SUPERSONIC INVISCID FLOWS WITH THREE-DIMENSIONAL INTERACTION OF SHOCK WAVES IN CORNERS FORMED BY INTERSECTING WEDGES Y.P. Goonko, A.N. Kudyavev, and R.D. Rakhmov Inue of Theoecal and Appled Mechanc SB RAS, 630090 Novbk, Rua Reul of numecal and analycal nvegaon of nvcd eady upeonc flow n cone fomed by neecng wedge ae peened. Depe he gea numbe of pape dealng wh uch flow, many feaue of he fomaon of complex yem of paally neacng hock wave n cone confguaon have no been adequaely uded. Some of hee poblem ae nvegaed n he peen pape by condeaon of he genec cone flow, whch ae concal and ymmec abou he beco plane of a cone. Thee flow ae numecally calculaed by olvng hee-dmenonal eady Eule equaon wh a machng mehod. Effec of he angle of nclnaon of he cone uface o he fee eam decon, weep angle of he leadng edge and he dhedal angle of he cone ae dcued. Tanon fom egula o egula eflecon of hock wave In he cone flow condeed, plana hock wave geneaed by cone wedge neec and eflec a he beco plane. Boh egula eflecon (RR) and egula (IR) o Mach eflecon (MR) of hee hock wave ae poble [1-7]. I vey mpoan, boh fo heoy and applcaon, o deemne he boundae of doman of he egula and egula eflecon. In pacula, h poblem fo he cae of concal flow n cone wa condeed n [1-3], bu he daa obaned ae conadcoy. Dffeen cea of he anon fom egula o Mach eflecon (TRM) n wodmenonal flow ae well known (ee [8]). F, h he deachmen ceon (DC) deemned by he maxmal angle (θ w ) D of flow deflecon behnd he ncden hock wave (-hock wave), a whch he flow behnd he efleced hock wave (-hock wave) can ll eun o he flow decon n fon of he -hock wave. Regula eflecon heoecally mpoble fo wedge angle θ w > (θ w ) D. The econd, named onc, ceon (SC) deemned by he condon ha he flow velocy behnd he -hock wave onc. The coepondng wedge angle (θ w ) S dffe fom (θ w ) D by no moe han facon of a degee whn he ene ange of Mach numbe. The hd ceon, he o-called mechancal equlbum o von Neumann ceon (vnc), popoed (ee [8]) fo eflecon condon poble only a Mach numbe geae han a cean value, М > М * 2.2. The coepondng wedge angle (θ w ) N deemned by he condon ha he peue behnd he -hock wave equal o he peue behnd he nomal -hock wave. Two oluon, RR and MR epecvely, ae heoecally poble fo М > M * and (θ w ) N < θ w < (θ w ) D. In uneady flow wh eflecon of a movng hock wave fom a wedge, he RR oluon unambguouly ealze n h doman, ha, ehe he S o D-cea deemne anon o MR n h cae [8]. I ha been ecenly eablhed (ee evew [10]) he TRM n wo-dmenonal eady flow a М > M * accompaned by a hyee. In expemen, he anon o an MR occu a a cean θ w beween (θ w ) N and (θ w ) D a he wedge angle nceae, dependng on he level of flow dubance n he e facly, bu he back anon o an RR occu nea (θ w ) N a he wedge angle deceae. In a wnd unnel wh a low level of flow dubance o n a numecal expemen wh no dubance a all, he anon o Mach eflecon occu n accodance wh he DC o SC. Y.P. Goonko, A.N. Kudyavev, and R.D. Rakhmov, 2002 79
Repo Documenaon Page Repo Dae 23 Aug 2002 Repo Type N/A Dae Coveed (fom... o) - Tle and Suble Supeonc Invcd Flow Wh Thee-Dmenonal Ineacon of Shock Wave n Cone Fomed by Ineecng Wedge Conac Numbe Gan Numbe Pogam Elemen Numbe Auho() Pojec Numbe Tak Numbe Wok Un Numbe Pefomng Oganzaon Name() and Adde(e) Inue of Theoecal and Appled Mechanc Inukaya 4/1 Novobk 530090 Rua Sponong/Monong Agency Name() and Adde(e) EOARD PSC 802 Box 14 FPO 09499-0014 Pefomng Oganzaon Repo Numbe Spono/Mono Aconym() Spono/Mono Repo Numbe() Dbuon/Avalably Saemen Appoved fo publc eleae, dbuon unlmed Supplemenay Noe See alo ADM001433, Confeence held Inenaonal Confeence on Mehod of Aeophycal Reeach (11h) Held n Novobk, Rua on 1-7 Jul 2002 Abac Subjec Tem Repo Clafcaon unclafed Clafcaon of Abac unclafed Clafcaon of h page unclafed Lmaon of Abac UU Numbe of Page 7
