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Journal of Computatonal Physcs 230 (2011) 5328 5352 Contents lsts avalable at ScenceDrect Journal of Computatonal Physcs journal homepage: www.elsever.com/locate/jcp Realzable hgh-order fnte-volume schemes for quadrature-based moment methods V. Vkas a,.j. Wang a,, A. Passalacqua b, R.O. Fox b a Department of Aerospace Engneerng, 2271 Howe Hall, Iowa State Unversty, Ames, IA 50011, USA b Department of Chemcal and Bologcal Engneerng, 2114 Sweeney Hall, Iowa State Unversty, Ames, IA 50011, USA artcle nfo abstract Artcle hstory: Receved 12 May 2010 Receved n revsed form 5 February 2011 Accepted 21 March 2011 Avalable onlne 27 March 2011 Keywords: Knetc equaton Quadrature method of moments Number densty functon Realzabllty Fnte-volume scheme Unstructured grds Dlute gas partcle flows can be descrbed by a knetc equaton contanng terms for spatal transport, gravty, flud drag and partcle partcle collsons. However, drect numercal soluton of knetc equatons s often nfeasble because of the large number of ndependent varables. An alternatve s to reformulate the problem n terms of the moments of the velocty dstrbuton. Recently, a quadrature-based moment method was derved for approxmatng solutons to knetc equatons. The success of the new method s based on a moment-nverson algorthm that s used to calculate non-negatve weghts and abscssas from the moments. The moment-nverson algorthm does not work f the moments are non-realzable, whch mght lead to negatve weghts. It has been recently shown [14] that realzablty s guaranteed only wth the 1st-order fnte-volume scheme that has an nherent problem of excessve numercal dffuson. The use of hgh-order fnte-volume schemes may lead to non-realzable moments. In the present work, realzablty of the fnte-volume schemes n both space and tme s dscussed for the 1st tme. A generalzed dea for developng realzable hgh-order fnte-volume schemes for quadrature-based moment methods s presented. These fnte-volume schemes gve remarkable mprovement n the solutons for a certan class of problems. It s also shown that the standard Runge Kutta tme-ntegraton schemes do not guarantee realzablty. However, realzablty can be guaranteed f strong stablty-preservng (SSP) Runge Kutta schemes are used. Numercal results are presented on both Cartesan and trangular meshes. Ó 2011 Elsever Inc. All rghts reserved. 1. Introducton Knetc equatons occur n mesoscopc models for many physcal phenomena, such as rarefed gases [7,10,11,24,35,47], plasmas [8,29,55], multphase flows [14,44,48,54], optcs [3,15,21,42,43], and quantum physcs [22,25,26], to name just a few. In ths work, we wll use the knetc equaton descrbng dlute gas partcle flows as an example applcaton. However, the proposed numercal schemes can easly be extended to treat a wde range of knetc equatons descrbng other applcatons. Gas partcle flows occur n many engneerng and natural systems such as fludzed-bed reactors, catalytc crackers, volcanc ash transport n the atmosphere, and helcopter brown-out. Currently, there exst several dfferent approaches for smulatng the knetc equaton descrbng the partcle phase and ts couplng to the gas phase. In general, all approaches use the same type of flow solver for the gas phase, but they dffer n the way n whch the knetc equaton s treated: Correspondng author. Tel.: +1 515 294 1614; fax: +1 515 294 3262. E-mal addresses: vvkas@astate.edu (V. Vkas), zjw@astate.edu (.J. Wang), albertop@astate.edu (A. Passalacqua), rofox@astate.edu (R.O. Fox). 0021-9991/$ - see front matter Ó 2011 Elsever Inc. All rghts reserved. do:10.1016/j.jcp.2011.03.038
V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 5329 () drect solver that dscretzes the velocty phase space of the partcle number densty functon [4,36], () Lagrangan solver that tracks all the partcles ndvdually [6], () hydrodynamc models wth knetc-theory moment closures [16], (v) quadrature method of moments (QMOM) that solves for moments of the partcle number densty functon wth quadrature-based closures [14,17,31,37]. In many applcatons, the drect soluton of the knetc equaton s prohbtvely expensve due to the hgh dmensonalty of the space of ndependent varables, whle Lagrangan solvers are computatonally very expensve, snce the number of partcles to be tracked s very large. On the other hand, hydrodynamc models are developed assumng that the Knudsen number of the partcle phase s very small, whch s equvalent to assumng a Maxwellan (or nearly Maxwellan) equlbrum partcle velocty dstrbuton. Ths, however, s not correct n relatvely dlute gas partcle flows, where the Knudsen number can be hgh, the collson frequency s small and phenomena lke partcle trajectory crossng can occur. In partcular, Desjardns et al. [14] showed that the assumpton that a gas partcle flow can be descrbed accountng for only the mean momentum of the partcle phase leads to ncorrect predctons of all the velocty moments, ncludng the partcle number densty, showng the need of usng a mult-velocty method, n order to correctly capture the physcs of the flow. Smlar observatons were made for moderately collsonal gas partcle flows by Sakz and Smonn [44]. QMOM for gas partcle flow [17 19] s based on the dea of trackng a set of velocty moments of arbtrarly hgh order, provdng closures to the source terms and the moment spatal fluxes n the moment transport equatons by means of a quadrature approxmaton of the number densty functon. The key step of the approach s an nverson algorthm that allows one to unquely determne a set of weghts and abscssas from the set of transported moments. The condton for the nverson algorthm to be appled s that the set of moments s realzable, meanng t actually corresponds to a velocty dstrbuton. Ths condton s not generally satsfed by the tradtonal fnte-volume methods used n computatonal flud dynamcs. Desjardns et al. [14] recently showed that realzablty s guaranteed only wth the 1st-order fnte-volume scheme. The use of any other hgh-order fnte-volume scheme may lead to non-realzable moments, thereby resultng n negatve weghts. Weghts are representatve of partcle densty and hence cannot be negatve. Ths lmtaton n turn leads to the use of a hghly refned mesh for computaton as a 1st-order fnte-volume scheme produces hghly dffused solutons on a coarse mesh. For ths reason, mproved fnte-volume schemes are sought that could provde less-dffused solutons and smultaneously guarantee the realzablty of moments. In the present work, a generalzed dea for developng mproved fnte-volume schemes for quadrature-based moment methods s presented. The realzablty of the new mproved fnte-volume schemes s guaranteed under sutable realzablty crtera. These new schemes gve remarkable mprovement n the solutons for the class of problems where the velocty abscssas are constant over a range of cells. The present work also shows that the standard Runge Kutta tme-ntegraton schemes do not guarantee realzablty. However, the realzablty can be guaranteed f strong stablty-preservng (SSP) Runge Kutta schemes [23] are used. The remander of ths paper s organzed as follows. In Secton 2, QMOM s revewed. Secton 2 also dscusses fnte-volume schemes and ther realzablty propertes. Thereafter, n Secton 3, new realzable hgh-order fnte-volume schemes are presented. In Secton 4, mult-stage tme-ntegraton s dscussed and realzablty propertes of the standard RK2 and RK2SSP schemes are presented. Secton 5 presents some numercal results ncludng accuracy studes. Conclusons from the present study are summarzed n Secton 6. Fnally, n Appendx A we present an extenson of the realzable schemes to velocty-ndependent densty functons. 