Citation
Effects of Compressibility on the Unsteady Inviscid Flow over a Cylinder Close to a Wall

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Title:
Effects of Compressibility on the Unsteady Inviscid Flow over a Cylinder Close to a Wall
Creator:
Kiran, Rohitashwa
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:
Acceleration ( jstor )
Aerodynamic coefficients ( jstor )
Aerodynamic lift ( jstor )
Compressible flow ( jstor )
Cylinders ( jstor )
Drag coefficient ( jstor )
Mach number ( jstor )
Mass ( jstor )
Mechanical engineering ( jstor )
Velocity ( jstor )
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre:
Electronic Thesis or Dissertation
born-digital ( sobekcm )
Mechanical Engineering thesis, M.S.

Notes

Abstract:
Unsteady forces on a cylinder near a plane, rigid wall were studied in subcritical compressible flow. In an incompressible flow, the presence of the wall serves to raise the drag and inertial coefficients exponentially with increasing proximity to the wall. The unsteady, inviscid force is then the added mass force, which in essence is the force required to displace a quantity of fluid equal to the volume of the cylinder. When we consider a compressible flow, the unsteady force rises gradually, and reaches a constant value after a certain time determined by the acoustic propagation speed. In this region, an unsteady force coefficient can be defined which does not depend on the value of the acceleration, so long as the acceleration does not significantly change the Mach number of the flow. The Euler equations were solved in a frame of reference attached to the cylinder, for different free stream Mach number values and separations from the wall. Presence of the wall raises this coefficient to values greater than those in the absence of a wall. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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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.
Statement of Responsibility:
by Rohitashwa Kiran.

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University of Florida
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University of Florida
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Copyright Kiran, Rohitashwa. 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.
Resource Identifier:
664680106 ( OCLC )
Classification:
LD1780 2008 ( lcc )

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Iamincrediblygratefultomyadvisor(Dr.AndreasHaselbacher,AssistantProfessorofMechanicalandAerospaceEngineeringattheUniversityofFlorida),forhisconstantguidanceandsupport.ImustacknowledgeDr.S.Balachandar(ChairoftheMechanicalandAerospaceEngineeringDepartment),forhisgeneroussupportatseveraltimes.IthankDr.RenweiMeiforhishelpfulcriticismsandsuggestionsasamemberofthesupervisorycommitteeforthisthesis.IamalsodeeplyindebtedtomylabmatesattheComputationalMultiphysicsGroup,notablyManojParmar,fortheircontinuoushelpandeverybitoftheirvaluableinput.SincerethanksarealsoduetoCharlesTaylorandtheUniversityofFloridaHighPerformanceComputingCenterforenablingmetorunallthecomputationsinvolvedecientlyandsmoothly. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 6 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 10 2INCOMPRESSIBLEFLOWANALYSIS ...................... 13 2.1ComplexPotential ............................... 13 2.2PressureCoecient ............................... 16 2.3DragandLiftCoecients ........................... 16 3NUMERICALSIMULATIONS ........................... 21 3.1RocuPackage ................................. 21 3.1.1GoverningEquations .......................... 21 3.1.2BoundaryConditions .......................... 22 3.1.3ForceComputation ........................... 25 3.2GridGeneration ................................. 26 3.3SolutionMethod ................................ 26 4RESULTS ....................................... 30 4.1GridIndependence ............................... 30 4.2SteadyStateSolution .............................. 30 4.2.1DragCoecient ............................. 30 4.2.2LiftCoecient .............................. 30 4.2.3PressureCoecient ........................... 31 4.2.4FlowField ................................ 31 4.3AcceleratingPhase ............................... 31 4.3.1DragForce ................................ 31 4.3.2LiftForce ................................ 31 5CONCLUSION .................................... 42 REFERENCES ....................................... 43 BIOGRAPHICALSKETCH ................................ 45 5

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Table page 3-1Matrixoftestcases .................................. 28 6

