A new integrated-rbf-based domain-embedding scheme for solving fluid-flow problems

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1 Home Search Collectons Journals About Contact us My IOPscence A new ntegrated-rbf-based doman-embeddng scheme for solvng flud-flow problems Ths artcle has been downloaded from IOPscence. Please scroll down to see the full text artcle IOP Conf. Ser.: Mater. Sc. Eng ( Vew the table of contents for ths ssue, or go to the journal homepage for more Download detals: IP Address: The artcle was downloaded on 06/07/2010 at 08:09 Please note that terms and condtons apply.

2 IOP Conf. Seres: Materals Scence and Engneerng 10 (2010) IOP Publshng do: / x/10/1/ A new ntegrated-rbf-based doman-embeddng scheme for solvng flud-flow problems K. Le-Cao 1, N. Ma-Duy 1, C.-D. Tran 2 and T. Tran-Cong 1 1 Computatonal Engneerng and Scence Research Centre Faculty of Engneerng and Surveyng, The Unversty of Southern Queensland, Toowoomba, QLD 4350, Australa 2 CSIRO, Geelong, VIC, Australa E-mal: KhoaCao.Le@usq.edu.au Abstract. Ths paper presents a new doman embeddng numercal scheme for the smulaton of flows of a Newtonan flud n multply-connected domans. The governng equatons are taken from the stream functon-vortcty formulaton. The problem doman s converted nto a smply-connected doman that s then dscretsed usng a Cartesan grd. Radal-bass-functon networks, whch are constructed through ntegraton rather than the usual dfferentaton, are employed on grd lnes to approxmate the feld varables. Each feld varable s assumed to vary over nteror holes accordng to approprate polynomals that satsfy the boundary condtons. Pont collocaton s appled to dscretse the governng equatons. Several lnear and nonlnear problems, ncludng natural convecton n the annulus between square and crcular cylnders are smulated to verfy the proposed technque. 1. Introducton Solvng the Naver-Stokes equatons n rregularly shaped domans presents a challenge n CFD. The concept of doman embeddng or fcttous doman s known to provde an effcent way to handle complex geometres. The basc dea of doman embeddng methods/fcttous doman methods s to extend the problem defned on a geometrcally-complex doman to that on a larger, but smpler shape doman. The obtaned fcttous doman allows the use of a regular grd/mesh that can be fxed to represent the computatonal doman, and one can thus use fast drect solvers for the resultant algebrac system. All gven boundary condtons must be mposed n order to match the soluton on the fcttous doman wth that on the orgnal doman. Fcttous-doman technques have been very successful n solvng complcated engneerng problems. Glownsk et al. [2] have presented a famly of fcttous-doman technques whch are based on the explct use of Lagrange multplers defned on the actual boundary and assocated wth the boundary condtons for Drchlet ellptc problems. Snce then, the Lagrange multpler/fcttous-doman methods have become ncreasngly popular. Typcal examples nclude ncompressble vscous flows (e.g. [5]), flud/rgd-body nteractons (e.g. [3]) and flud/flexble-body nteractons (e.g. [13]). The basc equatons governng the moton of a flud can be wrtten n dfferent dependent varables, e.g. the velocty - pressure, stream functon - vortcty and stream functon formulatons. Each formulaton has some advantages over the others for certan classes of c 2010 Publshed under lcence by IOP Publshng Ltd 1

3 IOP Conf. Seres: Materals Scence and Engneerng 10 (2010) IOP Publshng do: / x/10/1/ problems. For the stream functon - vortcty formulaton, one has to derve boundary condtons for the vortcty whose accuracy strongly affects the overall soluton. There are many dscretsaton methods, ncludng those based on a fnte-element mesh, a fnte-volume mesh, a Cartesan grd or a set of unstructured ponts, to reduce the PDEs to sets of algebrac equatons. Among them, generatng a Cartesan grd can be seen to be the most straght forward task. The use of Cartesan grds for solvng problems defned on rregular domans has receved much ncreased attenton n recent decades. In ths study, we report a numercal collocaton technque ncorporatng 1D-IRBFNs on grd lnes for the smulaton of heat transfers and flud flows n multply-connected domans. The technque combnes strengths of the three approaches, namely 1D-IRBFNs, Cartesan grds and fcttous domans. It should be emphassed that conventonal RBFN methods lead to fully populated matrces that tend to become ll-condtoned quckly wth ncreasng numbers of RBFs. Instead of usng conventonal schemes, 1D-IRBFN approxmaton schemes [7,8] are utlsed n the present work. In the case of natural convecton, an effectve formula for computng the vortcty boundary condton on a Cartesan grd s derved. Frst dervatves of the stream functon along the boundares are ncorporated nto the computatonal vortcty boundary values by means of ntegraton constants. The present IRBFN approxmatons are constructed to satsfy all boundary condtons dentcally. Through fcttous domans, the proposed technque s able to work for domans of dfferent shapes n a smlar fashon. Unlke Glownsk et al. [2], the feld varables at nteror holes are presently replaced by approprate polynomals that satsfy the boundary condtons. Results obtaned are compared well wth avalable numercal data n the lterature. The remander of the paper s organsed as follows. Secton 2 gves a bref revew of the governng equatons. In Secton 3, we descrbe the proposed doman embeddng technque. A formula for handlng vortcty boundary condtons at boundary ponts that are not grd nodes s gven n Secton 4. Numercal results are presented n Secton 5. Secton 6 concludes the paper. 2. Governng equatons The stream functon - vortcty - temperature formulaton s used here. The non-dmensonal basc equatons for natural convecton under the Boussnesq approxmaton n the Cartesan x y coordnate system can be wrtten as (e.g. [9]) 2 ψ x ψ = ω, (1) y2 ω t + u ω x + v ω ( P r 2 ) y = ω Ra x ω y 2 T x, (2) T t + u T x + v T ( y = 1 2 T RaP r x T y 2 ), (3) where ψ s the stream functon, ω the vortcty, T the temperature, t the tme, u and v the velocty components, and P r and Ra the Prandtl and Raylegh numbers defned as P r = ν/α and Ra = βg T L 3 /αν, n whch ν s the knematc vscosty, α the thermal dffusvty, β the thermal expanson coeffcent and g the gravty, respectvely. In ths dmensonless scheme, L, T (temperature dfference), U = glβ T and (L/U), are taken as scale factors for length, temperature, velocty and tme, respectvely. It s noted that the velocty scale s chosen here n a way that the buoyancy and nertal forces are balanced (e.g. [9]). The velocty components are defned n terms of the stream functon as u = ψ/ y and v = ψ/ x. The gven velocty boundary condtons, u and v, can be transformed nto two boundary condtons on the stream functon and ts normal dervatve ψ = γ and ψ/ n = ξ, 2

4 IOP Conf. Seres: Materals Scence and Engneerng 10 (2010) IOP Publshng do: / x/10/1/ where n s the drecton normal to the boundary, and γ and ξ prescrbed functons. In the case of fxed concentrc cylnders, non-slp boundary condtons usually lead to γ = 0 and ξ = One dmensonal IRBFN-based doman embeddng technque Fgure 1. Computatonal domans and dscretsatons. It s noted that the real doman s the regon between nner crcular cylnder and outer square cylnder. Consder a square doman wth a crcular hole located at the center. Ths physcal doman s extended to a square one that can be then convenently represented by a Cartesan grd of m m (Fgure 1). It can be seen that 1D-IRBFN expressons on the x and y grd lnes have smlar forms. In the followng, only a horzontal grd lne s consdered. The second-order dervatve of the feld varable f along a grd lne can be decomposed nto RBFs 2 f(x) x 2 = m w g (x) = =1 m =1 w I (2) (x), (4) { } where m s the number of RBFs, {g (x)} m =1 I (2) m (x) the set of RBFs, {w } m =1 the set of =1 weghts to be found and f represents ψ, ω and T. Approxmate expressons for the frst-order dervatve and the feld varable are then obtaned through ntegraton f(x) x f(x) = = m =1 m =1 w I (1) (x) + c 1, (5) w I (0) (x) + c 1 x + c 2, (6) where I (1) (x) = I (2) (x)dx and I (0) (x) = I (1) (x)dx and (c 1, c 2 ) are the constants of ntegraton. Collocatng (6) at the nodal ponts yelds f = Î(0) ŵ c 1 c 2, (7) 3

5 IOP Conf. Seres: Materals Scence and Engneerng 10 (2010) IOP Publshng do: / x/10/1/ where f = (f(x 1 ), f(x 2 ),, f(x m )) T, ŵ = (w 1, w 2,, w m ) T, Î (0) = I (0) 1 (x 1) I (0) 2 (x 1) I m (0) (x 1 ) x 1 1 I (0) 1 (x 2) I (0) 2 (x 2) I m (0) (x 2 ) x I (0) 1 (x m) I (0) 2 (x m) I m (0) (x m ) x m 1 Solvng (7) for the coeffcent vector ncludng the two ntegraton constants results n ŵ c 1 (Î(0) ) 1 = f, (8) c 2 (Î(0) ) 1 where s the generalsed nverse. The values of the frst and second dervatves of f wth respect to x at the nodal ponts are thus computed n terms of nodal varable values and where f x = Î(1) ( Î (0)) 1 f = D1x f, (9) 2 f x 2 = Î(2) ( Î (0)) 1 f = D2x f, (10) ( f x = f(x1 ) x, f(x 2) x,, f(x ) T m), x 2 ( f 2 ) T x 2 = f(x 1 ) x 2, 2 f(x 2 ) x 2,, 2 f(x m ) x 2, Î (1) = Î (2) = I (1) 1 (x 1) I (1) 2 (x 1) I m (1) (x 1 ) 1 0 I (1) 1 (x 2) I (1) 2 (x 2) I m (1) (x 2 ) I (1) 1 (x m) I (1) 2 (x m) I m (1) (x m ) 1 0 g 1 (x 1 ) g 2 (x 1 ) g m (x 1 ) 0 0 g 1 (x 2 ) g 2 (x 2 ) g m (x 2 ) , g 1 (x m ) g 2 (x m ) g m (x m ) 0 0 and D 1x, D 2x are the frst- and second-order dfferentaton matrces n the physcal space. In the case that a horzontal grd lne crosses the nner hole (Fgure 2), there are two nterfaces as shown at x b2 and x b3. The doman can be dvded nto two dfferent parts: the regon betweens the two nterfaces (extended doman) and the remanng regons (real doman). The extended doman thus represents a hole nsde a square cylnder. The solutons n the extended and real domans are denoted as f f and f r, respectvely. Because of the contnuty of the soluton, one has f f = f r on the nterfaces. We assume that the soluton on the extended doman s, 4

6 IOP Conf. Seres: Materals Scence and Engneerng 10 (2010) IOP Publshng do: / x/10/1/ Fgure 2. Ponts on a grd lne consst of nteror ponts x ( ) and boundary ponts x b ( ). known and can be descrbed by a polynomal. Ths polynomal can be constructed as follows. Snce there are four boundary ponts on the grd lne, one can use a polynomal of thrd order, ax 3 + bx 2 + cx + d, whose coeffcents are determned as where P = a b c d = P 1 f b1 f b2 f b3 f b4 x 3 b1 x 2 b1 x b1 1 x 3 b2 x 2 b2 x b2 1 x 3 b3 x 2 b3 x b3 1 x 3 b4 x 2 b4 x b4 1 (11) The soluton n the extended doman s thus computed as f f = ax 3 + bx 2 + cx + d, where x b2 x x b3. For llustraton purposes, the formulaton s presented n detal for the Posson equaton 2 f = b subject to Drchlet boundaty condtons. Usng (10) and tensor products, the PDE reduces to A f = b, (12) where A = D 2x Ĩ + Ĩ D 2y. In whch, Ĩ s the dentty matrx of dmensons of m m. In (12), the grd nodes are numbered from left to rght and bottom to top. Ths system can be rearranged for the unknown values of f n the real doman as A(dr, dr) f(dr) = b(dr) A(dr, db) f(db) A(dr, df) f(df), (13). where dr, db and df are the sets whose elements are the ndces of nodes n the real doman, on the outer boundary and n the extended doman, respectvely. 4. A new formula for computng vortcty boundary condtons It can be seen that boundary condtons are over-prescrbed for the stream-functon equaton (1) and under-prescrbed for the vortcty equaton (2). We use normal dervatve boundary condtons for the stream functon to derve boundary condtons for the vortcty. The values of the vortcty on the boundares can be computed va ω b = 2 ψ b x ψ b y 2, (14) 5

