Second-Order Accurate Godunov Scheme for Multicomponent Flows on Moving Triangular Meshes

Transcription

1 J Sc Comput (2008) 34: DOI /s Second-Order Accurate Godunov Scheme for Multcomponent Flows on Movng Trangular Meshes Guoxan Chen Huazhong Tang Pngwen Zhang Receved: 2 November 2006 / Revsed: 3 Aprl 2007 / Accepted: 19 September 2007 / Publshed onlne: 24 October 2007 Sprnger Scence+Busness Meda, LLC 2007 Abstract Ths paper presents a second-order accurate adaptve Godunov method for twodmensonal (2D) compressble multcomponent flows, whch s an extenson of the prevous adaptve movng mesh method of Tang et al. (SIAM J. Numer. Anal. 41: , 2003) to unstructured trangular meshes n place of the structured quadrangular meshes. The current algorthm solves the governng equatons of 2D multcomponent flows and the fnte-volume approxmatons of the mesh equatons by a fully conservatve, second-order accurate Godunov scheme and a relaxed Jacob-type teraton, respectvely. The geometrybased conservatve nterpolaton s employed to remap the solutons from the old mesh to the newly resultng mesh, and a smple slope lmter and a new montor functon are chosen to obtan oscllaton-free solutons, and track and resolve both small, local, and large soluton gradents automatcally. Several numercal experments are conducted to demonstrate robustness and effcency of the proposed method. They are a quas-2d Remann problem, the double-mach reflecton problem, the forward facng step problem, and two shock wave and bubble nteracton problems. Keywords Adaptve movng mesh method Fnte volume method Godunov scheme Mult-component flows Unstructured mesh 1 Introducton The hydrodynamcs of the mxture of dfferent fluds s of great nterest n a wde range of physcal flows. Among them some fundamental ssues are the dynamcs and stablty of bubbles and nterfaces, mxng processes, bubbly flows, and lqud suspensons, etc. Such flud G. Chen H. Tang ( ) P. Zhang LMAM and CCSE, School of Mathematcal Scences, Pekng Unversty, Bejng , People s Republc of Chna e-mal: hztang@math.pku.edu.cn G. Chen e-mal: gxchen@math.pku.edu.cn P. Zhang e-mal: pzhang@math.pku.edu.cn

2 J Sc Comput (2008) 34: flows gve rse to challengng problems n both theory and numercal smulaton. Recent two decades have seen a growng nterest n developng numercal methods for compressble multcomponent flows and the nvestgaton of the physcal phenomena n complex flud flows, see e.g. [1, 3, 21, 23 25, 29, 33, 35]. It s well-known that conservatve computatons of such flows run nto unexpected dffcultes commonly due to oscllatons generated at materal nterfaces. To overcome those dffcultes, many authors studed varous models (the γ -model, the mass fracton model, and the level-set model etc.) and proposed some (locally) nonconservatve schemes, see [2] and references theren. Recently, Abgrall and Karn n [2] revewed some of the recent models and numercal algorthms that had been proposed and ponted key deas that they had n common. Although t has been proved that the nonconservatve schemes are very successful n smulatng compressble mult-component flows, they wll take a rsk of producng ncorrect results due to non-conservaton, see [16]. Up to now, there also exsts some work on conservatve schemes for compressble multcomponent flows, see e.g. [22, 28, 46]. The man objectve of ths paper s to extend the adaptve movng mesh method developed n [39] to two-dmensonal compressble multmateral flows. The governng equatons wll be solved by a fully conservatve second-order Godunov scheme wth exact Remann solver on unstructured trangular meshes. Although hgher-order accurate Godunov-type schemes on trangular meshes have been studed n many lteratures, see e.g. [4, 17, 20, 43], t would be more nterestng to see them n smulatng multcomponent flows on adaptve movng meshes. Locally clusterng mesh ponts n the regons of the materal nterface wll effectvely reduce possble errors (or oscllatons) produced by a fully conservatve Godunov-type scheme at the materal nterface. Adaptve movng mesh methods have mportant applcatons for a varety of scentfc and engneerng areas such as sold and flud dynamcs etc., where sngular or nearly sngular solutons are developed dynamcally n farly localzed regons of shock waves, boundary layers, and detonaton waves etc. Numercally nvestgatng these phenomena requres extremely fne meshes over a small porton of the physcal doman to resolve the large soluton varatons. Successful mplementaton of an adaptve strategy can ncrease accuracy of the numercal approxmatons and decrease the computatonal cost. Up to now, there have been many mportant progresses n adaptve movng mesh methods for partal dfferental equatons, ncludng grd redstrbuton approach based on the varatonal prncple of Wnslow [44] and Brackbll [6], and Ren and Wang [32]; movng fnte element methods of Mllers [30], and Davs and Flaherty [10]; movng mesh PDEs methods of Russell et al. [5, 7, 19]; and movng mesh methods based on the harmonc mappng of Dvnsky [12], L et al. [11, 26, 27, 38, 47], and Cenceros and Hou [8]. Computatonal costs of movng mesh methods can be possbly saved wth locally varyng tme steps [36], but at the cost of ncreasng the algorthm complexty. Recently, Chertock and Kurganov n [9] proposed a conservatve locally movng mesh method for one-dmensonal multflud flows. It wll be more challengng to conduct research n adaptve movng mesh methods for twoand three-dmensonal multcomponent flows. The paper s organzed as follows. The governng equatons for multcomponent flud flows are ntroduced n Sect. 2, and approxmated n Sect. 3 by a second-order accurate Godunov scheme on a fxed and unstructured trangular mesh, where a smple slope-lmter s used to avod numercal oscllatons. Secton 4 dscusses the teratve mesh redstrbuton. The conservatve varables are remapped onto the newly resultng meshes by usng a hgh-resoluton geometry-based conservatve nterpolaton. Full soluton procedure wll be outlned n Sect. 5. Secton 6 gves numercal experments to valdate the robustness and effcency of the proposed adaptve algorthm. Fnally, we conclude ths work n Sect. 7.

