Order of Accuracy Study of Unstructured Grid Finite Volume Upwind Schemes

Transcription

1 João Luz F. Azevedo et al. João Luz F. Azevedo Comando-Geral de Tecnologa Aeroespacal Insttuto de Aeronáutca e Espaço IAE São José dos Campos, SP, Brazl Luís F. Fguera da Slva lusfer@esp.puc-ro.br Pont. Unv. Católca do Ro de Janero PUC-Ro Ro de Janero, RJ, Brazl Danel Strauss danel@tecss.com.br Unversdade de São Paulo USP Escola Poltécnca São Paulo, SP, Brazl Order of Accuracy Study of Unstructured Grd Fnte Volume Upwnd Schemes A detaled numercal study s presented of the order of accuracy of some proposed cell centered, fnte volume schemes used for the soluton of the 2-D gasdynamc equatons on trangular unstructured grds. The schemes studed are based on a MUSCL-type lnear reconstructon of nterface propertes, whch seeks to acheve 2nd-order accuracy n space. They are also nomnally flux-vector splttng-type schemes, and the results here presented use Lou s AUSM+ algorthm. The basc aspects effectng the scheme s order of accuracy are the form n whch the reconstructon process s desgned and the form n whch the lmtng process s performed. Two basc concepts are tested wth regard to the reconstructon process, namely the use of 1-D-type and gradent-based reconstructon. The lmter can also be constructed as a 1-D-type lmter or as a truly mult-dmensonal lmter. The schemes are tested on a lnear convecton-lke model equaton and the numercal solutons are compared to the analytcal soluton, for dfferent mesh szes, n order to assess the scheme s order of accuracy. For comparson purposes, the results obtaned wth a centered scheme are also presented. Second-order accuracy s shown to be only obtaned for the centered scheme. The nomnally 2nd-order upwnd algorthms lead to actual orders of accuracy, whch vary from 0.9 to 1.5. Keywords: CFD, unstructured grd methods, fnte volume, upwnd schemes, order of accuracy. Introducton 1 Snce the poneerng work of Barth and Jespersen (1989), varous upwnd, nomnally second order accurate, fnte volume schemes have been proposed n the lterature (see, for nstance, Durlofsky, Engqust and Osher, 1992; Ln, Wu and Chn, 1993; Venkatakrshnan, 1995; Aftosms, Gatonde and Tavares, 1995; Slegh et al., 1998; Fguera da Slva, Azevedo and Korzenowsk, 1999). In these papers, spatal second order accuracy s sought by some form of gradent evaluaton wthn the control volume, followed by extrapolaton of the cell centered propertes up to the cell nterfaces. Upwndng s acheved va flux vector splttng or flux dfference splttng technques. Lmtng procedures are, then, used n order to guarantee soluton monotoncty. Assessment of the effectve order of accuracy for unstructured fnte volume methods s not a straghtforward task, when compared to classcal structured fnte dfference/volume technques. In these latter cases, an order of accuracy study may be performed va Taylor seres expansons of the dfference scheme (Hrsch, 1988). Such a study s not applcable when the mesh pont connectvty s varable, as t s the case for fnte volume unstructured grd methods. Therefore, numercal computatons of model problems seem to be the only avenue to pursue, f one seeks to determne the order of accuracy of such methods. For nstance, Aftosms, Gatonde and Tavares (1995) studed the convergence and accuracy of dfferent cell vertex, unstructured mesh, and fnte volume algorthms by comparson wth a steady analytcal soluton of the Euler equatons. Another model problem that can be used for these purposes s the two-dmensonal lnear advecton problem (Durlofsky, Engqust and Osher, 1992), whch also allows the assessment of the unsteady behavor of the soluton. The present work presents a detaled numercal study of the order of accuracy of some proposed cell centered, fnte volume schemes typcally used for the soluton of the 2-D gasdynamc equatons on trangular unstructured grds (Barth and Jespersen, 1989; Fguera da Slva, Azevedo and Korzenowsk, 1999). The schemes studed are nomnally second order accurate, based on a Paper accepted August, Techncal Edtor: Arsteu Slvera MUSCL-type lnear reconstructon of nterface propertes, and they are also nomnally flux-vector splttng schemes. The basc aspects effectng the scheme s order of accuracy are the form n whch the reconstructon process s desgned and the form n whch the lmtng process s performed. Most of the results here presented consder the mnmod lmter, although the superbee lmter s also used (Hrsch, 1990). Two basc concepts are tested wth regard to the reconstructon process. The frst concept essentally attempts to create a onedmensonal stencl normal to the control volume edge of nterest and, then, t uses ths 1-D stencl n a very straghtforward fashon n order to reconstruct nterface propertes. The other approach s based on computng cell averaged property gradents and usng these n order to obtan lnear reconstructed nterface propertes. The lmter can also be constructed as a 1-D-type lmter or as a truly multdmensonal lmter. The schemes are tested on the lnear convecton-lke model equaton (Durlofsky, Engqust and Osher, 1992) and the numercal solutons are compared to the analytcal soluton, for dfferent mesh szes, n order to assess the scheme s order of accuracy. The results obtaned wth the upwnd schemes are also compared to those computed wth a centered scheme (Jameson and Baker, 1983; Jameson and Mavrpls, 1986). Nomenclature a = constant advecton velocty for model problem a x, a y = cartesan components of constant advecton velocty C = convecton operator CFL = Courant number d = vector poston D = artfcal dsspaton operator e = specfc nternal energy ε = total energy per unt of volume E, F = flux vector for the Euler equatons, j = unt vectors n Cartesan coordnates l = length of control volume edge n = unt vector normal to control volume edge p = pressure Q = vector of conserved propertes for the Euler equatons r = rato of consecutve gradents 78 / Vol. XXXII, No. 1, January-March 2010 ABCM

