A stable, robust and high order accurate numerical method for Eulerian simulation of spray and particle transport on unstructured meshes

Transcription

1 Center for Turbulence Research Annual Research Brefs A stable, robust and hgh order accurate numercal method for Euleran smulaton of spray and partcle transport on unstructured meshes By A. Larat, M. Massot AND A. Vé 1. Motvaton and objectves The general framework of the present contrbuton s the numercal smulaton of physcal phenomena where a dscrete cloud of dense nclusons, lqud or sold, also called the dspersed phase, s transported wthn a contnuous flud phase. The large range of applcatons ncludes domans of multphase combuston and alumna partcles n rocket engnes, as well as pollutant partcle dsperson or cosmology. Ths paper addresses the dynamcs of the dscrete partcle phase. A common and wdespread way of modelng ths phase s to descrbe t at the level of each ncluson, consdered as a pont partcle: the mcroscopc level. Each partcle nteracts wth ts local surroundng, and ts state s then smply governed by a set of ordnary dfferental equatons. Ths method, referred to as Lagrangan Dscrete Partcle Smulaton, has quckly ganed n popularty for ts relatve smplcty. It s however no longer sutable when a large number of partcles has to be tracked to reach statstcal convergence, as needed n most realstc applcatons. Furthermore, as the partcle phase can become hghly nhomogeneous, overcomng ths ssue by means of parallelzaton s complcated because of load balancng (Garca 009). These drawbacks of the Lagrangan method explan why we are nterested here n a mesoscopc approach, referred to as Euleran. The partcle dstrbuton s represented by a Number Densty Functon (NDF), whose values depend on dfferent parameters: space-tme poston, droplet sze, velocty and temperature. The NDF varatons are then governed by an evolutonary PDE n the parameter space, also called the phase space, whch s known as the Wllams-Boltzmann equaton (Wllams 1958). Because the phase space s hgh-dmensonal, the drect resoluton of the Wllams-Boltzmann s hardly consdered, and Euleran methods nstead solve for a fnte set of moments, whch are ntegrated quanttes over the phase space. Because some nformaton s lost durng ths last process, a closure model has to be gven n order to compensate for the unknown nformaton about hgher order moments. To deal wth the complex dynamcs of partcles n turbulent flows, several closures have been proposed n the lterature (see Vé et al. 01 and references theren). The man pont s that, f the Stokes number based on the Kolmogorov tme scale s close to one, hgh segregaton effects occur, leadng to the formaton of depleton zones and stff accumulatons regons. Such effects can be reproduced usng a monoknetc closure (Laurent & Massot 001; de Chasemartn 009; Fréret et al. 01), whch leads to the well-known Pressureless Gas Dynamcs system of equatons (PGD). Ths system s weakly hyperbolc and s known to generate δ-shocks,.e., strong accumulatons, whch often coexst wth vacuum neghborng regons. Such mathematcal sngulartes are dffcult to handle, and partcular care has to be gven to

2 06 A. Larat, M. Massot and A. Vé the numercal method. We are nterested here n smulatng ths system because, whle beng predctve for low Stokes number, t s a challengng test case. For sprays on structured grds, de Chasemartn (009) has proposed a second order knetc scheme based on the work of Bouchut et al. (003). It uses the underlyng knetc equaton to obtan an exact evaluaton n tme of the fluxes at each nterface, whch leads to a mnmal level of numercal dsspaton. The fnal scheme has the mportant property of ntrnscally preservng the natural propertes at PDE level for the PGD system: the postvty of the number densty and a dscrete maxmum prncple on the velocty, whch s not the case wth classcal lnear hgh order fnte volume schemes. Ths