3 66 J Sc Comput (2008) 34: Governng Equatons Multcomponent flows we consder are a subset of multphase flows where the dfferent flud components, characterzed by ther respectve (constants) rato of specfc heats, are mmscble. Moreover, we neglect dffusve effects, surface tenson and cavtaton, and assume that the flud conssts of two components. The governng equatons for such multcomponent flows may be wrtten by usng a sngle velocty and a sngle pressure functon as follows ρ t + (ρu) + (ρv) = 0, x y (ρu) t (ρv) t E + t + (ρu2 + p) x + (ρuv) x (u(e + p)) x + (ρuv) y + (ρv2 + p) y + = 0, = 0, (v(e + p)) y = 0, (2.1) where ρ, u = (u, v), p, ande = ρe ρ(u2 + v 2 ) are the densty, the velocty vector, the pressure, and the total energy, respectvely, e denotes the nternal energy. Four equatons n (2.1) express conservaton of mass, momentum, and energy of the flud mxture. Besdes specfyng the equaton of state (EOS) for the effectve thermodynamcs p = p(ρ,e), the multcomponent flow descrpton s completed by provdng an addtonal equaton that descrbes the dynamcs of the flud composton. For the rest of the paper, the varable φ s used to descrbe the flud composton. Varous choces of φ have been consdered n the lteratures, dependng on the model assumptons. For example, t s taken to be the rato of specfc heats, the mass fracton, or the level-set functon. For all these models, the governng equaton for the varables φ may be wrtten n a conservatve form of (ρφ) t + (ρφu) x + (ρφu) y = 0. (2.2) Ths work s restrcted to the perfect gases and the γ -model. Thus the thermodynamc propertes of the flud mxture s descrbed by the deal EOS p = (γ 1)ρe, and the nterface between two fluds s represented by φ = 1 γ 1,whereγ denotes the effectve rato of specfc heats of the flud mxture and depends on the flud composton for two flud flows. It turns out that ths partcular choce of φ offers clear advantages.

4 J Sc Comput (2008) 34: We wll use the Godunov scheme [14] wth the ntal data reconstructon [42] to solve (2.1) and(2.2),.e. U t + F (U) x + G(U) y = 0, (2.3) wth ρ ρu U = ρv, F (U) = E ρφ ρu ρu 2 + p ρuv u(e + p) ρuφ, G(U) = ρv ρuv ρv 2 + p v(e + p) ρvφ. The detaled procedure for the exact Remann solver of (2.3) s completely smlar to that for the one-component flows gven n the lterature, see e.g. [41]. 3 Second-Order Accurate Godunov Scheme on Trangular Meshes Our adaptve multcomponent flow calculaton s formed by two ndependent parts: the evoluton of the governng equaton and the teratve mesh redstrbuton. The frst part s dscussed n ths secton, whle the second part wll be ntroduced n Sect. 4. In the followng, we begn to ntroduce second-order accurate Godunov scheme of (2.3) on a fxed, unstructured trangular mesh. Gve a trangulaton of the physcal doman p, denoted by T ={E 0,E 1,...,E ne }, where E s the th trangle of the trangulaton. For the trangle E,wedenotex j = (x j,y j ) ts jth vertex, E j ts jth neghborng element, l j ts jth edge, j = 1, 2, 3, and n j the outward unt normal vector on l j,seefg.1 for a detaled schematc dagram. We also assume that a partton of the tme nterval [0,T] s gven as {t n = t n 1 + t n t n > 0,n N},where the tme step sze t n should be determned by the stablty condton n practce. Fg. 1 Schematc dagram of the trangle element E

