Solution evolutionary method of compressible boundary layers stability problems

Transcription

1 S. A. Gapoov A. N. Smov Itratioal Joural o Mathmatical ad Computatioal Mthods Solutio volutioary mthod o comprssibl boudary layrs stability problms S. A. GAPONOV A. N. SEMENOV Khristiaovich Istitut o hortical ad Applid Mchaics Novosibirsk 639 RSSIA gapoov@itam.sc.ru smov@itam.sc.ru Abstract: I th papr th solutio volutioary mthod o hydrodyamic stability problms is ord. h ssc o this mthod cosists that a arbitrary iitial disturbac is dscribd by o wav with th gratst icrmt o larg tims which varis accordig to th law xp( i ). I ordr to vriy th w mthod ad to work out th umrical schm th stability calculatios wr carrid out also o th bas o th classical thory. h volutioary mthod is usd to study th cts o th gas ijctio dirctio through a porous surac o stability o a suprsoic boudary layr at th Mach umbr M=. It was stablishd that with rductio o a slop agl o a gas ijctio to th lat plat th stability o a boudary layr icrass ad th tagtial blowig iluc o a boudary layr stability o is poorly Ky-Words: volutioary mthod comprssibl boudary-layr hydrodyamic stability gas ijctio umrical schm Itroductio Porous coolig is a ctiv mthod or a thrmal protctio o hat-strssd lmts o tchical apparatus [-]. h basic mchaism o a porous coolig cosists i absorptio o th thrmal rgy o th hot gas by a cold gas which is ijctd through a prmabl surac. Whil th dirctio o a blowig cold gas rlativly stramlid suracs may b dirt (rom ormal to tagtial). I additio to th hat protctio thr is aothr importat problm. It is coctd with th cotrol o th lamiar-turbult trasitio. It is kow that with a icras o a gas dsity ar th wall th boudary layr stability icrass. o rais dsity ar th walls it is possibl to blow havy gas through a porous wall. For a cas o a subsoic boudary layr th possibility o its stabilizatio o th basis o a havy gas ijctio was coirmd i [3]. Normal ilow rlativly a stramlid surac promots to a apparac o a ilctio poit i a vlocity proil that lads to th dstabilizatio o low [4-6]. For rductio o this ct it is possibl to blow a gas udr som agl to th mai low dirctio. h tagtial ilow is th limitig cas th vctord ijctio. h boudary layr stability with th gas ijctio udr agls ot qual to π to th surac has ot b ivstigatd so ar. It srvd as motivatio o th ral papr i which th cas o th uiorm gas ijctio is cosidrd. As or mthods o th stability charactristics calculatio it is cssary to otic that as a rul authors us th stadard mthod o lmtary wavs ladig to th solutio o th igvalu problm o th homogous systm o ordiary dirtial quatios with homogous boudary coditios [4.7]. h lack o this mthod is i th diiculty to id th wavs with th highst icrmt. Its sarch coms to th d succssully udr a coditio i its approximat valu is kow. hror growig i tim wavs giv th rot dirctio ad wav umbr dpdig o th icomig valus o mai low such as Mach umbr Ryolds umbr ad othrs ar calculatd o th bas o small chags o dtrmiativ paramtrs. Howvr th wav with th maximum icrmt or som basic trms will ot b th dtrmiativ o (with a maximum growth actor) or th othr low paramtrs. hror it is dsirabl to hav such a calculatio mthod which would guarat uiquly obtaiig o th wav with th highst icrmt. For liar problms this ca b achivd by a volutioary mthod by th itgratio ovr tim o partial dirtial quatios. Bcaus ay disturbac satisyig uiorm boudary coditios ca b dcomposd ito th sum o th wavs with dirt icrmts th wav with th largst icrmt will domiat at larg tims. his mthod ca b calld by th usual trm - th stablishig mthod. I cotrast to th grally accptd mthod o stablishig wh th solutio gos to th costat i our cas th solutio gos to th xpotial dpdc o tim. I th hydrodyamic stability thory thr ar th tmporary istability (wav umbr o uiorm spatial coordiats ar ral) ad th spatial istability (wh prturbatios with ral rqucis growth i th spac). At low ISSN: X 75 Volum 6

2 S. A. Gapoov A. N. Smov Itratioal Joural o Mathmatical ad Computatioal Mthods ampliicatio rat i th spac ad tim which is charactristic or th boudary layrs tmporal ad spatial icrmts ar associatd with th simpl approximat rlatio: th ampliicatio rat i spac quals th gativ tmporary dividd by th wav group vlocity [4.8]. I cssary th mor prcis valu o th spatial ampliicatio rat ca b obtaid by th classical mthod. I this papr w ivstigat thortically iluc o th gas blowig dirctio through th porous surac th suprsoic boudary layr stability usig th classical mthod o lmtary wavs ad volutioary mthod. Basic quatios h iitial quatios o th disturbacs volutio i a suprsoic boudary layr ar wll kow Navir - Stoks cotiuity rgy ad stat [9]: p v dv = - grad - grad div +Div(μ S) dt 3 d div( v ) dt d dp c div p S v dt dt 3 c p div grad Pr p R. Hr v vlocity with compots u v w i x y z dirctios p prssur dsity ad tmpratur c P spciic hat at costat prssur R gas costat S vlocity tsor p Pr c thrmal coductivity dyamic viscosity. I this papr th disturbac i suprsoic boudary layrs o a lat plat at high Ryolds umbrs R u x is xplord whr x u - vlocity dsity ad dyamic viscosity o xtral bordr o boudary layr x - distac rom th rot dg o th plat. I this cas th mai low is idpdt o th trasvrs z coordiat wakly dpdts o x - coordiat ad vlocity i y - dirctio is low. hror th mai (statioary) low ca b cosidrd as a plaparalll. All its paramtrs dpd o th o coordiat y oly vlocity i th x-dirctio u y is uqual to zro. W hav itroducd th dimsiolss: coordiats tim ad low paramtrs i th orm: dx dx dy dz dz d u dt v v u p p p whr x u - th boudary layr thickss idx idicats that th valu is tak at th outr dg o th boudary layr. Vlocity dsity prssur ad tmpratur o th comprssibl gas i th boudary layr ca b rprstd i th orm: u Y u v v w w Y Y p P Y p whr Y P Y vlocity prssur ad tmpratur i th uprturbd lamiar boudary layr. h prturbd paramtrs ar markd by th prim which dpd o X Y Z ad. Equatios or liar disturbacs i th approximatio o Daa-Li Alksv [7 ] or th two-dimsioal boudary layr hav th orm []: u u p u v X Y M X R Y v v p X M Y w w p w X () M Z R Y d u v w v X X Y Z d v X p p X PrR Y p P. h systm () should b solvd with boudary coditios [4]: d u v c d at Y. () h classical thory o stability oudd o th mthod o lmtary wavs a pa Y πxpi X Z. Hr compots th vctor a h ar amplituds o prturbatios u v w. Equatios () ar giv to systm o th liar ordiary dirtial quatios: d i d i c M R ISSN: X 76 Volum 6

3 S. A. Gapoov A. N. Smov Itratioal Joural o Mathmatical ad Computatioal Mthods i d h i ch M R d i c (3) M d d i c i ih d d i c i ih d PrR π P ζ θ. From () it is possibl to rciv w boudary coditios: d c d приy. (4) Wav umbrs ad ar ral at th tmporary istability ad rqucy is complx-valud which is a rsult o solvig th igvalus problm o homogous quatios with homogous boudary coditios. h low i th boudary layr is ustabl or positiv valus o th imagiary part o r ii. I gral th umbr o igvalus is iiit or at last larg. Howvr w ar itrstd primarily i rqucy with th highst valus o th imagiary part. h sarch o such rqucis is a challg. h volutioary mthod or idig o such rqucis is proposd ad ralizd i this papr or th irst tim. h ssc o this mthod cosists that a arbitrary iitial disturbac is dscribd by o wav with th gratst icrmt o larg tims which varis accordig to th law xp( i ). For disturbacs X Y xpiz a a quatios () ad boudary coditios () tak th orm: π X Y M X R Y h h h i X M R Y X M Y (5) r d ih X Y X Y X Y ih X Y PrR Y r. d h c d h. y y y y y 3 h computatioal domai ad th umrical schm h problm was solvd or th priodic prturbatio i th coordiat x i.. a Y X a ( Y X L ) ad moochromatic coditios o th latral coordiat z. h rgio o a itgratio i th ormal dirctio was closd i th itrval <Y<Y. W took ito accout th coditios o quality to zro disturbacs at Y. Valu Y was accptd rathr larg that its additioal icras did ot lad to sstial chag o disturbacs icrmts. For th itgratio o th systm (5) w usd - stp iit-dirc schm []. h irst stp: i i i j j hy M hx R h h i i i R hy i h h h M i i M hy d j j ih hx h P r (6а) i i y ISSN: X 77 Volum 6

4 S. A. Gapoov A. N. Smov Itratioal Joural o Mathmatical ad Computatioal Mthods d j j ih hx h i i y j j ( hx ih i i i i i hy Pr R hy ). h scod stp: a a aj aj h x r. (6б) P h schm is stabl; th approximatio ordr is x y O h h. Valus r h o th ( ) layr wr obtaid rom ach quatio i th appropriat ordr. kow valus at th boudary wr obtaid by itrpolatig o thr adjact poits. h valu was dtrmid by th ormula: ( ) ( in l N ). Calculatios wr prormd util its valu was costat with th accptabl accuracy. I this cas th ral or imagiary q Y X wr chagd accordig to th part o c c rlatio: qr i Y X ar isi X r i. h valu o m L whr m- th umbr o priods stackd o th calculatig rag o L. W usd a rctagular msh with 4 poits i th X- coodriat ad 4 poit i Y- coordiat with th tim stp.. 4 Boudary layr quatios ad thir solutio. I sl-similar variabls boudary layr quatios hav th orm []: d d d g d d g d M d Pr dg. Hr C C - ratio o spciic hats p V M u a - Mach umbr ad a - soud vlocity at th xtral bordr o boudary layr. At a uiorm gas blowig through a wall at a agl λ to th mai low dirctio th vlocity compots o a V Gsi wall ar did as ollows: Gcos g RV. Du to th act that w [3] it is possibl to gt g GRsi w. Lt us th paramtr C RG charactrizs th itsity o th q w suctio or blowig through th surac. I this cas boudary coditios o thrmally isulatd surac ca b writt as: w d at Y : g Cqsi Cqcos R ; at Y :. Itroducig th w variabls: d d z z g z3 z4 z5 ; Pr boudary layr quatios ar writt as a systm o irst ordr quatios: dz z dz dz3 z dz5 Prz4 g (7) dz4 Prz4 F g M. Boudary coditios ar rwritt i th orm: z Cqsi 3 w z Cqcos 4 R z z3 z5. h systm (7) is itgratd by th Rug-Kutta mthod rom wall to Y. Ncssary valus a z b z 5 coditio that z Y 3 m m 5 m ad Y m ar dtrmid durig th itratios basd o Nwto's mthod ad a z Y. h dpdc o o tmpratur was adoptd i a accordac with th Suthrlad's law which i dimsiolss orm ca b writt as ollows: 3 s whr s K- Suthrlad's costat - tmpratur o th boudary layr dg. I wid tuls without hatig at a costat stagatio tmpratur M. It was ac- cptd 3 K i this papr. s ISSN: X 78 Volum 6

5 S. A. Gapoov A. N. Smov Itratioal Joural o Mathmatical ad Computatioal Mthods 5 Rsults h rsults o calculatios o statioary valus i th boudary layr at Mach umbr M = ar prstd i igurs -3. Distributios o th logitudial vlocity tmpratur ad dyamic viscosity ar show i Fig. or th ijctio paramtr =. It should b otd that all th statioary low paramtrs com to uit approximatly at Y = 8. h calculatios rsults o logitudial vlocity proils or dirt valus o th paramtr ar prstd i Fig. Not that approachig to valu o vlocity to uit is slowd with icrasig ijctio rats. hus o ca clarly s that ormal blowig lads to icrasig o boudary layr thicks. Furthrmor a ilctio poit is appard i th vlocity proil which ca cotribut to dstabilizatio o th boudary layr Fig. Proils o statioary low paramtrs M= = Y = =-. =-. =-.3 =-.4 = Y Fig. Distributio o logitudial vlocitis or various valus o th paramtr Iluc o blowig dirctio i th distributio o logitudial vlocity is show i Fig. 3. h vlocity distributio without blowig is markd by symbols. From ths data it ollows that th statioary low paramtrs ar dpdt o th tagtial ijctio wakly. h ormal vlocity compot plays a dcisiv rol i this rspct Fig. 3 Dpdcs o logitudial vlocitis o th ormal coordiat or th dirt λ =-.5 M= = Fig. 4 h dpdc o th ral part o th prssur prturbatio ar th walls ovr tim. r r r 3 A M= R=5 L=4 R=5 M=Y= = =3 Fig. 5 h dpdc o th ral part o th prssur prturbatio o th coordiat X. B X Calculatios rsults o prturbatio paramtrs i th suprsoic boudary layr ar prstd i Fig. 4-8 icludig th maximum dgr o thir Y r ISSN: X 79 Volum 6

