Conceptual Linearization of Euler Governing Equations to Solve High Speed Compressible Flow Using a Pressure-Based Method

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1 Concptual Linarization of Eulr Govrning Equations to Solv High Spd Comprssibl Flow Using a Prssur-Basd Mthod Masoud Darbandi, 1 Ehsan Roohi, 1 Vahab Mokarizadh 2 1 Dpartmnt of Arospac Enginring, Sharif Univrsity of Tchnology, Thran, P.O. Box , Iran 2 Enrgy & Environmntal Rsarch Cntr, Niroo Rsarch Institut, Thran, P.O. Box , Iran Rcivd 26 Fbruary 2006; rvision 30 Novmbr 2006; accptd 4 May 2007 Publishd onlin in Wily IntrScinc ( DOI /num Th main objctiv of th currnt work is to introduc a nw concptual linarization stratgy to improv th prformanc of a primitiv shock-capturing prssur-basd finit-volum mthod. To avoid a spurious oscillatory solution in th chosn collocatd grids, both th primitiv and xtndd mthods utiliz two convcting and convctd momntum xprssions at ach cll fac. Th xprssions ar obtaind via a physical-basd discrtization of two inclusiv statmnts, which ar constructd via a novl incorporation of th continuity and momntum govrning quations. Ths two xprssions in turn provid a strong coupling among th Eulr consrvativ statmnts. Contrary to th primitiv work, th linarization in th currnt work rspcts th dfinitions and ssnc of physics bhind driving th Eulr govrning quations. Th accuracy and fficincy of th nw formulation ar thn invstigatd by solving th shock tub as a problm with moving normal and xpansion wavs and th convrging-divrging nozzl as a problm with strong stationary normal shock. Th rsults show that thr is good improvmnt in prformanc of th primitiv prssur-basd shock-capturing mthod whil its suprior accuracy is not dtrioratd at all Wily Priodicals, Inc. Numr Mthods Partial Diffrntial Eq 00: , 2007 Kywords: collocatd grid; comprssibl flow; finit volum mthod; Nwton Raphson linarization schm; prssur-basd approach; shock-capturing tchniqu I. INTRODUCTION Th ky rol of dnsity in high spd flow rgims has rsultd in choosing it as a major dpndnt variabl in dvloping numrical algorithms to solv ithr Eulr or Navir Stoks govrning quations. As a prliminary stp, th Godunov mthod was stablishd basd on th important rol of dnsity changs in high spd flow rgims [1]. Sinc th innovation, this primitiv mthod has Corrspondnc to: M. Darbandi, Dpartmnt of Arospac Enginring, Sharif Univrsity of Tchnology, Thran, P.O. Box , Iran (-mail: darbandi@sharif.du) 2007 Wily Priodicals, Inc.

2 2 DARBANDI, ROOHI, AND MOKARIZADEH bn widly improvd from diffrnt prspctivs including accuracy, consrvativity, stability, and consistncy,.g., s Rfs. [2 5]. For xampl, to captur shock mor prcisly, th primitiv mthod has bn incorporatd with high rsolution schms. Morovr, to rduc th complxity in th high rsolution formulations, thr hav bn attmpts to compact th computational stncil whil maintaining th sam ordr of accuracy [6]. Dspit such progrsss in dnsity-basd mthods, thr is major difficulty to solv low Mach numbr flows fficintly [7 9]. Th difficulty can b ovrcom using low Mach numbr prconditioning, which consquntly incrass th computational cost. Thrfor, thr has bn srious dmand to xtnd nw mthods capabl of solving not only high spd comprssibl flow but also low Mach numbr flow with similar fficincis. In this rgard, th prssur-basd mthods hav bn practicd to lssn th difficultis ncountrd in solving low Mach flows. As it is known, prssur prforms no dficincy in solving low Mach numbr flow [10 14]. To bnfit from this advantag, Rossow [15] dvlopd an altrnativ mthod to low spd prconditioning for th computations of narly incomprssibl flows using a blndd prssur/dnsity basd mthod. Th volution of th prssur-basd mthods is dscribd shortly. From th consrvation prspctiv, th advantags of finit volum mthod hav promotd th computational fluid dynamics workrs to mploy it widly. Thr ar two basic choics to writ th finit volum formulation if a prssur-basd algorithm is chosn. On choic is th typ of primary dpndnt variabls utilizd in th computational algorithm and th othr on is th rlativ locations of th dpndnt variabls on th computational grids. Th primitiv finit-volum work of Patankar and Spalding [16], which is known as SIMPLE, considrs th continuity quation as a constraint quation for prssur. Howvr, SIMPLE and its variants suffr non-physical oscillatory prssur and vlocity filds. Staggrd grid arrangmnt has bn mployd as a gnral rmdy to ovrcom th drawbacks. This arrangmnt stors th dpndnt variabls at two diffrnt locations, which ar displacd with rspct to ach othr [12, 17]. Boundary condition implmntation difficulty and xcssiv book-kping ar two major objctions to staggrd grid approachs. In addition, th vlocitis that satisfy mass do not ncssarily consrv momntum in th sam control volum. Morovr, mor smaring is pronouncd around discontinuitis [13]. Ths drawbacks bcom mor crucial in th curvilinar coordinat systms to solv ithr laminar [18, 19] or turbulnt [20, 21] flow rgims. Thrfor, th gomtrically simplicity of th collocatd grid arrangmnt is vry attractiv and it will b significant if th caus of wavy non-physical prssur fild is rmovd [22]. Thr ar diffrnt approachs to supprss th chckrboard problm on a collocatd grid. Rhi and Chow [23] usd additional momntum-basd intrpolation to trat th cll fac vlocitis in th continuity quation. Millr and Schmidt [24] showd that th ida of Rf. [23] would prdict spurious cll vlocitis whn local variation of prssur dpartd considrably from linarity. Askoy and Chn [25] similarly mployd momntum wightd intrpolation approach to supprss th chckrboard problm in a finit-analytic schm. Rahman t al. [26] modifid th approach of Rf. [23] by incorporating a non-prssur gradint sourc trm in th fac approximations. Dat [27] drivd a nw prssur corrction quation, which in turn rquird a furthr corrction calld smoothy prssur corrction. Darbandi and Schnidr [28] drivd both cll fac vlocitis via incorporating th mass and momntum govrning quations. Lin [11] adoptd th collocatd storag arrangmnt for all variabls and liminatd th chckrboard oscillations by using a prssur-wightd intrpolation mthod, similar to that of Rhi and Chow [23]. As is sn, th abov brif litratur rviw indicats that th dual rols of vlocity has bn long practicd in collocatd schms. Ths two vlocitis can b classifid as convctd and convcting vlocitis. Thy ar appropriatly substitutd in th linarizd form of th govrning quations. Th substitution ar normally prformd only for th activ vlocity componnts in th formulations.