2. Quadrature method of moments 2.1. Knetc equaton Dlute gas partcle flows can be modeled by a knetc equaton [9,10,46] of the form: @ t f þ v @ x f þ @ v ðf FÞ ¼C; ð1þ where f(v,x,t) s the velocty-based number densty functon, v s the partcle velocty, F s the force actng on an ndvdual partcle, and C s the collson term representng the rate of change n the number densty functon due to partcle partcle collsons. The collson term can be descrbed usng the Bhatnagar Gross Krook (BGK) collson operator [5]: C ¼ 1 s ðf eq f Þ; ð2þ where s s the characterstc collson tme, and f eq s the Maxwellan equlbrum number densty functon gven by M 0 f eq ðvþ ¼q jv! exp U pj 2 ; ð3þ ffffffffffffffffffffffffffffffffffffff ð2pr eq Þ 3 2r eq
5330 V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 n whch U p s the mean partcle velocty, r eq s the equlbrum varance and M 0 ¼ R fdv s the partcle number densty. In gas partcle flows, the force term s gven by the sum of the gravtatonal contrbuton (F g ) and the drag term (F d ) exerted from the flud on the partcles: F ¼ F g þ F d : ð4þ For dlute gas partcle flows, the drag force on a partcle can be approxmated by F d ¼ 3m pq g 4q p d p C d ju r ju r ; where U r = U g U p s the relatve velocty between two phases, U g s the gas velocty, U p s the partcle phase local mean velocty, q g and q p are gas and partcle denstes, respectvely, and d p s the partcle dameter. The drag coeffcent C d s gven by the Schller Naumann correlaton [45]: C d ¼ 24 1 þ 0:15Re 0:687 p ; ð6þ Re p n whch Re p = q g d p ju g U p j/l g,l g beng the dynamc vscosty of gas phase. 2.2. Moment transport equatons In the quadrature-based moment method of Fox [17 19], a set of moments of the number densty functon f s transported and ts evoluton n space and tme s tracked. Each element of the moment set s defned through ntegrals of the number densty functon. For the frst few moments the defnng ntegrals are: M 0 ¼ fdv; M 1 ¼ v fdv; M 2 j ¼ v v j fdv; M 3 jk ¼ v v j v k fdv: ð7þ In these equatons, the superscrpt of M represents the order of correspondng moment. Moment transport equatons are obtaned by applyng the defnton of moments to (1). The transport equatons for moments n (7) can be wrtten as: @M 0 @t þ @M1 @x ¼ 0; ð5þ @M 1 @t @M 2 j @t @M 3 jk @t þ @M2 j @x j ¼ g M 0 þ D 1 ; þ @M 3 jk ¼ g @x k M 1 j þ g j M 1 þ C 2 j þ D2 j ; þ @M 4 jkl ¼ g @x l M 2 jk þ g jm 2 k þ g km 2 j þ C3 jk þ D3 jk : In (8), g, g j, g k are the components of gravty, D 1 ; D2 j ; D3 jk are due to the drag force and C2 j ; C3 jk are due to collsons. ð8þ 2.3. Quadrature-based closures Usng the BGK model, the collson terms n (8) can be closed. However, the set of transport equatons n (8) s stll unclosed because of the spatal flux and drag terms. Each equaton contans the spatal fluxes of the moments of order mmedately hgher. In quadrature-based moment methods, quadrature formulas are used to provde closures to these terms n the moment transport equatons, by ntroducng a set of weghts and abscssas. The number densty functon f s wrtten n terms of the quadrature weghts (n a ) and abscssas (U a ) usng a Drac delta representaton: f ðvþ ¼ Xb n a dðv U a Þ: The method based on (9) s called b-node quadrature method. The moments can be computed as a functon of quadrature weghts and abscssas by usng the above defnton of f n (7): ð9þ M 0 ¼ Xb n a ; M 1 ¼ Xb n a U a ; M 2 j ¼ Xb n a U a U ja ; The source terms due to drag and gravty are computed as: D 1 ¼ Pb D 2 j ¼ Pb D 3 jk ¼ Pb n a m p F a ; na m p n a m p F a U ja þ F ja U a ; F a U ja U ka þ F ja U ka U a þ F ka U a U ja : M 3 jk ¼ Xb n a U a U ja U ka : ð10þ ð11þ
V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 5331 The detals of the computaton of the force terms F a, F ja and F ka, can be found n [17]. In order to ensure that the moments reman realzable and that the dscretzed fluxes are hyperbolc, the spatal flux terms are closed accordng to ther knetc descrpton [13,14,38,41]. Each moment nvolved n the spatal dervatve s decomposed nto two contrbutons, as shown n (12) for the zero-order moments: M 1 ¼ 0 v 1 fdv j dv k dv þ þ1 v 0 fdv j dv k dv : In a 1-D case, the left ntegral wll account for partcles gong from left to rght and the rght ntegral wll account for partcles gong from rght to left at the face. Usng (9), (12) can be wrtten as: M 1 ¼ Xb n a mnð0; U a Þþ Xb n a maxð0; U a Þ: For a 1-D case, the left ntegral/summaton n (12)/(13) s evaluated usng the values on left sde of the face and rght ntegral/summaton s evaluated usng the values on rght sde of the face. In order to solve the abovementoned moment transport equatons, boundary condtons are needed. These boundary condtons can be specfed ether n terms of moments or n terms of the weghts and abscssas. The latter approach s more convenent and s followed n the present work. In ths work, three types of boundary condtons are used: Drchlet, perodc and wall-reflectve. At a Drchlet boundary, the weghts and abscssas are specfed. Perodc boundary condtons copy the weghts and abscssas from the outgong perodc boundary cell to the correspondng ncomng perodc boundary cell. The boundary condtons at the walls are set so that a partcle that colldes wth the wall s specularly reflected. Ths condton corresponds to changng the sgn of the velocty component of the partcle along the drecton perpendcular to the wall. The mplementaton of ths boundary condton n the quadrature-based algorthm s done by changng the sgn of the abscssas n the approprate drecton [37]. If = 0 ndcates the poston of the wall, perpendcular to the second drecton of reference frame, and = 1 ndcates the neghborng computatonal cell, the boundary condton can be wrtten as: 0 1 0 1 n a n a =e w U a U a B C ¼ B C ; ð14þ @ V a A @ e w V a A W a ¼0 W a ¼1 where e w s the partcle wall resttuton coeffcent. All the boundary condtons are appled usng a ghost-cell approach. ð12þ ð13þ 2.4. Fnte-volume method The moment transport equatons n (8) contan convecton, drag and collson terms. The three terms are treated separately usng an operator-splttng technque. The soluton algorthm nvolvng all the terms s gven later. The collson and the force terms do not create non-realzablty problems. Hence for all the analyss, these terms are dropped. For smplcty, a one-dmensonal case wth two quadrature nodes s dscussed here. A general 3-D case s presented n a later secton. For the 1-D case, the set of moment transport equatons after droppng collson and drag terms can be wrtten as: where @W @t þ @HðWÞ ¼ 0; @x W ¼½M 0 M 1 M 2 M 3 Š T and HðWÞ ¼½M 1 M 2 M 3 M 4 Š T : ð16þ For a 2-node quadrature there are two weghts (n 1,n 2 ) and two abscssas (U 1,U 2 ). Let the set of weghts and abscssas be denoted as N =[n 1 n 2 U 1 U 2 ] T. The frst four moments can be wrtten n terms of these weghts and abscssas as: ð15þ M 0 ¼ n 1 ðu 1 Þ 0 þ n 2 ðu 2 Þ 0 ; M 1 ¼ n 1 ðu 1 Þ 1 þ n 2 ðu 2 Þ 1 ; M 2 ¼ n 1 ðu 1 Þ 2 þ n 2 ðu 2 Þ 2 ; M 3 ¼ n 1 ðu 1 Þ 3 þ n 2 ðu 2 Þ 3 : In the frst two equatons for M 0 and M 1, powers of U 1 and U 2 are redundant. The conserved moments and moment fluxes n (16) can be wrtten n terms of the number densty functon: W ¼ KðvÞf ðvþdv; HðWÞ ¼ vkðvþf ðvþ dv; ð17þ ð18þ where KðvÞ ¼½1 vv 2 v 3 Š T : ð19þ