7 IOP Conf. Seres: Materals Scence and Engneerng 10 (2010) IOP Publshng do: / x/10/1/ Fgure 3. A curved boundary. where the subscrpt b s used to ndcate the boundary quanttes. The handlng of ω b thus nvolves the evaluaton of second-order dervatves of the stream functon n both x and y drectons. For regular boundary ponts (also grd nodes), one can apply (14) drectly. The x and y grd lnes passng through those ponts can be used for computng / x 2 and / y 2, respectvely. However, n general, the boundary ponts do not concde wth the grd nodes and hence they le on ether x or y grd lnes. Informaton about ψ s thus gven n one coordnate drecton only. A great challenge here s how to compute second dervatves of ψ n (14) wth respect to the drecton wthout a grd lne. A new formula to overcome ths dffculty s proposed below. Consder a curved boundary, along whch the values for ψ and ψ/ n are prescrbed (Fgure 3). It can be seen that the values of ψ/ x and ψ/ y on the boundary can then be obtaned n a straghtforward manner. Let s be the arclength of the boundary. By ntroducng an nterpolatng scheme (e.g. 1D-IRBFNs), one s able to derve dervatves of ψ/ x and ψ/ y wth respect to s such as 2 ψ/ x s and 2 ψ/ y s. A tangental dervatve of a functon F at a boundary pont can be computed usng the followng formula F s = F x t x + F y t y (15) where t x and t y are the two x and y components of the unt vector t tangental to the curve (t x = x/ s and t y = y/ s). Replacng F wth ψ b / x, one has or where / x s s consdered as a known quantty. Smlarly, takng F as ψ b / y results n x s = 2 ψ b x 2 t x + 2 ψ b x y t y, (16) x y = 1 t y ( 2 ψ b x s 2 ψ b x 2 t x), (17) x y = 1 t x ( 2 ψ b y s 2 ψ b y 2 t y), (18) where / y s s a known value. From (17) and (18), one can derve the relatonshp between 2 ψ/ x 2 and 2 ψ/ y 2 at a boundary pont 1 ( 2 ψ b t y x s 2 ψ b x 2 t x) = 1 ( 2 ψ b t x y s 2 ψ b y 2 t y). (19) 6

8 IOP Conf. Seres: Materals Scence and Engneerng 10 (2010) IOP Publshng do: / x/10/1/ Consder a x grd lne. The nterpolatng scheme employed along ths lne does not facltate the computaton of second-order dervatve of ψ wth respect to the y coordnate. However, such a dervatve at a boundary pont can be found by usng (19) y 2 = (t x t y ) 2 2 ψ b x 2 + q y, (20) where q y s a known quantty defned by q y = t x t 2 y x s + 1 t y y s. (21) By substtutng (20) nto (14), a boundary condton for the vortcty at a boundary pont on a horzontal grd lne wll be computed by ω b = [ 1 + ( tx t y ) 2 ] x 2 + q y, (22) where only the approxmatons n the x drecton are needed. In the same manner, on a vertcal grd lne, a boundary condton for the vortcty at a boundary pont wll be computed by ω b = where q x s a known quantty defned by [ 1 + ( ty t x ) 2 ] y 2 + q x, (23) q x = t y t 2 x y s + 1 t x x s. (24) The boundary condtons for the vortcty are thus wrtten n terms of second dervatve of the stream functon wth respect to x or y only. 5. Numercal examples All multply-connected domans are extended to rectangular domans. Calculatons are performed on unform Cartesan grds. The IRBFN approxmatons are mplemented wth the multquadrc functon, where the RBF wdth s chosen to be a grd sze. Three examples are employed to study the performance of the present technque Example 1: Posson equaton wth analytc soluton Ths example problem s governed by 2 f x f = b(x, y), (25) y2 on a doman as shown n Fgure 1 wth Drchlet boundary condtons. The exact soluton of ths problem s taken as f e = 1 sn(πx) sn(πy), (26) π2 7

9 IOP Conf. Seres: Materals Scence and Engneerng 10 (2010) IOP Publshng do: / x/10/1/ from whch the drvng functon b(x, y) and the boundary condtons can be derved analytcally. Fve grds from to are used. The accuracy of an approxmaton scheme s measured by means of the dscrete relatve L 2 error defned as M =1 Ne = (f e f ) 2 M (27) =1 (f e) 2 where M s the number of unknown nodal values of f, and f e and f are the exact and approxmate solutons, respectvely. Results of N e and the condton number of the system matrx are dsplayed n Table 1. Table 1. Example 1: Errors and condton numbers of the system matrx. Grd Error Cond(A) e-3 1e e-4 8e e-4 2e e-4 4e e-4 6e Example 2: Heat transfer n a mult-hole doman Fgure 4. Contour plot. Ths example s chosen to llustrate the capablty of the proposed technque n handlng geometrcally-complex problems usng a Cartesan grd. Consder the Posson equaton 2 f = 1 defned on a unt square wth 10 holes of the same radus 0.01 and subject to Drchlet boundary condtons f = 0. Fgure 4 presents a contour plot of the soluton f usng a grd of Example 3: Concentrc annulus between a square outer cylnder and a crcular nner cylnder Ths example s concerned wth natural convecton between a heated nner crcular cylnder and a cooled square enclosure (Fgure 1). An aspect rato of L/2R = 0.26 (L: the sde length of 8