3 CONCEPTUAL LINEARIZATION OF EULER EQUATIONS 3 Howvr, th linarization procdur of th govrning quations may rsult in svral laggd vlocitis, which nd to b tratd vry cautiously. As it is known, th SIMPLE algorithm was originally dvlopd for solving incomprssibl flow. Howvr, thr hav bn major attmpts to xtnd it for comprssibl flow tratmnt as wll. Darbandi t al. [12, 29 31] hav shown that th SIMPLE algorithm can b radily xtndd to solv comprssibl thrmobuoyant filds prforming high dnsity variation with and without mploying th Boussinsq assumption. Thy hav also shown that th xtndd staggrd-basd grid is capabl of solving subsonic Eulr flow rgims [14]. In th rout of xtnding incomprssibl flow algorithms to trat high comprssibl flow rgims, Van Doormal t al. [32] also xtndd SIMPLE and its variants to th solution of comprssibl flows in th staggrd grid contxt. Thy usd a Nwton Raphson linarization stratgy [33] to linariz th mass flux in th continuity quation. Thir linarization schm considrs activ rol for both dnsity and vlocity in th drivd formulations. Karki and Patankar [10] providd a prssur-basd algorithm using SIMPLE procdur. Thy wr not abl to captur shock prcisly in suprsonic flow. On of th famous variants of SIMPLE is PISO algorithm, which includs on prdictor and two corrctor stps [34]. This algorithm and its variants hav bn succssfully utilizd for trating high spd comprssibl flow rgims [35, 36]. As was mntiond, Lin [11] dvlopd a prssurbasd algorithm on collocatd grid incorporating th basic ida of Rf. [23] in his algorithm. van dr Hul t al. [13] xtndd an improvd markr-and-cll schm to trat flow in both comprssibl and incomprssibl flows. Darbandi and Schnidr [28] also dvlopd a fully implicit prssur-basd algorithm and solvd flow at all spds on a collocatd grid arrangmnt. Contrary to Rf. [23], thy considrd altrnativ rols for vlocity to supprss th spurious solutions. Additionally, thy mployd th Nwton Raphson linarization stratgy to linariz th nonlinar convction trms in th momntum quations. Sinc thr ar dual rols for th two vlocity componnts in th momntum convction trms, th linarization stratgy would b constructd in a mannr, which maintains th individual rol and concptual maning of ach vlocity in th formulation. This is a ky point which has not bn takn into account in th prcding collocatd algorithms. Th major contribution of th currnt work is to corrct th linarization procdur takn in a primitiv prssur-basd shock-capturing mthod [37 39] via mploying a novl concptual linarization, which fully rspcts th ssnc of physics bhind stablishing th flow govrning quations. This by itslf is a contribution and th nw schm dos not ncssarily nd xhibiting a bttr prformanc than th primitiv mthod. Howvr, our invstigation shows that th fficincy of th xtndd mthod is considrably highr than th primitiv on whil its accuracy is th sam as th primitiv on. This outcom can b countd as th scond contribution of th currnt work. To prsnt th achivd outcoms, th xtndd formulation is valuatd against th primitiv on by solving th shock tub and convrging-divrging nozzl problms. Th accuracy and fficincy of th nw algorithm ar thn compard with thos of th primitiv work. II. GOVERNING EQUATIONS In shock-capturing tchniqus, it is vry customary to prsnt th prformanc of th nwly dvlopd mthods by trating th stady quasi-1d and unstady 1D problms. Similar to many past invstigators, who hav chosn ithr stady flow in convrging-divrging nozzl problm [40 42] or unstady flow in shock tub problm [5, 13, 43], w choos th unstady quasi-1d govrning quations to quantify th prformanc of our proposd linarization. W hav targtd

4 4 DARBANDI, ROOHI, AND MOKARIZADEH both th stady quasi-1d flow and th unstady Rimann problms in our study. Th unstady quasi-on-dimnsional form of th Eulr govrning quations is writtn as q τ + B(q) = s (2.1) x whr th solution vctor q, th convction flux vctor B(q), and th sourc trm vctor s ar rspctivly givn by q = (ρa, ρua, ρa) T (2.2) B = (ρua, ρu 2 A, ρua + pua) T (2.3) s = (0, A( p/ x),0) T (2.4) In th arlir quations, τ, ρ, p, u, and A rprsnt tim, dnsity, prssur, vlocity, and th cross-sction ara, rspctivly. If w nglct th chang in potntial nrgy, th total nrgy pr unit mass for a prfct gas is givn by = c v t +u 2 /2, whr t is tmpratur and c v is th spcific hat valu at constant volum. Considring this xprssion, th quation of stat p = ρrt rlats th vlocity and prssur filds to th tmpratur fild in comprssibl flows. Th discrtizd govrning quations ar simultanously solvd for prssur p, tmpratur t, and momntum componnt f( ρu) variabls of which th lattr on is chosn instad of th vlocity variabl. Past xprinc has shown that th us of momntum componnt as a dpndnt variabl can rsult in svral important outcoms in a prssur-basd shock-capturing mthod. For xampl, it provids a strong analogy btwn th comprssibl and incomprssibl govrning quations. This in turn nabls th solution of comprssibl flow using incomprssibl mthods [12, 37]. It simplifis th rquird linarization procdur [38]. It supprsss th oscillations occurrd passing through a discontinuity [44]. It can also improvs th prformanc of th prssur-basd shock-capturing mthods [39]. III. DOMAIN DISCRETIZATION Figur 1 illustrats th grid distribution in a convrging-divrging nozzl. Th control volums ar locatd btwn th crosss, which ar calld intgration points. Th grid nods ar locatd at th gomtric cntrs of control volums. Thy ar shown by circls. Th subscripts E and W ar usd to dnot th nodal quantitis associatd with th control volum to th ast and wst of th control volum cntrd at nod P. Similarly, and w indicat th ast and wst facs of th sam control volum. In this study, uppr cas lttrs such as P, U, T, and ϱ ar associatd with FIG. 1. Th nomnclatur usd for th cll facs and thir nighboring clls.

