Supersonic boundary layer stability with vectored mass transfer through a porous surface

Transcription

1 Suprsonic boundary layr stability with vctord mass transfr through a porous surfac S. A. GAPONOV and N. M. TEREKHOVA Khristianovich Institut of Thortical and Applid Mchanics Novosibirsk, , RUSSIA gaponov@itam.nsc.ru trkh@itam.nsc.ru Abstract: Th study continus th cycl of invstigations concrnd with th modling of th mthods of controlling flow rgims in comprssibl boundary layrs. Th ffct of distributd hat and mass transfr on th stability paramtrs of a suprsonic boundary layr is considrd at a modrat suprsonic Mach numbr M =. Emphasis is placd on th modling of both th normal injction, whn only th V componnt of th man vlocity is nonzro, and injction in othr dirctions, including th tangntial injction, whn only th U componnt is nonzro on th wall. Th formulation of th problm is similar with that of th gas curtain influnc on th small fluctuation dvlopmnt. It is assumd that th ffct of th injction of a similar gas with diffrnt tmpraturs is analogous to th injction of a gas with diffrnt dnsitis, namly, th cold gas injction mimics th havy gas injction, and vic vrsa. For this rason, in this study this modling is ralizd by mans of varying th tmpratur factor (wall hating or cooling). Th cas, in which th so-calld cutoff rgim is ralizd, that is, th vlocity disturbancs on a porous surfac can b takn to b zro, is also considrd. Ky-Words: comprssibl boundary layr, hydrodynamic stability, laminar-turbulnt transition. Introduction In numrically invstigating th mthods of controlling th rgims of suprsonic flow past bodis considrabl attntion has bn givn to th distributd mass transfr ffct [ 3]. Th us of injction or suction has a considrabl influnc on both th proprtis of th original flow and its stability. Th suction ffct on th stability and laminar-turbulnt transition is wll studid for a subsonic boundary layr [4]. Th mploymnt of mass transfr producs crtain charactristics of boundary layrs that nsur th givn disturbanc paramtrs on th rang of th loss of stability and th transition. Th suction withdraws low vlocity gas masss from wall rgions; as a rsult, th man vlocity profils bcom mor inflatd and hav gratr transvrs gradints in th wall rgions, which lad to an incras in thir stability, th ris of th critical Rynolds numbrs, and transition dlay. Th boundary layrs bcom thinnr with th rsult that th tndncy to transition in th turbulnt stat rducs [5]. Contrariwis, gas injction lads to th boundary layr thickning and a dcras in its inflation with th tndncy to th apparanc of local inflctions which acclrats th disturbanc growth, rducs th stability margin, and maks th transition narr. Nvrthlss, gas injction through a prmabl wall is widly applid in th practic. First of all, this is du to th us of cold gas injction as a mans of thrmal protction of thrmally strssd lmnts of nginring dvics. Its main mchanism consists in th absorption of th thrmal nrgy of th hot gas by a coolr arriving through th prmabl surfac. In this cas, th dirction of th cold gas injction rlativ to th surfac in a flow can b diffrnt, from normal to tangntial. In [6] many aspcts of using th slot injction in th form of gas curtains ar prsntd and thir ffct on hat transfr and th thrmal paramtrs of boundary layrs is discussd. Typical of th gas curtains is that th cold gas is injctd along th surfac. Th thortical modling of th slot injction rquirs th us of th complt Navir Stoks quations which maks th problm solution considrably mor difficult. Th problm can b considrably simplifid