Three-dimensional instability mechanism in a four-vortex ...
25 pages
English

Three-dimensional instability mechanism in a four-vortex ...

Le téléchargement nécessite un accès à la bibliothèque YouScribe
Tout savoir sur nos offres
25 pages
English
Le téléchargement nécessite un accès à la bibliothèque YouScribe
Tout savoir sur nos offres

Description

Technical Note
(For internal use by AWIATOR partners and Commission only)
Three-dimensional instability mechanism
in a four-vortex system; Benchmark
computations.
Prepared by: H.L.C. Moet (CERFACS)
Work Package: WP1
Task: 1.1.1
Document No.: AW-CERFACS-111-001
Version: 1 (Draft)
Issued by: CERFACS
Date: July 17, 2003
Approved by:
Task Manager:
Issueing organisation: CERFACS Three-dimensional instability mechanism
in a four-vortex system; Benchmark
computations.
Henri Moet
CERFACS, Centre Europ¶een de Recherche et de Formation Avan»c¶ee en Calcul Scientiflque,
42, Av. G. Coriolis, 31057 Toulouse Cedex, France
Abstract. To compare difierent numerical methods, a "benchmark" computation has been
deflned within Subtask 1.1.1 "Design and Manufacture". The benchmark con-
sists in simulating the unstable behaviour of a four-vortex system composed of two counter-
rotating vortex pairs. An energy diagnostic is applied to analyse the evolution of the kinetic
energy in the vortex o w governed by the instability mechanism with the purpose of esti-
mating the artiflcial/numerical viscosity efiects. It is essential to demonstrate that numerical
efiects are within acceptable limits which allows the use of the numerical methods for pre-
dicting and selecting the desired vortex o w conflgurations.
July 17, 2003 Table of Contents
1 Introduction............................................. 6
2 The numerical model and initial condition ................... 7
2.1 The numerical code NTMIX3D ........ ...

