetryundercontrolifthephenomenonisstable.Theobservationthatasmallperturbation(physicalornumerical)brokethesymmetryandshiftedtheoweldtowardstransversaloscillationsledPandolandLaroccatoconcludethatthecongurationofcirculatingbubbleseventhoughsymmetric,wasunsteadyandalsohighlyunstable. TheworkofPandolandLaroccaledtosomeimportantinsightsintothebehavioroftransonicowsaboutcylinder.Theyfoundsymmetricowsattheinitialstagesintime.Figures 2-1 and 2-2 showhistoryofcoecientsoftheliftanddragcomputedbyPandolandLaroccaforM1=0:5andM1=0:6.Itcanbeseenthatthestructure 11

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BuildingonPandolandLarocca'swork,Botta[ 4 ]alsostudiedtwo-dimensionalinviscidtransonicowaboutacircularcylinder.PandolandLaroccainvestigatedowsspecicallywithMachnumbersM1=0:5andM1=0:6ows.Bottafurtheredthisresearchtotheentiretransonicregime.Hisresultswereextremelyusefulinunderstandingthebehaviorofowintransonicregime.TheEulerequationswereintegratedwithatime-dependenttechniqueandwerebasedonahigh-resolutionnite-volumeupwindmethod[ 5 ].Botta'sspacediscretizationconsistedofpolargridsof32128,64256and128512cellsintheradialandcircumferentialdirectionsrespectively.Cellsonthecylindersurfacehadunityaspectratio.Bottaimplementedthehigh-resolutionnite-volumemethoddescribedin[ 6 ]insuchawaythatthesolutionwassymmetricforsymmetricaldata.Thecodewasthenusedtoinvestigatestabilityoftheowbyslightlyperturbingtheinitialcondition.Perturbationwasachievedbychangingthevaluesofdensityinoppositecellsadjacenttothexaxisontheouterboundary. Bottaanalyzedtheowatveryshort,short,andlargetimes.Suchadivisionwaspossibleduetotheintroductionoftimescales.ThedivisionwasperformedbyusingtimestsandtcrasseeninFigure 2-3 .Duetounitvaluesofpressureanddensity,thetimesseeninthegureshouldbedividedbyafactorof280togetphysicaltimeunderstandardconditions.Therstintervalbetweent=0andt=ts,ischaracterizedbyafastdecreaseofthedragcoecientintime.Thisintervalwascalledasveryshorttimes.Duringthesecondtimeintervalbetweentsandtcr,thedragcoecientisalmostconstantorslowlyoscillatestowardssomemeanvalue.Thisintervalisdesignatedasshorttimes.Att=tcr,thebreakdownofsymmetryoccursandtimet>tcrwasknownaslargetimes.Botta 12

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BottaobservedthatasMachnumberisincreased,thesolutionundergoestwotransitionsapartfromthebreakdownofsymmetry.Throughthersttransition,theregularperiodicowentersachaoticregime.Thistransitionisgermanetotheentiretransonicrange.Throughthesecondtransition,thechaoticowcomesbacktoanalmoststationarystate.Botta[ 4 ]alsoobservedthatthetimeofrsttransitionreduceswithincreaseinMachnumberoftheow.Also,owaboutthecylinderdependsqualitativelyonthevalueofMachnumberwhereasthebehaviorofowinshortandveryshorttimesdoesnotshowqualitativedependenceonMachnumber.ThevaluesoftwocriticalMachnumbersatwhichthetwotransitionsoccurredwerefoundtobeM1=0:6andM1=0:9,respectively.AninterestingobservationwasthatthesecondtransitionwasaccompaniedbyasuddenincreaseinStrouhalnumber.TheStrouhalnumberisadimensionlessnumberdescribingoscillatoryowsandcanbedenedasSt=fL Vwhere,fisthefrequencyofvortexshedding,ListhecharacteristiclengthandVisthevelocityofow.Figure 2-4 showsthehistoryofdragandliftcoecientsalongwiththerespectivespectraldiagrams.Itcanbeseenthatthedragcoecientremainsclosetozeroandtheliftcoecientiszerobeforetheonsetofasymmetry.Botta[ 4 ]discussedissuesregardingsensitivitytodiscretization,vortex-shock,andvortex-vortexinteractions.Hediscussedphysicalaspectsofirregular,apparentlyaperiodicanderraticbehavior.Thereisnosignicantworkonanalysisoftransonicowsaboutasphereandsuchananalysisisthecentralgoalofourstudy. 13