5 CONCEPTUAL LINEARIZATION OF EULER EQUATIONS 5 quantitis at main nods, whil lowr cas lttrs such as p, u, t, and ρ rfr to quantitis at th cll facs (or intgration points). IV. COMPUTATIONAL MODELLING Equation (2.1) can b intgratd ovr an arbitrary control volum, s th shadd volum in Fig. 1. Using th divrgnc thorm, intgration of th continuity quation yilds (ρa) da + (ρua)ds = 0 (4.1) A τ S whr S indicats th intgration ovr th cll facs. Sinc th currnt mthod is fully implicit, th scond trm is valuatd at th advancd tim and th transint trm is approximatd using a mass-lumpd approach. Th lattr tratmnt yilds (ρa) da x (ϱ P ϱo) P A P (4.2) A τ τ whr ϱ o indicats th lumpd dnsity of th cll cntrd at P and its magnitud is obtaind P from th prcding tim stp. Th scond trm in Eq. (4.1) is simply intgratd ovr th boundary of th cll. It rsults in (ρua)ds (ρua) (ρua) w (4.3) S Evntually, th discrtizd form of th mass quation can b writtn as x(ϱ P ϱp o) + j (ρu) j w (ρu) w = 0 (4.4) τ Th paramtrs j = A /A P and j w = A w /A P rprsnt th ratios of th ast and wst cll fac aras rspctivly to th ara at th cll cntr. Sinc dnsity is considrd as a scondary unknown in our prssur-basd algorithm, th transint trm in Eq. (4.4) nds to b linarizd furthr. A simpl linarization ida is suggstd as ϱ = (1/R T)P, which considrs an activ rol for P and a passiv rol for T. Altrnativly, w may mploy a Taylor sris of that to considr activ rols for both P and T, i.., ϱ ϱ + ϱ P (P P)+ ϱ T (T T) (4.5) whr ϱ P = 1 ϱ R T T = P (4.6) R T 2 Th intgration of th momntum quation ovr th chosn control volum yilds [ (ρua) (pa) da + (ρu 2 A) ds + p A ] da = 0 (4.7) A τ S A x x Th us of a mass-lumpd approach for th transint trm rsults in (ρua) da x[(ϱu) P (ϱu) o P ] A P (4.8) τ τ A

6 6 DARBANDI, ROOHI, AND MOKARIZADEH Additionally, th intgration of th momntum convction trm ovr th cll facs yilds (ρu 2 A) ds (ρu 2 A) (ρu 2 A) w (4.9) S Th last trm in Eq. (4.7) can b approximatd by [ (pa) p A ] da (pa) (pa) w P p (A A w ) (4.10) x x A Evntually, th discrtizd form of th momntum quation can b prsntd by x[(ϱu) P (ϱu) o P ] + j u (ρu) j w u w (ρu) w + j p j w p w P P (j j w ) = 0 (4.11) τ To b abl to us a linar algbraic solvr in our fully implicit algorithm, th nonlinar convction trms in Eq. (4.11) nd to b linarizd proprly. Assuming th momntum componnt ρu as a primary dpndnt variabl, a simpl linarization schm suggsts ρuu ū(ρu) (4.12) Howvr, a mor sophisticatd linarization, which considrs th activ impact of th two variabls, is obtaind using th Nwton Raphson Linarization Schm, NRLS [33]. Th us of NRLS rsults in u(ρu) ū(ρu) + (ρu)u ρuu (4.13) If NRLS is furthr applid to u = ρu/ρ in th scond trm on th RHS, Eq. (4.13) bcoms ρuu 2ū(ρu) ū 2 ρ (4.14) A gnral xprssion to includ both Eqs. (4.12) and (4.14) can b dfind as ρuu 2k 1 ū(ρu) k 2 ū 2 ρ (4.15) whr k 1 and k 2 ar two constants. If k 1 = k 2 = 1, it rsults in NRLS, i.., Eq. (4.14). On th othr hand, if k 1 = 1/2 and k 2 = 0, it yilds a simpl linarization, i.., Eq. (4.12). Equation (4.14) has bn tstd in 1D invstigation with succss [38, 45]. Th xprinc has shown that NRLS would gnrally prform bttr than th simpl linarization schm. Th nxt stp is to trat th nrgy quation. This quation involvs mor nonlinar trms than th prcding quations. Th nrgy quation can b similarly intgratd ovr th chosn control volum. It yilds (ρa) da + A(ρu + pu) ds = 0 (4.16) A τ S Similar to th mass and momntum quations, th intgrals can b tratd proprly. Th consrvabl form can b vntually writtn as x[(ϱ E) P (ϱ E) o P ] τ + (ρuj) (ρuj) w + (puj) (puj) w = 0 (4.17)

