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

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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 ........ ...

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TechnicalNote (ForinternalusebyAWIATORpartnersandCommissiononly) Three-dimensionalinstabilitymechanism inafour-vortexsystem;Benchmark computations. Preparedby:H.L.C.Moet(CERFACS) WorkPackage:WP1 Task:1.1.1 DocumentNo.:AW-CERFACS-111-001 Version:1(Draft) Issuedby:CERFACS Date:July17,2003 Approvedby: TaskManager: Issueingorganisation:CERFACS

Three-dimensionalinstabilitymechanism inafour-vortexsystem;Benchmark computations. HenriMoet CERFACS,CentreEurope´endeRechercheetdeFormationAvanc¸e´eenCalculScientifique, 42,Av.G.Coriolis,31057ToulouseCedex,France Abstract. Tocomparedifferentnumericalmethods,a”benchmark”computationhasbeen definedwithinSubtask1.1.1”DesignandManufacture”.Thebenchmarkcomputationcon- sistsinsimulatingtheunstablebehaviourofafour-vortexsystemcomposedoftwocounter- rotatingvortexpairs.Anenergydiagnosticisappliedtoanalysetheevolutionofthekinetic energyinthevortexflowgovernedbytheinstabilitymechanismwiththepurposeofesti- matingtheartificial/numericalviscosityeffects.Itisessentialtodemonstratethatnumerical effectsarewithinacceptablelimitswhichallowstheuseofthenumericalmethodsforpre- dictingandselectingthedesiredvortexflowconfigurations. July17,2003

TableofContents 1 2 3 4 Introduction............................................. Thenumericalmodelandinitialcondition................... 2.1ThenumericalcodeNTMIX3D........................ 2.2Definitionofthebenchmarkinitialcondition............. Presentationofthesimulationresults....................... 3.1Globaldynamics..................................... 3.2SpectralAnalysis..................................... Conclusion.............................................. 6 7 7 21 41 14 91 32

1Introduction 1Introduction Insomeoccasionstheextendednearfieldorearlyfarfieldwakeiscom- posedofmorethanonecounter-rotatingvortexpair.Dependingonthe configuration(acounter-orco-rotatingpaironeachsideofthesymme- tryplane,theseparationdistancebetweeneachvortexandtheindivid- ualstrengthofthevortices)severalinstabilitymechanismsmaydevelop, namelyalongorshortwavelengthinstability. Crouch[4]presentedresultsofastabilityanalysisfortwotrailingvor- texpairs(inco-rotatingconfiguration)andshowedtheexistenceofthree differentgrowthmechanisms,thelongwavelength,theshortwavelength andatransientgrowthmechanism.Thesemechanismsrevealedtobevery relevanttothevortexbehaviourbehindaircraftinflaps-downhigh-lift configurationsrelatedtoaccelerateddecay.Rennichetal.[11,12]devoted hisPhDstudiestoafour-vortexsystem(incounter-rotatingconfigura- tion)andfoundthatinboardvorticescanbeusedtoenhancethedevel- opmentofCrowinstabilityintheoutboardvortices. Ortegaetal.[10]werethefirsttoconductexperimentswithmultiple vortexpairs(incounter-rotatingconfiguration).Theydemonstratedan importantreductionintheinducedrollingmomentforavortexsystem generatedbehindatriangularwinginatowingtankcomparedtotheone obtainedforaclassicalrectangularwing.Themainmechanismresponsi- blewastherapiddevelopmentofaninstabilityinthesecondaryvortices andsubsequentlyinthemainvortices.Notethattheirconfigurationwas characterizedbyaratioofthecirculationofthesecondaryvorticesand themainvorticesthatissubstantiallylargerthanwouldbefoundfora morerealisticconfigurationrepresentingacivilaircraft. Fabreetal.[6]haveconductedaninvestigationthatcoveredarange ofparameterscorrespondingtorealisticaircraftconfigurations(afour vortexsystemincounter-rotatingconfiguration)anddiscoveredthatthe mostunstablemodeshaveshortwavelengthscomparedtoforexample thewavelengthofCrowinstability.Themostunstablemodehasawave- lengththatisoftheorderoftheseparationdistancebetweenthemain vortices.Theseshortwavelengthinstabilitiesareseentohaveadestruc- tiveeffectonthesecondaryvorticesbutdonotleadtoasignificantdecay inthemainvortices.Themaximumdecreaseincirculationofthemain vorticesdoesnotexceedtheratio|Γ 2 /Γ 1 |whichisrathersmallforreal- isticaircraft.Fabreetal.thenproposedtoacceleratethedevelopment ofCrowinstabilityinthemainvorticesbyexertingaforcingcoming 6

