A Cartesian grid finite-difference method for 2D incompressible viscous flows in irregular geometries

Transcription

1 Journal of Computatonal Physcs 204 (2005) A Cartesan grd fnte-dfference method for 2D ncompressble vscous flows n rregular geometres E. Sanmguel-Rojas a, J. Ortega-Casanova b, C. del Pno b, R. Fernandez-Fera b, * a Unversdad Poltécnca de Cartagena, E.T.S. Ingeneros Industrales, Cartagena, Murca, Span b Unversdad de Málaga, E.T.S. Ingeneros Industrales, Málaga, Span Receved 23 Aprl 2004 receved n revsed form 11 October 2004 accepted 12 October 2004 Avalable onlne 11 November 2004 Abstract A method for generatng a non-unform Cartesan grd for rregular two-dmensonal (2D) geometres such that all the boundary ponts are regular mesh ponts s gven. The resultng non-unform grd s used to dscretze the Naver Stokes equatons for 2D ncompressble vscous flows usng fnte-dfference approxmatons. To that end, fntedfference approxmatons of the dervatves on a non-unform mesh are gven. We test the method wth two dfferent examples: the shallow water flow on a lake wth rregular contour and the pressure drven flow through an rregular array of crcular cylnders. Ó 2004 Elsever Inc. All rghts reserved. 1. Introducton The numercal smulaton of flows wth rregular geometres s a problem of ncreasng nterest. In partcular, much effort have been dedcated n recent years to the use of Cartesan grds whch does not conform to the rregular boundares [1 6]. In relaton to the conventonal structured-grd approach wth curvlnear grds that conforms to the boundares, ths approach has the man advantage of ts smplcty, both n the grd generaton and n the governng equatons. In addton, the transformaton of the governng equatons to a curvlnear coordnate system that conforms to very complcated boundares s not an easy task, and usually affects to the stablty, convergence, and accuracy of the numercal solver. In ths paper we present a technque for Cartesan grd generaton that conforms to rregular twodmensonal (2D) boundares. It has the advantage of workng wth a Cartesan mesh n whch all the * Correspondng author. E-mal address: ramon.fernandez@uma.es. (R. Fernandez-Fera) /$ - see front matter Ó 2004 Elsever Inc. All rghts reserved. do: /j.jcp

2 E. Sanmguel-Rojas et al. / Journal of Computatonal Physcs 204 (2005) boundares nodes are regular nodes of the grd, thus avodng the usual complcated nterpolatons needed for the Cartesan cells cut by mmersed boundares, or the use of artfcal body forces, or other artfces such as a dstrbuton of vortcty sources, to mpose the boundary condtons (see, for example [1 6]). The prce one has to pay s that the Cartesan grd s non-unform. For ths reason we develop second-order accurate fnte-dfference approxmatons of the dervatves for non-equdstant grd ponts that substtutes the usual fnte-dfference approxmatons for unform grds (some of these expressons are already reported n the lterature see, e.g. [7]). Thus, we are led to dscrete equatons wth the same level of complexty than the Cartesan equatons dscretzed on a unform grd, but conformng to an rregular geometry. One dsadvantage n relaton to the nterpolaton technques gven n some of the references cted above s that the present method s not suted for movng nterfaces. However, second-order nterpolaton technques [4] are not easly appled to Neumann boundary condtons. The structure of the paper s the followng. Secton 2 ntroduces the grd generaton technque on a generc, rregular 2D doman. The expresson for the fnte-dfference approxmatons of the Cartesan dervatves on non-unform grds are gven n Secton 3. Sectons 4 valdates the method wth two examples qute dfferent to each other: the 2D shallow water, wnd-drven flow on a lake wth rregular contour, and the 2D ncompressble, pressure-drven flow around an rregular array of crcular cylnders. Some conclusons are drawn n the last secton. 2. Cartesan grd generaton that conforms to an rregular 2D doman Consder the 2D rregular doman of Fg. 1. Our objectve s to generate a Cartesan grd where all the boundary ponts are regular mesh ponts. Ths means that all the nteror ponts have to be collocated n relaton to a set of selected boundary ponts, and that the resultng Cartesan grd wll not be unform. In order to smplfy the storage of the grd ponts locaton n matrx form, what we propose here s a ray tracng technque. One starts at a gven boundary pont (marked wth a crcle n Fg. 1), and generates a set of boundary ponts by ÔCartesan reflectonsõ of the ray (squares n Fg. 1). In order to avod an nfnte regress, the process ends when the ray reaches a boundary perpendcular to t (.e., when t reaches a secton of the boundary parallel to one of the Cartesan axs), or when a boundary node s generated very close to a prevous one (ther separaton s less than a gven tolerance). It s mportant to detect frst the man boundary ponts or ponts of ntersecton between the several sectons of the boundary (crcles n Fg. 2). The frst rays wll start from these man ponts, dvdng the doman, and the boundary, n a number sectons. Each of these sectons s then dvded usng a number of ponts on each boundary secton whch depends on the desred precson. The resultng mesh (see Fg. 2(b)) concentrates the nodes wth the desred precson at the dfferent sectons of the boundary. Programng ths technque s relatvely easy and the storng n matrx form of the resultng grd ponts locatons s also straghtforward. The technque s a lttle more complcated n domans wth concave boundary sectons lke that depcted n Fg. 3. We have traced three rays startng from the three crcled ponts. These rays generate a seres of mesh ponts, both on the boundary and nsde the doman, before stopng at a boundary parallel to the x-axs. The last three nodes on that boundary have no correspondng ones on the lower boundary. Thus, n order to facltate the storage of the nodes n matrx form,.e., n order to have an structured grd, t s convenent to contnue these rays tll the lowest boundary (dashed lnes n Fg. 3(a)). Though we store all the nodes, we can dsregard the three trangled nodes n Fg. 3(b), beng the effectve node to the rght of node j that labelled wth j + 4. Once the grd s generated, one may create an ndrect access wth a new ndex, say jj, such that jj + 1 corresponds to j + 4. Thus, the fnal non-structured grd s stored wthn a structured grd form of n m nodes (Fg. 3(c)). The ndrect access through the ndexes (,jj) tell us whether a node of the structured grd s an actual node of the computatonal non-structured grd.