θw 5 4 3 2 z M α b ν y θ w θ wn Fg. 1. A kech of a cone confguaon. Deachmen ceon Sonc ceon ν=90 χ=0 α =0 1 0 1 2 3 4 5 6 7 8 9 10 Fg. 2. Cone wedge angle coepondng o he S and D-cea of TRM. χ P Beco plane m b M x The ame wo-dmenonal TRM-cea may be aumed o be vald n poblem of paal neacon of plana hock wave geneaed by wedge n he cone confguaon condeed. In he peen wok, he coepondng o hee cea angle of flow deflecon a eflecon of hock wave a he beco plane of a cone ae dee-mned by oluon [10] of a poblem on a plana hock wave geneaed by a yawed wedge n a unfom upeonc flow. Th hee-dmenonal poblem ha been educed n [10] o an equvalen poblem fo he wo-dmenonal flow n a plane nomal o he leadng edge of he condeed yawed wedge. The Mach numbe of he equvalen wo-dmenonal flow deemned by a pojecon of he fee-eam velocy on he nomal plane. The poblem of neacon of wo plana hock wave n a pace o yawed eflecon of a plana hock wave a a plane uface amoun o oluon of wo equenal poblem on deemnaon of a hock wave geneaed by a yawed wedge. A kech of a cone confguaon wh defnon of paamee hown n Fg. 1. A confguaon whch ha he leadng edge of a weep angle χ = 0 and an angle beween hem ν = 90 (n he plane zy ) condeed uually. The fee eam aumed o be deced along he x-ax (wh no angle of aack n he beco plane α b = 0, ee Fg. 1), and he change n flow egme deve fom he change n he wedge angle θ w (θ w = θ wn a χ = 0, hee he angle θ wn defned n a plane nomal o he leadng edge). Noe, a change n θ w nvolve a change n he dhedal angle beween wedge uface and, epecvely, condon fo he cone flow a a whole ae changed. The eul of analycal calculaon fo he ypcal cone confguaon how ha he Mach numbe componen M n nomal o he lne of neecon of he hock wave doe no exceed M * even n he lm, a М. Thu, he anon fom egula o Mach eflecon n h cae hould occu n accodance wh he S o D-ceon. The wedge angle (θ w ) D and (θ w ) S obaned fom hee cea ae ploed n Fg. 2 a funcon of he Mach numbe. A well a he wo-dmenonal flow, he anon boundae deemned by he S and D-cea fo he cone flow dffe by facon of a degee. The maxmum value (θ w ) D 4.185 eached fo a Mach numbe abou М 2.8. Noe, he bounday obaned ung he D-ceon dffe fom ha deemned n [1]. In pacula, baed on he daa of [1], he angle (θ w ) D, whch maxmum fo a Mach numbe М 3, equal o (θ w ) D 8. Fo М 5, h bounday cloe o ha obaned n [3], bu hee a dffeence n he neval М = 1.5-4, whee he maxmum value of he anon wedge angle eached. In [3], h value appoxmaely by a degee geae han n he peen wok. The analycal udy of TRM boundae wa upplemened by dec numecal compuaon. Fo he cone flow condeed, hee-dmenonal eady Eule equaon wee olved by a machng mehod. A hgh-ode TVD cheme decbed n [11] wa ued. The fluxe a he face beween he cell wee compued ung he HLLE Remann olve, and negaon along he longudnal (machng) coodnae wa pefomed by he Runge Kua mehod. Compuaon wee pefomed fo М = 1.5, 3, 6, and 10 wh vayng he wedge angle n a vcny of anonal value wh a 0.5 ep. The one of an MR-ype flow egme wh nceang he wedge angle and, hence, a anon wedge angle wa deemned, f, vually on 80
