Thermochemical Non-Equilibrium Reentry Flows in Two-Dimensions: Seven Species Model Part II

Transcription

1 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment Thermochemcl Non-Equlbrum Reentry Flow n Two-Dmenon: Seen Spece Model Prt II EDISSON SÁVIO DE GÓES MACIEL () nd AMILCAR PORTO PIMENTA () IEA Aeronutcl Engneerng Don ITA Aeronutcl Technologcl Inttute Prç Ml. do Ar Edurdo Gome, 50 Vl d Acác São Joé do Cmpo SP BRAZIL () edo@edono.eng.br () nd () mlcr@t.br Abtrct: - Th work, econd prt of th tudy, preent numercl tool mplemented to multe ncd nd cou flow employng the recte g formulton of thermochemcl non-equlbrum. The Euler nd Ner-Stoke equton, employng fnte olume formulton, on the context of tructured nd untructured ptl dcretzton, re oled. Thee rnt llow n effecte compron between the two type of ptl dcretzton mng erfy ther potentlte: oluton qulty, conergence peed, computtonl cot, etc. The eropce problem of the hyperonc hot g flow round double ellpe nd round reentry cpule, n two-dmenon, re multed. The recte multon wll nole n r chemcl model of een pece: N, O, N, O, NO, NO + nd e -. Eghteen chemcl recton, nolng docton, recombnton nd onzton, wll be tuded. The Arrhenu formul wll be employed to determne the recton rte nd the lw of m cton wll be ued to determne the ource term of ech g pece equton. Key-Word: - Euler nd Ner-Stoke equton, Recte formulton, Thermochemcl nonequlbrum, Hyperonc flow, Vn Leer lgorthm, Fnte Volume, Untructured dcretzton. Introducton A hyperonc flght ehcle h mny pplcton for both mltry nd cln purpoe ncludng reentry ehcle uch the Spce Shuttle nd the Automted Trnfer Vehcle (ATV) of the Europen Spce Agency (ESA). The extreme enronment of hyperonc flow h mjor mpct on the degn nd nly of the erodynmc nd therml lodng of reentry or hyperonc crue ehcle. Durng hyperonc flght, the pece of the flow feld re brtonlly excted, docted, nd onzed becue of the ery trong hock we whch creted round ehcle. Becue of thee phenomen, t necery to conder the flow to be n therml nd chemcl non-equlbrum. In hgh peed flow, ny djutment of chemcl compoton or thermodynmc equlbrum to chnge n locl enronment requre certn tme. Th becue the redtrbuton of chemcl pece nd nternl energe requre certn number of moleculr collon, nd hence certn chrctertc tme. Chemcl non-equlbrum occur when the chrctertc tme for the chemcl recton to rech locl equlbrum of the me order the chrctertc tme of the flud flow. Smlrly, therml non-equlbrum occur when the chrctertc tme for trnlton nd rou nternl energy mode to rech locl equlbrum of the me order the chrctertc tme of the flud flow. Snce chemcl nd therml chnge re the reult of collon between the conttuent prtcle, non-equlbrum effect prel n hghpeed flow n low-denty r. In chemcl non-equlbrum flow the m conerton equton ppled to ech of the conttuent pece n the g mxture. Therefore, the oerll m conerton equton replced by mny pece conerton equton the number of chemcl pece condered. The umpton of therml non-equlbrum ntroduce ddtonl energy conerton equton one for eery ddtonl energy mode. Thu, the number of goernng equton for non-equlbrum flow much bgger compred to thoe for perfect g flow. A complete et of goernng equton for nonequlbrum flow my be found n [-]. Anly of non-equlbrum flow rther complex becue () the number of equton to be oled much lrger thn the Ner-Stoke equton, nd () there re ddtonl term lke the pece producton, m dffuon, nd brtonl energy relxton, etc., tht pper n the goernng equton. In typcl flght of the NASP flyng t Mch 5, onzton not expected to occur, nd E-ISSN: Iue, Volume 8, Jnury 03

2 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment 5-pece r dequte for the nly (ee [3]). Snce the rottonl chrctertc temperture for the conttuent pece (nmely N, O, N, O, nd NO) re mll, the trnltonl nd rottonl energy mode re umed to be n equlbrum, where the brtonl energy mode umed to be n non-equlbrum. [4] h mplfed the thermodynmc model by umng hrmonc oclltor to decrbe the brtonl energy. Ionc pece nd electron re not condered. Th mplfe the et of goernng equton by elmntng the equton goernng electron nd electronc exctton energy. [4] h tken the complete et of goernng equton from [], nd mplfed them for fe-pece two-temperture r model. The problem of chemcl non-equlbrum n the hock lyer oer ehcle flyng t hgh peed nd hgh lttude n the Erth tmophere he been dcued by eerl netgtor ([5-8]). Mot of the extng computer code for clcultng the nonequlbrum rectng flow ue the one-temperture model, whch ume tht ll of the nternl energy mode of the geou pece re n equlbrum wth the trnltonl mode ([7-8]). It h been ponted out tht uch one-temperture decrpton of the flow led to ubtntl oeretmton of the rte of equlbrum becue of the eleted brtonl temperture ([6]). A three-temperture chemcl-knetc model h been propoed by [] to decrbe the relxton phenomen correctly n uch flght regme. Howeer, the model qute complex nd requre mny chemcl rte prmeter whch re not yet known. A comprome between the three-temperture nd the conentonl one-temperture model, twotemperture chemcl-knetc model h been deeloped ([5-6]), whch degnted heren the TT model. The TT model ue one temperture T to chrcterze both the trnltonl energy of the tom nd molecule nd the rottonl energy of the molecule, nd nother temperture T to chrcterze the brtonl energy of the molecule, trnltonl energy of the electron, nd electronc exctton energy of tom nd molecule. The model h been ppled to compute the thermodynmc properte behnd norml hock we n flow through contnt-re duct ([9-0]). Rdton emon from the non-equlbrum flow h been clculted ung the Non-equlbrum Ar Rdton (NEAIR) progrm ([-]). The flow nd the rdton computton he been pckged nto ngle computer progrm, the Shock-Tube Rdton Progrm (STRAP) ([0]). A frt-tep ement of the TT model w mde n [0] where t w ued n computng the flow properte nd rdton emon from the flow n hock tube for pure ntrogen undergong docton nd wek onzton (onzton frcton le thn 0.%). Generlly good greement w found between the clculted rdton emon nd thoe obtned expermentlly n hock tube ([3-5]). The only excepton noled the brtonl temperture. The theoretcl tretment of the brtonl temperture could not be ldted becue the extng dt on the brtonl temperture behnd norml hock we ([5]) re thoe for n electronclly excted tte of the moleculr ntrogen on N nted of the ground electronc tte of the neutrl ntrogen molecule N whch clculted n the theoretcl model. The meured brtonl temperture of N w much mller thn the clculted brtonl temperture for N. Th work, econd of th tudy, decrbe numercl tool to perform thermochemcl nonequlbrum multon of recte flow n twodmenon. The [6] cheme, n t frt- nd econd-order eron, mplemented to ccomplh the numercl multon. The Euler nd Ner-Stoke equton, on fnte olume context nd employng tructured nd untructured ptl dcretzton, re ppled to ole the hot g hyperonc flow round double ellpe nd round reentry cpule n two-dmenon. The econd-order eron of the [6] cheme obtned from MUSCL extrpolton procedure (detl n [7]) n context of tructured ptl dcretzton. In the untructured context, only frtorder oluton re obtned. The conergence proce ccelerted to the tedy tte condton through ptlly rble tme tep procedure, whch h proed effecte gn n term of computtonl ccelerton (ee [8-9]). The recte multon nole n r chemcl model of een pece: N, O, N, O, NO, NO + nd e -. Eghteen chemcl recton, nolng docton, recombnton nd onzton, re multed by the propoed model. The Arrhenu formul employed to determne the recton rte nd the lw of m cton ued to determne the ource term of ech g pece equton. The reult he demontrted tht the mot correct erodynmc coeffcent of lft obtned by the [6] cheme wth frt-order ccurcy, n cou formulton, to recte condton of thermochemcl non-equlbrum wth fe nd een pece model, to the reentry cpule. E-ISSN: Iue, Volume 8, Jnury 03

