Comparison of different approaches in the Trefftz method for analysis of fluid flow between regular bundles of cylindrical fibres

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1 Joural of Physics: Coferece Series PAPER OPEN ACCESS Compariso of differet approaches i the Trefftz method for aalysis of fluid flow etwee regular udles of cylidrical fires To cite this article: ierzwiczak et al 06 J. Phys.: Cof. Ser Related cotet - A ew RBF-Trefftz meshless method for partial differetial equatios Leilei Cao, Qig-Hua Qi ad Nig Zhao - A simple ad efficiet method of aalyzig mechaical ehaviors of multi-layered orthotropic plates i rectagular shape Byug Chai Lee ad Eu Sok Kim - EDE for solvig elliptic oudary value prolems with aular domais J Aarão, B H Bradshaw-Hajek, S J iklavcic et al. View the article olie for updates ad ehacemets. This cotet was dowloaded from IP address o 30//07 at 0:3

2 XXII Fluid echaics Coferece (KKP06) Joural of Physics: Coferece Series 760 (06) 009 doi:0.088/ /760//009 Compariso of differet approaches i the Trefftz method for aalysis of fluid flow etwee regular udles of cylidrical fires ierzwiczak, J K Graski ad J A Kołodziej Poza Uiversity of Techology, Istitute of Applied echaics, Jaa Pawla II 4, Poza, Polad Astract. I the paper three differet approaches for the Trefftz method are compared i aalysis of the fluid flow etwee regular udles of cylidrical fires. The approximate solutio is a liear comiatio of such trial fuctios which fulfil exactly the goverig equatios. The trial fuctios ca e defied i the Cartesia coordiate system (the first approach), i the cylidrical coordiate system (ad ca fulfil also some oudary coditios the secod approach) or e defied as a fudametal solutios (the third approach the method of fudametal solutios).the average velocity ad the product of the frictio factor ad the Reyolds umer f Re are compared for selected parameters of a cosidered regio.. Itroductio The Trefftz method (T) was proposed i 96 y Erich Trefftz []. I this method a approximate solutio is a liear comiatio of trial fuctios which fulfil exactly the goverig equatio of the cosidered prolem. The trial fuctios i the method are called the Trefftz fuctios or the T- fuctios. However real developmet of this method came oly with developmet of computer techiques. Oe of the versio of the Trefftz method is the oudary collocatio techique. I the method the goverig equatio is fulfilled exactly y the trial fuctios while the oudary coditios are satisfied i a approximate way. Differet applicatios of the method ca e foud i []. Sometimes the T-fuctios fulfil also some of the oudary coditios. I such case the method is ofte called the Trefftz method with the special purpose Trefftz fuctios (SPTF) [3]. The the oudary collocatio techique is applied oly for ufulfilled oudary coditios. This approach was successfully applied for differet prolems i applied mechaics, e.g. coductive heat flow [4], fluid flow i coduits with polygoal cross-sectio [5] or elastic torsio of ars [6]. Aother versio of the Trefftz method is the method of fudametal solutios (FS). The FS was proposed i 964 y Kupradze ad Aleksidze [7]. The umerical implemetatio of the FS was give y atho ad Johsto [8]. I the FS the approximate solutio is a liear comiatio of fudametal solutios which are fuctios of distace etwee the poits ad the source poits. The fudametal solutios fulfil exactly the goverig equatio ad the ukow coefficiets of the approximate solutio are calculated usig the oudary collocatio techique []. The source poits are located outside the cosidered regio which has similar shape to the oudary of the domai. Some iterestig review of applicatios of the FS ca e foud i [9-]. The purpose of the preset paper is to compare these three approaches i aalysis of the logitudial fluid flow etwee udles of cylidrical fires. The filtratio velocity ad the product of the frictio factor, f ad the Reyolds umer, Re are compared for a differet arragemet of parallel cylidrical fiers. Cotet from this work may e used uder the terms of the Creative Commos Attriutio 3.0 licece. Ay further distriutio of this work must maitai attriutio to the author(s) ad the title of the work, joural citatio ad DOI. Pulished uder licece y Ltd