2Thenumericalmodelandinitialcondition fromtherapiddevelopmentofaninstabilityinthesecondaryvortices withawavelengththatmatchestheCrowwavelength.Theobtainedper- turbationoftheforcingappearedtoresultinasignificantgaininthe amplificationofCrowinstability. High-ReynoldsnumbersimulationsperformedbyLaporte[7]evenshowed thedevelopmentoftheellipticinstabilityinsystemscomposedofmul- tiplevortexpairs.Animportantaspectrelatedtotheinvestigationinto thestabilityofthesemultiplevortexpairsystemsisthatinexperiments theReynoldsnumberissignificantlylowerwhichprimarilyaffectsthe evolutionoftheellipticinstability. Thisreportpresentsthenumericalresultsrelatedtotheinstabilitymech- anismssusceptibleofdevelopinginafour-vortexsystemcomposedoftwo counter-rotatingvortexpairs.Forthispurposeabenchmarkcomputation wasdefinedasproposedbyCapartetal.[3]whichconsidersafour-vortex systemcomposedofamainvortexpairresultingfrommergerofthetip vortexandtheoutboardflapvortexandasecondaryvortexpairthat maycomefromtheinboardflapsandhorizontaltailplane.Thestability ofthissystemhasbeeninvestigatedfollowingtheLarge-EddySimulation (LES)approach. Section2describesthenumericaltoolandtheappliedsubgrid-scalemod- ellingapproachthathasbeenusedforthepresentstudy.Thesectionends withthedescriptionofthebenchmarkcomputationthatdeterminesthe initialconditionforthecomputations.Thenumericalresultswhichhave beenobtainedarepresentedinSection3byconsideringtheglobaldy- namicsaswellasanenergydiagnostic.Someconcludingremarksare giveninSection4aboutthemainoutcomesofthepresentstudy. 2Thenumericalmodelandinitialcondition 2.1ThenumericalcodeNTMIX3D ThecodeusedforthesestudieswillbeNTMIX3Dwhichhasbeendevel- opedbytheCentredeRecherchesurlaCombustionTurbulente(CRCT) oftheInstitutFrancaisduPe´trole(IFP).Theparallelcodesolvesthe Navier-Stokesequationsfor3DcompressibleflowonaregularCarte- siangrid.ThesolverisdevotedtoDirectNumericalSimulationsand Large-EddySimulations.AnondimensionalformulationoftheNavier- Stokesequationsisusedandahighaccuracyofthesolutionisguaran- teedbya6 th ordercompactschemeforthediscretisationinspace,and 7

2Thenumericalmodelandinitialcondition timeintegrationisperformedwitha3 rd orderRunge-Kuttamethod. Thesubgrid-scalemodelusedtotakeintoaccounttheinfluenceofthe nonresolvedscalesontheresolvedscaleswillbetheFilteredStructure Functionmodel,whichprovedtoprovideexcellentresultsforinstability studies.Inparticular,themodeldoesnotprovideanyenergydissipation forwell-resolvedflowscorrespondingtothestagesbeforetheinstability saturationandtransitiontoturbulence.Thecodehasbeenrunonthe CompaqSC40computerofCERFACS. Governingequations Theconservationofmass,momentumandenergyofathree-dimensional unsteadycompressibleviscousflowisdescribedbytheNavier-Stokes equations.Ina3DCartesiancoordinatesystem(x,y,z),theNavier- Stokesequationsinconservativeformcanbeexpressedasfollows: ∂∂∂W∂ +(f−f v )+(g−g v )+(h−h v )=0(1) ∂t∂x∂y∂z ThestatevectorWisdefinedas )2( ρ  uρ wρ W=  ρv   Eρ withρthedensity,u,v,warethe3Dvelocitycomponents,pthepressure andEthetotalenergy.Theconvectivefluxesf,gandhinthedirections x,yandzaredefinedas: ρuρvρw  ρu 2 +pρvuρwu f=ρuv,g=ρv 2 +p,h=ρwv(3)   ρuw     ρvw     ρw 2 +p   u(ρE+p)v(ρE+p)w(ρE+p) Theviscousfluxescanbeexpressedas: 8

2Thenumericalmodelandinitialcondition 0 τ xx 0 τ yx 0 τ zx  f v =τ xy ,g v =τ yy ,h v =τ zy (4)   τ xz     τ yz     τ zz   u j τ xj −q x u j τ yj −q y u j τ zj −q z whereq i arethecomponentsoftheheatfluxvectordeterminedby Fourier’slawofheatconduction.Furthermore,τ ij istheReynoldsstress tensorlinkedtoviscosityaspostulatedinStokes’hypothesis.Suther- land’sviscositylawisusedtorepresentthevariationofviscositywith temperature.Thepressureislinkedtothestatevectorbytheequation ofstateforaperfectgas.InNTMIX3Danondimensionalizedformula- tionoftheNavier-Stokesequationshasbeenused.Thisformulationis basedonreferencequantities(L ref ,a ref ,ρ ref ,ν ref )inamannerthatthe nondimensionalquantities(denotedby ∗ )aredefinedas u i ∗ ∙a ref =u i (5) L t ∗ ∙ ref =t(6) a fer ρ ∗ ∙ρ ref =ρ(7) ν ∗ ∙ν ref =ν(8) Subgrid-scalemodelforLES Thesubgrid-scalemodelusedistheFilteredStructureFunctionmodel. TheNavier-Stokesequationsarethereforefilteredinphysicalspaceby aconvolutionintegralwithaconvolutionkernel(filterfunction)char- acteristicforthefilter,whichdependsdirectlyonthemeshsize,anda secondfilterhasbeenappliednamelytheFavre’sfilter(forcompressible flows)toeliminateevenmorestructuresfromtheflow.Subsequently, theBoussinesqhypothesisisusedtomodelthesubgrid-stresstensor τ ij =−(ρu i u j −ρu˜ i u˜ j ),withuthefilteredvariablebyconvolutionwith thefilterfunction,andu˜=ρu/ρobtainedafterapplyingFavre’sfilter. Thishypothesisdefinesµ t =ρν t 1 ³ ∂u˜ i ∂u˜ j 2∂u˜ k ´ τ ij − 3 τ kk δ ij =µ t ∂x + ∂x − 3∂x δ kk (9) kij 9

