Various Speed Ratios of Two-Sided Lid-Driven Cavity Flow using Lattice Boltzmann Method

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1 Penerbt and Thermal Scences ISSN (onlne): Varous Speed Ratos of Two-Sded Ld-Drven Cavty Flow usng Lattce Boltzmann Method Nor Azwad Che Sdk *,a and St Asyah Razal b Department of Thermo-Fluds, Faculty of Mechancal Engneerng, Unverst Teknolog Malaysa, Skuda, Johor Bahru, Malaysa *,a azwad@fkm.utm.my, b ctecah@gmal.com Artcle Info Abstract In the present study, the flow confguraton of two-sded ld-drven cavty has Receved 13 August 2014 been nvestgated usng the Lattce Boltzmann Receved n revsed form 7 September 2014 method. Frst, the code was valdated aganst the Accepted 10 September 2014 numercal results taken from prevous study of flud flow n a sngle-ld drven cavty. The nfluence of varous speed ratos whch vary from 0 to 1 and several Reynolds number (100, 400, and 1,000) on the flow confguraton of the cavty were analyzed. The results show that the ncrease n both speed rato and Reynolds number gves an effect on flow confguraton of the cavty. Copyrght 2014 Penerbt - All rghts reserved. Keywords: Lattce Boltzmann method, Two-sded, Ld-drven cavty, Parallel wall moton 1. INTRODUCTION The problem of cavty flow of movng boundary has been a major topc for research studes due to ts smplcty n geometry n the last four decades. It has also been wdely used frequently n ndustral and technologcal applcatons whch nclude coatng system [1-2], mxng [3], dryng technologes [4], polymer processng [5] and ceramc tape castng [6]. Numerous nvestgatons of flow feld n a sngle-sded ld-drven cavty flow have been conducted ether by expermental or numercal studes. A great number of papers on lddrven cavty flow can be found n avalable lterature [7-12]. An extended study of two-sded ld-drven cavty from sngle-sded ld-drven cavty flow problem was done by Kuhlmann and other nvestgators [13], where ther nvestgaton specfed that the cavty aspect rato and the Reynolds number wll lead to the exstence of non-unque two-dmensonal steady flow, whch s determned by the wall veloctes. At low Reynolds number, the flow conssts of separate co-rotatng vortces next to each of the movng walls. As the veloctes of the wall ncrease, a jump transton occurs, and the two vortces partally merge to generate a flow pattern whch resembles cat s eyes. At hgh Reynolds number, the flow of the cat s eye becomes unstable and transforms nto a steady three-dmensonal cellular flow. One of the alternatve methods n CFD s the lattce Boltzmann method (LBM). LBM s a relatvely new smulaton technque for complex flud systems and has attracted nterest from researchers n many felds. Unlke tradtonal CFD methods, whch solve conservaton equatons of macroscopc propertes (.e. mass, momentum, and energy) numercally, LBM models flud that contans fctve partcles, and such partcles perform successve propagaton and collson processes over a dscrete lattce mesh. Due to ts partculate nature and local dynamcs, LBM has several advantages over other conventonal CFD methods, especally n 11

