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

Permanent Link: http://ufdc.ufl.edu/UFE0024102/00001

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Title: Effects of Compressibility on the Unsteady Inviscid Flow over a Cylinder Close to a Wall
Physical Description: 1 online resource (45 p.)
Language: english
Creator: Kiran, Rohitashwa
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: 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: 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.
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 Rohitashwa Kiran.
Thesis: Thesis (M.S.)--University of Florida, 2008.
Local: Adviser: Haselbacher, Andreas.
Local: Co-adviser: Balachandar, Sivaramakrishnan.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0024102:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024102/00001

Material Information

Title: Effects of Compressibility on the Unsteady Inviscid Flow over a Cylinder Close to a Wall
Physical Description: 1 online resource (45 p.)
Language: english
Creator: Kiran, Rohitashwa
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: 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: 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.
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 Rohitashwa Kiran.
Thesis: Thesis (M.S.)--University of Florida, 2008.
Local: Adviser: Haselbacher, Andreas.
Local: Co-adviser: Balachandar, Sivaramakrishnan.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0024102:00001


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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