5332 V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 The moments n (15) can be advanced n tme usng a fnte-volume scheme. If a sngle-stage explct tme-ntegraton scheme s used, the updated set of moments can be wrtten as: h W nþ1 ¼ W n Dt Dx G Wn þ1=2;l ; Wn þ1=2;r G W n 1=2;l ; Wn 1=2;r ; ð20þ where superscrpts n and n + 1 denote tme levels, G s the numercal flux functon evaluated at cell nterfaces, and l and r denote the left and rght states at the nterfaces respectvely. Henceforth, the varables wth subscrpt wll denote the cell-averaged values and the ones wth ( + 1/2) or ( 1/2) as subscrpt wll denote the reconstructed values at the nterfaces. G s defned as: where GðW l ; W r Þ¼ v þ Kf l dv þ v Kf r dv; v þ ¼ 1 ðv 2 þjvjþ and v ¼ 1 ðv jvjþ: ð22þ 2 Ths corresponds to a splttng between partcles gong from left to rght (frst term) and partcles gong from rght to left (second term). Insertng the expresson for f n (21) yelds: ð21þ GðW l ; W r Þ¼H þ ðw l ÞþH ðw r Þ; ð23þ wth 0 1 0 1 1 1 U H þ 1l ðw l Þ¼n 1l maxðu 1l ; 0ÞB C @ A þ n U 2l 2l maxðu 2l ; 0ÞB C @ A ; U 2 1l U 2 2l U 3 1l 0 1 0 1 1 1 U H 1r ðw r Þ¼n 1r mnðu 1r ; 0ÞB C @ A þ n U 2r 2r mnðu 2r ; 0ÞB C @ A : U 2 1r U 3 2l U 2 2r ð24þ U 3 1r U 3 2r Before dscussng the 1st-, 2nd- and 3rd-order fnte-volume schemes, t s worth notcng that, n order to calculate the fluxes at the nterface, varables need to be reconstructed at the faces of each cell. However, there s an ambguty n the choce of varables to be chosen for reconstructon. Two choces are possble: () reconstructng the moments (H(W)=[M 1 M 2 M 3 M 4 ] T ) and () reconstructng the weghts and abscssas (N =[n 1 n 2 U 1 U 2 ] T ). For the 1st-order fnte-volume scheme both choces are equvalent. However, for second or any hgh-order fnte-volume schemes, these two choces are n general dfferent. The former approach often leads to non-realzable moments, especally n the problems nvolvng dscontnuous velocty felds. For ths reason, the second approach s adopted n ths work. In the sectons below, any reconstructon essentally refers to reconstructon of the weghts and abscssas. For example, W +1/2,l refers to moments at the left sde of nterface + 1/2 calculated usng the reconstructed values of the weghts and abscssas. 2.4.1. Frst-order fnte-volume scheme The 1st-order fnte-volume scheme for solvng moment transport equatons uses a pecewse constant approxmaton and s descrbed n [14]. The weghts and abscssas are assumed to be constant over a cell: N n 1=2;r ¼ Nn ; Nn þ1=2;l ¼ Nn : ð25þ 2.4.2. Second-order fnte-volume scheme In the 2nd-order fnte-volume scheme, a pecewse lnear reconstructon for the weghts and abscssas s used. The pecewse lnear reconstructon s obtaned usng a mnmod slope lmter [27]. For the th cell, ths can be wrtten as: N n 1=2;r ¼ Nn Dx 2 @N ; N n þ1=2;l ¼ Nn þ Dx 2 @N ; ð26þ where @N ¼ mnmod Nn N n 1 ; Nn þ1 Nn : ð27þ Dx Dx The mnmod functon s defned as: mnmodðx; yþ ¼sgnðxÞ 1 þ sgnðxyþ mnðjxj; jyjþ: 2 ð28þ
V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 5333 2.4.3. Thrd-order fnte-volume scheme In the 3rd-order fnte-volume scheme, a pecewse parabolc reconstructon for the weghts and abscssas s used. For the th cell, the reconstructed weghts and abscssas can be wrtten as: N n 1=2;r ¼ Nn 1 3 Nn N n 1 þ 1 6 Nn þ1 Nn ; N n þ1=2;l ¼ Nn þ 1 6 Nn N n 1 þ 1 3 Nn þ1 ð29þ Nn : The 3rd-order MUSCL reconstructon wthout a lmter s the same as (29). However, n the case of a dscontnuous soluton a lmter s essental. A lmted verson of 3rd-order MUSCL can be found n [27]. 2.5. Soluton algorthm The fnte-volume schemes presented n the above secton are used to solve for the spatal transport part of the moment transport equatons. However, the complete moment transport equatons have collson and force terms as well. As mentoned earler, these two terms are treated usng an operator-splttng technque. A detaled soluton algorthm nvolvng all the terms can be found n [14,18,37]. Here a bref overvew of the steps nvolved n the soluton procedure s presented, assumng a sngle-stage explct tme-ntegraton: 1. Intalze weghts and abscssas n the doman. 2. Apply boundary condtons to weghts and abscssas. 3. Compute moments usng weghts and abscssas. 4. Compute tme-step sze Dt. 5. Reconstruct weghts and abscssas at cell faces. 6. Compute spatal flux terms at cell faces. 7. Advance moments by Dt due to spatal flux terms usng a fnte-volume approach. 8. Compute weghts and abscssas from moments usng the moment-nverson algorthm. 9. Advance weghts by Dt due to force terms (drag and gravty). 10. Compute moments usng weghts and abscssas. 11. Advance moments by Dt due to collson terms. 12. Compute weghts and abscssas from moments usng the moment-nverson algorthm. 13. Apply boundary condtons to weghts and abscssas. 14. Repeat steps (4) through (l3) at each tme step. 2.6. Non-realzablty problem At each tme step, the weghts and abscssas need to be recovered from the moments. The moment-nverson algorthm computes the set of weghts and abscssas from the correspondng set of moments by solvng a set of nonlnear equatons. In the moment-nverson algorthm M 0, M 1, M 2, M 3 are known and n 1, n 2, U 1, U 2 are computed, by solvng (17) n reverse drecton usng the product-dfference (PD) algorthm [32,53,40]. However, the set of moments cannot be consttuted by arbtrary values of each moment, but they have to conform to the defnton of the non-negatve number densty functon. The applcaton of the PD algorthm to a set of realzable moments leads to a set of weghts and abscssas that satsfy the propertes of Gaussan quadrature. In partcular the weghts are always postve. Durng the development of the numercal schemes, whch s the object of ths work, a set of weghts and abscssas wll be sad to represent realzable moments f weghts are postve and abscssas le n the nteror of the support of f. Because of the non-lnearty of the nverson problem, t s extremely dffcult to determne n advance whether a gven set of moments s realzable. However, Desjardns et al. [14] descrbed how any fnte-volume scheme that could guarantee non-negatvty of the effectve number densty functon wll always keep the moments n realzable space. The updated set of moments can be wrtten as: W nþ1 ¼ Khdv; ð30þ where h ¼ f n k v þ f n þ1=2;l þ v f n þ1=2;r v þ f n 1=2;l v f n 1=2;r ¼ f n kv þ f n þ1=2;l kv f n þ1=2;r þ kv þ f n 1=2;l þ kv f n 1=2;r ð31þ n whch k = Dt/Dx. In(31), h s the effectve number densty functon and has dfferent forms for dfferent fnte-volume schemes as the nterface values are reconstructed n dfferent ways. Desjardns et al. [14] stated that any fnte-volume scheme that guarantees the non-negatvty of h for all v, s realzable. Usng ths proposton, they derved the realzablty crteron for the 1st-order fnte-volume scheme. In the sectons below, the realzablty of the 1-D verson of 1st-, 2nd- and hgh-order fnte-volume schemes s dscussed.
