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PSE

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PSE

    PSE

App1.Math.Mech.?Eng1.Ed.,2008,29(1):1-8

    ;DOI10.1007/s10483008-01017

    ;~EditorialCommitteeofApp1.Math.Mech.and

    ;SpringerVerlag2008

    ;AppliedMathematics

    ;andMechanics

    ;(EnglishEdition)

    ;PSEasappliedtoproblemsofsecondaryinstabilityinsupersonic

    ;boundarylayers

    ;ZHANGYong-ming(张永明),ZHOUHeng(周恒),

    ;(1.DepartmentofMechanics,TianjinUniversity,Tianjin300072,P.R.China ;2.LIUHuiCenterofAppliedMathematicsofNankaiUniversityand

    ;TianjinUniversity,Tianjin300072,P.R.China)

    ;(ContributedbyZHOUHeng)

    ;AbstractParabolizedstabilityequations(PSE)approachisusedtoinvestigateprob- ;lemsofsecondaryinstabilityinsupersonicboundarylayers.Theresultsshowthatthe ;mechanismofsecondaryinstabilitydoesworkwhetherthe2.Dfundamentaldisturbance ;isofthefirstmodeorsecondmodeT-Swave.Thevariationofthegrowthratesofthe ;3_Dsubharmonicwaveagainstitsspan-wisewavenumberandtheamplitudeofthe2

    D

    ;fundamentalwaveisfoundtobesimilartothosefclundinincompressibleboundarylayem. ;Butevenastheamplitudeofthe2-Dwaveisaslargeastheorder2%themaximum ;growthrateofthe3-Dsub-harmonicisstillmuchsmallerthanthegrowthrateofthe ;mostunstablesecondmode2.DT.Swave,Consequentlysecondaryinstabilityisunlikely ;themaincauseleadingtotransitioninsupersonicboundarylayers.

    ;Keywordsparabolizedstabilityequations,secondaryinstability,fundamentaldistur- ;bances,sub-harmonicwaves

    ;ChineseLibraryClassification0357?41

    ;2000MathematicsSubjectClassificatlon76E09

    ;Introduction

    ;Theproblemoflaminar-turbulenttransitionisofgreatinterest.Forincompressibleflows, ;transitionmaystartfromthelinearamplificationofasmall2

    Ddisturbance.Whentheampli

    ;tudeof2-Ddisturbancebecomeslarger.3

    Ddisturbancesmaybegeneratedduetononlinear

    ;effectandstarttoplayanimportantroleintransition.1eadingtoturbulentflowsfinally. ;Oneofthepossiblemechanismsforthegenerationof3

    Ddisturbancesisthesecondaryin

    ;stabilitymechanism,asintroducedbyHerberttl-31.Thismechanismhasbeenconfirmedb

y

    ;experiments[4J,manifestedintheappearanceofAshapedstructuresshownbvflowvisualiza-

    ;tion.Thevariationofthegrowthrateofthe3

    Dsub-harmonicwaveagainstitsspan-wisewave

    7. ;numberhasalsobeenmeasured[5

    ;Theproblemconcernedinthispaperis.whetherthesamemechanismworksforsupersonic ;boundary1ayers.Sofar.therewereonlyafewworksinthisregard.Onewastheexperimental ;.Received.Dec.4,2007

    ;ProJectsupportedbytheNationalNaturalScienceFoundationofChina(Nos

    10632050.,90716007):

    ;andtheFoundationofLIUHuiCenterofAppliedMathematicsofNankaiUniversityandTianjin

    ;University

    ;CorrespondingauthorZHOUHeng,Professor,E-mail:hzhoul@tju.edu.cn ;

    ;2ZHANGYong-mingandZHOUHeng

    ;QMkinmhichsubhar

    ;confi~duetosecondary

    ;anemg~:ThIBoth

    ;numericalsimulations

    ;erwasthe

    ;monichasbeendetectedinthetransition.butonecanhardly

    ;instability,becauseitwasonlysignalsdetectedbyhotwire

    ;numericalstudyofDong&Zhou[.