3 304 E. Sanmguel-Rojas et al. / Journal of Computatonal Physcs 204 (2005) (a) (b) Fg. 1. Generc 2D doman (a) and llustraton of grd generaton by ray tracng (b). In very rregular domans t may happen that ths ray tracng method may generate very small cells near some boundares, or even nsde the doman. To avod ths we use a lower lmt for the cell sze, n such a way that nodes that generate cells wth sze less than ths lmt are dscarded, n a smlar way to what s done n concave boundares, as descrbed above. 3. Fnte-dfference approxmaton on non-unform meshes In order to dscretze the flow equatons n the non-unform grd developed n the above secton, one has to use fnte-dfference approxmatons on a non-unform mesh. In ths secton we develop these fntedfference expressons for all the spatal dervatves appearng n the Naver Stokes equatons. Some of them have been prevously reported by Turkel [7]. In partcular, Turkel provdes the frst dervatve, and the centered form of the second dervatve wth frst-order truncaton error. All the expressons we gve below (some of them are gven n Appendx A) are second-order accurate, and we nclude forward and backwards expressons for the second-order dervatves, whch are needed at the boundares. It has been shown [,9] that second-order accuracy can be obtaned (on non-unform grds) even though local truncaton

4 E. Sanmguel-Rojas et al. / Journal of Computatonal Physcs 204 (2005) Fg. 2. Man ponts (crcles), and resultng non-unform grd (b). errors are of lower order. However, ths s vald only for non-unform grds n whch the varaton of the mesh sze s very small and for lnear equatons. Snce we want to apply the fnte-dfferences method to arbtrary non-unform meshes, and to the non-lnear Naver Stokes equatons, we need second-order truncaton errors to reach second-order accuracy. Ths s mportant n problems wth a long tme evoluton, such as the examples gven below, where second-order accuracy s needed at both nner and boundary nodes. Consder a 1D non-unform grd wth nx + 1 dscrete ponts (0 6 6 nx) located arbtrarly on the unt length (Fg. 4). If the value of a generc functon f(x) and ts dervatves are known at the pont th, x = Dx, Dx =1/nx, the values of f at the ponts ± 1 and ± 2 can be approxmated usng Taylor expansons. Indcatng wth a subscrpt the grd pont, and wth prmes the dervatves wth respect to x, the unknown values f ±1 and f ±2 can be wrtten as f þ1 ¼ f þ h þ1 f 0 þ h2 þ1 2 f 00 þ h3 þ1 6 f 000 þ Oðh 4 þ1þ ð1þ f 1 ¼ f þ h 1 f 0 þ h2 1 2 f 00 þ h3 1 6 f 000 þ Oðh 4 1Þ ð2þ