approach has been used for the smulaton of turbulent partculate flows, and has also been adapted to more complex moment methods for polydsperse flows (Kah et al. 01). For unstructured grds, a scheme has been proposed (Kah et al. 01), but t s based on an approxmate evaluaton n tme of the fluxes usng a Runge-Kutta method, and hence no longer mnmzes the dsspaton; ths mposes to reach hgher order for the numercal scheme. Classcal centered schemes, ether fnte element or fnte volume, requre artfcal dffuson to stablze the computaton. Ths approach needs to optmze several parameters, and does not formally prevent the occurrence of negatve number densty, or hgh velocty oscllatons. In ths paper, we present a Dscontnuous Galerkn (DG) formulaton for the smulaton of PGD. Based on the work of Zhang & Shu (010) and Zhang et al. (01), ths scheme s bult to preserve the mean state wthn each element n a convex space of admssble states wthout destroyng the overall arbtrarly hgh order accuracy of the scheme. The convex constrants on the mean states are essental for preservng the postvty of the number densty and the maxmum prncple on the velocty. The content of the paper s the followng: Secton gves a global overvew of the modelng process, n order to understand the key features of PGD that our numercal scheme has to respect. In Secton 3, we descrbe the numercal method n detal, frst n one dmenson of space, for the sake of legblty of our contrbuton, then extended to two dmensonal trangular meshes. Once ths has been done, the extenson to any unstructured mesh s straghtforward. Fnally, numercal results and computaton effcency are dscussed n Secton 4.. Mesoscopc knetc modelng of the dynamcs of partcles We consder a dsperse phase composed of partcles, small enough to neglect ther volume and allow a pont-partcle approxmaton. The statstcs of such a dsperse phase can be represented by a Number Densty Functon (NDF) f (t, x, ϕ) where t s the tme, x the poston and ϕ the phase space composed of as many dmensons as the number of parameters needed to descrbe the state of a partcle. For example, ϕ = ( c, S, T) represents the space of admssble states n terms of velocty, sze and temperature. In ths phase space, the dynamcs of f s governed by the Wllams-Boltzmann equaton (Wllams 1958): t f + x ( c f ) + c ( F f ) + S (R S f ) + T (K f ) = Γ + Q, (.1) where the frst two terms represent the free transport of the spray, F = d t c s the acceleraton due to the drag force comng from the underlyng carrer flud, R S = d t S the rate of change n the droplet sze (evaporaton), K = d t T the rate of change of the droplet temperature and Γ and Q are addtonal source terms to take collson and secondary break-up nto account. By ntroducng a characterstc partcle relaxaton tme

3 A convex state preservng scheme on unstructured meshes 07 τ p, we can further model the drag force nteracton as: F = ( u g c)/τ p, wth u g beng the carryng gas velocty. Gven the hgh number of dmensons of the phase space, the drect resoluton of knetc equaton (.1) s hardly consdered. It s nterestng to ntegrate ths equaton over whole drectons of the phase space and to deduce the equatons governng the evoluton of the varous moments of the NDF, f. In order to focus on transport and drag force, we consder a unque sze for all partcles, and neglect collsons, secondary break-up, evaporaton and heat transfer, thus cancelng the four last terms of (.1). In fact, modelng strateges have already been proposed for each term (de Chasemartn 009; Dosneau et al. 01; Rmbert et al. 01) and we envson takng them nto account usng a specfc splttng strategy, see for example Dosneau et al. (01). If we consder a monoknetc assumpton, we can now wrte the NDF as: f (t, x, c) = ρ(t, x)δ( c u( x, t)) where u(t, x) s the mean velocty of the dsperse phase. By ntegratng Eq. (.1) over the velocty space, t yelds n a straghtforward