in th cas of gas injction through a prmabl surfac. Howvr, th tchniqus of tangntial mass transfr through prmabl walls wr for a long tim absnt. In th fw rcnt yars a crtain progrss in this dirction has bn achivd which maks it possibl to raliz injction/suction at diffrnt angls to th surfac. By way of illustration, w will not th papr [7] in which an insrt nsuring th tangntial injction is dscribd. In this connction, a thortical invstigation of th boundary layr in th prsnc of gas injction through porous walls at diffrnt angls to th surfac has bcom particularly topical. Injction of a havy or light forign gas can b usd for th purpos of controlling th friction drag and hat fluxs. In [8] it was thortically shown that injction of a coold similar gas influncs th drag and th hat fluxs in th sam ISSN: Volum, 06

2 fashion as injction of a havy forign gas. Along with th thrmal protction problm and th control of th drag and th hat fluxs, thr xists anothr important problm concrnd with th control of laminar-turbulnt transition. With incras in th gas dnsity nar th wall th boundary layr stability incrass which is achivd by havy gas injction through a porous wall. In th cas of a subsonic boundary layr th thortical possibility of its stabilization by mans of havy gas injction normal to th wall was confirmd in [9]. In th prsnt study gass with diffrnt dnsitis ar modld by varying th tmpratur factor (wall hating or cooling) [0, ]. Unfortunatly, th possibility of th disturbanc supprssion using this mans has its own rstrictions. Th injction normal to th surfac favors th apparanc of an inflction point in th vlocity profil which lads to flow dstabilization. To rduc this ffct an attmpt to injct th gas at a crtain angl to th main flow dirction can b mad. Hr, th joint ffct of th injction and th tmpratur factor is studid in th cas of flow past a prmabl porous surfac, whos por radii ar much smallr than th boundary layr thicknss scals. In this cas, th so-calld cutoff rgim [] is ralizd, whn th vlocity disturbancs on th body surfac can b takn to b zro. Emphasis is placd on th modling of distributd injction and an analysis of its ffct on th disturbanc dvlopmnt scnario at a modrat suprsonic Mach numbr M=. Both th normal injction, whr only th man vlocity componnt V prpndicular to th surfac is nonzro, and injction in diffrnt dirctions, including th tangntial injction, whr only th longitudinal componnt U is nonzro, ar modld. This formulation can b similar with th problm of th gas curtain influnc on th small fluctuation dvlopmnt scnario. In th cas of comprssibl gas flow thortical invstigations ar mad mor difficult by th ncssity of taking th tmpratur and dnsity disturbancs into account. At M = only vortx disturbancs, or travling Tollmin Schlichting wavs (first mod), ar considrd. At prsnt it is wll stablishd that th normal injction dstabilizs both vortx and acoustic disturbancs. At th sam tim, in th cas of flow past a surfac with hat transfr th influnc of this factor on disturbancs can b diffrnt: in cooling th vortx disturbancs ar stabilizd (th rat of thir stramwis growth diminishs), whras th acoustic disturbancs ar dstabilizd. Th sam diffrnc in th injction ffct xists in th cas of hating: th vortx disturbancs ar dstabilizd, whil th acoustic disturbancs ar stabilizd. Th situation is mad considrably mor complicatd in th cas of th joint influnc of mass and hat transfr. Thus, th purpos of th study is an invstigation of th influnc of injction and hat transfr on th man vlocitis, th critical Rynolds numbrs of vortx disturbancs, and thir frquncy