An improved Barth-Jesperson limiter for Flux-Reconstruction method and its applications

Transcription

1 Tnth Intrnatonal Confrnc on Computatonal Flud Dynamcs (ICCFD10), Barclona, Span, July 9-13, 018 ICCFD10-xxxx An mprovd Barth-Jsprson lmtr for Flux-Rconstructon mthod and ts applcatons Zhqang H *, Zhongzhou Guo *, Wnwn Zhao * and Wfang Chn * Corrspondng author: wwzhao@zju.du.cn *Collg of Aronautcs and Astronautcs, Zhjang Unvrsty, Hangzhou 31007, Chna 1 Introducton Th scond-ordr CFD mthods hav bn playng an mportant rol n th dsgn of arcraft. Howvr, t s unabl to captur structur dtals whl solvng complx unstady flows du to th xcssv dsspaton ntroducd through th spatal dscrtzaton. At th sam tm, hgh-ordr mthods can obtan mor flow dtals at a lowr computatonal cost. DG (Dscontnuous Galrkn Mthod) s vry popular ths yars for ts ablty to gt arbtrary ordr accuracy wth unstructurd grds. But ts hug computatonal cost rstrcts ts scop of applcaton. Flux Rconstructon (FR) mthod was frst ntroducd by Huynh n 007 [1], whch s capabl to rcovr svral hgh-ordr schms for lnar fluxs, ncludng spctral dffrnc (SD) and nodal dscontnuous Galrkn (NDG). Not lk DG, FR mthod mploys dffrnc quatons rathr than ntgral quatons. It dscrtz th numrcal soluton through xpansons usng pcws hgh-ordr Lagrang polynomal bass functons, thn choos a corrcton polynomal, for xampl, Raudu polynomal, to rconstruct a globally contnuous flux from th pcws dscontnuous flux n ach cll. Study n papr [nvstgaton of 3d ntrnal flow usng nw flux-rconstructon hgh ordr mthod] show that FR s 8 tms fastr than DG for p1, p and p3. Rcntly, flux rconstructon mthod has bn succssfully appld to aroacoustcs [], larg ddy smulaton [3, 4] and so on, and thus gan wdr attnton. Howvr, thr ar stll som ssus whn FR s appld n transonc and suprsonc flows, whr shock-capturng s stll not fully satsfactory. Artfcal vscosty and lmtng ar two common stratgs to supprss spurous non-physcal oscllatons around shock. On mportant advantag of th artfcal vscosty mthod s that only th nformaton of currnt cll s ndd, so t kps th compact of th FR mthod. Howvr, such a mthod can potntally chang th ordr of th quatons, and thus rduc th tm stp. Th othr drawback of artfcal vscosty s that t s problm-dpndnt. It rqurs som xprncs to gt approprat vscosty. Othrws, xcssv vscosty wll affct th prcson of th soluton, or too small on cannot supprss spurous non-physcal oscllatons. Artfcal vscosty was usd to captur th shock succssfully at 1.6 Ma for NACA001 arfol n papr [5]. Lmtng s wdly usd n FVM wth much succss. Lmtng lmts th soluton to satsfy TVD or TVB proprts, and can totally supprss oscllatons around shocks. Snc FR mthod s vry smlar to nodal DG, t would b natural to apply lmtrs dsgnd for nodal DG to FR. MLP (hrarchcal mult-dmnsonal lmtng procss) dsgnd for DG was frstly ntroducd to FR n papr [6], whch s abl to captur th flow structurs n dtal. Howvr, t rqurs wdr stncls than tradtonal lmtrs for scond-ordr CFD mthods, and thus maks t mor dffcult to program. WENO schms wr also ntroducd to FR n papr [7]. It lmnats th oscllaton whl kpng hgh ordr accuracy. But t s qut dffcult to apply t to hybrd grd solvr. In ths papr, w ntroduc an mprovd Barth-Jsprson lmtr to FR mthod, and mploy a shock ndcator, to mantan hgh ordr accuracy n smooth rgons whl supprssng oscllatons around shocks. Th papr s organzd as follows. In th scond scton, th govrnng quatons and FR mthod 1