5 68 J Sc Comput (2008) 34: Integratng (2.3) over the trangle E T,wehave E d dt U E (t) = F n (U) ds = E 3 j=1 l j F nj (U) ds j, (3.1) where E s the area of the element E,ds and ds j stand for the surface element measure, and F nj = F (U)n x j + G(U)ny j s the flux functon n the n j = (n x j,ny j ) drecton, n s the outward unt normal vector of the trangle boundary E := l 1 l 2 l 3.HereU E (t) denotes the cell average of the conservatve varable U on the trangle E,defnedby U E (t) = 1 U(x,t)dx. (3.2) E E We use the mdpont ntegraton formula to approxmate the ntegraton n (3.1), and replace the exact soluton U at the mddle pont of the edge l j by the approxmate soluton (.e. the pecewse polynomal reconstructed by usng the cell averages n the fnte volume method) and the flux F nj (U) by any two-pont Lpschtz numercal flux F nj (U L l j, U R l j ), respectvely. Based on those, (2.3) can be approxmated as d dt U E (t) = 1 E 3 F nj (U L l j, U R l j ) l j, (3.3) where l j denotes the length of edge l j. Ths work employs the Godunov flux wth the exact Remann solver [14] n(3.3): j=1 F nj (U L l j, U R l j ) = F nj (ω (0; U L l j, U R l j )), (3.4) where ω (x/t; U L l j, U R l j ) denotes the exact soluton of the one-dmensonal Remann problem U + F n j (U) = 0, t x { U L lj, x <0, (3.5) U(x, 0) = U R l j, x >0. We refer the readers to the lteratures, e.g. [41], for ts detaled dervaton. If set U L l j := U E, U R l j := U Ej, then (3.3) s only a frst-order accurate sem-dscrete scheme of (2.3). To get a second-order accurate spatal dscretzaton, the ntal reconstructon technque [42] s used to reset U L l j and U R l j on the edge l j,seefg.2, wheree and E j are neghborng each other and have a common edge l j. To do those, we frst compute the cell vertex approxmate values of the soluton by usng ts cell averages as follows U 3 := 1 U trangle surroundng x3, ne 3 U j2 := 1 ne j2 U trangle surroundng xj2,

6 J Sc Comput (2008) 34: Fg. 2 Schematc dagram of the ntal data reconstructon where ne 3 (or ne j2 ) denotes the number of trangles surroundng x 3 (or x j2 ). Next, we defne U L l j and U R l j on the jth edge of the trangle E by U L l j = U E (U E U 3, U Ej U E ), (3.6) U R l j = U Ej 1 2 (U E j U E, U j2 U Ej ), (3.7) where (, ) s a nonlnear lmter functon whch s used to suppress the possble pseudooscllaton. If (, ) 0, then the edge values U L l j and U R l j become correspondng cell averages and the sem-dscrete scheme (3.3) degenerates to frst order accurate spatal approxmaton. In our computatons, we use van Leer s slope lmter [42] ab ψ = ψ (a, b) = (sgn(a) + sgn(b)) a + b +ε, where ψ s the th component of = (ψ 1,...,ψ 5 ) T, ε s a small postve postve number, 0 <ε 1, whch s used to avod that the denomnator becomes zero. System (3.3) may be approxmated by any stable tme dscretzaton. For example, we use an explct second-order accurate TVD Runge-Kutta method [34] to evolve solutons of the governng equatons (2.3) from t n to t n+1 : U = U n + tl(u n ), (3.8) U n+1 = 1 2 U n (U + tl(u )), (3.9) where L(U) denotes the term at the rght hand sde of (3.3). Here we have used U n E to stand for an approxmaton of U E (t n ) at tme t n.