dpndncs (frquncy cuts). Basic quations and mthods of solution. Equations for Disturbancs. Th mthod of dtrmining th disturbanc paramtrs in comprssibl suprsonic boundary layrs is basd upon th classical prturbation mthod. W will prsnt crtain ncssary facts. Th flow fild is rprsntd in th form of th sum of th man and fluctuating quantitis [5] u U( Y) ε u,ε v,ε w, ρ( Y) ε ', P( Y) εp, T( Y) ε, p / P ς / ρ Θ / T, whr ε is th fluctuation fild scal. W will considr th disturbd filds of th comprssibl gas vlocity, dnsity, prssur, and tmpratur in th dimnsionlss Cartsian coordinat systm X, Y, Z = (x,y,z)/ δ, whr δ is th scal lngth, δ xν / U. Th primd and primlss quantitis ar th fluctuating and man componnts of th corrsponding quantitis dividd by thir valus at th outr dg of th boundary layr (U, ρ, T, P ). Th wav solutions ar sought in th form: Z ' Z( Y)xp( iθ), (.) whr Z u, v, w, p, ar th prturbations of th longitudinal, normal, and transvrs vlocitis, th prssur, and th tmpratur, θ αx βz ωt, α αr iαi, αi 0 is th growth rat, th ral frquncy ω π f ; and th wavnumbrs α, β and th frquncy ar rlatd by th disprsion quation α=α(ω,β) in accordanc with th linar thory. Th spctral paramtrs and th structural forms of th prturbations ar dtrmind from th Dan Lin systm: ρ( Gu U v) iα p / γm μ u / R 0, Y YY ρ β / γm μ / R 0 Gw i p w YY, ρ Gv p Y / γm 0 (.) Gς ρ v ρ( iαu v iβ w) 0, Y Y μγ ρ( G TY v) (γ )( iαu vy iβ w) YY 0, σr ( p / P / T), G i( U). Hr M is th Mach numbr at th outr dg, γ = c p /c v is th adiabatic xponnt, σ = c pμ /k is th ISSN: Volum, 06

3 Prandtl numbr and k is th tmpratur conductivity cofficint. All th paramtrs ar nonimnsionlizd on thir valus at th outr dg of th boundary layr and th following normalization of th ignfunctions is chosn: vy ( k ). Th homognous systm (.) is th basic systm for dtrmining th ignvalus α at givn β and ω and th Rynolds numbrs R u δ / ν, and for constructing th amplitud ignfunctions of th linar wavs (.). Th systm was intgratd using th orthogonalization mthod [5]. As notd abov, th problm was formulatd for th cutoff rgim (r/δ << ), whn, in viw of th smallnss of th por radii r, th comprssibility ffcts in th por itslf can b nglctd. Thn th following conditions hold: u, v, w, Θ = 0, Y = 0; (.3) at th outr dg and in th far fild (Y = ) th usual conditions of dcay (bounddnss) of th amplitud functions ar prassignd u, v, w, Θ = 0 (.4) Th solution of th ignvalu problm for systm (.) undr th boundary conditions (.3) and (.4) closs th stag of constructing small linar prturbations (.).. Avrag Paramtrs Th boundary layr quations for th dimnsional astriskd quantitis ar usd in th Cartsian coordinat systm (x, y) u u u ρ ( u v ) μ x y y y T T u T ρ Cp ( u v ) μ k x y y y y * * * * ρ u ρ v 0 p p x y, R T In accordanc with [], w introduc th scal lngth δ x ν / U and th dimnsionlss slfsimilar coordinat Y y/ δ. In this systm th di- mnsionlss quations of th boundary layr go ovr into th systm of ordinary diffrntial quations d du du μ g 0 dy dy dy d μ dt g dt (γ )M μ du dy σ dy dy dy (.5) dg U = dy T Whn a similar gas is injctd through th wall at an angl φ, th curtain angl is clockwis masurd from th dirction opposit to that of th main flow. Th schmatics of th injction ar shown blow. Thvlocity componnts at th wall ar dtrmind as follows: V(0)=G sin φ, U(0)=-G cos φ, whr G is th absolut valu of th injctd gas vlocity. In viw of th fact that g(0) = V(0)R/Tw [5], w obtain g(0) = GR/Tw sin φ. W will introduc th paramtr Cq = GR/Tw charactrizing th injction/suction intnsity. In this cas, th boundary conditions can b writtn in