Citation
Inviscid Transonic Flow around a Sphere

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

Thesis/Dissertation Information

Degree:
Master's ( M.S.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering
Mechanical and Aerospace Engineering
Committee Chair:
Haselbacher, Andreas
Committee Co-Chair:
Balachandar, Sivaramakrishnan
Committee Members:
Mei, Renwei
Graduation Date:
12/19/2008

Subjects

Subjects / Keywords:
Aerodynamic coefficients ( jstor )
Amplitude ( jstor )
Autocorrelation ( jstor )
Cylinders ( jstor )
Drag coefficient ( jstor )
Entropy ( jstor )
Mach number ( jstor )
Shock waves ( jstor )
Transonic flow ( jstor )
Velocity ( jstor )
3d, behavior, cfd, euler, flow, inviscid, mach, modelling, rocflump, shocks, simulation, sphere, thesis, transonic, vortex, wake
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre:
Electronic Thesis or Dissertation
born-digital ( sobekcm )
Mechanical Engineering thesis, M.S.

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. ( en )
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.
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
Statement of Responsibility:
by Parag Karanjkar.

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Source Institution:
University of Florida
Holding Location:
University of Florida
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Copyright Karanjkar, Parag. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
12/31/2010
Resource Identifier:
664684913 ( OCLC )
Classification:
LD1780 2008 ( lcc )

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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