RSF N" 23/1123 AY Page 15 O.N.E.R.A. N93- 28745' _29 =,/o50 KNNEXE A DEVELOPPEMENT D'UN CODE EULER MULTIDOMAINE 3D Par Ph. GUILLEN et M. DORMIEUX RSF N" 23/1123 AY Page 16 DESIGN OF A 3D MULTIDOMAIN EULER CODE Ph.Guillen ONE.RA.Ae,o_asamic¢Deparontat 29Avrau__ItlaDivisioLazclzrc-92J20CHA TTLLON (Fraace) md M. Dormieux AE_OSPATIAL_. Division £ntls_ Tac_qut, 2d18R_,Biro_a_r-92320CHATILLON [Fral_e) ABSTRACT A MultidomainEulerCodehasbeendevelopedto numericallysimulateflowsoflastsof different nature aroundcomplex configurations with an emphasison supersonic and hypersonic flows._ main choices concerning computational and numerical aspects ate described. The code hasbeenwriuen in order to allow easy implunentaxion o( new boundary _-eaum-.nourfun_r extensiontsomoreccxnplex_u .ofequationsS.ome typicalresults concernincglassicsahlapes_.4_rsonkandhypersoniHcermes shutdeandu'znsverstexxjets Ireshown. EqTRODUCTION The developmen[ofindustrinaulmericalcodestosolvetheequationsoffluidmechanics representsan imponam investmentwith some uncertain prospectivechoicesto bemade. Tbes__ years quiteanimponzm numberof _n'lctive methods havebeenproposed for the simulation of perfect fluid flows around 3D conflllurations: finite volume or finite element memods,us/ngsuuctm'edor_u_d Fids, havingacenteredoranon<entered numerical scheme,etc... Theselection of themost promising meLhodseemsto bean hazardousand d/ffkult choice., monetofcompe:_nisebetweendiffetem¢ormderarionswhicharegene_lly conn'-_licxx'y. Moreover son_ imponamtelements ofchoice, such astheavailablecomputer I_.hnolog7 ate difficutlotbeestimatedanditb ckm' IJ_ttheevaluationofthealgorithm ¢_icieneyisquitedifferen_t¢¢ordin|tothekindofprocesslngs:calarv,ectororparallel, Fromanindusu_xplointofviewoncethedesirekdvelofac_ Isre.acbebdyth_method, themostimp:)ctaqnutalitoyfthecod=isrobustneswshich canlead to selecallgorithmnsot very computadoaallycfficic_LAnotherimportantpointto beconsideredisd_ extension of thecodeu)amorecompk:xsetofequationsofd_csamefamily.Forinstancenumerical codes solvingtheEulersetofequationfsoraperfecgtaswillprobablyhavetobeextendedtomore complexsuueequations ortomulti-speciesgassoon_ orlater.Theeasyimplementationof new boundar)v,eacrnenubymeansofamodu_ coding.M wellasugonomicconsiderations atealsoveryimportanrtnacterfsorfuturedevelopment,riduseofthecode.Theaimofthis paperistcpresenat3D Eulercodedevelopedforthenumericalsimulatioonfflowsofgases ofdifferenntaturekeepinginmindall_hepointsstatedbefore_. differengtasesconsidered ateped'ecgtas.realgasa[equilibriunmndnon.reactivte*o-s'pecimeisxtureI.nafirsptan, thechoicesfeudingtotheotchitectuot4¢"thecodeatedescribeTdh.esecondpande',dwsiththe numericalschemeanditsimplementationL.asdythethirdpanexhibiusome firsrtesults ob_ncd. RSFN" 23/1123 AY Pagp 17 I. ARCH]TECTURE OFTTi_CODE:GENI_RALCONSIDERA_ONS The mrchitec_uteo( the code hasbeen dicmu_dby consta'aintsconcemin| |eome_cal _radons, computadomdaspectsandthespecif_nanzR_ theflow. _al corL_idcratic¢_ The _..auT_nt of compk:x geometries "hasled us to_$opl a mukiblock grid mm:Mof sevend Slructured, poss/bly ovedappinI orpincheddommns.Thischoiceco_sideJ'ablysimplifiesthe meshcoeum_tionandallowsIJ_same|en¢ndityIs unso_lxn'edIFid.s.Suchmu]6block I_d SmUCCiesa_ cutrtndy _inll devek:q:zdatOHER.A [I] andAF.ROSPATIALE [2]and will be implementedwith thecode.Another interestingpossibilityhasbeenintroducedto enable di_ent kux_ofbounda_conditionsoni IEivcackxnam Tbe consvain:sconcemin| the compu,,,Uon_lupect$ ,re clearly linked ,.otheavaitable computer tochnololfy. FromthispointOfview. vectcr andpm-allelprocessinghavebeen consi<k:rcdA. naturalideainmu]tidomaincodesistodism'bu,,"eachdomainonsprocessor. butthiscouldbeno_soinmresx.inpgracticallysincethedomainshaw verydifirutnt numbers points inusualsituationsandmosxoftheprocessorswill havetowait for chosedealin$ with _ $,rtates_ domains. From this synchronizltion waiting.dine point of view a coc_ basedon thecomputationof s_:_u'atedplanesseemsmoR inmrcsdn$. Anotheafavor'ab_e computadonaJ_p¢ct ofaptm_ su'uctu_ isthst thewc_tr.in| Ifrsys for _ num_cal scheme are_ldressed by two indicesand henceno( very expensiveincore for I/_ pfcs_rlt time computers.This can bevery interesdn| especiallyin I/_ cue of implicit allp_rithms. Of cour_ this way of limitin| d_ arntys indexedbythreeindicesRstricts the choiceof 3D aiip_ithrns. For instanceAD[ inthe3direcdoe.s is no more possible.Wid_this choiceand for meshesof acurrent;ndusui_lsized'mtis tOsayabout20(XXX)nodes it is possible to wodc without anymassiveI/Co)naccxn_ter withafew mepwoNs ofmemory co_. Foesulxrson< I_ws itisvet,/impo,-umt totake_=lle of thehypcrbolk:propertyof the sxe.adyEulerequations (s¢¢[3] for in._ance)inIJ_main direcUo_by usin| spacenum:hin| u:chniqueswheneverpossible. Doing sotheCPU dinerequiredd_reates by _morderof ma_itu_. Thefact that, plate of adomaincanbecompu_l separately allows to m_c multiblock space.mm'_hin| computationsplan_ by plane without any additional efforL Anotherconsequenceofthissupersonicnmun:is that forsv._y probk:msthe_mre ofthe computationcanbedifferenctlc_ndinlolnthedomain.A blockwithasubsonicpocketsuch asthebluntbodypurrwill haveto becalculau_dinanuns_y w,y, _ do_'nsutam panof Now with a spuce-mm_:hinl[ strategy. Possiblysome real gaseffects will have to be considereidnsomepunsof the1low_ no( inod_nLTheorderinwhu:hthedifferenbtioc_ have m becomputedis no(obvious,andit isnomor_possiblem i_rate similarlyinall domainsasit canbedonefoetransoniccomputadous.Thecompul,IUontUl.Stobe managed byacommandint=rpreu:rmallow ace.r_inflexibility. Itresulu ina code organisatiobnuiltaround3 keyunits:a command interpretwehrich assumestheuserinterfacea,planemonitoringunitwhich decidesofthetypeofthe computauon,andap"lanperocessoirncludintghenumericalscheme,TheplaneIXOCeSSOi_s describetdhoroughliynthesecondpan.We nowdetaiallitdmcored_ twofirsutniu, The command interpreter is a language, this means_t the input file is interpreted dynamically.Asalangua$¢it