he ba of he compued co-flow feld. Theoecally, lp lne hould come ou fom ple pon of he Mach wave confguaon. In numecal calculaon, hee lp lne ae manfeed a laye wh a dac anvee change n deny. A n [12], he flow paen wee vualzed by he numecal chleen echnque, whch capue deny gaden. The compued co-flow paen wee ued o deemne he chaacec ze of he Mach em. I wa pecfed, n a co econ x = con, by he angle ϕ beween he adu-veco of he ple pon P of he Mach confguaon and he beco. Fom he daa on ϕ value obaned fo a ee of angle θ w a a gven Mach numbe, he anon wedge angle wa alo deemned by exapolang he ϕ value o ϕ = 0, ha, by he neecon of he cuve appoxmang he ϕ value and he θ w -ax. A cubc polynomal appoxmaon by he lea-quae echnque wa pefomed. A a whole, he compued anon wedge angle coepond o he analycal dependence, hough he accuacy of h numecal deemnaon of he anon doe no how defnely whch, he S o D-ceon, eponble fo he anon (he dffeence beween hem le han 0.3 ). Anohe cone confguaon, whch had been eale expemenally uded n [2], wa alo condeed n he peen wok. Th confguaon dffe fom he pevou ypcal one by he fac ha ha θ w 0 and he cone flow n deemned by he fee-eam angle of aack α b n he beco plane (ee Fg. 1) ahe han changng wedge angle. Theewh, he dhedal angle of he cone eman he ame. The eul of analycal calculaon of he angle α b coepondng o he S o D-ceon of TRM fo uch a cone confguaon wh ν = 90, χ = 0 ae ploed n Fg. 3. The ame fgue how he expemenal daa of [2] obaned fo Mach numbe М = 2.03, 3.02, and 4.03 n he T-313 wnd unnel baed a ITAM. The expemenal anon value of α b wee obaned wh an eo up o ±2, hey ae hghe han he calculaed value by 2-6, ae monooncally nceang, and have no maxmum whn he Mach numbe ange examned. One of he eaon of he above dageemen he dplacng effec of he bounday laye, whch wa afcally pped n he expemen of [2]. A mgh be deduced fom he daa [2], he effec of hock wave dplacemen n RR flow egme, a compaed o hock wave poon n he nvcd flow, equvalen o an nceae n he angle α b by 2 3. In addon, behavo of expemenal anon angle α b dependng on he Mach numbe elaed o he chaace of neacon of efleced hock wave wh he bounday laye on he plane uface. Thee hock wave ae glancng elave o he uface and can nduce hee-dmenonal (oblque) epaaon of he bounday laye. The numecal eul how ha he neny of he efleced hock wave lowe han he ccal value fo oblque epaaon of he ubulen bounday laye fo М = 2 and exceed fo М = 3 and 4. Type of upeonc concal flow wh egula eflecon of hock wave The numecal udy conduced fo flow n cone wh ν = 90, χ = 0 how ha ype of egula eflecon of hock wave fomng n co flow n hee cone ae he ame, whch ae ypcal of egula eflecon n wo-dmenonal peudo-eady flow [8]. Namely, von Neumann eflecon (vnr), ngle (SMR), anonal (TMR), and double (DMR) Mach eflecon occu. Noe, n cona o ha, only he SMR ype poble n wo-dmenonal eady flow [8]. Numecal calculaon of SMR-ype concal flow n cone have been known fo a long me [4-6]; TMR and DMR ype have been alo obaned numecally n [7]. Howeve, hee ude nclude no analy of he flow egme wh he ue of hock pola, α b 14 12 10 8 6 4 2 Sonc ceon ν=90,χ=0 Expemenal daa of Dem' yanenko Deachmen ceon 0 1 2 3 4 5 6 7 8 9 10 Fg. 3. Angle of aack n he beco plane coepondng o he S and D-cea of TRM. M 81