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ResultforliftcoecientobtainedbyPandolandLarocca[ 3 ]forfullcylinderinM1=0:5andM1=0:6 Figure2-2. ResultfordragcoecientobtainedbyPandolandLarocca[ 3 ]forfullcylinderinM1=0:5andM1=0:6 14

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Divisionintothreetimeintervals;dragcoecientforcircularcylinderforowsM1=0:5andM1=0:98 15

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AnalysisofCdandClplotsforM1=0:5byBotta[ 4 ] 16

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7 ]whichhasbeenusedinourstudysolvesthree-dimensionaltime-dependentcompressibleEulerequationsonunstructuredgrids.TheEulerequationarestatedas @xZ~Wd+I@~FcdS=0(3{1) Vectoroftheconservativevariables: Vectorofinvisciduxes: whereisthedensityofuid,u,vandwarethecomponentsofvelocityinx,yandzdirectionrespectively,Eisthetotalenergyperunitmass,V=~V~nisthevelocityalignedwithn=fnx;ny;nzgt,nx,nyandnzarethecomponentsofunitnormalvectorinx,yandzdirectionrespectively.pisthestaticpressure,Histotal(stagnation)enthalpy, SpatialRoe'sdiscretizationiscarriedoutusingthenite-volumemethod[ 8 9 ].InvisciduxesareapproximatedbyGodunov-basedupwindschemestoallowforcapturingofshockwavesandcontactdiscontinuities[ 10 ].FacestatesfortheuxcomputationareobtainedfromaWENOextrapolationbasedonleastsquaresgradientoperators 17

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9 ].Explicitstencilsarecreatedforthegradientoperators.InRocuMP,stencilsforgradientoperatorsarecreatedusingOctree-basedmethod[ 7 ].Thisisinitializedusingcellcentroidcoordinates,andqueriedwithlocationsatwhichthegradientoperatorsaretobeconstructed[ 8 ].UnsteadyowsarecomputedwiththefourthorderRunge-Kuttamethod. Toimposeboundaryconditions,itishelpfultoconverttheEulerequationsinanfnsrgreferenceframe: @t+@un @n(3{5) @s(3{6) @r(3{7) @t+@Hun whereaboundaryisconsideredsuchthatnormaldirectionnisperpendiculartothefaceands;rdirectionsaretangentialtotheboundary.Atanypointontheboundary,asystemofequationswithone-dimensionalinviscidrelationscanbespeciedlocallywhichusethederivativesoftheowvariablesinface-normaldirection.Timederivativesofconservativevariablesareexpressedintermsofwaveamplitudes(incomingoroutgoingwave)andspacederivativesintransversedirections.Foreachboundaryconditionspecied,thecorrespondingequationiseliminated.Usingone-dimensionalrelations,incomingwaveamplitudescanbeexpressedintermsofoutgoingwaveamplitudes.Characteristicamplitudescorrespondingtooutgoingamplitudesarecomputedusingone-sideddierence.Usingthesecharacteristicamplitudesandthenon-eliminatedEulerrelations,conservativevariablesnotimposedbyboundaryconditionsarefound.ThisisexplainedingreaterdetailintheRocuMPbook[ 7 ]. Eachboundaryiscomposedoffaces.Forcesonaboundaryarecomputedfromthesumofforcesandmomentsonthefacesofthatboundary.Forceonafaceiwithunit 18

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whereiisthedensity,viisthevelocityvector,piisthepressureandprefisthereferencepressure.Thecomponentsoftheforceare Non-dimensionalforcecoecientsaredenedby 2refV2refSref=(Cp;inx;i)Si 2refV2refSref=(Cp;iny;i)Si 2refV2refSref=(Cp;inz;i)Si whereCp;iisthepressurecoecientwhichisdenedasCp;i=pipref 2refV2ref. Forcecoecientsforanentireboundaryaresimplygivenbythesummationoftheforcecoecientsforthefacesonthatboundary: ThecodehasbeenrunonnumeroustestcasesconsideredfromBotta's[ 4 ]case.ManojParmar,researchassistantintheComputationalMultiphysicsgroupatUniversity 19