th form: on th body surfac (Y = 0) g =Cq sin φ, U =CqTw/Rcos φ,t' = 0 or T = Tw, (.6) whr th formr tmpratur condition corrsponds to a thrmally insulatd wall and th lattr condition to a constant wall tmpratur. on th outr dg (Y= ) U = ; T = (.7) Th systm of quations (.5) with th boundary conditions (.6) and (.7) was intgratd using th Rung Kutta mthod from th wall to Ym, whr Ym is crtainly gratr than th boundary layr thicnss. Th tmpratur dpndnc of μ was takn in acordanc with th Suthrland law. For any rgim th dimnsionlss paramtr of th wav frquncy F rlatd with th frquncy by th quation F= ω/r and th dimnsionlss wav paramtr b = β 0 3 /R wr introducd. Both twodimnsional (plan) wavs with β = 0 and thrdimnsional (obliqu) wavs with β 0 wr considrd. For vortx disturbancs th thrdimnsional componnts ar most growing and thir growth rats ar considrably gratr than thos of two-dimnsional wavs. 3 Discussion of th rsults Calculations hav bn prformd in th assumption of a prfct gas with constant valus of spcific hat ratio γ=.4 and Prandtl numbr σ=0.7. It is hlpful to rcall how th distributd injction/suction influncs th vortx disturbanc paramtrs at M = []. In Fig. th volution of th disturbanc growth rats is shown for four frquncis; th cas of th imprmabl wall corrsponds to th Cq = 0 lin. In th cas of suction (Cq > 0) th disturbancs of all th frquncis stabiliz and bcom dcaying at Cq ISSN: Volum, 06

4 = 0.5, whras in th cas of injction (only th normal injction with U(0) = 0 is considrd) all th disturbancs bcom or rmain growing. Th valu Cq = 0.3 was th last valu for which th disturbanc paramtrs could b obtaind. At this valu th man vlocity profils with an inflction point ar formd and thr aris som difficultis in solving th boundary valu problm for Cq < th 0 φ < π/ rang to th countr-stram injction, and th π/ < φ π rang to th classical stramwis injction. Hr, to liminat variant radings, th limiting positions φ = 0 and π ar rspctivly calld th countrstram and stramwis curtains. Clarly visibl is that th growth rats strongly dcras only in th vicinitis of ths limiting valus. Whil th valu Cq = 0.3 is limiting for th normal injction, in th cas of th curtains Cq can incras almost without bounds. In Fig. it is also wll sn that th diffrnc btwn th growth rats of th two diffrnt curtains incrass with C q, th stramwis curtain favoring th gratr stabilization of th vortx disturbancs Cq 0.05 i Fig.. Growth rats of th thr-dimnsional vortx wavs at M =, R = 600, and b = 0.5 as functions of th mass transfr intnsity Cq for th frquncis F = (0.9, 0.38, 0.57, 0.76) 0 4 (curvs ( 4)) Cq Fig. 3. Effct of th injction intnsity Cq on th growth rats of th thr-dimnsional vortx wavs at M =, R = 600, b = 0.5, and F = for th stramwis φ= π () and countrstram φ = 0 () curtains Fig.. Effct of th angl φ of inclination of th injctd gas vlocity on th growth rats of th thr-dimnsional vortx wavs at M =, R = 600, b = 0.5, and F = ; ( 8) rlat to φ= 0,.57, 3.0, 3.04, 3.08, 3., 3.4, and 3.46; in th sam figur th schmatics of th arrangmnt of th longitudinal U and normal V vlocity componnts ar prsntd. In Fig. injction is schmatically rprsntd. Th angl φ is masurd from th wall, whn th injctd-gas and main-flow vlocity vctors ar oppositly dirctd. In Fig. it can b sn how th disturbanc growth rats vary with th angl φ th valu φ = π / corrsponds to th normal injction, In Fig. 3 this diffrnc is illustratd for fairly larg injctions. Th man vlocitis nar th wall can b radily valuatd from Eq. (.6). Sinc in th cas of th stramwis curtain th gas with only th vlocity componnt U is tangntially