5 306 E. Sanmguel-Rojas et al. / Journal of Computatonal Physcs 204 (2005) (a) (b) (c) Fg. 3. (a) and (b) Fcttous grd ponts n a doman wth concave boundary sectons. (c) Fnal non-structured grd (squares). Fg. 4. Nonunform grd wth nx + 1 grd ponts dstrbuted arbtrarly.

6 E. Sanmguel-Rojas et al. / Journal of Computatonal Physcs 204 (2005) f þ2 ¼ f þ h þ2 f 0 þ h2 þ2 2 f 00 þ h3 þ2 6 f 000 þ Oðh 4 þ2þ ð3þ f 2 ¼ f þ h 2 f 0 þ h2 2 2 f 00 þ h3 2 6 f 000 þ Oðh 4 2Þ ð4þ where the last terms n these expressons ndcate the order of the truncaton error of the approxmaton, and h þ1 ¼ x þ1 x h 1 ¼ x 1 x h þ2 ¼ x þ2 x h 2 ¼ x 2 x : ð5þ These expansons can be used to approxmate the nth dervatve of f at the pont up to any order of the truncaton error, provded that they are convenently combned. Wth second-order accuracy, the fntedfference approxmaton n the centered form for the frst and second dervatves of f at the pont are (for hgher dervatves, or hgher order for the desred truncaton error, more ponts than those consdered n (1) (4) are needed): h m ¼ þ1 h þ1 h 1 þh 2 1 h f 0 n ¼ 1 ¼ m f 1 þ s f þ n f þ1 þ E h þ1 h 1 þh 2 þ1 ð6þ s ¼ ðm þ n Þ >: E ¼ 1 h 6 þ1h 1 f 000 f 00 ¼ m f 1 þ s f þ n f þ1 þ p f þ2 þ E f 00 ¼ p f 2 þ m f 1 þ s f þ n f þ1 þ E m ¼ n ¼ p ¼ 2ðh þ1 þh þ2 Þ ðh þ1 h þ2 h 1 h þ1 þh 2 1 h 1h þ2 Þh 1 2ðh 1 þh þ2 Þ ð h þ1 h 1 þh 1 h þ2 þh 2 þ1 h þ1h þ2 Þh þ1 2ðh þ1 þh 1 Þ ðh þ1 h þ2 h 1 h þ1 h 2 þ2 þh 1h þ2 Þh þ2 s ¼ ðm þ n þ p Þ >: E ¼ 1 m 12 h 4 1 þ n h 4 þ1 þ p h 4 þ2 f v m ¼ n ¼ p ¼ 2ðh þ1 þh 2 Þ ðh þ1 h 2 h 1 h þ1 þh 2 1 h 1h 2 Þh 1 2ðh 1 þh 2 Þ ð h þ1 h 1 þh 1 h 2 þh 2 þ1 h þ1h 2 Þh þ1 2ðh þ1 þh 1 Þ ðh þ1 h 2 h 1 h þ1 h 2 2 þh 1h 2 Þh 2 s ¼ ðm þ n þ p Þ >: E ¼ 1 p 12 h 4 2 þ m h 4 1 þ n h 4 þ1 f v where E s the truncaton error of each approxmaton. Ths truncaton error s a functon of the separaton between the local ponts around. Thus, n a non-unform grd, expressons (6) () wll be more accurate as the grd ponts becomes closer. Note that two dfferent expressons for the second dervatve are gven, (7) and (), nether of them fully centered on the pont. Ths s because one needs four grd ponts to have a second-order truncaton error n the second dervatve, so that two dfferent expressons result dependng on whether we use the pont + 2, or the pont 2. ð7þ ðþ