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11 ],amulti-block,structuredgridgeneratortoolwasusedtogeneratesphericalgridsoverthesphericalbody.Twotwelve-blockgridswerepreparedforcomputation,onebeingmorethanfourtimesnerthantheother.Thediameterofboundaryisthirtytimesascomparedtothediameterofsphereunderinvestigation.Figures 3-1 and 3-2 showboththegrids. Inbothcases,thesphericalbodyhasadiameterof2mandtheouterboundaryisaspherewithadiameterof60m.G1isacoarsemeshwith33,034cellsonthefaceofsphericalbodywhereasG2has154,210cells.Cellsonthespheresurfacewerecubesofdimension0.0105mand0.0209mrespectivelyinG1andG2.Thestretchingratiointhenormaldirectionwasapproximately1.1inbothcasesandthenumberofintervalsinradialdirectionwere52forG1and59forG2.Reasonfortherelativelylowstretchingratiowastomaintainaccuracynearthesphere.ThetotalnumberofcellsinG1is1,717,768andinG2is9,098,390.ComputationswhichwererunusingG1approximatelyrequired40hoursofcomputationaltimetocomplete2secondsofphysicaltime.G2consumedmorethanaweekofcomputationaltimetocomplete2secondsofphysicaltime.Hence,agridnerthanG2couldnotberunduetothecomputertimerestrictions. Thegridswerepartitionedinto256regionstohavebettercomputingeciency.WiththehelpofHigh-PerformanceComputingCenterattheUniversityofFloridathe 20

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3-1 showsthevaluesoftimestepusedforcomputationofallcases. 4 ]. Outersphericalboundaryisspeciedasafareldboundaryandhasstaticpressure,statictemperature,angleofattackandsideslipanglespecied.Flowisuniformandangleofattack(angleofowrespecttoxdirection)isthevariablechangedtosetinperturbationintheowasdescribedinsection3.4.Boundaryconditionusedforthesphericalbodyholdstheinviscidowconditionswithvelocitiesnormaltothesurfacebeingzero. 3 ]thatthesymmetriccongurationswithcirculationbubblesareunstableinnature.Anysmallperturbationtendstodestroysymmetryandshiftstheoweldtotransversaloscillations.Botta[ 4 ]alsoperturbedtheowtostudyunsymmetricalconditionbychangingvaluesofdensityinadjacentcells.Theideaofinducingperturbationtoobtainasymmetryinowhasbeencontinuedinourstudy.Perturbationisappliedaftertheinitialtransientsarecompletelygoneandmeanvaluesoftheforcecoecientsarereached.Thistypicallyhappensafter0.2seconds.Flowisthenperturbedbychangingtheangleofattack.Figure 3-3 showsthevaluesof 21

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Coarsegridusedinthecomputation:GridG1with33,034cellsonsphere angleofowwithrespecttoxaxis.At0.3seconds,theangleofattackischangedfrom0degreesto5degreeswithrespecttoxaxis.Thisowconditioniskeptconstantfor0.025seconds.From0.325secondsto0.35seconds,theowconditionisreversed,i.e,theangleofuniformowwithrespecttoxaxiswaschangedfrom5degreesto-5degrees.Hence,owisperturbedfor0.05secondsandhastwodierentvaluesofanglesofattackduringthisperiod.Angle5degreeswaschosenasarelativelysmallbutnottoosmallavalue.Durationoftheperturbationwasdecidedas0.05secondssincethatisthetimeitrequirestopropagatefromtheboundaryandoverthesphere.Thetimelengthsensurethattheentiresphereisexposedtoperturbation. 22

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Finergridusedinthecomputation:GridG2with154,210cellsonsphere Figure3-3. Symbolicrepresentationofperturbationinducedinthiswork 23

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Timestepsforthecomputationofallcases MachGridTimestep(s) 0.6G11.3e-050.7G11.25e-050.8G11.21e-050.8G24.8e-060.95G11.17e-050.95G24.65e-06 24

