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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?

    ;+2+16~j+130+16~j12

    ;12?

    ;Forthepointsnexttotheboundary,theyareTeplacedbyrespectivelowerorderones ;3Resultsanddiscussions

    ;(13)

    ;(14)

    ;Twocasesarestudiedinthispaper.withthe2-Dfundamentaldisturbancesbeingthefirst ;andsecondmodeT-Swaves,respectively.

    ;3.1CasewithfirstmodeT.Swaveas2-Dfundamentaldisturbance

    ;WetaketheBlasiussimilaritysolutionas.thebasicflow.withMachnumberM=4.5 ;andRe0=10471,andstartwitha2.DT.Swaveanda3.Dwaveattheentranceofthe ;cornputationaldomain.Fortheconvenienceofreference.thefundamental2.DwaveiSassigned

    ;asmodef2,0)wave,whilethe3.DSUb-harmonicwaveiSassignedasmode(1,1)wave.The ;2.DwaveiSafirstmodeT.SwavewithamildgrowthratewithfrequencyCd2d=0.0533and ;stream-wisewavenumbera2d0.0642,anditsshapefunctioniSsolvedfromthecorresponding

    ;Orr.SommerfeldequationinthelinearstabilitytheoryfLST).Thefrequencyofthe3.Dwave

    ;ll=嘞叼

    ;

    ;PSEaNappliedtoproblemsofsecondaryinstabilityinsupersonicboundarylayers5 ;iShalfofthatforthe2Done,SOiSitsstream-wisewavenumber,i.e.d=0.0267,and

    ;OL3d=0.0321.ItsshapecanbeanarbitraryfunctionofYsatisfyingtheboundaryconditions.

    ;DifferentinitialamplitudesA2dofthe2

    Dwaveoftheorder1%havebeentried,andtheinitial

    ;amplitudeA3dofthe3DwaveiS0.01%.

    ;ForA2d=1.2%andspanwisewavenumberofthe3Dwave=0.2327,Figure1shows

    ;thevariationoftheshapefunctionof3Dwavealongthestream-wisedirection.Theshape

    ;functionkeepschanginguntilX=430,duringwhichthelocationofthemaximumI

    11Imoves

    ;fromY5.8toY=5.3.Furtherdownstream.itnolongerchanges.Figure2showsthevariation ;oftheamplitudeofthe3

    Dwaveinstream-wisedirection.Itvariesdramaticallyfromentrance

    ;toX=430.andthengrowsslowlyfromthereon.AsshowninFig.3.thevariationofthe ;growthrateofthe3DwaveiSsimilartothatoftheamplitude.Aftertheadjustmentnear ;theentranceregion,anunstable3

    Dsub’harmonicwave(>0)iSfoundinthesupersonic

    ;boundarylayer,confirmingthatthemechanismofsecondaryinstabilitydoeswork. ;TomakesurethattheSOfoundgrowthrateofSUb-harmonicswavesiSreliable,A3dtakes3 ;differentvalues,i.e.0.01%,0.001%and0.001%whiletheotherparameterskeepunchanged.

    ;Theresultingamplitudeandgrowthratevariationsofthe3

    DwaveareshowninFigs.4and5.

    ;respectively.Thethreegrowthratesagreeperfectlywitheachother,whilethethreeamplitu

des

    ;differfromeachotherinorderofmagnitude.butallofthemaresmallenough.thusconfirmin

    g

    ;thatthegrowthratesof3Dwavesobtainedherearereliable.

    ;O

    ;llll:0

     ;2E-05?4E05

    ;l:430,llll:860

    ;6E-05

    ;4EO5

    ;2E05

    ;0

    ;-

    ;2E-05

    ;-

    ;3000300600900l2oo

    ;Fig?lProfile

    ;.

    ;s

    ;.0fDsub-harmonicFig.2Amplitudeof3-DwaveatY:5.3wavell1lat=0,430,860

    ;0

    ;b0.02

    ;

    ;0.04

    ;-

    ;30003006009001200

    ;Fig.3Growthrateof3-DwavebasedonFig.4Amplitudesof3-DwavesatY5.3

    ;amplitudeatY:5.3

    ;

    ;6ZHANGYong-mingandZHOUHeng ;b0

    ;

    ;O

    ;Fig.

    ;

    ;3oo03006009001200

    ;Growthratesof3Dwavesbasedon

    ;amplitudesaty=5.3

    ;Fig.6Sllbharmonicgrowthratesagainst ;spanwisewavenumbersfordifferent ;amplitudesof2-Dfundamentalwave ;SamecomputationhasbeencarriedoutfordifferentA2dand,andtheresultsareshown

    ;inFig.6.whereisthegrowthrateofthe3.Dwaveat=860andAtheamplitudeof2D ;wave.alsoat=860.FromFig.6thefollowingconclusionscanbedrawn:

    ;(11Asvaries,therei8amaximumofforeachA,allat=0.2327.Becausethevariation ;ofisstepwise,andthecurvesareobtainedsimplybyconnectingthemwithstraight:lines,so

    ;actually,theymaynot.correspondexactlyt.o=0.2327.buttosomedifferentvaluesnotfar

    =0.2327. ;from

    ;f21AsAincreases,themaximumof0.alsoincreases.