a compaon o IR ype ang n wo-dmenonal flow dd no caed ou, and he compued flow egme wee no denfed a ndcaed above. Typcal flow paen obaned by he numecal chleen echnque ae pcued below n he concal coodnae,. Flow paen ae accompaned wh analycal hock pola peened n he coodnae (p/p, θ n ), whee p/p he peue behnd he ncden () o efleced () hock wave nomalzed o he fee-eam peue p, and θ n he angle of flow deflecon n he plane pependcula o he lne of neecon of he - and -hock wave. The --pola combnaon ae gven fo wo plane of hock wave eflecon o neacon: f, fo he beco plane (ϕ = 0), and econd, fo he medan plane (ϕ > 0) pang hough he x- ax and he ple pon P of he IR confguaon obaned numecally. The --hock pola fo ϕ > 0 ae gven becaue he followng eaon. If one conde a cone flow n he concal coodnae, he ple pon, wh he evoluon of an IR confguaon, move away fom he beco bu eman on he -hock wave, and he adu-veco of h pon nceae. Theewh, he co-flow velocy upeam of he ple pon and he lope of he -hock wave o h velocy deceae. Conequenly, by vue of he cone flow concy, he behavo of he --pola combnaon, deemnng whch IR confguaon come no beng, fo a cae of ϕ > 0 dffe fom hoe fo ϕ = 0. An example of a flow wh he egula eflecon of vnr-ype hown n Fg. 4a fo М = 1.5 θ w = 5 ; he hock wave pola ae ploed n Fg. 4b. In boh cae of ϕ = 0 and ϕ = 5.5, he -hock pola on he gh of he p/p ax and neve neec he -hock pola. The uaon mla o ha fo he von Neumann eflecon of weak hock wave n wodmenonal peudo-eady flow [8]. The ame vnr-ype of egula eflecon obeved n he condeed cone flow f he -hock pola neec he -hock pola on he gh of he -hock pola cenelne, a ake place n wo-dmenonal peudo-eady flow. An example of a flow wh he SMR-ype of hock wave eflecon and --pola combnaon ae hown n Fg. 5а and 5b fo М = 3, θ w = 10. The -hock and -hock pola neec on he lef of p/p 1.8 1.6 1.4 1.2 ϕ=0 ϕ = 5.5 1 0 2 4 6 Fg. 4. Co flow paen of vnr-ype (a) and hock wave pola (b) fo M = 1.5, θ w = 5, ν = 90, χ = 0. m P a) m a) p/p 4 3 2 Fg. 5. Co flow paen of SMR-ype (a) and hock wave pola (b) fo M = 3, θ w = 10, ν = 90, χ = 0. ϕ=4.1 ϕ=0 P P P P b) θ n 1 0 5 10 15 20 deg b) θ n deg he cenelne of he lae. Fo he MR confguaon obaned numecally (ϕ = 4.1 ), he pon of neecon (he ple pon P ) le on he -hock pola hghe han he pon P, fo whch he po-hock velocy componen, whch nomal o he lne of neecon of he - hock and -hock wave, onc. Th coepond o condon fo wodmenonal flow n whch he SMR occu [8]. Noe, he analy of he popee of SMR confguaon fomng n cone flow how ha, n a co econ, he Mach em can be convex, concave, and alo eclnea, n cona o wo-dmenonal eady flow wh SMR, whee he Mach em alway concave. Fo nance a М = 3, he eclnea Mach em fom fo θ w 13. TMR- and DMR ype of hock wave eflecon n wo-dmenonal peudo-eady flow aound a wedge fom f he flow behnd he efleced hock wave nea he ple pon P 82
become onc o upeonc, and ome poon of h hock wave emanang fom he ple pon eclnea. A ypcal feaue of hee eflecon ype he fac ha he lp lne emanang fom he ple pon P ncden on he wedge; appoachng he wedge, he lp lne un coune he fee eam and col up n pal cul. In he DMRflow, deceleaon of he upeonc flow behnd he efleced -hock wave eul n he fomaon, a a cean dance fom he ple pon of he man Mach em m, of anohe hee hock wave yem wh a ple pon P 1. Th wave yem nclude he efleced -hock wave, he ong hock wave whch ncden on he colng lp lne, and he econday Mach em m 1. In he TMR-flow, a ceneed compeon wave fom nead of he -hock wave. The ucue of DMR-ype cone flow mla. The numecal udy how ha a ahe wde ange of 0 10 20 30 40 θ n deg flow egme nemedae beween SMR and DMR ype hould be clafed a cone concal flow wh he eflecon of ncden hock wave of TMR-ype. In hee flow, he eam behnd he efleced -hock wave alo anveely upeonc n a cloe vcny of he ple pon bu become ubonc a a dance fom, and he -hock wave become moe and moe concave. Th cuvng he -hock