Use of Proper Closure Equations in Finite Volume Discretization Schemes

Transcription

1 Us of ropr Closur Equations in Finit Volum Discrtization Schms 3 A. Ashrafizadh, M. Rzvani and B. Alinia K. N. Toosi Univrsity of Tchnology Iran 1. Introduction Finit Volum Mthod (FVM) is a popular fild discrtization approach for th numrical simulation of physical procsss dscribd by consrvation laws. This articl prsnts som advancs in finit volum discrtization of hat and fluid flow problms. In th cas of thrmo-fluid problms, dcision rgarding th choic of unknown variabls nds to b takn at th continuous lvl of formulation. A popular choic, among many availabl options, is th prssur-basd primitiv variabl formulation (Acharya, 2007), in which th flow continuity quation is considrd as a constraint on fluid prssur. rssur-basd finit volum mthods, in turn, can b catgorizd basd on th computational grid arrangmnt usd for th discrtization of solution domain. In a staggrd grid arrangmnt, vlocity componnts and othr unknown scalars ar dfind at diffrnt nodal points (atankar, 1980). On th contrary, all unknown variabls ar dfind at th sam nodal locations in a co-locatd grid arrangmnt. Th discussion of finit volum discrtization schms in this articl is limitd to prssur-basd formulations on co-locatd grids. At th continuous lvl of formulation in a finit volum mthod, an intnsiv variabl is constraind by an intgral balanc quation for a control volum CV boundd by th control surfac CS as follows: In Eq. (1) J t dv J nda S dv (1) CV CS CV rprsnts th flux of across th control surfac, S stands for th rat of gnration of within th control volum and n is th outward unit normal vctor to th control surfac. In most of th nginring problms of intrst, thr ar two mchanisms for th transport of, i.. advction and diffusion, and th flux function is mathmatically dscribd as follows: J V In Eq. (2) V is th advctiv mass flux and is th diffusion cofficint associatd with variabl. By proprly dfining, J and S, Eq. (1) can b usd as a gnric quation to (2)

2 62 Finit Volum Mthod owrful Mans of Enginring Dsign mathmatically dscrib th transport of mass, momntum and nrgy in fluid flow problms (atankar, 1980). A typical two dimnsional discrt domain covrd by contiguous control volums (clls) is shown in Fig. 1. Each cll rprsnts a nodal point and is boundd by a numbr of panls which compris th control surfac. Typically, ach panl includs on intgration point. Thr isn't any limitation on th shap and th numbr of panls of a cll. Hr w us a background finit lmnt msh to dfin th control volums. Figur 1 shows structurd and unstructurd two-dimnsional grids in an lmnt-basd finit volum mthod (EBFVM) contxt (Ashrafizadh t al., 2011). Th objctiv of any finit volum discrtization schm is to us th CV balanc quations, similar to Eq. (1), to obtain a st of algbraic quations which constrains th unknown nodal valus throughout th domain. This objctiv is achivd by carrying out th discrtization in two stags. At th first stag of discrtization, hr calld lvl-1 approximation, Eq. (1) is convrtd to a smi-discrt form comprising of both nodal and intgration point variabls. Lvl-1 approximat form of Eq. (1) for an intrnal finit volum in Fig. 1 can b writtn as follows: ( ) t np N N ip ip V ( V nda ) ( A ) ip n S V (3) ip ip1 ip1 In Eq. (3), subscripts np and ip stand for nodal and intgration point variabls rspctivly. Closur quations ar now ndd to rlat ip variabls to np variabls. At th scond stag of discrtization, hr calld lvl-2 approximation, Eq. (3) is convrtd to a fully-discrt form, i.. an algbraic quation which constrains th variabl at nodal point. Th computational molcul obtaind at this stag of discrtization is writtn in th following convnint form: N nb a a S (4) nb nb nb1 Hr, subscript nb stands for th nighbor nods and suprscript N nb rfrs to th numbr of influntial nighbor nods. Influnc cofficints a and a nb and th sourc trm S contain trms and paramtrs which modl all physical ffcts rlvant to th valu of at th nodal point. If Eq. (1) is a nonlinar quation, linarization also bcoms ncssary and should b carrid out during th discrtization. In principl, th quation can b linarizd bfor or aftr ach of th approximation lvls and this might affct th proprtis of th final discrt modl. In most nginring problms, including hat and fluid flow ons, a coupld st of nonlinar partial diffrntial quations nds to b discrtizd and solvd. In such cass, th intr-quation couplings should also b numrically modld. Two-dimnsional, incomprssibl Navir-Stoks quations, for xampl, can b writtn in a form similar to Eq. (1) in which is qual to 1 in th mass balanc quation, u in th x-momntum balanc quation, v in th y-momntum balanc quation and (spcific nrgy) in th nrgy