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Thissectionelaboratesonresultsobtainedthroughthecomputations.ItisknownthatcriticalMachnumberforthesphereisM10:6.ComputationswereconductedatMachnumbersM1=0:6,M1=0:7,M1=0:8andM1=0:95.Twogridswerepreparedforthecomputationsasexplainedinsection3.2.CoarsegridG1wasusedforaninitialinvestigationoftheow.FinergridG2wasusedforadeeperanalysisoftheowandit'seects.GridG2wasusedinthecasesofMachnumberM1=0:8andM1=0:95asinitialinvestigationwithG1showedsomeinterestingresults.Computationswerecarriedoutforaphysicaltimeoftwosecondswithperturbationasexplainedinsection3.4.Tables 4-1 4-2 and 4-3 comparethestatisticaldataofcoecientofforcesinallthefourcases.Statisticalmeasuresconsideredareexplainedasfollows, Firstmomentisdenedas CD=1 Secondmomentofabsolutevaluesisdenedas C2D=1 Secondmomentofuctuationsisdenedas 4-1 wasfoundtobethelowestamongstallcases.Figure 4-1 showstheforcecoecientsagainsttimeforthiscomputation.Perturbationhasaneectinyaxisbutitsteadiesafterawhileandthecaseisaperfectlysymmetricone.Figure 4-2 showstheauto-correlationfunctionofthesethreeplots.Thefactthatthesefunctionsarenearingzeroshowsthenonoscillatorynatureof 25

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4-3 showstheMachcontoursofowtakenatt=1:7sandFigure 4-4 showsthecorrespondingentropycontours.Symmetricnatureoftheowcanbeseen.Shockisveryweakinnaturewiththeextentofdisturbanceandwakeisverysmall.Thepointcorrespondingtot=1:7sisblackenedinFigure 4-1 .Computationshavebeenrunuptotwosecondsofphysicaltime.TocomparethistimewithBotta'sscalingthetimehastobemultipliedbyafactorof287. 4-5 showsthehistoryofforcecoecients.Thereisaslightriseintheamplitudeofoscillationsalongwithriseinthemeanvalueofdragcoecient.Tables 4-1 4-2 ,and 4-3 allshowthisriseinvalue.Theriseinforcecoecientinzaxisismoresignicantthaninydirectionindicatingasymmetry.SteadinessinoscillationsofforcecoecientsandlowmeanvaluestogethershowthattheshocksittingonsphereremainsmostlystationaryanddoesnotpossesswaveringcharacteristicsasseeninhigherMachnumbercases.Figures 4-7 and 4-8 showtheMachandentropycontoursrespectively.Also,theshockstandsatalmost90degreesfromstagnationpointandismuchsharperthanthepreviouscase.SymmetrycanbeobservedandwakedoesnotextendfarinthedownstreamregioneventhoughtheextentismuchlargerascomparedtoM1=0:6case. 4-1 .Forcecoecientsacrosstheothertwoaxesremainclosetozerointheirmeanvaluesthoughtheamplitudeofoscillationincreasedgreatly.Figures 4-9 and 4-10 showthehistoryofforcecoecientsincomputationswithG1andG2.HighervalueofthedragcoecientshowsdevelopingunsteadyforcesinxdirectionandpresenceofastrongershockasalsoseeninFigures 4-13 and 4-14 .Valuesofothertwoforcecoecientsbeingsolow,showthesimilarityinnatureoftheowinyandzaxes.Higheramplitudeof 26

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4-11 and 4-12 enhancesthefactthatthereisbetterperiodicityinowascomparedtolowerMachnumbercases.Figures 4-15 and 4-16 showthatthereisnosignicantvortexsheddingeventhoughaslightmovementofshockinwaveringfashionisobservedthroughtheuctuationinvaluesofforcecoecients.Figures 4-13 and 4-14 showthattheradialextentofwakeismuchlargerascomparedtohigherMachnumbercases.ThewakeislargerascomparedtolowerMachnumbercasesandmoreexpansiveascomparedtohigherMachnumbercases.ReferringtoTable 4-1 ,meanvalueofforcecoecientinzdirectionismorethanthemeanvalueofforcecoecientinydirectionindicatingthattheshockpositionisnotsymmetrical.MeanpositionoftheshockislocatedfurtherdownstreamascomparedtotheM1=0:7case.Figures 4-14 and 4-16 showMachandentropycontoursforthecomputationwiththenergrid.Table 4-1 showsthatthemeanvalueofCddoesnotchangemuchwiththechangeofgrid.ItmightbethecasethatG2isnotneenoughtohaveaccurateanalysisbutlookingatthesmalldierenceintwovaluesitcanbefairlyconsideredtobethecorrectsolution. 4-1 showsthatitreachesunityvalues.Figures 4-17 and 4-18 showtheincreaseinamplitudesofoscillationofforcecoecients.Table 4-3 showstheriseinvaluesofsecondmomentofuctuationsforliftcoecientsinyandzaxes.Theotherforcecoecientsremaininvicinityofzerobutamplitudesinthesecoecientswereobservedtobehigherascomparedwithpreviouscases.Figures 4-17 and 4-18 showthehistoryofforcecoecientsforcomputationsusinggridsG1andG2respectively.Valuesofforcecoecientsinyandzdirectionsshowthatthedierenceinrstmomentsofabsolutevalues(means)offorcecoecientsinyandzdirectionswasmuchlargerforthenergrid.Figures 4-19 and 4-20 showtheautocorrelationfunctionforforcecoecientsfor 27