7 70 J Sc Comput (2008) 34: Remark 3.1 When we evolve the soluton wthn E j,seefg.2, the edge values U L l j U R l j on the th edge of the trangle E j are set as and U R l j = U E 1 2 (U 3 U E, U E U Ej ), U L l j = U Ej (U E U Ej, U Ej U j2 ). If the lmter functon satsfes the property of that ( V, W) = (V, W ), then we have U L l j = U R l j, U R l j = U L l j. 4 Adaptve Mesh Redstrbuton Ths secton extends the adaptve mesh redstrbuton of Tang et al. [39, 40] to unstructured trangular meshes. It s an teratve procedure: redstrbute or move trangular mesh ponts by teratvely solvng Euler-Lagrange equatons n the logcal doman l and at the same tme remap the physcal varables onto the resultng new mesh. 4.1 Mesh-Redstrbuton Based on Varatonal Methods Let l be the logcal doman wth the orthogonal coordnates ξ = (ξ, η) and a Delaunay trangulaton, denoted by T l, whose data structure s same as that of the trangulaton T of the physcal doman p.letv denote the dual cell assocated wth the vertex ξ n l, whch s delmted n jonng the barycenter of all the trangles surroundng ξ, see the left plot n Fg. 3. SnceT l s assumed to be a Delaunay trangulaton of l, the Vorono dagram or Drchlet tessellaton s a natural choce of the dual parttonng of l. Fg. 3 Schematc dagram of the dual cell V and the trangle elements of l (left) and p (rght)

8 J Sc Comput (2008) 34: We denote ℵ() the number of trangles surroundng ξ, for example, ℵ() = 6nthe schematc n Fg. 3, andζ j the mdpont of the edge l j connectng nodes ξ and ξ j. Let ˆl j be a lnear segment connectng centrods of two neghborng trangles surroundng ξ and ξ j. A one-to-one coordnate transformaton from the logcal or computatonal doman l to the physcal doman p s denoted by x = x(ξ), ξ l. (4.1) We lmt our attenton to the case of that the physcal doman p s convex and the map (4.1) s to fnd the mnmzer of the followng functonal [8, 39, 40] Ẽ(x) = =1 l ( x ) T G x dξ, (4.2) where =( ξ, η ) T,andG ( = 1, 2) are gven symmetrc postve defnte matrces called montor functons. In general, the montor functons depend on the soluton or ts dervatves of the underlyng governng equatons. The smplest choce of the montor functons s G = ωi, = 1, 2, see [44], where I denotes the dentty matrx and ω s a postve weght functon. More terms can be added to the above functonal to control other aspects of the mesh such as orthogonalty and algnment wth a gven vector feld, see e.g. [6, 18]. Usng the choce of Wnslow, we deduce the Euler-Lagrange equatons of the functonal (4.2) to (ω x) = 0. (4.3) In ths study, ω = ω( ξ U) =: ω(u), whch wll be defned n Sect. 6. We begn to gve a fnte volume approxmaton of (4.3) subject to boundary condtons x p, f ξ l. Integratng (4.3) over the dual cell V as shown n Fg. 3 and usng the dvergence theorem gves 0 = w x ℵ() V n ds = j=1 ˆl j w(u) x n ds, (4.4) where n = (n ξ,n η ) s the unt outward normal vector of V. Usng the numercal ntegraton formula and approxmatng the term x n (ζ j ) by gves the followng dscrete mesh equaton x n (ζ j ) x j x ξ ξ j, ℵ() w j ˆl j x j x ξ j ξ j=1 = 0, (4.5) where w j = w(u(ζ j )).Herewehaveusedtheassumptonthatˆl j les on the perpendcular bsector of the edge l j connectng ξ and ξ j. In practcal computaton, ths assumpton may be relaxed.

9 72 J Sc Comput (2008) 34: Generally, (4.5) s a nonlnear algebrac system, due to dependence of w j on the soluton. To avod ths dffculty, we lnearze (4.5) and then use a relaxed teraton method to solve t as follows: for ν = 0, 1,...,where x [ν+1] ℵ() ˆx = W j x [ν] W j, (4.6) j=1 j / ℵ() j=1 = μ ˆx + (1 μ )x [ν], (4.7) { W j = τ ℵ() } V w j ˆl j / ξ j ξ, and μ = max W j,σ. Here τ and σ are two artfcal parameters that control the qualty of the mesh movement. Obvously, f 0 max { ℵ() j=1 W j },σ 1, then the teraton (4.6, 4.7) s postve-preservng. But f μ s too bg, e.g. near 1, then t s easy that the mesh may be badly dstorted; conversely, f μ s near 0, the mesh ponts move too slow. In ths work, the mesh teraton s contnued untl x [ν] x [ν+1] < 10 6 or ν<5, and we take max { ℵ() j=1 W j }= 1 2 and σ = 0.3, whch ensure bascally that x [ν+1] wll be wthn the convex hull of the mdponts of the edge x x j, j = 1, 2,...,ℵ(), see the shaded regon n the rght plot of Fg. 3. Remark 4.1 (Boundary mesh redstrbuton) The boundary mesh ponts should be redstrbuted smultaneously along wth the nner mesh movement because the dscontnutes may nteract wth the boundary of the physcal doman p at some fnte tme. For convenence, we assume that l x := {y = 0,x a x x b } s part of p, and mapped to part of l,denoted by l ξ := {η = 0,ξ a ξ ξ b }. The mesh ponts on l x are redstrbuted by solvng a one-dmensonal mesh equaton ( ) x w lξ = 0, ξ l ξ, (4.8) ξ ξ subject to the Drchlet boundary condtons j=1 x(ξ a ) = x a and x(ξ b ) = x b. (4.9) Here, the montor w lξ s specfed as value of the montor over ts adjacent nner element. 4.2 Conservatve Interpolaton on New Meshes After each teratve step of (4.6) and(4.7), we need to remap the approxmate solutons onto the newly resultng mesh {x [ν+1] } or {E [ν+1] } from the old mesh {x [ν] } or {E [ν] }. Tang and Tang [39] proposed a conservatve and upwnd nterpolaton formula on the structured quadrangular mesh. Recently, Han and Tang [15] gave a smplfed geometrcal nterpolaton approach on the same mesh whch also preserves conservaton property of the conservatve

