1 Introduction. * Rocket Systems Division/ IMI, Israel ** School of Mathematics/Tel-Aviv University.

Transcription

1 Seeth Iteratoal Coferece o Cotatoal Fld Dacs (ICCFD7), Bg Islad, Haa, Jl 9-, ICCFD7-94 Fast Iterate Methods for Naer-Stoes Eqatos th SS rblece Model ad Chestr O. eles *, E. rel ** ad S. Ya * Corresodg athor: eltrel@gal.co * Rocet Sstes Dso/ IMI, Israel ** School of Matheatcs/el-A erst. Abstract: he stead state coressble Naer-Stoes eqatos are soled th a elct Rge-Ktta schee accelerated b ltgrd ad a lct recodtoer. Etesos are ade to clde a -/SS trblece odel ad checal reactos. he lct soother eables the se of hgh CFL bers eldg fast coergece. We reset alcatos to trblet soltos for flos abot gs ad les ad for flos th seeral checal reactos. Keords: Naer-Stoes, coergece accelerato, SS trblece, chestr. Itrodcto he coressble stead state Naer-Stoes sste of eqatos s a olear ed herbolcelltc-arabolc sste. Classcal ethods for solg these eqatos se secod order accrate ethods sace. he stadard techqe s to arch the te deedet eqatos (sedo) te tl a stead state s acheed. he te archg s doe b ether a elct ethod (La- Wedroff, Rge-Ktta etc.), ADI or else b a fll Neto ethod coled th a Krlo techqe for the resltat large sarse lear eqatos. hs s freqetl sleeted th a ltgrd (MG) accelerato. I ste of all these techqes, the coergece to the stead state reqres a teratos freqetl ragg fro seeral hdred MG ccles to thosads of elct te stes. hs slo coergece becoes ee orse for hgh Reold s ber flos he a hghl stretched esh s sed the bodar laer reqrg sall te stes. Local te steg ad lct ethods alleate bt do ot elate ths slodo. Rosso ad later Saso ad rel [,,] trodced a lct recodtoer hch allos for rad coergece to a stead state, th CFL, that s tcall abot 5 tes faster, C te, tha reos otal solers. Freqetl, coergece to egeerg accrac s obtaed tes of MG ccles ad coergece to ache accrac abot MG ccles sg a stage elct Rge-Ktta schee. he orgal reslts ere obtaed for the stead state coressble Naer-Stoes eqatos coled th a algebrac (Bald-Loa) trblece ode. I ths aer e eted these reos reslts seeral drectos. We cosder etesos fro a algebrac trblece odel to a lt-eqato trblece odel. As a tcal eale e hae chose the - SS eqatos. I order to adace the trblece eqatos th a large CFL e trodce a lct recodtoer slar to that sed for the fld eqatos. hs recodtoer aga allos ch larger te stes tha sall sed for these eqatos allog a rad coergece to a stead state. I addto, the recodtoer creases the robstess of the coergece.

2 I aother drecto of etesos e cosder the addto of flds th reactats. hs, addto to the fld eqatos there are a checal reactos. hese reactos add a stff sorce ter that seerel redces the alloable te ste for the fld. call, dstral calclatos se CFL.5 for these calclatos. he trodcto of the lct soother allos a ch larger te ste ad aga a faster ad ore robst coergece to a stead state. he -/SS Eqatos I reos or th the accelerato techqe ol algebrac trblece odes (Bald-Loa) ere cosdered. hs, the lct recodtoer as aled ol to the fld eqatos. We o cosder the eteso of ths techqe to lt-eqato trblece odels (see [4]). For the trblece odel eqatos, freqetl, the ADI ethod s sed to adace the eqatos sedote to a stead state at each flo te ste. hs solto algorth decoles the fld ad trblece eqatos. he fld eqatos are soled for a ge trblet coeffcet µ t ad the trblece eqatos are soled searatel to date µ t. Hoeer, ADI hle forall codtoall stable, ractce, allos oe to crease the CFL ol b a factor of abot 5- oer the elct te restrcto. hs les that the te ste sed for the trblece odel s ch saller tha that sed for the fld eqatos, hch ca be tes larger tha the elct CFL becase of the lct recodtoer. Here, e trodce the RK lct soother to the -/SS odel hch eables s to effcetl sole the eal coled sste of eqatos of the Naer-Stoes eqatos th the to eqato trblece odel. We frst descrbe soe detals of the trblece odel ad ts ercal aroato. he trblet scost s defed as µ a t ν t a a, SF here S s the ortct ad - s the dstace fro the all. F ad F are defed as: F F 4 5ν 4σ.9 CD tah a,, 5ν tah a, * β he the SS trblece eqatos are ge b: r * β ( ν σ ν ) t j j r α S β ( ν σ ν ) ( F ) σ t j j j j