manner to the PGD system of equatons wth source terms: t ρ + x ρ u = 0, (.) ( ) ρ( u g u) t ρ u + x ρ u u =. (.3) τ If we consder the free transport part of ths system, t has three man propertes (see Bouchut 1994 for detals): (1) t s weakly hyperbolc, n the sense that the Jacoban of the flux has n each drecton a unque egenvalue of multplcty d, the space dmenson, but s non-dagonalzable. Ths mples the ablty of the system to generate δ-shocks,.e., stff accumulatons of partcles, that need to be handled by the numercal scheme; () ρ 0 to ensure that the par of moments (ρ, ρ u) s always lnked to f (t, x, c) 0 (realzablty condton); and (3) u must satsfy a maxmum prncple. Such propertes defne a convex space of vald moment sets, called the realzable space. Therefore convex constrants have to be held on the numercal soluton states to deal wth sngular solutons, such as δ- shocks. Consderng the treatment of the source terms, as the current study s focused on the free transport part, we evaluate the source terms usng the mean value of the state nsde each cell. A specfc treatment adapted to DG can be envsaged, but s out the scope of the present work. 3. A hgh order realzable space preservng numercal scheme on unstructured grds Ths secton summarzes the man deas of Zhang & Shu (010) and Zhang et al. (01), and extends ther framework to the partcular case of the weakly hyperbolc Pressureless Gas Dynamc system One dmensonal analyss In one dmenson of space, the system of conservaton laws can be wrtten: W t + F (W) x = 0, x [0, 1], t [0, T], (3.1) where W(x, t) s the unknown state and F (W) the conservatve flux. In the case of the PGD system, one has W = (ρ, ρu) and F (W) = (ρu, ρu ). A problem wth ntal condton W 0 and perodc boundary condton s consdered: W(0, t) = W(1, t), 0 < t < T, W(x, 0) = W 0 (x), x [0, 1].

4 08 A. Larat, M. Massot and A. Vé Wthout loss of generalty, [0, 1] s dscretzed nto N regular sub-ntervals. Let x 1 = 1 N, x = 1 N, x + 1 =, = 1,.., N, N and C be the nterval ]x 1, x + 1 [ Classcal DG Formulaton If k N s the desred order of the method, let ϕ j (x), j = 1,.., k + 1 be k + 1 bass functons of polynomals of order k wthn C. Wthout further explanaton (see Cockburn & Shu 1998), the Dscontnuous Galerkn approxmaton of equaton (3.1) s the pecewse polynomal soluton W h (x, t) = W j (t)ϕ j (x)χ C (x),, j where subscrpt h stands for a mesh characterstc sze and χ C s C ndcator functon, of the followng dfferental system: ( ) C (M) jk d t W k + F ϕ j + 1 (x + 1 ) F ϕ j 1 (x 1 ), (3.) = F (W h (x)) x ϕ j dx, = 1,.., N, j = 1,.., k, C where M s the dscrete problem mass matrx gven by (M) jk = ϕ j C (x)ϕk (x) dx and F and F are the numercal fluxes at x and x + 1, respectvely. Ths numercal scheme n space has a truncaton error of order h k+1 = N k 1 for regular enough solutons. Furthermore, when a (k + 1) th -order accurate scheme n tme s used to ntegrate each equaton of system (3.), the scheme s globally (k + 1) th -order n space and tme. Ths s always gong to be the case when usng a k + 1-step Runge-Kutta (RK) method Realzablty preservng hgh order scheme In the rest of ths secton, we consder a forward Euler approxmaton n tme, whch s also a 1-step Strong Stablty Preservng (SSP) RK method. Ths restrcton s only for the sake of smplcty and all the results generalze to any SSP tme ntegrator. In the followng, we explan how to generate a realzablty preservng hgh order numercal scheme that satsfes the realzablty condtons defned n Secton for the PGD system. At frst order, the general Lax-Fredrchs scheme wrtes: F + 1 = F (W +1) + F (W ) α +1 W +1 W, (3.3) where α +1 s greater than the egenvalues (veloctes) of the Jacoban of F at W +1, W. The state n C s updated by: W n+1 = W n t C = ( 1 β +1 wth β j defned by ( F F β 1 ) ) W + β +1 Wn +1 F (Wn +1) α +1 = tα j C. β j + β 1 Wn 1 + F (Wn 1) α 1, (3.4)

5 A convex state preservng scheme on unstructured meshes 09 In the case of PGD, we can defne the followng abstract state W,± = W j ± F (W ( j) = (1 ± u j )ρ j, (1 ± u ) j )(ρu) j, j = ± 1, α j α j α j so that f the densty of W j s strctly postve, the resultng state W,± also has a strctly postve densty and the same velocty as W j. Therefore, when the followng CFL condton holds: t(α +1 + α 1 ) 1, C W n+1 has a strctly postve densty and ts velocty s a convex combnaton of those of W 1, W and W +1, hence satsfyng the maxmum prncple. Let us now go to hgher order. Usng an approprate Gauss-Lobatto formula, the followng statement s exact, provded m 3 k: W = W h (x)dx = C m ω q W h (x q ), (3.5) where x q, q = 1,.., m are the coordnates of the Gauss-Lobatto quadrature ponts wthn C and ω q ther assocated strctly postve weghts. For later purposes, we extend these quadrature ponts to q = 0,.., m + 1, settng x 0 (resp. x m+1 ) as the coordnate of the rght (resp. left) quadrature pont n the left (resp. rght) neghborng cell. Then by summng Eq. (3.) over all the degrees of freedom j of one cell C, assumng the tme dervatve terms have been approxmated by a forward Euler scheme, we obtan: W n+1 = W n t C q=1 ( F F ), (3.6) where the numercal fluxes F and F depend on the reconstructed states on each sde of the cell boundary and one can wrte: F + 1 m F = F (x ) F (x 1 ) = F (x q+1 ) F (x q ). (3.7) Combnng (3.6), (3.5) and (3.7), we fnally get m ( W n+1 = ω q W h (x q ) t ( F (x q+1 ) F (x q ) )). (3.8) ω q C q=0 All quadrature weghts beng strctly postve, ths s a convex combnaton of frst order abstract updates at quadrature ponts. Thus, W n+1 has a strctly postve densty and ts velocty s a convex combnaton of the veloctes at quadrature ponts, f the densty s strctly postve at all the quadrature ponts and the stronger CFL constrant t.α q+1 q C q=1 mn q ω q, (3.9) holds everywhere. Even though the last restrcton on the tme step could appear as very strong, t s n fact close enough to the CFL constrants of the current, wdely used, RK-DG methods (see Zhang et al. 01 for more detal).

6 10 A. Larat, M. Massot and A. Vé Mean value and accuracy preservng realzablty space projecton The last thng s now to ensure that the numercal soluton W h respects the constrants at the quadrature ponts. To acheve ths, we need a smart projecton at each tme step that ensures the last property, wthout changng the cell mean value and wthout destroyng the soluton accuracy. The dea s that for any quadrature state W q lyng outsde the space of constrants, there exst a unque θ [0, 1] such that W q = θ q W q + (1 θ q )W s at the boundary of the contrant space. Wthn each cell C, we can now redefne the numercal soluton as W h (x) = θ (W h (x) W ) + W, θ = mn q=1,..,m θ q. Frst, the mean value of the updated numercal soluton s obvously W. Next, f the contnuous soluton s smooth enough, W h W = O(h) and W h W h = O(h k+1 ) f θ 1 = O(h k ). Ths last property s thoroughly demonstrated n Zhang (011), whch ends the proof that W h s a k + 1 th -order approxmaton of W h, respectng the convex contrants at all the quadrature ponts of the mesh. 3.. Realzablty preservng DG method n two dmensons What mostly prevents the straghforward generalzaton of the prevous reasonng to two dmensonal trangular meshes s how to defne an exact quadrature wth postve weghts on the trangles such that an equvalent form of Eq. (3.7) can be wrtten. We frst recall the DG varatonal formulaton of a D conservaton law. Let F (W) = ( f (W), g(w)) be a conservatve flux and F (W, W j, n j ) an assocated numercal flux across an edge wth normal n j, separatng two states W and W j. F s chosen to be realzablty preservng, lke the general Lax-Fredrchs flux n (3.4). If T s a trangle of the mesh and j a degree of freedom of ths trangle wth ϕ T j (x) ts assocated kth -order bass functon, the scheme reads: T (M) jk d t W T j + F (W T ext (s), W nt (s), n(s))ϕ T j (s) ds = F (W(x)). T ϕ T j dx. (3.10) As n the prevous subsecton, we now sum over all the degrees of freedom j of T and get the equaton governng the evoluton of the mean value n T: d t W T + F (W ext (s), W nt (s), n(s)) ds = 0. (3.11) T The contour ntegral n the prevous equaton can be estmated at the accuracy of the scheme usng the approprate Gauss quadrature. Let then Nq T denote the number of Gauss quadrature ponts used per edge and Nq T the number of quadrature ponts used to compute the rght hand sde of (3.10) on the whole element. If we want to generalze formula (3.8) to two dmensons n order to propagate the convex constrant from one tme step to another, we need to fnd an element quadrature formula wth the followng propertes: t s exact for polynomal of order k; all ts quadrature weghts are strctly postve; ts restrcton to the edges of the element are the Gauss quadrature ponts used to ntegrate the numercal flux on T. A general procedure to acheve such a quadrature s gven n Zhang et al. (01). The man dea s that such a quadrature naturally exsts on quadrangles by the tensoral

7 A convex state preservng scheme on unstructured meshes 11 product of 1D Gauss quadrature ponts n one drecton and 1D Gauss-Lobatto quadrature ponts n the other drecton. Such a D quadrature can then be carred on trangles usng the three non one-to-one Q 1 transformatons of the plan whch send the top edge of the reference quadrangle onto each of the three vertces of the trangle respectvely and the remanng three edges of the quadrangle on the three edges of the trangle. Zhang et al. (01) descrbe thoroughly the transformatons, the three resultng set of quadrature ponts and the quadrature weghts of the new quadrature: the superposton of the three obtaned quadrature. It s then shown that the new quadrature fulflls the three above requrements. Fnally, when a SSP scheme n tme s used to approxmate the tme dervatve n Eq. (3.11), e.g., forward Euler, when the D new quadrature s used to replace the mean state n T at tme step n (W n T), and all these terms are smartly rearranged, the mean state n T at tme step n + 1 (W n+1 T ) can be expressed as a convex combnaton of the states n the nteror of T and convex constrants preservng fluxes between the states at the Gauss quadrature ponts on the edges of T and ts neghbors (see Zhang et al. 01 for more detals). Thus, f the states at all the quadrature ponts le n the space of constrants, automatcally does n too. The fnal process s then to ensure that the constrants are verfed at each quadrature pont of each trangle of the mesh and ths s done usng the same algorthm as the one descrbed at the end of the 1D subsecton. W n+1 T 4. Numercal results Ths secton now presents results on PGD test cases n one and two dmensons of space. However, n order to assess the accuracy of the method, a smple lnear advecton problem wth a relatvely smooth ntal condton s frst consdered D test cases In order to valdate our understandng and the performance of the general scheme, we have frst appled t on one dmensonal test cases. The frst one s a lnear scalar advecton of a regularzed hat functon. The goal s to show that the scheme behaves well n the presence of pseudo-dscontnutes, that no oscllatons occur and that we stll keep the desred order of accuracy. Next, we proceed to a more complex case n the Pressureless Gas Dynamc (PGD) framework. We study the collson of two symmetrc one dmensonal droplet clouds. The soluton s known to produce a Drac sngularty n fnte tme on the densty component, what s called a δ-shock The regularzed hat lnear advecton Let us consder the smple one dmensonal lnear advecton equaton u t + a u x = 0, (x, t) [0, 1] [0, 1 ], (4.1) a where u s a scalar unknown and a s the constant advecton speed (a = 1 wthout loss of generalty). The perodc boundary condton s ensured by u(0, t) = u(1, t), t. Fnally, the ntal condton s gven n the form of a regularzed hat-shaped functon. Gven a

8 1 A. Larat, M. Massot and A. Vé e e-06 1e e Fgure 1. Numercal smulatons of problem (4.1) wth ntal condton (4.). ε = Left: numercal soluton after one complete rotaton ( ) and exact soluton (sold). N = 50 and thrd order DG scheme. Rght: convergence curves of 1st (+), nd ( ) and 3rd (+ ) order formulatons. Slopes of sold lnes are respectvely 0.65,.08 and thckness parameter 0 < ε < 0.5, we prescrbe for all x [0, 1]: 0.5 ( 1 + tanh ( ε 0.5 x + ε 0.5+ε x)), 0.5 x ε, 1, ε x 0.75 ε, u(x, 0) = u 0 (x) = 0.5 ( 1 + tanh ( ε 0.75 x ε 0.75 ε x)), 0.75 ε x 0.75, 0, otherwse. (4.) Because the ntal condton u 0 takes ts value between 0 and 1, we want u h (x, t) to stay between these two bounds. Ths s here our constrant. We then perform a seres of smulatons wth dfferent values of h = δx = 1/N and 0 < ε < 0., and compare the soluton at tme t = 1 (after t has completed a full rotaton around [0, 1]), wth the ntal condton. Results are presented n Fgure 1. On the top left mage, we can apprecate the good behavor of the numercal soluton even when the exact soluton s stff: ε = No slope lmter has been used here and the CFL constrant s 0.9 tmes the theoretcal stablty constrant. Ths means (see 3.9) t < 0.9h at frst order, t < 0.45h at second order and t < 0.15h at thrd order. We dd not use 1.0 because we wanted to avod the exact soluton obtaned at frst order. The varaton of the mean values of the cells s perfectly monotonc. Moreover, when lookng at the convergence curves, the order of accuracy s very satsfyng. The convergence slope at thrd order n space and tme meets the theoretcal order of accuracy, even for small values of ε. We observe two dstnct behavors: when dx = 1/N < ε/10 the scheme converges at lower order, but t recovers ts theoretcal order as soon as enough ponts le n the stff regon. The convergence on coarser meshes s however much better than the expected frst order when the numercal scheme sees a shock and t fully recovers the hgher order when the representaton of the exact soluton s smooth enough The PGD Drac sngularty We now present a more complex problem. We consder the one dmensonal Pressureless Gas Dynamc system, wth ρ the densty and ρu the frst order velocty momentum: ρ t + ρu x = 0, ρu t + ρu x (4.3a) = 0. (4.3b)

9 1 A convex state preservng scheme on unstructured meshes Fgure. Snapshots of the soluton of problem (4.3) wth ntal condtons (4.4)-(4.5). Left: Intal soluton, ρ and u = ρu. Rght: ρ and u at t = 0.5, thrd order DG. Vaccum has been created far from the ρ δ-shock regon and ρ s close to machne zero. Ths s why u = ρu s undefned outsde nterval [0.4, 0.58]. ρ We are stll nterested n the unty doman [0, 1] wth perodc boundary condtons. The frst equaton s a transport equaton at advecton speed u. Next, when ρ and u are both regular enough, partal dervatves of Eq. (4.3b) can be expanded and by subtractng the frst relaton we get the Burgers equaton on u. Intal condton { a x 0.5 u(x, 0) = u 0 (x) = a x > 0.5, (4.4) s known to be a steady soluton of the Burgers equaton. We set a = 1 and ntalze ρ wth a functon whch s very regular on [0, 1] and symmetrc wth respect to 0.5, let us say ρ(x, 0) = ρ 0 (x) = sn 4 (πx). (4.5) The symmetrc perturbatons n ρ are then propagatng toward each other and the densty s rreversbly concentratng at x = 0.5. The soluton of system (4.3) wth ntal condton (4.4)-(4.5) (whch s C 1 when consderng the conserved state W = (ρ, ρu)) becomes sngular and steady at tme t = 0.5 wth ρ = m δ 0.5, u = 1 f x < f x = f x > 0.5, (4.6) where m denotes the conserved total mass, and δ x the Drac dstrbuton. These fgures show the evoluton of the partcle densty. The constrant space s preserved all along the smulaton. At tme t = 0.5, all the matter s concentrated n a sharp needle at x = 0.5 and ρ s postve everywhere. The δ-shock s contaned wthn cells on each sde of the sngularty and ths stays true when refnng the mesh. Detal of the velocty shows that the condton 1 < u < 1 stll apples, and u vares monotoncally. Furthermore, vacuum has been created outsde of the nterval [0.4, 0.58], where u = ρu ρ s undefned. Ths proves the low level of numercal dffuson close to transton to vacuum zones. 