2 ar brfly rvwd. In th thrd scton, shock dtctor and an mprovd Barth-Jsprson lmtr ar dscrbd n dtal. In th fnal scton, w draw a conclud and summarz som futur works. Govrnng quatons and Flux Rconstructon mthod.1 Eulr quatons Th oscllatons around shock wll b surpassd wth lmtr functons. Howvr, th soluton n smooth rgons wll also b lmtd and pollutd. It s ssntal to turn off lmtr n smooth rgons f w want to mantan hgh ordr accuracy. A smoothnss ndcator from th artfcal vscosty mthod s mployd as a shock dtctor []. Th unstady Eulr quatons can b wrttn n consrvatv form as u( xt, ) Fu ( ) 0 (1) t whr consrvatv stat vctor and flux ar dfnd as u j u u,f uu j p j () u j ( p) n whch, p, dnot dnsty, total nrgy and prssur, u dnot vlocts. For a prfct gas, th prssur s gvn by 1 p ( 1)( u ju) (3) Th spcfc hat rato s assumd to b constant and quals to Flux Rconstructon mthod Consdr 1D hyprbolc law u x, t F u 0 (4) t x Frstly, dscrt th computatonal doman x x nto N non-ovrlappng lmnts, [ xn, xn1 ], n 0,1,, N 1. W transform ach lmnt to transformd spac s [ 1,1] for both mathmatcal smplcty and computatonal ffcncy. For 1D, mappng functon for ach lmnt s 1 1 x xn xn1 (5) Th govrnng quaton Eq(4) n transformd spac bcoms uˆ 1 Fˆ 0 (6) t J x u u(, t) F F, t, J. whr, ˆ, ˆ Thn, dfn k 1 soluton ponts n s, so th soluton uˆ and flux F ˆ can b approxmatd by followng K-th ordr Lagrang ntrpolat polynomals whr uˆ Fˆ D k 1 uˆ l 0 k 1 0 ˆ F l ( ) 0, N l s Lagrang ntrpolat bass dfnd at soluton ponts n (7) s. Th flux F ˆ s gnrally

3 dscontnuous across cll ntrfacs. To satsfy th consrvaton law, corrcton functons g L and g R wr ntroducd to kp th flux contnuous at cll ntrfacs. VCHJ corrcton functons ar th most frquntly usd corrcton functons [8]. Thy ar k 1 k Lk 1 Lk 1 Lk 1k 1 L L gr Lk 1k whr L k s th Lgndr polynomal of dgr p and g L k k 1 k 1 1! c k akk k (9) (8) Fgur 1. Fgur shows th flux corrcton procdur n computatonal doman. Th paramtr c s calld th VCJH paramtr and w can rcovr svral xstng schms wth spcfc c, such as nodal DG, SD. Th total contnuous flux s wrttn as whr ˆ C F k 0 ˆ ˆ C ˆ D ˆ ( ˆ I ) ˆ D ˆ I ˆ D F F F F l F L F L gl F R F R gr (10) s corrcton componnt of th contnuous flux, ˆ C ˆ I ˆ D ˆ I ˆ D F F F g F F g (11) L L L R R R ˆ I F L and ˆ I F R ar fluxs at lft and rght ntrfacs computd wth Rmann solvr, ˆ D F L dscontnuous fluxs at cll boundary. Th drvatv of th contnuous flux s ˆ k F ˆ dl ˆ I ˆ D dgl ˆ I ˆ D dgr F FL FL FR FR ( ) d d d 0 and ˆ D F R (1) Fnally, Eq(1) s usd n Eq(6) to obtan an ODE whch s thn tm-advancd usng a hghordr tm ntgraton schm, lk RK45 n ths papr. 3 Shock dtctor and an mprovd Barth-Jsprson lmtr 3.1 Shock dtctor ar 3