3 XXII Fluid echaics Coferece (KKP06) Joural of Physics: Coferece Series 760 (06) 009 doi:0.088/ /760//009. Formulatio of the prolem Cosider the steady, fully developed, lamiar, isothermal flow of a icompressile viscous fluid drive y a costat pressure i a system of regular parallel fiers. The flow is logitudial with respect to fiers which are arraged i a regular hexagoal (Fig. a), square (Fig. ) ad triagular (Fig. c) array. The radius of the fiers is equal to a, ad the distace etwee the fiers is equal to. a) ) c) Figure. Three differet kids of a regular array of cylidrical rods (a) L = 6 hexagoal array, () L = 4 square array, ad (c) L = 3 triagular array. The equatio of motio is reduced to a sigle partial differetial equatio i the form w w dp + = i x y µ dz Ω f () where w is the velocity compoet i the z directio, dp/dz is the costat pressure gradiet, µ is the dyamic viscosity, ad Ω f is the repeated elemet of a fluid array (Fig a). Equatio (3) is solved with the followig oudary coditios i a repeated elemet array (Fig. a) w for x + y = a, () y for y, a x, (3) for x π x =, 0 y cot, (4) L π π for y = x cot, a si x. (5) L L It is coveiet to itroduce the followig dimesioless variales

4 XXII Fluid echaics Coferece (KKP06) Joural of Physics: Coferece Series 760 (06) 009 doi:0.088/ /760//009 x X =, y Y =, a E =, Now, the goverig equatio (3) has the dimesioless form µ w = dp dz, (6) X + Y = i Eq. (9) is solved with oudary coditios i a dimesioless form X Ω f, (7) for X + Y = E, (8) Y for Y, E X, (9) π for X =, 0 Y cot, (0) L π π for Y = X cot, E si X. () L L The o-liear goverig Eq. (9) with oudary coditios (0-3) yields a micro structural oudary value prolem (BVP) i a repeated elemet of a array of fiers. y ( a ) ( ) Γ 3 collocatio poits Figure. The repeated elemet a) the formulatio of the oudary value prolem, ) the distriutio of source, collocatio ad iterpolatio poits 3. ethod of solutio of micro structural oudary value prolem The BVP prolem ca e solved usig the meshless method. The approximate solutios ca e expressed as a sum of geeral ad particular solutio The particular solutio for Eq. (9) has a form a Γ 4 w Ω y f π L Γ L = { 3,4,6} x Γ x source poits ( X, Y ) ( X, Y ) + ( X Y ) = () g p, s 3

5 XXII Fluid echaics Coferece (KKP06) Joural of Physics: Coferece Series 760 (06) 009 doi:0.088/ /760//009 ( X, Y ) ( X Y ) p = + (3) 4 The geeral solutio g (X,Y) fulfills the Laplace equatio ad ca e solved usig oe of the three Trefftz method. The detailed differeces are preseted i this sectio. 3.. The Trefftz method - T The approximate geeral solutio usig the Trefftz method ca e writte i the followig form g ( X Y ) = c F ( X, Y ) + = 0, d G ( X, Y ) (3) where F (X,Y) ad G (X,Y) are the trial fuctios defied i the Cartesia coordiate system = k F ( X, Y ) = ( ) ; ( X, Y ) = ( ) ( k )! ( k )! k= 0 X k Y k k= 0 k k+ k X Y G. (4)! ( k )! ( k + ) The aove trial fuctios fulfill the Laplace equatio i D. The ukow coefficiets { { } d = c } 0 = ad are calculated y fulfillig the oudary coditios (8-) i the collocatio sese []. I this way we otai a system of liear equatios. The umer of collocatio poits must e greater or equal to the umer of ukows coefficiets ( +). 3.. The Trefftz method with special purpose Trefftz fuctios - SPTF As the geeral solutio of the Laplace equatio i D the followig expressio i the polar coordiate system ca e used g 0 3 si = = λ λ λ λ ( R, θ ) = A + A θ + A θ l R + A l R + ( B R + C R ) cos( λ θ ) + ( D R + E R ) ( λ θ ) Applyig it for the specific regio, some of the oudary coditios ca e fulfilled exactly y calculatig some costats i the aove solutio. After that the approximate solutio fulfils the goverig equatio ad some of the oudary coditios (8, 9, ). The costats { g A } = L( ) ( R ) = E + A + A l R (5) L( ) R E, θ ( ) ( ( ) ) L cos L θ. (6) 4 E = R are calculated from Eq. (0) usig the oudary collocatio techique []. The umer of collocatio poits located o Γ must e greater or equal The method of fudametal solutios - FS I the FS the approximate solutio is assumed as a liear comiatio of fudametal solutios. The fudametal solutio fulfils exactly the goverig equatio ad it is a fuctio of distace etwee the poit iside the cosidered regio ad the source poit. Usig the FS the approximate solutio for the Laplace equatio takes the form where { c } j j= g ( X Y ) = c j l( rj ),, (7) j= are ukow coefficiets, r j is the distace etwee the poit (X, Y) ad the j-th source poit (Xs j, Ys j ). To avoid the sigularities sources poits are located outside the cosidered domai o a fictitious oudary (see Fig. ). The distace etwee the source oudary ad the real oudary is 4