7 30 E. Sanmguel-Rojas et al. / Journal of Computatonal Physcs 204 (2005) If the grd were unform, h h +1, h 1 = h, h +2 =2h, h 2 = 2h, expressons (6) () are, obvously, the standard centered second-order fnte-dfference approxmaton for the frst and second dervatves: f 0 ¼ m f 1 þ s f þ n f þ1 þ E m ¼ 1 2h n ¼ 1 2h s ¼ 0 >: E ¼ 1 6 h2 f 000 ð9þ m ¼ 1 h 2 n ¼ 1 h 2 f 00 ¼ m f 1 þ s f þ n f þ1 þ p f 2 þ E p ¼ 0 ð10þ s ¼ 2 : h >: 2 E ¼ 1 12 h2 f v Now the grd ponts ± 2 do not appear n the approxmaton to the second dervatve because p =0. From (1) (4) one can obtan not only centered approxmatons, but also forward or backward ones. These expressons, whch are needed at the boundares, are gven n Appendx A wth second-order accuracy. We also gve there fnte-dfference expressons needed to apply Neumann boundary condtons wth second-order accuracy on a non-unform grd. 4. Results 4.1. Wnd-drven flow n a lake One of the man ntended applcatons of the present technque s the smulatons of 2D envronmental flows, such as the flows n shallow lakes and reservors, whch usually have very rregular geometres. For ths reason, as a frst example of a 2D flow n a complex doman we consder the wnddrven flow n Lake Belau (Northern Germany), for whch Podsetchne and Schernewsk [10] reported numercal and expermental results whch can be used to compare wth. The bathymetry of the lake s gven n Fg. 5. Snce the lake s not very deep, one may use the vertcally ntegrated equatons of contnuty and momentum on the horzontal plane (x, y), or shallow water approxmaton (see, for example, [11]): of þrv ¼ 0 ot ð11þ ov ot þr vv h þ ghrf m r 2 v 2rf r v v r 2 f f ^ v kw j W jþ gv j v j ¼ 0: ð12þ H H H C 2 2 H In these equatons, v =[u(x, y, t), v(x, y, t)] s the depth-averaged horzontal velocty, v Z f h v h dz wth v h (x,y,z,t) the local horzontal velocty, H(x,y,t) h(x,y) +f(x,y,t) s the total water depth, wth h the depth below the horzontal reference plane (z = 0) and f the water surface elevaton above z = 0, $ = o/ox + o/oy, g. 9.1 m/s 2 s the acceleraton due to gravty, f = fe z, wth f s 1 s the Corols parameter, m s the averaged, horzontal eddy vscosty, and C s the Chezy coeffcent. ð13þ

8 E. Sanmguel-Rojas et al. / Journal of Computatonal Physcs 204 (2005) Fg. 5. Dgtal contour and bathymetry (n meters) of Lake Belau such as they are used n the computatons (taken from Fg. 2 n [10]). The flow wll be drven by a wnd of velocty W =(W x,w y ), through the term kwjwj, where k = q a C W /q, wth q and q a the denstes of water and ar, respectvely, and C W the wnd drag coeffcent. The numercal values of the parameters that wll be used to solve these equatons are the followng [10]: m = 0.01 m 2 / s, C =40 m 1/2 /s, C W = 0.002, q =10 3 kg/m 3, q a = kg/m 3, and a spatally unform south-westerly wnd (a headng of 220 ) wth a speed jwj of 6 m/s. The boundary condtons at the contour of the lake are u = v =0. A mesh of 7517 grd ponts has been generated usng the technque of Secton 2 (see Fg. 6). The equatons have been dscretzed n ths non-unform grd usng the fnte-dfference approxmatons gven n Secton 3. In partcular, we have used ArakawaÕs grd of the type C [12], where the water elevaton f s evaluated at the grd ponts, whle the averaged velocty components are evaluated at the md ponts of ther respectve cell sdes (see Fg. 7). An explct, two-step, second-order accurate, predctor corrector scheme has been used to advance n tme. If one wrtes Eqs. (11) and (12) schematcally as of/ot = A(v), ov/ot = B(f, v), these two steps are gven by

9 310 E. Sanmguel-Rojas et al. / Journal of Computatonal Physcs 204 (2005) y (m) 250 y (m) x (m) x (m) Fg. 6. Two detals of the dscretzed geometry of the lake showng some of the grd ponts. Fg. 7. ArakawaÕs scheme used n the computatons, where f s evaluated at the grd ponts (trangles), u s evaluated at the squares, and v at the crcles.