5334 V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 2.6.1. Frst-order fnte-volume scheme For the 1st-order fnte-volume scheme (25) mples f n 1=2;r ¼ f n ; f n þ1=2;l ¼ f n : Usng ths n (31), the effectve number densty functon for the 1st-order fnte-volume scheme can be wrtten as: h ¼ð1 kjvjþf n þ kv þ f n 1 kv f n þ1 : As the moments at tme level n are assumed to be realzable, the non-negatvty of the number densty functon at tme level n s guaranteed,.e., f n P 0; f n 1 P 0 and f n þ1 P 0 for all v. Also, v+ P 0 and v 6 0. Hence, the non-negatvty of h wll be guaranteed f (1 kjvj) P 0. When wrtten n terms of the abscssas, ths condton becomes: 1 k 6 max ; U n : ð34þ U n 1 2 Ths s the realzablty crteron for the 1st-order fnte-volume scheme [14]. 2.6.2. Second- and hgh-order fnte-volume schemes For the 2nd-order and, n general, any hgh-order fnte-volume scheme, the reconstructed values at the cell nterfaces are dfferent from the cell-averaged values. The equalty of reconstructed values and cell-averaged values s a specal property that holds only for the 1st-order fnte-volume scheme. For 2nd and hgh-order fnte-volume schemes, the effectve number densty functon h can be obtaned from (31) by puttng n the reconstructed values. An nterestng thng to notce n (31) s that on the rght-hand sde, out of the fve terms, only three are non-negatve. The second and ffth terms are always nonpostve. As was stated n [14], realzablty can only be guaranteed f the effectve number densty functon s non-negatve for all veloctes. Clearly ths does not hold, n general, for 2nd and hgh-order fnte-volume schemes. Hence, realzablty cannot be guaranteed for any fnte-volume scheme other than 1st-order. ð32þ ð33þ 3. Improved realzable fnte-volume schemes 3.1. Basc dea As was stated earler, the two non-postve terms on the rght-hand sde of (31) mght lead to non-realzablty problems. The non-negatve terms present a physcal vew of the number densty functon. An mportant thng to notce s that, despte the presence of the two non-postve terms, the 1st-order fnte-volume scheme s stll realzable under the restrcton of a realzablty crteron. Ths s possble because n the 1st-order fnte-volume scheme, the nterface values are the same as the cell-averaged values f n ¼ f n þ1=2;l ¼ f n 1=2;r, thereby allowng a groupng of the frst term wth the two non-postve terms on rght-hand sde of (31). Ths dea of groupng the terms s essental to the realzablty of the 1st-order fnte-volume scheme and t also forms the bass of the development of new mproved realzable fnte-volume schemes. For hgh-order fnte-volume schemes, n general: N n N n þ1=2;l Nn 1=2;r ; ð35þ.e. n n ;a nn þ1=2;a;l nn 1=2;a;r and U n ;a Un þ1=2;a;l Un 1=2;a;r : ð36þ Consder a specal reconstructon where n n ;a nn þ1=2;a;l nn 1=2;a;r and U n ;a ¼ Un þ1=2;a;l ¼ Un 1=2;a;r : ð37þ For ths specal reconstructon, realzablty can always be guaranteed wth a sutable constrant on the tme-step sze. Ths s the subject of the followng theorem. Theorem 1. Let b; p 2 N and a 2 {1,2,...,b}. Also let the cell-averaged and reconstructed values of the weghts satsfy n n ;a > 0 and n n þ1=2;a;l ; nn 1=2;a;r P 0 8a. If a fnte-volume scheme usng a sngle-stage Euler explct tme-ntegraton scheme s devsed that uses a pth-order reconstructon for weghts and 1st-order reconstructon for abscssas, the non-negatvty of the effectve number densty functon (31) n the th cell can always be guaranteed under an explct constrant on tme-step sze ðdt 2 R þ Þ. Proof. Usng (31), the effectve number densty functon, regardless of the fnte-volume scheme used, can be wrtten as: h ¼ f n kv þ f n þ1=2;l þ kv f n 1=2;r þ nþ : In the above expresson, the frst non-negatve term and the two non-postve terms have been represented explctly. The other two terms are always non-negatve and have been grouped under n +. For a b-node quadrature, usng (9), the expresson for h becomes: ð38þ
h ¼ Xb h n n ;a d v Un ;a kv þ n n þ1=2;a;l d v Un þ1=2;a;l þ kv n n 1=2;a;r d v Un 1=2;a;r þ n þ : If a 1st-order reconstructon s used for the abscssas, then the nterface values of the abscssas wll be the same as the cellaveraged values: U n ;a ¼ Un þ1=2;a;l ¼ Un 1=2;a;r : V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 5335 ð39þ ð40þ Puttng ths n (39), the effectve number densty functon becomes: h ¼ Xb n o n n ;a kvþ n n þ1=2;a;l þ kv n n 1=2;a;r d v U n ;a þ n þ : ð41þ For Dt 2 R þ satsfyng the condton: 0 n n ;a k ¼ mn a2f1;2;...;bg @ A; ð42þ n n max þ1=2;a;l Un ;a ; 0 n n mn 1=2;a;r Un ;a ; 0 1 h s non-negatve for all v. Ths concludes the proof. h Clearly, Theorem 1 guarantees the realzablty of the specal reconstructon n (37), for all cases except for the one where n n ;a ¼ 0. However ths case turns out to be trval f a mnmod lmter s used to lmt the reconstructed values. A mnmod lmter guarantees that whenever n n ;a ¼ 0, then nn þ1=2;a;l ¼ nn 1=2;a;r = 0, thereby automatcally droppng the two non-postve terms n (39). For p = 1,(42) reduces to the same realzablty crteron as n (34): 0 1 B 1 k ¼ mn a2f1;2;...;bg C @ A: ð43þ U n ;a Ths new reconstructon uses a hgh-order reconstructon for the weghts, but a 1st-order reconstructon for the abscssas. To remove the ambguty, t s worth clarfyng that all the fnte-volume schemes dscussed earler, n whch the same order of reconstructon was used for the weghts and abscssas wll be termed standard fnte-volume schemes. For example, a standard 2nd-order fnte-volume scheme s the one dscussed n Secton 2.4.2, where a 2nd-order reconstructon s used for both the weghts and abscssas. More generally, a standard pth-order fnte-volume scheme uses a pth-order reconstructon for both the