10 J Sc Comput (2008) 34: Fg. 4 Movement of the control volume E [ν] to E [ν+1] varables U n the sense of that E [ν] U [ν] E = E [ν+1] U [ν+1], Ẽ wherewehaveredefnedẽ := E [ν+1] and E := E [ν] n the subscrpt. In ths work, we extend the geometrcal approach of Han and Tang [15] to the unstructured trangular mesh. Let D j denote the regon scanned by the edge l [ν] j after one teratve step of (4.6) and(4.7), j = 1, 2, 3, see Fg. 4. We remap the conservatve varables as Ẽ U [ν+1] = E Ẽ U [ν] E + 3 S j, (4.10) where S j s the ntegral of the approxmate soluton U over the doman D j. Followng thedean[15], we may smplfes the calculaton of S j.taked 1 shown n Fg. 4 as an example, we frst compute D 1 by j=1 D 1 := 1 2 ((x[ν+1] 3 x [ν] 2 )(y[ν] 3 y[ν+1] 2 ) (y [ν+1] y [ν] 3 2 )(x[ν] 3 x[ν+1] 2 )). It s obvous that D 1 s the sgned area functon whch means that D 1 s the area of D 1 f x [ν] 2, x[ν+1] 2, x [ν+1] 3,andx [ν] 3 are located by counter-clockwse order, and s the nverse of area of D 1 f the above four ponts are located by clockwse order. Then S 1 can be approxmately calculated as S 1 = max{ D 1, 0}U R l 1 + mn{ D 1, 0}U L l 1, (4.11) where U L l 1 and U R l 1 are the reconstructed left and rght states on the edge l 1 by {U [ν] E },see (3.6) and(3.7).

11 74 J Sc Comput (2008) 34: Soluton Procedure Our soluton procedure s formed by two ndependent parts: evoluton of the governng equatons and an teratve mesh redstrbuton. The frst part s a second-order accurate Godunov method on fxed unstructured trangular meshes, see Sect. 3. In each teraton of the second part, the trangular mesh ponts are frst redstrbuted by the relaxed teraton method (4.6) and(4.7) n Sect. 4.1, and then the conservatve varables U are updated on the resultng new meshes by the conservatve-nterpolaton formula (4.10) and(4.11) as well as (3.6) and(3.7), see Sect The soluton procedure can be llustrated by the followng flowchart: Algorthm 1 Step 1 Gve ntal quas-unform trangulatons of the physcal doman p and the logcal doman l, denoted by {x 0 } and {ξ } respectvely. Compute the cell average of the conservatve varables U denoted by U 0 E. Step 2 For n = 0, 1,...,set x [0] := x n, U [0] E := U n E, and do the followng steps. Step 3 For ν = 0, 1, 2,...,μ 1, do the followng: (1) Move mesh ponts x [ν] to x [ν+1] by solvng (4.6)and(4.7). (2) Update the conservatve varables U [ν+1] E on the new mesh {x [ν+1] } accordng to (4.10) and (4.11). Step 4 Set x n := x [μ], U n E := U [μ] E, and evolve the governng equatons (2.3) on the adaptve mesh {x n } by usng the second-order accurate fnte volume Godunov method, gven n Sect. 3, to obtan the numercal approxmaton U n+1 E at the tme level t = t n+1. Step 5 If t n+1 <T, then go to Step 2; otherwse output the computed results and stop run. 6 Numercal Experments In ths secton, we apply the proposed adaptve mesh algorthm to several two-dmensonal problems to valdate ts effcency and performance. Throughout our computatons, the CFL number s taken 0.25 unless stated otherwse, the ntal mesh s generated by the free software EASYMESH [31], and the montor functon s taken as w E = 1 + α ρ w E 2 (β ρ,ρ)+ α s w E 2 (β s,s)+ α γ w E 2 (β γ,γ) 2, (6.1) where s = p/ρ γ, α q and β q (0, 1] (q = ρ,ors,orγ ) are some problem-dependent postve parameters, and wll be ascertaned n each example. Here w E s defned by w E (β q,q)=: mn{1, ξ q E / }, = β q max{ ξ q E }. E Example 6.1 (The double-mach reflecton problem) Ths problem was studed extensvely by Woodward and Colella n [45] and later by many others, e.g. [39]. We use exactly the same setup as n [39, 45],.e., the same ntal and boundary condtons and same soluton doman p =[0, 4] [0, 1]. The CFL number s 0.4. Intally a rght-movng Mach 10 shock n ar s postoned at x = 1 6,y = 0 and makes a 60 angle wth a horzontal wall from x = 1 6