3 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment Moreoer, the hock poton cloer to the geometry ung the recte formulton of fe nd een pece model, n ther econd-order eron. It w erfed n the ncd nd cou ce. Error le thn 3% were obtned wth the [6] lgorthm n the determnton of the tgnton preure t the body noe, for both double ellpe problem, where n the reentry cpule problem, error le thn 0% were found, emphzng the correct mplementton nd good reult obtned from the recte formulton. The mn contrbuton of th work to extend the tructured formulton of een (7) pece chemcl model to n untructured formulton, clong, n th wy, the tructured nd untructured een (7) pece model mplementton n twodmenon. Formulton to Recte Flow n Thermochemcl Non-Equlbrum. Recte Equton n Two-Dmenon The recte Ner-Stoke equton n therml nd chemcl non-equlbrum were mplemented on fnte olume context, n the two-dmenonl pce. In th ce, thee equton n ntegrl nd conerte form cn be expreed by: t V dv S F nds V S CV E E F F j dv, wth F, () e where: the ector of conered rble, V the olume of computtonl cell, F the complete flux ector, n the unt ector norml to the flux fce, S the flux re, S CV the chemcl nd brtonl ource term, E e nd F e re the conecte flux ector or the Euler flux ector n the x nd y drecton, repectely, E nd F re the cou flux ector n the x nd y drecton, repectely. The nd j unt ector defne the Crten coordnte ytem. Eleen () conerton equton re oled: one of generl m conerton, two of lner momentum conerton, one of totl energy, x of pece m conerton nd one of the brtonl nternl energy of the molecule. Therefore, one of the pece bent of the terte proce. The CFD ( Computtonl Flud Dynmc ) lterture recommend tht the pece of bgget m e frcton of the geou mxture hould be omtted, mng to reult n mnor numercl ccumulton error, correpondng to the bgget mxture conttuent (n the ce, the r). To the preent tudy, n whch choen chemcl model to the r compoed of een (7) chemcl pece (N, O, N, O, NO, NO + nd e - ) nd eghteen (8) chemcl recton, beng ffteen (5) docton recton (endothermc recton), two () of exchnge or recombnton, nd one () of onzton, th pece cn be the N or O. To th work, t w choen the N. The ector, E e, F e, E, F nd S CV cn, hence, be defned follow ([4]): u e, e V E u u p u Hu u u 4u 5u 6u u 7 e Vu e, F u p H e ; () e V x xx xy xxu xy q f,x q,x x E x ; (3) Re q,x x 5x 6x 7x n whch: the mxture denty; u nd re Crten component of the elocty ector n the x nd y drecton, repectely; p the flud ttc preure; e the flud totl energy;,, 4, 5, 6, 7 re dente of the N, O, O, NO, NO + nd e -, repectely; H the mxture totl enthlpy; e V the um of the brtonl energy of the molecule;,x E-ISSN: Iue, Volume 8, Jnury 03

4 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment the re the component of the cou tre tenor; q f,x nd q f,y re the frozen component of the Fourer-het-flux ector n the x nd y drecton, repectely; q,x nd q,y re the component of the Fourer-het-flux ector clculted wth the brtonl therml conductty nd brtonl temperture; y xy yy xyu yy q f,y q,y y F, (4) y Re S CV mol e *, q e,y , y 5y 6y 7y ,y mol e,, (5) x nd y repreent the pece dffuon flux, defned by the Fck lw; x nd y re the term of mxture dffuon;,x nd,y re the term of moleculr dffuon clculted t the brtonl temperture; the chemcl ource term of ech pece equton, defned by the lw of m cton; * e the moleculr-brtonl-nternl energy clculted wth the trnltonl/rottonl temperture; nd the trnltonl-brtonl chrctertc relxton tme of ech molecule. The cou tree, n N/m, re determned, ccordng to Newtonn flud model, by: u u x 3 x y xx, u u xy nd yy, y x y 3 x y (6) n whch the flud moleculr coty. The frozen component of the Fourer-het-flux ector, whch conder only therml conducton, re defned by: q f,x T k f nd x q f,y T k f, (7) y where k f the mxture frozen therml conductty, clculted conform preented n [0]. The brtonl component of the Fourer-het-flux ector re clculted follow: q,x T k nd x q,y T k, (8) y n whch k the brtonl therml conductty nd T the brtonl temperture, wht chrcterze th model of two temperture: trnltonl/rottonl nd brtonl. The clculton of T nd k demontrted n [0]. The term of pece dffuon, defned by the Fck lw, to condton of therml non-equlbrum, re determned by ([4]): YMF, x D nd x YMF, y D, y (9) wth referent to gen pece, Y MF, beng the molr frcton of the pece, defned : Y MF, n M (0) M k nd D the pece-effecte-dffuon coeffcent. The dffuon term x nd y whch pper n the energy equton re defned by ([]): x n n xh nd y k k h, () y E-ISSN: Iue, Volume 8, Jnury 03

