(IREMOS) On New Solutions for Non-Newtonian Visco-Elastic Fluid in Pipe by N. Moallemi, I. Shafieenejad, H. Davari, A. Fata

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1 ISSN Vol. 4 N. 4 August Internatonal Revew on Modellng and Smulatons Contents: (contnued from Part B) (IREMOS) PART C Emprcal Models for the Correlaton of Global Solar Radaton Under Malaysa Envronment by H. A. Rahman, K. M. Nor, M. Y. Hassan, M. S. Majd Mult-Objectve Sngle Faclty Locaton Problem: a Revew by Vashal Wadhwa, Deepak Garg Free Vbraton Functonally Graded Materal Crcular Cylndrcal Shell wth Varous Volume Fracton Laws Under Symmetrcal Boundary Condtons by M. Setareh, M. R. Isvandzbae On New Solutons for Non-Newtonan Vsco-Elastc Flud n Ppe by N. Moallem, I. Shafeenejad, H. Davar, A. Fata Integraton of a Recent Profle Reducton Method n Fnte Element Program for Movement Smulaton wth Movng Band Method by R. Saraou, Y. Boutora, N. Benamrouche, M. Ounnad Thermal-Hydraulc Modelng of a Radant Steam Generator Usng Relap5/Mod3. Code by A. L. Deghal Cherd, A. Chaker Response to Customer Relablty Rurements wth Reserve Market Management by M. Najaf, M. Smab, M. Hosenpour, R. Ebrahm Lattce Boltzmann Smulaton of Cavty Flows at Varous Reynolds Numbers by M. A. Mussa, S. Abdullah, C. S. Nor Azwad, R. Zulkfl Transent Analyss Vbraton of Two Type FGM Crcular Cylndrcal Shell Based on Thrd Order Theory Usng Hamlton's Prncple wth Smply Support-Smply Support Boundary Condtons by M. R. Isvandzbae Effect of Shear Theory on Analyss Free Vbraton of Two Knds Functonally Graded Materal Hollow Crcular Cylndrcal Shell Accordng to a 3D Hgher-Order Deformaton Theory wth Free-Smply Support Boundary Condtons by M. R. Isvandzbae (contnued on nsde back cover)

2 Internatonal Revew on Modellng and Smulatons (IREMOS) Edtor-n-Chef: Santolo Meo Department of Electrcal Engneerng FEDERICO II Unversty Claudo - I85 Naples, Italy santolo@unna.t Edtoral Board: Maros Angeldes (U.K.) Brunel Unversty M. El Hachem Benbouzd (France) Unv. of Western Brttany- Electrcal Engneerng Department Debes Bhattacharyya (New Zealand) Unv. of Auckland Department of Mechancal Engneerng Stjepan Bogdan (Croata) Unv. of Zagreb - Faculty of Electrcal Engneerng and Computng Cecat Carlo (Italy) Unv. of L'Aqula - Department of Electrcal and Informaton Eng. Ibrahm Dncer (Canada) Unv. of Ontaro Insttute of Technology Guseppe Gentle (Italy) FEDERICO II Unv., Naples - Dept. of Electrcal Engneerng Wlhelm Hasselbrng (Germany) Unv. of Kel Ivan Ivanov (Bulgara) Techncal Unv. of Sofa - Electrcal Power Department Jn-Yuh Jang (Tawan) Natonal Cheng-Kung Unv. - Department of Mechancal Engneerng Heuy-Dong Km (Korea) Andong Natonal Unv. - School of Mechancal Engneerng Marta Kurutz (Hungary) Techncal Unv. of Budapest Baodng Lu (Chna) Tsnghua Unv. - Department of Mathematcal Scences Pascal Lorenz (France) Unv. de Haute Alsace IUT de Colmar Santolo Meo (Italy) FEDERICO II Unv., Naples - Dept. of Electrcal Engneerng Josua P. Meyer (South Afrca) Unv. of Pretora - Dept.of Mechancal & Aeronautcal Engneerng Bjan Mohammad (France) Insttut de Mathématques et de Modélsaton de Montpeller Pradpta Kumar Pangrah (Inda) Indan Insttute of Technology, Kanpur - Mechancal Engneerng Adran Traan Pleşca (Romana) "Gh. Asach" Techncal Unversty of Ias Ľubomír Šooš (Slovak Republc) Slovak Unv. of Technology - Faculty of Mechancal Engneerng Lazarus Tenek (Greece) Arstotle Unv. of Thessalonk Lxn Tan (Chna) Jangsu Unv. - Department of Mathematcs Yoshhro Tomta (Japan) Kobe Unv. - Dvson of Mechancal Engneerng George Tsatsarons (Germany) Technsche Unv. Berln - Insttute for Energy Engneerng Ahmed F. Zobaa (U.K.) Unv. of Exeter - Camborne School of Mnes The Internatonal Revew on Modellng and Smulatons (IREMOS) s a publcaton of the Prase Worthy Prze S.r.l.. The Revew s publshed bmonthly, appearng on the last day of February, Aprl, June, August, October, December. Publshed and Prnted n Italy by Prase Worthy Prze S.r.l., Naples, August 3,. Copyrght Prase Worthy Prze S.r.l. - All rghts reserved. Ths journal and the ndvdual contrbutons contaned n t are protected under copyrght by Prase Worthy Prze S.r.l. and the followng terms and condtons apply to ther use: Sngle photocopes of sngle artcles may be made for personal use as allowed by natonal copyrght laws. Permsson of the Publsher and payment of a fee s rured for all other photocopyng, ncludng multple or systematc copyng, copyng for advertsng or promotonal purposes, resale and all forms of document delvery. Permsson may be sought drectly from Prase Worthy Prze S.r.l. at the e-mal address: admnstraton@praseworthyprze.com Permsson of the Publsher s rured to store or use electroncally any materal contaned n ths journal, ncludng any artcle or part of an artcle. Except as outlned above, no part of ths publcaton may be reproduced, stored n a retreval system or transmtted n any form or by any means, electronc, mechancal, photocopyng, recordng or otherwse, wthout pror wrtten permsson of the Publsher. E-mal address permsson ruest: admnstraton@praseworthyprze.com Responsblty for the contents rests upon the authors and not upon the Prase Worthy Prze S.r.l.. Statement and opnons expressed n the artcles and communcatons are those of the ndvdual contrbutors and not the statements and opnons of Prase Worthy Prze S.r.l.. Prase Worthy Prze S.r.l. assumes no responsblty or lablty for any damage or njury to persons or property arsng out of the use of any materals, nstructons, methods or deas contaned heren. Prase Worthy Prze S.r.l. expressly dsclams any mpled warrantes of merchantablty or ftness for a partcular purpose. If expert assstance s rured, the servce of a competent professonal person should be sought.