injctd into th boundary layr and this componnt naturally incrass with Cq, th ovrall ffct lads to an incras in th wall vlocity. To som xtnt this is quivalnt to th situation occurring in th cas of suction, whn nar-wall low-vlocity layrs vanish. Anothr xplanation of this ffct is that in th wall rgion th diffrnc btwn th outr flow and boundary layr vlocitis diminishs, which lads to th rduction in th ffctiv Rynolds numbr Rff and th growth rats. Th incras in th growth rats in th cas of th countrstram curtain can also b xplaind. In crtain situations (whn th gas is tangntially injctd in a limitd rgion) th flow pattrn can b tratd as a flow with boundary layr sparation, which lads to an in- ISSN: Volum, 06

5 cras in R ff. Th profils of th man longitudinal vlocitis and tmpraturs for diffrnt curtains ar plottd in Fig. 4. In th cas of th countrstram curtain an incras in Cq lads to th transvrs lngthning of th rgion of ngativ U. It is clarly visibl that in th cas of th stramwis curtain th man tmpraturs in th boundary layr dcras, which favors th disturbanc stabilization, and, contrariwis, in th cas of th countrstram curtain ths tmpraturs incras. It is worthwhil to rmind that th tmpratur of th thrmally insulatd imprmabl wall Tw = F R.0 U 0.5 а Fig. 5. Nutral curvs of th vortx mod on thrmally insulatd surfacs for diffrnt Cq in th cas of th normal injction φ=π/, Cq = 0.3 (curv ) and th stramwis curtain φ=π, Cq = 0.3, 50, and 00 ( 4). 0.0 I II Y i T.0 б I II F Y Fig. 4. Effct of th curtain angl and th injction intnsity Cq on th profils of th man longitudinal vlocity (a) and th man tmpratur (b) for Cq = 50 (I) and 00 (II); (, ) corrspond to φ= π and 0. Th ffct of injction and curtain rgims is wll illustratd in Fig. 5 for th nutral curvs on th thrmally insulatd surfac and in Fig.6 for th frquncy cuts. In th cas of th normal injction with Fig. 6. Frquncy cuts illustrating th growth rats of thr-dimnsional disturbancs (b = 0.5) at R = 600 and th sam paramtrs as in Fig. 5 Cq = 0.3 th critical Rynolds numbr is vry small (R c 80) and considrably smallr than R c for th tangntial injction (Rc = 90 for th stramwis and countrstram curtains and for th imprmabl wall). It is ncssary to rcall that in th Cq = 0.3 cas undr considration th variations in U and th man tmpratur ar so small that all th disturbanc paramtrs coincid in th plots. With incras in Cq in th cas of th stramwis curtain th disturbancs ar considrably stabilizd, th critical Rynolds numbr Rc incrass, and th hazardous frquncy rang sharply shrinks. As shown in Fig. 6, in th cas of th normal injction growing fluctuations xist within an ISSN: Volum, 06

6 xtrmly broad frquncy band from 0 to 90 khz, with vry high growth rats; in th cas of th stramwis curtain with th sam Cq this rang rducs down to 0 to 40 khz, whil th maximum growth rats dcras by a factor of thr. Th incrasingly strongr variations ar obsrvabl at highr Cq. In th cas of cooling at a givn Cq th longitudinal vlocity componnt of th injctd gas diminishs for both curtains du to th wall tmpratur dcras (in accordanc with Eq. (.6)), which must hav a dstabilizing (stabilizing) ffct on th disturbanc in th cass of stramwis (countrstram) curtains. Thr appars th compting influnc on th disturbancs which can manifst itslf in crtain rgims..0 U a Y.0 U б Fig. 8. Tmpratur (Tw) dpndnc of th thrdimnsional disturbanc growth rats (b = 0.5) at R = 600 and F = in th cas of th normal injction (Cq = 0.3 and φ=π/, curv ) and at Cq = 0.3 and 50 for th stramwis (, 3) and countrstram (4, 5) curtains Y Fig. 7. Effct of th tmpratur factor on