weghts and abscssas. It s worth reteratng that the realzablty of the standard pth-order fnte-volume scheme s not guaranteed. Correspondng to the standard pth-order fnte-volume scheme another scheme can be developed based on the specal reconstructon dscussed above. Ths new scheme wll use pth-order reconstructon for the weghts and 1st-order reconstructon for the abscssas. Realzablty of ths new scheme s guaranteed by the constrant, also known as realzablty crteron, n (42). Because a 1st-order reconstructon s used for the abscssas, the new scheme s less accurate than the standard pth-order fnte-volume scheme and henceforth wll be termed a quas-pth-order fnte-volume scheme. The standard 1st-order fnte-volume scheme and the quas-1st-order fnte-volume scheme are the same and wll be smply referred to as the 1st-order fnte-volume scheme. Two mportant facts about the new schemes are: () they are better than 1st-order fnte-volume schemes as far as accuracy s concerned and () realzablty s guaranteed under the realzablty crteron that has an explct form. Ths marks a sgnfcant mprovement n soluton methods for quadrature-based moment methods. Over the years, many hgh-order fnte-volume schemes have been developed [1,2,12,28,49 52] for convecton-domnated problems n the feld of flud dynamcs. However, quadrature-based moment methods have not benefted from these hgh-order schemes because of the non-realzablty lmtaton. However, wth the new approach, all the already exstng knowledge about hgh-order fnte-volume schemes can now be utlzed for solutons usng quadrature-based moment methods. In general, the new quas-pth-order realzable fnte-volume scheme wll be less accurate compared to the standard pthorder fnte-volume scheme but for the problems where the veloctes are constant over a range of cells, the dfference n accuracy wll be neglgble. Ths fact s further demonstrated n Secton 5. Next the realzablty crteron for dfferent dmensons s presented, and we explan the way n whch t s appled. In Appendx A we show how quas-pth-order fnte-volume schemes can also be wrtten for the advecton of velocty-ndependent densty functons. 3.2. Realzablty crteron for 1-D cases The relaton gven n (42) s the realzablty crteron for 1-D problems. It can also be wrtten as: n o n n ;a k maxðu a; 0Þn n þ1=2;a;l þ k mnðu a; 0Þn n 1=2;a;r P 0 8a 2f1; 2;...; bg: ð44þ Ths smple realzablty crteron can be used for the calculaton of k and hence Dt for 1-D cases wth b-node quadrature.
5336 V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 3.3. Realzablty crteron for 2-D/3-D cases h In (44), maxðu a ; 0Þn n mnðu þ1=2;a;l a; 0Þn n 1=2;a;r s the total outgong flux from the th cell for the ath weght. Hence, (44) can also be wrtten as: n n n ;a k X o Outgong Flux n ;a P 0 8a 2f1; 2;...; bg; ð45þ where the summaton s over the faces of the th cell. Ths form of the realzablty crteron can be used for 2-D and 3-D cases. Although (45) was obtaned usng an analogy from (44), t can be derved drectly from the expresson for fnte-volume scheme n 2-D or 3-D. Consder a 3-D case. Let X and ox denote the th cell and ts boundary respectvely. Also let e 2 ox be a face of the th cell, A e be ts area and ^n e ¼½n e;x n e;y n e;z Š be the outward unt normal at ths face. The fnte-volume expresson for the snglestage Euler explct tme-ntegraton, analogous to (20) can be wrtten as: X n W nþ1 ¼ W n Dt jx j e2@x G W n e;l ; Wn e;r ; ^n e A e o ; ð46þ where jx j denotes the volume of the th cell and W n e;l and W n e;r represent the reconstructed value of W on the left and rght sdes of face e. The outward normal vector ^n e defned above, ponts from the left sde to the rght sde of face e. The form of W analogous to the one n (18) s: W ¼ Kðu;v; wþf ðu;v; wþdudv dw: ð47þ The numercal flux functon can be wrtten as: G W n e;l ; Wn e;r ; ^n e ¼ v þ n Kf e;l dudv dw þ v n Kf e;r dudv dw; ð48þ where v þ n ¼ maxðn e;xu þ n e;y v þ n e;z v; 0Þ; v n ¼ mnðn e;xu þ n e;y v þ n e;z v; 0Þ: Substtutng (47) and (48) n (46), the effectve number densty can be wrtten as: h ¼ f n k X v þ n f e;la e þ v n f e;ra e : ð50þ e2@x In (50), k = Dt/jX j. For b-node quadrature, f has the same form as n (9). Usng (9) and the specal reconstructon U n e;a;l ¼ Un e;a;r ¼ Un ;a ; h can be further wrtten as: ( h ¼ Xb n n ;a k X ) ( v þ n nn e;a;l A e þ v n nn e;a;r A e d v U n ;a ¼ Xb n n ;a k X ) v þ n nn e;a;l A e d v U n ;a þ n þ ; ð51þ e2@x where agan all the non-negatve terms except for the frst one have been grouped under n +. Non-negatvty of h can be guaranteed f: ( n n :a k X ) v þ n nn e;a;l A e P 0 8a 2f1; 2;...; bg: ð52þ e2@i e2@x ð49þ Ths s exactly the same as (45). It s worth reteratng that the summaton n (45) and (52) s for the outgong fluxes only. Hence, for calculatng these fluxes only the reconstructed weghts on the nteror sdes (sde towards the th cell) of faces should be used, the ones on the opposte sde should be set to zero,.e., the flux comng n from neghborng cells should not be accounted for. Consder a smple 2-D Cartesan case shown n Fg. 1. For the sake of smplcty, subscrpt a and superscrpt n wll be dropped and only 1-node quadrature wll be demonstrated. The cell n the center, X 0, has four neghbors X 1, X 2, X 3, X 4 and the reconstructed values of the weghts on nner sdes of the correspondng faces are n 01, n 02, n 03, n 04, respectvely. The cell-averaged weght for cell 0 s n 0 and the correspondng X-drecton and Y-drecton abscssas are U 0 and V 0, respectvely. The realzablty condton for X 0 can be wrtten as: fn 0 k½n 01 A 1 maxðu 0 ; 0Þþn 02 A 1 maxðv 0 ; 0Þ n 03 A 1 mnðu 0 ; 0Þ n 04 A 4 mnðv 0 ; 0ÞŠg P 0; ð53þ where A 1, A 2, A 3, A 4 are the areas of the four faces. In (53), k = Dt/jX 0 j. All the analyss to ths pont has been done for a number densty functon that depends on the velocty. However, many populaton balance models nvolve densty functons that do not depend on velocty. For such cases, each moment can be advected ndependently usng a 1st-order fnte-volume scheme wthout volatng realzablty of the moment set. On the other hand, the use of any hgh-order fnte-volume scheme may lead to unphyscal or non-realzable moment sets