3 Us of ropr Closur Equations in Finit Volum Discrtization Schms 63 Fig. 1. Structurd and unstructurd lmnt-basd finit volum grids. balanc quation. In gnral, th computational molcul associatd with ach quation of a coupld st includs mor than on variabl to rflct th fact that th quations and th variabls ar intr-rlatd. Thrfor, in th cas of a st of quations with two coupld variabls and, th following modifid form of Eq. (4) is usd to constrain at th nodal point : Nnb Nnb nbnb nbnb nb1 nb1 (5) a a a a S In this articl, w focus on lvl-2 approximation of Navir-Stoks quations in th contxt of co-locatd, prssur-basd finit volum mthod. This lvl of approximation is particularly important bcaus it dals with th modling of physical influncs such as upwind and

4 64 Finit Volum Mthod owrful Mans of Enginring Dsign downwind ffcts and th couplings btwn variabls. A numbr of famous lvl-2 approximations ar rviwd and th mthod of propr closur quations (MCE), proposd by th first author, is introducd. To avoid th complxitis associatd with multi-dimnsional problms and to focus on th natur of th approximations in a simpl stting, on dimnsional quations and tst cass ar discussd in dtails. Two-dimnsional rsults ar only brifly prsntd to show th applicability of th MCE in multi-dimnsional problms. 2. Th smi-discrt form of quations in a 1D contxt A on-dimnsional variabl-ara duct is shown in Fig. 2. Th complt st of govrning quations for unstady 1D-inviscid comprssibl flow in this duct, using th momntum variabl formulation, is as follows (Chtrntal, 1999): 1 da m f S f adx 1 da u f fu p fu S x a dx f pu 1 da pu adx Whr is tim, f u is th momntum variabl, u is vlocity, p is prssur and a is th cross sctional ara. Not that no artificial nrgy sourc trm is usd in th nrgy quation. In ordr to clos th abov st of quations, an auxiliary quation is ndd. Hr, th idal gas quation of stat is usd: (6) p Rt (7) Whr t is tmpratur and R is th gas constant. Nglcting th potntial nrgy trm, th total nrgy pr unit mass ( ) of an idal gas can b writtn as follows: Whr cv is th spcific hat at constant volum. 1 2 cvt u (8) 2 By intgrating Eq. (6) ovr th control volum in Fig. 2 and aftr using divrgnc thorm, th smi-discrt forms of th govrning quations ar obtaind as follows: m V SV fa fwaw V p F F fu p w ( w a) a fu p w a a a S V V EE f pu a fu a w w 0 (9) Not that capital lttrs (lik F and ) ar usd for th nodal valus and small lttrs (lik f and p ) ar usd for intgration point variabls. Hr, V stands for th volum of cll. Closur quations for th intgration point variabls ar discussd nxt.

5 Us of ropr Closur Equations in Finit Volum Discrtization Schms 65 Fig. 2. Grid arrangmnt in a on-dimnsional variabl ara duct. 3. Closur quations for intgration point variabls Aftr obtaining th smi-discrt form of govrning quations and linarizing thm, closur quations ar ndd to approximat intgration point (ip) quantitis in trms of th nodal valus. By mploying ths closur quations, th smi-discrt quations ar convrtd to fully-discrt quations. Ths ip-quations play a critical rol in th robustnss and accuracy of a collocatd schm. Thrfor, diffrnt solution stratgis for dfining ipquation ar brifly dscribd in this sction. Hr, w only discuss th closur quations for ast intgration point of a typical cll around nod. Th closur quations at th wst ip ar obtaind similarly. 3.1 Mthods basd on gomtrical intrpolation Simpl symmtric intrpolation is th most trivial choic for an ip-valu in a uniform grid: On non-uniform grids, wightd intrpolation is usd as follows: E (10) 2 (1 ) (11) E Whr is th wight factor for ast ip, calculatd by th following formula: x X E X X (12)

6 66 Finit Volum Mthod owrful Mans of Enginring Dsign This schm is 2nd-ordr accurat, but is unboundd so that non-physical oscillations may appar in rgions with high gradints. To show th problm associatd with this schm, considr th 1-D inviscid incomprssibl flow in a constant ara duct shown in Fig. 3. Th smi-discrt form of continuity and momntum quations in this cas ar as follows: u 0 Au (13) 0 w mu u Ap p (14) w w Whr A is th cross sctional ara of th duct and m is th mass flow rat basd on th most rcnt availabl valu of vlocity. Fig. 3. A 1-D Co-locatd grid in a constant cross sctional ara duct. Using simpl symmtric intrpolation, Eqs. (13) and (14) ar convrtd to th following fully-discrt forms: A U E U W 0 (15) 2 m A 2 2 U U 0 E W E W Combining th abov quations, on obtains th following constraint on th prssur at nodal point : E W (16) (17) Not that prssur at nod is absnt in this quation and this constraint cannot distinguish btwn a uniform prssur fild and th wiggly prssur fild shown in Fig Mthods basd on taylor-sris xpansion It is logical to assum that convctd quantitis in a thrmo-fluid problm ar influncd by th upstram condition. Thrfor, assuming th flow dirction from to E, th following truncatd Taylor sris xpansions ar both valid approximations for : (18) φ x x X (19)