5 beng h the pecfc enthlpy (enble) of the chemcl pece. Detl of the clculton of the pecfc enthlpy, ee [-3]. The moleculr dffuon term clculted t the brtonl temperture,,x nd,y, whch pper n the brtonl-nternl-energy equton, re defned by ([4]): mol, x,x h nd mol, y,y h, () wth h, beng the pecfc enthlpy (enble) of the chemcl pece clculted t the brtonl temperture T. The um of Eq. (), lo thoe preent n Eq. (5), conder only the molecule of the ytem, nmely: N, O, NO nd NO +.. Chemcl nd brtonl model The chemcl model employed to th ce of thermochemcl non-equlbrum the een pece model of [34], ung the N, O, N, O, NO, NO + nd e - pece. Th formulton ue, n the clculton of the pece producton rte, temperture of recton rte control, ntroduced n the plce of the trnltonl/rottonl temperture, whch employed n the clculton of uch rte. Th procedure m couple between brton nd docton. Th temperture defned : rrc T T T, where T the trnltonl/rottonl temperture nd T the brtonl temperture. Th temperture T rrc replce the trnltonl/rottonl temperture n the clculton of the pece producton rte, ccordng to [4]. The brtonl model decrbed n [0]. 3 Untructured [6] Algorthm to Thermochemcl Non-Equlbrum Conderng the two-dmenonl nd untructured ce, the lgorthm follow tht decrbed n [, 5], conderng, howeer, the brtonl contrbuton ([0, 6]) nd the eron of the twotemperture model to the frozen peed of ound, preented n [6]. Hence, the dcrete-dynmcconecte flux defned by: L R R L H u H u H u H u M S R (3) y x 0 p S p S 0, (3b) the dcrete-chemcl-conecte flux defned by: L R R L M S R, (4) nd the dcrete-brtonl-conecte flux determned by: L R R L e e e e M S R, (5) where the fluxe re clculted t the nterfce l =, for exmple, ccordng to [6] notton. The me defnton preented n [6] re ld to th lgorthm. The tme ntegrton performed employng the Runge-Kutt explct method of fe tge, econd-order ccurte, to the three type of conecte flux. To the dynmc prt, th method cn be repreented n generl form by: (k) ) (n ) (k k (0) (k) (n) (0) V R t, (6) to the chemcl prt, t cn be repreented n generl form by: (k) ) (n ) (k C ) (k k (0) (k) (n) (0) S V R t, (7) where the chemcl ource term S C clculted wth the temperture T rrc. Fnlly, to the brtonl prt: WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment E-ISSN: Iue, Volume 8, Jnury 03

6 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment (0) (k) (n) (n) (0) n whch: t (k) mol k TV, (k) (k) R V S, (8) S q S e ; (9) C, mol wth k =,...,5; = /4, = /6, 3 = 3/8, 4 = / nd 5 =. The [7] het flux due to trnltonlbrtonl relxton, q T-V,, defned n [0, 6]. Th cheme frt-order ccurte n pce nd econd-order ccurte n tme. The econd-order of ptl ccurcy obtned by the MUSCL procedure (detl n [7]). The cou formulton follow tht of [8], whch dopt the Green theorem to clculte prmte rble grdent. The cou ector re obtned by rthmetcl erge between cell (,j) nd t neghbor. A w done wth the conecte term, there need to eprte the cou flux n three prt: dynmcl cou flux, chemcl cou flux nd brtonl cou flux. The dynmcl prt correpond to the frt four equton of the Ner-Stoke one, the chemcl prt correpond to the followng x equton nd the brtonl prt correpond to the lt equton. The ptlly rble tme tep technque h proded excellent conergence gn demontrted n [8-9] nd mplemented n the code preented n th work. 4 Reult Tet were performed n Dul-Core Intel proceor notebook wth.3ghz of clock nd.0gbyte of RAM. A the nteret of th work tedy tte problem, t necery to defne crteron whch gurntee the conergence of the numercl reult. The crteron dopted w to conder reducton of no mnml three (3) order of mgntude n the lue of the mxmum redul n the clculton domn, typcl CFD communty crteron. The redul of ech cell w defned the numercl lue obtned from the dcretzed conerton equton. A there re eleen () conerton equton to ech cell, the mxmum lue obtned from thee equton defned the redul of th cell. Hence, th redul compred wth the redul of the other cell, clculted of the me wy, to defne the mxmum redul n the, clculton domn. In the multon, the ttck ngle w et equl to zero. 4. Intl Condton The ntl condton re preented n Tb. nd. The Reynold number obtned from dt of [9]. The boundry condton to th problem of recte flow re detled n [30]. Tble Intl condton to the problem of the double ellpe. Property Vlue M kg/m 3 p 794 P U 5,08 m/ T 300 K T, 300 K Alttude 50,000 m c N 0-9 c O c O c NO c 0.0 NO c 0.0 e L 5.0 m Re.574x0 6 Tble Intl condton to the problem of the reentry cpule. Property Vlue M kg/m 3 p 3,885 P U 4,68 m/ T 473 K T, 473 K Alttude 40,000 m c N 0-9 c O c O c NO c 0.0 NO c 0.0 e L 3.0 m Re 3.468x0 6 The frt geometry under tudy double ellpe, where the econd one reentry cpule. E-ISSN: Iue, Volume 8, Jnury 03

7 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment Thee confgurton re decrbed n Fg nd nd the computtonl mehe, to ncd nd cou ce, re preented n Fg. 3 to 6. The dmenonle employed n the Euler nd Ner- Stoke equton n th tudy re lo decrbed n [30]. 4. Decrpton of the Geometrcl Confgurton nd Employed Mehe Fgure nd exhbt the geometrcl confgurton of the geometre under tudy n th work. In Fgure preented the double ellpe confgurton nd n Fg. preented the reentry cpule confgurton. Fgure 3 nd 4 preent the ncd nd cou tructured double ellpe mehe, where Fg. 5 nd 6 preent the ncd nd cou tructured reentry cpule mehe. The untructured mehe re obtned trnformng meh of qudrlterl n meh of trngle nd the connectty, neghborng, ghot nd node coordnte tble re generted n pre-proceng tge of the computton. The double ellpe meh compoed of 3,58 rectngulr cell nd 3,650 node n the tructured ce nd of 7,056 trngulr cell nd 3,650 node n the untructured ce. The reentry cpule compoed of 3,36 rectngulr cell nd 3,50 node n the tructured ce. The untructured ce to the reentry cpule problem w not multed. Fgure 3. Double ellpe meh (Incd ce). Fgure. Double ellpe confgurton. Fgure 4. Double ellpe meh (Vcou ce). Fgure. Reentry cpule confgurton. Fgure 5. Reentry cpule meh (Incd ce). E-ISSN: Iue, Volume 8, Jnury 03