3 * th τ, β j j here s the stress tesor. I a alcatos τ j s relaced b j µ S. CD σ φ φ F φ ( F ) a, 5 α, α.44 9 * 9 β, β.88, β 4 σ.85, σ σ.5, σ.856 Nercal Schee for SS he trblece odel eqatos are soled sg a sedo te tegrato th a lo storage Rge- Ktta schee: For each R-K stage e hae: ( q) () ( q) αq ( q) () ( q) αq q..ber of rge tta stages, ad t ( F F ) ( q) ( q) ( q) c S tr Vol all faces are soothed ales, here: ad slarl for. sg the terolog of the RK/lct soother, the eqatos are: Fc, F, R S S t all faces F c, all faces F, R he ercal schee for the soother s: I ε t Vol ( q) ( q) ( A A ) S R (, ) all faces q ε t Vol ( q) ( A A ) S NB all faces local

4 here,...,, s the ber of soothg stes, ad ε s a lct araeter. I addto, A I, A - - I, Aν s the scos fl Jacoba: q A ν S ν σ ν A Vol ν σ ν υ NB A υ R s the Jacoba of the sorce ter. Ol the decag ters are sed the sorce fl Jacoba: R R * β (, ) R R * β β R hs sste s soled th a Gass-Sedel ethod sg a sall ber of teratos. 4 Reslts for SS rblece Model We cosder the cobed fld ls trblece set of eqatos. he to sets are eal coled, ad the lct soother s aled, searatel, to both the fld ad trblece eqatos. At reset, ltgrd s sed ol for the fld eqatos. We descrbe the SS- trblet trasoc calclatos for the flofeld arod a RAE8 g ad the trblet flofeld of a rocet otor. 4. rblet trasoc flofeld arod RAE8 g he free strea codtos are: M.7, α.79. Fgre shos the coergece hstor (destred, -gree) of the calclato of the flo arod a RAE8 g. he calclato ses seqecg of to coarse grds ad three leels of ltgrd o the fest grd, the fld ol. A secod order d schee th a Seb lter s sed. he fld CFL s, ad trblet CFL s,; ache accrac s obtaed. Fgre shos the coergece ad fgre shos the coarso betee FLDYNS (a -hose code) solto ad the eeret.. Fgre : Dest (red) ad trblet etc eerg (gree) coergece hstor

5 Fgre : Coarso betee FLDYNS solto ad eeret 5 Chestr rblet reacte flo s ortat for aros egeerg robles. For eale, desgg of cobsto chabers ad odelg of rocet otor ad rocet otor les. I rocet otor les, for eale, trblet flo s essetall dfferet fro laar flo ad real les ca t be odeled thot cosderg trblece. he checal reactos also add secfc heoea as after-brg he the cobsto rodct eets the atosherc oge starts to br aga. Relable odelg of reacte flo les s er ortat, for eale, to ealate the IR sgatre of the le [7]. We aga eted the dea of the RK/lct soother to reacte flo. he challege of ercal slato of reacte flo s dealg th the stffess of sste, both the trblece odel (K- SS ths or) ad esecall the stffess fro the checal reactos. hs stffess s the reslt of a er de rage of the eolto te scales of the reactats. 5. Goerg eqatos he checal reacto sorce ters of the hoogeeos Naer-Stoes eqatos are: (,,,...,, ). he gas tre dffso ters are added to the seces ass coserato eqatos. he coserate arables are: r,, ( E ) Let r (,, ) the N N E e ( ) e ( ) e ( ) he ea teral eerg ass ts s e( ) e ( ). N

6 5.. Sorce ters We se the rte arables q (,, ) arables (,, ) eght d W e. We also defe aother set of rte R. he eqato of state s R th the ea oleclar W. he Jacoba atr fro rte to coserate arables s ( ) e ( ) e ( ) he checal sorce ter s (,,, ), odel s descrbed b the reacto eqato: υ ' χ <> υ' ' χ c dq Jdq. Each reacto of Nr reactos the checal here υ ' ad υ' ' are the forard ad bacard stochoetrc coeffcets resectel. Nr υq υ υ' ' υ ' he rogress arable q s ge b q υ ' ( ) K r, ( ) K f, υ '' f E / R A β e / r, f, c,, A, ad E β are the Arrhes costats: eoet ad E s the actato eerg, A s the rate costat, β s the teeratre K ( S H ) ( ) e ad r, R R K at R K υ S υ S H υ H e ( ) K K c

7 So, coserate arables the sorce ters are: c e e e E dt d 5.. Dffso ters A addtoal dffso ter µ Sc aears the RHS of the ass coserato eqatos hch s the reslt of the gas tre. O the RHS sde of the eerg coserato eqato aears Sc µ here µ s the scost ad Sc s the Schdt ber for trblet flo. Sc µ s relaced b t t l l Sc Sc µ µ here the sbscrts l ad t refer to the laar ad trblet scostes ad Schdt bers resectel. 5. RK/Ilct Soother for Real gas For each ot, e sole the sste faces all NB j j faces all ds Vol t ds Vol t I,, ~ ~ δ δ δ δ δ here δ are the resdals before soothg, δ % are the resdals the -th ste of the soothg, ad the sbscrt NB deotes eghbor cells. hs eqato s lct ad e sole t teratel sg the Gass-Sedel ethod. ± ± here W A W c ad Λ ad s the atr of egeectors, Λ s a dagoal atr cotag the egeales. s the Jacoba atr fro the rtes δ to the coserates W δ. s the Jacoba of the erse trasforato. If F A r ad A F F A c r r, A, / / / / / /

8 Π NN N I e e e A.. gas real for e R Π r NN I / / / NN I c c c c c c c c c c c / / / /, sg / c r r 5.4 RK/Ilct Soother for reacte flo For reacte flo, e eed to add the sorce ter Jacoba to the RHS of the soother NB faces all q NB c q local q faces all q local c S A A Vol t R S A A Vol t I ) ( ) ( ) ( ε ε I rte arables e se ol art of the Jacoba etres:

9 { },, J R he leetato FLDYNS reqres the Jacoba { },, arables ge b { } R R R R W W W W R W W W W R W W W W R J,, here R 5.5 Deterato of the teeratre fro the teral eerg For a ge teral eerg e, e at to detere the teeratre (ad/or the ressre). Sce for o-deal gases the teral eerg s a o-lear fcto of the teeratre, e hae to sole t ercall sg the Neto-Rahso ethod. We sole the eqato e e f teratel: f f f f ' ' sce c d de e get c e e c 5.6 Reslts D Stead State Reacte Flo

10 Or frst leetato s a test case for a -d stead state, reacte flo. he test roble has bee tae fro NASA S NARC aldato ebste. he bodar codtos ad sle (D) geoetr are sho fgre. hs eale cotas Hdroge-Oge-Water reactos ( radcals). Fgre : NARC aldato test geoetr ad BC. he aal dstrbto of teeratre ad, ater ass fracto are sho fgres 4-5. I these fgres e also coare the reslts to those obtaed fro the CFD code. he te dela betee the soltos of the test code ad fro CFD ca be elaed b a ercal gto dela. he coergece hstor for aros CFL bers ad lct factors ε s sho fgre 6. I the NARC ebste ths eale as coted th CFL.5. Wth the recodtoer the astotc CFL sed as 6. he coergece roes as e decrease the lct factor ε. For ε belo.4 the cotato becae stable. Fgre 4: eeratre dstrbto. CFD gree, est codes red.

11 Fgre 5: ater (HO) dstrbto. CFD gree, est codes red. Fgre 6: Coergece hstor. CFL red, CFL gree, CFL 4 ble, CFL 8 ageta, CFL 6-lght ble (all th ε). CFL 6; ε.8 bro, CFL 6; ε.6 ello, CFL 6; ε.4 orage Rad easo dffser

12 I ths roble, a hgh Mach stea s jected to a ole-le dece. he detaled geoetr ad bodar descrtos are sho fgre 7. hs eale has bee tae fro [8]. he gas tre cotas Hdroge-Oge-Water reactos ( radcals). Fgre 8 resets the dest coergece hstor. he checal reactos create a shoc close to the etrace of the ole. hs shoc does ot occr the o-reacte case. Fgre 9 resets the cotor a of the teeratre, Mach ber ad soe of the seces ass fracto. Good agreeet betee or reslts ad those reseted [8] are obtaed. Fgre 7: roble defto Fgre 8: coergece hstor for CFL thot ltgrd

13 5.6. Blt rojectle Fgre 9: eeratre, Mach, seces ass fracto cotors We et cosder a blt rojectle s flg a stochoetrc tre of hdroge ad oge. he coergece hstor s reseted fgre. he rse of the teeratre behd the bo shoc cases the gto. he dest cotors a s sho fgre. he aal dstrbto of the seces ass fracto ad the teeratre are sho fgre ad ad coared to the reslts fro [9].

14 Fgre : Coergece hstor thot ltgrd Fgre : dest cotors a

15 A B Fgre : Aal dstrbto of teeratre. A FLDYNS. B Sheffer 998

16 A B Fgre : Aal dstrbto of seces ass fracto. A FLDYNS. B Sheffer Rocet otor le We o calclate the rocet otor le etg fro the otor ole to a lo Mach ber free strea flo. he le bodar codtos are defed o the ole throat here the flo eloct

17 s soc ad the seces ass fractos are ge. he seces sed for ths roble are: H, O, OH, H, O, CO, CO, HO, HCL ad N. he reactos are descrbed [7]. Fgre 4 shos the coergece hstor of the dest (red) ad the trblet etc eerg (gree). Fgres 5-8 sho the Mach cotors, teeratre cotors, cotors ad seces ass fracto, resectel. he calclato ses seqecg of to leels of coarse grds ad three leels of ltgrd o the fest leel. A secod order d schee th a Seb lter s sed. he fld CFL s, ad trblet CFL s. Fgre 4: Coergece hstor of the dest (red) ad the trblet etc eerg (gree) Fgre 5: Mach cotors of the le

18 Fgre 6: eeratre cotors of the le Fgre 7: rblet etc eerg cotors of the le

19 Fgre 8: CO ad OH ass fracto cotors of the le 6 Coclsos he RK/Ilct soother has bee eteded to sole the trblece -/SS odel eqatos. We soled for flo arod a trasoc RAE8 g ad a sersoc jet le sg the cobed fld ad SS-trblece eqatos. We obtaed ecellet coergece rates ad a accrate solto. For reacte flo the sorce ters trodce stffess to the Naer-Stoes eqatos. We trodced the sorce ter Jacoba to the RK/Ilct soother to sole the roble. he algorth ad colcated cases sch as reacte rad dffser ad reacte blt rojectle are reseted for scos reacte flo ad the trblet, reacte rocet otor le th the -/SS trblece odel s reseted ad er good reslts ere obtaed. Refereces []. C-C. Rosso, "Coergece Accelerato for Solg the Coressble Naer-Stoes Eqatos", AIAA J. 44: 45--5, 6. []. R. C. Saso, E. rel, C. -C. Rosso ad V.N. Vatsa, "Coergece Accelerato for Mltstage e-steg Schees", AIAA []. R. C. Saso, E. rel ad C. -C. Rosso, "Coergece Accelerato of Rge-Ktta Schees for Solg the Naer-Stoes Eqatos", J. Co. hscs 4:65--88, 7. [4]. O. eles, S Ya ad E. rel, Coergece Accelerato of Rge-Ktta Schees sg RK/Ilct Soother for Naer-Stoes Eqatos th SS rblece,roceedgs of 5d Israel Aal Coferece o Aerosace Sceces,. [5]. A. Jaeso, W. Schdt ad E. rel, "Nercal Soltos of the Eler Eqatos b Fte Vole Methods sg Rge-Ktta e-steg Schees", AIAA aer 8l-l59, 98.

20 [6]. C. Saso ad E. rel, "O Cetral Dfferece ad d Schees", J. Co. hscs 9--6, 99. [7]. G. Atal, Y. Cohe, L. Gass, Y. Kaelba, J. Macales, B. rea, S. Ya, M. Le, J. Strcer, A. Sterleb, "Eeretal ad Cotatoal Std of Ifrad Esso fro dereaded Rocet Ehast les", Joral of erohscs ad Heat rasfer, Vol. 5, No. 4, October-Deceber. [8].. J. Chg, Cotatoal Fld Dacs, Ca- brdge erst ress, Cabrdge, [9]. S. G. Sheffer, L. Martell, A. Jaeso, A Effcet Mltgrd Algorth for Coressble Reacte Flos, Joral of Cotatoal hscs 44, (998)