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ForcecoecientsforM1=0:6;blackcirclerepresentst=1:7s,thetimeatwhichsolutioncontoursareshownasingure 4-3 and 4-4 computationsofbothgridsanddirectmoretowardsperiodicowcharacteristicsthananyothercase.UnlikeM1=0:8case,thiscaseproducedlargervaluesofforcecoecientsinydirectionthaninzdirection.Figures 4-21 and 4-22 showthatthelengthofshockismuchlargerinradialdirectionascomparedtotheM1=0:8case.Theshockpositionisverysharp,atasteepangleandevenfurtherdownstreamascomparedtothelowerMachnumberows.Also,thewakeismuchmoresharplydenedandlongerascomparedtolowerMachnumbercases. 28

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AutocorrelationfunctionoftheforcecoecientsforM1=0:6 Figure4-3. FreestreamMachnumberofM1=0:6,G1,Machcontoursatt=1:7s

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FreestreamMachnumberofM1=0:6,G1,entropycontoursatt=1:7s ForcecoecientsforM1=0:7;blackcirclerepresentst=1:7s,thetimeatwhichsolutioncontoursareshownasingure 4-7 and 4-8 30

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AutocorrelationfunctionoftheforcecoecientsforM1=0:7 Figure4-7. FreestreamMachnumberofM1=0:7,G1,Machcontoursatt=1:7s

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FreestreamMachnumberofM1=0:7,G1,entropycontoursatt=1:7s ForcecoecientsforM1=0:8computedusingG1;blackcirclerepresentst=1:7s,thetimeatwhichsolutioncontoursareshownasingure 4-13 and 4-15 32

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ForcecoecientsforM1=0:8computedusingG2;blackcirclerepresentst=1:7s,thetimeatwhichsolutioncontoursareshownasingure 4-14 and 4-16 Figure4-11. AutocorrelationfunctionoftheforcecoecientsforM1=0:8,computedusingG1 33

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AutocorrelationfunctionoftheforcecoecientsforM1=0:8,computedusingG2 Figure4-13. FreestreamMachnumberofM1=0:8,G1,Machcontoursatt=1:7s

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FreestreamMachnumberofM1=0:8,G2,Machcontoursatt=1:7s FreestreamMachnumberofM1=0:8,G1,entropyContoursatt=1:7s

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FreestreamMachnumberofM1=0:8,G2,entropyContoursatt=1:7s ForcecoecientsforM1=0:95computedusingG1;blackcirclerepresentst=1:7s,thetimeatwhichsolutioncontoursareshownasingure 4-21 and 4-23 36

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ForcecoecientsforM1=0:95computedusingG2;blackcirclerepresentst=1:7s,thetimeatwhichsolutioncontoursareshownasingure 4-22 and 4-24 Table4-1. Meanvaluesforthecoecientofforces MachGridCdCyCz Table4-2. Secondmomentsofabsolutevaluesfortheforcecoecients MachGridCdCyCz 37

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AutocorrelationfunctionoftheforcecoecientsforM1=0:95,computedusingG1 Figure4-20. AutocorrelationfunctionoftheforcecoecientsforM1=0:95,computedusingG2 38

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FreestreamMachnumberofM1=0:95,G1,Machcontoursatt=1:7s FreestreamMachnumberofM1=0:95,G2,Machcontoursatt=1:7s

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FreestreamMachnumberofM1=0:95,G1,entropycontoursatt=1:7s FreestreamMachnumberofM1=0:95,G2,entropycontoursatt=1:7s

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Secondmomentsofuctuationsfortheforcecoecients MachGridCdCyCz 41