3 Internatonal Revew on Modellng and Smulatons (I.RE.MO.S.), Vol. 4, N. 4 August Lattce Boltzmann Smulaton of Cavty Flows at Varous Reynolds Numbers M. A. Mussa, S. Abdullah, C. S. Nor Azwad, R. Zulkfl Abstract The lattce Boltzmann method () s a numercal method evolved from the statstcal approach that has been well-accepted as an alternatve numercal scheme for computatonal flud dynamcs (CFD). In comparson to other numercal schemes, the lattce Boltzmann method () can be regarded as a bottom-up approach that derves the Naver- Stokes uaton through statstcal behavor of partcle dynamcs. Hence, ths paper presents the smulaton of ld-drven cavty for deep and shallow flow usng the lattce Boltzmann method where the effect of the Reynolds number on the flow pattern at aspect ratos of.5,.5,.5 and 4. was studed. These types of flow exhbt a number of nterestng physcal features but are scarcely smulated usng the scheme. The source code was establshed based on the BGK model on rectangular lattce geometry. The comparson of the results was n excellent agreement wth those gathered from the lterature even wth relatvely coarse grds appled to the numercal calculaton. Copyrght Prase Worthy Prze S.r.l. - All rghts reserved. Keywords: Lattce Boltzmann Method, Dstrbuton Functon, Mcroscopc Velocty, Ld-Drven Cavty Flow, BGK Model Nomenclature c Mcroscopc velocty f Dstrbuton functon f Equlbrum dstrbuton functon H Characterstc heght L Mean free path of partcle P Pressure dstrbuton t Tme u Velocty U Centerlne velocty at ext W Heght of mcrochannel x, y Co-ordnates Greek symbols ν Vscosty of flud ρ Densty τ Tme relaxaton Ω Collson operator I. Introducton The lattce Boltzmann method can be consdered as a numercal method to solve the Boltzmann uaton n dscrete phase space and dscrete tme. In the Boltzmann uaton, the velocty space of the partcle s contnuous, whle n the lattce Boltzmann method the velocty space of the partcle s dscrete. In the lattce Boltzmann method, as n the Boltzmann uaton, the partcle s n one of the two processes: the frst, movng wth dscrete velocty to the neghborng node and the second s the collson process. The magnary propagaton and collson actons of flud partcles are reformulated durng the development of the code. These processes are represented by the evoluton of partcle dstrbuton functon, f(x,t) whch descrbes the statstcal populaton of partcles at locaton x and tme t. The most mportant features of are the smplcty of formulaton, sutablty to work on parallel computng and ease n dealng wth complex boundary condtons, and n addton, and because t s bult on the bass of knetc theory, s more effectve n the handlng and analyss of complex systems such as flud flow mult-component, multphase flow []-[3], flow n porous meda [4], the flow of suspensons [5], and turbulent flow [6]. The advantages of nclude smple calculaton procedures, sutablty for parallel computatons, ease and robust handlng of multphase flow, complex geometres, nterfacal dynamcs and others [7]. A few standard benchmark problems have been smulated usng and the results are found to agree well wth the correspondng Naver-Stokes solutons [8]. The ld-drven cavty flow s one of the most mportant benchmarks for new numercal method to be developed. It represents the flow of a rectangular or square geometry where the flow s drven by a tangental moton wth constant velocty of a sngle ld, representng the Drchlet boundary condtons. Moreover, the drven-cavty flow exhbts a number of nterestng physcal features [9], []. However, on the Manuscrpt receved and revsed July, accepted August Copyrght Prase Worthy Prze S.r.l. - All rghts reserved 99