8 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment 4.4 Reult n Thermochemcl Non- Equlbrum Double Ellpe Ce Fgure 6. Reentry cpule meh (Vcou ce) Incd, tructured nd frt order ccurte ce Fgure 7 exhbt the preure contour round the double ellpe geometry clculted t the computtonl domn by the [6] cheme, n t frt-order eron, n thermochemcl nonequlbrum. The non-dmenonl preure pek equl to, unte nd locted t the confgurton noe. The hock we t the confgurton noe norml. The econd hock we weker thn the frt one, obered n the preure contour. The oluton preent good hock we repreentton n both ellpe. 4.3 Studed Ce Tble 3 preent the tuded ce of the double ellpe problem n th work, the meh chrctertc nd the order of ccurcy of the [6] cheme. An exponentl tretchng w employed n the cou multon mng to cpture the mn cou flow chrctertc, for ntnce: crculton bubble formton, detchment nd rettchment of the boundry lyer, wke formton, etc. Tble 4 preent the tuded ce of the reentry cpule problem n th work, the meh chrctertc nd the order of ccurcy of [6]. Tble 3 Studed ce, meh chrctertc nd ccurcy order (Double ellpe problem). Fgure 7. Preure contour. Ce Meh Accurcy Order Incd D 73x50 Frt Vcou D 73x50 (7.5%) c Frt Incd D 73x50 Second Vcou D 73x50 (7.5%) Second Incd D 73x50 Frt b Vcou D 73x50 (7.5%) Frt b Structured ptl dcretzton; b Untructured ptl dcretzton; c Exponentl tretchng. Tble 4 Studed ce, meh chrctertc nd ccurcy order (Reentry cpule problem). Ce Meh Accurcy Order Incd D 65x50 Frt Vcou D 65x50 (7.5%) Frt Incd D 65x50 Second Vcou D 65x50 (7.5%) Second Fgure 8. Mch number contour. Fgure 8 how the Mch number contour clculted t the computtonl domn. Regon of ubonc flow re formed behnd the norml hock we, t the frt nd econd ellpe. The hock we deelop normlly: norml hock we t the E-ISSN: Iue, Volume 8, Jnury 03

9 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment two ellpe, decyng to oblque hock we nd fnlly rechng, fr from the double ellpe, the Mch we. In relty, the econd ellpe hock we nterct wth the frt ellpe hock we nd retrd th decyng to oblque hock we. Th occur t the upper urfce. At the lower urfce, the hock we more cured, ndctng the nturl phycl tendency to rech oblque hock we behour. Fgure 9 preent the contour of the trnltonl/rottonl temperture dtrbuton clculted t the computtonl domn. The trnltonl/rottonl temperture reche pek of 8,863 K t the confgurton noe nd determne n pproprted regon to docton of N nd O. Along the double ellpe, the trnltonl/rottonl temperture ume n pproxmted lue of 6,000 K, wht lo repreent good lue to the docton frtly of O nd, n econd plce, of the N. Fgure 0 exhbt the contour of the brtonl temperture clculted t the two-dmenonl computtonl domn. It pek reche lue of 6,90 K nd lo contrbute to the docton of N nd O, nce the employed temperture to the clculton of the forwrd nd bckwrd recton rte (recton-rte-control temperture, T rrc ) n the thermochemcl non-equlbrum equl to T.T V, the qure root of the product between the trnltonl/rottonl temperture nd the brtonl temperture. Hence, the effecte temperture to the clculton of the chemcl phenomen gurntee the couple between the brtonl mode nd the docton recton. In th confgurton noe regon, the temperture T rrc reche, n the tedy tte condton, the pproxmted lue of 7,80 K, gurnteeng tht the docton phenomen decrbed boe occur. Fgure preent the elocty ector feld round the double ellpe confgurton. A cn be een, the tngency condton well tfed by the lgorthm. The hock we profle lo well cptured. Fgure 9. Trnltonl/rottonl temperture contour. Fgur. Velocty ector feld. Fgure 0. Vbrtonl temperture contour. Fgure how the m frcton dtrbuton of the een chemcl pece under tudy, nmely: N, O, N, O, NO, NO + nd e -, long the geometry tgnton lne. A cn be obered from th fgure, enough docton of N nd O occur, wth the conequent menngful ncree of N, of NO nd of O n the geou mxture. A mentoned erly, th behour expected due to the effecte pek temperture reched t the clculton domn. The NO preented the bgget bolute ncree n t formton, where the N preented the bgget relte ncree. E-ISSN: Iue, Volume 8, Jnury 03

10 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment multon. The hock we preent the expected behour: norml hock we t the confgurton noe, oblque hock we nd Mch we fr from the double ellpe. Fgure. M frcton dtrbuton t the double ellpe tgnton lne Vcou, tructured nd frt order ccurte ce Fgure 3 exhbt the preure contour to the cou recte flow round double ellpe, n two-dmenon, clculted t the computtonl domn. The non-dmenonl preure pek reche,863 unte, more eere thn tht obtned wth the ncd ce. The preure feld lo more eere n the cou ce thn n the ncd one. The hock cloer to the geometry due to the meh exponentl tretchng nd the cou recte effect of the multon. The regon of the preure pek lo better defned. Fgure 4. Mch number contour. Fgure 5 exhbt the dtrbuton of the trnltonl/rottonl temperture clculted t the computtonl domn. The pek of trnltonl/rottonl temperture reche the pproxmted lue of 9,30 K t the confgurton noe. Fgure 5. Trnltonl/rottonl temperture contour. Fgure 3. Preure contour. Fgure 4 how the Mch number contour clculted t the computtonl domn. The ubonc flow regon, whch re formed behnd the norml hock, re well cptured nd propgte by the lower nd upper geometry wll, due to the trnport phenomen condered n the cou Fgure 6 preent the brtonl temperture dtrbuton clculted t the computtonl domn. It pek, t the confgurton noe, reche n pproxmted lue of 6,83 K. The effecte temperture to the clculton of the docton nd recombnton recton, T rrc, equl pproxmtely to 8,800 K, whch gurntee tht procee of docton of O nd N cn be cptured by the employed formulton. Th lue of effecte temperture to the cou recte E-ISSN: Iue, Volume 8, Jnury 03

11 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment multon uperor to tht obtned n the ncd ce. Fgure 6. Vbrtonl temperture contour. Fgure 7 preent the elocty ector feld round the double ellpe confgurton. A cn be een, the dherence nd non-permeblty condton re well tfed by the cou formulton. The hock we profle lo well cptured. Fgure 8. M frcton dtrbuton t the double ellpe tgnton lne Incd, tructured nd econd order ccurte ce Fgure 9 how the preure contour obtned by the ncd multon performed by the econdorder [6] cheme employng mnmod non-lner flux lmter. The non-dmenonl preure pek pproxmtely equl to,594 unte, lghtly nferor to the repecte pek obtned by the frtorder oluton. Th preure pek occur t the confgurton noe. Fgure 7: Velocty ector feld. Fgure 8 exhbt the m frcton dtrbuton of the een chemcl pece under tudy long the geometry tgnton lne. A cn be obered, enough docton of the N nd O occur, wth the conequent menngful ncree of the N, of the O, nd of the NO n the geou mxture. Th behour expected due to the temperture pek reched n the clculton domn. The bgget bolute ncree n the formton of pece w due to the NO, whle, n relte term, w due to the N. The electron formton dcrete, chrcterzng wek onzton flow. Fgure 9. Preure contour. Fgure 0 preent the Mch number contour obtned t the computtonl domn. The ubonc regon whch re formed behnd the norml hock we re well chrcterzed t the frt nd econd ellpe. The hock we orgnted t the confgurton noe preent the expected behour, png from norml hock t the confgurton tgnton lne to Mch we fr from the double ellpe. E-ISSN: Iue, Volume 8, Jnury 03