    ;(3)Unstable3-DwaveswouldappearonlyifA1.09%.ForAsmallerthanthisthreshold, ;onlystable3Dwaves(<0)exist.

    ;(4)Forthelargestamplitudewehavecomputed,Le.A=1.91%,themaximumgrowthrate ;is8,23×105,muchsmallerthanthegrowthrate5.25×100Ofthemostunstablesecond

    ;mode2DTSwave..

    ;3.2CasewithsecondITIOdeTSwaveas2Dfundamentaldisturbance

    ;Insteadoftakingafirstmode2DT.Swaveasthefundamentalwave.wenowtakea2

    ;DsecondmodeT.Swaveasthefundamentalwave,otherconditionsremainthesamea8in ;previoussection.The2

    DsecondmodeT-SwavehasafrequencyCO2d=0.2791andawave

    ;number2d=0.3145.Accordingly,theparamet.ersof3

    DwavearetakenasOY3d=0.1396and

    ;a3d=0.1572.

    ;Thereliabilityofthegrowthrat.eofthesub.harmonicwavehasalsobeencheckedinthe ;sameway.Thatis,A3dtakesthreedifferentvalues0.01%,0:001%and0,001%respectively, ;withA2d=0.1%and=0.6981.Figures7and8showthevariationsoftheamplitudesand ;growthrat.esofthe3-Dwavesat’Y=9.5,respectively.Similartotheresultsinsection3.1,the

    ;threegrowthrat.esagreedperfectlywitheachother,withthethreeamplitudesdifferfromeach

    ;other.inorder

    f’magrtitu~thusconfirmingthatthegrowthratesobtainedherearereliable.

    ;Again,thecomputationhasbeencarriedout.fordifferentA2dand,andtheresultsare ;showninFig9whereisgrowthrat.eobtainedatx=455andAtheamplitudeof2-D ;waveatthesamelocation.FromFig.9thefollowingconclusionscanbedrawn: ;f11Themechanismofsecondaryinstabilityalsoworksforsecondmodefundamenta1wave,

    ;asunstable3Dsu,).harmonicwavesdoexistforcertainValueofAand8. ;f21Asvaries,therearemaximumsforthegrowthrate,againclusteraroundacertain ;valueof8;0.3490

    ;

    ;PSEasappliedtoproblemsofsecondaryinstabilityinsupersonicboundarylayers7 ;Fig.7Amplitudesof3-DwavesatY=9.5

    ;O.O5

    ;0.04

    ;0.O3

    ;b0.O2

    ;0.0l

    ;0

;

    ;O.O1O

    ;Fig.8Growthratesof3-Dwavesbase,

    ;don

    ;amplitudesatY=9.5

    wisewavenumbersfordifferent ;Fig.9Subharmonicgrowthratesagainstspan

    ;amplitudesof2-Dwaves

    ;f3)AsAincreases,themaximumofalsoincreases,

    ;f41Unstable3.Dwaveswouldappearonly?ifA>0.86%.ForAsmallerthanthethreshold,

    ;onlystable3Dwavesexist.

    ;(5)ForthelargestamplitudeA=1,73%wehavecomputed,themaximumgrowthrateis ;6.388×10.againappreciablysmallerthanthegrowthrate5.448×10

    .ofthemostunstable

    ;secondmode2DTSwave.

    ;Unfortunately,thereiSnodetailedexperimenta1resultthatcanbeusedtocheckourresults. ;AlthoughMaslovdiddetectsub

    harmonicsignalsincertaintransitionexperiments.noreliable

    ;quantitativedatahasbeengiven,andonecannotbesureifthedetectedsignalreallycame ;from3Dwavegeneratedbysecondaryinstability:

    ;4Conclusions

    ;(1)ThePSEmethodcan’beappliedtostudythesecondaryinstabilitymechanisminsuper

    ;sonicboundarylayer.atleastforMachnumber4.5.

    ;(2)Themechanismofsecondaryinstabilitydoeswork,nomatterthefundamentalwaveis ;firstmodeorsecondmodeTSwave.

    ;(3)Asthespanwisewavenumberofthe3Dwavevaries,thereisamaximumforthe

    ;

    ;8ZHANGYong-mingandZHOVHeng

    ;growthrate.

    ;(4)Astheamplitudeofthe2Dfundamentalwaveincreases,themaximumgrowthrateof ;the3.Dwavealsoincreases.

    ;(5)Unstablesub-harmonicwaveswillappear,onlyiftheamplitudeofthefundamentalwave

    ;exceedsacertainthreshold.

    ;f6)Forthecasesweinvest~ated,eventheamplitudeofthe2Dfundamentalwaveisas ;largea8oftheorder2%,themaximumgrowthrateforthe3Dsubharmonicwaveisstillmuch ;smallerthanthegrowthrateofthemostunstablesecondmode2DTSwave.Inviewofthis, ;secondaryinstabilitymechanismisunlikelyanimportantmechanismleadingtotransition.This

    ;maybeaQfOrtheeffectivenessoftheemethodinpredictingthetransitionlocation,3.8 ;suggestedbyCebecietall13J,basedontheirownexperience.

    ;}

    ;References

    ;

1JHerbertTh.Secondaryinstabilityofplanechannelflowtosubharmonicthreedimension

aldistur

;bances[J.PhysicsofFluids,1983,26(4):87

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