7 Us of ropr Closur Equations in Finit Volum Discrtization Schms 67 Fig. 4. A typical wiggly prssur fild. Equation (18) is calld th simpl or Fist Ordr Upwinding (FOU) and provids highly stabl and unconditionally boundd numrical schms. Howvr, it is only first ordr accurat. Equation (19) provids a Scond Ordr Upwinding (SOU) stratgy. On can show that a numrical schm which mploys Eq. (10) for intgration point prssurs and Eqs. (18) or (19) for intgration point vlocitis in Eqs. (13) and (14), would not b abl to dtct th chckrboard prssur filds. Such numrical schms ar pron to unphysical prssurvlocity dcoupling symptom. 3.3 Mthods basd on momntum intrpolation Rhi and Chow (Rhi, 1983) mployd a co-locatd grid to solv th flow fild around an airfoil. To avoid nonphysical, wiggly numrical solutions, thy cam to th conclusion that diffrnt closur quations had to b usd for th convcting or mass-carrying ( û ) and convctd or transportd ( u ) vlocity componnts at th intgration points. Th convcting vlocity, usd in th smi-discrt continuity quation, is linkd to th prssur through a momntum intrpolation procdur. Hnc, this discrtization schm is calld th Momntum Intrpolation Mthod (MIM). As compard to th classical schms on staggrd grids, on may assum that nodal momntum balancs ar concptually staggrd to obtain closur quations for convcting vlocity at intgration points. Th discrt momntum balanc for nod can b writtn as follows: bnbunb V au anbunb V 0 U (20) x a a x Th convcting vlocity at is obtaind by staggring th stncil of th computational molcul to th intgration point: Which rsults in: nb nb ˆ u a U V a a x (21) U ˆ UE d u x x x E (22)

8 68 Finit Volum Mthod owrful Mans of Enginring Dsign Whr 1 V VE d 2 a ae (23) Th closur for th convctd vlocity, and any othr transportd variabl, coms from an upwind schm. Th simplst option at th intgration point would b th FOU: u U if m 0 (24) Th intrpolation formula, Eq. (22), is not uniqu as xplaind in (Acharya, 2007) and on might succssfully implmnt th ida using modifid forms of this approach. Th ky point, howvr, is to us diffrnt closur quations for th convctd and convcting vlocity componnts. Th convcting vlocity should b proprly rlatd to th prssur fild. Th closur quation for p has not bn a mattr of concrn, at last for incomprssibl flows. Taking into account th lliptic bhavior of th prssur, th following closur quation is usd in th MIM: E (25) 2 This tchniqu guarants th rquird physical prssur-vlocity coupling in th numrical modl and prvnts any wiggly solutions (Ashrafizadh t al., 2009). 3.4 Mthods basd on physical influncs Th plasur of working with all variabls on a singl nodal position, as opposd to th pain of working with staggrd grids limitd to simpl gomtris, has mad th MIM vry popular. Thrfor, attmpts hav bn mad to mploy and improv th MIM for th solution of flows at all spds. Schnidr and Raw (Schnidr & Raw, 1987a, 1987b) proposd th physical influnc schm to unify th intgration point vlocity intrpolation formulas in co-locatd grids. Thy rtaind a unifid dfinition for th intgration-point vlocity componnts, but argud that th closur for u should not b obtaind solly by purly mathmatical upwind schms. Thy suggstd th non-consrvativ form of th momntum quation to provid a closur for u : u u u u 0 x x x (26) Th mthod, calld th physical influnc schm, provids th following closur quation in th contxt of th 1-D tst cas: x ˆ u u Up 2u x E (27)

9 Us of ropr Closur Equations in Finit Volum Discrtization Schms 69 Whr th ovr bar rfrs to th most rcnt availabl valu in th nonlinar itrations. Equation (25) is usd as th closur for p in this schm. Whil succssfully implmntd in a numbr of multidimnsional viscous problms, th physical influnc schm fails to prvnt prssur chckrboard problm in inviscid incomprssibl flows (Bakhtiari, 2008). This failur rinforcs th blif that two dfinitions for convctd and convcting vlocitis at intgration points ar ncssary whn th calculations ar carrid out on a co-locatd grid. 3.5 Mthods basd on modifid physical influncs To fix th problm associatd with th physical influnc schm, Karimian and Schnidr (Karimian, 1994a) accpt th notion of dual vlocitis at th intgration points and mploy a combination of momntum and continuity quations to provid a closur for th convcting vlocity ( û ): Which rsults in: U ˆ UE x u Momntum Error ucontinuity Error 2 2u (28) uˆ U UE E (29) 2 2u Thy also us Eq. (27) for convctd vlocity ( u ) and Eq. (25) as th closur for p. Numrical xprimnts with th 1-D tst cas show that this mthod works wll and succssfully supprsss any oscillatory numrical solution vn in tim-dpndd all spd flows (Karimian, 1994b). This mthod has also bn xtndd to 2D flows on structurd (Alisadghi t al., 2011a) and unstructurd (Alisadghi t al., 2011b) grids. Darbandi (Darbandi t al., 1997) proposd a prssur-basd all spd mthod similar to th Karimian's mthod, in which th momntum componnts ( f u ) ar usd as th flow dpndnt variabls instad of th vlocity componnts ( u ). Thy usd th following form of th momntum quation as th physical constraint on th intgration point convctd vlocity: f f u u f 0 x x x (30) Whr f u is th momntum variabl. Appropriat discrtization of Eq.(30) lads to: 1 f F E (31) 4u In this mthod, a combination of momntum and continuity quations is usd to constrain th intgration point convcting vlocity: f f u f u f u 0 x x x x (32)