10 predctor step: v ¼ v n þ Dt 2 Bðfn v n Þ f ¼ f n þ Dt 2 Aðv Þ E. Sanmguel-Rojas et al. / Journal of Computatonal Physcs 204 (2005) corrector step: v nþ1 ¼ v n þ DtBðf v Þ ð15þ f nþ1 ¼ f n þ DtAðv nþ1 Þ where the superscrpts denote the nstant of tme, and Dt s the tme step. The numercal computatons are started at t = 0 wth the flud at rest and f = 0. We use Dt = 1 s. The results for the velocty feld at t =3h are plotted n Fg.. These results compares very well wth those gven n Fg. 4(a) of Podsetchne and Schernewsk [10], who used a fnte-element method on a trangular mesh created wth a commercal 1000 ð14þ N W S Wnd E 600 y (m) Velocty scale 10 cm/s x (m) Fg.. Averaged velocty feld at t =3h.

11 312 E. Sanmguel-Rojas et al. / Journal of Computatonal Physcs 204 (2005) software package to solve the shallow water equatons. These authors also checked ther numercal results wth expermental measurements Pressure-drven flow through an array of crcular cylnders As a second example we have selected the 2D ncompressble flow around an rregular array of three crcular cylnders nsde a channel (see Fg. 9). In partcular we have consdered the pressure drven flow [13] orgnated by a gven pressure dfference set between the nlet and the outlet. The dmensonless equatons are rv ¼ 0 ð16þ ov ot þ v rv ¼ rpþ 1 Re r2 v ð17þ where v =(u,v) and p are the dmensonless velocty and pressure, respectvely. To non-dmensonalze these equatons we have used the dameter D of the cylnders as the length scale, and p a characterstc velocty based on the the pressure dfference Dp c between the nlet and the outlet, V c ¼ ffffffffffffffffffffffff Dp c =q, where q s the flud densty the Reynolds number s based on ths velocty [13] Re ¼ V cd m ¼ sffffffffffffff Dp c q D m where m s the knematc vscosty of the flud. The moton of the flud s set by the boundary condtons pðx ¼ 0 y ¼ 10 tþ ¼1 pðx ¼ 20 y ¼ 10 tþ ¼0: ð19þ The remanng boundary condtons are v = 0 on the cylnders, and at the channel walls, y = 0 and y = 10. The equatons are solved numercally wth the second-order (both n space and tme) projecton method descrbed n [13], usng a fnte-dfference scheme on a non-unform grd generated wth the method ð1þ y x Fg. 9. Geometry of the channel flow through three crcular cylnders.

12 E. Sanmguel-Rojas et al. / Journal of Computatonal Physcs 204 (2005) y Fg. 10. Mesh ponts concentrated around the cylnders. x Re q t Fg. 11. Reynolds number based on the ext flow rate as a functon of tme.

13 314 E. Sanmguel-Rojas et al. / Journal of Computatonal Physcs 204 (2005) descrbed above. In partcular we have used a grd of mesh ponts, and Dt = A detal of the grd near two of the cylnders s depcted n Fg. 10. To solve numercally the Posson equaton for the pressure we use an ADI based technque, and standard solvers for band matrces wth LU factorzaton from Blas and Lapack packages. The fact that we have now, n general, a non-structured grd does not affect to the effcency of these Posson solvers because what s suppled to them are the actual computatonal nodes and ther correspondng dscretzed equatons through the ndrect access mentoned n Secton 2. Results for Re = 50 are plotted n Fgs. 11 and 12. The flow evolves n tme from rest untl t reaches a quas-perodc state. Ths s shown n Fg. 11, where we plot as a functon of tme the Reynolds number based on the flow rate at the ext of the channel (x = L = 20), Re q q H Re Z H qðtþ ¼ uðx ¼ L y tþdy 0 where H = 10 s the wdth of the channel. Fnally, streamlnes and sobars at t = 106 are plotted n Fg. 12. ð20þ Fg. 12. Streamlnes (a), and sobars (b), at t = 106.

14 E. Sanmguel-Rojas et al. / Journal of Computatonal Physcs 204 (2005) y (a) x y (b) x Fg. 13. Flow nsde a channel wth a sngle crcular cylnder for Re q = 10. (a) Streamlnes (all the lengths are scaled wth the dameter of the cylnder). (b) Detal of the wake behnd the cylnder. The above results show that the current method has no dffculty n resolvng the complex flow pattern and recrculaton regons behnd the cylnders. However, snce no prevous results are reported for ths partcular flow, we consder next the case of a sngle cylnder nsde a channel (see Fg. 13). Ths flow, wth the dmensons gven n Fg. 13(a), has been studed expermentally and numercally [14]. It has been reported that the length l of the axsymmetrc wake behnd the cylnder for moderately small Reynolds number vares lnearly wth Re q. Thus, for nstance, for Re q = 10 (whch n our smulaton corresponds to Re. 17), the length of the wake behnd the cylnder reported s l (tmes de dameter of the cylnder) [14]. Fg. 13(b) shows a detal of the streamlnes behnd the cylnder obtaned numercally by our numercal code, showng that l. 0.21, n close agreement wth the prevous result. 5. Conclusons We have presented here a fnte-dfference method n a non-unform Cartesan grd whch allows us to solve 2D unsteady vscous flows n rregular geometres. It has all the advantages of solvng numercally the flow equatons n a Cartesan mesh (smplcty, accuracy, stablty), and has the mportant pecularty that all the boundares are ftted to the regular ponts of the mesh, so that complcated nterpolatons at the boundares are avoded. We have developed a smple method for generatng such a non-unform

15 316 E. Sanmguel-Rojas et al. / Journal of Computatonal Physcs 204 (2005) Cartesan mesh n complex geometres, and provded second-order fnte-dfference approxmatons n non-unform meshes, whch allow us to solve the flow equatons wth second-order accuracy n all the flow doman, ncludng the boundares. To check the valdty of the method n flows wth both rregular external boundares and complex mmersed boundares, we have selected two typcal examples where we thnk ths method wll be of most nterest: the 2D (shallow water) flow n a lake wth a very complcated external boundary, and a vscous flow wth a complex nternal boundary such as the flow through an rregular array of cylnders between parallel plates. These flows are solved wth second-order tme accuracy usng two dfferent numercal schemes, and the results show that the non-unform Cartesan mesh s able to accurately smulate these flows wth easy. Acknowledgement Ths research has been supported by the COPT of the Junta de Andalucía (Contract No. 07/ ). The numercal smulatons have been made n the computer facltes at the SAIT (U.P. Cartagena) and at the SCAI (U. Málaga). Appendx A. In ths appendx we gve some addtonal expressons for fnte-dfference approxmatons on a non-unform grd that we have used n our computatons. For example, to apply Drchlet boundary condtons we need forward or backward expressons for the frst and second dervatves. Wth second-order accuracy, these one sded approxmatons are: f 0 ¼ s f þ m f 1 þ p f 2 þ E f 0 ¼ s f þ n f þ1 þ p f þ2 þ E for the frst dervatve, and f 00 ¼ s f þ m f 1 þ p f 2 þ q f 3 þ E h m ¼ 2 h 1 ðh 1 h 2 Þ h p ¼ 1 h 2 ðh 1 h 2 Þ s ¼ ðm þ p Þ >: E ¼ 1 h 6 1h 2 f 000 h n ¼ þ2 h þ1 ðh þ1 h þ2 Þ h p ¼ þ1 h þ2 ðh þ1 h þ2 Þ s ¼ ðn þ p Þ >: E ¼ 1 h 6 þ1h þ2 f 000 m ¼ p ¼ q ¼ 2ðh 2 þh 3 Þ ð h 2 h 1 þh 2 h 3 þh 2 1 h 3h 1 Þh 1 2ðh 1 þh 3 Þ ð h 3 h 1 þh 2 h 1 h 2 2 þh 2h 3 Þh 2 2ðh 1 þh 2 Þ ð h 3 h 1 þh 2 h 1 þh 2 3 h 2h 3 Þh 3 s ¼ ðm þ p þ q Þ >: E ¼ 1 m 12 h 4 1 þ p h 4 2 þ q h 4 3 f v ða:1þ ða:2þ ða:3þ