hasto ownclassicMcontrol instructionssuchasDOloopsof s_temenLs, ]u mainfunctions arethefollowings. RSF N" 23/1123 AY Page 18 Monitori,_ tAecoee: The memoryallocatloa ismadesothatthearraysindexedbythe whole 3D d_ta ate reducedto theminimum, namely for unsmadyapplicationsthethree coordinatesplus thenecessary conservative variables(in numberequ,l to tha_of the equations ofthesys_m). Thiskind ofdam_ su)axltequentiaJlydomaina/_udomaininthe cote by chcmeanof poinu_rs.Monit_tnl thecoreconsistsin tm'ibutm| poinm'swhena new ck)n'_n iscrr..a_tof reorpnizinl kwh=n ado_a_ hatsbeencompuu_l_ isnomore w_,fuL The samekind olrmonilcrin| is made for thedamn_cessarytocouplingboundary co_Sdonswhere 4_incar lnu_o_n| damhavembek_x for_ch node invo4ved [4]. D_t _,mo4y._,_: _u : Accordinl m @_etemdymmic condition4s_Rc_t_t_ o_ systemso(unitscanbeap_r_ For Sowsuperloniccompuuttioasit mightbeinu_tsdn| m_fme _c variablerselatively © criticcaolndi_ @uuisIou_ vMucs ckfmedfor asonic flow. For hy_rsonic now the _ o4'Moiliettableskadsm Stunits.Fromanethic poim of view it hasaLsoseemed inmn_sdn| m&1_ lhcoppo_uniW mthe usa ,o use a sitoLpossiblcways04"dcfinm| chetmodymimic Def_.'da!sche,,ep_'a,nerers _ _o,_w7 co,_uw_ : Forins_mo=_c Cour4ncnumb_ :_ nam_ of _ boundar_ condiuo_ l_i_li:iI_dom_iavas_blts: Many o_tiort_s beused.Initialization c=nbem_Ic by mear_ of formatted oeunforma::e_dIcs. with umi[oem states previously _fined. Ifnecessary it canbe done onapnnofa dommnonly. f.z_b_ a_:a: This includesno_only resuhtiers tobeinteqxtted trotalso intefmedta_ printingsandcr_aUonofPINGS metafilcs coaudnial gntptw._data. C_I_ l _Arpirate,,,on_;o, : And_ook)inI c_lMI [he8umerk_ scheme. As me c,_culauon is b,wd on thecompuu_km o( se_ planes, di_en_m kinds of czkuLmonscanbcmack_ccon_nI mthewayU_cs=planes_ computed: U_s_,_ws : Thetime ssc'_isthesameforevet,ycelloLc'_trydomain. P._do _,,.steaa_,rio,v, : The soluuonis..4vmccd inu unne.ady w,y bu_wtt._out takinI o( the phys,c.almeaning 04'theflow befo_ _ot_ver_enc_ 5p_cem_'cA_.I compu_on : Thesolution ona_ysic_ _ iSadvtnced tillconvergence before comp'alinli ll_ next physical p.l_nc. Since there are basically two kinds of calculation, space marching and unsteady compuumons,d_m are_Jsotwo kinds o/' multidommn sult_|ies. A plane multidomain su'lte_y wh¢_ all theplanesof different domainsr.oa_Ji:x_dinl to a physicalplane are calculated u_gethct, and amore classical way where the solution is advtnced on a whole domainbeforecons,dcnn|anotherone. 2._ NUN_RICAL SCHEb_. Formulation Theun.,_ady3D Eukeequations_ "m'it_n inconservationform: Wt*Fx*Gy*Hz=O where, forinstance, foraone species I_rfecl g_ w=t(p, pu. Or.pw. e) RSF N" 23/1123 AY Page 19 F=t(pu. p*pu2,puv.puw.(e÷p)u) G=t_v. puv. p_v2, pv.. (e÷p)v) H=t_w. puw, pvw. _,¢w2. (e÷p)w) P"('r"IXe-l/2.0.