Sujets

Informations

Publié par
Nombre de lectures 43
Langue English

Exrait

TechnicalNote(ForinternalusebyAWIATORpartnersandCommissiononly)Three-dimensionalinstabilitymechanisminafour-vortexsystem;Benchmarkcomputations.Preparedby:H.L.C.Moet(CERFACS)WorkPackage:WP1Task:1.1.1DocumentNo.:AW-CERFACS-111-001Version:1(Draft)Issuedby:CERFACSDate:July17,2003Approvedby:TaskManager:Issueingorganisation:CERFACS
Three-dimensionalinstabilitymechanisminafour-vortexsystem;Benchmarkcomputations.HenriMoetCERFACS,CentreEurope´endeRechercheetdeFormationAvanc¸e´eenCalculScientifique,42,Av.G.Coriolis,31057ToulouseCedex,FranceAbstract.Tocomparedifferentnumericalmethods,a”benchmark”computationhasbeendefinedwithinSubtask1.1.1”DesignandManufacture”.Thebenchmarkcomputationcon-sistsinsimulatingtheunstablebehaviourofafour-vortexsystemcomposedoftwocounter-rotatingvortexpairs.Anenergydiagnosticisappliedtoanalysetheevolutionofthekineticenergyinthevortexflowgovernedbytheinstabilitymechanismwiththepurposeofesti-matingtheartificial/numericalviscosityeffects.Itisessentialtodemonstratethatnumericaleffectsarewithinacceptablelimitswhichallowstheuseofthenumericalmethodsforpre-dictingandselectingthedesiredvortexflowconfigurations.July17,2003
TableofContents1234Introduction.............................................Thenumericalmodelandinitialcondition...................2.1ThenumericalcodeNTMIX3D........................2.2Definitionofthebenchmarkinitialcondition.............Presentationofthesimulationresults.......................3.1Globaldynamics.....................................3.2SpectralAnalysis.....................................Conclusion..............................................6772141149132
1Introduction1IntroductionInsomeoccasionstheextendednearfieldorearlyfarfieldwakeiscom-posedofmorethanonecounter-rotatingvortexpair.Dependingontheconfiguration(acounter-orco-rotatingpaironeachsideofthesymme-tryplane,theseparationdistancebetweeneachvortexandtheindivid-ualstrengthofthevortices)severalinstabilitymechanismsmaydevelop,namelyalongorshortwavelengthinstability.Crouch[4]presentedresultsofastabilityanalysisfortwotrailingvor-texpairs(inco-rotatingconfiguration)andshowedtheexistenceofthreedifferentgrowthmechanisms,thelongwavelength,theshortwavelengthandatransientgrowthmechanism.Thesemechanismsrevealedtobeveryrelevanttothevortexbehaviourbehindaircraftinflaps-downhigh-liftconfigurationsrelatedtoaccelerateddecay.Rennichetal.[11,12]devotedhisPhDstudiestoafour-vortexsystem(incounter-rotatingconfigura-tion)andfoundthatinboardvorticescanbeusedtoenhancethedevel-opmentofCrowinstabilityintheoutboardvortices.Ortegaetal.[10]werethefirsttoconductexperimentswithmultiplevortexpairs(incounter-rotatingconfiguration).Theydemonstratedanimportantreductionintheinducedrollingmomentforavortexsystemgeneratedbehindatriangularwinginatowingtankcomparedtotheoneobtainedforaclassicalrectangularwing.Themainmechanismresponsi-blewastherapiddevelopmentofaninstabilityinthesecondaryvorticesandsubsequentlyinthemainvortices.Notethattheirconfigurationwascharacterizedbyaratioofthecirculationofthesecondaryvorticesandthemainvorticesthatissubstantiallylargerthanwouldbefoundforamorerealisticconfigurationrepresentingacivilaircraft.Fabreetal.[6]haveconductedaninvestigationthatcoveredarangeofparameterscorrespondingtorealisticaircraftconfigurations(afourvortexsystemincounter-rotatingconfiguration)anddiscoveredthatthemostunstablemodeshaveshortwavelengthscomparedtoforexamplethewavelengthofCrowinstability.Themostunstablemodehasawave-lengththatisoftheorderoftheseparationdistancebetweenthemainvortices.Theseshortwavelengthinstabilitiesareseentohaveadestruc-tiveeffectonthesecondaryvorticesbutdonotleadtoasignificantdecayinthemainvortices.Themaximumdecreaseincirculationofthemainvorticesdoesnotexceedtheratio|Γ21|whichisrathersmallforreal-isticaircraft.Fabreetal.thenproposedtoacceleratethedevelopmentofCrowinstabilityinthemainvorticesbyexertingaforcingcoming6
2ThenumericalmodelandinitialconditionfromtherapiddevelopmentofaninstabilityinthesecondaryvorticeswithawavelengththatmatchestheCrowwavelength.Theobtainedper-turbationoftheforcingappearedtoresultinasignificantgainintheamplificationofCrowinstability.High-ReynoldsnumbersimulationsperformedbyLaporte[7]evenshowedthedevelopmentoftheellipticinstabilityinsystemscomposedofmul-tiplevortexpairs.AnimportantaspectrelatedtotheinvestigationintothestabilityofthesemultiplevortexpairsystemsisthatinexperimentstheReynoldsnumberissignificantlylowerwhichprimarilyaffectstheevolutionoftheellipticinstability.Thisreportpresentsthenumericalresultsrelatedtotheinstabilitymech-anismssusceptibleofdevelopinginafour-vortexsystemcomposedoftwocounter-rotatingvortexpairs.ForthispurposeabenchmarkcomputationwasdefinedasproposedbyCapartetal.[3]whichconsidersafour-vortexsystemcomposedofamainvortexpairresultingfrommergerofthetipvortexandtheoutboardflapvortexandasecondaryvortexpairthatmaycomefromtheinboardflapsandhorizontaltailplane.ThestabilityofthissystemhasbeeninvestigatedfollowingtheLarge-EddySimulation(LES)approach.Section2describesthenumericaltoolandtheappliedsubgrid-scalemod-ellingapproachthathasbeenusedforthepresentstudy.Thesectionendswiththedescriptionofthebenchmarkcomputationthatdeterminestheinitialconditionforthecomputations.ThenumericalresultswhichhavebeenobtainedarepresentedinSection3byconsideringtheglobaldy-namicsaswellasanenergydiagnostic.SomeconcludingremarksaregiveninSection4aboutthemainoutcomesofthepresentstudy.2Thenumericalmodelandinitialcondition2.1ThenumericalcodeNTMIX3DThecodeusedforthesestudieswillbeNTMIX3Dwhichhasbeendevel-opedbytheCentredeRecherchesurlaCombustionTurbulente(CRCT)oftheInstitutFrancaisduPe´trole(IFP).TheparallelcodesolvestheNavier-Stokesequationsfor3DcompressibleflowonaregularCarte-siangrid.ThesolverisdevotedtoDirectNumericalSimulationsandLarge-EddySimulations.AnondimensionalformulationoftheNavier-Stokesequationsisusedandahighaccuracyofthesolutionisguaran-teedbya6thordercompactschemeforthediscretisationinspace,and7
2Thenumericalmodelandinitialconditiontimeintegrationisperformedwitha3rdorderRunge-Kuttamethod.Thesubgrid-scalemodelusedtotakeintoaccounttheinfluenceofthenonresolvedscalesontheresolvedscaleswillbetheFilteredStructureFunctionmodel,whichprovedtoprovideexcellentresultsforinstabilitystudies.Inparticular,themodeldoesnotprovideanyenergydissipationforwell-resolvedflowscorrespondingtothestagesbeforetheinstabilitysaturationandtransitiontoturbulence.ThecodehasbeenrunontheCompaqSC40computerofCERFACS.GoverningequationsTheconservationofmass,momentumandenergyofathree-dimensionalunsteadycompressibleviscousflowisdescribedbytheNavier-Stokesequations.Ina3DCartesiancoordinatesystem(x,y,z),theNavier-Stokesequationsinconservativeformcanbeexpressedasfollows:W+(ffv)+(ggv)+(hhv)=0(1)txyzThestatevectorWisdefinedas)2(ρuρwρW=ρvEρwithρthedensity,u,v,warethe3Dvelocitycomponents,pthepressureandEthetotalenergy.Theconvectivefluxesf,gandhinthedirectionsx,yandzaredefinedas:ρuρvρwρu2+pρvuρwuf=ρuv,g=ρv2+p,h=ρwv(3)ρuwρvwρw2+pu(ρE+p)v(ρE+p)w(ρE+p)Theviscousfluxescanbeexpressedas:8
