Simulation of single bubble rising in liquid using front tracking method

Transcription

1 Advances in Fuid Mechanics VI 79 Simuation o singe bubbe rising in iquid using ront tracking method J. Hua & J. Lou Institute o High Perormance Computing, #01-01 The Capricorn, Singapore Abstract Front tracking method is improved to simuate the rising and deorming o a bubbe in quiescent viscous iquid under various ow regimes. The simuation resuts demonstrate that the current agorithm is more robust in modeing a muti-uid system with wider ranges o the Reynods number (1<Re<200) and the Bond number (1<Bo<200) under a high density ratio (ρ /ρ b =1000) and a viscosity ratio (µ /µ b =100). The new ront tracking agorithm is aso appied to investigate bubbe rising and deorming behaviour in the various ow regimes o the air bubbe-water soution system under eects o the Reynods number and the Bond number, which have been expored experimentay in iterature. The simuation resuts are compared with the experiments in aspects o bubbe shape, termina veocity and bubbe wake ow pattern. Reasonabe agreements are achieved in most ow regimes. Keywords: bubbe rising, ront tracking method, numerica simuation, mutiphase ow. 1 Introduction Muti-uid systems pay an important roe in many natura and industria processes such as combustion / chemica reacting, petroeum reining, boiing, etc. The rising o singe bubbe driven by buoyancy orce in viscous iquid is one o such typica muti-uid systems. A comprehensive understanding o the ow behaviour and mechanism o such muti-uid systems in u ow regimes has not been we deveoped so ar. In the present work, ront tracking method, a hybrid approach o ront capturing and tracking technique proposed by Unverdi and Tryggvason [1] or WIT Transactions on Engineering Sciences, Vo 52, ISSN (on-ine) doi: /afm06009

2 80 Advances in Fuid Mechanics VI modeing muti-uid system was examined. In this method, a stationary, ixed grid is used or the uid ow, and a set o adaptive eements on the ront is used to represent the interace. A singe set o governing equations (Navier-Stokes equation and continuity equation) or the whoe computationa domain is soved or the two phases by treating the dierent phases as one uid with variabe materia properties. The uid properties such as density and viscosity are cacuated based on the position o the interace, and updated basing on the motion o the bubbe ront. Hence, this method coud avoid numerica diusion, and capture sharp interaces. Interacia source terms such as surace tension are computed on the ront and transerred to the ixed background grid using a δ (dirac-deta) distribution unction rom the interace between the phases. The ront tracking method has been appied to various interacia ow probems [1]. It has been noted that the capabiity o conventiona agorithm is imited to ower density ratios, ower Reynods numbers and ower Bond numbers [2]. A urther extension o this method is proposed in this paper to make it appicabe to wider ow regimes. As benchmarking tests, the new agorithm is used to mode singe air bubbe rising in viscous iquid, which actuay has been treated as typica cases to vaidate numerica methods or mutiphase/interacia ows. Most o previous vaidation works are perormed under conditions o ower density ratio, the simuation resuts are vaidated against experimenta observations about bubbe shape under some typica ow regimes [3, 4]. In this paper, the ront tracking method is used to systematicay investigate air bubbe rising in water soutions under various ow regimes. The simuation resuts are compared with experimenta resuts in bubbe shape, termina veocity and wake ow pattern, and good agreement can be obtained in dierent ow regimes. 2 Mathematica ormuation and numerica method 2.1 Governing equations In this study, we investigate the rise o a bubbe in a quiescent iquid. It is normay reasonabe to treat both iquid and bubbe phases as incompressibe uid. Hence, the mass conservation equation on the whoe domain (both uid phases and the interace) may be expressed as, u = 0 (1) The Navier-Stokes equation, governing the momentum baance in each uid domain and the interace, may be expressed as, ( ρu) + ρuu = p + [ µ ( u + t T u)] + σ κ n δ ( x x ) + ( ρ ρ ) g where, p is the pressure in the uid domain, σ is the surace tension, κ is the curvature o the interace. δ ( x x ) is a deta unction that is zero everywhere except at the interace, i.e., at x = x. g is the gravitationa acceeration, and (2) WIT Transactions on Engineering Sciences, Vo 52, ISSN (on-ine)

3 Advances in Fuid Mechanics VI 81 subscript reers the ront / interace. ρ reers the density o uid, and density o iquid phase. We introduce the oowing dimensioness characteristic variabes, x x = D u (gd) ; u = ; 1/ 2 t g τ = ; 1/ 2 1/ 2 D ρ ρ = ; ρ p p ; = ρ gd µ ; µ µ = ρ the κ =. D κ 1 1/ 3 where D is the eective diameter o a bubbe and deined as D = ( 6V b / π ) and V is the bubbe voume. The subscripts and b stand or the iquid and gas b bubbe phases, respectivey. So, the non-dimensiona Navier-Stokes equation may be re-expressed as, ( ρ u) + ρ u u = p + t 1 Re T [ µ ( u + u)] + 1 Bo κ n δ ( x x ) + ( ρ 1) g, (3) in which the superscript is omitted or convenience, whie the non-dimensiona Reynods number, Bond number (aso known as Eotvos number) and Morton number are deined as oowing, ρ g D 1/ 2 3/ 2 Re = ; µ Bo 2 ρ gd = ; σ In the experimenta work [5], the oowing dimensioness parameters, Morton (M) and Reynods numbers (Re), are aso used to characterise the uid ow, ρ DU Re = ; µ 4 3 gµ Bo M = = ρ σ Re Treatment o the discontinuities across the ront The novety o the ront tacking method proposed by Unverdi and Tryggvason [1] is that the ront is considered to have a inite thickness o the order o the mesh size instead o zero thickness. In the transition zone around the interace, the uid properties change smoothy and continuousy rom the vaue on one side o the interace to the vaue on the other side. The materia property ieds over whoe domain may be reconstructed using an indicator unction I( x, t), which has the vaue o one in the bubbe gas phase and zero in the iquid phase at a given timet. b( x, t) = b + ( b b ) I ( x, t) ; I( x, t) = δ ( x x ) dv (4) Ω b in which b stands or either uid density or viscosity. The indicator unction can be written in the orm o an integra over the whoe domain Ω(t) bounded by the phase interace Γ (t). δ ( x x ) is a deta unction that has a vaue o one where x = x and zero everywhere ese. In this study, the deta unction is ( t ) WIT Transactions on Engineering Sciences, Vo 52, ISSN (on-ine)