th man vlocity profils at Cq = 50 for diffrnt stramwis (a) and countrstram curtains (b); () thrmally insulatd wall, () cooling, and (3) hating. In considring th tmpratur factor influnc (modling gass with diffrnt dnsitis) it is worthwhil to mak th rfrnc to [0, ], whr th natur of th variations in th man vlocitis and tmpraturs undr hating and cooling ar considrd in dtail. Th common fatur is th thickning of th boundary layr, both dynamic and thrmal, undr hating and its thinning undr cooling and th gratr (smallr) inflation of th U profils undr cooling (hating). In th cas of th tangntial injction som intrsting faturs appar in th proximity of th wall, as illustratd in Fig. 7. Th cas is possibl, in which th tmpratur is so low that it, as it wr, blocks th injction, th initial valu bing U(0)=0. This intrsting wall tmpratur ffct on both th man charactristics and th small fluctuations is confirmd in Fig. 8 in which th dpndncs of th growth rats on th wall tmpratur Tw ar plottd. At Tw < th dpndncs on both th typ of th curtain and th injction intnsity dgnrat, whil th disturbanc dcay rats ar th sam in all th rgims and compltly coincid with th dcay rats on th imprmabl wall. It can also b notd that in th cas of hating th domains of a wak Tw influnc on th growth rats appar. Th strongst influnc can b obsrvabl on th low tmpratur rang. Th similar α i (Tw) dpndncs xist for th normal injction which is also prsntd in th figur. In Fig. 9 th wall tmpratur ffct on th nutral curv positions is prsntd for th stramwis curtain. Clarly visibl is that th hating and, thrfor, a dcras in th injctd gas dnsity, lads to th vortx disturbanc dstabilization; th critical Rynolds numbr dcrass and th hazardous frquncy rang broadns. Contrariwis, th cold gas ISSN: Volum, 06

7 injction (high dnsity) lads to th disturbanc stabilization, an incras in Rc, and th hazardous frquncy rang shrinking. It should b notd that th natur of th dpndncs is vry rgular, without any jumps. 0 4 F Fig. 9. Nutral curvs of th vortx mod on noninsulatd surfacs for th stramwis curtains at Cq = 50 in th cas of hating (Tw =.6, ) and cooling (Tw =.4, 3) in comparison with th cas of th thrmally insulatd wall (Tw =.5, ) i R F Fig. 0. Frquncy cuts illustrating th growth rats of thr-dimnsional disturbancs (b = 0.5) at R = 600 and Cq = 50; for th stramwis curtain th sam paramtrs as in Fig. 9 and for th countrstram curtain thrmally insulatd wall with Tw =.889 (4). This rgularity is also clarly visibl on th frquncy cuts (Fig. 0) plottd for th sam tmpraturs. For th lowr tmpraturs th shrinking of th domain of xistnc of growing fluctuations is accompanid by a considrabl dcras in th growth rats. This is in complt agrmnt with th infrncs mad in [9, 3]. For th sak of comparison in th sam figur th frquncy cut of th countrstram curtain is also plottd. Clarly that in this 3 rgim th hazardous frquncy rang is considrably widr, th growth rats ar high, and thir maximum is displacd toward th lowr frquncis (about 5 khz in th cas undr considration. 4 Conclusions Th mthods of controlling flow rgims in comprssibl boundary layrs by mans of distributd injction ar modld. Th variations in th man paramtrs of th boundary layrs ar studid for diffrnt vrsions of th tangntial injction. It is shown that in th cas of th stramwis curtain th nar-wall vlocitis incras, which lads to considrabl stabilization of th vortx mods. At larg injction intnsitis th disturbancs can bcom dcaying. Contrariwis, in th cas of th countrstram curtain a rvrs flow can b ralizd in th wall rgions, which considrably dstabilizs small