(u2*-v2÷w2)) To solvethemweuseanimplicitupwindTVD finite volumeschemeotrVanLeer MUSCL type. To obtain_ maximumbenefit from thestructmtd oqlantsttion of thegrid andthe oqanisation b7planeof thecode,theimplicit perto0esistsinanADI like inversionine_ch planecou#edwithaGam;-S4:icle.Ilikerelaxation Lathethirddirectioe.Basicallythescheme comprises the 3 folLowin| Ui.a. j+_.. k÷_ • Uij,k * o giijk * _,IJijk *I_lkijk whereU=P"I(W) isasetofvariables.To ixtservestabilityneardiscoetinuitiesitisnecessary mintroducelimitersineachdirection: if o.i+1/2- Ui.l j.k.Uij.k #ijk'limiter(oi+ 1/2._. 1/2) Many limite_ hzve _ implemen_d tmon$ _ u_ "minmod"[5],Van Leer [6],Van ,Ml_la [TI. and "supcr_" [51formulauons. The secot v_ial_s canbe chosen amonI u_ coeservtdve,rite p_mitive _o.u,v.w,p)orthecharacteristic_ Thentheschemefulfills a monodime_ "r'vD 2) ¢onWa:_ioa _zhe _/_it p_'r _W_ p= .&_eVol(Fi. i/2.Fi. I/2.,.G_, I/2.Gj. l/2*Hk÷ I/'2"Hk.1/2). where Fi,,.l/2 (rtsp. G, H) istn rv_du_Uonof thefluxesat aninterfaceofthecell control volumebymeansofanapproximateRiemannSolverbetweenthetwoseamsoneachs_dcof theinterface cak:utau:dinthefirst step. Many _roxirn_te Rierrumn5avers havebeentested: Forperfect gas_TheVan Leer($]. Roe[9]andOsher[1O]formulations Fcx amixture of nonreactivetwo species |as: Ab_tll [11] extension of Roe fluxes and AblgraH-Montagn_112]extension of Osherfluxes Foerealll:aswithanequilibrium assumlxi_: Vinokur.Montagn_ [13]ex_'nsion of VanLeer and Roe scheme, and AbgrtU-Montagn_ [12] extensim ofOsher scheme. 3)Co#_a._oa _tht implicit I_rt _ tack plane A.,_W-_Wex p where A canbeseenasanappmxtmalkm of whereF isanapproximation ofFx+Gy*H z Two approximationshavebeentested:thelinearizedconscrv=tiveimplicit formulation of Steger.Warmin$ and the linearized non conservative formulation of Harten.Yee [14] or _vat_y 1151. RSF N" 23/1123 AY Pag_ 20 Foeboundm'yconditionmsany v_unentshavebeenconsideredf,romVivia_.Veuillo[t16] compatibility rcl_tionsto moreclas_ca[fluxm_m'_nu andtheir_plicimions. Asitcanbeseen,Van Lee_sMUSCL approachpresentsseveraaldvzntsge:s The schemeisTVD andsowellsuitedforflowswiths_ongdiscontinuities. The extension to mo_ complicated stateequidons or mul_sl_cies coevecuve no_ so'mghfforwardbyimplementingthecorrespondinlnew approximateR_nann solver new jicobizn of the Nuxes. The rnod_Jc_locls _ rnm_ verylocally in_ cornpuu_c'odeInd theoverallarchitectuirsefullkyept.Moceoverellthemodification_s beaddedsothatwe have• undue source forall u'_ diffe_m kindso_flows. Formosx of the variantsofthescheme,no _ haw m befixedbythe user. To implementthisnumericalschemethefollowingstepsatecodedintheplaneprocessor whichadvancesthesoluUononaplan=koft dommn. 1.Re_ch o/ ime,_ecti, t bo_utary co_':io_. This sxepdeu:rmincasmong the bo4zndaryconditions thathave beendecl:urtehdosewhicharc co¢_ernodwiththisplane. 2. Treamera of geomttric s_t_"ies. In thoseroutinesboundarynodesincludingfictitious points mightIx moved inordertou_at some [eomevical singulm'tdeslike axesor somespecialboundaryve_unen_slike those ck_alingwithhalfcontrolvolumecell(forinstancewalllikeco_iuo_s) orsymmemy. 