10 70 Finit Volum Mthod owrful Mans of Enginring Dsign Whr is a usr-dfind wight factor. This quation can b discrtizd to obtain th following formula: ˆ 1 f F E (33) 2 u In this mthod, lik all othr mthods discussd so far, Eq. (25) is usd as th closur for p. Numrical rsults hav shown that this st of closur quations rmovs th possibility of a chckrboard solution and provids strong coupling btwn prssur and vlocity filds. Anothr analysis has shown that th us of momntum-variabl formulation, instad of primitiv variabl formulation, improvs th stability and accuracy of th solution spcially in th solution of comprssibl flows with shock wavs (Darbandi, 2004). 3.6 Mthod of propr closur quations (MCE) On-dimnsional incomprssibl flow Ashrafizadh t al. (Ashrafizadh t al., 2009) hav shown that it is possibl to dvlop a colocatd numrical schm without rsorting to th convcting and convctd vlocity concpts, originally proposd by Rhi and Chow. Th proposd mthod, calld th mthod of propr closur quations (MCE), mploys a propr st of physically rlvant quations to constrain th vlocity and prssur at intgration points. It has alrady bn succssfully implmntd and usd in th solution of stady 1-D inviscid incomprssibl and comprssibl flow problms (Ashrafizadh t al., 2008, 2009) and stady 2-D viscous incomprssibl flow problms on both structurd and unstructurd grids (Alinia, 2011; Bakhtiari, 2008). In this mthod, following th physical-basd approach proposd by Schnidr and Raw (Schnidr t al., 1987) and modifications proposd by Karimian and Darbandi (Darbandi, 1997; Karimian, 1994a), a combination of non-consrvativ form of momntum and continuity quations is usd to obtain an intrpolation formula for th ip-vlocity: which rsults in: momntum Eq. u continuity Eq. 0 u (34) u u 2 1 da u u 1 da m u u u S uu u S 0 x x Adx x x Adx UDS 1 CDS CDS 2 CDS (35) u m Hr S and S ar artificial momntum and mass sourc trms, and a and u ar th ipcross sctional ara and a schm control paramtr rspctivly. Thr ar options availabl to th usr at this stag. Using UDS or CDS Schms for trms 1 and 2 andu 0, 1, 2, as discussd in (Rzvani, 2008), diffrnt formulas for u can b obtaind. Th most common formula is as follows: E x u m u U ( S us ) (36) 2u 2u

11 Us of ropr Closur Equations in Finit Volum Discrtization Schms 71 As mntiond arlir, th classical closur choic for th intgration point prssur has bn th linar approximation, i.. Eq.(25). Howvr, th MCE mploys rlvant govrning quations to obtain th rquird closurs for all intgration point quantitis, including th prssur. Th momntum quation is a natural choic for obtaining a propr closur quation for p. Howvr, only th prssur gradint appars in th momntum quation and CDS-basd discrtization, physically propr for th prssur, rsults in wiggly solutions. As discussd in (Rzvani, 2010), by taking th divrgnc of th momntum quation, a prssur poisson quation for th prssur is obtaind. Th prssur oisson quation is an appropriat constraint quation for th prssur discrtization. Thrfor, as compard to Eq. (34), th following closur quation is proposd for th intgration point prssur: Divrgnc of 4u Continuity Eq. 0 Momntum Eq. x (37) Which rsults in: U U A A S 2 2 E u u x u u m E E a 4 x x (38) Whr 0 is a schm control paramtr. Numrical tst rsults hav shown that th mthod works wll with any nonzro positiv valu for (Rzvani-2010) On-dimnsional comprssibl flow Extnsion of th MCE to comprssibl flow problms rquirs two additional tasks. First, th nrgy quation should also b solvd in ordr to find th tmpratur fild. Scond, dpnding on th chosn unknown variabls, propr linarization is crucially important. Following approachs prsntd in (Darbandi t al., 2007; Karimian, 1994a), th Nwton- Raphson linarization stratgy is mployd to linariz th nonlinar trms in th balanc quations. For xampl, trm pu in th nrgy quation is linarizd as follows: pu Rt u u Rt R ft R ft tf ft (39) Closur quation for f can b asily obtaind using th unstady-comprssibl form of Eq. (34). First ordr upwind diffrnc schm is usd for th discrtization of momntum variabl gradint trms. Us of u 1 in th gnral form of th closur for th momntum variabl at th ast fac of th cll, rsults in th following formula for f : 2C C f C x C x f F S u S 12C u 12C 12C 12C u 12C E u m (40) Whr, C is th Courant numbr dfind as C u x and suprscript stands for th valus from prvious tim stp.