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Thissectionconcludesthestudyoffourcasessimulated.Meanvalueofthedragcoecient(Table 4-1 )isobservedtoriseastheMachnumberisincreased.Tables 4-2 and 4-3 showthatamplitudesofoscillationincreaseinthecaseofdragcoecientwithincreaseintheMachnumber.Amplitudesofoscillationofliftcoecientinyandzdirectiondonotchangemuchascomparedtothedragcoecient.Hence,theunsteadyforcesexperiencedbyspherearelargerincaseoftransonicowswithhigherMachnumbers. PositionoftheshockmovesfurtherdownstreamasMachnumberofthetransonicowisincreased.TheshockisobservedtobeginsignicantwaveringmovementfromM1=0:8onwardsasseenbytheoscillationinliftcoecients.RadialextentoftheshockincreaseswithMachnumberofow.Also,thereisnospeciclocationfortheshockonspherewhichisreasonablesincethebodyunderconsiderationiscompletelysymmetric.ExtentofwakeincreasesbuttheoutwardspreadofwakereduceswiththeincreaseinMachnumberofow.IncaseofM1=0:6andM1=0:7theregionofwakeiswidespreadbutveryshort.IncaseofM1=0:95,theregionofwakeislongandverysharpandrestrictedbehindthesphere. Thechangesingridallowtohaveanerobservation.However,thenatureofbehaviordoesnotchange.PeriodicnatureoftheowwasapparentinhigherMachnumbercases.TheautocorrelationplotsinFigure 4-20 showsmuchbetterperiodicnatureascomparedtotheplotinFigure 4-19 .Thisshowsthattheresultscomputedusinganergridaremoretowardsthepredictednatureofow.Hence,GridG2isabettergridtounderstandthebehaviorofowascomparedtogridG1. Furtherworkcanbeachievedbyusingnergridsintheanalysis.G2isabettergridbutduetothetimeconstraintsthatcomewithnergrids,itwasnotpossibletorunanergrid.Itisshownthatnergridhelpstounderstandtheowbetter.Anevenner 42

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TheanalysiscanalsobewidenedtoentiretransonicrangebystudyingmorecasesofdierentintermediateMachnumbers.TransformationsthattheowcharacteristicsundergointherangebetweenM1=0:7andM1=0:95canbestudiedingreaterdetail.Thiswillprovideanexhaustiveanalysisofthetransonicowaboutasphereanddeepentheunderstandingofthetransonicregime. 43

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[1] H.W.LiepmannandRoshko,Elementsofgasdynamics(1985),Doverpublications,Inc.,NewYork:Mineola. [2] J.D.Anderson,Jr.,ComputationalFluidDynamics(1995),McGrawHillinternationaleditions,Singapore. [3] M.PandolandF.Larocca,Transonicowaboutacircularcylinder,ComputersandFluids,17(1)(1989)205-220. [4] N.Botta,Theinviscidtransonicowaboutacylinder,J.FluidMech,301,(1995)225-250. [5] E.F.Toro,RiemannSolversandNumericalMethodsforFluidDynamics(1997),Springer,Berlin,Germany [6] N.Botta,Numericalinvestigationsoftwo-dimensionalEulerows:cylinderattransonicspeed,SwissFederalInstituteofTechnology,Zurich,Diss.ETHNo.10852. [7] A.HaselbacherTheRocuMPBook.1.13.0(2008)DepartmentofMechanicalandAerospaceEngineering,UF,Gainesvillle,Florida. [8] A.Haselbacher,OnConstrainedReconstructionOperators,AmericanInstituteofAeronauticsandAstronautics,(2006)1274. [9] A.Haselbacher,AWENOReconstructionalgorithmforunstructuredgridsbasedonexplicitstencilconstruction,AmericanInstituteofAeronauticsandAstronautics,(2005)0879. [10] A.HaselbacherandS.Balachander,S.W.KieerAmericanInstituteofAeronauticsandAstronauticsJournal,45(8)(2007). [11] Programdevelopmentcompany,Gridpro,Agridgeneratingtool(2008),v4.5,NewYork,USA,www.gridpro.com. 44

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ParagVasantKaranjkarwasborninPune,Indiain1985.HegraduatedfromJnanaPrabodhiniHighSchoolin2001andAbasahebGarwareJuniorCollegein2003.HeobtainedhisBachelorofEngineeringinmechanicalengineeringfromBirlaInstituteofTechnology,Ranchi,India,inMay2007.HeenteredthegraduateprogramattheUniversityofFloridainAugust2007.HeworkedunderDr.AndreasHaselbacherintheDepartmentofMechanicalandAerospaceEngineeringtoobtainhisMasterofSciencedegree. 45