4 other hand, the smulaton of ld-drven flow nsde cavty s scarcely performed usng. Cavty flow s so mportant to researchers due to ts relaton to a lot of ndustral applcatons and ts smlarty to many of common flow phenomena such as corner vortces, longtudnal vortces and turbulence [9], []. The numercal study whch was done by Burggraf [] was among the earler research of steady state flow n a cavty, where he numercally solved the full Naver- Stokes uatons for the Reynolds number up to 4. Furthermore, he showed that at ths Reynolds number there wll be a two secondary vortces present n the bottom corners of the cavty n addton to the prmary large vortex n the flow core. Schrebera and Keller [3] supported ths fndng at hgh Reynolds number where there were vortces formed near the bottom corners. Mehta and Lavan [4] numercally studed the ld drven cavty flow for the aspect ratos of.5, and wth the Reynolds numbers of,,, and they concluded that the strength of the large vortex ncreases wth the ncrease n the Reynolds number wth no vortces at the bottom corners due to low number of nodes and low Reynolds number. Cheng and Hung [5] also confrmed Burggraf results for the vortex structure of a two-dmensonal vscous flow n a lddrven cavty of rectangular secton, where they found that the major feature of flow n a deep rectangular cavty was that the doman was flled wth counterrotatng large vortces and ther sze, the central poston and the number of the vortces depend on both the Reynolds number and the aspect rato. Up to date, only a few studes have dealt wth the nfluence of the heght of cavty on the flow nsde. In comparson wth the flow nsde a square cavty, one new parameter has to be taken nto account, the aspect rato of the cavty, K = W/H, W beng the cavty wdth and H the depth. McNamara and Zanett [6] were the frst group of researchers whch used the lattce Boltzmann uaton as a numercal model, where they suggested t as an alternatve technque to the lattce-gas automata for the study of hydrodynamc propertes. Ths approach completely elmnates the statstcal nose that plagues the usual lattce-gas smulatons and therefore permts smulatons that demand much less computer tme, whch was thought to be more effcent than the lattce-gas automata for ntermedate to low Reynolds number. Bhatnagar et al. [7] smplfed the collson operator by ntroducng a new model whch appled a sngle relaxaton tme approxmaton, and ths model s referred to as the Lattce Boltzmann Bhatnagar-Gross-Crook (BGK) model. As shown by Hou et al. [9] and Shen and Floryan [8], the physcs of drven flow n a cavty ndcated the exstence of crtcal aspect rato at whch the corner eddes merged and formed a prmary eddy. The dependence of the vortex structure to the aspect rato at dfferent Reynolds numbers were nvestgated by Pan and Acrvos [9], where a flow vsualzaton experment was performed to study effects of nerta force n the flow structure n the Reynolds number range of Re 4. Due to expermental lmtaton, the shallow cavty flow experment usually demonstrated a sngle nvscd core of unform vortcty, whle the vortex structure below the prmary vortex n the deep cavty could not be captured. Based on the revew, t s necessary for the formaton of eddes and vortces to be nvestgated further for a certan type of flows whch has been used as benchmark for valdaton of CFD codes so that the outcome of the smulaton wll be stll the same, regardless of the method used. Therefore, the objectves of the present artcle are to utlze the smulatons for cavty flows and to nvestgate and nterpret the results wth a partcular focus on the structure of prmary and secondary eddes. II. Numercal Methododology Hstorcally, s the logcal development of lattce gas automata (LGA) method []. Lke n LGA, the physcal space s dscretzed nto unform lattce nodes. Every node n the network s then connected wth ts neghbours through a number of lattce veloctes to be determned through the model chosen. The lattce Boltzmann uaton s gven by: f f + c =Ω t ( f ), α xα where f s the dstrbuton functon for partcles wth velocty c,α at poston x α and tme t. Equaton () conssts of two parts; propagaton (left-hand sde) whch refers to the propagaton of dstrbuton functon to the next node n the drecton of ts probable velocty, and collson Ω ( f ) (rght-hand sde) whch represent the collson of the partcle dstrbuton functon. In, the magntude of c,α s set so that n each tme step t, every partcle denoted by approprate dstrbuton functon propagates n a dstance of lattce nodes spacng x. Ths wll ensure that the partcles under consderaton arrve exactly at the lattce nodes after t and smultaneously collde wth other adjacent partcles. There are a few versons of collson operator Ω ( f ) avalable n lterature. However, the most well accepted verson was the Bhatnagar Gross Krook collson model (BGK) [7] due to ts smplcty and effcency [], []. The uaton that represents ths model s gven by: ( xc ) ( xc ) f,,t f,,t Ω ( f ) = τ where f s the ulbrum dstrbuton functon and τ s the tme to reach ulbrum condton durng collson process and s called relaxaton tme. Equaton () () Copyrght Prase Worthy Prze S.r.l. - All rghts reserved Internatonal Revew on Modellng and Smulatons, Vol. 4, N. 4 9