12 72 Finit Volum Mthod owrful Mans of Enginring Dsign Similarly, Closur quation for Eq. (37) as follow: p can b obtaind using th unstady-comprssibl form of p E p fu p u F F A A a E E 2 x u u m p 2 p S ux 4 x x 2 Th non-consrvativ form of th continuity quation in Eq. (6), in th absnc of its sourc trm, is proposd as a propr closur quation for th dnsity (Darbandi t al., 1997; Karimian, 1994a). Hr, in ordr to control th stability and accuracy of th cod in both subsonic and suprsonic rgims, a smart blnding factor ( ) is usd which maks th quation hyprbolic in suprsonic flow rgions [Karimian, 1994a]: (41) x F f A x 1 2u x a x x 2 (42) m m M M 1 whr M is local Mach numbr at th ast intgration point and m=2 is suggstd for th suprscript m. Th final rquird closur quation is an intrpolation formula for th intgration point tmpratur. Hr, th non-consrvativ form of th nrgy quation in th absnc of th sourc trm is a natural candidat (Darbandi t al., 1997) and on obtains th following xprssion: 2C Cp F FE C p A AE t t T 12C fc v1 2C E 12C cva 12C Th final discrt computational molculs for solving vlocity, prssur and tmpratur filds ar obtaind aftr closur quations ar substitutd in Eq. (9): pp pp pf pf pt pt p nb nb nb nb nb nb a a a F a F a T a T b (43) (44) (45) (46) ff ff fp fp ft ft f nb nb nb nb nb nb a F a F a a a T a T b (47) tt tt tf tf tp tp t nb nb nb nb nb nb at a T a F a F a a b ff fp In this linar algbraic st, a, a,... ar th influnc cofficints and b, b,... ar right hand sid constant trms. Subscript nb stands for th immdiat nighbors of nod (i.. W and E ). Th first suprscript rfrs to th rlvant quation and th scond suprscript points to th rlvant physical variabl. Th assmbld linar quations can b solvd using any dirct or itrativ solvr. Hr, th Gauss limination mthod is usd to solv th st of simultanous quations. Th coupld st of th algbraic quations might also b solvd using a smi-implicit (sgrgatd) solution stratgy. p f

13 Us of ropr Closur Equations in Finit Volum Discrtization Schms Two-dimnsional incomprssibl flow Th MCE has also bn usd to solv 2D, incomprssibl laminar flows on both structurd and unstructurd grids (Bakhtiari, 2008; Alinia, 2011). Th implmntation follows th basic philosophy alrady dscribd in a 1D contxt. Howvr, mor complx intrpolation formulas ar rquird to tak into account th ffct of cll shap and orintation on balanc quations. Implmntation dtails ar not discussd hr for th sak of brvity. 4. On-dimnsional tst cass 4.1 Incomprssibl flow rformanc of th MCE for solving incomprssibl and comprssibl flows is xamind hr through two diffrnt tst cass. Stady incomprssibl flow in a constant ara duct is th first tst cas which is usd to xamin th prssur-vlocity coupling in MCE. Quasi 1D flow in a convrgnt-divrgnt nozzl is anothr tst cas usd to chck th accuracy of th proposd mthod in inviscid incomprssibl flow problms Stady incomprssibl flow in a constant ara duct Th 1D incomprssibl flow in a constant ara duct is usd to s if oscillatory solutions appar whn th MCE quations govrn th discrt flow fild. For this purpos, artificial lmnt and volum sourcs/sinks ar usd in continuity and momntum quations. In a domain with 30 nods, th artificial lmnt sourc trms ar locatd at th 10th and 20th cll-facs and control volum sourc trms ar activatd at th 10th and 20th nodal positions. Numrical rsults show that th ffcts of sharp variations in vlocity and prssur du to th artificial sourc trms ar quickly dampd out and MCE works wll vn with p 0. Figurs 5 and 6 show th ffcts of control-volum artificial sourcs on th numrical solution. Ths rsults show that numrical oscillations ar dvlopd in both vlocity and prssur filds for p 0. Howvr, oscillations ar strongly dampd for any non-zro valu of p. Not that th notation N-D in ths figurs stands for Non-Dimnsional and vlocity is normalizd with th inlt vlocity and prssur is normalizd using th xit prssur. Fig. 5. Vlocity and prssur filds in a constant ara duct with control-volum mass sourc/sink.