7 CONCEPTUAL LINEARIZATION OF EULER EQUATIONS 7 Th nonlinar ϱe in th transint trm can b linarizd with rspct to ϱ and E using NRLS. On th othr hand, th intrnal nrgy variabl E can b linarizd to E = c v T + (Ū/2 ϱ)(ϱu). Our xprinc shows that ths linarizations provid mor robust convrgnc. Using ths linarization stratgis, th transint trm in Eq. (4.17) is approximatd by x[(ϱ E) P (ϱ E) o P ] τ x [Ū τ 2 (ϱu) + Ē ( R T P + ϱc v ( ϱē) ) ] T (ϱe) T o P (4.18) Similarly, using NRLS for ρu trm, linarizing it with rspct to and ρu, and utilizing th linarizd form suggstd for E (or ) finally yild ) ρu (ē + ū2 f + (c v f)t (f ) (4.19) 2 Th two last trms in th LHS of Eq. (4.17) can b convrtd to Rtf. This can b achivd via th quation of stat. Additionally, Rtf can b linarizd with rspct to f and t using ithr simpl linarization or NRLS. Th lattr choic rsults in Th combination of Eqs. (4.19) and (4.20) is givn by up = Rtf (c p cv)( tf ft) Rft (4.20) ρu + up (c p t + u 2 )f + (c p f)t (ē + R t) f (4.21) Th substitutions of Eqs. (4.18) and (4.21) in Eq. (4.17), th discrtizd nrgy quation is writtn as [Ū x τ 2 (ϱu) + Ē ( R T P + ϱc v ( ϱē) ) ] T (ϱe) T o +[(c p t + u 2 )f + (c p f)t (ē + R t) f ]j P {[(c p t + u 2 )f + (c p f)t (ē + R t) f ]j} w = 0 (4.22) By this drivation, th discrtization of th Eulr govrning quation is finishd. Th nxt stp is to approximat th magnituds at cll facs in trms of th magnituds at th nods. At this stag, it is worth to mntion that thr ar many diffrnt choics to linariz th nonlinar trms in th consrvativ statmnts. Howvr, our main objctiv in this work is not to xamin th impact of diffrnt possibl linarizations but to xtnd our concptual linarization, which is applicabl to nonlinar convction trms in th momntum quations, s Sction C. A. Intgration Point Equations To mak th algbraic systm of quations wll-posd, this stag of our modlling rquirs to prsnt th major dpndnt variabls at cll facs in trms of nodal variabls, s f (or ρu)in Eq. (4.4), p in Eq. (4.11), and t in Eq. (4.22). Thrfor, it is ncssary to driv suitabl xprssions for momntum componnt, prssur, and tmpratur at th cll facs. Upwind, QUICK, and HYBRID schms can b nominatd as suitabl mathmatical intrpolations [17]. Altrnativly, thr ar mor advancd schms, which includ mor physics of flow. For xampl, Prakash and Patankar [46] prsntd profils which wr attmpting to includ th rlvant physics into th intrpolation functions. Schnidr and Raw [47] mployd Physical Influnc Schm PIS, which

8 8 DARBANDI, ROOHI, AND MOKARIZADEH considrd th flow govrning quations, to driv th intgration point xprssions in incomprssibl flow simulations. Darbandi and Schnidr [28, 37] xtndd this modl to comprssibl flow simulations. Th fundamntal concpts of PIS will b mployd in this work as wll. Th xprssions for th momntum componnts can b drivd from th momntum quation. In this rgard, th momntum quation givn in Eq. (2.1) is xpandd to A f τ + u (fa) + fa u x x + A p x = 0 (4.23) Th trms in this quation ar diffrncd in crtain mannrs, which rspct th corrct physics of flow. To achiv this purpos, thy ar approximatd by A f f f o τ A τ u (fa) (Af ) (AF ) P x u x/2 fa u x f A (f / ρ F P / ϱ P ) x/2 A p P E P P x A x (4.24) (4.25) (4.26) (4.27) Consistnt with thir physics, th convction trms ar tratd in an upwind mannr. Anothr possibl form for Eq. (4.26) is f A u/ x A u/ x f. Howvr, this schm rsultd in poor convrgnc of th mthod. Th substitutions of th discrtizd trms into Eq. (4.23) and its rarrangmnt finally rsult in an xprssion for th momntum componnt at intgration point. A compact form of that can b writtn as f = 2C (A P /A + ρ / ϱ P ) 1 + 4C F P + C ū (1 + 4C ) (P P P E ) + f o 1 + 4C (4.28) whr th Courant numbr is dfind as C =ū t/ x. Equation (4.28) indicats that th us of a physical influnc schm producs a strong connction btwn th intgration point variabl at fac and its nighboring nodal variabls locatd at P and E. It can b shown that th substitutions of f and f w in th mass Eq. (4.4) and momntum Eq. (4.11) quations provid rliabl coupling btwn prssur and vlocity filds. Th xprssion for th tmpratur at intgration point can b obtaind by suitabl discrtization of th nrgy quation givn by Eq. (2.1). This quation is rwrittn as t ρc v τ + ρuc t v x + p u x + pu [ln(a)] = 0 (4.29) x Th transint trm is discrtizd similar to Eq. (4.24). Th upwind and cntral diffrncs ar utilizd for th scond and third trms, rspctivly. Th last trm is tratd as a sourc trm.

9 CONCEPTUAL LINEARIZATION OF EULER EQUATIONS 9 Ths considrations yild t ρc v t t o τ ρ c v (4.30) τ t ρuc v x t T P f c v (4.31) x/2 p u U E U P x p (4.32) x pu [ln(a)] [ln(a)] x p ū x (4.33) Using th arlir approximations, th tmpratur xprssion at intgration point is obtaind from t ( 2C (1 + 2C ) T C p FP p + F ) E ( ρū) c v (1 + 2C ) ϱ P ϱ E xc p ρ c v (1 + 2C ) [ln(a)] x + t o (1 + 2C ) (4.34) Furthrmor, th unknown dnsity at intgration points can b calculatd from th quation of stat. In this rgard, th quation of stat can b firstly linarizd with rspct to prssur and tmpratur using Eq. (4.5). Scondly, th prssur (s th nxt paragraph) and tmpratur, Eq. (4.34), xprssions ar substitutd in that. Schnidr and Raw [47] usd th prssur Poisson quation as an xplicit quation and showd that th prssur fild would b strongly lliptic in incomprssibl flow. Similarly, w us a linar intrpolation to dtrmin th prssur variabl at intgration points, i.., p (P P + P E )/2, if th local flow is subsonic. Howvr, if th flow is suprsonic w utiliz an upwind schm to approximat th prssur at th cll facs. B. Chckrboard Problm and Rmdy In th prcding sction, w obtaind f, p, t, and ρ xprssions at th cll facs. Th substitution of ths xprssions in Eqs. (4.4, 4.11, 4.22) liminats th unknowns at th cll facs in our drivations. Howvr, it is ncssary to invstigat th chckrboard problm in our xtndd collocatd formulation. Darbandi and Bostandoost [44] launchd thir incomprssibl invstigation and showd that th substitution of th drivd momntum xprssion, Eq. (4.28), in both mass Eq. (4.4) and momntum Eq. (4.11) quations might produc unralistic wavy prssur and momntum solutions. Unfortunatly, th spurious solutions fully satisfy th govrning quations and thir imposd boundary conditions. This instability in th solution is similarly rportd by othr finit-volum prssur-basd invstigators [22]. To supprss such non-physical zigzag solutions, thy suggst and mploy a nw statmnt to driv th scond xprssion for th momntum componnt at th intgration point. Thir primary purpos has bn to includ th rol of continuity quation in driving th scond momntum componnt xprssion. Thus, th scond xprssion should b obtaind in a mannr, which taks into account th rols of not only th mass but also th momntum govrning quations. To achiv thir purpos, thy suggstd [(Momntum Eq. Error) u(mass Eq. Error)] =0 (4.35)