2 Order of Accuracy Study of Unstructured Grd Fnte Volume Upwnd Schemes S = boundary of control volume t = tme u = conserved property for model problem u, v = velocty components n Cartesan coordnates V = area of control volume x, y = Cartesan coordnates Greek Symbols α 1 α 5 = coeffcents n Runge-Kutta tme marchng scheme γ = tme step value Δt = tme step value Δr = vector poston wth respect to the cell centrod = gradent operator ρ = densty ψ = lmter ξ = 1-D type gradent of conserved varable Subscrpts k k L R Ω current control volume nterface between control volumes and k neghbor f -th control volume left state rght state control volume for gradent calculaton Superscrpts l Runge-Kutta stage counter n current tme level n 1, n 2 end ponts of k-th edge + rght state left state The Model Problem In order to be able to analyze the order of accuracy of unstructured fnte volume methods, one needs to numercally solve a model problem, the choce of whch s dctated by several constrants. Obvously, the model problem must have a known analytcal soluton at all tmes. Another desrable feature s that ths soluton should be contnuous, so that the order of accuracy can be measured by comparson wth the computed result va successve refnements of the computatonal mesh. Moreover, t s essental that the numercal results are not nfluenced by the condtons arsng at the boundares of the computatonal doman. Wth these restrctons n mnd, and followng the work of Durlofsky, Engqust and Osher (1992), the authors have chosen as model problem the twodmensonal, perodc, lnear advecton problem, usng as ntal condtons for the scalar quantty a sne curve n both drectons. The computatonal doman s a square wth unty sde and the mesh s composed of regular trangles. A sketch of the computatonal doman and of the analytcal soluton to the model problem at the ntal condton and, thus, also at the end of a full perod of ntegraton s shown n Fg. 1. The procedure starts wth a coarse mesh, n whch the computatons are run through one advecton perod and the L 1 norm of the error s calculated. The mesh s halved successvely, and the computatons rerun. The slope of the best lne ft, n a least squares sense, through a plot of the logarthm of the L 1 norm as a functon of the logarthm of the characterstc sze of the mesh, gves a measure of the actual order of accuracy of the method. Fgure 1. Sketch of the computatonal doman and of the model problem soluton. Mathematcal Formulaton For the classes of problems of nterest to the authors, the approprate theoretcal formulaton for the flud dynamc problems would be based on the 2-D Euler equatons. For the present paper, however, a two-dmensonal lnear advecton problem s consdered. The authors emphasze that the paper mantans a parallel between the problem actually beng solved n the present case and the 2-D Euler equatons as an attempt to facltate the nterpretaton of the procedures developed. Clearly, the motvaton for the development here dscussed s the soluton of the Euler equatons for aerospace applcatons. The authors, however, are applyng the methodology to a model advecton equaton as a form of performng a deeper analyss of order of accuracy of the proposed approaches. Therefore, the governng dfferental equaton for the present case can be wrtten as u + = t ( au) 0. (1) Here, u(x, y, t) s the dependent varable, ( ) s the dvergent operator and the constant advecton velocty, a, can be wrtten as a = a x + a y j, (2) where and j represent the unt vectors n the Cartesan drectons. As prevously dscussed, the scalar problem s subjected to an ntal condton of the form (,,0) sn ( 2π ) sn ( 2π ) u x y = x y, (3) and the boundary condtons are perodc on all four sdes of the square computatonal doman. For the sake of completeness, the Euler equatons are presented, snce the same general nomenclature s used when dealng wth the model problem. The 2-D Euler equatons for an deal gas can be wrtten n ntegral form for a 2-D Cartesan coordnate system as: J. of the Braz. Soc. of Mech. Sc. & Eng. Copyrght 2010 by ABCM January-March 2010, Vol. XXXII, No. 1 / 79

3 João Luz F. Azevedo et al. t V ( ) 0 S Qdxdy + Edy F dx =. (4) Here, V represents the area of the control volume and S s ts boundary. For a statonary mesh, the vector of conserved quanttes Q and the convectve flux vectors are gven by ρ ρu ρv 2 u p uv ρu ρ + ρ. (5) Q=, E=, F= 2 ρv ρuv ρv + p ε ( ε + pu ) ( ε + pv ) The nomenclature used here s the standard one, such that ρ s the densty, u and v are Cartesan velocty components, p s the pressure and ε s the total energy per unt of volume. Equaton (4) must be supplemented by the equaton of state for deal gases, whch can be wrtten as ( ) p = γ 1 ρe, (6) where γ s the rato of specfc heats and e s the specfc nternal energy, whch can be obtaned from the total energy by the expresson ε = ρ 1 e ( u + v ). (7) The governng equatons are dscretzed n a cell centered context, n whch the dscrete vector of conserved varables for the -th cell s defned as 1 Q = dxdy V Q, (8) V where V s the volume of the -th cell. The Euler equatons can, then, be rewrtten for each -th control volume as t ( V ) + ( dy dx) 0 S Q E F =. (9) It should be observed that, for a cell centered approach, the control volume used for the ntegraton of the governng equatons s formed by each trangular cell tself (Batna, 1991). The role of the spatal dscretzaton algorthm s to approxmate the surface ntegral n Eq. (9). Ths aspect s dscussed n detal n the forthcomng paragraphs. For now, t s suffcent to defne the convectve operator, C(Q ), whch s responsble for ths spatal dscretzaton. Hence, the Euler equatons, fully dscretzed n space and assumng a statonary mesh, can be wrtten as dq dt 1 = ( ) ( ) V CQ DQ (10) Here, D(Q ) represents the artfcal dsspaton operator whch s requred when a centered scheme s used, and t s dentcally zero, f an upwnd spatal dscretzaton s employed. In order to use the formulaton developed for the Euler equatons n the analyss of the scalar problem, one must dentfy the vector of conserved varables and the flux vectors wth ther approprate defntons for the present case. Hence, the defnton of these vectors n the scalar case becomes Q u, E au x, F au. y (11) It s, probably, mportant to further emphasze that the test case, whch s actually beng solved n the present work, s lnear and smooth. However, the numercal methods that are mplemented rely on lmters n order to acheve the desred behavor for real lfe problems. These lmters, on the other hand, are typcally nonsmooth and non-dfferentable nonlnear functons. Therefore, t would be correct to state that the present study s actually assessng how such nonlneartes affect the order of accuracy of the resultng schemes. As the work n the paper demonstrates, they clearly have a detrmental effect. However, one cannot do away wth such nonlneartes, because they are precsely the ngredents, whch allow the resultng schemes to perform adequately for the gasdynamc problems of nterest n aerospace engneerng. Tme Dscretzaton Algorthm The present work uses a well-tested, fully explct, 2nd-order accurate, 5-stage Runge-Kutta tme-steppng scheme (Mavrpls, 1988) to advance the governng equatons n tme. The tme ntegraton scheme can be wrtten as ( 0) n Q = Q, Δt Q = Q C Q, l = 1,...,5, Q ( ) () l ( 0) ( l 1) αl V n+ 1 ( 5) = Q, (12) where the superscrpts n and n+1 ndcate that these are property values at the begnnng and at the end of the n-th tme step. The values used for the α l coeffcents are (Mavrpls, 1988) α1 =, α2 =, α3 =, α4 =, α5 = 1. (13) Spatal Dscretzaton Schemes The prmary nterest n the present work s to dscuss order of accuracy ssues assocated wth unstructured upwnd schemes. In partcular, the emphass s on trangular grds and flux-vector splttng schemes. Hence, a scalar verson of Lou s AUSM + scheme (Lou, 1996) s presented both as a nomnally 1st-order scheme and as a nomnally 2nd-order scheme. The 2nd-order verson uses a MUSCL reconstructon (van Leer, 1979), and the reconstructon process optons mplemented are dscussed n detal n the forthcomng sectons. For comparson purposes, a centered scheme s also mplemented and ts detals are presented n the next secton. Note that all schemes presented are mplemented usng an edge based data structure (Azevedo and Mtchell, 1995; Azevedo and Korzenowsk, 1996). Centered Scheme The spatal dscretzaton procedure used s equvalent to a central dfference scheme. The convectve operator, C(Q ), whch approxmates the surface ntegral n Eq. (9), s defned as 80 / Vol. XXXII, No. 1, January-March 2010 ABCM

4 Order of Accuracy Study of Unstructured Grd Fnte Volume Upwnd Schemes 3 ( ) = ( k)( k k ) C Q EQ y y k = ( k )( xk xk ) F Q 2 1 (14) In ths expresson, Q k s the arthmetc average of the conserved propertes n the cells, whch share the k nterface (Jameson and Mavrpls, 1986),.e., 1 Qk = ( Q + Q k ). (15) 2 Moreover, n the prevous expresson ( xk, y ) and 1 k ( x ) 1 k, y 2 k2 are, as shown n Fg. 2, the coordnates of the vertces, whch defne the nterface between cells and k. These ponts are always ordered n the counterclockwse drecton for each -th control volume. Note that, n the case of the Euler equatons, artfcal dsspaton terms must be added n order to control nonlnear nstabltes (Mavrpls, 1990). In the present lnear scalar case, artfcal dsspaton terms are not requred and they were not used n the smulatons here reported. Fgure 2. Sketch of the extrapolaton stencl used for prmtve varable lnear reconstructon n the 2nd-order upwnd scheme. AUSM+ Scheme The convectve operator for the AUSM + scheme, n the present unstructured grd context (Azevedo and Korzenowsk, 1998), and for the scalar advecton problem, can be wrtten as where 3 CQ ( ) = ( Fkl k ), (16) k= 1 + Fk = au k L + au k R = 1 1 = ak ul + ur ak ur ul 2 2 ( ) ( ) (17) and l k denotes the length of the k nterface. The current nomenclature consders that s the trangle to the left of the k nterface (see Fg. 2), snce all nterface segments are assumed orented as prevously ndcated. Therefore, for a 1st-order scheme, the left state, L, s dentfed wth the propertes of the -th trangle whereas the rght state, R, s dentfed wth those of the k-th trangle. The AUSM + scheme performs a splttng of the u ± a egenvalues of the Euler equatons usng the M ± 1 base functons. Usng the nomenclature prevously ntroduced, one can wrte the convectve velocty n the drecton normal to the edge under consderaton as a = an = an + an. (18) k x xk y yk The above equaton clearly uses the fact that, n the present case, a s a constant. Actually, f one follows the usual nomenclature of usng subscrpts L and R to denote left and rght states, respectvely, at a gven nterface, t s also possble to wrte that, n the present case, ak = al = a. (19) R Hence, the splt convectve speeds at the k nterface can be defned as 1 a ± k = ( ak ± ak ). (20) 2 In order to obtan second order accuracy, and keepng wth a MUSCL approach, the left and rght states at the nterface must be somehow lnearly reconstructed at the nterface. As prevously mentoned, the extenson to 2nd-order accuracy s obtaned n the present work wth the applcaton of the MUSCL approach (van Leer, 1979; Anderson, Thomas and van Leer, 1986). Therefore, the second order scheme follows exactly the same formulaton, except that the left and rght states are obtaned by a MUSCL extrapolaton descrbed n the followng secton. Reconstructon Methods Two basc reconstructon procedures are tested n the present work. The frst one defnes a one-dmensonal stencl normal to the control volume edge of nterest and, then, t uses ths 1-D stencl n a very straghtforward fashon n order to reconstruct nterface propertes. The other approach s based on computng cell averaged property gradents, and usng these n order to obtan lnear reconstructed nterface propertes. It s stll mportant to note that the latter approach s also mplemented n two qute dfferent forms n the present work. Ths dfference s assocated wth the gradent calculaton, whch s computed n the standard fnte volume fashon n whch the dervatves are transformed nto ntegrals along the boundares of the control volume. Hence, n one approach, the control volumes used for the cell averaged gradent calculatons are the trangles themselves, whle the other approach uses an extended control volume (Barth and Jespersen, 1989). One-Dmensonal Reconstructon Ths procedure s based on buldng a 1-D calculaton stencl normal to a partcular edge under consderaton. Once ths stencl s defned, lnear reconstructon s performed n the most straghtforward way as f one were dealng wth a 1-D problem. Ths approach s nspred n the work of Lyra (1994), whch s based on a fnte element technque. The major dfference between the present constructon and the one used n Lyra (1994) les n the drecton n whch the one-dmensonal stencl s constructed. In the cted reference (Lyra, 1994), the stencl for extrapolaton s constructed along the drecton of the edge. Here, snce a cell centered fnte volume method s of nterest, the extrapolaton stencl s constructed n a drecton normal to the edge. In an attempt to renterpret the 1-D deas n the present unstructured grd context, a lne s drawn normal to the edge passng J. of the Braz. Soc. of Mech. Sc. & Eng. Copyrght 2010 by ABCM January-March 2010, Vol. XXXII, No. 1 / 81

5 João Luz F. Azevedo et al. through the center of the nscrbed crcle to that trangle. As llustrated by the sketch shown n Fg. 2, a thrd pont s located over ths lne, and away from the edge under consderaton, at a dstance from the center of the nscrbed crcle equal to the dameter of the crcle. The control volume wthn whch ths 3rd pont les s dentfed, and the propertes of ths trangle used n the lnear reconstructon. In order to make the nomenclature clear, the two trangles, whch are adjacent to the edge under consderaton, are denoted as and k. The second trangle dentfed by the prevously descrbed process and assocated wth trangle s denoted as l. The correspondng one assocated wth k s denoted as trangle m. Ths s also llustrated n Fg. 2. Once trangles l and m are dentfed, the reconstructon of left, L, and rght, R, states at the k nterface can be performed as a lmted extrapolaton of u n the form 1 ul = u + ψ ( u ul) ur = uk ψ um u k 2 ( ) (21) In these expressons, ψ ± represent the lmter, whch s dscussed n the next secton. Note that the search of l and m trangles consttutes a pre-processng operaton. Therefore, although ths can be a costly operaton, t s performed only once durng a gven run, provded that the grd topology remans fxed. Clearly, f grd adaptaton s performed, the operaton would have to be repeated. The addtonal nformaton regardng the l and m trangles, on the other hand, must be added to the edge-based data structure, whch mples n the need for extra storage. Gradent Reconstructon The approach followed n ths case conssts on attrbutng cell averaged propertes and gradents to the control volume centrod, whch allows the lnear reconstructon of propertes at any pont wthn the cell. The basc expresson can be wrtten as (, ) u( x, y ) u x y = + u Δ r, (22) where (x, y) denotes the coordnates of a generc pont wthn the control volume, (x, y ) s the poston of the -th cell centrod, u s the gradent of the u property and Δr s the vector poston of the (x, y) pont wth respect to the cell centrod. Therefore, there are two aspects that need to be dscussed. The frst one concerns the fact that t s necessary to nclude a lmter n order to avod the creaton of new extrema. The lmter constructon s dscussed n the next secton. The other aspect concerns the calculaton of the gradent tself. Gradents are computed n the present work usng Green s theorem (Swanson and Radespel, 1991) and, therefore, transformng dervatve calculatons nto lne ntegrals around approprate control volumes (n the 2-D case). In ths context, the cell averaged dervatves for the -th control volume can be wrtten as u 1 u 1 = dv = udy x V S V Ω x V Ω Ω Ω u 1 u 1 = dv = udx y V V Ω y V SΩ Ω Ω (23) The basc queston that remans s: whch control volume, V Ω, s used for ths ntegraton? The present work uses two dfferent forms of defnng ths control volume. The frst approach conssts n defnng V Ω = V,.e., the trangles themselves are used as the control volumes for the gradent calculaton. Ths s the smplest approach possble, but t s usually crtczed n the lterature (Barth and Jespersen, 1989) because t cannot recover the correct gradent of a lnear functon. The second opton conssts n defnng the control volume for the gradent calculaton as the polygon formed by connectng the centrods of all trangles whch have an edge or a vertex n common wth the -th trangle. Ths s the approach recommended by Barth and Jespersen (1989), snce t satsfes the crtera enforced by these authors for gradent calculatons: 1. One must obtan the exact soluton for the gradent of the functon when the functon has a lnear varaton; 2. The gradent must be defned for arbtrary meshes. However, ths approach requres more memory usage to store the new extended control volume areas and t also requres addtonal computatonal tme, because more complcated control volume yelds addtonal operatons n order to form the gradents. Once the cell averaged gradents have been computed, the k nterface propertes can be lnearly reconstructed n the standard fashon as u u L R = u + ψ u Δr + = u ψ u Δr k k k (24) Here, u and u k are the gradents computed for trangles and k, respectvely, ψ ± are the lmters, and Δr and Δr k are the vector postons of the nterface mdpont wth respect to the centrod of each of the two trangles whch share the nterface. Lmtng Procedures In order to avod oscllatons, the extrapolated states must be lmted (Hrsch, 1990). The majorty of the results dscussed here uses the mnmod lmter. Another aspect concerns the fact that the lmter constructon s clearly connected to the approach used for the lnear reconstructon. Hence, the form used to defne the lmter assocated wth the one-dmensonal approach for property reconstructon s dfferent from the one used when the gradent reconstructon s adopted. For the 1-D reconstructon case, the lmter s also constructed as a one-dmensonal lmter, snce the stencl for the calculaton s already set up. Hence, the ψ ± functons that appear n Eqs. (21) can be wrtten as ψ ( r ) = ψ, (25) ± ± where the ratos of consecutve gradents are gven by r uk u uk u =, r =. (26) u u u u + l m k For the mnmod lmter, the ψ(r) functon can be wrtten as 0, f r < 0 ψ ( r) = mn mod ( 1, r) = r, f 0 r < 1 1, f r 1 (27) Correspondng expressons could be wrtten for the other lmters, e.g., the superbee lmter, and these expressons can be easly found n the lterature (Hrsch, 1990). For the cases n whch the reconstructon process uses gradents, two lmter desgns are used. The frst one attempts to buld a lmter whch s also sort of one-dmensonal, whereas the other approach 82 / Vol. XXXII, No. 1, January-March 2010 ABCM

6 Order of Accuracy Study of Unstructured Grd Fnte Volume Upwnd Schemes follows the work of Barth and Jespersen (1989) and consders a truly multdmensonal lmter. In order to make the nomenclature clear for the dscusson of the 1-D-type lmter wth gradent reconstructon, the centrods of the and k trangles are assumed to have coordnates (x, y) and (x, y) k, respectvely, and the k nterface mdpont has coordnates (x, y) h. The followng vector postons can be defned d d d ( x x ) ( y y ) j ( x x ) ( y y ) j ( x x ) ( y y ) = + = + = + k k k h h h hk k h k h and t s also convenent to defne j (28) d = d, d = d, d = d. (29) k k h h hk hk Wth these defntons, the followng one-dmensonal-type gradents can be computed: uk u ξk =, dk 1 u u ξ = + dh x y 1 u u ξ = + dhk x k y k ( x x ) ( y y ) h h h,. ( x x ) ( y y ) hk k h k h The r ± ratos are, then, defned as r (30) ξk ξk =, r =, (31) ξ ξ + h hk and the lmters can be wrtten as n Eq. (25). In ths case, nstead of usng equatons (24) for the actual reconstructon, t would be more approprate to use ul = u + ψ ξhdh + ur = uk ψ ξhkdhk (32) snce the varous terms are avalable due to the lmter calculaton. Ths form of lmter has both theoretcal and practcal drawbacks, although t s very straghtforward to mplement and qute nexpensve from a computatonal cost pont of vew. The theoretcal concern s assocated wth the fact that the property gradents are computed usng nformaton from all neghbors of a gven trangle whereas the lmter only uses nformaton along a pseudo-one-dmensonal stencl normal to the partcular edge. In other words, the reconstructon process uses nformaton, whch s mult-dmensonal, whereas the lmter s one dmensonal. Nevertheless, for the model problem, ths lmter constructon does not cause any dffcultes and the results obtaned, n terms of order of spatal accuracy for the scheme, are the best the authors were able to acheve throughout ths nvestgaton. Unfortunately, and ths s the practcal drawback, when the authors attempted to use ths lmter for nvscd flow smulatons at hgh Mach numbers, the numercal solutons nvarably dverged for all cases tested. Numercal solutons could actually be obtaned for the Euler equatons at low supersonc Mach numbers, of the order of 2 or 3, for flows over a wedge. However, the computatons would fal f one attempted the hgher Mach numbers of nterest to the authors (Fguera da Slva, Azevedo and Korzenowsk, 1999). Snce the same hgh Mach number cases could be computed wth the other combnatons of lmter and type of reconstructon, whch are dscussed here, the authors concluded that ths 1-D-type lmter s not the most adequate for the sake of smulatng compressble flows. The results wth ths lmter for the model problem are, however, presented here for completeness. The problems observed wth the 1-D-type lmter together wth the gradent reconstructon prompted the use of a truly multdmensonal lmter. Ths procedure closely follows the work of Barth and Jespersen (1989). The lmter defnton starts by attrbutng the cell averaged value of the conserved varable, u, to the -th trangle centrod. In other words, t assumes that the property at the centrod has a value u = u(x, y ). Then, the values mn and max u are defned such that ( ) ( ) u = mn u, u, k = 1,2,3 u = max u, u, k = 1, 2,3 mn k max k u (33) where u k, k = 1, 2, 3, denote the neghbors of the -th trangle. For each j-th vertex of the -th trangle, the procedure computes u j = u(x j,y j ), j = 1, 2, 3, usng Eq. (22),.e., u j = u + u Δr (34) j where Δr j = (x j x ) + (y j y ) j. For each node of the -th control volume, a prelmnary lmter value s defned as max u u mn 1,, f uj u > 0 uj u mn u u ψ j = mn 1,, f uj u < 0 uj u 1, f uj u = 0 (35) Fnally, the value of the lmter, whch wll be used for the reconstructon usng propertes of the -th trangle, s ( 1 2 3) ψ = mn ψ, ψ, ψ. (36) Ths lmter constructon s essentally equvalent to a multdmensonal verson of the mnmod lmter. In prncple, one could desgn mult-dmensonal lmter constructons, whch would mmc the van Leer, superbee or other lmters. However, ths s not attempted n the present work and the only verson of a multdmensonal lmter tested s the one ndcated n the prevous equatons. Furthermore, when defnng the u j node values, the authors use the gradents already calculated for the control volumes. However, one could also thnk of defnng such node values smply by a weghted averaged of the centrod values of all trangles, whch share that partcular node. The authors, however, have not expermented wth ths form of defnng property node values. Mesh Generaton and Boundary Treatment The model problem consdered nvolves an extremely smple geometry and, therefore, one would thnk that trangular mesh generaton n ths case would be a trval task. Ths would ndeed be true, but some addtonal nformaton, beyond the usual unstructured grd nformaton, s necessary n order to mplement the perodc boundary condtons. Furthermore, there s nterest n havng control J. of the Braz. Soc. of Mech. Sc. & Eng. Copyrght 2010 by ABCM January-March 2010, Vol. XXXII, No. 1 / 83

7 João Luz F. Azevedo et al. over the orentaton of the trangles, because the work also nvestgates the grd orentaton effect on the order of accuracy of the schemes tested. One should observe that, snce the computatonal doman s a square wth unt sde length, t s natural to dvde the doman nto squared control volumes based on the number of subdvsons n each sde of the doman. Hence, the trangular grd could be obtaned by dvdng each quadrlateral volume by one of ts dagonals, yeldng two trangles. The mplementaton adopted allows the user to select the dvson wth dagonals orented wth +45 deg. and 45 deg. wth respect to the x axs or a truly unstructured grd (Lo, 1985), n whch the orentaton of the dagonals s somewhat random. Examples of the possble grd types are shown n Fg. 3. The grds shown n Fg. 3 have the coarsest resoluton used n the present work, namely wth 10 subdvsons along each sde of the squared doman or a characterstc length of 0.1. Ths yelds a grd wth 200 trangular control volumes, regardless of the mesh topology adopted. The nvestgaton also consders grds wth 20 20, and dvsons. These yeld, respectvely, a total of 800, 3200 and control volumes and characterstc lengths of 0.05, and n dmensonless unts. Regardless of the grd topology adopted, enough nformaton s stored n order to allow an exact mplementaton of perodc boundary condtons. For each segment along the boundary, the dentfcaton of the correspondng segment along the boundary on the other sde of the computatonal doman s stored. Boundary condtons are mplemented through the use of ghost volumes n the present code, and the procedure adopted conssts n forcng the ghost volume assocated wth a gven trangle at the boundary to receve the property values of the other prevously dentfed nternal trangle of the par. Results and Dscusson Test Cases and General Informaton The work consders a smple 2-D, lnear, scalar advecton problem, as prevously descrbed. Even n such a smple stuaton, there are qute a few parameters, whch can nfluence the results. Clearly, the man objectve of the work s to dentfy how the error n the computed soluton decreases as the mesh s refned. Hence, all smulatons compute the model problem for one perod of the soluton and, then, compare the numercal soluton wth the exact soluton for the problem. The global error s measured n terms of the L 1 norm of the dfference between exact and numercal results. Ths can be expressed as Fgure 3. Grd topologes used n the present nvestgaton: (a) truly unstructured grd; (b) dagonals orented wth 45 deg.; and (c) dagonals orented wth +45 deg. I = 1 Fgure 3. (Contnued). I 1 L1 = uexact x y unum x,y (, ) ( ), (37) where the pont wth coordnates (x, y ) ndcates the centrod of the -th control volume. One should observe that ths s the natural defnton of the error for a cell centered fnte volume scheme. However, one mght obtan somewhat dfferent results, n terms of order of accuracy, f some form of averagng of the computatonal results s performed pror to computng the error (Durlofsky, Engqust and Osher, 1992). Hence, a calculaton of the L 1 norm of ths averaged error s also performed. In the present case, ths averagng s performed by obtanng (averaged) numercal results at the nodes of the control volumes. Therefore, the L 1 norm of ths averaged soluton at the nodes s computed usng an expresson smlar to Eq. (37), but wth exact and numercal values of u evaluated at the nodal pont locatons. The averaged numercal values of the functon at the nodes are obtaned smply by an arthmetc average of the dscrete propertes of all control volumes, whch share a gven node. Ths s suffcent for the present case, snce all control volumes have the same area. The procedure used here to verfy the order of accuracy of the proposed schemes conssts n runnng the problem for the four meshes prevously defned, wth ncreasng refnement, and plottng the logarthm of the L 1 norm of the error as a functon of the logarthm of the mesh spacng. A best ft straght lne, n the least square sense, s passed through these ponts and the lne slope determnes the order of accuracy of the method. Thus, theoretcally, a 1st-order method should yeld a lne wth unt slope, whereas a 2nd-order method should yeld a lne wth slope equal to 2. Four major cases for the nomnally 2nd-order upwnd scheme are consdered. These could be classfed as (a) one-dmensonal reconstructon wth a 1-D-type lmter, (b) gradent reconstructon wth a smplfed ntegraton control volume and a 1-D-type lmter, (c) gradent reconstructon wth a smplfed ntegraton control volume and a mult-dmensonal lmter, and (d) gradent reconstructon wth an extended ntegraton control volume and a multdmensonal lmter. These results are compared to the nomnally 1st-order upwnd scheme, descrbed n the secton that dscusses the AUSM + scheme, and to the standard 2nd-order centered scheme, also prevously dscussed. It should be emphaszed that the four cases 84 / Vol. XXXII, No. 1, January-March 2010 ABCM

8 Order of Accuracy Study of Unstructured Grd Fnte Volume Upwnd Schemes lsted above test the most relevant aspects dscussed n the paper, whch are the form n whch reconstructon s performed, the process used to defne the lmter and, for the case of gradent reconstructon, the control volume used for property gradent evaluaton. All tests are performed for a constant CFL of 0.1, except for a sngle test for whch the CFL s 0.01 n order to make sure the order of tme accuracy of the scheme has no effect n the results. It should be emphaszed that such small CFL numbers are used n order to guarantee that the tme ntegraton method has no effect n the subject matter of the present study, whch s to assess the spatal accuracy of the schemes under nvestgaton. Both the cases wth lnear advecton n the x-drecton as well as advecton along a 45 deg. drecton wth the x-axs are consdered, whch correspond to the advecton velocty a = (a x,a y ) = (1, 0) and (1, 1), respectvely. The mportance of testng these two cases s assocated wth an evaluaton of the effect of the mesh orentaton on the fnal order of accuracy for the schemes. For some of the trangular grds consdered n ths nvestgaton, an advecton velocty a = (1, 1) s ether algned wth or perpendcular to a large number of grd edges. A smlar analyss for a grd wth dagonals orented 45 deg. wth respect to the x-axs, and stll consderng a = (1, 0), s presented n Fg. 5. The 2nd-order scheme uses the mnmod lmter. Moreover, both cases n whch the error s calculated wth and wthout averagng of numercal property values are presented n ths fgure. The slopes of the least square fts for these cases are ndcated n Table 1. One can see that the grd orentaton has essentally no effect on the 1st-order scheme n ths case, and t has a small effect on the 2nd-order scheme. Moreover, the soluton averagng pror to the error calculaton mproves the measured order of accuracy for the 2nd-order scheme, whch s consstent wth the results reported n Durlofsky, Engqust and Osher (1992). However, t has a small detrmental effect n the measured order of accuracy for the 1st-order scheme. In any event, t s clear from these results that the nomnally 2nd-order scheme s qute far from yeldng true 2nd-order accuracy. One-Dmensonal Reconstructon Results The ntal tests use the one-dmensonal-type of property reconstructon at nterfaces for the nomnally 2nd-order scheme. The frst test case consders a truly unstructured mesh and the convecton velocty a = (1, 0). The results are shown n Fg. 4 for the nomnally 1st-order scheme and the 2nd-order scheme wth the mnmod and superbee lmters. The actual orders of accuracy obtaned numercally n each case are 0.82, 0.94 and 0.65, respectvely. The L 1 norm of the error for these results s calculated wthout any averagng procedure,.e., the error s calculated for propertes evaluated at the actual control volume centrod. It s clear from these results that none of nomnally 2nd-order case s even close to true 2nd-order accuracy. Actually, calculatons wth the superbee lmter yeld results wth an order of accuracy smaller than that of the 1st-order scheme. Moreover, the 1st-order scheme s also somewhat worse than true 1st order, and the 2nd-order scheme wth the mnmod lmter gves a slghtly better order of accuracy than the 1st-order scheme. It s also ntrgung that calculatons wth the mnmod lmter gve better order of accuracy than those wth the superbee lmter, snce the former s supposed to be much more dsspatve. Note that all 1st-order results presented hereafter do not acheve actual 1st-order accuracy. Ths s also observed by Aftosms, Gatonde and Tavares (1995), who attrbuted such a dscrepancy to the non-ortogonalty of the cell nterfaces wth respect to the computed fluxes. Fgure 4. L 1 norm of the error for unstructured grd wth one-dmensonal reconstructon for the case a x = 1 and a y = 0. Lnes are least square fts. Fgure 5. L 1 norm of the error for grd wth dagonals orented 45 deg., wth one-dmensonal reconstructon, for the case a x = 1 and a y = 0. Lnes are least square fts. Table 1. Order of accuracy for grd wth dagonals orented 45 deg., wth one-dmensonal reconstructon, for the case a x = 1 and a y = 0. Method L 1 Norm L 1 Norm wthout Averagng wth Averagng AUSM + 1st order AUSM + 2nd order Although these results are dscouragng and the authors dd not analyze any other cases usng the one-dmensonal reconstructon procedure, t s nterestng to try to understand what caused such poor performance. The frst dea that comes to mnd s the fact that there are reasons to attrbute cell averaged values of the propertes to the cell centrod. However, the same s not true for attrbutng cell averaged values to the center of the nscrbed crcle, whch s essentally what the present procedure does. Moreover, the order of accuracy of the scheme could probably be mproved f an nterpolaton s performed n order to obtan the propertes at the second ponts used for the reconstructon,.e., the ponts that le n trangles l and m, as ndcated n Fg. 2. The current procedure smply attrbutes to these ponts the cell averaged values of the propertes. It seemed, at the tme, that t was more effectve to nvest n the reconstructon process usng gradents and, hence, no other tests wth the one-dmensonal reconstructon were performed. As an a posteror thought, however, the one-dmensonal reconstructon technque probably deserves further nvestgaton. J. of the Braz. Soc. of Mech. Sc. & Eng. Copyrght 2010 by ABCM January-March 2010, Vol. XXXII, No. 1 / 85

9 João Luz F. Azevedo et al. Results for Smplfed Gradent Calculaton wth 1-D Lmter The results dscussed n ths secton use property gradent-based reconstructon, but the lmter s stll constructed usng 1-D-type deas. Moreover, property gradents are calculated usng the trangular cells themselves as control volumes for ntegraton. The plots for the L 1 norm of the error for a test case wth advecton velocty gven by a x = 1 and a y = 0, and usng the grd wth dagonals orented wth 45 deg., are presented n Fg. 6 both for the cases n whch no averagng of the results s performed before computng the error and for the cases n whch averagng of the numercal results s made pror to the error calculaton. Ths fgure shows results for the 1st-order upwnd scheme and for the 2nd-order upwnd scheme wth the mnmod lmter and wthout any lmter at all. Moreover, for comparson purposes, the fgures also show the L 1 norm of the error for the calculatons wth a 2nd-order centered scheme. The orders of accuracy actually obtaned n each case are summarzed n Table 2. Obvously, the 1st-order upwnd scheme results are the same reported n the prevous secton, snce the form of reconstructon does not affect the 1st-order scheme. The actual order of accuracy obtaned wth the nomnally 2ndorder upwnd scheme should be compared, for nstance, wth the results n Table 1. It s clear from ths comparson that the use of gradent reconstructon yelds better orders of accuracy than the 1-D reconstructon process. However, there are some strange features n the results shown n Table 2. For nstance, t s not clear why the calculaton wthout any lmter at all yelds an actual order of accuracy whch s smaller than that obtaned when the calculatons are performed wth the mnmod lmter. The model problem has a smooth soluton, whch means that a lmter should, n the worst case, be clppng the smooth peaks and valleys of the smooth model functon, f t perceves the gradents as too hgh. But, n any case, there s no apparent reason to obtan better results wth the mnmod lmter than wthout any lmter. Moreover, the actual values of the L 1 norm of the error for the calculatons wth the mnmod lmter are smaller than those for the unlmted extrapolaton case. Furthermore, attempts to run ths case wth the superbee lmter resulted n numercal nstablty for the fner grds, although a soluton could be obtaned wth the coarsest grd consdered. Fgure 6. L 1 norm of the error wthout (top) and wth (bottom) averagng for grd wth dagonals orented 45 deg., smplfed gradent reconstructon and 1-D-type lmter, for the case a x = 1 and a y = 0. Lnes are least square fts. Fgure 6. (Contnued). Table 2. Order of accuracy for grd wth dagonals orented 45 deg., wth smplfed gradent and 1-D-type lmter, for the case a x = 1 and a y = 0. Method L 1 Norm wthout Averagng L 1 Norm wth Averagng AUSM + 1st order AUSM + 2nd order wth mnmod AUSM + 2nd order, no lmtng nd order centered scheme The calculatons summarzed n Table 2 also ndcate that even the best results obtaned wth the 2nd-order upwnd scheme were stll qute far from true 2nd-order accuracy as dsplayed by the centered scheme. It s also nterestng to observe that the averagng of the soluton pror to the error calculaton consstently mproves the numercal order of accuracy of the 2nd-order upwnd scheme. However, ths s not true ether for the 1st-order upwnd scheme or for the 2nd-order centered scheme. These results are n contrast wth those reported n Durlofsky, Engqust and Osher (1992), where the averagng always mproves the computed order of accuracy. It should be emphaszed, however, that the averagng s performed n a dfferent fashon n the present work, when compared to the cted reference. Here, the averaged value of the soluton s computed at the nodes of the mesh from the dscrete cell-averaged values calculated at the cells by the present cell-centered scheme. In Durlofsky, Engqust and Osher (1992), ths averaged value s computed at the center of the squared cells formed by two adjacent trangles. A study s also performed to nvestgate the effect of the CFL number on the present results. For that, a grd wth dagonals orented wth +45 deg. s used, together wth an advecton velocty gven by a x = 1 and a y = 0. The AUSM + scheme wth gradent reconstructon and the 1-D-type lmter, wth the mnmod lmter, s used and the test case s run wth CFL = 0.1 and The order of accuracy obtaned n the varous cases analyzed s presented n Table 3. For comparson purposes, the orders of accuracy ndcated n Table 3 are calculated usng only the results for the three coarsest meshes. Ths s done because the ntal results already ndcated that there was no CFL number nfluence n the order of accuracy of the methods and the cost of runnng the fnest grd wth CFL = 0.01 was very hgh, of the order of 10 CPU hours n the equpment beng used by the authors at the tme, namely HP-9000/720 workstatons. Moreover, for the cases n whch values of the L 1 norm of the error are avalable for the four grds,.e., for the cases wth CFL = 0.1, the order of accuracy obtaned usng the results for the four grds s ndcated wthn parentheses n Table / Vol. XXXII, No. 1, January-March 2010 ABCM

10 Order of Accuracy Study of Unstructured Grd Fnte Volume Upwnd Schemes Table 3. Effect of the CFL number on the order of accuracy. Calculatons used grd wth dagonals orented +45 deg., wth smplfed gradent and 1-D-type mnmod lmter, for the case a x = 1 and a y = 0. Method CFL L 1 Norm L 1 Norm wthout Averagng wth Averagng 1st order (0.82) 0.69 (0.77) 1st order nd order (1.37) 1.59 (1.55) 2nd order The effect of grd orentaton s nvestgated by consderng grds wth a +45 deg. and a 45 deg. orentaton for the quadrlateral dagonals used to construct the trangular meshes. The AUSM + scheme s used for these tests, wth 2nd-order reconstructon usng gradents computed on the trangular cell tself and wth the 1-Dtype of lmter constructon. The mnmod lmter s also used n these cases. In order to make any grd effects more evdent, the lnear advecton problem wth a x = a y = 1 s selected for ths test case. Moreover, a CFL number of 0.1 s used n the tests. The plots for the L 1 norm of the error are presented n Fg. 7 both for the cases n whch no averagng of the results s performed before computng the error and for the cases wth averagng of the numercal soluton pror to the error calculaton. Ths fgure shows results for both the 1st-order and 2nd-order schemes. The orders of accuracy actually obtaned n each case are summarzed n Table 4. Fgure 7 and Table 4 are ndcatng that, at least for the 2ndorder scheme, ths test case shows very lttle effect of the grd orentaton on the scheme order of accuracy. Moreover, for the advecton velocty wth a x = a y = 1, there s also very lttle dfference between the orders of accuracy obtaned wth and wthout averagng the soluton before the error calculaton for the nomnally 2nd-order scheme. Ths stuaton s n drect contrast wth what one sees for the 1st-order scheme. For the 1st-order scheme, there s clearly a mesh orentaton effect on the results. One can observe that, for the mesh wth +45 deg. orentaton, the order of accuracy obtaned s ndependent of averagng and ts value s somewhat the average between those obtaned wth and wthout averagng for the grd wth 45 deg. orentaton. Fgure 7. L 1 norm of the error wthout (top) and wth (bottom) averagng, smplfed gradent reconstructon and 1-D-type mnmod lmter, for the case a x = a y = 1. Lnes are least square fts. Fgure 7. (Contnued). Table 4. Effect of grd orentaton on the order of accuracy. Calculatons used 2nd-order reconstructon wth smplfed gradent and 1-D-type mnmod lmter, for the case a x = a y = 1. Method Grd Orentaton L 1 Norm wthout Averagng L 1 Norm wth Averagng 1st order -45 deg st order +45 deg nd order -45 deg nd order +45 deg The overall concluson one can draw from the results wth gradent reconstructon wth the smplfed gradent calculaton and wth 1-D-type lmtng procedures s that the orders of accuracy acheved are qute a lot better than those obtaned wth the onedmensonal-type reconstructon dscussed n the prevous secton. Moreover, even though the orders of accuracy acheved for the present cases are stll far from true 2nd order, they are much hgher than that delvered by the correspondng 1st-order scheme calculatons. In partcular, for the cases wth advecton along a 45 deg. drecton wth the x-axs, the 2nd-order results are about three tmes better than those wth the 1st-order scheme n terms of the actual order of accuracy one can actually obtan n the numercal calculatons. However, as dscussed n connecton wth the results n Table 2, some aspects of the soluton behavor wth the procedure emphaszed n ths secton are, at least, strange. The worst problem, however, wth the use of gradent reconstructon and a 1-D lmtng procedure occurred when the authors attempted to extend the capablty n order to perform smulatons of nvscd flows at hgh Mach numbers. The code extended to run cases for the Euler equatons s able to obtan solutons for low supersonc Mach numbers as, for nstance, for the flow over a wedge wth freestream Mach number 2. However, the same test case results n numercal nstablty f the freestream Mach number s ncreased to, for nstance, 8. These results lead to authors to conclude that ths 1-D-type lmter s not the most adequate for these applcatons. Apparently, these problems arse from the fact that, when gradents are used, the reconstructon process s truly mult-dmensonal, usng nformaton from all neghbors of the trangle under consderaton. On the other hand, the 1-D-type lmtng procedure does not use nformaton from all neghbors and, hence, t actually does not provde an adequate lmtng at all and leads to numercal nstablty. Therefore, wth a gradent-type reconstructon, a truly mult-dmensonal lmtng procedure seems to be requred n order to avod numercal problems. J. of the Braz. Soc. of Mech. Sc. & Eng. Copyrght 2010 by ABCM January-March 2010, Vol. XXXII, No. 1 / 87