12 J Sc Comput (2008) 34: Fg. 5 Example 6.1: Adaptve mesh and contour plots of the densty at t = 0.2 to 4. The gas densty ahead of the shock s 1.4, and the pressure s 1; the densty behnd the shock s 8. More precsely, the ntal data are { (8, , , ) T, for y h(x, 0), U = (1.4, 0, 0, 2.5) T, otherwse, where h(x, t) = ( 3 x 1 ) 20t 6 s the poston of the rght-movng shock, the output tme t = 0.2. The reflectve boundary condton s specfed on the wall. The flud varables are specfed as the left state of the ntal shock on the rest of the bottom boundary, whle the left and rght states of the rghtmovng shock at ( (1 + 20t)/ 3, 1) on the whole top boundary, respectvely. The nflow and outflow boundary condtons are used on the left and rght boundares, respectvely. In ths example, we take α ρ = 50, β ρ = 0.1, α s = α γ = β s = β γ = 0, and an ntal quasunform trangulaton of the physcal doman p as well as the logcal doman l s generated wth the horzontal (and vertcal) boundary partton of 134 (and 34) segments. Such quas-unform trangulaton s formed by 5417 nodes, edges, and elements, whose edge length and element area are approxmately equal to and , respectvely. Fgure 5 shows the adaptve mesh and the densty contours at t = 0.2, whch are obtaned by usng the proposed method. Note that the densty contours are only shown wth 30 equally spaced contour lnes n a part of the physcal doman: 0 <x<3. At the output tme t = 0.2, the smallest and largest element areas are respectvely and , and the smallest and largest edge lengths are and , respectvely. Comparng them wth the ntal quas-unform mesh, we see obvously that the mesh ponts are well-redstrbuted, and effcently clustered n farly localzed regons of shock waves and a small porton of the physcal doman contanng the large soluton varatons. The dense jet along the wall has been resolved whch s generally senstve to the numercal dsspaton and one of the crtcal components n ths problem.

13 76 J Sc Comput (2008) 34: Fg. 6 Example 6.2: Adaptve mesh and contour plots of the densty at t = 4 Example 6.2 (The forward facng step problem) Ths problem was frst studed by Emery n [13], and then consdered extensvely by many other researchers, e.g. [37, 45]. The problem begns wth unform Mach 3 flow n a wnd tunnel contanng a step. The wnd tunnel s 1 length unt wde and 3 length unts long. The step s 0.2 length unts hgh and s located 0.6 length unts from the left-hand end of the tunnel. Intally the wnd tunnel s flled wth a gamma-law gas, wth γ = 1.4, whch everywhere has densty 1.4, pressure 1, and velocty 3. Along the walls of the tunnel reflectng boundary condtons are appled. The n-flow and out-flow boundary condtons are specfed at the left- and rght-hand ends of the tunnel. Fgure 6 shows the adaptve mesh and the densty contours at t = 4. The smallest and largest element areas and edge lengths of the fnal mesh are , , and , respectvely. Ths example takes α ρ = 50,β ρ = 0.1, α s = α γ = β s = β γ = 0, and an ntal quas-unform trangulaton of the physcal doman p as well as the logcal doman l s generated by 3424 nodes, edges, and 6578 elements, whose edge length and element area are approxmately equal to and , respectvely. The sonc gltch phenomenon s almost nvsble, Those results are comparable results n [37] obtaned on the unform structure mesh wth x = y = 1/200. Example 6.3 (The quas-2d Remann problem) It s a genune multmateral flow calculaton, but a smple two-dmensonal extenson of the Remann problem of Abgrall [1]. We take the computatonal doman p as [0, 1] [0, ] and specfy the ntal flud varables as { ( , 0, 0, , 1.67) T, for x 0.5, U = ( , 0, 0, , 1.4) T, otherwse, because of the dffculty n preventng the numercal oscllaton of the pressure and velocty, even n one dmensonal case, ths Remann problem s very nterestng and are attractng many researchers attenton. In ths example, we take α ρ = α γ = 60,α s = 0, β s = β γ = 2β ρ = 1, and an ntal quas-unform trangulaton of the physcal doman p as well as the logcal doman l s generated by the horzontal (and vertcal) boundary partton of 150 (and 6) segments. Such quas-unform trangulaton s formed by 1078 nodes, 2915

