Home Search Collectons Journals About Contact us My IOPscence A Cartesan-grd ntegrated-rbf method for vscoelastc flows Ths artcle has been downloaded from IOPscence. Please scroll down to see the full text artcle. 2010 IOP Conf. Ser.: Mater. Sc. Eng. 10 012210 (http://opscence.op.org/1757-899x/10/1/012210) Vew the table of contents for ths ssue, or go to the journal homepage for more Download detals: IP Address: 139.86.2.14 The artcle was downloaded on 06/07/2010 at 08:19 Please note that terms and condtons apply.
A Cartesan-grd ntegrated-rbf method for vscoelastc flows D. Ho-Mnh, N. Ma-Duy and T. Tran-Cong Computatonal Engneerng and Scence Research Centre (CESRC), Faculty of Engneerng and Surveyng (FoES), The Unversty of Southern Queensland (USQ), Toowoomba, QLD 4350, Australa E-mal: hod@usq.edu.au Abstract. Ths paper s concerned wth the use of ntegrated radal-bass-functon networks (IRBFNs), Cartesan-grds and pont collocaton for numercally solvng 2D flows of vscoelastc fluds. In the proposed method, RBFNs, whch are constructed through ntegraton, are employed to approxmate the feld varables ncludng stresses, and the governng equatons are dscretsed by means of pont collocaton. Advantages ganed from the ntegraton process over dfferentated RBFNs process are () better accuracy () more straghtforward mplementaton of boundary condtons and () more stable solutons. The method s verfed through several test problems, e.g. planar Poseulle flow and fully-developed flow n a square duct, wth dfferent types of vscoelastc fluds, namely Oldroyd-B and CEF. The obtaned results show that () The proposed method s easy to mplement; () Its pre-processng s smple and () Accurate results are obtaned usng relatvely-coarse grds. 1. Introducton Radal-bass-functon networks (RBFNs) have been shown to be a powerful numercal tool for the soluton of partal-dfferental equatons (PDEs). The frst report on ths subject was made by Kansa [1]. For Kansa s method, a functon s frst represented by an RBFN whch s then dfferentated to obtan approxmate expressons for ts dervatve functons. On the other hand, to avod the reducton n convergence rate caused by dfferentaton, Ma-Duy and Tran-Cong [2] proposed an ndrect/ntegrated RBFN (IRBFN) approach n whch the hghest-order dervatves n the PDE are frst decomposed nto RBFs, and ther lower-order dervatves and the functon tself are then obtaned through ntegraton. Prevous studes (e.g. [2]) showed that IRBFN collocaton methods yeld better accuracy than dfferentated RBFN (DRBFN) ones for both the representaton of functons and the soluton of PDEs. Snce the global RBF nterpolaton matrx s fully populated and ts condton number grows rapdly wth respect to the ncrease of RBF centres and/or wdths [3], several RBF technques based on local approxmatons have been proposed. In the context of IRBFNs, collocaton schemes, based on one-dmensonal (1D) IRBFNs and Cartesan grds, for the soluton of 2D ellptc PDEs were reported n, e.g. [4]. The RBF approxmatons at a grd node nvolve only ponts that le on the grd lnes ntersected at that pont rather than the whole set of nodes. As a result, the constructon process s conducted for a seres of small matrces rather than for a large sngle matrx ( local approxmaton). c 2010 Publshed under lcence by Ltd 1