10 10 DARBANDI, ROOHI, AND MOKARIZADEH As is sn, this suggstion taks into account th ffct of both continuity and momntum quation rrors in th nw xprssion. Th justification bhind dfining and using this spcial form is to provid an inclusiv statmnt, vry similar to Eq. (4.28), to approximat our scond momntum componnts at th cll facs. To achiv this, w nd starting from th basic govrning quations, vry similar to th on givn by Eq. (4.23). In this rgard, w suggst a nw rlation, which is mor maningful for th comprssibl flow applications. Th nw rlation is dfind as [ A f τ + u (fa) + fa u ] [ x x + A p u A ρ x τ + (fa) ] = 0 (4.36) x Th abov statmnt indicats that two typs of rrors ar incorporatd in xtnding th scond cll-fac xprssion. In fact, if a nonxact solution is substitutd into th mass and momntum quations, it will rsult in rrors or rsiduals for both of thm. Additionally, if th nonxact solution dos not satisfy only th mass quation, th impact is subsquntly appard in th scond cll fac xprssion but not th first on. Th discrtization of th first brackts in Eq. (4.36) is xactly similar to what was fulfilld for th momntum intgration point quation, s Eqs. (4.24) (4.27). Howvr, th trms in th scond brackts ar approximatd using ua ρ ρ τ ū A τ (4.37) u (fa) A f A P F P x ū (4.38) x/2 Th substitutions of Eqs. (4.24) (4.27) and Eqs. (4.37) (4.38) in Eq. (4.36) and its suitabl rarrangmnt finally yild fˆ = 2C ρ F p + C (P P P E ) + f o + C x ρ 1 + 2C ϱ P 1 + 2C 1 + 2C 1 + 2C τ (4.39) W call this nw xprssion convcting momntum and rfr to th prcding xprssion givn in Eq. (4.28) as convctd momntum. W hav lablld th convcting momntum with a hat to distinguish it from th convctd on. Th substitution of th convcting momntum into th continuity quation Eq. (4.4) ntirly supprsss th possibility of a zigzag prssur fild in th domain [44]. Morovr, Eq. (4.4) is usd to solv th prssur fild now. In anothr words, although th continuity quation in its original form has no trac of activ prssur variabl, th substitution of Eq. (4.39) in Eq. (4.4) prmits to solv Eq. (4.4) for th prssur fild now. C. Concptual Linarization Stratgy Figur 2 shows an infinitsimal volum with a finit lngth of δx takn from Fig. 1. Th inlt and outlt mass flow rats in addition to thir rsulting momntum forcs ar indicatd at th lft and right facs of th lmnt. Sinc w ar concrnd on th rol of mass flux, w avoid prsnting th prssur forcs around this volum. To mphasiz th critical rol of momntum componnt in th continuity quation (and subsquntly in our formulations), w hav usd mass flux componnt f instad of th multiplication of ρ and u, i.., ρu, in this figur. As it is known,

11 CONCEPTUAL LINEARIZATION OF EULER EQUATIONS 11 FIG. 2. forcs. An infinitsimal lmnt illustrating th inlt and outlt mass fluxs and thir rsulting momntum th flow of mass f through th cll fac rsults in momntum on that fac, i.., u(f). Considring this point, th balancs of mass and momntum for th stady conditions bcom d(f A)/dx = 0 (4.40) d[u(f A)]/dx + A(dp/dx) = 0 (4.41) Th f componnt in ths two quations can b rplacd with ρu. Howvr, to rtain th original rol of u in th mass flux componnt, w idntify it with a hat. Thrfor, th prcding coupld quations ar upgradd to d(ρûa)/dx = 0 (4.42) d[u(ρûa)]/dx + A(dp/dx) = 0 (4.43) Ths two quations contain two vlocity componnts, i.., u and û. Th vlocity componnt which appars in th mass flux componnts is namd convcting vlocity or mass consrving vlocity. This nam indicats th spcific rol of th vlocity componnt in mass balanc quations, i.., Eq. (4.40) [or Eq. (4.42)]. On th othr hand, th componnt of u in Eq. (4.41) [or Eq. (4.43)] is calld th convctd vlocity. This vlocity convcts th mass flux (or transports th scalars) through th control volum. To distinguish ths two vlocitis from ach othr, th convcting componnt has bn alrady idntifid by a hat. Th dnsity in th convction trm of th momntum quation Eq. (4.43) can b coupld with u instad of û without jopardizing th individual maning of th convctd and convcting vlocitis. Th aformntiond displacmnt rsults in d(ρûa)/dx = 0 (4.44) d[û(ρua)]/dx + A(dp/dx) = 0 (4.45) With this knowldg, w can simply dfin our convctd f and convcting componnts and rvis Eqs. (4.44) (4.45) to ˆ f momntum d( fˆ A)/dx = 0 (4.46) d[û(f A)]/dx + A(dp/dx) = 0 (4.47)