12 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment Fgure 0. Mch number contour. Fgure preent the contour of the brtonl temperture dtrbuton clculted t the computtonl domn. The brtonl temperture pek pproxmtely equl to 6,578 K nd obered t the confgurton noe. The effecte temperture to clculton of the recton rte (recton rte control temperture, T rrc ) pproxmtely equl to 7,56 K, whch repreent temperture ble to cpture the docton phenomen of N nd O. Fgure 3 preent the elocty ector feld round the double ellpe confgurton. A cn be een, the tngency condton well tfed by the ncd formulton. The hock we profle lo well cptured. Fgure exhbt the contour of the trnltonl/rottonl temperture dtrbuton clculted t the computtonl domn. The trnltonl/rottonl temperture pek occur t the confgurton noe nd pproxmtely equl to 8,588 K. Fgure 3. Velocty ector feld. Fgure. Trnltonl/rottonl temperture contour. Fgure 4. M frcton dtrbuton t the double ellpe tgnton lne. Fgure. Vbrtonl temperture contour. Fgure 4 exhbt the m frcton dtrbuton of the fe chemcl pece under tudy, nmely: N, O, N, O nd NO, long the geometry tgnton lne. A cn be obered, conderble docton of N nd O occur, wth conequent menngful E-ISSN: Iue, Volume 8, Jnury 03

13 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment ncree of the N, of the O nd of NO n the geou mxture. Th behour expected due to the effecte temperture pek reched t the computtonl domn to the clculton of thermochemcl non-equlbrum nd to econdorder numercl formulton. Interetng enough the NO oluton whch, oppoed to the me nly n [0], preent decreed behour cloe to the geometry Vcou, tructured nd econd order ccurte ce Fgure 5 exhbt the preure contour clculted t the computtonl domn to the tuded confgurton of double ellpe. The nondmenonl preure pek pproxmtely equl to,837 unte, le thn the repecte lue obtned by the frt order oluton. The hock potoned cloer to the blunt body due to the meh tretchng nd the employed-cou-recte formulton. Fgure 6 how the Mch number contour obtned t the computtonl domn. The ubonc regon behnd the norml hock we well chrcterzed nd propgte long the lower nd upper geometry urfce, due to the conderton of the trnport phenomen (coty, therml conductty nd pece dffute). The expected behor to the hock we rtfed: norml hock, oblque hock nd Mch we. The hock we behour lo the expected: norml hock t the geometry noe, oblque hock we cloe to the confgurton nd Mch we fr from the geometry. Fgure 7. Trnltonl/rottonl temperture contour. Fgure 5. Preure contour. Fgure 8. Vbrtonl temperture contour. Fgure 6. Mch number contour. Fgure 7 exhbt the trnltonl/rottonl temperture dtrbuton clculted t the computtonl domn. The temperture pek t the confgurton noe reche 9,5 K. Fgure 8 how the brtonl temperture dtrbuton clculted t the computtonl domn. The temperture pek t the noe nd long the lower nd upper urfce of the geometry equl to 7,67 E-ISSN: Iue, Volume 8, Jnury 03

14 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment K. The effecte temperture to the clculton of the recton rte, T rrc, w of 8,43 K, uperor to tht obtned wth the frt order oluton, whch repreentte to the clculton of the N nd O docton. Fgure 9 preent the elocty ector feld round the double ellpe confgurton. A cn be een, the dherence nd non-permeblty condton re well tfed by the cou formulton. The hock we profle lo well cptured. Fgure 30 preent the m frcton dtrbuton of the een chemcl pece under tudy, nmely: N, O, N, O, NO, NO + nd e -, long the geometry tgnton lne. A cn be obered, good docton of N nd O occur, wth conequent good ncree of N, O nd NO n the geou mxture. Th behour expected due to the effecte temperture pek reched t the computtonl domn to the clculton of thermochemcl non-equlbrum nd to econdorder numercl formulton, whch behe n more conerte wy ([3]), prodng mjor docton of N nd O. In other word, th oluton proded by the econd-order [6] cheme, een n other ce, tend to orgnte bgger docton of N nd O. A th oluton more prece (econd-order), t hould be condered tndrd to compron wth other cheme. Agn, t poble to obere tht the NO oluton preent plteu cloe to the noe confgurton Incd, untructured nd frt order ccurte ce Fgure 3 preent the preure contour obtned by the ncd multon t the computtonl domn. The non-dmenonl preure pek, pproxmtely,73 unte, bgger thn tht obtned wth the ncd, frt-order ccurte n pce, tructured oluton. Th preure pek occur t the confgurton noe. The econd hock t the mnor ellpe well cptured. Fgure 9. Velocty ector feld. Fgure 3. Preure contour. Fgure 30. M frcton dtrbuton t the double ellpe tgnton lne. Fgure 3. Mch number contour. E-ISSN: Iue, Volume 8, Jnury 03

15 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment Fgure 3 exhbt the Mch number contour clculted t the computtonl domn. The ubonc regon behnd the norml hock well chrcterzed, lo t the econd hock, t the mnor ellpe. The hock deelop nturlly. Fgure 33 how the trnltonl/rottonl temperture dtrbuton clculted t the computtonl domn. The temperture pek t the noe nd t the confgurton mnor ellpe reche pproxmtely 8,90 K. The hock we n both fgure preent the expected behor: png from norml hock to oblque hock nd fnhng wth Mch we. Fgure 35 how the elocty ector feld to n ncd flow. The flow tngency condton well gurnteed by the employed ncd formulton. Fgure 35. Velocty ector feld. Fgure 33. Trnltonl/rottonl temperture contour Vcou, untructured nd frt order ccurte ce Fgure 36 exhbt the preure contour clculted t the computtonl domn. The non-dmenonl preure pek, pproxmtely,88 unte, lghtly uperor to the repecte one obtned from the tructured, frt-order ccurte oluton. The preure pek occur t the confgurton noe, where the norml hock occur. The econd hock t the mnor ellpe well cptured. Fgure 34. Vbrtonl temperture contour. Fgure 34 preent the brtonl temperture dtrbuton clculted t the computtonl domn. The temperture pek t the geometry noe pproxmtely equl to 6,67 K. The effecte temperture to the clculton of the recton rte (recton rte control temperture, T rrc ) w of 7,709 K, whch relent to the clculton of N nd O docton. Th effecte temperture, T rrc, mller thn tht obtned to the ncd, tructured, frt-order ccurte ce nd, therefore, mnor number of N nd O docton re expected. Fgure 36. Preure contour. Fgure 37 preent the Mch number contour clculted t the computtonl domn. The ubonc flow regon, behnd the frontl hock, well chrcterzed long the geometry, due to E-ISSN: Iue, Volume 8, Jnury 03

16 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment effect of trnport phenomen (coty, therml conductty nd geou dffuon). The hock behor lo the expected: norml hock deelopng to oblque hock we nd fnhng n Mch we. Fgure 39. Vbrtonl temperture contour. Fgure 37. Mch number contour. Fgure 38 how the trnltonl/rottonl temperture dtrbuton clculted t the computtonl domn. The temperture pek t the confgurton noe reche pproxmtely 9,30 K. Fgure 39 exhbt the brtonl temperture dtrbuton clculted t the computtonl domn. The temperture pek t the geometry noe pproxmtely equl to 7,3 K. The effecte temperture to the clculton of the recton rte, T rrc, w of 8,0 K, whch conderble to the clculton of the N nd O docton. Th effecte temperture le thn tht obtned to the cou, tructured, frt-order ccurte ce nd, therefore, mnor number of N nd O docton re expected. The hock we n both fgure preent the expected behor: norml hock, oblque hock we nd Mch we. Fgure 40 exhbt the elocty ector feld to cou flow formulton. A mll detchment nd rettchment of the boundry lyer t the mnor ellpe occur. The dherence nd mpermeblty condton re gurnteed by the cou formulton. Fgure 40. Velocty ector feld. 4.5 Reult n Thermochemcl Non- Equlbrum Reentry Cpule Ce Fgure 38. Trnltnl/rottonl temperture contour Incd, tructured nd frt order ccurte ce Fgure 4 exhbt the preure contour to the reentry cpule confgurton clculted t the computtonl domn. The non-dmenonl preure pek pproxmtely equl to,70 unte. The oluton preent good ymmetry chrctertc. Fgure 4 preent the Mch number contour clculted t the computtonl domn. A ubonc flow regon formed behnd the norml hock we t the geometry tgnton lne. The E-ISSN: Iue, Volume 8, Jnury 03

17 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment hock we deelop normlly: p from norml hock we, t the geometry tgnton lne, to oblque hock we, fnhng wth Mch we. Good ymmetry chrctertc re obered. condton well tfed. The elocty ector tngent long ll the confgurton wll. Fgure 43. Trnltonl/rottonl temperture contour. Fgure 4. Preure contour. Fgure 44. Vbrtonl temperture contour. Fgure 4. Mch number contour. Fgure 43 how the trnltonl/rottonl temperture dtrbuton clculted t the computtonl domn. The temperture pek occur t the confgurton noe nd t the confgurton trlng edge. Th pek of trnltonl/rottonl temperture pproxmtely equl to 7,88 K. Fgure 44 exhbt the brtonl temperture dtrbuton clculted t the computtonl domn. The pek of brtonl temperture occur t the confgurton noe nd pproxmtely equl to 6,538 K. The recton rte control temperture, T rrc, pproxmtely equl to 6,903 K, whch conderble to the O docton, mnly, nd to the N. Both oluton, Fgure 43 nd 44, preent good ymmetry properte. Fgure 45 preent the elocty ector feld to the ncd ce. A cn be obered, the tngency Fgure 45. Velocty ector feld. Fgure 46 exhbt the m frcton dtrbuton of the een chemcl pece tuded n th work, nmely: N, O, N, O, NO, NO + nd e -, long the E-ISSN: Iue, Volume 8, Jnury 03

18 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment geometry tgnton lne. Good docton of N nd O occur cloe to the confgurton noe. Good formton of N, n relton to t ntl lue, nd of O occur. There lo good ntl formton of NO t the tgnton lne, t the pot-hock regon, preentng the bgget bolute rton mong the formed pece. Howeer, cloe to the blunt noe the NO formton reduced nd preent mller ncree n relton to the O t th regon. The e - well the NO + formton re dcrete. Menngful formton were reched wth n effecte temperture lttle boe 6,903 K. Th the behor preented to the tructured, ncd, frtorder ccurte n pce, oluton. formton of crculton bubble re predomnnt. The hock deelop normlly: norml hock t the geometry tgnton lne, oblque hock we long the confgurton nd Mch we fr from the geometry. Good chrctertc of ymmetry re obered n the oluton. Fgure 47. Preure contour. Fgure 46. M frcton dtrbuton t the reentry cpule tgnton lne Vcou, tructured nd frt order ccurte ce Fgure 47 exhbt the preure contour clculted t the computtonl domn to the reentry cpule problem ubmtted to cou flow. A tructured, frt-order formulton employed n th multon. The non-dmenonl preure pek equl to,388 unte, bgger thn tht obtned n the ncd, tructured, frt-order oluton. The preure feld n th ce more eere thn tht of the ncd one. Good ymmetry properte re obered. Fgure 48 how the Mch number contour clculted t the computtonl domn. The oluton preent ubonc flow regon t the confgurton noe, behnd the norml hock. The extenon of th regon depend of the flow properte (Mch number, preure, etc.) nd of the geometry chrctertc. Th ubonc flow regon propgte round ll geometry due to the trnport phenomen orgnted by the cou formulton. At the geometry trlng edge pper cou wke, where gret lo of flow energy nd the Fgure 48. Mch number contour. Fgure 49. Trnltonl/rottonl temperture contour. E-ISSN: Iue, Volume 8, Jnury 03

19 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment Fgure 49 preent the contour of the trnltonl/rottonl temperture dtrbuton clculted t the computtonl domn. The pek of the trnltonl/rottonl temperture occur cloe to the confgurton noe nd t the cou wke t the geometry trlng edge. The pek lue of th temperture pproxmtely equl to 7,88 K. Fgure 50 exhbt the contour of the brtonl temperture dtrbuton clculted t the computtonl domn. The brtonl temperture pek occur cloe to the confgurton noe. Th pek pproxmtely equl to 6,75 K. The effecte temperture to be employed n the chemcl model, T rrc, pproxmtely equl to 7,70 K, whch ndcte menngful N nd O docton phenomen. Fgure 49 nd 50 preent good ymmetry chrctertc. Thee bubble re well cptured by the oluton. In th regon there gret lo of energy. Th energy dpted through the crculton bubble tructure. The dherence nd mpermeblty condton re gurnteed by the cou formulton. Good ymmetry chrctertc re obered. Fgure 5 preent the m frcton dtrbuton of the een chemcl pece tuded n th work, nmely: N, O, N, O, NO, NO + nd e - long the geometry tgnton lne. Good docton of N nd O occur wth both rechng contnt lue cloe to the geometry noe. Good formton of N, n relton to t ntl lue, nd of O. It lo een good formton of the NO t the tgnton lne, t the pot-hock regon, preentng the bgget bolute rton mong the pece tht were formed. Moreoer, the NO formton reche pek hed of the noe regon nd t formton reduced cloe to the blunt noe. A poble reon to th behor cn be the contncy n N nd O docton n th regon. The e - nd NO + formton re dcrete. Menngful formton were reched wth n effecte temperture of 7,70 K. Th the behor preented to the tructured, cou, frt-order ccurte n pce, oluton. Fgure 50. Vbrtonl temperture contour. Fgure 5. M frcton dtrbuton t the reentry cpule tgnton lne. Fgure 5. Velocty ector feld nd tremlne. Fgure 5 how the elocty ector feld to th cou flow. A mentoned erler, there the formton of boundry lyer eprton regon, wth the formton of pr of crculton bubble Incd, tructured nd econd order ccurte ce Fgure 53 exhbt the preure contour clculted t the computtonl domn to the ncd, tructured, econd-order ccurte n pce, ce. Th econd-order ccurcy obtned by the [6] cheme employng TVD MUSCL procedure, wth mnmod non-lner lmter. The nondmenonl preure pek pproxmtely equl to,3 unte, uperor to the repecte pek obtned by the ncd, frt-order ccurte, E-ISSN: Iue, Volume 8, Jnury 03

20 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment oluton. The preure feld lo more eere thn the repecte feld of the ncd, frt-order ccurte, oluton. The preure pek occur t the confgurton noe. Good ymmetry chrctertc re obered n the fgure. Fgure 53. Preure contour. Fgure 54 how the Mch number contour clculted t the computtonl domn. A t regon of ubonc flow formed t the confgurton noe, behnd the norml hock. Th regon formed by the hock ntenty nd t extenon depend on the flow properte (Mch number, preure, etc.) nd on the geometry chrctertc. The hock we deelop normlly: norml hock we t the ymmetry lne, oblque hock we long the confgurton nd Mch we fr from the geometry. Good ymmetry chrctertc re obered n the oluton. Fgure 55 preent the trnltonl/rottonl temperture dtrbuton clculted t the computtonl domn. The trnltonl/rottonl pek occur t the confgurton noe nd pproxmtely equl to 7,09 K. Fgure 56 exhbt the brtonl temperture dtrbuton clculted t the computtonl domn. The brtonl temperture pek occur t the confgurton noe nd pproxmtely equl to 6,537 K. The effecte temperture to be condered to the chemcl formulton, T rrc, pproxmtely equl to 6,809 K, whch menngful to cpture the N nd O docton. Th temperture mller thn the repecte one obtned by the ncd, frt-order ccurte, oluton nd ndcte tht mnor N nd O docton hould occur. Fgure 54. Mch number contour. Fgure 56. Vbrtonl temperture contour. Fgure 55. Trnltonl/rottonl temperture contour. Fgure 57 preent the elocty ector feld to th ncd flow ce clculted wth the econdorder ccurte [6] cheme. The tngency condton gurnteed by the ncd formulton. Good ymmetry chrctertc re lo obered. Fgure 58 exhbt the m frcton dtrbuton of the een chemcl pece tuded n th work long the geometry tgnton lne. Good N nd O docton occur, wth the N rechng contnt E-ISSN: Iue, Volume 8, Jnury 03

21 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment lue cloe to the confgurton noe. Good formton of N, n relton to t ntl lue, nd of O nd of NO, wth the O preentng the bgget rton mong the formed pece, occur. Wth t, t poble to ffrm tht menngful formton were reched wth n effecte temperture of 6,809 K (mller thn n the ncd, frt-order ccurte, ce). Th the behor preented to the tructured, ncd, econd-order ccurte n pce, oluton. globlly le eere. Good ymmetry chrctertc re obered n the oluton. Fgure 59. Preure contour. Fgure 57. Velocty ector feld. Fgure 60 exhbt the Mch number contour clculted t the computtonl domn. A ubonc flow regon, obered n the other oluton too, formed t the confgurton noe, behnd the frontl hock. Th ubonc flow regon propgte round the reentry cpule geometry untl the trlng edge. The extenon of the ubonc regon t the confgurton noe depend on the flow properte nd on the geometry chrctertc. At the confgurton trlng edge, cou wke formed, where the formton of crculton bubble occur, een erler. Th regon typclly of gret lo of energy nd preure. Good ymmetry chrctertc re obered. Fgure 58. M frcton dtrbuton t the reentry cpule tgnton lne Vcou, tructured nd econd order ccurte ce Fgure 59 preent the preure contour clculted t the computtonl domn to the tructured, cou, econd-order ccurte, ce. The nondmenonl preure pek reche n pproxmted lue of,35 unte, nferor to the repecte one obtned n the cou, frt-order ccurte, oluton. The preure pek locted t the confgurton noe. The preure feld lo Fgure 60. Mch number contour. Fgure 6 how the contour of the trnltonl/rottonl temperture dtrbuton clculted t the computtonl domn. The trnltonl/rottonl temperture pek ume E-ISSN: Iue, Volume 8, Jnury 03

22 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment the pproxmted lue of 7,786 K, cloe to the geometry noe. Fgure 6 preent the brtonl temperture dtrbuton clculted t the computtonl domn. The brtonl temperture pek occur lo cloe to the confgurton noe nd reche the pproxmte lue of 7,534 K. The effecte temperture to be condered n the chemcl clculton, T rrc, reche the pproxmted lue of 7,659 K, whch n pproprted temperture to promote the N nd O docton cloe to the reentry cpule geometry. Th temperture uperor to tht obtned to the cou, frt-order ccurte, ce nd, wth t, more N nd O docton re expected. Fgure 63 exhbt the elocty ector feld clculted t the computtonl domn. A cn be obered, the boundry lyer detchment t the trlng edge regon of the reentry cpule nd the formton of pr of crculton bubble due to the cou phenomen nd due to the dere preure grdent n th regon occur. Good ymmetry chrctertc re obered n the oluton. Fgure 63. Velocty ector feld nd tremlne. Fgure 64 how the m frcton dtrbuton of the een chemcl pece tuded n th work, nmely: N, O, N, O, NO, NO + nd e - long the geometry tgnton lne. A menngful N nd O docton occur cloe to the confgurton noe. Good formton of N, n relton to t ntl lue, nd good formton of O nd NO occur. A dcrete formton of e - nd NO + obered. It lo een good formton of NO long the tgnton lne, t the pot-hock regon. The ncree t the formton of NO would be bgger thn tht of the O, whether reducton n relton to t ntl behor h not occurred. Th reducton n the NO m frcton cloe to the geometry noe due to the contncy reched n th regon by the N nd O docton nd due to the ncree n the formton of the N nd of O. Fgure 6. Trnltonl/rottonl temperture contour. Fgure 64. M frcton dtrbuton t the reentry cpule tgnton lne. Fgure 6. Vbrtonl temperture contour. Een o, menngful formton were reched wth n effecte temperture lttle bgger thn 7,650 K, E-ISSN: Iue, Volume 8, Jnury 03

23 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment lthough wth mnor N nd NO formton thn thoe obtned wth the cou, frt-order ccurte, ce. In th lt ce, bgger effecte temperture, T rrc, w reched. Th the behor preented by the cou, tructured, econd-order ccurte n pce, oluton. 4.6 Shock Poton of the Double Ellpe Ce (Shock t the Noe) In th ecton preented the behor of the hock poton n thermochemcl non-equlbrum condton conderng the model of fe (5) nd een (7) pece. Frt- nd econd-order oluton re compred. The detched hock poton n term of preure dtrbuton, n the ncd ce nd frt- nd econd-order ccurte oluton, exhbted n Fg. 65. A cn be obered, the een pece model yeld moother profle to the hock we thn the fe pece model. Second-order oluton predct the hock poton cloer to the blunt noe thn the frt-order oluton, n both fe nd een pece model. Th reult the expected one due to the mot ccurte oluton wth the econd-order cheme. Fgure 66. Shock poton (Vcou ce). 4.7 Shock Poton of the Reentry Cpule Ce (Shock t the Noe) In th ecton preented the behor of the hock poton n thermochemcl non-equlbrum condton conderng the model of fe (5) nd een (7) pece. Frt- nd econd-order oluton re compred. Fgure 65. Shock poton (Incd ce). The detched hock poton n term of preure dtrbuton, n the cou ce nd frt- nd econd-order ccurte oluton, exhbted n Fg. 66. A cn be obered, greeng wth the ncd reult, frt-order oluton re longer thn econdorder oluton n relton to the blunt noe. The dfference n ech chemcl model re dcrete n relton to ther ncd counterprt. The hock profle predcted by ech model re moother thn the repecte one to the ncd ce. Fgure 67. Shock poton (Incd ce). The detched hock poton n term of preure dtrbuton, n the ncd ce nd frt- nd econd-order ccurte oluton, exhbted n Fg. 67. A cn be obered, the een pece model yeld the me hock profle tht the fe pece model, n both frt- nd econd-order ccurte reult. Second-order oluton predct the hock poton cloer to the blunt noe thn the frt-order oluton, n both fe nd een pece model. Th reult the expected one due to the mot ccurte oluton wth the econd-order cheme. E-ISSN: Iue, Volume 8, Jnury 03

24 WSEAS TRANSACTIONS on APPLIED nd THEORETICAL MECHANICS Edon Sáo De Góe Mcel, Amlcr Porto Pment The detched hock poton n term of preure dtrbuton, n the cou ce nd frt- nd econd-order ccurte oluton, exhbted n Fg. 68. A cn be obered, greeng wth the ncd reult, frt-order oluton re longer thn econdorder oluton n relton to the blunt noe. The dfference n ech chemcl model re enble n relton to ther ncd counterprt. The frt-order oluton preent pek t the hock plteu. Moreoer, the hock profle predcted by ech model, n the econd-order ce, re moother thn the repecte one to the frt-order ce. Fgure 68. Shock poton (Vcou ce). 4.8 Aerodynmc Coeffcent of Lft nd Drg Tble 5 exhbt the erodynmc coeffcent of lft nd drg obtned by the problem of the double ellpe, wth tructured dcretzton, to the recte formulton. Both recte formulton of thermochemcl non-equlbrum wth fe pece ([6]) nd thermochemcl non-equlbrum wth een pece ([30]) re condered n th compron. Thee coeffcent re due to the preure term lone. The contrbuton of the frcton term w not condered. Tble 5 Aerodynmc coeffcent of lft nd drg to the tructured double ellpe ce. Studed Ce c L c D t -Order/Incd/5 Spece t -Order/Vcou/5 Spece nd -Order/Incd/5 Spece nd -Order/Vcou/5 Spece t -Order/Incd/7 Spece t -Order/Vcou/7 Spece nd -Order/Incd/7 Spece nd -Order/Vcou/7 Spece To the problem of the double ellpe, there not ymmetrc geometry n relton to the x x. Hence, dfferent lue of zero expected to the lft coeffcent. By Tble 5, t poble to note tht the mot eere lue to c L w tht of the [6] cheme wth frt-order ccurcy, n n ncd formulton, employng the thermochemcl nonequlbrum model of een pece. The mxmum c D w obtned by the oluton of the [6] cheme, frt-order ccurte, n cou formulton, employng the thermochemcl non-equlbrum model of een pece. Tble 6 exhbt the erodynmc coeffcent of lft nd drg obtned by the problem of the reentry cpule, wth tructured dcretzton, to the recte formulton. Thee coeffcent re due to the preure term lone. The contrbuton of the frcton term w not condered. To the problem of the reentry cpule, ymmetrc geometry n relton to the x x, zero lue expected to the lft coeffcent. By Tble 6, t poble to note tht the oluton cloet to th lue to c L were thoe of the [6] cheme wth frtorder ccurcy, n cou formulton, employng thermochemcl non-equlbrum model of fe n een pece. The mxmum c D were lo obtned by the oluton of the [6] cheme, frtorder ccurte, n cou formulton, employng thermochemcl non-equlbrum model of fe nd een pece. Tble 6 Aerodynmc coeffcent of lft nd drg to the tructured reentry cpule ce. Studed Ce c L c D t -Order/Incd/5 Spece 4.x t -Order/Vcou/5 Spece -.x nd -Order/Incd/5 Spece.4x nd -Order/Vcou/5 Spece -6.x t -Order/Incd/7 Spece 4.x t -Order/Vcou/7 Spece -.x nd -Order/Incd/7 Spece.4x nd -Order/Vcou/7 Spece -6.x unttte Anly In term of qunttte reult, the preent uthor compred the recte reult wth the perfect g oluton. The tgnton preure t the double ellpe nd the reentry cpule noe w eluted umng the perfect g formulton. Such prmeter clculted t th wy not the bet compron, but n the bence of prctcl recte reult, thee conttute the bet lble reult. To clculte the tgnton preure hed of the E-ISSN: Iue, Volume 8, Jnury 03