    ;TheycarriedoutdireCt

    ;(DNS)sothatsecondaryinstabilitymechanismwasconfirmedandthe

    ;variationsofthegrowthratesofsub

    harmonicwavesagainstitsspan.wisewavenumberandthe

    ;amplitudeofthe2

    Dfundamentalwerediscovered.However,DNSistimeconsuming.sothat ;thenumberofcasesinvestigatedissurelyverylimited.Asaresult.thethresholdamplitudeof ;thefundamentalwavesforproducingunstablesub-harmonicwaveswasnotgivenbyDong&

    ;Zhou.

    ;Ithepast.fewyears,PSEapproachforthecompressibleflowshasbeendeveloped[9-~1]as ;itscomputationaltimerequiredissignificantlyless.Zhang&Zhou[12Jverifiedtheeffectiveness

    ;ofPSEmethodforcompressibleboundarylayersandfoundthatresultsfromPSEagreed ;quitewellwiththosefromDNS.Consequently,PSEcouldbeareliabletooltoinvestigatethe ;evolutionofdisturbancesinsupersonicboundarylayers.

    ;Inthispaper,PSEmethodisusedtoinvestigatethesecondaryinstabilityinsupersonic ;boundarylayers.

    ;1Governingequations

    ;StartingfromthefullNavierStokesfNS1equations,thefulldisturbanceequationscanbe ;obtained.Then.stabilityequationswerederived.withthecharacteristicsofdisturbancesbeing

    ;takenintoaccount.Finally,equationsareparabolizedtobecomePSE,undertheassumption ;thatthegrowthoftheboundarylayerthicknessisslowinthestream.wisedirection. ;With~llesuperScriptdenotingthedimensionalquantities.thefullNSequationsin3D

    ;Cartesiancoordinatesare

    ;+t()=0

    ;

    ;[?]=×]+2]

    ;?

    ;??+西

    ;p=pRT,

    ;(1)

    ;whereV=(,v,w)Tisthevelocityvector,tthetime,Pthedensity,Pthepressure, ;Tthetemperature,Cpthespecificheat,thethermalconductivity,thefirstcoefficient ;ofviscosity,thesecondcoefficientofviscosity,andRthegasconstant,theviscous ;disgipationfunCtion,expressedas

    ;=2s:s+(.V).(2)

    ;wheresisthestrainratetensor.

    ;T0maketheequationsnondimensional,f=4~eXo/Ueisusedasthereferencelengthscale, ;whereisthedistancefromtheleadingledgetotheentranceofthecomputationaldomain, ;thekinematicViscOsity,:thefreestreamveloCity,andthesubscript”e’’impliesvaluetakenat

    ;theouteredgeoftheboundarylayer.0therreferencequantitiesarethefreestreamvelocityue ;andfreestreamtemperature.Theresultingnondimensionaldistancefromtheleadingled

    ge

    ;totheentranceofthecomputationaldomainisz0=/=,///,theCorrespOnding

    ;ReynoldsnumberisReo=f/7e=4UeX0/=X0.Theresultingnon-dimensionalNrS ;equationsarethenobtained,withthesereferencequantitiesbeingused. ;

    ;PSEasappliedtoproblemsofsecondaryinstabilityinsupersonicboundarylayers3 ;Theflowvariablesaredecomposedasthesumofthesteadybasic’flOWandthedisturbance

    ;u=+u

    ;P=+P

    ;.

    ;+

    ;V=+V

    ;T=f+T

    ;=+

    ;W=+W

    ;P=+P

    ;=+.