16 E. Sanmguel-Rojas et al. / Journal of Computatonal Physcs 204 (2005) f 00 ¼ s f þ n f þ1 þ p f þ2 þ q f þ3 þ E n ¼ p ¼ q ¼ 2ðh þ2 þh þ3 Þ ð h þ2 h þ1 þh þ2 h þ3 þh 2 þ1 h þ3h þ1 Þh þ1 2ðh þ1 þh þ3 Þ ð h þ3 h þ1 þh þ2 h þ1 h 2 þ2 þh þ2h þ3 Þh þ2 2ðh þ1 þh þ2 Þ ð h þ3 h þ1 þh þ2 h þ1 þh 2 þ3 h þ2h þ3 Þh þ3 s ¼ ðn þ p þ q Þ >: E ¼ 1 n 12 h 4 þ1 þ p h 4 þ2 þ q h 4 þ3 f v ða:4þ for the second dervatve (h ±3 = x ±3 x ). To apply Neumann boundary condtons we need the second-order dervatve n terms of the the frstorder one. Wth second-order accuracy, these expressons (forward and backward) are: f 00 ¼ r f 0 þ s f þ n f þ1 þ p f þ2 þ E f 00 ¼ r f 0 þ s f þ n f 1 þ p f 2 þ E r ¼ 2ðh þ1þh þ2 Þ h þ1 h þ2 s ¼ 2ðh2 þ1 þh þ1h þ2 þh 2 þ2 Þ h 2 þ1 þh2 þ2 n ¼ 2h þ2 p ¼ >: ðh þ1 h þ2 Þh 2 þ1 2h þ1 ðh þ1 h þ2 Þh 2 þ2 E ¼ 1 h 12 þ1h þ2 f v r ¼ 2ðh 1þh 2 Þ h 1 h 2 s ¼ 2ðh2 1 þh 1h 2 þh 2 2 Þ h 2 1 þh2 2 n ¼ 2h 2 p ¼ >: ðh 1 h 2 Þh 2 1 2h 1 ðh 1 h 2 Þh 2 2 E ¼ 1 12 h 1h 2 f v : ða:5þ ða:6þ References [1] T. Ye, R. Mttal, H.S. Udaykumar, W. Shyy, An accurate cartesan grd method for vscous ncompressble flows wth complex mmersed boundares, J. Comput. Phys. 156 (1999) [2] D. Calhoun, R.J. LeVeque, An cartesan grd fnte-volume method for the advecton dffuson equaton n rregular geometres, J. Comput. Phys. 157 (2000) [3] E.A. Fadlun, R. Verzcco, P. Orland, J. Mohd-Yusof, Combned mmersed-boundary fnte-dfference methods for threedmensonal complex flow smulatons, J. Comput. Phys. 161 (2000) [4] F. Gbou, R. Fedkw, L.T. Cheng, M. Kang, A second-order-accurate symmetrc dscretzaton of the Posson equaton on rregular domans, J. Comput. Phys. 176 (2002) [5] D. Calhoun, A cartesan grd method for solvng the two-dmensonal streamfuncton vortcty equatons n rregular regons, J. Comput. Phys. 176 (2002) [6] A. Glmanov, F. Sotropoulos, E. Balaras, A general reconstructon algorthm for smulatng flows wth complex 3D mmersed boundares on cartesan grds, J. Comput. Phys. 191 (2003) [7] E. Turkel, Accuracy of schemes wth nonunform meshes for compressble flud-flows, J. Appl. Numer. Math. 2 (196) [] T.A. Manteuffel, A.B. Whte, The numercal soluton of second-order boundary value problems on nonunform meshes, Math. Comput. 47 (196)

17 31 E. Sanmguel-Rojas et al. / Journal of Computatonal Physcs 204 (2005) [9] H.O. Kress, T.A. Manteuffel, B. Swartz, B. Wendroff, A.B. Whte, Supra-convergent schemes on rregular grds, Math. Comput. 47 (196) [10] V. Podsetchne, G. Schernewsk, The nfluence of spatal wnd nhomogenety on flow patterns n a small lake, Wat. Res. 33 (1999) [11] T. Wejan, Shallow Water Hydrodynamcs, Elsever, Amsterdam, [12] A. Arakawa, V. Lamb, Computatonal desgn of the basc dynamcal processes of the UCLA general crculaton model, Methods Comput. Phys. 17 (1977) [13] R. Fernandez-Fera, E. Sanmguel-Rojas, An explct projecton method for solvng ncompressble flows drven by a pressure dfference, Comput. Fluds 33 (2004) [14] J.H. Chen, W.G. Prtchard, S.J. Tavener, Bfurcaton for flow past a cylnder between parallel planes, J. Flud Mech. 24 (1995)