V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 5337 Fg. 1. Cells wth faces algned along Cartesan axes. [33,34,39,56]. As demonstrated n Appendx A, ths problem can be avoded f QMOM s used wth the quas-pth-order fntevolume schemes descrbed above. Before movng onto the next secton, here s a bref dscusson on the Courant Fredrchs Lewy (CFL) crteron vs. the realzablty crteron. The realzablty crteron s analogous to a CFL crteron. However, ther purposes are dfferent. A CFL crteron guarantees the stablty of the soluton whle the realzablty crteron guarantees the physcal nature of the soluton. An explct form of the CFL crteron exsts only for 1-D cases, whch s equvalent to the realzablty crteron. For 2-D and 3-D problems, there s no explct form for the CFL crteron. However, there exsts an explct form for the realzablty crteron even for 2-D and 3-D cases as shown above. It also turns out from numercal experments that the realzablty crteron s a more strct constrant on Dt as compared to the CFL crteron. Also the realzablty crteron correspondng to a pth-order (p > 1) fnte-volume scheme s n general strcter than the one for the 1st-order fnte-volume scheme. 4. Mult-stage tme-ntegraton For smplcty, all the prevous sectons employed sngle-stage Euler explct tme-ntegraton. However, sngle-stage Euler explct tme-ntegraton s only 1st-order accurate n tme. To mprove tme-accuracy, mult-stage explct tme-ntegraton schemes are used n practce. Nevertheless, not all mult-stage tme-ntegraton schemes guarantee realzablty. Here, two dfferent tme-ntegraton schemes are presented: RK2 and RK2SSP [23]. The former s the standard 2nd-order two-stage Runge Kutta scheme, whle the latter s the 2nd-order two-stage strong-stablty-preservng (SSP) Runge Kutta scheme. 4.1. RK2 scheme The standard 2nd-order two-stage Runge Kutta scheme can be wrtten as: W ¼ W n Dt h 2Dx G Wn þ1=2;l ; Wn þ1=2;r G W n 1=2;l ; Wn 1=2;r ; ð54þ W nþ1 ¼ W n Dt h Dx G W þ1=2;l ; W þ1=2;r G W 1=2;l ; W 1=2;r : ð55þ 4.1.1. Frst stage The 1st stage s the same as the sngle-stage Euler explct tme-ntegraton scheme, the only dfference beng the factor of (1/2) n Dt Dt.Ifk ¼, realzablty can be guaranteed for 1st-order and quas-pth-order fnte-volume schemes subject to condton 2Dx 2Dx (42). 4.1.2. Second stage The set of moments for the th cell after the 2nd stage can be wrtten as: ¼ Kh nþ1 dv; where W nþ1 h nþ1 ¼ f n ¼ f n k v þ f þ1=2;l þ v f þ1=2;r vþ f 1=2;l v f 1=2;r ¼ f n kv þ f þ1=2;l kv f þ1=2;r þ kv þ f 1=2;l þ kv f 1=2;r kv þ f þ1=2;l þ kv f 1=2;r þ nþ ; ð56þ ð57þ
5338 V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 n whch k = Dt/Dx. For the 1st-order and quas-pth-order fnte-volume schemes: U þ1=2;a;l ¼ U 1=2;a;r ; ð58þ but n general U n ;a U þ1=2;a;l and U n ;a U 1=2;a;r : ð59þ Hence, groupng of terms s not possble and realzablty cannot be guaranteed for the 2nd stage. The RK2 scheme does not guarantee realzablty of the moment set. 4.2. RK2SSP scheme The 2nd-order two-stage strong stablty-preservng Runge Kutta scheme can be wrtten as: W ¼ W n Dt h Dx G Wn þ1=2;l ; Wn þ1=2;r W nþ1 ¼ 1 2 Wn þ W Dt n Dx G W þ1=2;l ; W þ1=2;r G W n 1=2;l ; Wn 1=2;r ; ð60þ o : ð61þ G W 1=2;l ; W 1=2;r 4.2.1. Frst stage The 1st stage s exactly the same as the sngle-stage Euler explct tme-ntegraton scheme and realzablty can be guaranteed for 1st-order and quas-pth-order fnte-volume schemes subject to condton (42). 4.2.2. Second stage The set of moments for the th cell after the 2nd stage can be wrtten as: where W nþ1 ¼ Kh nþ1 dv; ð62þ Table 1 L 1 error and order of accuracy of schemes usng 1-node quadrature. Grd sze L 1 error Order 1st-order scheme 25 0.504533 50 0.347427 0.54 100 0.207609 0.74 200 0.114032 0.86 Standard 2nd-order scheme 25 0.133869 50 0.055688 1.27 100 0.016926 1.72 200 0.004903 1.79 Quas-2nd-order scheme 25 0.133869 50 0.055688 1.27 100 0.016926 1.72 200 0.004903 1.79 Quas-3rd-order scheme (wthout lmter) 25 0.010411 50 0.001318 2.98 100 0.000165 2.99 200 0.000024 2.77 Quas-3rd-order scheme 25 0.019774 50 0.004261 2.21 100 0.000885 2.27 200 0.000180 2.30
V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 5339 h nþ1 ¼ 1 h 2 f n ¼ 1 h 2 f n ¼ 1 h 2 f þ f þ f k v þ f þ1=2;l þ v f þ1=2;r v þ f 1=2;l v f 1=2;r kv þ f þ1=2;l kv f þ1=2;r þ kvþ f 1=2;l þ kv f 1=2;r kv þ f þ1=2;l þ kv f 1=2;r þ w þ ; ¼ 1 h 2 f n þ f kv þ f þ1=2;l þ kv f 1=2;r þ n þ ð63þ n whch k = Dt/Dx and w þ ¼ n þ þ 1 2 f n. For the 1st-order and quas-pth-order fnte-volume schemes, groupng of the frst three terms s possble because U ;a ¼ U þ1=2;a;l ¼ U 1=2;a;r : ð64þ After groupng: h nþ1 ¼ Xb n o n ;a kvþ n þ1=2;a;l þ kv n 1=2;a;r d v U ;a þ w þ : ð65þ Table 2 L 1 error and order of accuracy of schemes usng 2-node quadrature. Grd sze L 1 error Order 1st-order scheme 25 0.210375 50 0.148604 0.50 100 0.091184 0.70 200 0.051804 0.82 Standard 2nd-order scheme 25 0.071737 50 0.028923 1.31 100 0.011792 1.29 200 0.004584 1.36 Quas-2nd-order scheme 25 0.071198 50 0.028921 1.30 100 0.011793 1.29 200 0.004584 1.36 Quas-3rd-order scheme 25 0.022921 50 0.009345 1.29 100 0.002946 1.67 200 0.000990 1.57 Fg. 2. Grd convergence study for dfferent schemes.