2 Penerbt and Thermal Scences ISSN (onlne): dealng wth complex boundares, ncorporaton of mcroscopc nteractons, and parallelzaton of the algorthm. The ld-drven cavty flow s also the subject of numerous lattce Boltzmann studes. For example, Nor Azwad et al. [14] performed a numercal nvestgaton of ld-drven cavty flow based on two dfferent methods: lattce Boltzmann method and splttng method. In ther study, the results from unform and stretched form of splttng method were compared wth the results from lattce Boltzmann method. Ld-drven cavty problem at varous Reynolds numbers was used as a numercal test case. Meanwhle, Predrag et al. [15] used lattce Boltzmann method n order to explore ncompressble flud flow nsde a two-sded ld-drven staggered cavty. They also presented the characterstcs of flow pattern for a varety of Reynolds numbers (50 3,200) for parallel moton of lds. The authors had also obtaned an asymmetrc steady-state flow pattern for parallel moton of lds. Recently, Perumal and Dass [16] presented a result of a numercal study n a two-sded lddrven square cavty by the LBM. They found that for parallel moton of the walls, there was a par of counter-rotatng secondary vortces of equal sze near the center of the wall. The man am of the present study s to nvestgate the effects of speed rato and Reynolds number on the development of vortex n cavty usng LBM. 2.0 MATHEMATICAL FORMULATION 2.1 Lattce Boltzmann Method The results presented n ths paper are obtaned usng D2Q9 lattce Boltzmann model (where the numbers followng the letters D and Q refer to the model s dmensonalty and number of lattce speeds, respectvely). The startng pont for lattce Boltzmann smulaton s the evoluton equaton for a set of dstrbuton functons f whch s dscrete n both space and tme: f [ ] F eq ( x + e, t + 1) f ( x, t) = f ( x, t) f ( x, t) + τ f 1 (1) Where e s the partcle s velocty, τ s the relaxaton tme for the collson, f eq s an equlbrum dstrbuton functon and = 0, 1,, 8 for two-dmenson nne-velocty model (D2Q9). Noted that the rght hand sde of Eq. 1 s the collson term where the Bhatnagar- Gross-Krook (BGK) approxmaton has been appled [17]. The dscrete velocty s expressed as e = (0, 0) for = 0, e = (cos ( 1)π/4, sn ( 1)π/4) for = 1, 3, 5, 7 and e = 2 1/2 (cos ( 1)π/4, sn ( 1)π/4) for = 2, 4, 6, and 8. Macroscopc densty ρ and velocty u of the flud are determned by the followng velocty moments of the dstrbuton functon: f = ρ (2) eq eq e, α f = ρu α (3) 12

3 Penerbt and Thermal Scences ISSN (onlne): The equlbrum dstrbuton functon, f eq s chosen such that the contnuum macroscopc equatons approxmated by the evoluton equaton correctly descrbe the hydrodynamcs of the flud. For D2Q9 model, f eq s defned as: f ( e u) 2 2 e u 3u ρω (4) c 2c 2c eq = 2 where c = (3RT) 1/2 and the weghts are ω 0 = 4/9, ω 1,3,5,7 = 1/9 and ω 2,4,6,8 = 1/36. Through multscalng expanson, the mass and momentum equatons can be derved from D2Q9 model as follows: u = 0 (5) 1 + u u = + υ t ρ p u 2 u (6) The vscosty, υ can be related to the tme relaton n lattce Boltzmann equaton as follows: 1 τ = 3 υ + (7) Code Valdaton To valdate the present numercal method, the LBM code was used to compute the sngle lddrven flow for Re = 1,000 on a lattce sze. A ld velocty of U = 0.1 was consdered n ths work. Fg. 1 shows the comparson of steady-state u-velocty profle along a vertcal lne and v-velocty profle along a horzontal lne passng through the geometrc center of the cavty at Re = 1,000 wth the benchmark solutons of Gha et al. [9]. The excellent match of the present results wth the present study demonstrates ts valdty for the smulaton. Fgure 1: u-velocty profle (left) and v-velocty profle (rght) at Re = 1,000 13

4 Penerbt and Thermal Scences ISSN (onlne): To ensure grd dependence, LBM code was smulated usng three dfferent grds; , and All reported results n the present study converged to a maxmum resdual of RESULTS AND DISCUSSION A comprehensve analyss has been conducted for a square cavty wth L = H. Most of the prevous nvestgatons were carred out by consderng equal magntude of the velocty of the opposte walls rrespectve of the drecton of moton. Fgure 2: Geometry of two-sded parallel wall moton Therefore, ths present study attempts to examne the change n the flud moton. The magntude of the ld veloctes and Reynolds number were vared (Re = 100, 400, 1,000) for partcular parallel wall moton where the top and bottom ld moved n the same drecton but dfferent veloctes as shown n Fg.2. To track varous flow transformatons arsng n a cavty, speed rato ( S = U U B T ) was vared from 0 to 1. Fg. 3 sgnfes classcal cavty flow drven by the unform moton of one of the lds whle all other lds are statonary. In ths stuaton, one prmary vortex and one secondary corner vortces were observed n the rght bottom cavty. As S ncreased from 0, the secondary corner vortces started to ncrease. The vortces combned and formed another prmary vortex, counter rotatng wth respect to the already exsted vortex at the bottom of the cavty. Wth the ncrease of S, the sze of the bottom vortex ncreased and fnally became equal to the top vortex wth a free shear layer n between at S = 1. Meanwhle, t was observed that the locaton of the center of the prmary vortex at the top remans more or less fxed n spte of the change n S. However, the center of the vortex at the bottom sde slowly moved up as the speed rato ncreased. Fgs. 4 and 5 also show the smlar streamlne pattern where the secondary vortces or bottom vortex combned and ncreased untl the vortces became symmetrcal wth te top prmary vortex. Nevertheless, t can also be seen that the locaton of the top and bottom vortces was nfluenced by the varaton of Reynolds number. Increased Reynolds number made the prmary vortex cores moved towards the centers of the top and bottom halves of the cavty. It 14

5 Penerbt and Thermal Scences ISSN (onlne): can also be observed that wth the ncrease of Reynolds number, the secondary vortex par grew n sze at the rght sde of the cavty. S= 0 S = 0.05 S = 0.25 S = 1.0 Fgure 3: Streamlne plots for varous speed ratos of the ld for parallel moton at Re = 100. S= 0 S = 0.25 S = 0.05 S = 1.0 Fgure 4: Streamlne plots for varous speed ratos of the ld for parallel moton at Re =

6 Penerbt and Thermal Scences ISSN (onlne): S= 0 S = 0.05 S = 0.25 S = 1.0 Fgure 5: Streamlne plots for varous speed ratos of the ld for parallel moton at Re = 1,000. Both Tables 1 and Table 2 show the locaton of the center of vortces for two-sded ld-drven cavty flow at Reynolds numbers of 100, 400 and 1, speed ratos (S) were consdered n ths study (.e. 0, 0.05, 0.25, 0.5, and 1.0). S s the rato between the veloctes of the bottom ld to the velocty of the top ld.. From both tables, t can be determned that at Reynolds number 100, 400, 1,000, and S=1.0, the prmary vortex center and the secondary vortex center n the x/h drecton had smlar values, whch ndcated these vortces became symmetrcal. Table 1: Locatons of the prmary vortex center for parallel wall moton for Reynolds number of 100, 400, and 1,000. Re Prmary Vortex Center S=0 S=0.05 S=0.25 S=0.5 S= x/h y/h x/h y/h ,000 x/h y/h

7 Penerbt and Thermal Scences ISSN (onlne): Table 2: Locatons of the secondary vortex center for parallel wall moton for Reynolds number of 100, 400, and 1,000. Re Secondary Vortex Center S=0 S=0.05 S=0.25 S=0.5 S=1.0 Left Rght Left Rght Left Rght 100 x/h y/h x/h y/h ,000 x/h y/h CONCLUSION The present study numercally smulated an ncompressble two-dmensonal lamnar flow nsde a square cavty. Two-sded and parallel motons were consdered. In ths nvestgaton, the am was to examne the effects of dfferent speed ratos and Reynolds numbers on the development of vortex n cavty usng lattce Boltzmann method. For varous speed ratos, the bottom vortex ncreased wth the ncreased of speed rato and became symmetrcal between the top and bottom vortces. Meanwhle, the varaton of Reynolds number also gave an effect on flow confguraton, whch produced prmary vortex at dfferent locaton. A par of counter-rotatng secondary vortces symmetrcally placed about the centerlne parallel to the moton of the walls was also observed. REFERENCES [1] Z. Cao, M.N. Esmal, Numercal study on hydrodynamcs of short-dwell paper coaters, AIChE J. 41 (1995) [2] N.G. Trantafllopoulos, C.K. Adun, Relatonshp between flow nstablty n shortdwell ponds and cross drectonal coat weght non unformtes, TAPPI J.73 (1990) [3] C.W. Leong, J.M. Ottno, Experments on mxng due to chaotc advecton n a cavty, Journal of Flud Mechancs 209 (1989) [4] N. Alleborn, H. Raszller, F. Durst, Ld-drven cavty wth heat and mass transport, Internatonal Journal of Heat and Mass Transfer 42 (1999) [5] P.H. Gaskell, J.L. Summers, H.M. Thompson, M.D. Savage, Creepng flow analyses of free surface cavty flows, Theoretcal Computatonal Flud Dynamcs 8 (1996) [6] H. Hellebrand, Tape Castng, n: R.J. Brook (Ed.), Processng of Ceramcs, Part1, VCH Verlagsgesellschaft mbh, Wenhem. 17 (1996) [7] OR Burggraf, Analytcal and numercal studes of the structure of steady separated flows, Journal of Flud Mechancs 24 (1966)

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