1D-IRBFNs were successfully ntroduced nto the pont-collocaton (e.g. [4]) and Galerkn formulatons (e.g. [5, 6]). Numercal results showed that they yeld accurate results and hgh rates of convergence. In addton, the latter approach can obtan smlar levels of accuracy for both types of boundary condtons (.e. Drchlet only and Drchlet-Neumann). Snce ntegraton s requred for dscretsaton, hgher computng resources may be requred when solvng largescale problems. In ths paper, the IRBFN collocaton technque, whch s based on 1D-IRBFNs and Cartesan grds, s appled to smulate fully developed flows between two parallel planes and n rectangular ducts. Results obtaned show that good agreement wth data avalable n the lterature s acheved usng relatvely coarse grds. The remander of ths paper s organsed as follows. Governng equatons are summarsed n Secton 2. Secton 3 presents the proposed ntegrated-rbf method. Two test problems are smulated n Secton 4. Secton 5 concludes the paper. 2. Governng equatons In the case of an ncompressble flud, the equatons for the conservaton of momentum and mass take the forms ( ) v ρ t + v v = σ + f x Ω, (1) v = 0 x Ω, (2) where v s the velocty vector, f the body force per unt volume of flud, ρ densty, σ the stress tensor (Cauchy stress tensor), t the tme, x the poston vector and Ω the doman of nterest. The stress tensor can be decomposed nto σ = pi + τ, (3) where p s the pressure, I the unt tensor and τ the extra stress tensor. In ths paper, the workng fluds are the CEF and the Oldroyd-B wth the extra stress tensors defned as wth τ = 2η (d)d ψ 1 d +4ψ2 d.d (CEF model), (4) τ = 2η s ( λ 2 λ 1 )d + τ v (Oldroyd B model), (5) τ v + λ 1 τ v = 2η v (1 λ 2 λ 1 )d, (6) where d s the rate of deformaton tensor d s the scalar magntude of d d = 1 2 ( v + ( v)t ); (7) d = 2tr (d.d); (8) n whch tr denotes the trace operaton; Ψ 1 and Ψ 2 the frst and the second normal stress coeffcents, respectvely; η s the solvent vscosty; τ v s the extra stress due to vscoelastcty; λ 1 and λ 2 are characterstc relaxaton and retardaton tmes of the flud, respectvely; η v s the zero shear polymer vscosty and [ ] the upper convected dervatve defned as [ ] = [ ] t + v [ ] ( v)t.[ ] [ ] v. (9) 2
3. Proposed 1D-RBFN technque In the present technque, pont collocaton s appled to reduce the PDEs to sets of algebrac equatons. The problem doman s represented by a Cartesan grd. Snce the governng equatons are of second order, the ntegral RBF scheme of second order s ustlsed here. IRBFN expressons on the x and y grd lnes have smlar forms. In the followng, only a x grd lne s consdered. On the grd lne, a functon f and ts dervatves wth respect to x can be represented as follows d 2 f (x) dx 2 = df (x) dx = f(x) = =1 =1 =1 w g (x) = =1 w I (2) (x), (10) w I (1) (x) + c 1, (11) w I (0) (x) + c 1 x + c 2, (12) where s the number of nodes on the grd lne, {w } Nx =1 the set of network weghts, { } {g (x)} Nx =1 I (2) Nx (x) the set of RBFs, I(1) (x) = I (2) (x)dx, I (0) (x) = I (1) (x) dx, =1 and c 1 and c 2 are the constants of ntegraton. Evaluaton of (10) - (12) at the grd nodes leads to Î (2) = d 2 f dx 2 = Î(2) α, (13) df dx = Î(1) α, (14) f = Î(0) α, (15) where the superscrpt (.) s used to denote the order of the correspondng dervatve functon; I (2) 1 (x 1 ), I (2) 2 (x 1 ),, I (2) (x 1 ), 0, 0 I (2) 1 (x 2 ), I (2) 2 (x 2 ),, I (2) (x 2 ), 0, 0........ ; (16) I (2) 1 (x Nx ), I (2) 2 (x Nx ),, I (2) (x Nx ), 0, 0 Î (1) = I (1) 1 (x 1 ), I (1) 2 (x 1 ),, I (1) (x 1 ), 1, 0 I (1) 1 (x 2 ), I (1) 2 (x 2 ),, I (1) (x 2 ), 1, 0........ ; (17) I (1) 1 (x Nx ), I (1) 2 (x Nx ),, I (1) (x Nx ), 1, 0 Î (0) = I (0) 1 (x 1 ), I (0) 2 (x 1 ),, I (0) (x 1 ), x 1, 1 I (0) 1 (x 2 ), I (0) 2 (x 2 ),, I (0) (x 2 ), x 2, 1........ ; (18) I (0) 1 (x Nx ), I (0) 2 (x Nx ),, I (0) (x Nx ), x Nx, 1 α = (w 1,w 2,,w Nx,c 1,c 2 ) T ; (19) 3