6 S. A. Gapoov A. N. Smov Itratioal Joural o Mathmatical ad Computatioal Mthods tmporary growth. Mai rsults wr obtaid o th basis o quatios (3). Stability calculatios wr carrid out by th classical thory or procssig o th sttlmt schm (4). As alrady mtiod at larg tims th solutio is dscribd by a xpotial dpdc o th tim rgardlss o th iitial data. hror w will ot dwll o th iitial data which wr st arbitrarily h R=5 =.5 M=. Y Fig. 6 Dpdcs o amplituds o th vlocity dsity tmpratur ad prssur disturbacs o th ormal coordiat. Fig. 4 shows th tim variatio o th ral part o th prssur amplitud ar th wall Y wh Y =4. I th graph B th rsult is show i th tim itrval 3 Iitial valus o π r o th graph B icrasd i o thousad tims ar show o th top graph A rom which o ca s that i th iitial tim momts thr ar svral rqucis. Howvr ovr tim th most growig rqucy is allocatd which chags udr th law cos( r ) xp i. Fig. 5 shows th distributio o th ral part o th prssur amplitud at th two tims aalogously to Fig. 4. It is s that at larg tims th spatial dpdc is dscribd by a harmoic dpdc with th wav umbr L rathr wll. h rsultig icrmt ω i is ot dirt rom th valu o th classical thory i act. Proils o th absolut valus o th amplituds o th prturbatio ar show i Fig. 6. hy corrspod to th tim wh th solutio cam to a xpotial dpdc. Charactristically that logitudial vlocity has th largst amplitud. hror all amplituds wr ormalizd o th maximum logitudial vlocity valu. It is also should b otd that th ormal vlocity amplitud prturbatio rachs th maximum valu at th outr dg. h cssary computatioal domai is dtrmid mpirically by comparig o th growth rats or dirt valus o Y with th data o th classical thory. From ig. 7 it is clarly visibl that at th thickss Y 4 rsults o umrical modlig dir rom data o th classical thory a littl Y= Y=3 Y=4 Classic thory R=5 M= Fig. 7 Dpdcs o th icrmts o th wav umbr or dirt thicksss = R= M= Fig. 8 Dpdcs o th grow rats o th paramtr or Cq.5 adcq Fig. 8 shows a chag o grow rats dpdig o th wav umbr or various ijctio dirctios o at Cq.5. h rd li corrspods to tagtial blowig ad gr circls marks rprst th rsults without blowig. It is s that th boudary layr stability icrass with dcrasig o th agl λ ad tagtial blowig ( ) dos ot act th boudary layr stability. At th sam tim ormal blowig ca icras th rat ampliicatio i svral tims. ISSN: X 8 Volum 6

7 S. A. Gapoov A. N. Smov Itratioal Joural o Mathmatical ad Computatioal Mthods 4 Coclusios. I th papr th w mthod o stability problm solvig o th boudary layr is proposd which is basd o a volutioary prturbatios dvlopmt i tim.. Iluc o th gas blowig dirctio through a porous surac o th suprsoic boudary layr stability was studid or th irst tim. I th cotrast to th strog iluc o ormal blowig o th boudary layr stability tagtial blowig has a littl ct o it. 3. h dvlopd mthod will b usd i problms o th suprsoic boudary layr stability with blowig o orig gass ad th umrical schm will b work or modlig o oliar problms o th lamiar-turbult trasitio. problm at suprsoic spds SAGI issu 4 97 (i Russia). [] V. M. Kovja Dirtial mthods o th solutio o multidimsioal tasks. Novosibirsk NG 4 (i Russia). [] Mack L. M. Computatio o th stability o th lamiar comprssibl boudary layr. Nw York: Acadmic Prss i "Mthods i Computatioal Physics" (B. Aldr d.) 965 Vol his papr has b supportd by Russia Foudatio or Basic Rsarch (projct No a). Rrcs: [] S.S. Kutatladz A.I. Lotiv Hat-mass xchag ad a rictio i a turbult boudary layr Moscow Ergoizdat 985 (i Russia). [] E.P. Volchkov Wall gass curtais Novosibirsk Nauka 983 (i Russia). [3] J. O Powrs. G. Hich ad S. F. Sh h Stability o Slctd Boudary-Layr Proils NOLR [4] S.A. Gapoov ad A.A. Maslov Disturbac Dvlopmt i Comprssibl Flows Novosibirsk Nauka 98 (i Russia). [5] S.A. Gapoov ad N.M. rkhova hr- Wav Itractios btw Disturbacs i th Hyprsoic Boudary Layr o Imprmabl ad Porous Suracs. Fluid Dyamics 9 Vol. 44 No. 3 pp [6] S.A. Gapoov ad N.M. rkhova Stability ad hr- Wav Itractio o Disturbacs i a Suprsoic Boudary Layr with Mass rasr o th Wall ploiz. Aromkh. Vol. 9 3 P [7] D.W. Du C.C. Li O th stability o th lamiar boudary layr i a comprssibl luid J.Aro.Sci. Vol. No pp [8] M. Gastr A ot o a rlatio btw tmporally icrasig ad spatially icrasig disturbacs i hydrodyamic stability J. Fluid Mch. 96 Vol. 4. P. -4. [9] L.G. Loitsjasky Mchaics o Liquid ad Gas Moscow Nauka. 973 (i Russia). [] M.A. Alksv O asymptotic approximatios i th lamiar boudary layr stability ISSN: X 8 Volum 6