    ;whereasuperscript””denotesthebasicflow.andaprime’thedisturbance.Substituting

    ;(3)intothenondimensionalNSequationsandsubtractingtheequationscorrespondingtothe

    ;steadybasicflow,oneobtainsthedisturbanceequations: ;02~02~V

    ;x

    ;02~

    ;+

    ;(r0~AO~+B+c.)

    ;whereisthedisturbancevector=(P,tz,u,,T)T.ThecoemcientsmatricesVxz,…2,

    ;,z,z,r,A,B,c,Daredecomposedintoalinearpartandanonlinearpart,donaotoerdby

    ;superscripts”l”and’’n”,respectively.Theresultingmatricesarewrittenas

    ;z=+,=

    ;=+,=.

    ;F=Fl+F”A=Al+A”.

    ;+.=+

    ;+,.=.+,

    ;B=B+B”,C=Cl+C”,

    ;D:DI+Dn

    ;Thedisturbanceequations(4)canthenberearrangedinthe.followingform

    ;02~

    ;+

    ;02~

    ;.

    ;102~

    ;+

    ;02~

    ;+.

    ;02~

    ;+.

    ;02~

    ;+

    ;(r++B++D1,?(5)

    ;wherevectorfunctionontherightsiderepresentsthenonlinear,termswithrespecttothe

    ;disturbance.Toaccountforthenonlinearproblem,thedisturbanceisassumedtohavethe

    ;followingform:

    ;(,Y,(,).mn()dn(6)

    ;wherem=(,m,n,0m,m,)donatestheshapefunction,&mft’,

    ;thestream-wise

    ;wavenumber,thespan-wisewavenumber,and?Tt02thefrequency~ ;Inpractice.onlya

    ;limitednumberoftermsintheaboveexpressionareretained:tnourcasem:Variedfrom

    ;to7andnvariedfrom3tO3.SubstitutingEq.(6)into(5),neglectingthetermsoforder

    ;1/Re,andcollectingtermswiththesameffequencyaridspanwisewavenumber,aparaboiized

;stabilityequationisobtainedforeachshapefunctionofmodefm,n)asfoli0wS:

    ;l+

    ;‘Yymn.

    ;02Stun

    ;Oy+

    ;(m=0,1,2,…?;=0,12,…

    ).Because(,y,z,)is;~rea]., ;whereFmnisthenonlinearternformode(,

    ;(m0)needtobeconsidered,othertermsaretheircomplexconjugates. ;(7)

    ;with

    ;

    ;4ZHANGYong-mingandZHOUHeng

    ;No-slipandisothermalboundaryconditionsareusedatthe,wall, ;timn=西mn=西mn=n=0,atY=0.

    ;Free-streamconditionsareusedoutsidetheboundarylayer ;timn=mn=西mn=n=0,YOO.

    ;Butformode(0,0),theyarereplacedbythecondition ;U00::wO0’:O0’:0,Y.._-U,_..

    ;

    ;toall0wthemeanfl0wdistortionadjutigitselfformassbalance ;2NumericalmethOd

    ;(10)

    ;Inthestream-wisedirection,uniformstepsizeisemployed,whichmustbesmallerthan

    ;one-twentiethofthefundamentaldisturbancewavelength.Non-uniformgridswithdenser

    grids

    ;nearthewallareemployedinthenormaldirection.Fortheapplicationofhighorderdifferen

    Ce

    ;schemeinthisdirection,thecoordinateYistransformedintoanewcoordinate

    byaproPer

    ;transformation:

    ;=()(11)

    ;suchthatinthenewcoordinatesystem,thegridpointsareuniformeddistributed.Andthe

    ;equationshavetobetransformedaccordingly.

    ;Thesecond.orderbackwardsdifference:

    ;Oxi=

    ;2Ax+

    ;()+()~i-2,一一一十一//(12)

    ;areusedinthestream.wisedirection.Thederivativeswithrespectto

    areapproximatedby

    ;fourth-ordercentraldifferenceschemes:

    ;+2+8~5+l8~51+2

    ;12?

    ;