14 74 Finit Volum Mthod owrful Mans of Enginring Dsign Fig. 6. Vlocity and prssur filds in a constant ara duct with control-volum momntum sourc/sink Th quasi-1-d tst cas Idal flow in a convrging-divrging nozzl is anothr tst cas to xamin th prformanc of MCE. Figur 7 compars th non-dimnsional vlocity and prssur distributions along th nozzl with th analytical solutions. Excllnt agrmnt is obsrvd. Fig. 7. Vlocity and prssur fild in a 1D convrging-divrging nozzl ( p 1 ). 4.2 Comprssibl flow For comprssibl flow, prformanc of MCE is xamind in thr diffrnt tst problms. Stady subsonic comprssibl flow in convrgnt-divrgnt nozzl, stady comprssibl flow with normal shock in convrgnt-divrgnt nozzl and unstady comprssibl flow with normal shock and xpansion wavs in a shock tub Stady subsonic comprssibl flow in a convrgnt-divrgnt nozzl Rsults of solving subsonic comprssibl flow in a symmtric convrgnt-divrgnt nozzl with aspct ratio 2 using 51 nods is shown in this sction. For th boundary condition implmntation, mass flow rat and th static tmpratur ar spcifid at th inlt and th

15 Us of ropr Closur Equations in Finit Volum Discrtization Schms 75 static prssur is providd at th outlt. In ordr to assss th accuracy of th proposd mthod, numrical solutions corrsponding to inlt Mach numbrs of 0.05, 0.1, 0.15, 0.2, and 0.25 ar prsntd in Fig. 8. Flow with th inlt Mach numbr of , for which th nozzl is chokd, is also considrd. In this tst cas, p 1 is usd and prssur, tmpratur, and dnsity ar nondimnsionalizd with th outlt prssur, inlt tmpratur, and inlt dnsity, rspctivly Stady flow with shock wavs In th divrgnt sction of th prvious tst cas, normal shock wavs appar for crtain inlt boundary conditions whil th position and th strngth of th shock can b controlld by rgulating th back prssur. Figur 9 prsnts th Mach numbr, non-dimnsional prssur, tmpratur and dnsity distributions along th nozzl for th prssur ratios ( outlt inlt ) of 0.65, 0.75 and 0.85 using 81 nods. It is sn that th normal shock is wll capturd using a fw nods in all cass and no numrical oscillation is dvlopd along th nozzl. Not that this rsults ar computd using m 2 in Eq.(43) and p taks valus lowr than 1 in th suprsonic parts of th solution domain and valus highr than 1 in th subsonic rgions. Fig. 8. Stady subsonic comprssibl flow in a convrgnt-divrgnt nozzl.

16 76 Finit Volum Mthod owrful Mans of Enginring Dsign Unstady comprssibl flow in a shock tub As th final unstady 1D tst cas with both comprssion and xpansion wavs, shock tub problm is discussd. Th gomtry shown in Fig.10 is a constant ara duct dividd into two rgions sparatd by a diaphragm at th middl of th duct. At 0 th gas tmpratur is 298 K, th prssur at th right sid of th mmbran is 100 Ka and th prssur at th lft sid of th mmbran is 1000 Ka. Th tim stp for th marching numrical solution is 6 qual to 4 10 sconds. Figur 11 shows th numrically calculatd distributions of Mach numbr, non-dimnsional prssur, tmpratur, and dnsity aftr sconds. Hr, th Courant numbr is which is computd using th maximum magnitud of vlocity in th domain. Th prssur, tmpratur and dnsity ar non-dimnsionalizd using th low prssur sid valus. Good agrmnt with th xact solution is obsrvd. Fig. 9. Stady flow with shock wavs in a convrgnt-divrgnt nozzl. 5. Two-dimnsional incomprssibl flow tst cass Th MCE has also bn usd to solv many 2D bnchmark tst problms. Hr, only th rsults for two standard tst cass ar prsntd.

17 Us of ropr Closur Equations in Finit Volum Discrtization Schms 77 Fig. 10. Gomtry and th flow pattrn for th shock tub problm. Fig. 11. Rsults of unstady comprssibl flow in shock tub

18 78 Finit Volum Mthod owrful Mans of Enginring Dsign 5.1 Flow in a 2D lid-drivn cavity Flow in a lid-drivn cavity, shown in Fig. 12, is a widly usd tst cas for 2D stady incomprssibl flows govrnd by Navir-Stoks quations. Numrical solution via MCE on an unstructurd grid with triangular lmnts, shown in Fig. 13, is compard to th numrical rsults of Erturk (Erturk, 2005). Flow pattrns at diffrnt Rynolds numbrs ar shown in Fig. 14. Excllnt agrmnt btwn th calculatd vlocitis at cavity cntrlins and th bnchmark rsults in Fig. 14 is also obsrvd. Fig. 12. Lid-drivn cavity boundary conditions with an schmatic of flow faturs. Fig. 13. Gnratd unstructurd msh for lid-drivn cavity tst cas. 5.2 Flow ovr a backward facing-stp Th backward stp flow is anothr commonly usd tst cas for numrical solution algorithms. Th gomtry of this tst cas is shown in Fig. 15. Forty nods ar usd across th channl, 300 nods ar usd from th stp to 30h and 50 nods ar usd from 30h to 50h along th channl. Nods ar distributd uniformly. Figur 16 shows th calculatd stramlins at R=800, in which a rcirculation zon appars at th uppr wall. Normalizd locations of th r-attachmnt point at th lowr wall (X1), th sparation point at th uppr