and d k ( f d k ) T f 1 dx k = dx k, dk f 2 dx k,, dk f Nx dx k, k = {1,2}, (20) f = (f 1,f 2,,f Nx ) T, (21) n whch d k f j / dx k = d k f (x j ) / dx k and f j = f (x j ) wth j = {1,2,, }. It s noted that f s used to represent the feld varables, e.g. the component of v, n (1) (6). 4. Numercal results Ths secton presents the verfcaton of the proposed method wth the smulaton of the planar Poseulle flow and fully-developed vscoelastc flow n rectangular ducts. Fluds models consdered are Oldroyd-B and CEF. Unform grds are used to represent the problem doman, and 1D-IRBFNs are mplemented wth the multquadrc (MQ) functon g (x) = (x c ) 2 + a 2, (22) where c and a are the centre and the wdth/shape-parameter of the th MQ-RBF. The MQ wdth s smply chosen to be the grd sze. 1.4 1.2 analytc x=0.5 1 0.8 v 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 y Fgure 1. Steady-state planar Poseulle flow usng Oldroyd-B model: the velocty on the mddle plane x = 0.5 wth 31 31 collocaton ponts. 4.1. Planar Poseulle flow Frst, planar Poseulle flow that has an analytcal soluton s consdered. equatons wth Oldroyd-B model n dmensonless form are gven by The governng v = 0, (23) v t + v v = 1 Re p + 1 Re τ v + β v, Re (24) τ v + We τ v = 2(1 β)d, (25) where We s the Wessenberg number, Re the Reynolds number and β the rato of retardaton and relaxaton tmes of the flud (ts value s taken to be 1/9 n ths work). 4
A unt doman, [0,1] [0,1], s taken as a computatonal doman. On the sold boundares (y = 0 and y = 1), non-slp condtons are mposed. At the steady-state condton, the velocty and stress felds are gven n the followng analytc forms v x = 4(1 y)y, (26) v y = 0, (27) ( ) 2 vx τ xx = 2We(1 β), (28) y τ yy = 0, (29) ( ) vx τ xy = (1 β). (30) y In solvng (23) (25), the materal dervatves are approxmated usng an Euler scheme G t + v G Gn+1 G n + v n G n (31) t n whch G represents v and τ, and the deformaton term s treated explctly v τ + τ ( v) T v n τ n + τ n ( v n ) T. (32) The soluton procedure can be outlned as follows Calculaton of ntermedate velocty v from the momentum equaton (24), where the pressure term s removed. At the begnnng, the nlet velocty s chosen to be zero or takes a parabolc profle as n (26), Calculaton of pressure from the contnuty equaton (23) p = Re t v, (33) Correcton of ntermedate velocty usng the computed pressure just obtaned v n+1 = v n t Re pn+1, (34) Calculaton of vscoelastc stress components from the consttutve equaton (25), Re-mposton of nlet velocty profle usng the obtaned outlet velocty profle, Repeatng the above steps untl there s no further change (above a gven small tolerance) n the outlet velocty profle. The results of smulaton are reported n Fgures 1 and 2. Fgure 1 shows the computed velocty profle on the mddle plane (x = 0.5) whch s n excellent agreement wth the analytc one. The shear stress and the frst normal stress dfference on the mddle plane at dfferent values of We number,.e 0.1, 0.5, 1, 2 and 4, are gven n Fgure 2. In comparson wth the analytc results, t can be seen that the error of the frst normal stress dfference (Fgure 2a) and the error of the shear stress (Fgure 2b) are very small. 5
a) b) 140 120 x=0.5 analytcal 4 3 x=0.5 analytcal 100 2 τ xx 80 We=4 τ xy 1 0 60 We=2 1 40 We=1 2 20 We=0.5 We=0.1 0 0 0.2 0.4 0.6 0.8 1 y 3 4 0 0.2 0.4 0.6 0.8 1 y Fgure 2. Steady-state planar Poseulle flow usng Oldroyd-B model: a) the frst normal stress dfference (N 1 = τ xx τ yy = τ xx ) and b) the shear stress on the mddle plane x = 0.5 wth 31 31 collocaton ponts at dfferent values of the We number. a) b) c) d) Fgure 3. Flow n rectangular ducts: Streamlnes of secondary flows n one quarter of the cross-secton wth the aspect ratos of a) 1, b) 1.56, c) 4 and d) 6.25 usng a grd of 41 41. 6
a) b) c) d) Fgure 4. Flow n rectangular ducts: Contour plots for the vortcty ω n one quarter of the cross-secton at four dfferent aspect ratos: a) 1, b) 1.56, c) 4 and d) 6.25 usng a grd of 41 41. a) b) c) d) Fgure 5. Flow n rectangular ducts: Contour plots for the prmary velocty v z n one quarter of the cross-secton at four dfferent aspect ratos: a) 1, b) 1.56, c) 4 and d) 6.25 usng a grd of 41 41. 7