5 () of the BGK collson model descrbes that /τ of nonulbrum dstrbuton relaxes to ulbrum state wthn tme τ on every collson. By replacng the BGK collson model nto the Boltzmann uaton, the BGK Boltzmann Equaton s obtaned: f f f f + cα = t x τ α The general form of the lattce velocty model s expressed as DnQm where D represents spatal dmenson and Q s the number of connecton (lattce velocty) at every node. In ths paper, the 9-mcroscopc velocty or 9-bt model (DQ9) s used. The lattce geometry s shown n Fg. and the ulbrum dstrbuton functon of the 9-bt model s [3]: (3) 9 ( ) ( ) 3 f = ρω c u c u u (4) where the weghts are: ω = 4, ω ~ 5 =, ω6~ 9 = (5) and the mcroscopc velocty components are: c = (6) The macroscopc quanttes can be calculated from: c 7 c 4 ρ = f (7) = f f u c (8) c 3 c 6 c c c c 8 c 5 9 Fg.. Lattce geometry (DQ9) II.. Dervaton of the -form of Flow Equatons In ths subsecton, the dervaton of contnuty and Naver-Stokes uatons usng the Chapman-Enskog expanson [4] s dscussed. If a two-dmenson nne-bt model s used, then the tme evoluton lattce Boltzmann uaton can be expressed as: ( α α ) ( α ) f x + c t,t+ t f x,t =, ( f f ) = τ Usng the Taylor seres expanson and retanng up to the second order terms, the left hand sde of (9) can be rewrtten as follows: ( c α ) + f + t, ( α α α ) + + c + :c c f t t,,, (9) () In order to relate the lattce Boltzmann uaton wth a macroscopc uaton, t s necessary to solate dfferent tme scales. Ths s to ndcate dfferent scales for dfferent physcal phenomena whch are treated separately n the fnal macroscopc uaton. Hence, the space and tme dervatves are expanded n terms of the Knudsen number ε [5] as follows: ( ) 3 t ε t ε t O ε = + + () ( ε ) = ε + O () Expandng the dstrbuton functon f about gves: where: ( ) 3 ε ε ε f f = f + f + f + O (3) n n f = f c,α = for n (4) Equaton (3) mples that the non-ulbrum n dstrbuton functons ( ) f do not contrbute to the local values of densty and momentum. Substtutng Equatons (), () and (3) nto Equaton () and regroupng the uaton up to the frst order of ε yelds: + = (5) τ ( t c, α ) f f Equaton (5) s then smplfed to the ε order usng Equaton (3), whch gves: + + = τ τ ( α ) tf t c, f f, α (6) Copyrght Prase Worthy Prze S.r.l. - All rghts reserved Internatonal Revew on Modellng and Smulatons, Vol. 4, N. 4 9

6 A summaton of Equaton (5) for all values of and α s then performed to produce the frst order contnuty uaton,.e.: ( ρ ) + u = (7) tρ Multplyng Equaton (5) wth c, α and carryng out summaton the same way Equaton (7) s produced leads to: where: ( ρ ) t u + Π = (8) ( α α ) Π = c c f (9),, s the momentum flux tensor. After some smple mathematcal manpulaton to satsfy Gallean nvarance and the property of an sotropc tensor, the fnal expresson for Π s: s Π = c ρδ + ρu u () βχ β χ Replacng Equaton () nto Equaton (8) leads to: t ( ρ ) ( ρ ) ( csρ) u + uu = () Equaton () s actually the Euler uaton, where the pressure can be obtaned from the rght-hand sde term,.e.: p c ρ () where c s s the speed of sound. Smlarly, the uaton for ρ and u can be obtaned from Equaton (8) for ε. Takng summaton for all and α gves: s ρ = (3) t Then, multplyng Equaton (6) wth c, α and takng the summaton as above yelds: where: t ( ρu ) + Π = (4) τ {( c 3 s ) γ ( uγ ) βχ 3 β ( uχ ) χ ( ρuβ ) uβ χ ( csρ) uχ β ( csρ) ( ρu u u )} Π = τ δ ρ δ + ρ γ β χ γ (5) Combnng these uatons for ( ) O ε and O( ε ) gves the correct form of the contnuty and momentum uatons for ncompressble flow, whch are respectvely as follows: t u = (6) ( ρuβ ) χ ( ρuβuχ ) ( cs ρ) ρ = τ = β + χ Sβχ 6 where Sβχ ( βuχ χuβ ) s (7) = +, p = c ρ and c s s gven by the followng relaton: s c = (8) 3 Fnally, the knematc vscosty of a flud ν s related to the tme relaxaton τ n mesoscopc scale as follows: τ = 3ν + (9) From the above dervatons, we can see that the evoluton of Equaton (9) can lead to the ncompressble Naver-Stokes uaton through the Chapman-Enskog expanson. Equaton (9) descrbes that the value of τ must be kept hgher than.5 n order to avod negatve value of knematc vscosty. Ths lmts our smulaton to low Reynolds number. However, ths lmtaton can partly be solved by usng hgh number of nodes but would lead to longer computatonal tme. Ths aspect wll be addressed n our future works. III. Results and Dscusson Ths secton demonstrates the applcaton of the scheme dscussed n the precedng secton to smulate the ld-drven cavty flow at dfferent Reynolds numbers. The smulatons were consdered to have reached a steady state condton when the r.m.s. change n horzontal and vertcal velocty decreased to a magntude of -6 or less. III.. Valdaton of the Scheme For the purpose of code valdaton, we carred out the smulaton of ld-drven flow n a square cavty, K = and compared wth benchmark results produced by Gha et al. [6] whch s theoretcal smulaton based on Naver-Stokes uatons and experments research conducted by Tsorng et al. [7]. Ld-drven flow n a square cavty s well-known as a standard test case for the numercal schemes of flud flows. Supposed that the ld s located at the top boundary y = H, and moves wth a constant speed U from left to rght. The Reynolds number for the system s gven by: Copyrght Prase Worthy Prze S.r.l. - All rghts reserved Internatonal Revew on Modellng and Smulatons, Vol. 4, N. 4 9

7 UL Re = (3) ν..5 Gha et al X Velocty Y Velocty (a) -. Fg.. Comparson of Velocty profles at md-heght (y-velocty) and md-wdth (x-velocty) of cavty between and Gha et al. [6] results for Re =. Gha et al.5 X Velocty Y Velocty (b) Fgs. 5. Comparson between (a) streamlnes and (b) Tsorng et al. [7] expermental study streamlnes for Re = Fg. 3. Comparson of Velocty profles at md-heght (y-velocty) and md-wdth (x-velocty) of cavty between and Gha et al. [6] results for Re = (a) (a) (b) Fgs. 4. Comparson of Velocty profles at md-heght (y-velocty) and md-wdth (x-velocty) of cavty between (a) and (b) Tsorng et al. [7] results for Re = 4 (b) Fgs. 6. Comparson between (a) streamlnes and (b) Tsorng et al. [7] expermental study streamlnes for Re = 4 Copyrght Prase Worthy Prze S.r.l. - All rghts reserved Internatonal Revew on Modellng and Smulatons, Vol. 4, N. 4 93

8 Ths s due to the effect of vscous effect produced by the prmary vortex. It s also notceable that the vortces strength ncreased as Reynolds number ncreased. Ths s n agreement wth the results of Cheng and Hung [5]. (a) (b) (a) Re = (b) Re = 4 Fgs. 7. Comparson between (a) streamlnes and (b) Tsorng et al. [7] expermental study streamlnes for Re = Smulatons were performed at Re =, 4 and, usng a grd sze of. Fgs. and 3 show the comparson of the velocty profle for the Reynolds numbers consdered between and Gha et al. [6] smulaton whle Fgs. 4 show the same comparson but wth Tsorng et al. [7] results for Re = 4. Plot of stream functon comparson have been showed n Fgs. 5,6 and 7. It s apparent that the flow structure and velocty dstrbuton are n good agreement wth the prevous work of Gha et al. [6] and Tsorng et al. [7]. III.. Deep Cavty Flow The numercal smulaton was performed to analyse flud flow n a deep cavty wth aspect ratos K =.5 and.5. The DQ9 lattce geometry wth the BGK collson model was used. Fgs. 8 show the streamlnes of a cavty wth aspect rato.5. These three fgures correspond to Re =, 4 and,. Ths smulaton employed a grd system. It can be seen from Fgs. 8 that for low Reynolds number smulaton Re =, a counter-rotatng vortex s formed below the movng ld. As the Reynolds number ncreases (Re = 4), the center of the prmary eddy begns to move downwards wth respect to the top ld. For the case of hgh Reynolds number (Re =,), the prmary eddy s formed at the center of the geometry. The secondary vortex can also be clearly seen at low Reynolds number and ntally formed at the lower rght of the geometry. As the Reynolds number ncreases, ths vortex shft to the center left of the test case geometry. (c) Re =, Fgs. 8. Streamlnes at steady state condton for K =.5 III.3. Shallow Cavty Flow In ths secton, the numercal soluton of the scheme for rectangular shallow cavtes s presented. Two sets of geometres were chosen whch represent aspect ratos of K =.5 and 4., respectvely. For the frst case where the aspect rato s K =.5, a 5 grd system was employed. Fgs. 9 show that for, the prmary vortex center descends to the center of cavty wth ncreased strength as the Reynolds number ncreases. Copyrght Prase Worthy Prze S.r.l. - All rghts reserved Internatonal Revew on Modellng and Smulatons, Vol. 4, N. 4 94

9 wth those obtaned from a commercal CFD software, namely. (a) Re = (a) Re = (b) Re = 4 (b) Re = 4 (c) Re =, Fgs.. Steamlnes at steady state condton for K = 4. (c) Re =, Fgs. 9. Steamlnes at steady state condton for K =.5 From the fgure, two secondary vortces of dfferent szes are vsble at the lower corners of the cavty wth the left vortex beng bgger than the rght vortex due to a flow drecton from the rght corner to the left corner. The dfference n sze becomes obvous wth ncreasng Reynolds numb..most of these results are n excellent agreement wth the results reported by Burggraf [] and Schrebera and Keller [3]. For an aspect rato of K = 4 whch represent a more shallow cavty, the smulaton employed a 4 grd system. As Reynolds number ncreases, Fgs. show that the prmary vortex began to splt nto two vortces. However, there s no secondary vortces observed n corners even when the Reynolds number ncreases. Hence, t can be sad that the enlarged cavty n the flow drecton wll suppress the generaton of vortces at the corners. III.4. Verfcaton of the Scheme wth the Results Obtaned from Ths secton dscusses the verfcaton and comparson between the results obtaned from the schemes The man dfference s that the scheme utlzes the concept of statstcal molecular dynamcs n smulatng flud flow, whle common CFD software employs a fnte volume method to smulate advancement of flud mass n the computatonal doman. For the purpose of comparson, the aspect ratos of K =.5,.5,.5 and 4 are selected and the flow veloctes are vared to produce Re =, 4 and. The comparson were shown for the x-velocty component at the centerlne of the cavty where the results obtaned from the scheme and from were plotted n the same graph for all cases. For K =.5, the comparson between both results s llustrated n Fg.. From the fgure, t s obvous that the flow at the locaton along the centerlne from y =.7 and above s dsturbed by the ld velocty, thus producng a strong prmary vortex. The hgher the Reynolds number, the stronger the vortex. For a deep cavty wth hgher aspect rato, namely K =.5, the x-velocty profles are depcted n Fgs. for all the three Reynolds numbers. From the fgures, t can be seen that stronger vortces were formed from y =.5 and above. Below y =.5, a weaker secondary vortex s generated n an opposte drecton, whch s dentcal to the prevous case of K =.5. These vortces can clearly be seen n Fgs. 8. Both cases represent deep cavtes (K =.5 and.5) and t was observed that the present results are n excellent agreement wth the results obtaned usng and the prmary vortex grew bgger towards the bottom of the cavty as the secondary vortex got smaller. Copyrght Prase Worthy Prze S.r.l. - All rghts reserved Internatonal Revew on Modellng and Smulatons, Vol. 4, N. 4 95