3. ComputationofmuetMc$. From thedataofeither thecellcenterosrthecell reflext,hett_bute_04'thecorttrolvolume as volume,diameter 04"the includesdphereo¢out'warndocmalvecu_ arecompuu_d. 4. Co_ou:ation of ume _ep. And thenrepeat _ following s_eps 5to 11forthe 3directions ij.k. 5.Treaunemo/boundarym_cle_. According totheboundaryu_am_m involvedc,onse:'v_vevaluesoffronder nodes(poss/bly fictitionuosdes)mighthavetobeassignebdeforethecomput,lti_onftheslofxaI.tconcerns nearly all the boundary conditions since the fictitio_ poinkt have at least tobeexwapolated. It is crucial fo¢ matching boundary conditiot%swhere the n<xJcsmight be interpolated into mother domain. 6. Compmation of Hopes. This is thepan I)ofmenumerical scl'.emcdescribed above. 7. Trea_nwmo/bow_ary imerfaces. [tconcernsboundarytreatmentfsorwhichthevalues_ Ln._ucodntheinterfacre_sr than _ thenodes. 8.Comp,.tatio_ o/fl_c This isthepun:2)ofthenumericalscheme. 9. Treaunem of bo,_ndaryfluzes. Here, boundary fluxes are possibly modified. For instance for a wall ueatment the flux computedbytheschemeisn:p[acebdyapressu_flux. 10.Comptaa_ion of implicit cO¢/ficient.¢. Thisispan3or _ numuic_l scheme. l1.Treaune,to/bou_da_ implicictot_c_eaz._ According to _hcboundaryconditions theimplicit malzices are modil'_l. 12.Explichresult. Oh•a/nedbysumming thefluxesitconstituttehserighthandsideoftheimpliciptart. 13.Resolution of _Aein_olicsiy!stem. ItconsistosfanADI likeinversiobnutseveraoltheroption,ssuchasaGauss-SeidelJ.acobi RSF N" 23/i123 AY Psge 21 algorithms canr_ily beimpMmentecL 14.Cotnl_i_lity relation teea:_ of awall bowsd_'y. it consisu in a modification o[ _ valuesg thefrontiers recording to compa[ibilizy I_dadons. This decompositiono[ thet/toridun codin| t/lows {heimplementationof atatgev&ie{y of u_,.tunents.To bemoreconcrete,letusudcethecas_of theimplementaUonof anexplicit wall westment condition and a machLng condition and see some of their possible implementations. For at%rstorder like cell mirror condition the f'_:ddousnode canbe imposed symmetrically to chcbo_ noclcmd sostop 5canbe used, For•secondorderlikecellmirro¢conditiontheoutwLrdin_rfacevaluecantximposed s'_nc_--'icaJlyto the innerinterface valueso_ step?isused. For acla.ssicaJ flux _enk f'_t t halfcontrol volume can bedefined next tothe wall and then thewill flux is _pliciUy computedas• pressurefl_. Su:p2and 9 areusedinthis case. For• compatibility relation u'cam_nt of aslip bcx_ (see [16] )cxdy s_ 14isused. Fog a matching condition amongmany possibilitiesthe two following ones have been irnplemencd: for anarea with asmoothflow, I secondorder ma,ehingconditioncanbe applied. The boundarynockanditsfictitious co_mterptnlureinterpolatedinthecouplingdomain.Then ordystep5isu._d. for an areawith suonggradientsa first orderroaching condition can beused.A connant distribu6on isintroducedin theboundarycell fortheoutw_d direction andtheexternal interfacevalueisinterlx3latedinthecouplingdomain.Thisctnbedonewithsteps5and7 3.CODE VALIDATION In thisFan someprclimintry rtsults showingsomepossibilitiesofthecodeareprcs_ntecL first ca.scpermitsthevalid_on