wave caued by compeon wave, bu he lae ae no ceneed a n a wo-dmenonal flow. Nevehele, he lp lne emanang fom he ple pon alo col up n pal cul. Example of TMR- and DMR-ype cone flow ae hown fo М = 6, θ w = 10, Fg. 6, and θ w = 15, Fg. 7. In boh cae, he flow behnd he efleced hock wave a he ple pon upeonc n he anvee decon. Bede, he -hock pola and -hock pola neec lowe han he pon P coepondng o he po-hock velocy whoe componen nomal o he lne of ne-econ of he hock wave onc. y / x Fg. 8. Co flow paen of MMR-ype fo M = 6, ν = 90, χ = 0, θ w = 20. The numecal udy of cone flow a hgh veloce М 6 how ha a vey of MR-ype eflecon ex, whch may be called MMR (Mul-hock-wave Mach Reflecon). An example of a cone flow of h ype hown n Fg. 8 fo М = 6, θ w = 20. Thee flow paen appea f he flow behnd he efleced hock wave anveely upeonc, bu he fee-eam velocy componen nomal o he lne of ne-econ of he -hock wave geae han n flow of DMR-ype. In a co flow of MMR-ype, flow deceleaon wh mulple eflecon of hock wave occu n he eam beween he ple pon P and P 1. Th mla o ha obeved n he cae of flow deceleaon n wo-dmenonal nle fom upeonc veloce a he enance, wh a anon hough anonc p/p 8 6 4 2 p/p 15 10 5 P 0 10 20 30 P P P ϕ = 6.3 P ϕ = 4.4 ϕ=0 ϕ=0 θn deg Fg. 6. Co flow paen of TMR-ype (a) and hock wave pola (b) fo M = 6, θ w = 10, ν = 90, χ = 0. m m P 1 m 1 a) P a) P Fg. 7. Co flow paen of DMR-ype (a) and hock wave pola (b) fo M = 6, θ w = 15, ν = 90, χ = 0. b) b) 83
θwn 45 40 35 30 25 No Neumann 20 30 ceon 15 20 10 5 χ=0 0 +10 1 2 3 4 5 6 7 8 9 10 M Fg. 9. Tanon wedge angle fo cone confguaon wh dffeen weep angle. χ=0,ν= va Sonc ceon Neumann ceon ν=45 25 60 20 15 ν=45 10 75 60 5 ν=90 No Neumann ceon 120 0 1 2 3 4 5 6 7 8 9 10 M θwn ν = 90, χ = va Sonc ceon Neumann ceon χ= 40 χ= 40 30 Fg. 10. Tanon wedge angle fo cone confguaon wh dffeen dhedal angle. veloce n he hoa, o ubonc veloce behnd he hoa. Type of upeonc concal flow fomng n cone wh vaou dhedal and weep angle The effec of he weep angle of he leadng edge and he dhedal angle of he cone on TRM boundae wee uded n he peen wok. Noe, ha he dhedal angle fo an abay cone confguaon defned by an angle ν beween he pojecon of he leadng edge on he plane zy, ee Fg. 1. Pevouly unknown paamee of cone confguaon wee deemned fo whch he anon n accodance wh he vn-ceon poble. The wedge angle (θ wn ) D and (θ wn ) N coepondng o he D- and vn-cea fo a numbe of weep and dhedal angle ae peened n Fg. 9 and 10 veu he Mach numbe. I hould be noed ha he anon n accodance wh vn-ceon poble begnnng fom a cean weepfowad angle χ < 20 fo ν = 90 and an acue dhedal angle ν < 75 fo χ = 0. Obvouly, hee a doman of vaou combnaon of cone weepfowad and acue dhedal angle povdng h anon. The man feaue of flow paen changng wh vayng he wedge angle θ wn unde he ad condon ae hown fo М = 6 ν = 90, χ = 30, θ wn = 10, 12.5, and 15, Fg. 11, and ν = 45, χ = 0, θ wn = 10 and 12, Fg. 12. In he cae of ν = 90, χ = 30, eflecon of he -hock wave of RR-ype fom a angle θ wn (θ wn ) N 10.61 and alo a a geae angle θ wn = 12.5, whch gnfcanly malle han he angle (θ wn ) D 27.42 deemned by he D- ceon. Iegula eflecon of DMR o MMR-ype fom a angle θ wn = 13 and 15, whch ae lage han he angle (θ wn ) N bu gnfcanly malle han he angle (θ wn ) D. In he cae of ν = 45, χ = 0, RR-ype eflecon of he -hock wave ae obeved a angle θ wn 11, wheea flow wh DMR o MMR-ype eflecon fom a an angle θ wn = 12, whch noceably malle han (θ wn ) N 15.67, and coepondngly, a geae angle θ wn > 12. The compuaon of cone flow fo paamee М = 6, χ = 0, ν = 60 how ha RR-ype θ wn = 10 o P m θ wn = 12.5 o θ wn = 15 o Fg. 11. Co flow paen fo wep cone confguaon ν = 90, χ = -30, M = 6. 84
eflecon of he -hock wave ae obeved a θ wn 11, wheea flow wh DMR o MMR-ype eflecon fom a θ wn = 12 cloe o (θ wn ) N 12 and, coepondngly, a geae angle. I hould be noed ha, n he condeed flow egme wh egula eflecon of -hock wave, a co flow paen occu whch nclude an nenal ong hock wave wh Mach em m ha fom behnd a poon of flow wh he egula eflecon of he -hock wave. One can ee h fo θ wn = 10 n boh cae of he wep cone confguaon wh χ = 30 a ν = 90, Fg. 11, and he confguaon wh a dhedal angle ν = 45 a χ = 0, Fg. 12. Snce hee eflecon nclude a egula eflecon of he man ncden hock wave geneaed by he wedge and an egula eflecon of po hock wave, hey may be called combned egula-egula eflecon o a po Mach eflecon. The compuaon how ha, a he wedge angle nceae, flow wh uch a eflecon of hock wave fom unl he nenal Mach em m eache he pon P of egula eflecon of he -hock wave. A geae wedge angle, h flow paen beak down, and DMR o MMR-ype eflecon of he ncden wedge-aached hock wave occu. The compuaon alo how ha, wh evee changng he weep and dhedal angle o cean value χ > 0 a ν = 90 o ν > 90 a χ = 0, one can oban he hock wave eflecon of SMR and vnr ype aleady dcued. Thu, a pecum of egula eflecon ype obaned fo flow n cone of weepfowad angle χ < 0 and acue dhedal angle ν < 90 moe boad n compaon o he cone confguaon of weepback angle χ 0 and ecangula o obue angle ν 90 bu nclude hoe one analyzed fo he lae cae. REFERENCES 1. Waon R.D., Wenen L.M. A Sudy of Hypeonc Cone Flow Ineacon // AIAA Jounal. 1971. Vol. 9. No. 7. P. 652-656. 2. Dem yanenko V.S. Expemenal Sudy of Thee-Dmenonal Supeonc Flow n Cone Beween Ineecng Plane Suface: Ph. D. The. Novobk: ITAM, 1973. (In Ruan). 3. Macon F. Supeonc, Invcd, Concal Cone Flowfeld // AIAA Jounal. 1980. Vol. 18. No. 1. P. 78-84. 4. We J.E., Kokedg R.H. Ineacon n he Cone of Ineecng Wedge a a Mach Numbe of 3 and Hgh Reynold Numbe // AIAA Jounal.. 1972. Vol. 10. No. 5P. 652-656. 5. Kule P. Supeonc Flow n he Cone Fomed by Two Ineecng Wedge // AIAA Jounal. 1974. Vol. 12. No. 5. P. 577-578. 6. Shanka V., Andeon D., Kule P. Numecal Soluon fo Supeonc Cone Flow // Jounal of Compu. Phy. 1975. Vol. 17. No. 2. P. 160-180. 7. Malo R. Vocal Soluon n Supeonc Cone Flow // AIAA Jounal. Vol. 31. No. 9. 1993. P. 1651-1658. 8. Ben-Do G. Shock Wave Reflecon Phenomena. N.Y. Beln Hedelbeg: Spnge-Velag, 1992. 9. Ivanov M.S., Vandomme D., Fomn V.M., Kudyavev A.N., Hadjadj A., Khoyanovky D. Tanon beween egula and Mach eflecon of hock wave: new numecal and expemenal eul // Shock Wave. 2001. Vol. 11. No. 3P. 199-207. 10. Goonko Y.P., Makelov G.N. The ue of equaon of an oblque hock wave n he wo-dmenonal flow fo oluon of poblem of neacon of plane hock wave oened hee-dmenonally n a pace: Pepn No. 15-89. Novobk: ITAM, 1989. (In Ruan) 11. Goonko Y.P., Khaonov A.M., Kudyavev A.N., Mazhul I.I, Rakhmov R.D. Eule mulaon of he flow ove a hypeonc convegen nle negaed wh a foebody compeon uface. Euop. Conge on Compu. Mehod n Appled Sc. and Eng.: CD-poceedng. Bacelona, Sep. 11-14, 2000. 12. Kudyavev A., Hadjadj A. Vualaon gafhque en mécanque de flude numéque // C.R. de 9 eme Colloque Fancophone de Vualaon e de Taemen d Imedge en Méchanque de Flude FLUVISU 2001. Rouen, [Fance], 2001. P. 55-62. m P θ w n = 10 o θ ω ν = 12 ο Fg. 12. Co flow paen fo cone confguaon wh a dhedal angle ν = 45, χ = 0, M = 6. 85