4.. D test case: a homogeneous spray over a frozen HIT In ths subsecton, we are nterested n testng the robustness of the algorthm on two dmensonal cases where sngulartes such as δ-shocks occur. The goal s to be able to capture these sngulartes accurately (possbly wthn one mesh cell), whle mantanng the postvty of the densty and ensurng a maxmum prncple on the advecton velocty. Ths last test case s a one way nteracton between a monoknetc spray whch s homogeneous at t = 0 and a steady homogeneously turbulent velocty feld. As explaned

10 14 A. Larat, M. Massot and A. V e Fgure 3. Number of partcles at steady state for the Frozen HIT problem. Top-Left: Reference soluton wth a nd -order FV-lke scheme on a 56 structured mesh. Top-Rght: Our DG soluton on the same mesh where quadrangles have been cut nto trangles. Bottom-Left: Our DG soluton on the structured 64 1 mesh cut nto trangles. Bottom-Rght: DG soluton on a completely unstructured mesh wth h = 64 at the boundary. n Secton, under the monoknetc assumpton the Wllams-Boltzmann equaton s equvalent to the PGD wth source term system (.), wth a zero pressure-lke tensor. Let us underlne that a refned ntegraton of the spatally varyng source term, even n the case of a non-lnear source, could be ntegrated wthout much dffcultes n the present framework, provdng a hgher order spatal couplng between the gaseous feld and the droplet repartton. However, we use here a smple ntegraton of the source term for the sake of comparson wth the FV method proposed n ths contrbuton. The rght hand sde s taken nto account wth a second order Strang splttng method. The underlyng gas velocty feld beng gven, steady and constant per element, the ODE gven by the splttng can be analytcally ntegrated, whch greatly smplfes the mplementaton. The smulaton begns wth a constant droplet densty throughout the doman, after whch the computaton s allowed to converge toward steady state. We compare the result gven by the new unstructured second order method wth the one gven by a structured second order TVD fnte volume method (de Chasemartn 009). Ths s shown n Fgure 3. Top-rght s the reference structured soluton. The structured mesh has frst been cut nto trangles and second order DG method has been run on the obtaned mesh. The DG result provdes much fner structures and droplet clusters than the FV result. Ths s

11 A convex state preservng scheme on unstructured meshes 15 due to smaller numercal dsspaton. On the other hand, the DG computaton requres a much hgher CPU tme, about 5 more tme per degree of freedom. We have then been runnng the same case on a coarser mesh. As one can see n Fgure 3, the result obtaned wth DG on a h = 1/64 unstructured mesh s qualtatvely comparable to the 56 structured FV reference result. In terms of computatonal tme, the FV computaton took 14.36s, when the unstructured DG on the coarser unstructured mesh took 13.08s. Of course, ths study s only qualtatve and further tools need to be developed to assess the compettveness of our DG method, but t stll s a very promsng comparson between FV and DG approaches. 