14 J Sc Comput (2008) 34: Fg. 7 Example 6.3: The 2D adaptve mesh at t = , and the edge length and the computed solutons (ρ, u, p, γ ) at the same tme along the lne y = 0 edges and 1838 elements, whose edge length and element area are approxmately equal to and , respectvely. Fgure 7 shows the two-dmensonal adaptve mesh at t = , and the edge length and the solutons at the same tme along the lne y = 0, where the sold lnes n the soluton plots denote the exact solutons. We see that the left-movng rarefacton wave, the materal nterface, and the rght-movng shock wave are well resolved; the mesh ponts are dstrbuted n farly localzed regons of three waves; and the pressure and velocty are oscllatory-free and constant around the materal nterface. At the output tme t = , the smallest and largest element area are respectvely and , the smallest and largest sde length are and , respectvely.

15 78 J Sc Comput (2008) 34: Fg. 8 Example 6.4: The adaptve meshes at t = 25, 50, 75, 100, 125 Example 6.4 (The shock wave and a Helum cylndrcal bubble nteracton) Ths problem has been extensvely studed by many authors, see e.g. [28]. We examne the nteracton of a M s = 1.22 planar shock wave, movng n the ar, wth a Helum cylndrcal bubble n the physcal doman p =[0, 325] [ 45, 45] wth the top and bottom reflectve boundares, the left nflow and rght outflow boundares. Here M s denotes the shock Mach number. The ntal flow s determned from the shock condton wth the gven shock Mach number. The bubble s assumed to be n both thermal and mechancal equlbrum wth the surroundng ar. More precsely, the ntal dmensonless data are (1, 0, 0, 1, 1.4), f 0 x 225, 44.5 y 44.5, W = (1.3764, 0.394, 0, , 1.4), f 225 <x 325, 44.5 y 44.5, (0.1358, 0, 0, 1, 1.67), f (x 175) 2 + y 2 25, where W = (ρ,u,v,p,γ).

16 J Sc Comput (2008) 34: Fg. 9 Same as Fg. 8 except for the schleren mages of the densty ρ In ths example, we take α ρ = 30,α s = 0,α γ = 20, β ρ = 0.02,β s = 0,β γ = 0.5, and an ntal quas-unform trangulaton of the physcal doman p as well as the logcal doman l s generated by the horzontal (and vertcal) boundary partton of 328 (and 90) segments. Such quas-unform trangulaton s formed by nodes, edges and elements, whose edge length and element area are approxmately equal to 1 and , respectvely. Fgures 8 and 9 show the adaptve meshes and the schleren mages of the densty ρ at t = 25, 50, 75, 100, 125, respectvely, where we have chosen a scalar functon ψ as ( ) ρ ψ = exp k, (6.2) ρ max wth k = 10 n the bubble and 60 otherwse, here ( ) ρ 2 ( ) ρ 2 ρ = +. x y

17 80 J Sc Comput (2008) 34: Fg. 10 Same as Fg. 8 except for denstes along y = 0

18 J Sc Comput (2008) 34: Fg. 11 Example 6.5:The adaptve meshes at t = 30, 60, 90, 120, 150 We see that the adaptve redstrbuton of the computatonal mesh mproves the quantty of the soluton successfully; the materal nterface s captured very well; at the same tme, some small wave structures are also resolved clearly. At the fnal output tme t = 125, the smallest and largest element area are and respectvely, and the smallest and largest sde length are and , respectvely. In Fg. 10, we gve a comparson of denstes at several output tmes projected onto a unform mesh for {x y = 0} obtaned by usng the second order accurate Godunov scheme wth adaptve movng mesh ( ) and wth fxed mesh of trangular elements (sold lne). The results show that the dscontnutes are resolved well and accurately. Example 6.5 (The shock wave and a R22 cylndrcal bubble nteracton) Ths problem s smlar to the above example, but the present gas R22 n bubble s heaver than the ambent ar, whereas Helum s an nert gas that s lghter than ar. The dfferences between Helum and R22 wll yeld dfferent flow patterns around the materal nterface after ts nteracton wth the shock. In ths example, we take the physcal doman p and ts ntal quas-unform trangulaton and boundary condtons same as those n the shock wave and the Helum

19 82 J Sc Comput (2008) 34: Fg. 12 Same as Fg. 11 except for the schleren mages of the densty ρ bubble problem. The ntal data are specfed as (1, 0, 0, 1, 1.4) T, f 0 x 225, 44.5 y 44.5, W = (1.3764, 0.394, 0, , 1.4) T, f 225 <x 325, 44.5 y 44.5, (3.1538, 0, 0, 1, 1.249), f (x 175) 2 + y Fgures 11 and 12 show the adaptve meshes and the schleren mages of the densty ρ at t = 30, 60, 90, 120, 150, respectvely, where we have chosen a scalar functon ψ defned n (6.2) wth k = 4 n the bubble and 80 outsde the bubble. Here we have taken we take α ρ = 30, α s = 0, α γ = 30, β ρ = 0.02, β s = 0, β γ = 0.5. The results show that the mesh ponts are well dstrbuted and mprove the quantty of the soluton effectvely; the wave patterns and the materal nterface are resolved very well. At the fnal output tme t = 150, the smallest and largest element area are and , respectvely, and the smallest and largest sde length are and , respectvely. We also gve a smlar comparson of denstes to Example 6.4 s gven n Fg. 13.Thecom-

20 J Sc Comput (2008) 34: Fg. 13 Same as Fg. 11 except for denstes along y = 0

21 84 J Sc Comput (2008) 34: Table 1 Example 6.5: estmated CPU tmes (mnutes) from t = 0 to 150 Algorthm Element number CPU tme (mnutes) Movng mesh Fxed mesh putatonal effcency comparson between the movng and fxed meshes s shown n Table 1, where the recorded CPU tmes on the Lenovo PC (Pentum IV, 3 GHz) under the Wndows envronment. 7 Conclusons Ths paper extended successfully the prevous adaptve movng mesh method developed by Tang and Tang [39] to two-dmensonal (2D) compressble multcomponent flows and unstructured trangular meshes. The proposed method solved the equatons governng 2D flows and the fnte-volume approxmatons of the mesh equatons by a fully conservatve, secondorder accurate Godunov scheme and a relaxed Jacob-type teraton, respectvely, and mplemented a smple and low-dsspatve slope lmter n each ntal reconstructon stage n order to get oscllaton-free solutons. In the mesh teratve redstrbuton, the geometrybased conservatve nterpolaton was employed to remap the solutons from the old mesh to the newly resultng mesh, and a new montor functon was carefully chosen to track and resolve both small, local, and large soluton gradents automatcally. As a result, the current adaptve mesh method s fully conservatve and non-oscllatory. Its robustness and effcency were demonstrated by several numercal experments. Our future work s to extend the current method to three-dmensonal multcomponent flow problems and nontrval domans. Acknowledgements The authors would lke to thank the anonymous referees for valuable comments and suggestons. Huazhong Tang was partally supported by the Natonal Basc Research Program under the Grant 2005CB321703, the Natonal Natural Scence Foundaton of Chna (No , ), SRF for ROCS, SEM, NCET, and Laboratory of Computatonal Physcs. Pngwen Zhang was partally supported by the Natonal Basc Research Program under the Grant 2005CB References 1. Abgrall, R.: How to prevent oscllatons n multcomponent flow calculatons: A quas conservatve approach. J. Comput. Phys. 125, (1996) 2. Abgrall, R., Karn, S.: Computatons of compressble multfluds. J. Comput. Phys. 169, (2001) 3. Abgrall, R., Saurel, R.: Dscrete equatons for physcal and numercal compressble multphase mxtures. J. Comput. Phys. 186, (2003) 4. Barth, T.J., Jespersen, D.C.: The desgn and applcaton of upwnd schemes on unstructured meshes. AIAA Paper No (1989) 5. Beckett, G., Mackenze, J.A., Robertson, M.L.: An r-adaptve fnte element method for the soluton of the two-dmensonal phase-feld equatons. Commun. Comput. Phys. 1, (2006) 6. Brackbll, J.U.: An adaptve grd wth drectonal control. J. Comput. Phys. 108, (1993) 7. Cao, W.M., Huang, W.Z., Russell, R.D.: An r-adaptve fnte element method based upon movng mesh PDEs. J. Comput. Phys. 149, (1999) 8. Cenceros, H.D., Hou, T.Y.: An effcent dynamcally adaptve mesh for potentally sngular solutons. J. Comput. Phys. 172, (2001)

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