a) b) c) d) Fgure 6. Flow n rectangular ducts: Contour plots for the second normal stress dfference (N 2 = τ xx τ yy ) n one quarter of the cross-secton at four dfferent aspect ratos: a) 1, b) 1.56, c) 4 and d) 6.25 usng a grd of 41 41. 4.2. Fully-developed vscoelastc flow n rectangular ducts The flow of a vscoelastc flud n a rectangular duct has receved a great deal of attenton because of ts fundamental and practcal mportance. Many studes were carred out on ths geometry wth dfferent consttutve models, (e.g. Rener-Rvln [7], CEF [8, 9] and modfed PTT (MPTT) [10]). Results by Gervang and Larsen [8], where both numercal and expermental nvestgatons were conducted, are often cted n the lterature. The governng equatons are taken n terms of stream-functon, vortcty and prmary velocty 2 ψ x 2 + 2 ψ + ω = 0, (35) y2 ( ψ ω ρ y x ψ ) ( ω 2 ) ω = η 0 x y x 2 + 2 ω y 2 + 2 S xy x 2 2 (S xx S yy ) 2 S xy x y y 2, (36) ( ψ v z ρ y x ψ ) v z = p ( 2 ) x y z + η v z 0 x 2 + 2 v z y 2 + S zx x + S zy y, (37) wth the homogeneous boundary condtons for ψ, ψ/ n and v z. A CEF flud s consdered. The computaton boundary condtons for the vortcty n (36) can be derved from the streamfuncton equaton. The process s smlar to that n [6]. Equatons (35) (37) are thus subject to Drchlet boundary condtons. The flows are generated by pressure drop p/ z. The stress components are drectly computed from the velocty feld usng (4). Unlke the planar Poseulle flow problem, a Pcard teratve scheme s appled here to solve the resultant system of nonlnear algebrac equatons. Fgures (3)- (6) show the patterns of the prmary and secondary flows together wth the second normal stress dfference for dfferent aspect ratos 1, 1.56, 4 and 6.25. 8
Snce the flow s symmetrc about the vertcal and horzontal centrelne, only one quarter of the cross-secton s plotted. Each quadrant has two vortces, whose pattern and strength strongly depend on the aspect rato for a gven mean prmary velocty. The two vortces are symmetrc about the dagonal plane n the case of rato 1. For aspect rato greater than 1, the vortex near the long wall moves towards the short wall whle the vortex near the short wall s reduced wth ncreasng aspect ratos. The results are smlar to those reported n the lterature (e.g. [8, 10]). 5. Concludng remarks Fully-developed flows of vscoelastc fluds between two parallel walls and n rectangular ducts were studed wth the IRBFN collocaton technque. The calculatons are carred out n both the prmtve varable and the stream-functon vortcty formulatons. The technque s easy to mplement. Results obtaned are n good agreement wth the exact soluton and those by other technques. Extenson of the proposed method to vscoelastc flows s underway and new results wll be reported n future. Acknowledgments Ths research s supported by Australa Research Councl and FoES-USQ. D. Ho-Mnh would lke to thank the CESRC, FoES and USQ for a postgraduate scholarshp. References [1] Kansa E J 1990 Computers and Mathematcs wth Applcatons 19 127 145 [2] Ma-Duy N and Tran-Cong T 2001 Neural Networks 14 185 199 [3] Schaback R 1995 Advances n Computatonal Mathematcs 3 251 264 [4] Ma-Duy N and Tran-Cong T 2007 Numercal Methods for Partal Dfferental Equatons 23 1192 1210 [5] Ma-Duy N and Tran-Cong T 2009 Engneerng Analyss wth Boundary Elements 33 191 199 [6] Ho-Mnh D, Ma-Duy N and Tran-Cong T 2009 CMES 44 219 248 [7] Green G and Rvln R 1956 Quaterfly of Appled Mathematcs 14 299 308 [8] Gervang B and Larsen P 1991 Journal of Non-Newtonan Flud Mechancs 39 217 237 [9] Ma-Duy N and Tanner R I 2007 Internatonal Journal of Numercal Methods for heat & flud flows 17 165 186 [10] Xue S C, Phan-Then N and Tanner R I 1995 Journal of Non-Newtonan Flud Mechanc 59 191 213 9