10 IOP Conf. Seres: Materals Scence and Engneerng 10 (2010) IOP Publshng do: / x/10/1/ Table 2. Example 3: Comparson of ψ max for Ra = 10 4, 10 5, 10 6 between the present technque and other methods. ψ max Ra Present [12] [1] 1e e e the outer square and R: the radus of the nner crcle), P r = 0.71 and Ra = {10 4, 10 5, 10 6 } are employed here. Numercal results are obtaned for two grds of and The soluton procedure nvolve the followng man steps. () Guess the dstrbutons of T, ω and ψ. () Dscretse the governng equatons n tme usng a frst-order accurate fnte-dfference scheme. () Dscretse the governng equatons n space usng 1D-IRBFNs. (v) Solve the energy equaton (3) for T. (v) Derve computatonal boundary condtons for ω. (v) Solve the vortcty equaton (1) for ω. (v) Solve the stream-functon equaton (2) for ψ. (v) Check to see whether the soluton has reached a steady state usng the followng convergence measure (CM) np ( ) ψ (k) ψ (k 1) 2 CM = ( ) < ɛ, (28) 2 =1 np =1 where n p s the number of nteror ponts n the real doman, k the tme level and ɛ the tolerance (n ths study, ɛ s taken to be ). ψ (k) (x) If t s not satsfed, advance tme step and repeat from step 4. computaton and output the results. Otherwse, stop the The obtaned results are shown n Table 2 and Fgure 5. Table 2 presents the comparson of the maxmum value of the stream functon between the proposed and other technques ([1,12]), showng a good agreement. Fgure 5 dsplays the velocty felds and sotherms for Ra = 10 4, Ra = 10 5 and Ra = 10 6, whose behavours are qualtatvely smlar to those n [1]. 6. Concludng remarks In ths artcle, a new doman embeddng scheme for the stream functon - vortcty formulaton usng Cartesan grds and 1D-IRBFNs s reported. Attractve features of the proposed technque nclude () The preprocessng s smple as the multple connected doman s converted nto a rectangular one, () The boundary condtons for the vortcty are mplemented n an effectve manner and () Numercal results show that the matrx condton number s relatvely small. The technque s verfed successfully through several test problems. 9

11 IOP Conf. Seres: Materals Scence and Engneerng 10 (2010) IOP Publshng do: / x/10/1/ Ra = 1e4 Ra = 1e5 Ra = 1e6 Fgure 5. Square-crcular cylnders: temperature (left) and velocty vector (rght) felds Acknowledgements Ths research s supported by the CESRC, Faculty of Engneerng and Surveyng, Unversty of Southern Queensland and Australa s Commonwealth Scentfc and Industral Research Organsaton. References [1] Dng, H and Shu, C 2005 Int. J. Computaton and Methodology, 47, [2] Glownsk, R. and Kuznetsov, Yu Computer Methods n Appled Mechancs and Engneerng, 96, [3] Glownsk, R., Pan, T.-W., Hesla, T.I., Joseph, D.D. and Praux, J J. of Comp. Phys., 169, [4] Glownsk, R., Pan, T.-W. and Praux, J Comp. M. n Appled Mechancs and Engneerng, 111, [5] Glownsk, R., Pan, T.-W. and Praux, J. 1994, Computer Methods n Appled Mechancs and Engneerng, 112, [6] Le Cao, K., Ma-Duy, N. and Tran-Cong, T Num. Heat Transfer, Part B, 55, [7] Ma-Duy, N., Le-Cao, K. and Tran-Cong, T Int. J. for Numercal Methods n fluds, 57, [8] Ma-Duy, N. and Tran-Cong, T Num. M. for Partal Dfferental Equatons,23(5), [9] Ostrach, S J. of Heat Transfer, 110, [10] Sarler, B Computer Modelng n Engneerng and Scences, 7(2), [11] Shu, C. and Zhu, Y.D Int. J. for Num. M. n Fluds, 38, [12] Wu, Y.L., Lu, G.R. and Gu, Y.T Num. Heat Transfer, Part B: Fundamentals, 48 5, [13] Yu, Z J. of Comp. Phys., 207,