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Figure page 2-1Referenceframefortwocylindersmovinginauidatrest ............ 18 2-2Notationusedfortheproblem ............................ 18 2-3Streamlinesforinviscidowovertwocylindersclosetoeachother ........ 19 2-4Variationofpressurecoecientwithseparationfromthewall .......... 19 2-5Variationofliftcoecientwithseparationfromthewall ............. 20 2-6Variationofaddedmasscoecientwithseparationfromthewall ........ 20 3-1Aviewofthenest2Dgridaroundthecylinder .................. 28 3-2Aviewofthenest2Dgridintheregionbetweenthecylinderandthewall .. 29 3-3Boundariesintheoweld ............................. 29 4-1Gridindependence:Dragcoecient(L=D=2,M1=0:2) ............ 32 4-2Gridindependence:Dragcoecient(L=D=2,M1=0:3) ............ 32 4-3Gridindependence:Liftcoecient(L=D=2,M1=0:2) ............. 33 4-4Gridindependence:Liftcoecient(L=D=2,M1=0:3) ............. 33 4-5Gridindependence:Pressurecoecient(L=D=2,M1=0:2) .......... 34 4-6Gridindependence:Pressurecoecient(L=D=2,M1=0:3) .......... 34 4-7Dragcoecientvariationwithseparationfromwall ................ 35 4-8Liftcoecientvariationwithseparationfromwall ................. 35 4-9Pressurecoecientvariationwithseparationfromwall .............. 36 4-10Pressurecontoursaroundthecylinder ....................... 36 4-11LocalMachnumbercontoursaroundthecylinder ................. 37 4-12Totalpressurecontoursaroundthecylinder .................... 37 4-13TimeevolutionoftheunsteadydragforcecoecientoverarangeofvaluesforL=D=2,M1=0:2 ............................... 38 4-14TimeevolutionoftheunsteadydragforcecoecientoverarangeofvaluesforL=D=1,M1=0:2 ............................... 38 7

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............................... 39 4-16TimeevolutionoftheunsteadydragforcecoecientoverarangeofvaluesforL=D=1,M1=0:3 ............................... 39 4-17TimeevolutionoftheunsteadyliftforcecoecientoverarangeofvaluesforL=D=2,M1=0:2 ................................. 40 4-18TimeevolutionoftheunsteadyliftcoecientoverarangeofvaluesforL=D=1,M1=0:2 ...................................... 40 4-19TimeevolutionoftheunsteadyliftforcecoecientoverarangeofvaluesforL=D=2,M1=0:3 ................................. 41 4-20TimeevolutionoftheunsteadyliftforcecoecientoverarangeofvaluesforL=D=1,M1=0:3 ................................. 41 8

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Unsteadyforcesonacylindernearaplane,rigidwallwerestudiedinsubcriticalcompressibleow.Inanincompressibleow,thepresenceofthewallservestoraisethedragandinertialcoecientsexponentiallywithincreasingproximitytothewall.Theunsteady,inviscidforceisthentheaddedmassforce,whichinessenceistheforcerequiredtodisplaceaquantityofuidequaltothevolumeofthecylinder.Whenweconsideracompressibleow,theunsteadyforcerisesgradually,andreachesaconstantvalueafteracertaintimedeterminedbytheacousticpropagationspeed.Inthisregion,anunsteadyforcecoecientcanbedenedwhichdoesnotdependonthevalueoftheacceleration,solongastheaccelerationdoesnotsignicantlychangetheMachnumberoftheow. TheEulerequationsaresolvedinaframeofreferenceattachedtothecylinder,fordierentfreestreamMachnumbervaluesandseparationsfromthewall.Presenceofthewallraisesthiscoecienttovaluesgreaterthanthoseintheabsenceofawall. 9