fluctuations. Th influnc of th tmpratur factor modling th injction of gass with diffrnt dnsitis is considrd. Along with th prviously invstigatd bhavior of th man tmpraturs and vlocitis undr hating and cooling, in th cas of th tangntial injction som intrsting faturs ar rvald in th wall rgion. At a givn injction intnsity in th cooling rgim th longitudinal nar-wall vlocitis of th injctd gas dcras for both curtains du to th wall tmpratur rduction. This wall tmpratur ffct on both th man paramtrs and th small fluctuations is confirmd by th dpndncs of th growth rats on th wall tmpratur Tw. At low tmpraturs, Tw <, th dpndncs on both th typ of th curtain and th injction intnsity dgnrat and th disturbanc dcay rats ar th sam in all th rgims. Th strongst Tw ffct is obsrvabl on th low tmpratur rang. Th wall tmpratur ffct manifsts itslf both in th nutral curv positions and in th form of frquncy cuts. For th lowr tmpraturs th domains of xistnc of growing fluctuations and incrasing disturbanc growth rats shrink. This has a positiv ffct on th stabilization of flow rgims. Th comparison with th frquncy cuts for th countr stram curtains shows that in this rgim th hazardous frquncy rang is considrably widr, th growth rats ar high, and thir maximum is displacd toward th lowr frquncis. Th gnral conclusion is that at suprsonic vlocitis th flow rgims can actually b controlld. Using th distributd tangntial injction th thrmal protction of th surfac in a flow can b ralizd with consrving th laminar natur of th flow. ISSN: Volum, 06

8 Th study was carrid out with th support of th Russian Foundation for Basic Rsarch (projct No ). Rfrncs: [] S.A. Gaponov and A.A. Maslov, Stability of a Suprsonic Boundary Layr with a Prssur Gradint and Suction, in; Disturbanc Dvlopmnt in Boundary Layrs, Novosibirsk, 979, p. 95 (in Russian). [] S.A. Gaponov and N.M. Trkhova, Stability and Thr-Wav Intraction of Disturbancs in a Suprsonic Boundary Layr with Mass Transfr on th Wall, Tplofiz. Aromkh. 0, 9, 30. [3] S.A. Gaponov and N.M. Trkhova, Controlling Suprsonic Boundary Layr Stability by Mans of Distributd Mass Transfr through a PorousWall, Fluid Dynamics 03, 48 (6), 76. [4] A.V. Boiko, G.R. Grk, A.V. Dovgal, and V.V. Kozlov, Turbulnc Gnration in Wall Flows,Novosibirsk, Nauka, 999 (in Russian). [5] S.A. Gaponov and A.A. Maslov, Disturbanc Dvlopmnt in Comprssibl Flows (in Russian), Novosibirsk, Nauka (980). [6] E.P. Volchkov, Nar-Wall Gas Curtains [in Russian], Novosibirsk, Nauka (983). [7] M.V. Protasov, T.F. Ivanov, and A.F. Polyakov, Distributd Injction Influnc on Flow past a Blunt Body with a Prmabl Lading Edg, in: th Scintific and Tchnical Conf. on th Optical Mthods of Flow Invstigation, Moscow, 0, (in Russian). [8] L.M. Albact andw.j. Glowacki, Skin Friction and Hat Transfr Charactristics of th Comprssibl Laminar Boundary Layr with Injction of a Light, Mdium, and Havy Gas, NOLTR 66-5, 967. [9] J.O. Powrs, G. Hich, and S.F. Shn, Th Stability of Slctd Boundary-Layr Profils, NOLTR 6-43, 963. [0] S.A. Gaponov and N.M. Trkhova, Stability and Thr-Wav Intraction of Disturbancs in a Suprsonic Boundary Layr with Cooling, Vstn. NGU. Sr. Fizika, 00, 5 (3), 5, (in Russian). [] S.A. Gaponov and N.M. Trkhova, Controlling Suprsonic Boundary Layr Stability by Mans of Distributd Mass Transfr through a PorousWall, Fluid Dynamics, 03, 48 (6), 76. [] W.H. Dorranc, Viscous Hyprsonic Flow, McGraw-Hill, Nw York, 96. [3] S.A. Gaponov and G.V. Ptrov, Stability of th Boundary Layr of a Nonquilibrium Dissociating Gas, 03, Novosibirsk, Nauka (in Russian). ISSN: Volum, 06