19 Us of ropr Closur Equations in Finit Volum Discrtization Schms 79 wall (X2), and th r-attachmnt point at th uppr wall (X3) ar compard with th rsults obtaind by Gartling (Gartling, 1990) for flow at R = 800. Comparison rsults ar shown in Tabl 1 and xcllnt agrmnt is obvious (Erturk, 2008). A comparison has also bn mad btwn numrical rsults via MCE and th bnchmark rsults in (Gartling, 1990) and shown in Fig. 18. Again, xcllnt agrmnt is obsrvd. Similar calculations and comparisons hav also bn mad for a backward-stp flow modl with an inlt channl. Excllnt rsults, not shown hr, ar also obtaind in this tst cas. (a) Stramlins at R = (b) Vlocity rofils at R = (c) Stramlins at R = (d) Vlocity rofils at R = Fig. 14. Schmatic viw of th backward facing-stp flow problm.

20 80 Finit Volum Mthod owrful Mans of Enginring Dsign Fig. 15. Calculatd stramlins in th backward-stp flow problm at R = 800. MCE Erturk X X X Tabl 1. Comparison of normalizd X1, X2 and X3 locations and lngth of th first rcirculating rgion on th uppr wall (X3 X2) for R = 800 X3 X Fig. 16. Horizontal vlocity profils across th channl at X/h = 14 and 30 for R = Conclusion Diffrnt prssur-basd finit volum discrtization schms for th solution of fluid flow problms ar xplaind and compard in th contxt of co-locatd grid arrangmnts and a nw schm, calld th Mthod of ropr Closur Equations (MCE), is proposd. Th proposd mthod is th only discrtization schm on co-locatd grids which mploys only on dfinition for th intgration point vlocity and succssfully solvs incomprssibl as wll as comprssibl flow problms. Simpl on-dimnsional tst cass ar prsntd to discuss diffrnt availabl schms and th building blocks of MCE. Implmntation of th MCE on 2D structurd and unstructurd grids has also bn carrid out but th dtails ar not rportd hr for th sak of brvity. Numrical solutions via MCE, howvr, ar prsntd for both 1D and 2D problms which clarly show th satisfactory prformanc of th MCE in th numrical solution of fluid flow problms.

21 Us of ropr Closur Equations in Finit Volum Discrtization Schms Rfrncs Acharya, S; Baliga, B. R; Karki, K.; Murthy, J. Y.; rakash, C. & Vanka, S.. (2007). rssur- Basd Finit-Volum Mthods in Computational Fluid Dynamics. Journal of Hat Transfr, Vol. 129, pp Alinia, B. (2011). Implmntation of th Mthod of ropr Closur Equations on Two- Dimnsional Unstructurd Grids for Stady, Incomprssibl Viscous roblms, M.Sc. Thsis, Dpartmnt of Mchanical Enginring, K.N.Toosi Univrsity of Tchnology, Thran, Iran, (In rsian) Alisadghi, H. & Karimian, S. M. H. (2011a). Diffrnt Modlings of Cll-Fac Vlocitis and Thir Effcts on th rssur-vlocity Coupling, Accuracy and Convrgnc of Solution, Intrnational Journal for Numrical Mthods in Fluids, Vol. 65, pp Alisadghi, H. & Karimian, S. M. H. (2011b). Comparison of Diffrnt Solution Algorithms for Collocatd Mthod of MCIM to Calculat Stady and Unstady Incomprssibl Flows on Unstructurd Grids, Computrs and Fluids, Vol. 46, Ashrafizadh, A.; Okhovat, S.; ourbagian, M. & Raithby, G. D. (2011). A Smi-Coupld Solution Algorithm in Arodynamic Invrs Shap Dsign, Invrs roblms in Scinc and Enginring, Vol. 19, No. 4, pp , ISSN Ashrafizadh, A.; Rzvani M.; & Bakhtiari, B. (2008). A Nw Co-locatd rssur-basd Finit Volum Mthod, 16th Annual Confrnc of th CFD Socity of Canada, Saskatoon, Saskatchwan, Canada Ashrafizadh, A.; Rzvani M.; & Bakhtiari, B. (2009). rssur-vlocity Coupling On Co- Locatd Grids Using th Mthod of ropr Closur Equations, Numrical Hat Transfr; art B, Vol. 56, pp Bakhtiari, B. (2008). Dcoupling of rssur and Vlocity in th Numrical Simulation of Incomprssibl Flows: A rviw. M.Sc. Thsis, K.N.Toosi Univrsity of Tchnology, Thran, Iran, (In rsian) Chtrntal, I. (1999). Two-Dimnsional Eulr Equations Solvr, M.Sc. Thsis, Graduat Dpartmnt of Arospac Scinc and Enginring, Univrsity of Toronto, Toronto, Canada Darbandi, M. & Schnidr, G. E. (1997). Momntum Variabl rocdur for Solving Comprssibl and Incomprssibl Flows. AIAA Journal, Vol. 35, pp Darbandi, M. & Mokarizadh, V. (2004). A Modifid rssur-basd Algorithm to Solv Flow Filds With Shock and Expansion Wavs, Numrical Hat Transfr, art B, Vol. 46, pp Darbandi, M.; Roohi, E. & Mokarizadh, V. (2007). Concptual Linarization of Eulr Govrning Equations to Solv High Spd Comprssibl Flow Using a rssur- Basd Mthod, Numrical Mthods for artial Diffrntial Equations, DOI , pp Erturk, E.; Cork, T. C. & Gökçöl, C. (2005). Numrical Solutions of 2-D Stady Incomprssibl Drivn Cavity Flow at High Rynolds Numbrs, Int. J. Numr. Mth. Fluids, Vol. 48, pp Erturk, E. (2008). Numrical Solutions of 2-D Stady Incomprssibl Flow Ovr a Backward Facing Stp, art I: High Rynolds Numbr Solutions, Int. J. Computrs & Fluids, Vol. 37, pp Gartling, D. K. (1990). A Tst roblm for Outflow Boundary Conditions Flow Ovr a Backward Facing Stp, Int. J. Numr. Mth. Fluids, Vol. 11, pp