2Thenumericalmodelandinitialcondition0τxx0τyx0τzxfv=τxy,gv=τyy,hv=τzy(4)τxzτyzτzzujτxjqxujτyjqyujτzjqzwhereqiarethecomponentsoftheheatfluxvectordeterminedbyFourier’slawofheatconduction.Furthermore,τijistheReynoldsstresstensorlinkedtoviscosityaspostulatedinStokes’hypothesis.Suther-land’sviscositylawisusedtorepresentthevariationofviscositywithtemperature.Thepressureislinkedtothestatevectorbytheequationofstateforaperfectgas.InNTMIX3Danondimensionalizedformula-tionoftheNavier-Stokesequationshasbeenused.Thisformulationisbasedonreferencequantities(Lref,arefrefref)inamannerthatthenondimensionalquantities(denotedby)aredefinedasuiaref=ui(5)Ltref=t(6)aferρρref=ρ(7)ννref=ν(8)Subgrid-scalemodelforLESThesubgrid-scalemodelusedistheFilteredStructureFunctionmodel.TheNavier-Stokesequationsarethereforefilteredinphysicalspacebyaconvolutionintegralwithaconvolutionkernel(filterfunction)char-acteristicforthefilter,whichdependsdirectlyonthemeshsize,andasecondfilterhasbeenappliednamelytheFavre’sfilter(forcompressibleflows)toeliminateevenmorestructuresfromtheflow.Subsequently,theBoussinesqhypothesisisusedtomodelthesubgrid-stresstensorτij=(ρuiujρu˜iu˜j),withuthefilteredvariablebyconvolutionwiththefilterfunction,andu˜=ρu/ρobtainedafterapplyingFavre’sfilter.Thishypothesisdefinesµt=ρνt1³∂u˜i∂u˜j2∂u˜k´τij3τkkδij=µt∂x+∂x3xδkk(9)kij9
2ThenumericalmodelandinitialconditionThesubgrid-scaleviscosityfortheFilteredStructureFunctionmodeliswrittenas)3(rνt(xc,t)=0.00084ΔcF2(xc,t)(10)whereΔc=π/kcisthecut-offlengthcorrespondingtothecut-offwavenum-berkc,whichischoseninspectralspace.TheexponentofF2indicatesthenumberoftimesthehigh-passfilterhasbeenappliedtotheresolvedflowfield.The2nd-orderstructurefunctionisdefinedbyF2(x,r,t)=hku(x+r,t)u(x,t)k2i(11)rr=kku,xandrdenotethevelocityvector,positionvectorandseparationvectorrespectively.hidenotesthestatisticalaverageovertheentirefluiddomain.Inpractice,thecut-offwavenumberischosenequaltothecut-offwavenumberimposedbythemeshsizeΔx:kc=kx=π/Δx.Thecut-offwavenumberissuchthatthefilteredvariableuˆinspectralspaceisgivenyb(uˆ=uˆsi|k|≤kc(12)0otherwisewithuˆtheFouriertransformofu.ThemodelisbasedontheequationgoverningtheevolutionoftheresolvedenergyspectrumE(k,t),depen-dantonthewavenumberkandtimet:+2νk2E(k,t)=T<kc(k,t)+TSG(k,t)(13)´³twiththeassumptionthatk<kc.T<kc(k,t)representstheenergytransferbetweentheresolvedscales(k<kc)andTSG(k,t)representstheenergytransferbetweenthesubgrid-scales.Thislasttermrequiresmodeling.TheturbulentviscosityinspectralspaceisdefinedbyTSG(k,t)νt(k,kc,t)=2k2E(k,t)(14)Whenthisexpressionisrewrittenwithrespecttothekineticenergyatthecut-offfrequency,oneobtainsvuE(k,t)kνt(k/kc,t)=f(k/kc)tc(15)c01
2Thenumericalmodelandinitialconditionwithfaknownfunction.Theexpression(15)showsthatthemodelwillnotdissipatewhilethereisnoenergyatthecut-off.Thisisthecaseforthesimulationsinthepresentstudy,wheretheinitialconditionshaveanegligibleenergylevelatthecut-off.Theenergylevelatthecut-offincreasesthroughthecourseofoursimulationwhenlocallytheflowmaytransitionandwhentheturbulentviscosityfollowsthisevolution.Thismodelprovestobewelladaptedforoursimulations.NumericalMethodsThenondimensionalizedNavier-Stokesequationsarediscretizedinspaceusingafinitedifferencemethod.ThecomputationalmeshesareCartesianandregularinthethreedirections.ThemeshsizeishandindirectionOx,thegridpointscanbegivenasxi=h(i1).Thetimeintegrationhasbeendonebya3rd-order3-stageRunge-Kuttamethod.WhenoneconsidersythesolutionofCauchy’sproblemy0=f(t,y),theintegrationfollowingRunge-Kutta’smethodgives:y(t+Δt)=y(t)+Δtfˆ(t,y)fˆ(t,y)=1/4K1+3/4K3K1=f(t,y)K2=f(t+Δt/3,y+Δt/3K1)K3=f(t+2Δt/3,y+2Δt/3K2))(16)Thespatialderivativesoforder1(convectiveterms)andorder2(dif-fusiveterms)arecomputedusinga6thcompactscheme[8](Pade´type).Forthefirstorderderivatives(du/dx)(xi)ofvariable/functionuatapointxi,weintroducetheapproximationui0,obtainedbyaPade´schemewhichinit’sgeneralformreads00000βui2+αui1+ui+αui+1+βui+2=cui+3ui3+bui+2ui2+aui+1ui1(17)6h4h2hInordertoobtainaschemewithatruncationerroroftheorderO(h6),onehastosatisfythefollowingconstrainta+24b+34c=25!(α+24β)(18)!411
  • Accueil Accueil
  • Univers Univers
  • Ebooks Ebooks
  • Livres audio Livres audio
  • Presse Presse
  • BD BD
  • Documents Documents