10 Ths secondary vortex started to break up nto two vortces at the bottom left and rght corners as the aspect rato approached. where the results for streamlnes and velocty profles can be referred to Fgs. -4. On the other hand, the smulaton for the shallow cavty produced dfferent phenomena. The cases for shallow cavtes are represented by geometres havng aspect ratos of K =.5 and 4.. For K =.5, t s llustrated n Fgs (a) Re = (a) Re = (b) Re = (b) Re = (c) Re =, Fgs.. The x-velocty at the centerlne for the scheme and for K = (c) Re =, Fgs.. The x-velocty at the centerlne for the scheme and for K =.5 Copyrght Prase Worthy Prze S.r.l. - All rghts reserved Internatonal Revew on Modellng and Smulatons, Vol. 4, N. 4 96

11 (a) Re = (a) Re = (b) Re = (b) Re = (c) Re =, Fgs. 3. The x-velocty at the centerlne for the scheme and for K =.5 From the fgures, the resultng velocty profles ndcated a prmary vortex domnated the central regon of the cavty, whle two smaller secondary vortces were shfted to the bottom corners as can be seen n Fgs. 9 for Re = 4 and,. For Re = however, the vortces formed were too weak to be vsble as streamlne (c) Re =, Fgs. 4. The x-velocty at the centerlne for the scheme and for K = 4. When the cavty became shallower, as can be seen n Fgs. 4 for K = 4., the weaker vortces dmnshed whle the domnant prmary vortex developed eccentrcty wthn tself especally for hgher Reynolds number. Ths s due to the domnant nerta force n a hgher Reynolds number flow as can be seen n Fgs. 7(b) and 7(c). Nevertheless, the comparson wth the results from produced an excellent agreement, whch s smlar to the cases for deep cavtes. Copyrght Prase Worthy Prze S.r.l. - All rghts reserved Internatonal Revew on Modellng and Smulatons, Vol. 4, N. 4 97

12 IV. Concluson The flow structure of a two-dmensonal vscous flow n ld-drven deep cavtes has been numercally studed and analysed usng the lattce Boltzmann scheme. The dynamc of the prmary and secondary vortces are well captured by these smulatons. For cavty flow there are a prmary vortcty formed n the core centre of the cavty, when the cavty s deep there wll be a two secondary vortces n the lower cavty corners, these two secondary vortces become larger when the Reynolds number ncrease. For shallow cavty the secondary vortces dsappear as the aspect rato ncrease. In addton, the present results are found to be n good agreement wth the results obtaned by a CFD software, namely verson 6.. It s also n good agreement wth the prevous studes by Gha et al. [6] and Tsorng et al. [7], whch proves that the lattce Boltzmann scheme s a powerfull alternatve tool for solvng complex flud flow problems wth hgh accuracy. However, the effects of boundary condtons n lateral drectons are not yet consdered snce ths paper only presents two-dmensonal case studes and threedmensonal cases wll be consdered n our future study. Acknowledgements The authors would lke to acknowledge the Mnstry of Scence, Technology and Innovaton, Malaysa for sponsorng ths work. References [] T. Seta, K. Kono, D. Martnez, and S. Chen, Lattce Boltzmann Scheme for Smulatng Two-Phase Flows, JSEM Internatonal Journal, Seres B, Vol. 43(): 35-33,. [] R. R. Nourgalev, T. N. Dnh, T. G. Theofanous, and D. Joseph, The lattce Boltzmann uaton method: theoretcal nterpretaton, numercs and mplcatons, Internatonal Journal of Multphase Flow, Vol. 9(): 7-69, 3. [3] D. Yua, R. Me, L.-S. Luo, and W. Shyya, Vscous flow computatons wth the method of lattce Boltzmann uaton, Progress n Aerospace Scences, Vol. 39(5): , 3. [4] N. S. Martys and H. Chen, Smulaton of Multcomponent Fluds n Complex Three-Dmensonal Geometres by the Lattce Boltzmann Method, Physcal Revew E, Vol. 53(): , 996. [5] A. Dupus and B. Chopard, Lattce Gas Modelng of Scour Formaton under Submarne Ppelnes, Journal of Computatonal Physcs, Vol. 78(): 6-74,. [6] H. Chen, et al., Extended Boltzmann Knetc Equaton for Turbulent Flows, Scence, Vol. 3(5633): , 3. [7] S. Chen and G. D. Doolen, Lattce Boltzmann Method for Flud Flows, Annual Revew of Flud Mechancs, Vol. 3(): , 998. [8] G. McNamara and B. Alder, Analyss of Lattce Boltzmann Treatment of Hydrodynamcs, Physca A: Statstcal Mechancs and ts Applcatons, Vol. 94 (-4): 8-8, 993. [9] S. Hou, et al., Smulaton of cavty flow by the lattce Boltzmann method, J. Comput. Phys., Vol. 8(): , 995. [] B. Kraloua and A. Hennad, CFD Smulaton and Expermental Verfcaton of the Flow Feld n a Centrfugal Separator Internatonal Revew of Electrcal Engneerng, Vol. 5(6): ,. [] S. Larb, T. Chergu, and A. Bouhdjar, Analyss of Flows Modellng and Energy Performances n Solar Chmneys Internatonal Revew on Modellng and Smulatons, Vol. (): 4 -, 8. [] O. R. Burggraf, Analytcal and Numercal Studes of the Structure of Steady Separated Flows, Journal of Flud Mechancs, Vol. 4: 3-5, 966. [3] R. Schreber and H. B. Keller, Spurous Solutons n Drven Cavty Calculatons, Journal of Computatonal Physcs, Vol. 49(): 65-7, 983. [4] U. B. Mehta and Z. Lavan, Flow n a two-dmensonal channel wth a rectangular cavty, Journal of Appled Mechancs, Vol. 36: 897-9, 969. [5] M. Cheng and K. C. Hung, Vortex Structure of Steady Flow n a Rectangular Cavty, Computers & Fluds, Vol. 35(): 46-6, 6. [6] G. R. McNamara and G. Zanett, Use of the Boltzmann Equaton to Smulate Lattce-Gas Automata, Physcal Revew Letters, Vol. 6(): , 988. [7] P. L. Bhatnagar, E. P. Gross, and M. Krook, A Model for Collson Processes n Gases. I. Small Ampltude Processes n Charged and Neutral One-Component Systems, Physcal Revew, Vol. 94(3): 5-55, 954. [8] C. Shen and J. M. Floryan, Low Reynolds number flow over cavtes, Physcs of Fluds, Vol. 8(): 39-3, 985. [9] F. Pan and A. Acrvos, Steady flows n rectangular cavtes, Journal of Flud Mechancs, Vol. 8(4): , 967. [] U. Frsch, B. Hasslacher, and Y. Pomeau, Lattce-Gas Automata for the Naver-Stokes Equaton, Physcal Revew Letters, Vol. 56: , 986. [] S. Chen, H. Chen, D. Martnez, and W. Matthaeus, Lattce Boltzmann model for smulaton of magnetohydrodynamcs, Physcal Revew Letters Vol. 67(7): , 99. [] Y. H. Qan, Lattce Gas and Lattce Knetc Theory Appled to Naver-Stokes Equaton, Ph.D. dssertaton, Unversty of Pars, Pars, 99. [3] C. S. N. Azwad and T. Tanahash, Three-Dmensonal Thermal Lattce Boltzman Smulaton of Natural Convecton n a Cubc Cavty, Internatonal Journal of Modern Physcs B, Vol. (): 87-96, 7. [4] S. Chapman and T. G. Cowlng, The mathematcal theory of nonunform gases : an account of the knetc theory of vscosty, thermal conducton, and dffuson n gases (Cambrdge Unversty Press, 99). [5] C. Cercgnan, Theory and Applcaton of the Boltzmann Equaton (Scottsh Academc Press, 975). [6] U. Gha, K. N. Gha, and C. T. Shn, Hgh-Re Solutons for Incompressble Flow Usng the Naver-Stokes Equatons and a Multgrd Method, Journal of Computatonal Physcs, Vol. 48(3): 387-4, 98. [7] S. J. Tsorng, et al., Behavour of macroscopc rgd spheres n lddrven cavty flow, Internatonal Journal of Multphase Flow, Vol. 34(): 76-, 8. Authors nformaton Department of Mechancal and Materals Engneerng, Unverst Kebangsaan Malaysa, 436 UKM Bang, Selangor. Faculty of Mechancal Engneerng, Unverst Teknolog Malaysa, 83 Skuda, Johor. M. Mussa was born n 975, n Baghdad, Iraq. He receved hs M.Sc. n Mechancal Engneerng from the Unversty of Baghdad. Also, he receved hs B.Sc. n Mechancal Engneerng from the Unversty of Baghdad. He has worked on dfferent topcs such as computatonal analyss of heat transfer, flud mechancs and mcrofludc, he has some papers were publshed n dfferent journals and conferences.. Copyrght Prase Worthy Prze S.r.l. - All rghts reserved Internatonal Revew on Modellng and Smulatons, Vol. 4, N. 4 98