5340 V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 Ths s exactly the same form as n (42) wth the superscrpt n replaced by. Hence, the realzablty condton can be guaranteed usng (42), by replacng superscrpt n wth. The RK2SSP scheme, combned wth quas-pth-order fnte-volume schemes, guarantees realzablty of the moment set. 4.2.3. Alternate form Another way to wrte RK2SPP s: W ð0þ ¼ W n ; W ð1þ W ð2þ ¼ W ð0þ ¼ W ð1þ h W nþ1 ¼ 1 2 Wð0Þ h Dt G Dx Wð0Þ ; þ1=2;l Wð0Þ þ1=2;r Dt Dx h G Wð1Þ ; þ1=2;l Wð1Þ þ1=2;r þ W ð2þ : G W ð0þ ; 1=2;l Wð0Þ 1=2;r G W ð1þ ; 1=2;l Wð1Þ 1=2;r ; ; ð66þ Ths form s more amenable to usng an operator-splttng technque for the collson and drag terms. 4.3. Calculaton of tme-step sze The global tme-step sze (Dt) should satsfy both the CFL and realzablty crtera to guarantee stablty and physcal nature preservaton. Usually, the largest value of CFL O(1) that gves a stable soluton s used as a CFL crteron for all 1-D/2-D/ 3-D cases. The defnton of the CFL vares, but a general form s: CFL ¼ Dt cell velocty magntude cell length scale : ð67þ Fg. 3. Comparson of schemes for 1-D case wth only convecton terms.
V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 5341 Let Dt CFL, and Dt realzable, be the tme-step szes n the th cell satsfyng the CFL and realzablty crtera, respectvely. An obvous way to calculate global tme-step sze s: Dt ¼ mn Dt CFL; ; Dt realzable; : ð68þ Fg. 4. Flud velocty for 1-D case. Fg. 5. Comparson of schemes usng mean densty for 1-D case wth convecton and drag terms.
5342 V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 However, n the present paper a slghtly dfferent approach s used for the calculaton of Dt. As stated earler, n general Dt realzable, < Dt CFL,. Ths mples that most of the tme Dt = mn (Dt realzable, ). To have a better tme accuracy, a mult-stage tme-ntegraton s used. For a mult-stage tme-ntegraton, Dt calculated for the 1st-stage s used n all the later stages. But mn (Dt realzable, ) may be dfferent for dfferent stages of tme ntegraton and n that case Dt calculated for the 1st-stage wll not satsfy the realzablty crteron for a later stage. In order to get rd of ths problem, the followng approach s used. Suppose, the problem under consderaton uses a quas-pth-order fnte-volume scheme. A value of the CFL O(1) s pre-specfed and Dt s calculated as Dt = mn (Dt CFL, ). Durng each stage of tme ntegraton, ths Dt s used as the global tme-step sze, and the realzablty condton correspondng to quas-pth-order fnte-volume scheme s checked n each cell. For the cells n whch the check succeeds, a quas-pth-order reconstructon s used for the weghts. And for the cells n whch the check fals, a 1st-order reconstructon s used for the weghts. Thereafter, n the faled cells, the realzablty crteron correspondng to 1st-order fnte-volume scheme s checked. Ths check succeeds on almost all occasons f the CFL s not very large, thereby satsfyng the realzablty crteron n each cell. However, f the last check fals, the whole process s re-ntated usng a smaller CFL. 5. Numercal results In ths secton several results are presented for 1-D and 2-D cases. For all the cases a 2nd-order RK2SSP scheme s used for tme-ntegraton and Dt s calculated usng CFL = 0.5 unless otherwse stated. Perodc boundary condtons are used for all 1- D cases whle for the 2-D cases a combnaton of wall, Drchlet and perodc boundary condtons s used. The doman for the 1-D cases s defned by x 2 [ 1,1]. The smulatons consder ether the spatal flux terms alone or n combnaton wth flud drag terms. Collsons are not ncluded. The drag force s calculated usng (5) and the abscssas are updated usng smple knematc relatons. More detals can be found n [17 19]. The drag terms are dependent on the Stokes number defned by Fg. 6. Comparson of schemes usng mean velocty for 1-D case wth convecton and drag terms.
V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 5343 St ¼ 1 q p d 2 p : ð69þ 18 l g For the two cases presented here that nclude the drag terms, St = 1 s used. For the 1-D cases, the numercal results usng the quas-2nd-order and quas-3rd-order schemes are presented. For the quas-2nd-order scheme, a lnear reconstructon wth the mnmod lmter as descrbed n Secton 2.4.2 s used whle for the quas-3rd-order scheme a MUSCL technque [27,49] s used. For the 2-D cases, results usng the quas-2nd-order scheme are presented. A 2nd-order least-squares reconstructon [1,50] s used for the weghts usng neghborng cell-averaged values. Moreover, a lmter [1,20,50] s appled to the least-squares reconstructon to avod spurous oscllatons. The results for the standard 2nd-order scheme are presented only for the smplest 1-D cases wth constant abscssas because of the nonrealzablty problem. In all the cases, the mean densty s computed usng P n a and the mean velocty s computed wth P ðna u a Þ= P n a. 5.1. Spatal accuracy study It was stated earler that although n general the new quas schemes have lower spatal accuracy compared to standard schemes, for problems where the veloctes are constant over a range of cells, both schemes have almost the same order of spatal accuracy. Here, the order of spatal accuracy of the standard and the new quas schemes s dscussed for cases where Fg. 7. Mean partcle densty for crossng partcle jets for dfferent schemes at two tmes.
5344 V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 the abscssas are constant over a range of cells. Results n ths secton are based on smple 1-D cases for whch analytcal solutons exst. L 1 errors are calculated by comparng the numercal soluton wth the analytcal soluton for mean densty. L 1 errors and order of convergence are presented for four schemes: 1st-order, standard 2nd-order, quas-2nd-order, quas- 3rd-order. Tables 1 and 2 show the results for 1-node and 2-node quadrature, respectvely. For both cases Dt = 0.001 s used. For 1-node quadrature, the ntal weght (n) and abscssa (U) are gven as: n ¼ 1:0 þ snðpxþ; U ¼ 1: ð70þ For 2-node quadrature, ntal weghts (n 1,n 2 ) and abscssas (U 1,U 2 ) are gven as: n 1 ¼jsnðpxÞj; n 2 ¼ 0; U 1 ¼ 1; U 2 ¼ 0 for x 2½ 1; 0Þ; n 1 ¼ 0; n 2 ¼jsnðpxÞj; U 1 ¼ 0; U 2 ¼ 1 for x 2½0; 1Š: ð71þ For 1-node quadrature, errors are calculated at t = 4, whle for 2-node quadrature errors are calculated at t = 1. It s observed that the formal order of accuracy can be obtaned for 1-node quadrature, but as the number of quadrature nodes s ncreased, the order of accuracy for all schemes decreases. Although the reason for ths loss of order of accuracy has not been studed extensvely, t can be attrbuted to the combned effects of an ncrease n the number of equatons and use of the momentnverson algorthm for ll-condtoned ponts. At many ponts the values of the weghts are very small and the non-lnear equatons solved usng the moment-nverson algorthm are often ll-condtoned for these ponts n the case of multple quadrature nodes. It can also be observed that the quas-2nd-order scheme always has approxmately the same order of convergence as the standard 2nd-order scheme, and the quas-3rd-order scheme s better compared to both. 5.2. Grd convergence study Grd convergence studes for the quas-2nd-order and quas-3rd-order schemes n 1-D are presented n Fg. 2. For both the schemes, the mean densty obtaned usng dfferent grd resolutons s compared wth the analytcal soluton. Four dfferent unform grds have been consdered wth the number of cells equal to 25, 50, 100, 200. The comparsons have been done for a 2-node quadrature case wth the same ntalzatons as n (71). Fg. 2(a) and (b) shows grd convergence for the quas-2ndorder and quas-3rd-order schemes, respectvely. As the number of grd cells s ncreased, the solutons usng both schemes converge towards the analytcal soluton. 5.3. Comparson of schemes for 1-D case wth only convecton terms For ths case, 2-node quadrature s used wth the weghts beng snusodal functons and the abscssas beng square functons. Ffty grd ponts are used and the results are shown n Fg. 3. The ntal (t = 0) condtons are same as n (71) and are shown n Fg. 3(a). Fg. 3(b) and (c) shows the fnal condtons for the mean densty and mean velocty, respectvely (t = 1). The weght dstrbuton s symmetrc about x = 0, wth the left wave movng towards the rght (negatve abscssa) and the rght wave movng towards the left (postve abscssa). The fnal tme has been chosen such that the waves coalesce at x = 0 and then separate agan. Four dfferent schemes have been compared: 1st-order, standard 2nd-order, quas-2nd-order, Fg. 8. Flud velocty for 2-D Taylor Green flow.