ofthemulddom_n space.mashingstrategy.T'ncsecond case exhibits some results o6cainedon I more complex shape,the Hermes shuttle, in supersonicandhypersoniacerodynamic confections. The lastcaseillustratesimult_block computationofanonr:accivtewospeciesgasflow withm adequatere_ncmem. O_ve-cylinder.Rareconfiguration Experimentalrr..sultsonthisconfi|uraocn areavailable in[17]. Intheselectedtest case the Machnumberis2.96andtheincidence4degrees.ThreespacernarchinEcomputations have beenmade. TheI'u-stonewithI monodomain grid,thesecondonewithaconlinuoustwo domain grid,the"lasotnewith adiscontinuoustwodomaingrid (figureI).Fii;utt 2shows theMath numbcrconto_,1Onthebodyandintheplaneof symmetry.It can bes_enthat resultsareverysimil,_,except asmallwaveissuedfrom_ intersectionofd'_ogivea_tache.d shockwithl.hccouplingftonder. Figure 3puts intOevidence the very '.vcakmlluence o4'the di[fer_n&tr*c_onb_.xlpyressureasndshowsarathegroodcomparisonofthccomputedrcsul_ withexperimcnud ones,The_nall diffetenc¢lbeAweenthecurvescome from _h¢intersection ofthepreviouswavewiththebody. Hermesshuulcconfiguration Two supersoniccomputations havebeenmadeforaMachnumbero4'2.5andincidencesof 0 and 10 degrees.The Math number contourson thebodyandin thesymmetry plane tre displayedon figur4e. comparisonof the lift with experimental dmah_sbccnfoundto be in verygoodagrccmcnLFigure5 showsdensitycontoursobtainedon Hcrrncsfurchodyfoean hypersonic{'loowfMach number12withoutinciden4:Teh.efluxesusedm thiscase_rethe Vim Leer.Vinokur.Mo_agn40aes. RSF N" 23/i123 AY Page 22 Ho_u'ans'_crsjcetc',k:ulatJon Thissimulutiondealswith theinter_K:do_occuringbe_n a_t en_ripng slTJightupfrom a t'_t pl:,teinto :,flow paralleltothis one [18] (seefilure 6 for theinitial _crodyncmic coc_itions). Ina peocucalsituationtheho__t Mrsaspecificheatr_o differenftromthe o_dlow. This canbesimulatednu_ly by considednl nonR.._Cdvemixtures of two speciesflOWS.An impor_n( expertmenud fe_ure of _es¢ flows is the exis..t.e.nce of a subsonic_'_ thatcould notbe _ on _ initia/_ _ Soinor_r tocatxm_this mbsOnic pocket a n_fineddomainhas beenadded_ itsmpposcd ioc_ion (_ the frame tik_e '7_mdthe compumdorudgridsliirum $). This dom_n Ms beeninitialized _th the solutioonbtainedinIJ".coanmgridandthenthecakuladonhasbeenachievodonlyonthe Rf'mcddomain(weak_ouplini0F.illUt9edx_vsd_ concenu'aJodnismbutionofthehot,let ind_en_fineadreaforthetwomcshe¢Ot_cansintehes_onl influencoefd_eIPid rtfmernent onthe_ shapeFigu_10showstl_Mach numb_ disuibutk_in6_e_owficld. Clearly mbso¢Ikpocket iSnicely C_lXUmdms_ _fmod _ (minimum_ numberof0.67). CONCLUSION A 3D multidomain Eulescode hasbee_developed, itsvery modularcodini_allows the implementationofvariousnumericalvm'ixncosfVan Lce_MUSCL schemeanda large v'uicty ofboundary conditions. The exm_sion to differensttateequations or muldspeclcs flowshasbeenshown tobe s:.,'Cillhd'oclwuaradb,ilittyohandleunpatchcdgridseases corts_dcr_btlhyemultiblock iriddin_,andpermitstomak_Rfinemcnu whereverwanzd. The codeismonitoredbyscommand inccrpR_rwhich possessesthenecessaryflexibilittoy tumdIesupersoniacndhypersoniccornpuu_ortg$roundcomplexshapc.s. REFERENCES 1, Jacquo_te, O.P. Generation, Optimization, and Adaptation of Multiblock Grids around Complex Configurltions, Proceedingsof AGARD 64_hFDP. Loen, 1989. 2. Ranoux. G.. Lordon. J.. Diet. J. Geom_uie et Maillaee de Confi;ur_tions Complexes pour les Calculs A_rodynamiques, Procecdin;s 0f AGARD 64th FDP, Locn, 1989. 3. Borrel, M., Montagnd,].L., Diet J., Guillen, Ph., Lordon, J. Upwind Scheme for Supersonic Flows around Tactical Missile, La Recherche A_rospauale. 1988-2. 4. Vuillot,A.M. A Multi.Domaln3D EulerSolvesforFlows inTurbomachines, Proceedings of the 9th ISABE Symposium, Athens, 1989. (to be published). 5. Roe,P.L. Some Contributiontso_ Model[inlolfDiscontinuousflows,Lectun_ inAppliedMathematicsv,ol.22,1985. 6. Van Leer.B. A. Second Order Sequel to Godunov Method, Journalof Computational Physics, vol. 23 pp.2"/6-299. 19T7, 7. Van Leer, B. lI Monotony and Conservation Combined in a Second Order Scheme.Journal ofComputationalPhysics,sol. 14, pp.361.370,1974. 8. Van Leer,B. FluxVecto¢Splitdnlof d_eEulu Equations.lcas¢Repor_82-30. 1982. 9. Roe. P.L. Approximate Ricmann Solvers. Pu_meter vectors, and Difference Schemes. Journal of Computational Physics. Vol. 43 pp. 357.372. 1982. I0. Osher,S.,Solomon F.,Upwind DifferenceSchemes forHypcrbollcSystemsof ConservationLaws, M_thematicsof Computation,Vol.38158 pp. 339-374, 1982. lI. Abgr,_llR.. G_n_ralisatiodnu Sch_rnade Roe pourleCalculd'Ecoulementsde M_lungcs dc Gaz JlConcentrationsVariables,La Recherche A_rospatiale. 1988-6. RSF N" 23/1123 AY Page 29 12. Abgrall. R.. Mo_c_gn_. J.L. G_n_raZisationdu Schema cl'Osncr pour ie Calcui d'Ecoulcmenc5 dc M_iam|es de Gaz k Concentrations Variabtcs et dc Gaz R6cls, To bepublish_l inLaP.,cche_h¢ Ai_ospadal©, 1989. 13. Montagn_. J.L,, Ye¢, H.C.. Vinokur. M, Comparative Study of High Resolution Shock Capturin| Scheraes for , Real Gas, Proceedings of ',h ?th GaminConl'cnmc_ onN_ M_'xx_ inFluidMechanics.1987. 14. Yc¢. H.C., Harten, A. Implicit TVD Schemes for Hyperbolic Conservation Laws inCurviUnearCooaJlnau_AIAA 85-153. I$. Chakravm,xhy. S.R.. Hi|h Re.solution Upwind Formulations for the Navier- Sa_JlccEsquaLio¢_VKI Lacuw¢19_._2. 16. Viviand. H.. Veuillot, |.P. IMd_ Pscu_lo Instationnaires pour le Calcul d'Ecoulcmcnts Transsonlques, Publication ONERA 19"/8.4 (English Tnmsl.at_onESA TI"561). 17. Lan4rum, EJ. Wind Tunnel Pressure Dam at MJch Number, from 1.63m 4.63 for • Series of Bodies of Revolution m Angles of Attack from _ m 60 Degrees, Nasa Tcchnic',dMcmocandum,LangleyP-,e,_ca_hCcnr_r,1977. 18. Dormicux. M., Mahd, C. C•Iculs Tridimensionnels de I'[nteraction d'un Jet Lateral avcc un E¢ou_ement Supersonique Externe, AGARD-CP n'43-/, L_.(1988). RSF N" 23/1123 AY Page 24 a)Mono4omai. l_id. b)Continuoustwo-4omain ip'ici. c)Discontinuous two-domain ip'id. Filp.tre1. Computation lp'lds ofLheogtve.cvltnclcr.fl:treconfi_tu"_tion.