2Thenumericalmodelandinitialcondition Thesubgrid-scaleviscosityfortheFilteredStructureFunctionmodelis writtenas )3( r ν t ( − x → ,Δ c ,t)=0.00084Δ c F 2 ( − x → ,Δ c ,t)(10) whereΔ c =π/k c isthecut-offlengthcorrespondingtothecut-offwavenum- berk c ,whichischoseninspectralspace.TheexponentofF 2 indicates thenumberoftimesthehigh-passfilterhasbeenappliedtotheresolved flowfield.The2 nd -orderstructurefunctionisdefinedby F 2 ( − x → ,r,t)=hk − u → ( − x → + − r → ,t)− − u → ( − x → ,t)k 2 i −→ (11) r r=kk − u → , − x → and − r → denotethevelocityvector,positionvectorandseparation vectorrespectively.hidenotesthestatisticalaverageovertheentirefluid domain.Inpractice,thecut-offwavenumberischosenequaltothecut-off wavenumberimposedbythemeshsizeΔ x :k c =k x =π/Δ x .Thecut-off wavenumberissuchthatthefilteredvariableuˆinspectralspaceisgiven yb ( uˆ= uˆsi|k|≤k c (12) 0otherwise withuˆtheFouriertransformofu.Themodelisbasedontheequation governingtheevolutionoftheresolvedenergyspectrumE(k,t),depen- dantonthewavenumberkandtimet: ∂ +2νk 2 E(k,t)=T <k c (k,t)+T SG (k,t)(13) ´³ t∂ withtheassumptionthatk<k c .T <k c (k,t)representstheenergytransfer betweentheresolvedscales(k<k c )andT SG (k,t)representstheenergy transferbetweenthesubgrid-scales.Thislasttermrequiresmodeling. Theturbulentviscosityinspectralspaceisdefinedby T SG (k,t) ν t (k,k c ,t)= 2k 2 E(k,t) (14) Whenthisexpressionisrewrittenwithrespecttothekineticenergyat thecut-offfrequency,oneobtains v u E(k,t) k ν t (k/k c ,t)=f(k/k c ) t c (15) c 01

2Thenumericalmodelandinitialcondition withfaknownfunction.Theexpression(15)showsthatthemodelwill notdissipatewhilethereisnoenergyatthecut-off.Thisisthecasefor thesimulationsinthepresentstudy,wheretheinitialconditionshave anegligibleenergylevelatthecut-off.Theenergylevelatthecut-off increasesthroughthecourseofoursimulationwhenlocallytheflowmay transitionandwhentheturbulentviscosityfollowsthisevolution.This modelprovestobewelladaptedforoursimulations. NumericalMethods ThenondimensionalizedNavier-Stokesequationsarediscretizedinspace usingafinitedifferencemethod.ThecomputationalmeshesareCartesian andregularinthethreedirections.Themeshsizeishandindirection Ox,thegridpointscanbegivenasx i =h∙(i−1). Thetimeintegrationhasbeendonebya3 rd -order3-stageRunge- Kuttamethod.WhenoneconsidersythesolutionofCauchy’sproblem y 0 =f(t,y),theintegrationfollowingRunge-Kutta’smethodgives: y(t+Δt)=y(t)+Δt∙f ˆ (t,y) f ˆ (t,y)=1/4K 1 +3/4K 3 K 1 =f(t,y) K 2 =f(t+Δt/3,y+Δt/3K 1 ) K 3 =f(t+2Δt/3,y+2Δt/3K 2 ))(16) Thespatialderivativesoforder1(convectiveterms)andorder2(dif- fusiveterms)arecomputedusinga6 th compactscheme[8](Pade´type). Forthefirstorderderivatives(du/dx)(x i )ofvariable/functionuata pointx i ,weintroducetheapproximationu i 0 ,obtainedbyaPade´scheme whichinit’sgeneralformreads 00000 βu i−2 +αu i−1 +u i +αu i+1 +βu i+2 = c u i+3 −u i−3 +b u i+2 −u i−2 +a u i+1 −u i−1 (17) 6h4h2h InordertoobtainaschemewithatruncationerroroftheorderO(h 6 ), onehastosatisfythefollowingconstraint a+2 4 b+3 4 c=2 5! (α+2 4 β)(18) !4 11