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Theaddedmassorhydrodynamicmassisdenedasthemassoftheuidaroundabodywhichisacceleratedduetotheactionofpressure.Forexample,theaddedmassforaplatebeingmovedinauidperpendiculartoitsplaneismuchgreaterthanwhenitismovedinitsownplane.Iftheaddedmassforabodyofmassmisdenotedbym0,theforceFtoacceleratethetotalmasswillbegivenbyEquation 1{1 whereaistheacceleration. ForacylinderofradiusR,movinginastilluidofdensity,farfromanyboundary,theaddedmassisgivenby Traditionally,wewritetheaddedmassintermsofanaddedmasscoecient,Cm. .Therefore,theaddedmasscoecient,foracircularcylinderfarfromanyboundaryis1. Inincompressibleow,alotofworkhasbeendoneregardingthesteadyandunsteadyaddedmassforceonacylinderinafreestreamintheinviscid(Re!1)andStokes(Re!0)limits[ 1 ].TheextensiontobodiesclosetoawallisdealtwithbyLamb[ 2 ].Lambdevelopedrelationsfortheaddedmassforceonthemotionoftwospheresmovinginthelineofthecentersinanincompressibleuid.Carpenter[ 3 ],studiedthemotionoftwocylindersinanymannerwiththeiraxesalwaysparallelinanidealuidwithoutvorticity.Heobtainedrelationsforthecomplexpotentialforacylindermovingparallelorperpendiculartoawallortouchingthewallinthepresenceofanambientow.Yamamotoetal.[ 4 ]usesuchrelationstondincompressibleexpressionsfortheaddedmassandliftcoecientsasacylinderisbroughtprogressivelyclosertoawall.Dalton 10

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5 ]givesrelationsfortheaddedmassandliftcoecientsforseveraldierentcylinderarrangementsinapotentialow.Bothoftheseworksindicatethattheaddedmassandliftcoecientsforacylinderinanincompressibleowcanbesignicantlygreaterwhenitisnearawall. Inthecompressibleregime,muchoftheworkhasbeendonetowardsndingthesteadydragforceonacylinderorasphere.Love[ 6 ]andTaylor[ 7 ]didtheearlyworkregardingthestudyofunsteadyforcesinthecompressibleregime.Miles[ 8 ]usedanacousticapproximationofthevelocitypotentialequationtoinvestigatethemotionofacylinderimpulsivelystartedfromrest.Hetreatedthecasesofmotiongeneratedbyaconstantforceappliedoveranitetimeintervalaswellasanimpulsivelyappliedvelocity.Longhorn[ 9 ]workedontheunsteadymotionofasphereonthesameapproximation.Theseworksindicatethelimitationsoftheconventionaladdedmassconceptindescribingtheinviscidforceincompressibleows. Suchlimitationsarecausedbytherelationbetweentheaddedmassforceandtheinstantaneousacceleration.Inanincompressibleow,theaddedmassforcedependsonlyontheinstantaneousacceleration.Inacompressibleow,howevertheinviscidforceincreasesonanacoustictimescaleofR=c1,whereRistheradiusofthecylinderandc1isthefreestreamspeedofsound.BothMilesandLonghornshowthatforc1&10,theinviscidforcereachesaconstantvalue.Brentner[ 10 ]performednumericalsimulationsofcompressibleowaboutanacceleratingcylinderfocusingonthepropagationofacousticenergyasthecylinderacceleratedfromresttoM1=0:4.Parmaretal.[ 11 ]numericallysolvetheEulerequationscastinaframeofreferenceattachedtothecylinder,prescribeitsmotionandcomputethedragcoecient.TheystudiedtheeectofMachnumberandthemagnitudeoftheaccelerationonthenon-monotonicevolutionandlongtimeconstantvalueoftheunsteadyaddedmassforceinresponsetoasuddenconstantacceleration. Thereappearstobenopriorworkoncompressibilityeectsontheaccelerationofacylinderclosetoawall.AnobjectiveofthisworkwastoextendtheworkofParmaret 11

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Inthischapter,theanalysisofincompressiblepotentialowaboutacylinderclosetoawallissummarized.Itwillbeusedasalimitingcaseforcomparisonwithourcompressibleanalysisinlaterchapters. 12 ]stipulatesthatacylinderclosetoawallwithafreestreamparalleltothewallisequivalenttotwocylindersinafreestream.Anaveanalysiswouldsuggestthatthecomplexpotentialofthesystemcouldberepresentedasacombinationofafreestreamandtwodoubletsofequalstrengthplacedatequaldistancesoneithersideofacoordinateaxis.However,onealsohastoconsiderthateachdoubletwoulddistorttheshapeoftheotherandwewouldnothavetwoperfectcylinders.Therefore,aseriesofdoubletsofdiminishingstrengthsisrequired.Inthefollowing,wemakeuseofpotentialowtheory,sinceanincompressibleandirrotationalowisbeingdealtwith.Apotentialowisavelocityeldwhichisdescribedasagradientofascalarfunction:thevelocitypotential,.Also,foranyow,astreamfunctionmaybedenedtosatisfythetwo-dimensionalcontinuityequation.Leturepresentthevelocityinthex-directionandvbethevelocityinthey-direction.Then,therelationsforpotentialfunctionandstreamfunctionare: @x=@ @y(2{1) @y=@ @x(2{2) Thephysicalplaneinwhichtheowoccursmayberepresentedbythecomplexvariablez=x+iy.Flowpropertiesmaybeexpressedascomplexfunctionsofz.ThecomplexpotentialW=W(z)isdenedas 13