12 12 DARBANDI, ROOHI, AND MOKARIZADEH It is worth to not that an arbitrary switch from th convctd componnt to convcting on or vic vrsa may jopardiz th original concpts on which th govrning quations ar foundd. Additionally, as was mntiond bfor, th past rsarchrs hav shown that th us of two diffrnt vlocitis in th continuity and momntum quations has th advantags of supprssing th possibl prssur-vlocity dcoupling phnomnon in collocatd solution domains [23]. Considring th abov physical-basd dfinitions, thr ar two major choics to linariz th convction trm in th momntum quation Eq. (4.11) with rspct to th momntum componnt. A simpl linarization choic for th convction trms of th momntum quation prmits an activ rol only for th convctd momntum componnt. This linarization schm rsults in ûf ûf (4.48) A scond linarization choic is to mploy a Nwton Raphson Linarization Schm (NRLS). This schm considrs mor activ rol for th individual componnts in th nonlinar trm. Back to Eq. (4.13), th convction trm in Eq. (4.47) can b linarizd to û(f A) ( ûa)f + ( fa)û ûf A (4.49) Sinc û is not a major unknown in this study, w us NRLS and linariz th convcting vlocity componnt shown in th scond trm on th LHS in trms of (ρû), i.., û = (ρû) ρ 1 ρ (ρû) û ρ ρ + û (4.50) Th substitution of Eq. (4.50) in Eq. (4.49) and prforming som mor simplifications finally rsult in ûf û(f ) +ū( ˆ f) uûρ (4.51) This linarization schm considrs activ rols for both convctd and convcting momntum componnts as wll as dnsity in th convction trm in Eq. (4.11). Th dnsity trm may b simply laggd and rplacd with th known dnsity of th prvious itration. W lav it in th rst of our formulations as it is. Considring th two dfinitions of th momntum componnts, w prsnt a gnral xprssion which includs both th simpl Eq. (4.48) and NRLS Eq. (4.51) cass. Th gnral form is suggstd as ûf ûf + k (ū ˆ f uûρ) (4.52) whr k = 0 rsults in a simpl linarization, i.., Eq. (4.48), and k = 1 rprsnts NRLS, i.., Eq. (4.51). If w ignor th individual concpts involvd in th convctd and convcting componnts, it yilds û = u and f ˆ = f. Thn, Eq. (4.52) can b rplacd with a nw on givn by (ρu u = ρûû) [(2k 1 ūf k 2 ū 2 ρ) = (2k 1 ûf k2 û2 ρ)] (4.53) whr k 1 and k 2 ar two constants, which mak th two linarizations possibl. Equation (4.53) is idntical with Eq. (4.15), which was drivd without th knowldg of convcting and convctd componnts. Similar to Eq. (4.15), th considration of k 1 = k 2 = 1 rsults in NRLS, i.., Eq. (4.14), and th considration of k 1 = 1 2 and k 2 = 0 rsults in a simpl linarization, i.., Eq. (4.12).

13 CONCEPTUAL LINEARIZATION OF EULER EQUATIONS 13 In th following sction, w valuat th prformanc of our nw dvlopd linarizations in solving unstady and stady flows using ithr Eq. (4.53) as Simpl Nwton Raphson Linarization Schm (SNRL) or Eq. (4.52) as Improvd Nwton Raphson Linarization Schm (INRL) to linariz th convction trms in th momntum quations, i.., Eq. (4.11). V. RESULTS AND DISCUSSION In this sction, th concptual linarization is applid to solv both stady and unstady flows with shock. In this rgard, th convrging-divrging nozzl and shock tub problms ar chosn to valuat th xtndd formulations. Th nozzl problm is known as a standard tst cas to xamin th prformanc of stady shock-capturing tchniqus in solving a solution domain with a wid rang of Mach numbrs including a strong normal shock. Altrnativly, th shock tub problm is known as a standard tst cas to valuat th unstady shock-capturing tchniqus. Both th transint flow faturs (i.., moving normal shock and xpansion wavs as wll as a contact discontinuity) and a wid rang of flow Mach numbrs (i.., subsonic, transonic, and suprsonic rgims) can b targtd in th lattr cas. To stop th itrations in ach tim stp, th convrgnc critrion ɛ at ach tim stp is chckd using max( (P i P i )/P i, (T i T i )/T i ) (ɛ = ) (5.1) whr th subscript i rprsnts th nod numbr. Th gas proprtis ar c v R = J/KgK, and γ = 1.4. = 720 J/KgK, A. Th Accuracy of th Extndd Schm At th first stag, w xamin th unstady shock tub problm. Th shock tub lngth is1mand a total of 201 nods is uniformly distributd along it. Th air prssurs ar 1,000 and 100 KPa in th high and low-prssur sids, rspctivly. Th initial tmpratur is 25 C in both sids. Following Rfs. [38, 45], th rsults ar normally prsntd at 500 µs aftr rupturing th sparating diaphragm. Figur 3 illustrats th distributions of dnsity, prssur, tmpratur, and Mach numbr using a Courant numbr of Th dnsity, prssur, and tmpratur ar nondimnsionalizd using thir rspctiv valus in th lowr prssur sid. Th figur prsnts th FIG. 3. Th numrical solutions in th shock tub problm using both SNRL and INRL schms and thir comparisons with th xact solutions.

14 14 DARBANDI, ROOHI, AND MOKARIZADEH FIG. 4. A comparison of th prsnt solution with that of van dr Hul t al. [13]. rsults of both INRL and SNRL schms. Thy ar compard with ach othr and thos of analytical solutions. Th rsults of both INRL and SNRL schms ar in good agrmnt with th analytical solutions. Additionally, th two INRL and SNRL schms prform similar accuracis. In anothr words, th numrical solutions ar indpndnt of th slctd linarization schm. This conclusion was dfinitly prdictabl bcaus th final rsults of th two schms should not b significantly affctd by th choic of schm to trat th nonlinar convction trms in th momntum quations. Although it is not a main concrn in this work, our study shows that th accuracy achivd in this unstady solution is comparabl and vn bttr than th accuracis prsntd by a fw othr Eulr flow solvrs. For xampl, comparing th currnt rsults with thos of th primitiv Godunov mthod [1], th artificially upstram flux vctor splitting of Sun and Takayama [48], and diffrnt wightd ssntially nonoscillatory schms of Titarv and Toro [49], th accuracy of th currnt solution is xcllnt. It should b notd that th currnt accuracy is obtaind without nforcing any additional filtrs and/or faturs. Sinc it is mor rational to compar our rsults with thos of prssur-basd mthods, Fig. 4 prsnts th currnt Mach distribution and compars it with that of van dr Hul t al. [13], who bnfit from th advantags of a prssur-corrction mthod. Thy solv Sod s Rimann problm using a fw diffrnt schms of which w hav chosn th on, which is mor consistnt with our formulation and its discrtization. Th Sod s Rimann problm is solvd in similar conditions for both cass. In th nxt stag, w study th convrging-divrging nozzl problm. Th nozzl profil is dfind as Ara(x) = 1 + m(x l/2) 2, whr m is a positiv intgr and l rprsnts th nozzl lngth. This quation provids symmtric profils with rspct to th nozzl throat. Dpnding on th flow conditions at th inlt and outlt, diffrnt boundary conditions ar rquird to b mployd at th boundaris. For th status with subsonic at th inlt, subsonic at th outlt, and a normal shock wav standing in th divrgnt part of th nozzl, th prssur, vlocity, and tmpratur ar spcifid at th inlt and th prssur is spcifid at th outlt. In fact, th back prssur dtrmins th shock position and its strngth in th divrgnt part. Figurs 5 and 6 illustrat th distributions of dnsity, prssur, tmpratur, and Mach numbr in th nozzl using two back

15 CONCEPTUAL LINEARIZATION OF EULER EQUATIONS 15 FIG. 5. Th prsnt rsults in a nozzl with P b = 85 kpa using INRL and SNRL schms and comparing thm with th xact solutions. prssurs of 85 and 45 kpa, rspctivly. Th rsults wr obtaind using a Courant numbr of Sinc th vlocity magnitud changs throughout th nozzl, th Courant numbr is calculatd basd on th maximum vlocity occurrd within th nozzl, i.., C = U max t/ x. Th dnsity, prssur, and tmpratur paramtrs ar non-dimnsionalizd with rspct to thir local stagnation point magnituds. Similar to th shock tub problm, th two INRL and SNRL schms xhibit similar accuracis and provid xcllnt agrmnts with that of th xact solution. Indd, a strongr shock dos not dtriorat th achivd accuracy at all. To quantify th accuracy of our rsults, Fig. 7 prsnts both th msh rfinmnt study and a comparison with othr solutions. Figur 7(a) illustrats th currnt Mach distributions for th msh with 80, 160, and 320 rsolutions. This plot dmonstrats that th msh with 160 nods is fin nough to provid rliabl accuracy. Thrfor, this msh rsolution is chosn to validat our rsults against th xact solution and th solution providd by Rossow [42], s Fig. 7(b). To hav a fair comparison, w hav prsntd th rsults of Rossow for a similar grid distribution. Similar to our cas, thr is not much diffrnc btwn th rsults of Rossow using ithr 160 or 320 grid rsolutions. According to Rossow, th rsults prsntd in Rf. [42] was not suitably dscribd thr. Th rsults of Rossow prsntd in Fig. 7(b) wr rcivd dirctly from him. Th comparison indicats that th accuracy of th currnt mthod is bttr than that of Rossow. Inspcting th currnt solutions givn in Figs. 3 7, it is obsrvd that thr ar no oscillations around ithr shocks or discontinuitis. Additionally, th solutions ar vry prcis in th rgions FIG. 6. Th prsnt rsults in a nozzl with P b = 45 kpa using INRL and SNRL schms and comparing thm with th xact solutions.

16 16 DARBANDI, ROOHI, AND MOKARIZADEH FIG. 7. Evaluating th currnt numrical solutions in solving th nozzl problm. (a) Th grid rfinmnt study. (b) Comparison with xact and Rossow [42] solutions. with smooth to modrat variations. Back to th Computational Modlling sction, w hav not nforcd any typs of limitr, xplicit damping function, xplicit artificial viscosity, and so on in our formulations to damp out possibl oscillations in our rsults. Dspit xcluding such ky faturs, which ar normally applid in shock-capturing mthods, no major or minor oscillations ar obsrvd around discontinuitis in our solutions. B. Prformanc of th Two Linarization Schms Figur 8 prsnts th avrag numbr of itrations pr tim stp κ to mt th spcifid convrgnc critrion at diffrnt Courant numbrs. Figur 8(a) compars th prformanc of SNRL with that of INRL in shock tub problm. Th problm is tstd for both isntropic and isothrmal conditions. W hav tstd th isothrmal condition in ordr to liminat th rol of nrgy quation from th systm of govrning quations in our nonlinar itrations. As is xpctd, th FIG. 8. Th avrag numbr of itrations pr tim stp vrsus Courant numbr. (a) Th shock tub problm. (b) Th nozzl problm.

17 CONCEPTUAL LINEARIZATION OF EULER EQUATIONS 17 avrag numbr of itrations pr tim stp normally incrass as Courant incrass. Our xprinc showd that th numbr of itrations at arlir tim stps would b much highr than th nxt ons. Additionally, th numbr of itrations may rduc to as low as 3 5 itrations pr tim stp in Courant numbrs lss than 0.5 if tim has lapsd nough. Irrspctiv of th typ of two schms, thir prformancs show considrabl diffrnc as Courant incrass. Howvr, th diffrncs diminish at lowr Courant numbrs. A carful comparison indicats that th choic of isothrmal cas rsults in a highr numbr of itrations with rspct to th corrsponding isntropic on. Thrfor, th advantags of INRL with rspct to SNRL ar gratr if th nrgy quation is coupld into th systm of govrning quations. On important point shown in Fig. 8(a) is that th rang of Courant numbr applicability is widr for INRL than that of SNRL in both isothrmal and isntropic conditions. Th diffrnc is mor pronouncd in isothrmal condition. As is sn, INRL convrgs to solution within a much widr rang of Courant numbrs at isothrmal condition. Th figur indicats that INRL convrgs to solutions for Courants clos to 1.5 in isothrmal condition; howvr, SNRL is rstrictd to a maximum Courant of about Thrfor, it is concludd that SNRL is mor rstrictd to th rang of larg Courant numbr mploymnt than INRL irrspctiv of mploying ithr isntropic or isothrmal conditions. Figur 8(b) similarly compars th prformanc of SNRL with that of INRL in solving th nozzl problm with two back prssurs of P b = 85 and 45 kpa. Similar to th shock tub cas, INRL gnrally xhibits gratr prformanc than SNRL. In th othr words, INRL schm rquirs lss numbr of itrations pr tim stp to mt th convrgnc critrion at idntical conditions. Comparing th rsults at th two chosn back prssurs, it indicats that th mthod is mor rstrictd to th rang of larg Courant mploymnt as th back prssur dcrass. It is bcaus th normal shock standing in th divrgnt part bcoms much strongr as P b dcrass. This ffct is similarly obsrvd using ithr INRL or SNRL schms. Howvr, SNRL is limitd to lowr Courants than INRL. As is sn, th numbr of itrations in cas with P b = 45 kpa drastically incrass as Courant approachs This ill prformanc is similarly obsrvd whn P b = 85 kpa as Courant approachs 0.8. Contrary to SNRL, INRL dos not prform such ill prformancs at th two back prssurs. Additionally, its rang of larg Courant mploymnt is much widr than SNRL. Figurs 8 compars th prformanc of INRL with that of SNRL at diffrnt conditions and a wid rang of Courant numbrs. Howvr, it dos not quantify th incras in prformanc. To quantify th qualitativ progrsss shown in this figur, th prcntag rduction in th numbr of itrations pr tim stp is dfind as ζ = 100 (κ SNRL κ INRL )/κ INRL (5.2) whr κ rprsnts th avrag numbr of itrations pr tim stp as illustratd in Fig. 8. Figur 9(a) dmonstrats th prcntag rduction vrsus Courant numbr. Morovr, Fig. 9(a) prsnts th rsults in solving th shock tub problm with ithr isntropic or isothrmal conditions. As is obsrvd, th prcntag rduction bhavior changs as Courant incrass. It prforms a dcras following a sharp incras. As is sn, th incras in prformanc is much mor pronouncd at th isothrmal condition than th isntropic on. Indd, th impact of using INRL is to caus a gratr improvmnt in th prformanc of th mthod irrspctiv of mploying ithr isothrmal or isntropic conditions. Th improvmnts ar as larg as 26% and 15% for th isothrmal and isntropic conditions, rspctivly. Similar to Fig. 9(a), Fig. 9(b) dmonstrats th achivd improvmnt in solving th nozzl problm. Th rsults ar prsntd for two back prssurs of 85 and 45 kpa. Indd, this figur

18 18 DARBANDI, ROOHI, AND MOKARIZADEH FIG. 9. Th prcntag rduction in th numbr of itrations vrsus Courant numbr. (a) Th shock tub problm. (b) Th nozzl problm. quantifis th improvmnt illustratd in Fig. 8(b). As is obsrvd, th improvmnt is xcllnt at both back prssurs. Figur shows that INRL prforms considrably bttr than SNRL in trating th stady comprssibl flow problm with shock, whr a strong discontinuity stands in th domain. Th figur also indicats that th prformanc will b boostd up drastically if th normal shock standing in th divrgnt part bcoms strongr. In anothr words, th improvmnt in prformanc of INRL is considrabl at lowr back prssurs. Th improvmnt is dramatic as Courant approachs th limiting Courants. Th improvmnts bcom as larg as 42% and 80% for th cass with th wak and strong normal shocks, rspctivly. To focus on a spcific Courant numbr in solving th shock tub problm and quantify th improvmnt in th prformanc, Fig. 10 prsnts th rsults for a shock tub with various diaphragm prssur-ratios. Indd, this study nabls us to valuat th prformanc of INRL FIG. 10. Th prformanc of INRL with rspct to SNRL in solving th shock tub problm considring diffrnt diaphragm prssur ratios, C = (a) Avrag numbr of itrations pr tim stp. (b) Prcntag rduction in numbr of itrations.

19 CONCEPTUAL LINEARIZATION OF EULER EQUATIONS 19 TABLE I. Improvmnt in th prformanc of th currnt collocatd shock-capturing mthod using INRL schm. Tst cas ζ max % η% Shock tub, isothrmal, r p = Shock tub, isntropic, r p = Shock tub, isothrmal, r p = N.A. Shock tub, isntropic, r p = N.A. Nozzl, P b = 85.0 KPa Nozzl, P b = 45.0 KPa in trating unstady problms with strongr discontinuitis. Figur 10(a) illustrats th avrag numbr of itrations pr tim stp rquird to convrg to th spcifid ɛ at C = Th rsults ar for both isothrmal and isntropic conditions. Th problm has bn solvd for diffrnt diaphragm prssur ratios from r p = 10 to 40. Th moving shock, xpansion wavs, and discontinuity bcom much strongr in th shock tub if r p incrass. Figur 10(a) indicats that th prformanc of INRL incrass as r p incrass. Th improvmnt is similarly obsrvd for both isntropic and isothrmal conditions. It is concludd that th prformanc of INRL is not dtrioratd as th discontinuitis bcom strongr in th domain. Figur 10(b) quantifis th prformancs shown in Fig. 10(a). It prsnts th prcntag rduction in th numbr of itrations pr tim stp vrsus th diaphragm prssur ratio. As is obsrvd, th improvmnt incrass as th prssur-ratio incrass. Th improvmnts ar as larg as 34% in isothrmal and 23% in isntropic conditions. Tabl I summarizs th outcoms achivd in this sction. Indd, it prsnts th maximum prcntag of improvmnt achivd by mploying INRL schm instad of th primitiv SNRL on. Th tabl is limitd to a fw cass, which wr studid and discussd in this sction. It provids two typs of prformancs. Th first on is th maximum prcntag rduction in th numbr of itrations pr tim stp ζ max. Th scond on is th progrss in th rang of larg Courant numbr applicability, which is dfind as η% = 100 (C INRL C SNRL )/C SNRL. Th tabl gnrally indicats that th improvmnts in th two valuatd paramtrs ar grat if INRL schm is usd in th collocatd shock-capturing mthods to solv both th stationary and nonstationary problms. Th tabl also indicats that th prformanc of INRL with rspct to SNRL incrass as th discontinuitis in th domain bcom strongr. VI. CONCLUSION A fully concptual linarization stratgy was suitably dvlopd to linariz th convction trms in th momntum quations. Th ssnc of stratgy rturns to th us of dual vlocity dfinitions at th cll facs in finit-volum-basd collocatd-grid mthods. Th concptual linarization maintains th original charactristics of th two vlocitis, which invitably appar in th consrvativ statmnts of th govrning quations. Th rsults show that thr will b considrabl improvmnt in th prformanc of th primitiv finit volum mthod if th concptual maning is fully authnticatd in th linarization. Indd, a widr rang of larg Courant numbr applicability in addition to bnfit from a fwr numbr of itrations indicat that th concptual linarization prforms numrously bttr than th primitiv linarization stratgy. Th conclusion was shown for both stady and transint flow problms. Th currnt invstigation showd that th improvmnt in th fficincy would b as larg as 80%. This prformanc can still incras if th discontinuitis in th domain bcom strongr. Th currnt rsults also indicat that th solution is accurat

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