V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 5345 quas-3rd-order. The standard 2nd-order and quas-2nd-order results are on top of each other. The quas-3rd-order scheme shows an mprovement over the quas-2nd-order scheme. 5.4. Comparson of schemes for 1-D case wth both convecton and drag terms For ths case, 4-node quadrature s used. Intal condtons (t = 0) for x 2 [ 0.8, 0.7] are gven as: n1 ¼ n2 ¼ n3 ¼ n4 ¼ snð10pðx þ 0:8ÞÞ; U 1 ¼ U 2 ¼ U 3 ¼ U 4 ¼ 0: ð72þ Everywhere else both the weghts and abscssas are zero. The partcle flow s drven by flud drag. The flud velocty s gven by 2 U g ¼ e x ; ð73þ and s shown n Fg. 4. The QMOM results usng three schemes 1st-order, quas-2nd-order, quas-3rd-order are compared wth the Lagrangan results. The results for the mean densty are presented at four dfferent tmes n Fg. 5. The quas-3rdorder and quas-2nd-order results are closer to the Lagrangan results as compared to the 1st-order ones. Fg. 6 shows Fg. 9. Grd resoluton study of mean partcle densty n 2-D Taylor Green flow at t = 4 wth 1st-order scheme.
5346 V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 comparsons for the mean velocty at the same tmes as n Fg. 5. It s evdent that the mean velocty s not constant but vares over a range of cells at each tme. Clearly n such scenaros as well, the new realzable schemes gve better solutons compared to the 1st-order scheme. 5.5. Comparson of schemes for 2-D case wth only convecton terms Here a dlute mpngng-jet problem n 2-D s presented. The doman conssts of a square (7 7) box wth two openngs on the bottom wall through whch partcle jets enter. As tme progresses, the jets cross each other, strke the wall and then rebound. These smulatons are done usng 4-node quadrature (n1, U1, V1), (n2, U2, V2), (n3, U3, V3), (n4, U4, V4). Intal (t = 0) condtons are gven as: n1 ¼ n2 ¼ n3 ¼ n4 ¼ 0:0001; U 1 ¼ 0:001; U 2 ¼ 0:001; V 1 ¼ 0:001; V 2 ¼ 0:001; U 3 ¼ 0:001; U 4 ¼ 0:001; V 3 ¼ 0:001; V 4 ¼ 0:001: Fg. 10. Grd resoluton study of mean partcle densty n 2-D Taylor Green flow at t = 4 wth quas-2nd-order scheme. ð74þ
V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 Fg. 11. Mean partcle densty n 2-D Taylor Green flow at t = 4 on trangular mesh wth dfferent schemes and cell numbers. 5347
5348 V. Vkas et al. / Journal of Computatonal Physcs 230 (2011) 5328 5352 Fg. 12. Partcle number densty n 2-D Taylor Green flow obtaned by Lostec et al. [30] usng a Lagrangan smulaton. Densty s proportonal to darkness. The values of the weghts and abscssas at the left nlet jet (Drchlet) are: n 1 ¼ n 2 ¼ n 3 ¼ n 4 ¼ 0:01; U 1 ¼ 1:001; U 2 ¼ 0:999; U 3 ¼ 1:001; U 4 ¼ 0:999; V 1 ¼ 1:001; V 2 ¼ 1:001; V 3 ¼ 0:999; V 4 ¼ 0:999: ð75þ And the values at the rght nlet jet (Drchlet) are: n 1 ¼ n 2 ¼ n 3 ¼ n 4 ¼ 0:01; U 1 ¼ 1:001; U 2 ¼ 0:999; U 3 ¼ 1:001; U 4 ¼ 0:999; V 1 ¼ 1:001; V 2 ¼ 1:001; V 3 ¼ 0:999; V 4 ¼ 0:999: ð76þ For elastc collsons wth the walls, e w = 1. Results are presented for the 1st-order and quas-2nd-order schemes n Fg. 7. The computatonal grd contans 2562 trangular elements. Fg. 7(a) and (c) show the mean densty usng the 1st-order and quas- 2nd scheme, respectvely, at t = 4 after the jets cross. Fg. 7(b) and (d) shows the mean densty at t = 7 after the jets bounce off the walls. The soluton obtaned usng the 1st-order scheme s more dffused. The mprovement n the soluton usng the quas-2nd-order scheme s clearly evdent. 5.6. Comparson of schemes for 2-D case wth both convecton and drag terms In ths case, the evoluton of partcles n a Taylor Green flow s presented. The doman conssts of a square (1 1) box. All the boundares are perodc. The flud velocty n the Taylor Green flow s gven by U gx ¼ snð2pxþ cosð2pyþ; U gy ¼ snð2pyþ cosð2pxþ; ð77þ and s shown n Fg. 8. Fgs. 9 and 10 present results for dfferent schemes on a structured mesh, and Fg. 11 presents results for unstructured meshes. Results are presented for the 1st-order and quas-2nd-order schemes at t = 4. For the structured mesh, four dfferent grd resolutons are used: 100 100, 200 200, 400 400, 800 800. Lostec et al. [30] presented results for the same problem usng both a Lagrangan smulaton and 1st-order QMOM. Here, the Lagrangan result obtaned n [30] s presented n Fg. 12. Note that the partcle number densty shown n Fg. 12 s proportonal to the mean partcle densty n our results. The results presented here clearly show that the quas-2nd-order scheme fares much better compared to the 1st-order scheme n resolvng the varous features obtaned n the Lagrangan smulaton n [30]. The unstructured mesh conssts of trangular cells. Three dfferent grd resolutons are used wth 5452, 21,830 and 89,558 trangular cells. The unstructured mesh results also confrm the mprovement n the solutons when the quas-2nd-order scheme s used. 6. Conclusons Htherto, the use of fnte-volume schemes for quadrature-based moment methods was lmted to the 1st-order scheme n order to guarantee realzablty. Over the years, an extensve research effort has been spent on developng hgh-order fnte-volume schemes n the feld of computatonal flud dynamcs. However, the ssue of non-realzablty has often acted