4 Th oscllatons around shock wll b surpassd wth lmtr functons. Howvr, th soluton n smooth rgons wll also b lmtd. It s ssntal to turn off lmtr n th smooth rgons f w want to mantan hgh ordr accuracy. A smoothnss ndcator from th artfcal vscosty mthod s mployd as shock dtctor [9]. Frstly, comput th L projcton to th lowr-ordr spac of th k-th ordr soluton n cll. Nxt, dfn smoothnss ndcator Aftr that, w ntroduc, S as S log k uˆ u L (13) 0 10 u uˆ d 1 S S0 1 S 1 S0 sn S0 S S0 (15) 0 S S0 whr 4, S0 C0log ( k), C0 3. Fnally, f 0.99, th cll s dntfd as a troubl cll, and ts solutons ar gong to b lmtd. 3. An mprovd Barth-Jsprson lmtr for FR As mntond n scton., th soluton n cll s a polynomal n th form Eq(7). Th Gbb s phnomnon would caus oscllatons around shock and unphyscal solutons. An mprovd Barth- Jsprson lmtr s ntroducd to FR mthod to avod ths at th avrag stp. All stps rqurd ar lstd blow as 1) Fnd all troubl clls wth th hlp of shock dtctor mntond n last scton. ) Evaluat th avrag solutons of troubl clls and thr nghbors sharng a fac, u and u nb. 3) Fnd th maxmum avrag valu u max and mnmum avrag valu u mn among troubl cll and ts nghbors. 4) Evaluat avrag gradnt, u of troubl cll. 5) For cll, dfn a lmtr functon umax u mn 1, Δ 0 Δ mncr 1 Δ 0 umn u mn 1, Δ 0 Δ whr Δ u Δr, cr, Δr, cr s th vctor from cll cntr to cll cornr. 6) Lmt th soluton at soluton ponts u d, j, j (14) (16) u u u r (17) 7) Th lmtd soluton s thn usd n FR procdur. W us cll cornr to avod valuatng soluton at vry flux pont. Stp 6 shows that soluton at troubl cll s lnarly dstrbutd so all solutons at soluton ponts and flux ponts hav th TVD proprty. 4 Numrcal rsults 4.1 1D sod shock tub problm 4

5 Th sod shock tub problm s a common on-dmnsonal Rmann problm to tst for th accuracy of computatonal flud cods. Th tst can b dscrbd wth Eulr quaton wth followng paramtrs, for lft and rght stats of an dal gas 1.0, 0.0,1.0 0 x 0.5, up, 0.15,0, x 1.0. (18). Th smulaton s stoppd at t=0. and wth 100, 00 and 400 lmnts. Th dnsty dstrbuton computd wth thrd-ordr FR schm wth shock dtctor and th mprovd Barth-Jsprson lmtr s shown n Fgur. Fgur : Dnsty dstrbuton wth dffrnt lmnts at t=0.s 4.1 Suprsonc flow n a convrgnt nozzl wth a ramp Th cas s to tst lmtr for stady problm. Its computatonal doman and boundary condtons ar shown n Fgur 3. Th ntal condtons ar as follows: 1.4, u.0, v 0, p 1.0 (19)..., Ma.0 Th vlocty of suprsonc nflow s also.0 Mach. Both structurd and unstructurd grds ar mployd. Th dnsty dstrbutons computd wth fourth ordr FR schm wth shock dtctor and th mprovd Barth-Jsprson lmtr ar shown n Fgur 4. Th shock capturd hr s vry sharp and almost no ovrshot. It shows that th mprovd lmtr hr nhrts advantags of Barth- Jsprson lmtr n scond-ordr FVM, and s abl to functon for both structurd and unstructurd grds. Fgur 3: Computatonal doman and boundary condtons of convrgnt nozzl problm. 5