22 82 Finit Volum Mthod owrful Mans of Enginring Dsign Karimian, S. M. H. & Schnidr, G. E. (1994a). Application of a Control-Volum-Basd Finit-Elmnt Formulation to th Shock Tub roblm, AIAA Journal, Vol.33 No.1, pp Karimian, S. M. H. & Schnidr, G. E. (1994b). rssur-basd Computational Mthod for Comprssibl and Incomprssibl Flows, Journal of Thrmophysics and Hat Transfr, Vol.8, No.2, pp atankar, S. V. (1980). Numrical Hat Transfr and Fluid Flow, Hmisphr ublishing Corporation, Washington, D. C. Rzvani, M. (2008). A Nw rssur-basd Computational Mthod for Solving Fluid flow at all spds, M.Sc. Thsis, Dpartmnt of Mchanical Enginring, K.N.Toosi Univrsity of Tchnology, Thran, Iran, (In rsian) Rzvani, M. & Ashrafizadh, A. (2010). Numrical Simulation of th Intr-Equation Couplings in All-Spd Flows via th Mthod of ropr Closur Equations, Numrical Hat Transfr; art A, Vol. 58, pp Rhi, C.M. & Chow, W. L. (1983). A Numrical Study of th Turbulnt Flow ast an Isolatd airfoil With Trailing Edg Sparation. AIAA Journal, Vol. 21, pp Schnidr, G. E. & Raw, M. J. (1987a). Control Volum Finit-Elmnt Mthod For hat Transfr and Fluid Flow Using Colocatd Variabls-1. Computational rocdur. Numrical Hat Transfr, part B, Vol. 11, pp Schnidr, G. E. & Raw, M. J. (1987b). Control Volum Finit-Elmnt Mthod For hat Transfr and Fluid Flow Using Colocatd Variabls-2. Application and Validation. Numrical Hat Transfr, part B, Vol. 11, pp

23 Finit Volum Mthod - owrful Mans of Enginring Dsign Editd by hd. Radostina trova ISBN Hard covr, 370 pags ublishr InTch ublishd onlin 28, March, 2012 ublishd in print dition March, 2012 W hop that among ths chaptrs you will find a topic which will rais your intrst and ngag you to furthr invstigat a problm and build on th prsntd work. This book could srv ithr as a txtbook or as a practical guid. It includs a wid varity of concpts in FVM, rsult of th fforts of scintists from all ovr th world. Howvr, just to hlp you, all book chaptrs ar systmizd in thr gnral groups: Nw tchniqus and algorithms in FVM; Solution of particular problms through FVM and Application of FVM in mdicin and nginring. This book is for vryon who wants to grow, to improv and to invstigat. How to rfrnc In ordr to corrctly rfrnc this scholarly work, fl fr to copy and past th following: A. Ashrafizadh, M. Rzvani and B. Alinia (2012). Us of ropr Closur Equations in Finit Volum Discrtization Schms, Finit Volum Mthod - owrful Mans of Enginring Dsign, hd. Radostina trova (Ed.), ISBN: , InTch, Availabl from: InTch Europ Univrsity Campus ST Ri Slavka Krautzka 83/A Rijka, Croatia hon: +385 (51) Fax: +385 (51) InTch China Unit 405, Offic Block, Hotl Equatorial Shanghai No.65, Yan An Road (Wst), Shanghai, , China hon: Fax: