A gradient smoothing method (GSM) for fluid dynamics problems

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1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluds 2008; 58: Publshed onlne 27 March 2008 n Wley InterScence ( A gradent smoothng method (GSM) for flud dynamcs problems G. R. Lu 1,2 and George X. Xu 3,, 1 Centre for Advanced Computatons n Engneerng Scence, Department of Mechancal Engneerng, Natonal Unversty of Sngapore, 10 Kent Rdge Crescent, Sngapore , Sngapore 2 Sngapore-MIT Allance (SMA), E , 4 Engneerng Drve 3, Sngapore , Sngapore 3 Insttute of Hgh Performance Computng, 1 Sngapore Scence Park Road, #01-01 The Caprcorn, Sngapore Scence Park II, Sngapore , Sngapore SUMMARY A novel gradent smoothng method (GSM) based on rregular cells and strong form of governng equatons s presented for flud dynamcs problems wth arbtrary geometres. Upon the analyses about the compactness and the postvty of coeffcents of nfluence of ther stencls for approxmatng a dervatve, four favorable schemes (II, VI, VII and VIII) wth second-order accuracy are selected among the total eght proposed dscretzaton schemes. These four schemes are successvely verfed and carefully examned n solvng Posson s equatons, subjected to changes n the number of nodes, the shapes of cells and the rregularty of trangular cells, respectvely. Numercal results mply us that all the four schemes gve very good results: Schemes VI and VIII produce a slghtly better accuracy than the other two schemes on rregular cells, but at a hgher cost n computaton. Schemes VII and VIII that consstently rely on gradent smoothng operatons are more accurate than Schemes II and VI n whch drectonal correcton s mposed. It s nterestngly found that GSM s nsenstve to the rregularty of meshes, ndcatng the robustness of the presented GSM. Among the four schemes of GSM, Scheme VII outperforms the other three schemes, for ts outstandng overall performance n terms of numercal accuracy, stablty and effcency. Fnally, GSM solutons wth Scheme VII to some benchmarked compressble flows ncludng nvscd flow over NACA0012 arfol, lamnar flow over flat plate and turbulent flow over an RAE2822 arfol are presented, respectvely. Copyrght q 2008 John Wley & Sons, Ltd. Receved 5 February 2007; Revsed 11 January 2008; Accepted 22 January 2008 KEY WORDS: numercal methods; CFD; FDM; GSM; GSD; stencl Correspondence to: George X. Xu, Insttute of Hgh Performance Computng, 1 Sngapore Scence Park Road, #01-01 The Caprcorn, Sngapore Scence Park II, Sngapore , Sngapore. E-mal: xuxg@hpc.a-star.edu.sg Copyrght q 2008 John Wley & Sons, Ltd.

2 1102 G. R. LIU AND G. X. XU 1. INTRODUCTION Fnte dfference method (FDM), fnte element method (FEM) and fnte volume method (FVM) are the three well-known numercal methods wdely used for solvng varous types of problems n scence and engneerng. Each of them possesses ts own advantages and dsadvantages. FDM s the oldest method among the three, beleved to have been ntroduced by Euler n the 18th century (see [1]). Technques on FDM were publshed as early as 1910 by Rchardson [2] even before the modern computers were avalable. FDM s smple and effcent and therefore t s wdely adopted n numercal smulatons [3 6]. In FDM, a dervatve s replaced wth an approxmate dfference formula that can generally be derved from a Taylor seres expanson, usng sngle- or multple-block structured mesh. The strong-form governng equatons are dscretzed onto nodes of a set of structured mesh resultng n a system of algebrac equatons wth a banded matrx of coeffcents. A number of effcent numercal technques can be used to quckly obtan the solutons for such a system of algebrac equatons [6]. However, t s not a trval process to generate structured mesh for arbtrary geometres, especally n the topologcal generaton of multple blocks [7]. Ths lmts the FDM beng vald only for smple geometres. Although FDM may be used for problems wth a slghtly complcated geometry, ssues related to the mappng from physcal doman to computatonal doman complcate the process n numercal mplementaton [3, 4]. FEM was earler used by Courant [8] for solvng a torson problem and was named by Clough [9]. It was studed to great extents and perfected n the 1960s and 1970s, mostly for analyzng structural mechancs problem [10, 11]. The FEM analyss of flud flow was developed n the md and late 1970s (see [12,13]). The classcal FEM dscretzaton s based on elements and a pecewse representaton of the soluton n terms of bass functons and the Galerkn weak form formulaton [11, 14]. To mprove numercal stablty, some more convncng fnte element procedures, such as the streamlne-upwnd Petrov Galerkn (SUPG) [15], the Galerkn/leastsquares (GLS) [16], bubble functons [17] and the varatonal multscale method [18], have been developed and wdely used nowadays n the fnte element communty for solvng convectondomnated transport problems. The strength of FEM s ts ablty to deal wth arbtrary geometry usng dfferent shapes and orders of elements that are usually formed based on a set of unstructured mesh. Generally, FEM can gve the hghest accuracy on coarse mesh among the three tradtonal numercal methods. It s very effectve for dffuson-domnated problems and free surface problems [13]. However, the codng of FEM s much more complcated than that of FDM, and FEM s relatvely costly n computaton especally when t s used for solvng turbulent flud flow problems. As depcted n [1, 19], the FVM s devsed to dscretze an ntegral form of the partal dfferental equaton (PDE) for a physcal law (e.g. conservaton of mass, momentum or energy). The problem doman s dvded nto a set of fnte volumes or cells, and the PDE s expressed n a form that can govern each fnte volume (or cell). The resultng system of equatons usually nvolves fluxes of the conserved varable, and thus the vtal FVM procedure s how to accurately calculate fluxes [20]. The basc advantage of ths method over FDM s that t s applcable for any type of mesh, both structured and unstructured. The resultng approxmate soluton for feld varables can be placed at cell centers or at nodes. The values of feld varables at non-storage locatons are obtaned usng nterpolaton. Further detals can be found n [19]. Early well-documented use of FVM was made by Evans and Harlow [21] and Gentry et al. [22]. In the late 1970s and the

3 GSM FOR FLUID DYNAMICS PROBLEMS 1103 early 1980s, t was further explored to be appled onto body-ftted mesh [23, 24]. By the early 1990s, unstructured FVM had been comprehensvely developed, e.g. [20, 25 27]. Comparatvely, FVM s conservatve even on coarse mesh [1]. FVM s now wdely adopted for solvng flud flow problems and mplemented n the well-known commercal CFD packages [19]. Smlar to FDM and FEM, FVM also has some dsadvantages. As addressed by Blazek [20], false dffuson often occurs n numercal predctons wth FVM, especally when smple numercs are engaged. It s also dffcult to develop schemes wth hgher than second-order accuracy for multdmensonal problems [27]. More recently, varous meshfree methods have been developed (see, for example, References [28 34]). Comparatvely, meshfree methods are mplemented on nodes at frst place nstead of meshes as done n tradtonal methods descrbed above. A survey paper wrtten by Babuska et al. [34] provdes the mathematcal foundaton of varous meshfree methods. The overvew of theoretcal, computatonal and mplementaton ssues related to varous meshfree methods about both weak and strong forms can be found n the monographs by Lu [30] and Lu and Gu [35]. By removng some of the restrctons owng to the use of elements, meshfree methods are more flexble and more sutable for adaptve analyses. They have been successfully appled to problems where element-based methods are often dffcult to gve satsfactory results. For example [35 37], meshfree methods are very attractve n the smulaton of cracks, underwater shock, exploson and free surface problems. In general, many meshfree methods, as mentoned above, are comprehensvely studed based on weak-form governng equatons. When governng equatons n strong form are treated, most of these meshfree methods exhbt nstablty [38, 39], and thus specal technques are requred to resolve t. Gradent smoothng operaton s often employed n stablzng the nodal ntegratons [40, 41] and nstallng lnear conformablty [42] for methods based on Galerkn weak forms. Enlghtened by the attractve merts of gradent smoothng operaton n Galerkn weak forms as mentoned above, we here ntend to examne the effectveness of gradent smoothng operaton as t s appled for strong-form (dfferental form) governng equatons. We call the method as gradent smoothng method (GSM). In GSM, all the unknowns are stored at nodes and ther dervatves at varous locatons are consstently and drectly approxmated wth gradent smoothng operaton based on relevant gradent smoothng domans (GSDs). Both regular and rregular grds are concerned n the development of GSM. A unfed and stable scheme regardless of the type of grd s pursued n ths study. As a whole, the mplementaton procedure of GSM s as smple as the tradtonal FDM. GSM can, however, be appled very effectvely wth excellent stablty for arbtrary geometres, as wll be shown n our studes n ths work. In the followng sectons, the theory of GSM s frst ntroduced. The GSM approxmatons to the gradents (frst-order dervatve) and Laplace operator (second-order dervatve) of a feld varable are presented n detal. Stencl analyses on coeffcents of nfluence correspondng to varous GSM schemes are then conducted. Important features of stencls for the dscretzed Laplace operator are dscussed. Numercal solutons to Posson s equatons are obtaned usng four favorable GSM schemes and nvestgated n detal to reveal the propertes on convergence and stablty. The computatonal effcency, accuracy n results and the robustness to the cell rregularty for GSM are also comprehensvely examned. Fnally, the GSM results for some benchmarked compressble flow problems, e.g. nvscd flow over a NACA0012 arfol, lamnar flow over flat plate and turbulent flow over an RAE2822 arfol, are presented. The convergence and accuracy of the GSM solutons are further demonstrated.

4 1104 G. R. LIU AND G. X. XU 2. GRADIENT SMOOTHING METHOD In GSM, dervatves at varous locatons, ncludng nodes, centrods of cells and mdponts of cell edges, are approxmated over relevant GSDs usng the gradent smoothng operaton. The detals about the theory, prncple and mplementaton procedure of GSM are ntroduced n ths secton wth the focus on the approxmaton of spatal dervatves Gradent smoothng operaton For smplcty, a two-dmensonal problem s consdered here to llustrate the gradent smoothng operaton. The gradents of a feld varable U at a pont of nterest at x n doman Ω can be approxmated n the form of [43, 44] U U(x ) U(x) w(x x )dx Ω (1) Integratng Equaton (1) by parts gves U U(x) w(x x ) n ds Ω U(x) w(x x )dv Ω (2) where s a gradent operator and w s a smoothng functon that can be chosen properly based on the requrement on the accuracy of the approxmaton [43]. Ω represents the external boundary of the GSD, and n denotes the unt normal vector on Ω, as shown n Fgure 1. In ths work, we use the followng pecewse constant functon as the smoothng functon: { 1/V, x Ω w = (3) 0, x / Ω where V s the area of Ω. x Fgure 1. Generc gradent smoothng doman.

5 GSM FOR FLUID DYNAMICS PROBLEMS 1105 Usng Equaton (3), the rght-hand sde second term n Equaton (2) vanshes and Equaton (2) then becomes U 1 V Ω U n ds (4) Equaton (4) gves an approxmaton of gradents at a pont usng a local smoothng doman. Such a gradent smoothng technque s often adopted n many meshfree methods, for example, n the wdely used smoothed partcle method [28, 43], to stablze the nodal ntegratons and ensure lnear conformablty [40, 41, 45, 46]. It has also been used n the development of the smoothed FEM [47, 48]. Analogously, by successvely applyng the gradent smoothng technque for second-order dervatves [47, 48], the Laplace operator at x can be approxmated as ( U ) 1 n U ds (5) V Hence, spatal dervatves at any pont of nterest can be approxmated usng Equatons (4) and (5) together wth properly defned smoothng domans that are dscussed n the followng subsecton Smoothng domans In our GSM, the problem doman s frst dvded nto a set of cells (regular or rregular) generated by connectng nodes, and the values of the feld functons are stored at nodes and a set of systems of algebrac equatons s formed by approxmatng the dervatves at nodes. Based on prmtve cells, a smooth doman can be constructed for any pont. Dependng on the locaton of the pont of nterest, we use dfferent types of smoothng domans based on the compact and conformal prncple. As shown n Fgure 2, three types of GSDs, whch are used for the approxmaton of spatal dervatves, are consttuted on prmtve unstructured trangular cells. The frst type of smoothng doman s the node-assocated GSD (ngsd) for the approxmaton of the dervatves at a node of nterest. It s formed by connectng relevant centrods of trangles wth mdponts of relevant cell edges. The second type s formed by a prmtve cell, whch s used for approxmatng the dervatves at the centrod of the cell, as n cell-centered FVM. It s called centrod-assocated GSD (cgsd) here. The thrd type s named mdpont-assocated GSD (mgsd), whch s used for the calculaton of the gradents at the mdpont of a cell edge of nterest. The preferred mgsd s formed by connectng the end nodes of the cell edge wth the centrods on both sdes of the cell edge, as shown n Fgure 2. Usng Equatons (4) and (5), spatal dervatves at any pont of nterest can be approxmated based on the correspondng smoothng doman descrbed above. Dfferent schemes are devsed for ths purpose, and the detals wll be elucdated further n the followng subsecton Spatal dscretzaton schemes We now need to accurately predct the ntegrals along the boundares of varous types of GSDs. In the current work, both the one-pont quadrature (rectangular rule) and the two-pont quadrature (trapezodal rule) are used and examned for the approxmaton of the dervatves at nodes. Only two-pont quadrature s used for the approxmaton of the gradents at both mdponts and centrods. As lsted n Table I, a total of eght dscretzaton schemes for spatal dfferental terms are developed usng dfferent types of quadrature and methods of approxmaton. Ω

6 1106 G. R. LIU AND G. X. XU Fgure 2. Illustraton of gradent smoothng domans and doman-edge vectors adopted n GSM. Table I. Spatal dscretzaton schemes for the approxmaton of dervatves. Approxmaton Approxmaton of gradents at of gradents at Drectonal Scheme Quadrature Type of GSD mdponts centrods correcton I One-pont ngsd Interpolaton (Not requred) No II One-pont ngsd Interpolaton (Not requred) Yes III Two-pont ngsd Interpolaton Interpolaton No IV Two-pont ngsd Interpolaton Interpolaton Yes V Two-pont ngsd, cgsd Interpolaton Gradent smoothng No VI Two-pont ngsd, cgsd Interpolaton Gradent smoothng Yes VII One-pont ngsd, mgsd Gradent smoothng (Not requred) No VIII Two-pont ngsd, mgsd, cgsd Gradent smoothng Gradent smoothng No In the one-pont quadrature schemes (I, II and VII), the ntegraton along a smoothng doman edge s approxmated wth the rectangular rule, where the ntegrand s evaluated only at the mdpont of a cell edge of nterest. In the two-pont quadrature schemes (III VI and VIII), the ntegraton s evaluated wth the trapezodal rule, where values of the feld varable and ts gradents at the two endponts of a smoothng doman edge of nterest (the mdpont of the cell edge and the centrod of the cgsd that are connected to the doman edge of nterest) are used. In ths work, both frst- and second-order dervatves at nodes are always approxmated wth the gradent smoothng operaton detaled n Secton 2.1. The gradents at the mdpont of a cell edge of nterest can be calculated n two ways: ether by smple nterpolaton usng the gradents at both end nodes of the cell edge (I VI) or by gradent smoothng technque usng Equaton (4) based on mgsd (VII and VIII) for the mdpont. Smlarly, the gradents at a centrod can be obtaned ether by smple nterpolaton usng the gradents at the nodes of the cgsd (III and IV) or by gradent smoothng over the correspondng cgsd (V, VI and VIII).

7 GSM FOR FLUID DYNAMICS PROBLEMS 1107 Note that when one-pont quadrature schemes are used, there s no need to approxmate the gradents at centrods, snce the ntegrands n Equatons (4) and (5) are evaluated only at the mdponts of cell edges Formulae for spatal dervatves Two-pont quadrature schemes. In GSM, spatal dervatves at any pont of nterest are approxmated usng the followng formulae based on prmtve trangular cells. Frst-order dervatves at nodes: Usng Equaton (4), at node, the frst-order dervatves of the feld varable U are gven by where U x 1 V U y 1 V n k=1 n k=1 { 1 2 (ΔS x) (L) [(U m ) jk +(U c ) Δjk j k+1 ]+ 1 2 (ΔS x) (R) [(U m ) jk +(U c ) Δjk j k 1 ]} (6) { 1 2 (ΔS y) (L) [(U m ) jk +(U c ) Δjk j k+1 ]+ 1 2 (ΔS y) (R) [(U m ) jk +(U c ) Δjk j k 1 ]} (7) (ΔS x ) (L) =ΔS (L) (n x ) (L) (8) (ΔS y ) (L) =ΔS (L) (n y ) (L) (9) (ΔS x ) (R) =ΔS (R) (n x ) (R) (10) (ΔS y ) (R) =ΔS (R) (n y ) (R) (11) where U,U m and U c denote the values of the feld varable U at nodes, mdponts of cell edges and centrods of trangular cells, respectvely. ΔS x and ΔS y are the two components of a doman-edge vector. n x and n y represent the two components of the unt normal vector of the doman edge. denotes the node of nterest and j k s the other end node of the cell edge lnked to node (see Fgure 2). Superscrpts (L) and (R) are ponters to the two doman edges assocated wth the cell edge of nterest,. The total number of supportng nodes wthn the stencl of node s denoted by n. Subscrpts Δ j k+1 and Δ j k 1 stand for the left-hand sde and rght-hand sde trangles of the cell edge. n denotes the total number of supportng nodes connected to node. These geometrcal parameters are calculated and stored before the ntensve calculaton starts. The values of the feld varable U at non-storage locatons,.e. at mdponts and centrods, are computed by smple nterpolaton of functon values at consttutve nodes, respectvely. Frst-order dervatves at mdponts and centrods: Analogous to the dscretzaton at nodes descrbed above, the gradents at the mdponts (( U m ) jk ) of cell edges and centrods (( U c ) Δjk j k+1 and ( U c ) Δjk j k 1 ) of cells can also be approxmated wth the gradent smoothng technque usng Equaton (4) but based on the related mgsds and cgsds, respectvely. Such a treatment s adopted for the approxmaton of the gradents at mdponts n Schemes VII and VIII, and gradents at centrods n Schemes V VIII. Smlarly, the geometrcal parameters ncludng the areas, doman-edge vectors and normal vectors of doman edges related to mgsds and cgsds should be predetermned and stored for use n the teratve process of solvng the algebrac equatons.

8 1108 G. R. LIU AND G. X. XU Alternatvely, the gradents at these non-storage locatons can be approxmated by smple nterpolaton of the gradents at relevant nodes. In Schemes I VI, the gradents at the mdponts of the cell edges are approxmated n ths manner, so are the gradents at centrods n Schemes III and IV. Second-order dervatves at nodes: In GSM wth the two-pont quadrature schemes, the second-order dervatves are obtaned usng Equaton (5) and gven n the followng form: 2 U x U y 2 1 Ω n k=1 [ + [ {[ x (U m) jk + ] x (U c) Δjk j k+1 (ΔS x ) (L) y (U m) jk + y (U c) Δjk j k+1 x (U m) jk + x (U c) Δjk j k 1 [ + y (U m) jk + y (U c) Δjk j k 1 ] (ΔS y ) (L) ] (ΔS x ) (R) ] } (ΔS y ) (R) All the frst-order dervatves used n Equaton (12) are approxmated as descrbed n prevous two subsectons One-pont quadrature schemes. In one-pont quadrature schemes (I, II and VII), t s assumed that and (12) (U c ) Δjk j k+1 =(U c ) Δjk j k 1 =(U m ) jk (13) ( U c ) Δjk j k+1 =( U c ) Δjk j k 1 =( U m ) jk (14) Therefore, Equatons (6), (7) and (12) can be smplfed as and where ( U) 1 Ω U x 1 Ω U y 1 Ω n k=1 n k=1 n k=1 (ΔS x ) jk (U m ) jk (15) (ΔS y ) jk (U m ) jk (16) [ x (U m) jk (ΔS x ) jk + ] y (U m) jk (ΔS y ) jk (17) (ΔS x ) jk =(ΔS x ) (L) +(ΔS x ) (R) (18) (ΔS y ) jk =(ΔS y ) (L) +(ΔS y ) (R) (19)

9 GSM FOR FLUID DYNAMICS PROBLEMS 1109 As shown n Equaton (15) (17), n one-pont quadrature schemes, only values for the feld varable and ts gradents at the mdponts of cell edges are needed n the approxmatons. Thus, the vectors for a par of doman edges connected wth the cell edge can be lumped together, whch n return reduces the storage space for geometrcal parameters. It should be noted that the gradents at mdponts n Equaton (17) are approxmated usng the gradent smoothng technque based on mgsds n Scheme VII, whereas they are approxmated usng the smple nterpolaton approach n Schemes I and II. The one-pont quadrature schemes are clearly smpler and much more cost-effectve n terms of both flops and storage. Schemes based on two-pont quadrature mpose extra requrements n computaton and storage for values of varables at centrods and doman-edge vectors for cgsds. When mgsds are used for predcton of gradents at the mdponts of cell edges, such demands become even hgher. However, theoretcally, schemes based on two-pont quadrature can gve more accurate results. Ths wll be verfed and dscussed later n the secton on numercal examples Drectonal correcton. In our study, we fnd that as the gradents at the mdponts of cell edges, whch are approxmated usng smple nterpolaton approach, are used for approxmatng the second-order dervatves, decouplng solutons (checkerboard problem) are attaned. To crcumvent such a problem, as proposed by Crumpton et al. [49], the approxmated gradents at mdponts are requred to be remeded wth the drectonal correcton technque. The drectonal correcton takes the form of [ ( ) ] ( Ũ m ) jk =( U m ) jk ( U m ) jk U t jk t jk (20) l where and ( ) U U j k U, l Δl jk t jk = r jk Δl jk, r jk =x jk x Δl jk = x jk x Here x and x jk denote the spatal locatons of nodes and j k, respectvely. Ths technque s adopted n Schemes II, IV and VI. Detals about the role of drectonal correcton wll be addressed n the followng secton on stencl analyses Dscretzaton of temporal terms For a transent or pseudo-transent problem, the governng equaton can be rewrtten n the form of U t = R (21)

10 1110 G. R. LIU AND G. X. XU where R represents the resdual dependent on the feld varable U and ts dervatves. In ths study, the temporal term ( U/ t) s dscretzed wth the followng explct fve-stage Runge Kutta (RK5) method [50]: U (0) = U n U (1) = U (0) α 1 ΔtR (0) U (2) = U (0) α 2 ΔtR (1) U (3) = U (0) α 3 ΔtR (2) U (4) = U (0) α 4 ΔtR (3) U (5) = U (0) α 5 ΔtR (4) where resdual R (k) s evaluated wth the values of the feld functon and ts dervatves approxmated wth the kth-stage RK soluton at node for every tme step. Δt denotes the tme step, and the coeffcents adopted n ths study are α 1 =0.0695, α 2 =0.1602,α 3 =0.2898,α 4 = and α 5 = Wth the RK5 method, only the 0th- and 5th-stage solutons at nodes should be stored n the memory. The RK5 method has been wdely used n the smulatons of many transent flud flow problems, because of ts satsfactory effcency and stablty Edge-based data structure The edge-based data structure recommended by Lohner [51] s adopted n our GSM, together wth the scatter gather approach as descrbed by Barth [26, 27]. The lnkages among cell edges, nodes, cells, GSDs, as well as doman-edge vectors and normals, are pre-determned. Our results prove that GSM wth the edge-based data structure s very effcent. (22) 3. ANALYSES OF DISCRETIZATION STENCIL Before conductng ntensve numerc excses, careful studes of the stencls of supportng nodes for varous schemes proposed for GSM are carred out. The coeffcents of nfluence for a node of nterest where dervatves are approxmated are derved usng GSM and analyzed. The objectves for stencl analyses are to select the most sutable schemes that satsfy the basc prncples of numercal dscretzaton. For clarty, the stencls for approxmatng the Laplace operator based on the cells n both square and equlateral trangle shapes are focused n ths study Basc prncples for stencl assessment In the stencl analyses, the followng fve basc rules are consdered to assess the qualty of a stencl resultng from a dscretzaton scheme: (a) Consstency at each nterface of two adjacent GSDs. (b) Postvty of coeffcents of nfluence. (c) Sum of the coeffcents of nfluence.

11 GSM FOR FLUID DYNAMICS PROBLEMS 1111 (d) Negatve-slope lnearzaton of the source term. (e) The compactness of the stencl. The frst four rules are summarzed by Patankar [52] wth consderaton of solutons wth physcally realstc behavor and overall balance. To satsfy Rule (a), t requres that the same expresson of approxmaton must be used on the nterface of two adjacent GSDs. Rule (b) requres that the coeffcent for the node of nterest and the coeffcents of nfluence must be postve, once the dscretzaton equaton s expressed n the form of a U = n k=1 a U jk +b. Rule (c) further mples that a = n k=1 a. Rule (d) relates to the treatment of the source terms. As addressed by Patankar [52], t s essental to keep the slope of lnearzaton to be negatve, snce a postve slope can lead to computatonal nstabltes and physcally unrealstc solutons. It s also necessary for a good dscretzaton stencl to satsfy Rule (e) for the concerns about numercal accuracy and effcency, as commented by Barth [27]. The very frst layer of nodes surroundng the node of nterest should be ncluded n the dscretzaton stencl. Moreover, as the stencl becomes larger, not only the computatonal cost ncreases, but eventually the accuracy decreases as less vald data from further away are brought nto approxmaton. In addton, Barth [27] has proposed a few lemmas to address the necessty of postvty of coeffcents to satsfy a dscrete maxmum prncple that s a key tool n the desgn and analyss of numercal schemes sutable for non-oscllatory dscontnuty (for example, shock). At steady state, non-negatvty of the coeffcents becomes suffcent to satsfy a dscrete maxmum prncple that can be appled successvely to obtan the global maxmum prncple and stable results. Hs statements reterate the mportance of Rule (b) as mentoned by Patankar [52]. In GSM, when the gradent smoothng technque s appled to GSDs, Rule (a) s automatcally satsfed, meanng that the local conservaton of quanttes s ensured and therefore for the global conservaton once proper boundary condtons are used. In ths study, Rules (b), (c) and (e) focus on the assessment of stencls for dfferent dscretzaton schemes Stencls for approxmated gradents The stencls for gradent approxmaton usng the eght types of dscretzaton schemes are derved. The coeffcents of nfluence based on the cells n square and equlateral trangle shapes wth unt length are shown n Fgures 3 and On cells n square shape. We fnd that the three one-pont quadrature schemes (I, II and VII) gve the same stencl when cells n square shape are used, as shown n Fgure 3(a). Ths stencl s also dentcal to that of the two-pont-based central-dfferencng scheme n FDM. We also observe that the stencl for all two-pont quadrature schemes s also the same, as shown n Fgure 3(b). Ths stencl s also dentcal to that of the sx-pont-based central-dfferencng scheme n FDM. Such fndngs confrm that when cells n square shape are used, GSM s dentcal to FDM. However, GSM works for cells n rregular shapes On cells n equlateral trangle shape. It s nterestng to know that the stencl for approxmated gradents based on the cells n equlateral trangle shape s found to be the same regardless of the GSM schemes, as shown n Fgure 4. Ths stencl s dentcal to that of the nterpolaton method usng sx surroundng nodes [30]. Note that for rregular trangular cells, the nterpolaton method can n general fal as addressed by Lu [30], but our GSM stll performs well, as wll

12 1112 G. R. LIU AND G. X. XU Fgure 3. Stencls for approxmated gradents based on the cells n square shape: (a) I, II and VII and (b) III VI and VIII. Fgure 4. The stencl for approxmated gradents based on the cells n equlateral trangle shape (dentcal for I VIII). be demonstrated n the secton on numercal examples. Ths s because of the crucal stablty provded by the smoothng operaton Stencls for approxmated Laplace operator On cells n square shape. The stencls for the approxmated Laplace operator wth GSM schemes based on the cells n square shape are derved and lsted n Fgure 5. Three schemes, I, III and V, as shown n Fgures 5(a), (c) and (e), result n wde stencls wth unfavorable weghtng coeffcents (zero and negatve) on cells n square shape. Wth such knds of stencls, unexpected decouplng solutons may be produced [20]. Ths s also confrmed n the present analyss and wll be llustrated n the followng secton on numercal examples. As mentoned by Moner [50], such knds of stencls cannot damp out hgh-frequency numercal errors. The stuaton becomes even

13 GSM FOR FLUID DYNAMICS PROBLEMS 1113 Fgure 5. Stencls for the approxmated Laplace operator on the cells n square shape: (a) I; (b) II and VII; (c) III; (d) IV; (e) V; and (f) VI and VIII. worse wthn the boundary layer regon where the vscous effect becomes domnant. Therefore, Schemes I, III and V are now labeled as unfavorable. It s found that such unfavorable stencls are resulted from the smple nterpolaton that s used to approxmate the gradents at the mdponts of cell edges n the three schemes. In Schemes II and VI where the drectonal correcton to the approxmated gradents at mdponts s made, relatvely compact stencls wth favorable coeffcents are obtaned, as depcted n Fgures 5(b) and (f). Scheme II s a fve-pont-based stencl and Scheme VI corresponds to a 9-node-based compact stencl, whch are the same as those for central-dfference scheme n FDM. However, as shown n Fgure 5(d), even wth drectonal correcton, an unfavorable stencl can stll occur n Scheme IV. Therefore, Schemes II and VI are found to be favorable, and Scheme IV s also labeled as unfavorable. The compact and favorable stencls are also obtaned usng Schemes VII and VIII where the gradents at the mdponts of cell edges are approxmated by applyng the gradent smoothng operaton to mgsds, as shown n Fgures 5(b) and (f). Ths mples that for approxmatng the gradents at the mdpont of a cell edge of nterest, the gradent smoothng operaton over the correspondng mgsd s a good alternatve to the smple nterpolaton wth drectonal correcton technque. From the ponts of vews of the consstency n the approxmaton of dervatves at dfferent locatons and the stablty feature of the gradent smoothng technque, Schemes VII and VIII are more preferable than Schemes II and VI.

14 1114 G. R. LIU AND G. X. XU On cells n equlateral trangle shape. The analyses are also conducted on cells n equlateral trangle shape and resulted stencls are shown n Fgure 6. It s found agan that Schemes I and III result n an unfavorable stencl, because the coeffcents of nfluence at the very frst layer of neghborng nodes are negatve as shown n Fgure 6(a), whch volates the basc Rule (b). Schemes IV and V, and Schemes II, VI, VII and VIII produce two sets of stencls wth favorable coeffcents, as shown n Fgures 6(b) and (c), respectvely. The stencl for Schemes IV and V conssts of two layers of surroundng ponts around the node of nterest. The stencl for Schemes II, VI, VII and VIII uses only the frst layer of neghborng nodes. Accordng to Rule (e), Schemes II, VI, VII and VIII are more favorable than Schemes IV and V, because of ther relatvely compact stencls. It should be mentoned that Rule (c) s satsfed n stencls for all schemes studed here. In summary, based on the stencl analyses, four GSM schemes,.e. II, VI, VII and VIII, are selected as favorable schemes and are further examned, because they consstently produce compact stencl wth favorable coeffcents on both shapes of cells of concern (square and equlateral trangle). Fgure 6. Stencls for the approxmated Laplace operator on the cells n equlateral trangle shape: (a) I and III; (b) IV and V; and (c) II, VI, VII and VIII. Table II. Truncaton errors n the approxmaton of frst dervatves n GSM. Scheme Shape of cells Truncaton error II and VII Square O x (h 2 )= h2 3 U j 6 x 3 + O(h3 ) O y (h 2 )= h2 3 U j 6 y 3 + O(h3 ) ( ) VI and VIII Square O x (h 2 )= h U j 24 x U j 2 x y 2 + O(h 3 ) ( ) O y (h 2 )= h U j 24 y U j 2 x 2 + O(h 3 ) y ( ) II, VI, VII and VIII Equlateral trangle O x (h 2 )= h U 24 x U 8 x y 2 + O(h 3 ) ( ) O y (h 2 )= h U 24 y U 8 x 2 + O(h 3 ) y

15 GSM FOR FLUID DYNAMICS PROBLEMS 1115 Table III. Truncaton errors n the approxmaton of the Laplace operator n GSM. Scheme Shape of cells Truncaton error ( II and VII Square O(h 2 )= h2 4 U j 12 x 4 ( VI and VIII Square O(h 2 )= h2 4 U j 12 x 4 II, VI, VII and VIII Equlateral trangle O(h 2 )= h2 16 ) + 4 U j y 4 + O(h 3 ) ) +3 4 U j x 2 y U j y 4 + O(h 3 ) ( ) 4 U x U x 2 y U y 4 + O(h 3 ) 3.4. Truncaton errors We next conduct analyses on truncaton errors n the approxmaton of frst- and second-order dervatves wth the four recommended schemes n GSM, and the results are summarzed n Tables II and III. They are derved based on the cells n square and equlateral trangle shapes, respectvely. It s clear that all these schemes are of second-order accuracy. The truncaton errors for Scheme VII are dentcal to those for Scheme II, and Schemes VIII and VI have the same truncaton errors. All these theoretcal fndngs wll be further conformed when these schemes are used to solve Posson s equatons. 4. APPLICATION AND VALIDATION OF GSM Through the stencl analyses for all the schemes, four schemes (II, VI, VII and VIII) are selected to use n the GSM, because of ther compact stencls wth favorable coeffcents. Numercal tests for soluton to two-dmensonal Posson s equatons are consequently conducted usng our GSM code. Dfferent spatal dscretzaton schemes are tested and compared wth one another n terms of numercal accuracy and computatonal effcency. The roles of drectonal correcton and gradent smoothng technque used for the approxmaton of the gradents at the mdponts of cell edges are numercally verfed. In addton, the effects of the shape, the densty and the rregularty of cells on the accuracy and stablty are ntensvely nvestgated. Fnally, GSM s successfully used to predct compressble flow over an RAE2822 arfol Solutons to Posson s equatons Governng equatons. Posson s equatons for a square computatonal doman are frst solved wth our GSM code. Posson s equatons govern many physcal problems, such as the heat conducton problems wth sources. Two problems wth varatons n source and boundary condtons are studed. In ths study, the Drchlet condtons are appled on the boundares,.e. the values of the feld functons at the boundares are prescrbed. The pseudo-transent approach s adopted for pursung steady-state solutons. The governng equatons under nvestgaton take the followng form: U t = 2 U x U f (x, y) (0 x 1,0 y 1) (23) y2

16 1116 G. R. LIU AND G. X. XU Fgure 7. Contour plots of exact solutons to the two Posson problems: (a) the frst Posson problem and (b) the second Posson problem. For the frst problem, the source and ntal condtons are prescrbed as } f (x, y,t)=13exp( 2x +3y), (0 x 1,0 y 1) (24) U(x, y,0)=0 As plotted n Fgure 7(a), the analytcal soluton to ths problem s known as Û(x, y)=e ( 2x+3y) (0 x 1,0 y 1) (25) For the second Posson problem, the source, ntal condtons and analytcal soluton are gven as follows: } f (x, y,t)=sn(πx)sn(πy), (0 x 1,0 y 1) (26) U(x, y,0)=0 Û(x, y)= 1 sn(πx)sn(πy) (0 x 1,0 y 1) (27) 2π2 The contour plot of the analytcal soluton to the second problem s shown n Fgure 7(b). Boundary condtons adopted n smulatons for both problems are consstent wth the correspondng analytcal solutons. These analytcal solutons are used for the evaluaton of numercal errors n the GSM solutons Evaluaton of numercal errors. Three types of numercal errors are evaluated n the study. The convergence error ndex, ε con, takes the form of ε con = nnode =1 / (U (n+1) U (n) nnode ) 2 =1 (U (1) U (0) ) 2 (28)

17 GSM FOR FLUID DYNAMICS PROBLEMS 1117 Fgure 8. Plot of the profle on convergence hstory. where U (n) denotes the predcted value of the feld varable at node at the nth teraton, and n node s the total number of nodes n the doman. The value of ε con s montored durng teratons and used to termnate the teratve process. In most smulatons, n order to exclude the effect owng to the temporal dscretzaton, computatons are not stopped untl ε con becomes stablzed, as ndcated n Fgure 8. The numercal error n a GSM soluton for the overall feld s defned usng the L 2 -norm of error as / nnode nnode error= (U Û ) 2 Û 2 (29) =1 where U and Û are predcted and analytcal solutons at node, respectvely. Ths type of error s used to compare the accuracy among dfferent schemes. The thrd type of error s the node-wse relatve error, whch s estmated n the form of =1 rerror = U Û / Û (30) The node-wse relatve errors dstrbuted over the computatonal doman are used to dentfy problematc regons n smulatons Cell types used n the study. Four types of cells varyng n shapes,.e. square, rght trangle, regular trangle and rregular trangle, as shown n Fgure 9, are used n GSM and nvestgated n ths study. The rregular trangles are desgned for the study of robustness of GSM to the rregularty of trangular cells. The numercal results for varous types of cells wth dfferent spatal dscretzaton schemes are dscussed n detal below The role of drectonal correcton. As shown n Fgure 10(a), the decoupled soluton s predcted usng Scheme I when t s appled onto cells n square shape n solvng the frst Posson

18 1118 G. R. LIU AND G. X. XU (a) (b) (c) (d) Fgure 9. Representatve cells under nvestgaton: (a) square; (b) rght trangle; (c) regular trangle; and (d) rregular trangle. Fgure 10. Contour plots of relatve errors on cells n square shape n the frst Posson problem: (a) Scheme I and (b) Scheme II. problem. The saw-toothed numercal errors (checkerboard problem) are generated and cannot be dampened out. Wth the drectonal correcton n Scheme II, such a problem s overcome, as shown n Fgure 10(b). Ths s consstent wth the fndngs n stencl analyses.

19 GSM FOR FLUID DYNAMICS PROBLEMS 1119 Note that wth drectonal correctons, the overall numercal error s sgnfcantly reduced, as shown n Table IV. Comparng to Scheme I, the magntudes of overall errors wth Scheme II decrease by 5 tmes or so. However, wth Scheme II, subjected to CFL condton, smaller tme step s needed for stablty requrement, resultng n more computatons, compared wth Scheme I. Ths becomes more obvous when fner meshes are used n smulaton Comparson among four favorable schemes. The four favorable schemes are comprehensvely tested on dfferent types of cells wth varous node resolutons for solutons to Posson s equatons. Evoluton of numercal error wth averaged node spacng for the frst Posson problem s plotted n Fgures 11 13, together wth ftted lnes. Note that the errors n these plots are evaluated usng Equaton (29), and the averaged node spacng, h, s evaluated usng the formula gven by Lu [30] n the form of V h = (31) nnode 1 where V and n node denote the area of the whole computatonal doman and the total number of nodes n the doman, respectvely. It s apparent that the averaged node spacng decreases as the number of nodes ncreases. On cells n square shape: As shown n Fgures 11(a) and (b), on cells n square shape, Scheme VII s as accurate as Scheme II, because the two schemes result n the same dscretzaton stencls, as already hghlghted n the stencl analyses. Ths s also true for Schemes VI and VIII. In addton, the two-pont quadrature schemes (VI and VIII) gve relatvely lower accuracy than the one-pont quadrature schemes (II and VII). They also result n hgher computatonal costs than the one-pont quadrature schemes. Therefore, based on the cells n square shape, Schemes II and VII are equvalent and they are superor to Schemes VI and VIII. The slope coeffcents of trendlnes confrm that these four schemes are of second-order accuracy, whch s consstent wth our fndng n the analyses of truncaton errors. On cells n rght trangle shape: Profles of numercal error aganst averaged node spacng based on the cells n rght trangle shape are shown n Fgures 12(a) and (b). Results reveal that Scheme VI gves a slghtly more accurate predcton than Scheme II. In addton, t s nterestng to fnd that based on the rght trangular cells, Scheme VII s as accurate as Scheme VIII. On cells n regular trangle shape: When regular trangular cells are used, both two-pont quadrature schemes (VI and VIII) produce a slghtly more accurate predcton than one-pont quadrature schemes, as shown n Fgures 13(a) and (b). Such a fndng s consstent wth what s observed when rght trangular cells are used. Table IV. Comparson of numercal errors wth Schemes I and II for the frst Posson problem. Scheme I Scheme II Number of nodes Error Iteraton Error Iteraton e e e e e e e e e e

20 1120 G. R. LIU AND G. X. XU Fgure 11. Evoluton of numercal error wth averaged node spacng based on the cells n square shape. Table V summarzes the numercal errors n number for dfferent dscretzaton schemes when regular trangular cells are used. It s clear that Scheme VII s slghtly more accurate than Scheme II, and Scheme VIII s more accurate than Scheme VI. Ths s also true when rght trangular cells are used n smulatons. Such dscrepancy n accuracy s related to the approxmaton of gradent at boundary nodes. In Schemes II and VI, the gradents at boundary nodes are needed to be approxmated, as they are desred by the nterpolaton approach for the approxmaton of gradents at mdponts of nternal cell edges that are lnked wth boundary nodes. Thus, addtonal errors are ntroduced due to the approxmaton. Comparatvely, n Schemes VII and VIII, subjected to Drchlet boundary condtons, approxmatons to gradents at the boundary nodes are entrely avoded, thanks to the gradent smoothng technques appled to the mgsds. In general, n terms of computatonal cost, Schemes VII and II are almost the same, and Schemes VIII and VI are very close to each other. Schemes VI and VIII requre roughly twce the computatonal tme as Schemes II and VII. Schemes VI and VIII are more accurate than Schemes II and VII, but the mprovement n accuracy s not sgnfcant. Therefore, to balance the numercal accuracy and computatonal effcency, the two one-pont quadrature schemes (II and VII) are preferred to be used n practce Robustness to rregularty of the cells. The consstent fndngs descrbed above are also observed n the solutons to the second Posson problem. For the second Posson problem, addtonal

21 GSM FOR FLUID DYNAMICS PROBLEMS 1121 Fgure 12. Evoluton of numercal error aganst averaged node spacng based on the cells n rght trangle shape. studes on the effects of rregularty of the trangular cells are carred out. It s well known that trangular cells have best adaptvty to complex geometres and can be generated automatcally n very effcent ways. Therefore, t s desrable to use trangular cells n GSM so that GSM can become a very robust method for engneerng problems wth complcated geometres. Our objectve s thus to further dentfy the senstvty of GSM to the cell qualty. Followng the fndngs as descrbed prevously, the two one-pont quadrature schemes (II and VII) are used n ths study (Table VI). To study ths n a systematc manner, we frst defne the rregularty for all trangular cells n the computatonal doman, γ, usng the followng formula: ne (a b ) 2 +(b c ) 2 +(c a ) 2 =1 a 2 γ= +b2 +c2 (32) n e where a,b and c, respectvely, denote the lengths of cell edges of a trangular cell, and n e stands for the total number of cells n the overall doman. Equaton (32) s derved from the formula proposed by Stllnger et al. [53] for a sngle trangle. Usng Equaton (32), the rregularty vanshes for equlateral trangles and s postve for all other shapes ncludng sosceles trangles.

22 1122 G. R. LIU AND G. X. XU Fgure 13. Evoluton of numercal error aganst averaged node spacng based on the cells n regular trangle shape. Table V. Comparson of numercal errors predcted on regular trangular cells for the frst Posson problem wth favorable schemes. Number of nodes Δt Scheme II Scheme VII Scheme VI Scheme VIII e e e e e e e e e e e e e e e e e e e e 5 Error Fgure 14 shows sx sets of trangular cells wth varous rregulartes but the same number of nodes. It s obvous that as the rregularty ncreases, the mesh s dstorted further. Note that when the rregularty s larger than 0.152, overlapped cells are found n the doman, as shown n Fgure 15, whch s an extreme that wll not normally happen n practce. Such an extreme case cannot be accurately reflected n the rregularty usng Equaton (32). Nevertheless, we stll use such extremely dstorted cells to test the robustness of GSM.

23 GSM FOR FLUID DYNAMICS PROBLEMS 1123 Table VI. Comparson of allowable maxmum tme step and numercal error for rregular trangular cells. Mesh Irregularty Maxmum tme step (Δt) Scheme VII Scheme II (a) (b) (c) (d) (e) (f) Error (a) (b) (c) (d) (e) (f) Fgure 14. Trangular cells wth varous rregulartes: (a) γ=0.021; (b) γ=0.028; (c) γ=0.048; (d) γ=0.079; (e) γ=0.118; and (f) γ= Convergent results for all rregular meshes are obtaned usng GSM wth Schemes II and VII. Contour plots of the predcted solutons wth Scheme VII are selectvely shown n Fgure 16. It s clear that the predcted results are reasonably accurate on all sets of rregular cells. Note that as the rregularty of the cells ncreases, the tme step (Δt) has to be reduced so as to guarantee stablty and convergence of the results, as shown n Table VI.

24 1124 G. R. LIU AND G. X. XU Fgure 15. Overlapped cells n the computatonal doman (γ = 0.16). It s nterestng to fnd that for cases wthout non-overlapped cells, the numercal errors predcted wth GSM do not vary so much as the rregularty of cells ncreases (see Fgure 17). Once overlapped cells occur n the doman, sudden jumps n numercal errors are observed. However, for Schemes II and VII, stable results are stll obtaned. Besdes, Scheme VII shows much better stablty and accuracy than Scheme II amongst all rregular cells examned here. Ths means that GSM wth Scheme VII s remarkably robust and nsenstve to cell rregularty. In other words, wth the proposed GSM, stable and accurate results can be obtaned even wth hghly dstorted trangular cells. Such an attractve feature can be attrbuted to the consstent use of smoothng technques n Scheme VII, whch provdes the crucal stablty and robustness to our GSM Solutons to Naver Stokes equatons The GSM procedure wth Scheme VII has also been successfully appled to solutons to some benchmarked compressble flows. The full Naver Stokes equatons for conservatve varables are solved accordngly. The well-known second-order Roe flux-dfferent slttng scheme [54] s adopted to evaluate the convectve fluxes, because of ts hgh accuracy for boundary layers and good resoluton of shocks [20]. The left and rght states are predcted wth the Barth and Jespersen method [25]. To avod oscllaton and non-physcal solutons, the Venkatakrshnan s lmter [55] s used. The flow turbulence s smulated wth the Spalart Allmaras one-equaton turbulence model [56] n our GSM code. In ths secton, the results for nvscd flow over a NACA0012 arfol, lamnar flow over a flat plate and turbulent flow over an RAE2822 arfol are presented, respectvely Invscd flow over an NACA0012 arfol. Euler solver wth the proposed GSM scheme dependent on multple GSDs has been developed and t s appled to the solutons to an nvscd flow

25 GSM FOR FLUID DYNAMICS PROBLEMS 1125 Fgure 16. Contour plots of solutons to the second Posson equaton dscretzed onto rregular trangular meshes: (a) γ=0.021; (b) γ=0.028; (c) γ=0.048; (d) γ=0.079; (e) γ=0.118; and (f) γ= Fgure 17. Numercal errors n GSM solutons (Schemes II and VII) to the second Posson problem wth respect to rregularty of cells.

26 1126 G. R. LIU AND G. X. XU over an NACA0012 arfol. In the testng case, for the freestream, T =288K, p = Pa, Ma=0.8 and the angle of attack α=1.25. Numercal results are selectvely presented below. Fgure 18 shows the unstructured grd used n the computaton. Plots of contours about the densty, the statc pressure and Mach number are dsplayed n Fgures 19 21, respectvely. It s Fgure 18. Irregular grd near the NACA0012 arfol. Fgure 19. Spatal dstrbuton of the predcted densty.

27 GSM FOR FLUID DYNAMICS PROBLEMS 1127 Fgure 20. Plot of contours of the predcted statc pressure. Fgure 21. Plot of contours of the predcted Mach number. obvous that the strong shock occurrng on the upper surface of the arfol s ncely captured, as well as the weak shock takng place on the lower surface. The results are qualtatvely agreeable wth the results publshed by Barth [26].

28 1128 G. R. LIU AND G. X. XU Lamnar flow over a flat plate. The thrd testng case s wth regard to the lamnar flow over a flat plate. The freestream s subject to the condtons Re=5000 and Ma=0.5. The angle of attack s zero for ths case. A rectangular computatonal doman was generated and grdded by rght trangles. Non-slp condtons are mposed onto the wall and symmetry condtons are appled to the regon ahead of the plate along the x-axs. Farfeld condtons (see [20]) are mposed at the external boundares. Fgure 22 shows the convergence hstory of the teratve process. The boundary layer effect s clearly ndcated n Fgure 23 about the vector plot near the boundary. The profle about the Fgure 22. Profle of the convergence hstory. Fgure 23. Velocty profles near the boundary and contour plot of Mach number.

29 GSM FOR FLUID DYNAMICS PROBLEMS 1129 Fgure 24. Comparson of predcted and analytcal wall frcton coeffcents. predcted skn frcton coeffcent C f vared wth the dstance x s compared wth analytcal Blasus soluton, as shown n Fgure 24. It s apparent that our numercal results show wonderful agreement wth the analytcal soluton Turbulence flow over an RAE2822 arfol. In ths testng case, the freestream corresponds to the followng condtons: T = K, p = Pa,Re= ,Ma=0.729 and α=2.31. Here, T, p, Re and Ma denote, respectvely, the temperature, statc pressure, Reynolds number and Mach number of the freestream, and α stands for the angle of attack of the RAE2822 arfol. The unstructured trangular cells shown n Fgure 25 are used n the smulaton. As seen n Fgure 26 on the contour plot of the Mach number, a shock occurrng on the upper sde of the arfol s captured. The predcted pressure coeffcent (C p ) dstrbuted on the arfol surface s plotted n Fgure 27, wth comparson to the results from the commercal CFD package FLUENT (FVM-based) [57], and expermental data [58]. Apparently, our GSM result s more accurate than the FLUENT result, especally near the leadng edge. The notceable dfference between our GSM soluton and expermental data, occurrng around the shock regon, may be due to the nsuffcently fne cells generated across the shock. It should be mproved once GSM s embarked wth soluton-based adaptve technques, whch wll be explored n the future. 5. CONCLUSION In ths study, a conservatve and effcent GSM formulated for the strong form of governng equatons s developed. GSM s vald for both regular and rregular cells so that t can be readly

30 1130 G. R. LIU AND G. X. XU Fgure 25. Unstructured trangular cells around the RAE2822 arfol. Fgure 26. Contour plot of Mach number for the flow over the RAE2822 arfol. appled to flud flow problems wth arbtrary geometry. Wth efforts n studyng eght proposed GSM schemes, t s found that Schemes II, IV, VI and VIII are favorable, because of ther compact stencls wth postve coeffcents of nfluence. Schemes VII and VIII that use the gradent smoothng technque on mgsds outperform Schemes II and VI n terms of robustness, stablty and accuracy.

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