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3 ].ConsidertwoparallelcylindersCandC0ofradiibandb0momentarilyatadistancefapartmovingnormaltotheiraxesinauidatrestatinnity.ChoosingthecoordinatesystemasinFigure 2-1 ,thecylindervelocitiesmaybeexpressedasUeiandU0ei0respectively.Thecomplexpotentialw0=0+i0duetoC,intheabsenceofC0isthatofadoubletwhoseaxisisinthedirection. wherezisthecomplexcoordinateposition.ThenormalvelocitiesintroducedonC0byw0arecancelledbyanimagedoubletwhoseaxisisinthedirection, where Similarly,thenormalvelocitiesinducedonCbyw1arecancelledbythesecondimagedoublet ff12ei where Continuinginthisfashion,aninnitesequenceofimagedoubletsofdecreasingstrengthcanbeobtained.Thegeneralexpressionforwkis where0=Ub2,1=Ub2(b0 14

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withf0=0. Asimilarsetofpotentialswk0canbeobtainedstartingwiththepotentialC0intheabsenceofC. Thusthecomplexpotentialforthemotionoftwocylindersinanidealuidis Ifthetwocylindersareparallelandidenticaltoeachother,b0=b=a(say),U0=U,and0===2.Theplaneequidistantbetweenthecylinderaxesisastreamlineandtheoverallowistheowoveracylindernearaplanewall.ThisarrangementisshowninFigure 2-2 .Notee=D=L=D1=2WealsoincludeatermUz,whereUisthefreestreamvelocityoftheuid.ThesummationcannowbewrittenelegantlyinthemannerofYamamotoetal.[ 4 ]andMuller[ 13 ]asbelow: where and 2L=aqn1;q0=0(2{15) inwhichaisthecylinderradiusandLthedistancebetweenthecenterofthecylinderandthewall.Extractingthestreamfunctionfromthecomplexpotentialabove,wecanplotstreamlinesforanyparticularvalueofaandL,asshowninFigure 2-3 fora=1andL=1:1. 15

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2{13 x2+(Laqjy)2+x x2+(Laqj+y)2(2{16) Wecannowwritethexandydirectionvelocitiesas: @x=U+Ua21Xj=0mjA2x2 @y=Ua21Xj=0mj2Sx A2+2Tx B2(2{18) where Thepressurecoecientcannowbegivenby AsimplecomputercodecanbewrittentondCpforanydesirednumberoftermsinthesummation.Figure 2-4 showstheincompressibleCpvariationaroundthecylinderfordierentL=Dratioswith40termsinthesummation. 16

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wherelisthewidthofthecylinderinthezdirection.Atthecylindersurface,wetaker=a=1andp(r;)canbeconvertedtoafunctionofCp().Now,byusingtherelationsfordragcoecientandliftcoecient,wecanreducethemto 2Z20Cp()cosd(2{26) 2Z20Cp()sind(2{27) ThevaluesofCp()obtainedfromequation 2{23 abovecanbenumericallyintegratedtoobtainvaluesofCDandCL.ItturnsoutthatCDisalwayszeroandCLalwayshasaspecicconstantnegativevalueforeverydistancefromthewall.Figure 2-5 showsthevariationinincompressibleCLfordierentseparationsfromthewall.Inthegure,eisthegapbetweenthecylinderandthewallasshowninFigure 2-2 Yamamotoetal.derivedgeneralrelationsforthedragandliftforceonacylinderbeingacceleratedparalleltoanearbywall.Theygotthefollowingresults: inwhich aqjqk;L>a.(2{31) 17

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2-6 .Aswegofartherfromthewall,CMtendstoone.CLchangeswithdistancefromthewallinthesamewayasinsteadyow,thevariationbeingshowninFigure 2-5 Referenceframefortwocylindersmovinginauidatrest Notationusedfortheproblem 18

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Streamlinesfortwocylindersclosetoeachotherinaninviscidow.a=1,L=1:1 Figure2-4. Variationofpressurecoecientwithseparationfromthewall 19

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Variationofliftcoecientwithseparationfromthewall Figure2-6. Variationofaddedmasscoecientwithseparationfromthewall 20

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Inthischapterthenumericalsimulationsusedaredetailedanddescribed.TheproblemissimulatedusingRocuMP[ 14 ]. @tZ~W@+I@~FcdS=0:(3{1) Thevectorofconservativevariablesis 21

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withtheface-normalvelocitygivenby whereVgisthegridspeed,i.e.,thegridvelocitynormaltothecontrol-volumeface. OverallapproachforCharacteristicBoundaryConditions TheEulerEquationsare @t+@u @x+@v @y+@w @z=0(3{5) @t+@uu @x+@uv @y+@uw @z=@p @x(3{6) @t+@vu @x+@vv @y+@vw @z=@p @y(3{7) @t+@wu @x+@wv @y+@ww @z=@p @z(3{8) @t+@Hu @x+@Hv @y+@Hw @z=0(3{9) Thesewouldremaininvariantinanycoordinatereferenceframe.Letusconsiderareferenceframe(nsr)insteadof(xyz).Flowvelocitiesinn,s,andrdirectionswouldnowbedenotedbyun,us,andurrespectively.TheEulerequationsarenow, @t+@un 22

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@n(3{11) @s(3{12) @r(3{13) @t+@Hun Consideraboundarysuchthatnormaldirectionnisperpendiculartofaceands;rdirectionsaretangentialtoboundary.UsingcharacteristicanalysistomodifytheHyperbolictermsofEulerequations 3{10 to 3{14 correspondingtowavespropagatinginthendirection,Eulerequationscanbewritteninthefollowingform, @t+d1+@us @s(3{17) @r(3{18) @t+1 2(u2n+u2s+u2r)+d2 Thevectord,whichcontainsnormalderivativeterms(@ @n),isgivenbycharacteristicanalysisasfollows: 2(L5+L1)1 2(L5+L1)1 2c(L5L1)L3L49>>>>>>>>>>=>>>>>>>>>>;=8>>>>>>>>>><>>>>>>>>>>:@un @nun@un @nun@us 23

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@nc@un @n@p @n)3@us @n+c@un wherei'saredenedas: Attheinteriorofthedomainnite-volumeformulationisused.Fluxintegrationforinteriordomainusescellaveragedquantity(Equations 3{23 to 3{24 ).Whileattheboundary,owequationsaresolvedusingnite-dierenceformulationandaresolvedforactualvariablesonboundaryface. @tZdV+I@v:nds=0(3{23) dt+I@v:nds=0(3{24) AtanypointontheboundarywecanobtainasystemofLocalOne-DimensionalInviscid(LODI)relationsbyconsideringthesystemofEquations( 3{15 to 3{19 )andneglectingtransverseterms(setting's'and'r'directiontermstozero).Thisisone-dimensionalEulerequationin'n'direction.Theserelationsmaybecombinedtoexpressthetimederivativesofallotherquantitiesofinterest(e.g.temperature,T). Characteristicsbasedboundaryconditions(NSCBC)areimplementedinthreesteps: 24

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3{15 to 3{19 ).likeifporTisimposed,thenthereisnoneedtosolveenergyequationatboundary. 2.Ateachboundarysomeofthecharacteristicsaregoingoutofdomainandsomearecomingin.CharacteristicamplitudesL0iscorrespondingtooutgoingwavescanbecomputedusingonesideddierencing.Incomingwaveamplitudescannotbecomputedusingonesideddierencing.UsingLODIrelationsandimposedboundaryconditions,theseincomingwaveamplitudescanbeexpressedintermsofoutgoingwaveamplitudes. 3.Usetheremainingconservationequationsofthesystem( 3{15 to 3{19 )combinedwiththevaluesoftheL0isobtainedfromStep2tocomputeallvariableswhicharenotimposedbyboundaryconditions. where,iisthedensity,viisthevelocityvector,piisthepressure.Thecomponentsoftheforcecanbegivenby: Non-dimensionalforcecoecientsaredenedby: 2refV2refSref=(Cp;inx;i)Si 2refV2refSref=(Cp;iny;i)Si 25

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2refV2refSref=(Cp;inz;i)Si whereCp;iisthepressurecoecient. 15 ]followedbyaproceduretosmoothenthemeshatthecorners.Tostudygridindependence,threegridswereusedwith160,320and640cellsrespectivelyaroundthecylindersurface.Thesehad67200,268800and1075200respectivelyintheentireoweld.Anaspectratioofabout1:1wasmaintainedatthecylindersurface.Thegridwaskeptnestaroundthecylinderandintheregionbetweenthecylinderandthewall.TheFigures 3-1 and 3-2 showthenestgrid.Thereisaportionofthegridatx=4andx=4wherethemeshiscoarseasseeninFigure 3-2 .However,theeectofthiscoarsenessdoesnotreachthecylindersurface.Thegridsweremadethree-dimensionalbyusingtheConupackagewhichconvertsthegridstoaformatcompatiblewithRocuMP.SinceRocuMPisathree-dimensionalcode,eventwo-dimensionalsimulationsasinthisworkrequireathree-dimensionalgrid.Thez-dimensionwaskepttoasinglecellofwidth0.01units. 16 ].Weseektoestablishtheeectsofcompressibilitywhenthecylinderisacceleratedparalleltothewall.Todoso,wefollowatwo-stepprocedure.Intherststep,asteadystatesolutionisobtained.Thesteadystatecanbeconsideredtohavebeen 26

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c21;(3{32) whereaistheabsolutevalueofacceleration,Ristheradiusofthecylinderandc1isthefreestreamspeedofsound.Thevaluesofmustsatisfytwocompetingrequirements.Firstly,itshouldbesmallenoughthattheresultantchangeinMachnumberissmall.Secondly,theaccelerationshouldnotbesosmallthattheresultingforceisweakwithlowsignal-to-noiseratio.IftheMachnumberisincreasedbytoomuch,ourobjectiveofgettingtheinviscidforceatadistinctMachnumberwouldnotbemet.Thedurationofacceleration,non-dimensionalizedintermsoftheacoustictimescale,ischosentobec1(tft0)=R=30,whichissucientfortheinviscidforcetoreachaconstantvalueandyetkeep,thechangeinMachnumberassmallaspossible. Thephysicalquantitiesusedinthesimulationsareobtainedfromtheisentropicrelationsforcompressibleow. 2M21(3{34) 2M21) 1(3{35) 27

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Matrixoftestcases Case 0:2 2 2 0:2 1 3 0:3 2 4 0:3 1 HereM1,T,p,,R,andUrefertoMachnumber,temperature,pressure,density,gasconstantandvelocityrespectively.Thesubscripts0and1respectivelyrepresentthestagnationandfreestreamconditions.istheratioofspecicheats. ThetreatmentofeachboundaryisshowninFigure 3-3 Figure3-1. Aviewofthenest2Dgridaroundthecylinder 28

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Aviewofthenest2Dgridintheregionbetweenthecylinderandthewall Figure3-3. Boundariesconsideredintheoweld 29

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4-1 4-6 ).Itisevidentfromtheseguresthatthecoarseandmediummeshesaretoocoarseandthenegridappearstobesuitableforfurthercomputations.Thedragcoecientshouldbeidenticallyzeroforsubsonicirrotationalow.Thenon-zerovaluesareduetonumericaldissipationthatisreducedwithdecreasingcellsize.Inthepressurecoecientplots,theincompressibletheoreticalvalueisalsoplottedforcomparison. 4.2.1DragCoecient 4-7 .Itcanbeseenthattheplotsalmostcollapseoneachotheratavalueclosetozero. 4-8 showstheliftcoecientforthesamecases.Thetheoreticalvalueoftheincompressibleliftcoecientis-0.0385and-0.3574atL=D=1andL=D=2respectively.Interestingly,themagnitudeoftheliftcoecientisgreaterthantheincompressiblevalueforL=D=1,butlesserthantheincompressiblevalueforL=D=2. 30

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4-9 showsthepressurecoecientvariationwithseparationfromthewall.ItisapparentthatMachnumberandseparationhaveasignicanteectatthe==2and==2positions. 4-10 4-11 ,and 4-12 respectivelyforoneofthecases(L=D=2,M1=0:2).ItistobenotedthattheMachnumberremainssubcriticalatallpointsandthevariationintotalpressureiseversoslight. 4-13 to 4-16 showtheevolutionofthisunsteadycoecientasafunctionofnon-dimensionaltime=c1(tt0)=R.Itcanbeseenthatprovidedissmall,thevalueofF0Disindependentoftheactualvalueof.Whenincreasesfurther,theincreaseinrelativeMachnumberoverthedurationoftheaccelerationhasanotableinuenceontheunsteadyforce,andtheresultcannotbeconsideredtocorrespondtoafrozenMachnumberof0.2and0.3respectively. 4-17 to 4-20 showthetimeevolutionofF0L.Theplotsalmostcoincidewitheachother,indicatingthatforsmall 31

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Figure4-1. Gridindependence:Dragcoecient(L=D=2,M1=0:2) Figure4-2. Gridindependence:Dragcoecient(L=D=2,M1=0:3) 32

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Gridindependence:Liftcoecient(L=D=2,M1=0:2) Figure4-4. Gridindependence:Liftcoecient(L=D=2,M1=0:3) 33

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Gridindependence:Pressurecoecient(L=D=2,M1=0:2) Figure4-6. Gridindependence:Pressurecoecient(L=D=2,M1=0:3) 34

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Dragcoecientvariationwithseparationfromwall Figure4-8. Liftcoecientvariationwithseparationfromwall 35

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Pressurecoecientvariationwithseparationfromwall Figure4-10. Pressurecontoursaroundthecylinder.(L=D=2,M1=0:2) 36

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Machnumbercontoursaroundthecylinder.(L=D=2,M1=0:2) Figure4-12. Totalpressurecontoursaroundthecylinder.(L=D=2,M1=0:2) 37

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TimeevolutionoftheunsteadydragforcecoecientoverarangeofvaluesforL=D=2,M1=0:2 Figure4-14. TimeevolutionoftheunsteadydragforcecoecientoverarangeofvaluesforL=D=1,M1=0:2 38

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TimeevolutionoftheunsteadydragforcecoecientoverarangeofvaluesforL=D=2,M1=0:3 Figure4-16. TimeevolutionoftheunsteadydragforcecoecientoverarangeofvaluesforL=D=1,M1=0:3 39

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TimeevolutionoftheunsteadyliftforcecoecientoverarangeofvaluesforL=D=2,M1=0:2 Figure4-18. TimeevolutionoftheunsteadyliftcoecientoverarangeofvaluesforL=D=1,M1=0:2 40

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TimeevolutionoftheunsteadyliftforcecoecientoverarangeofvaluesforL=D=2,M1=0:3 Figure4-20. TimeevolutionoftheunsteadyliftforcecoecientoverarangeofvaluesforL=D=1,M1=0:3 41

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Resultsshowthevariationoftheunsteadyforceinboththestreamwiseandstreamnormaldirectionsforaninstantaneousaccelerationoftheow.Inthestreamwisedirection,wecandeneaforcecoecientwhichisofthenatureoftheaddedmasscoecient.Thiscoecientrisesgraduallyandreachesconstantvaluesfortimesbeyondacertainacoustictimescale.Thepeakvalueofthisforcecanbeasmuchas2.3timestheincompressiblevalueforM1=0:3andL=Dratioof1.Thisunsteadyforcecoecientisnotdependentontheabsolutevalueoftheacceleration.ThevalueisslightlyincreasedfromthevaluesreportedbyParmaretal,duetothepresenceofthewall.Furthermore,asinthecasestudiedbyParmaretal,thenalvalueoftheunsteadyforceissignicantlyaectedbythefreestreamMachnumberoftheambientow.Also,insomecases,asecondpeakappearsintheunsteadydragforcecoecientplots.Thisisduetoreectionsfromthewall,whichcanbeassumedtoapplyasecondacceleratingeect. Inthestreamnormaldirection,theresultsindicatethatifwesubtractoutthequasi-steadyinuenceoftheaccelerationonincreasingthemagnitudeofthelift,thevalueofasimilarlydenedunsteadyliftforcecoecientisinvariant.Eveninincompressibleow,theliftcoecientforacylinderclosetoawallisdependentontheproximitytothewallonly.Theliftforceincreaseswhenanaccelerationisappliedbecausethevelocityoftheowisnowgreater.Thus,itappearsthatevenincompressibleow,theaccelerationservestoraisetheliftforceduringthedurationofitsapplication,butonceitisstoppedtheliftforceisgovernedbythenewincreasedvelocityoftheow. 42

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[1] L.D.Landau,andE.M.LifshitzFluidMechanics,2ndedn.,Butterworth-Heinemann(1987),London,UK. [2] H.Lamb,Hydrodynamics,6thedn.,CambridgeUniversityPress(1993),NewYork,USA. [3] L.H.Carpenter,OntheMotionofTwoCylindersinanIdealFluid,JournalofResearchoftheNationalBureauofStandards61(1958)83-87. [4] J.H.Nath,T.Yamamoto,andL.S.Slotta,WaveForcesonCylindersnearPlaneBoundary,JournaloftheWaterways,HarborsandCoastalEngineeringDivision100(1974)345-358. [5] C.Dalton,andR.A.Helnstine,PotentialFlowPastaGroupofCircularCylinders,JournalofBasicEngineering93(4)(1971)636-642. [6] A.E.H.LoveSomeillustrationsofmodesofdecayofvibratorymotions,Phil.Trans.RoyalSocietyA2(1904)88-113. [7] G.I.Taylor,Themotionofabodyinwaterwhensubjectedtoastrongimpulse,ThescienticPapersofG.I.Taylor(ed.G.K.Batchelor)3(1942)306-308. [8] J.W.Miles,Onvirtualmassandtransientmotioninsubsoniccompressibleow,Quart.J.Mech.Appl.Math.IV(4)(1951)388-400. [9] A.L.Longhorn,Theunsteady,subsonicmotionofasphereinacompressible,invisciduid,Quart.J.Mech.Appl.Math.V(1)(1952)64-81. [10] K.S.Brentner,Directnumericalcalculationofacoustics:solutionevaluationthroughenergyanalysis,J.FluidMech.254(1993)267-281. [11] M.Parmar,A.Haselbacher,andS.Balachandar,OntheUnsteadyInviscidForcesonCylindersandSpheresinSubcriticalCompressibleFlow,Phil.Trans.RoyalSocietyA336(2008)2161-2175. [12] R.W.Johnson,TheHandbookofFluidDynamics,CRCPress(1998),BocaRaton,Fl,USA. [13] W.vonMuller,SystemevonDoppelquelleninderEbenenStromung,insbesonderedieStromungumZweiKreiszylinder,ZeitschriftfurangewandteMathematikundMechanik9(3)(1929)200-213. [14] A.Haselbacher,TheRocuMPBook.1.13.0(2008),DepartmentofMechanicalandAerospaceEngineering,UF,Gainesville,Fl,USA. [15] J.F.Thompson,B.Soni,N.P.Weatherhill,HandbookofGridGenerationCRCPress(1999),BocaRaton,Fl,USA. 43

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G.Volpe,PerformanceofCompressibleFlowCodesatLowMachNumbers,AIAAJournal31(1)(1993)49-56. 44

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RohitashwaKiranwasborninMuzaarpur,India,in1985.HegraduatedfromDeNobiliSchool,Maithon,India,inMarch2001;andLaMartiniereCollege,Calcutta,IndiainMarch2003.HeobtainedaBachelorofTechnologydegreeinmechanicalengineeringfromNationalInstituteofTechnology,Hamirpur,India,inMay2007.HejoinedthegraduateprograminMechanicalEngineeringattheUniversityofFloridainAugust2007.HeworkedundertheguidanceofDr.AndreasHaselbacheroftheMechanicalandAerospaceEngineeringDepartment. 45