13 M. A. Mussa s member of Iraq Engneers Assocaton and member of the Assocaton of Unversty Lecturers, Iraq. S. Shahrr was born n 969, n Terengganu, Malaysa. He receved hs PhD n Mechancal Engneerng from Unversty of Wales Swansea, Unted Kngdom n 997, hs M.Sc. n Desgn and Economc Manufacture from Unversty of Wales Swansea, Unted Kngdom n 994, and hs B.Sc. n Mechancal Engneerng from Unverst Kebangsaan Malaysa n 99. He s currently a Professor at Unverst Kebangsaan Malaysa. He has authored several techncal papers n the feld of combuston and fuel engneerng, powertran engneerng, machne desgn, computaton theory and mathematcs and Numercal analyss. Hs current research focuses on appled scences and technologes, Nanotechnology, Mcro electro-mechancal system (MEMS). S. Abdullah s regstered engneer of Board of Engneers Malaysa, graduate member of Insttute of Engneers Malaysa, member of Socety of Automotve Engneers (SAE) and corporate member of Insttute of Engneers Malaysa. C. S. Nor Azwad was born n 977, n Kelantan, Malaysa. He receved hs PhD n Mechancal Engneerng from Keo Unversty, Japan n 7, hs M.Sc. n Thermal Power and Flud Engneerng from Unversty of Manchester Insttute Scence and Technology, Unted Kngdom n 3, and hs B.Sc. n Mechancal Engneerng and Materal Scence from Kumamoto unversty, Japan n. He s currently an Asst. Professor at Unverst teknolog Malaysa. He has authored several techncal papers n the feld of Multphase flow, Mcrofludc, Convectve Heat Transfer, Flud- Structure Interacton, Computatonal Methods, Lattce Boltzmann Method and Physcs of Flud. C.S. Nor Azwad s regstered engneer of Board of Engneers Malaysa. R. Zulkfl was born n 97, n Selangor, Malaysa. He receved hs PhD n Mechancal Engneerng from Unverst Kebangsaan Malaysa, Malaysa n, hs MSc n Advanced Engneerng Materals, Unversty of Lverpool, Unted Kngdom n 996, and B.Eng(Hons) n Mechancal Engneerng, Unversty of Lverpool, Unted Kngdom n 994. He s currently a senor lecturer at the Department of Mechancal and Materals Engneerng, Unverst Kebangsaan Malaysa. He has authored many techncal papers n the feld of Engneerng Scences, Mechancal Engneerng, Materal Scences, and Advanced Composte Materals. Copyrght Prase Worthy Prze S.r.l. - All rghts reserved Internatonal Revew on Modellng and Smulatons, Vol. 4, N. 4 99

14 Internatonal Revew on Modellng and Smulatons (IREMOS) (contnued from outsde front cover) Effect of the Inclned Vbratory Exctaton on Heat Transfer n a Space Flled wth a Nanoflud by S. Kadr, R. Mehdaou, M. Elmr Numercal Study of Heat and Mass Transfer n a Porous Membrane Used n Energy Recovery Devces by R. Sebaï, R. Choukh, K. Amara, A. Guzan Survey and Analyss of Mantenance System for Horseshoe-Shaped Secton Tunnel by K. Mosayeb, M. Kooh, M. Mohammad Dehcheshmeh, S. Fath, M. Ostadal Makhmalbaf Dynamc Analyss of Two Novel Mcro-Systems for Optcal Attenuaton and Modulaton by M. M. S. Fakhrabad, B. Dadashzadeh, V. Norouzfard, M. Dadashzadeh Vbratonal Analyss of Graphene Sheets Usng Molecular Mechancs by M. M. S. Fakhrabad, M. Dadashzadeh, V. Norouzfard, B. Dadashzadeh A New Fuzzy Approach on Plannng and Decson Makng n Engneerng and Constructon Industry by Parna Imannejad, Al Karmpour Numercal Smulaton of Stran Effect on the Extncton Rato and Inserton Loss Parameters n Asymmetrc Coupled Quantum Wells Electroabsorpton Optcal Modulator by Kambz Abed Analyss and Crcut Model of Optcal Injecton-Locked Semconductor Lasers by Kambz Abed, Mohammad Bagher Nasrollahnejad Operatng Temperature Correlaton wth Ambent Factors of Buldng Integrated Photovoltac (BIPV) Grd-Connected (GC) System n Malaysa by Hedzln Zanuddn, Sulaman Shaar, Ahmad Malk Omar, Shahrl Irwan Sulaman Comparson of Potental Power Plants n Jordan Usng Analytcal Herarchy Process by Adnan Mukattash, Ahmed Al-Ghandoor Delectrc Constant Computaton Model for N-Phase Porous Materals by L. Bouledjnb, S. Sahl Modelng and Control of a New LSRM for Shuntng the Ralways Channels by E. M. Barhoum, B. Ben Salah Development of a Transmsson Lne Scale Model for Evaluatng Electrc and Magnetc Felds by Adolfo Escobar, Gullermo Aponte Desgn and Analyss of Polymer Electrooptc Modulator Usng the Full Vectoral Fnte Element Method by Kambz Abed, Habb Vahd Artfcal Neural Networks Based Approach for Parametrc Optmzaton of CMOS Operatonal Amplfers by A. Jafar (contnued on outsde back cover) Abstractng and Indexng Informaton: Academc Search Complete - EBSCO Informaton Servces Cambrdge Scentfc Abstracts - CSA/CIG Elsever Bblographc Database SCOPUS Autorzzazone del Trbunale d Napol n. 78 del //8

15 (contnued from nsde back cover) EXTRACTED BY ICOMOS VIRTUAL FORUM ND INTERNATIONAL CONFERENCE ON MODELLING AND SIMULATIONS Calculaton and Vsualzaton of Electromagnetc Felds Usng EMP3: the Coaxal Lne Example by Konstantnos B. Baltzs 38 Ths volume cannot be sold separately by Parts A, B (8)4:4;-P