Simulation of Fluid Flow Inside a Back-ward-facing Step by MRT- LBM

Transcription

1 2012 Internatonal Conference on Flud Dynacs and Therodynacs Technologes (FDTT 2012) IPCSIT vol.33(2012) (2012) IACSIT Press, Sngapore Sulaton of Flud Flow Insde a Back-ward-facng Step by MRT- LBM Mohaad Pourtous +, Mohaad Razzaghan, Aran Safdar and Aer Nordn Darus Thero-flud Departent, Unverst Teknolog Malaysa, UTM Skuda, Johor, Malaysa Abstract. In ths paper, the lattce Boltzann ethod (LBM) s appled to sulate the two-densonal n copressble steady low Reynolds nuber backward-facng step flow n a channel. The selected Reynolds nuber s lted to a axu value of Re105.n ths Letter, we propose a nuercal schee to solve the velocty feld usng the ult-relaxaton-te (MRT-D2Q9).The an obectve of ths study s to analyze the flow feld n the back-ward-facng step whch s ncludng a blockage and also to study the shape of vortexes that are happened after the blockage. Ths odel s valdated by the nuercal sulaton of a classcal benchark proble. The results deonstrate the accuracy and effectveness of the proposed ethodology. Keywords: Lattce Boltzann ethod; Mult-relaxaton-te; Back-ward-facng step 1. Introducton Lattce Boltzann ethod wth Bhatnagar-Gross_Krook collson odel(lbgk)[3,6,7,10] also called sngle-relaxaton-te(srt)lb ethod has gotten notceable success n sulaton of hydrodynac ssues and the an portant advantages of the LBM can be classfed as ts explctly, splcty perforance, and natural to parallelze. there are any other factors that are addressed for provng the lattce Boltzan ethod such as capablty for coplex geoetry[2,9,10,13] applyng of the boundary condtons[5,12,14,15] and also sulaton of hgh Reynolds nuber. However, despte the presented advantages, soe weakness ponts of the LBGK odel are clear. For exaple ths ethod ay cause to nuercal nstablty when the densonless relaxaton te τ s close to 0.5.one way for solvng these weakness ponts of the LBGK odel s to use a ult-relaxaton-te(mrt),whch has been shown to ake stable soluton for hgh Reynolds nuber flows[4,8,11] and has the splcty and coputatonal of the LBGK ethod. One the an advantages of MRT-LB s that t has better nuercal stablty and also has ore degrees of freedo rather than SRT-LB odel. Followng the ethod of MRT-LB odel, an ncopressble MRT-LB(IMRT-LB) odel has been presented n ths Letter. The sulatng results agree well wth the analytcal solutons or other benchark data [1]. In addton, the nuercal results deonstrate that the presented (IMRT-LB) odel has better stablty n coparson to the LBGK odel. + Correspondng author. Tel.: + ( ). E-al address: (pohaad3@lve.ut.y). 130

2 2. Nuercal Method 2.1 Mult Relaxaton te Lattce Boltzann odel In ult-relaxaton ethod the collson operator s clearly classfed as f( x+ cδ tt, +Δt) f(,) xt Ω[ f(,) xt f (,)] xt (1) In ths uaton Ω s the collson step and ths ter s changed to oentu space and llustrated as 1 f( x+ cδ tt, +Δt) f( xt, ) M Sxt [ (, ) ( xt, )] (2) M presents Matrx transfor and S s defned as a dagonal atrx. (x,t) and are oents vectors. n ths odellng, Matrx transfor for D2Q9 s gong to be used lke followng; M The oentu vector here s ( ρ, e, ε,, q,, q, p, p ) T And ulbru of the oent s ρ 0 ρ ( x y ) ρ + x x y y 2 2 3( x y ) 2 2 x y x y Also s defned by followng uatons: x y x x y y xx xy (3) (4) 131

3 ρu f c x x x ρu f c y y y s dag(1,1.4,1.4, s3,1.2, s5,1.2, s7, s8) The acroscopc propertes such as densty and velocty are categorzed by f ρ, fe ρu On the other hand ulbru dstrbuton functon s calculated as followng 2 3 eu. 9( eu. ) 3. uu f wρ[1 + + ] c 2c 2c (5) (6) (7) (8) dx, dt and c are defned as lattce wdth, te step for lattce and lattce speed respectvely In backward-spacng flow proble c s assued 1 and dt dx Partcle speed e are deterned as e0 0 e (cos[ π( 1) / 2],sn[ π( 1) / 2]) c 1, 2, 3, 4 e (cos[ π( 4 1/2)/2],sn[ π( 4 1/2)/2]) 2c 5,6,7,8 (9) w has been presented weghtng factors that s classfed n three ters w0 4 / 9, w 1, 2,,3, 4 1/ 9, w 5,6,7,8 1/ 36 (10) The geoetrc paraeter of the nserted square blockage s w 1 3 H and the locaton s as plotted n Fg (1). Fg. 1.The geoetry of backward-facng step wth nserted square blockage 132

4 3. Results and Dscusson 3.1. Overvew of the velocty feld In ths letter a unfor rectangular esh (200*40) for sulatng of the flow was conducted. After and before the recrculaton zone, Unfor square lattce esh was appled. The Reynolds nuber of the flow 4U( H h) s presented as,where U s defned as the axu velocty n the nlet. 3υ Fg. 2. Velocty vector wth the nserted square blockage Fgure 2 shows the velocty vector feld wth the nserted square blockage after the enlargeent. 3.2 Influence of Reynolds nuber In ths letter dfferent Reynolds nubers (50,130,105) are used n order to analyze dfferent vortexes after the blockage and also to fgure out the reattachent area and deforaton of vortex after ths blockage. Accordng to analyzng dfferent velocty vector felds fgures t can be obtaned that, by perforng hgh Reynolds nubers shape of vortex wll be changed to bgger one as opposed to the sall one for low Reynolds nubers. Fg. 3. Velocty vector wth the nserted blockage for Reynolds nuber 50 Fg. 4. Velocty vector wth the nserted blockage for Reynolds nuber

5 Fg. 5.Velocty vector wth the nserted blockage for Reynolds nuber 105 By akng a coparson between the llustrated fgures for three Reynolds nubers whch are 50,130 and 105 respectvely, t s clear to note that by ncreasng the Reynolds nuber (Re130,105) the shape of the vortex after the blockage changes and ts sze turns to be bgger whereas ts sall sze for low Reynolds nuber(re50). Fg. 6. Velocty vector wthout blockage for Reynolds nuber 105 Fgure 6 clearly llustrates the velocty vector wthout the blockage. As can be seen when there s no nserted blockage n the channel the shape, sze and length of the vortex wll be changed to a bgger one as opposed to the channel ncludng the nserted blockage that were shown n Fgure 2,3,4and 5. 4.Concluson In ths study, dfferent Reynolds nubers backward-facng step flows are sulated usng ultrelaxaton-te odel based on D2Q9 LBM. The nuercal results obtaned for the velocty feld are n good agreeent wth the publshed nuercal results. When we appled hgh Reynolds nubers n the channel ncludng nserted blockage the shape, sze and length of the vortex wll have sgnfcant deforaton n coparson to the stuaton wth low Reynolds nuber. Fnally the channel wthout blockage s dscussed n order to fgure out the effect of nserted blockage on the vortex n dfferent aspects such as shape, sze and ts length. 5.Acknowledgents The authors would lke to thank Prof. Aer Nordn Darus for helpful dscussons. 6.References [1] C.-K. Chen, et al., "Lattce Boltzann ethod sulaton of backward-facng step on convectve heat transfer wth feld synergy prncple," Internatonal Journal of Heat and Mass Transfer, vol. 49, pp , [2] D.-J. Chen, et al., "Iersed Boundary Method Based Lattce Boltzann Method to Sulate 2d and 3d Coplex Geoetry Flows," Internatonal Journal of Modern Physcs C, vol. 18, pp

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