5. Fnal words In ths paper, we have presented a very promsng numercal method for the Euleran drect smulaton of partcle and spray dynamcs n turbulent flows. These dynamcs are descrbed n a Euleran pont of vew, through hyperbolc or weakly hyperbolc systems of conservaton laws for the evoluton of the moments of the NDF. A realzablty preservng arbtrary hgh order Dscontnuous Galerkn method has been desgned n order to solve the sngularty and vacuum generatng Pressureless Gas Dynamcs system coupled to Stokes drag force for monodsperse nclusons. Through the dfferent test cases consdered, we have apprecated the robustness and accuracy of the DG method, even when consderng sotropc unstructured grds and sngular soluton. Furthermore, for an equvalent qualty of the results, the computatonal tme of DG, even f desgned for unstructured grds, s comparable to the FV scheme wth optmal numercal dsspaton on structured grds. In the ultmate objectve of desgnng an accurate, robust and effcent scheme for unstructured meshes, the present DG scheme appears to be an nterestng frst step, as t fulflls all the realzablty and robustness propertes, whch are of prmary nterested for applcatons to realstc confguratons n complex geometres. Acknowledgments The authors would lke to thanks the support of the CTR for the stay of Adam Larat as well as for the techncal support durng the Summer Program 01, and the SAFRAN group, whch has sponsored the stay of Aymerc Vé. The support of the France-Stanford Center for Interdscplnary Studes through a collaboratve project grant (P. Mon/M. Massot) s also gratefully acknowledged. REFERENCES Bouchut, F On zero pressure gas dynamcs. Advances n knetc theory and computng, Bouchut, F., Jn, S. & L, X. 003 Numercal approxmatons of pressureless and sothermal gas dynamcs. SIAM J. Numer. Anal. 41 (1), de Chasemartn, S. 009 Euleran models and numercal smulaton of turbulent dsperson for polydsperse evaporatng sprays. PhD thess, Ecole Centrale Pars, France, avalable onlne at Cockburn, B. & Shu, C The Runge-Kutta dscontnuous Galerkn method for conservaton laws V - multdmensonal systems. JCP 141 (), Dosneau, F., Laurent, F., Murrone, A., Dupays, J. & Massot, M. 01 Euleran

12 16 A. Larat, M. Massot and A. Vé mult-flud models for the smulaton of dynamcs and coalescence of partcles n sold propellant combuston. J. Comp. Phys., In press, do: /j.jcp Dosneau, F., Sbra, A., Dupays, J., Murrone, A., Laurent, F. & Massot, M. 01 An effcent and accurate numercal strategy for two-way couplng n unsteady polydsperse moderately dense sprays: applcaton to Sold Rocket Motor nstabltes. J. Prop. Power, Submtted. Fréret, L., Thomne, O., Laurent, F., Revellon, J. & Massot, M. 01 Drect Numercal Smulaton of polydsperse evaporatng sprays n 3D jet confguraton usng euler-euler and euler-lagrange formalsms. In Proc. Summer Prog. 01. Center for Turbulence Research, Stanford Unversty. Garca, M. 009 Development and valdaton of the Euler-Lagrange formulaton on a parallel and unstructured solver for large-eddy smulaton. PhD thess, Unversté Toulouse III. Kah, D., Laurent, F., Massot, M. & Jay, S. 01 A hgh order moment method smulatng evaporaton and advecton of a polydsperse spray. J. Comput. Phys. 31 (), Laurent, F. & Massot, M. 001 Mult-flud modelng of lamnar poly-dspersed spray flames: orgn, assumptons and comparson of the sectonal and samplng methods. Combust. Theory & Modellng 5, Rmbert, N., Dosneau, F., Laurent, F., Kah, D. & Massot, M. 01 Twolayer mesoscopc model of bag-break-up n turbulent secondary atomzaton. In Proc. Summer Prog. 01. CTR, Stanford Unversty. Vé, A., Mas, E., Smonn, O. & Massot, M. 01 On the drect numercal smulaton of moderate-stokes-number turbulent partculate flows usng Algebrac-Closure- Based and Knetc-Based Moment Methods. In Proc. Summer Prog. 01. CTR, Stanford Unversty. Wllams, F. A Spray combuston and atomzaton. Phys. Fluds 1, Zhang, X. 011 Maxmum-prncple-satsfyng and postvty-preservng hgh order schemes for conservaton laws. PhD thess, Brown Unversty. Zhang, X. & Shu, C.-W. 010 On maxmum-prncple-satsfyng hgh order schemes for scalar conservaton laws. J. Comput. Phys. 9 (9), Zhang, X., Xa, Y. & Shu, C.-W. 01 Maxmum-prncple-satsfyng and postvtypreservng hgh order dscontnuous Galerkn schemes for conservaton laws on trangular meshes. J. Sc. Comput. 50 (1), 9 6.