6 XXII Fluid echaics Coferece (KKP06) Joural of Physics: Coferece Series 760 (06) 009 doi:0.088/ /760//009 equal to s. Fulfillig the oudary coditios (8-) i collocatio poits (Xc i, Yc i ) we otai a system of liear equatios j= j= c c j j l ( r ) = + ( Xc + Yc ), ij l X ( r ) l( r ) ij x + 4 Y where r ( Xc Xs ) + ( Yc Ys ) ij i j i j i ij i y = =, = [ x, y] i =, K, N + ( x Xc + y Yc ), i = N +, K, N i i (8) is a uit ormal vector to the oudary. The umer of collocatio poits N must e greater or equal to the umer of ukows coefficiets. After determied the velocity field, the filtratio velocity ca e calculated as, dω f, (9) Ω ( X Y ) = where Ω = ctg π is a total dimesioless area of the elemet, Ω L flow area. Used the Darcy-eisach equatio the Reyolds umer defiitio p = L f Ω f ρ w 4 a f π L = ctg π E L L is a dp, (0) dz aρw Re =, () µ ad the defiitio of the o-dimesioal variales (6), the product of frictio factor, f ad Reyolds umer, Re ca e express y a dimesioless form f 8E Re =. () For the firous porous medium the fier volume fractio is defied as where E is the o-dimesioal radius of the fier ad L = {3, 4, 6}. L π ϕ = πe ta. (3) L L 4. Numerical experimets I umerical experimets, the o-dimesioal field of the velocity for logitudial flow are calculated. The cylidrical fiers are arraged regularly i a triagular L = 3, square L = 4 ad hexagoal L = 6 arrays. At first we test the ifluece of the umer of trial fuctio N for a value of the error of fulfillig the oudary coditios. The calculatio was performed for a square L = 4 arrays of fiers, E = {0., 0.4, 0.6, 0.8} ad results are preseted o Figure 3. The smallest values of the error were otai for the SPTF. Figure 4 shows the value of the coditio umer of the collocatio matrix, CN. The coditio umer was calculated as a relatio of the maximal to the miimal sigular value of the matrix. The lowest value of CN was otaied for the T ad the greatest for the FS. 5

7 XXII Fluid echaics Coferece (KKP06) Joural of Physics: Coferece Series 760 (06) 009 doi:0.088/ /760//009 δ.e-0.e-0.e-03.e-04.e-05 (a) E. E.4 E.6 E.8 L = 4, T N E-0 (c) δ.e-03.e-04 δ.e-0.e-04.e-06.e-08.e-0.e-.e-4 () E. E.4 E.6 E.8 L = 4, SPTF N L = 4, FS.E-05.E-06.E-07 E. E.4 E.6 E N Figure 3. The error of fulfillig the oudary coditios for the square array of firous L = 4 ad for three methods: T, SPTF ad FS as a fuctio of N CN.E+06.E+04.E+0.E+00 (a) L = 4, T E. E.4 E.6 E N E+39.E+3.E+3 (c) CN L = 4, FS.E+7.E+3.E+09.E+05.E+0.E-03 () L = 4, SPTF E. E.4 E.6 E N CN.E+5.E+07.E-0 E. E.4 E.6 E N Figure 4. The coditio umer of collocatio matrix for the square array of firous L = 4 ad for three methods: T, SPTF ad FS as a fuctio of N 6

8 XXII Fluid echaics Coferece (KKP06) Joural of Physics: Coferece Series 760 (06) 009 doi:0.088/ /760// (a) FS, L = 6 SPTF, L = 6 T, L = 6 FS, L = 4 SPTF, L = 4 T, L = 4 FS, L = 3 SPTF, L = 3 T, L = () φ f Re Figure 5. The filtratio velocity ad the product f Re for three differet kid of firous array L = {3, 4, 6} ad three methods: T, SPTF ad FS as a fuctio of firous volume fractio φ. Figure 5 shows the value of the filtratio velocity, ad the product, f Re as a fuctio of the fier volume fractio, φ. The result otaied y the T, the SPTF ad the FS were show for hexagoal (L = 6), square (L = 4), ad triagular array (L = 3) of fiers. For all methods the results are the same. For the Trefftz method we used 00 collocatio poits ad = 6. For the SPTF we used the = 5, ad the umer of collocatio poits was also equal to 5. For the FS the umer of collocatio poit was equal to 86 while the umer sources poits was equal to 58. The distace etwee source cotour ad the oudary was equal s Coclusios Three differet approaches for the Trefftz method were compared i aalysis of the logitudial Newtoia fluid flow through the regular udles of cylidrical fires. The cylidrical fires were arraged regularly i a hexagoal, square ad triagular array. The umerical algorithm was ased o the oudary collocatio method. For a repeated elemet the Trefftz method, the special purpose Trefftz fuctios ad the method of fudametal solutios were used to determied the velocity field. Preseted methods are easy to implemet, accurate ad meshless. The filtratio velocity ad the product of the frictio factor ad the Reyolds umer were compared for those meshless methods. The results of f Re ad the filtratio velocity otaied for all methods were the same. The SPTF is more accurate ad less time cosumig, ecause some of the oudary coditios are fulfilled exactly y the trial fuctios. This method eed less umer of collocatio poits, hece the collocatio matrix has smaller size ad is etter coditioed. But we should otice that ot for all the oudary value prolem this method is appropriated. e ca use this method easy for a domai with symmetry lie ad with simple oudary at which the oudary coditio we ca fulfil exactly. 0.0 Ackowledgmets The paper is fiacially supported y the Grat 0//DSK/ FS, L = 6 SPTF, L = 6 T, L = 6 FS, L = 4 SPTF, L = 4 T, L = 4 FS, L = 3 SPTF, L = 3 T, L = φ 7

9 XXII Fluid echaics Coferece (KKP06) Joural of Physics: Coferece Series 760 (06) 009 doi:0.088/ /760//009 Refereces [] Trefftz E 96 Ei Gegestück zum Ritzsche Verfahre, Proceedigs of the d Iteratioal Cogress of Applied echaics (Zurich) Orell Fussli Verlag pp 3-37 [] Kołodziej J A ad Zieliński A P 009 Boudary collocatio techiques ad their applicatio i egieerig IT Press [3] Kołodziej J A ad Uściłowska A 997 Trefftz-type procedure for Laplace equatio o domai with circular holes, circular iclusios, corers, slits ad symmetry CAES: Computer Assisted ethods i Egieerig ad Sciece vol 4 pp [4] Kołodziej J A ad Stręk T 00 Aalytical approximatios of the shape factors for coductive heat flow i circular ad regular polygoal cross-sectios Iteratioal Joural of Heat ad ass Trasfer vol 44 pp [5] Kołodziej J A, Uściłowska A ad Ciałkowski 00 Semi-aalytical approximatios of the lamiar frictio coefficiets for flow i coduits with polygoal cross-sectio Acta echaica vol 58 pp 7-44 [6] Kołodziej J A ad Fraska A 005 Elastic torsio of ars possessig regular polygo i crosssectio usig BC, Computers ad Structures vol 84 pp 78-9 [7] Kupradze V D ad Aleksidze Q 964 The method of fuctioal equatios for the approximate solutio for certai oudary value prolems USRR Computatioal athematics ad athematical Physics vol 4 pp 8-6 [8] atho R ad Johsto R L 977 The approximate solutio of elliptic oudary value prolems y fudametal solutios SIA Joural of Numerical Aalysis vol 4 pp [9] Fairweather G ad Karageorghis A 998 The method of fudametal solutios for elliptic oudary value prolems Advaces i Computatioal athematics vol 9 pp [0] Fairweather G, Karageorghis A ad arti P A 003 The method of fudametal solutios for scatterig ad radiatio prolems Egierig Aalysis with Boudary Elemets vol 7 pp [] Karageorghis A, Lesic D ad ari L 0 A survey of applicatios of the FS to iverse prolems Iverse Prolems i Sciece ad Egieerig vol 9 pp