6 Fgur 4: Dnsty dstrbuton of convrgnt nozzl problm wth structurd and unstructurd grds. 4.3 Mach 3 wnd tunnl wth a forward stp In ths scton, a standard tst cas s smulatd to valuat FR schms wth th mprovd lmtr. Th problm contans a suprsonc Mach 3 unform nflow n a wnd tunnl wth a forward stp, multpl ntractons of shocks, xpanson fans and contact dscontnuts. Th computatonal doman and boundary condtons ar shown n Fgur 1. And a st of h=1/160 quadrlatral grds s mployd. Th ntal condtons ar lstd as follows: 1.4, u 3.0, v 0, p 1.0, Ma 3.0 (0). Fgur and Fgur 3 show th shock snsor and dnsty dstrbutons at t=4.0s computd wth th thrd and th fourth ordr FR schm wth th mprovd lmtr. Both shock snsors captur flow structurs accuratly and th hghr ordr schm ylds mor dtald faturs. Th tst cas also valdats th rsolvng capablty of th mprovd Barth-Jsprson lmtr for th Klvn-Hlmholtz nstablty from th shock trpl pont, whch ndcats that FR schm wth th nw lmtr s of low dsspaton. Fgur 5: Computatonal doman and boundary condtons of th forward stp Fgur 6: Shock snsor and dnsty dstrbuton for th 3 rd ordr FR schm at t=4s. 5 Concluson Fgur 7: Shock snsor and dnsty dstrbuton for 4 th ordr FR schm at t=4s. An mprovd Barth-Jsprson lmtr for FR schm s dvlopd n ths papr. It kps stncls 6

7 of th orgnal Barth-Jsprson lmtr for th scond-ordr FVM schm, and thus s asy to ralz basd on FVM cods. Som tst cass ar carrd out up to xamn th capablty of th mprovd lmtr n capturng suprsonc flow physcs. Rsults dmonstrat th proposd lmtr provds dtald flow structurs wthout oscllatons around shock rgon, and mantans th rqurd accuracy n smooth rgon at th sam tm. Rfrncs [1] Huynh H T. A flux rconstructon approach to hgh-ordr schms ncludng dscontnuous Galrkn mthods. AIAA papr, : 007. [] Vsbal M R, Gatond D V. Vry hgh-ordr spatally mplct schms for computatonal acoustcs on curvlnar mshs. Journal of Computatonal Acoustcs, 001, 9(04): [3] Lu Y, Lu K, Daws W N. Larg ddy smulatons usng hgh ordr flux rconstructon mthod on hybrd unstructurd mshs//aiaa Scnc and Tchnology Forum and Exposton (ScTch014), AIAA [4] Park J S, Wthrdn F D, Vncnt P E. Hgh-Ordr Implct Larg-Eddy Smulatons of Flow ovr a NACA001 Arofol. AIAA Journal, 017. [5] Lopz-Morals M, Bull J, Crabll J, t al. Vrfcaton and Valdaton of HFLES: a Hgh-Ordr LES unstructurd solvr on mult-gpu platforms. AIAA Papr, 014, [6] Park J S, Chang T K, Km C. Hghr-ordr mult-dmnsonal lmtng stratgy for corrcton procdur va rconstructon[c]//5nd Arospac Scnc Mtng. 014: [7] Du J, Shu C W, Zhang M. A smpl wghtd ssntally non-oscllatory lmtr for th corrcton procdur va rconstructon (CPR) framwork. Appld Numrcal Mathmatcs, 015, 95: [8] Vncnt P E, Castonguay P, Jamson A. A nw class of hgh-ordr nrgy stabl flux rconstructon schms. Journal of Scntfc Computng, 011, 47(1): [9] Prsson P O, Prar J. Sub-cll shock capturng for dscontnuous Galrkn mthods. AIAA papr, :