4 82 Advances in Fuid Mechanics VI approximated by the oowing distribution unction D(x) suggested by Peskin [6] are used or a two-dimensiona grid system, D( x x 2 (4h) ) = 0, 2 i= 1 π 1+ cos x x 2h, i x x < 2h otherwise. (5) where h is the background mesh grid size. The distribution unction deines the raction o the interace quantity can be distributed to nearby grid point across the artiicia thickness o the ront. 2.3 Numerica method Projection method or the integration o the Navier-Stokes equations (3) was used in the previous works o Unverdi and Tryggvason [1]. The diicuties in soving the above pressure equation have been reported. For exampe, a arge density ratio may ead to a probem in convergence [1, 2]. In order to overcome the diicuties in soving the pressure equation, an aternative approach is impemented in the present work. Here, the couping uid veocity and pressure is updated by soving the momentum equations and continuity equation using SIMPLE scheme [7], and simuation process is more robust even in case o arge density ratio because o the semi-impicit soving approach. In the muti-uid system, due to density jump over the phase interace, mass ux conservation in the contro voume crossing the ront interace is not vaid. Hence, voume ux conservation is adopted to modiy the SIMPLE agorithm. Based on this approach, SIMPLE agorithm is used to cacuate the correction vaue o pressure and veocity ater soving the momentum equation. 2.4 Front tracking Since the uid veocity is updated on the ixed grid, the node moving veocity on the ront shoud be computed by interpoating rom the ixed grid to ensure that the ront moves at the same veocity as the surrounding uids. In section 2.2, the distribution unction used to spread the uid property jump to the ixed points nearby the interace was discussed. Simiary, this unction can aso be used to interpoate ied variabes rom the ixed background grid to the ront using the oowing equation, u = D ( x x) u( x) (6) Then, the ront is advected aong its norma direction in a Lagrangian ashion, x n+ 1 x n = t u n (7) Ater the position o the ront is updated, the ront eements shoud be adapted to maintain the eement quaity. WIT Transactions on Engineering Sciences, Vo 52, ISSN (on-ine)

5 Advances in Fuid Mechanics VI Soution procedure With appropriate initia conditions or the uid ow and interace shape, the soution agorithm proceeds iterativey through the oowing steps: n n (1) Using the uid veocity ied ( u ) and the interace position ( x ), the n moving veocity o the ront marker points ( u ) is computed using equation (6). (2) Using the estimated norma interace veocity, the ront is advected to the n+1 new position ( x ). Subsequenty, the eements, representing the ront, are examined or adaptation and topoogy change. (3) At the new interace positions, the redistribution o the interace property is n+1 perormed with the reconstructed indicator unction I( x ). Hence, new n+ 1 n+ 1 uid property ied such as density ( ρ ), viscosity ( µ ), as we as the n+1 surace tension ( F ), are obtained st (4) With appropriate wa boundary conditions, the momentum equation and mass continuity equation can be soved using the modiied SIMPLE n+1 n+ 1 agorithm. This eads to update uid veocity ( u ) and pressure ( p ). (5) Repeat the soution steps rom (1) to (4) or the next time step cacuation. 3 Resuts and discussion 3.1 Experimenta observation o termina bubbe shape and rising veocity Experimenta studies on the rise o singe bubbe in quiescent iquid have been reported in the iterature [5, 8]. The termina bubbe shapes vary greaty in dierent ow regimes as a unction o the non-dimensiona parameters such as Eotvos number (Bond number), Reynods number and Morton number. The termina shapes o singe bubbe rising in quiescent iquid under a range o Reynods and Bound numbers ( Re < 200, Bo < 200) were shown in Figure 1. In genera, the rising bubbes have axisymmetric shapes when the Reynods and Bond numbers are not too higher ( Re < 200, Bo < 200). Hence, the singe bubbe rising in quiescent iquid under such ow regimes, which produce axisymmetric bubbes, coud be simuated using the ront tracking method in an axisymmetric co-ordinate system. 3.2 The eect o Reynods and Bond numbers on the bubbe shape The simuation predicted bubbe shapes in a wide range o Reynods and Bond numbers are summarised in Figure 2. In the regimes o ow Reynods and Bond numbers ( Re < 1and Bo < 5 ), the bubbes remain spherica whie they are rising in the iquid. With a sight increase in Reynods number ( Re = 10 ), the bubbe shape sti remains spherica or ow Bond number, where surace tension is higher. On the other end o high Bond numbers, the bubbe bottom becomes at or sighty dimped. With the urther increases in Reynods number ( 10 < Re < 20 ) and Bond number ( 5 < Bo < 20 ), bubbes in the shapes o WIT Transactions on Engineering Sciences, Vo 52, ISSN (on-ine)

6 84 Advances in Fuid Mechanics VI eiptic/obate eipsoid are observed. With urther increase in Reynods number ( 100 < Re < 200 ) and Bond number ( 20 < Bo < 100 ), the bubbe shapes range rom highy deormed eiptica-cap at ower Reynods number regime to the spherica-cap bubbe at higher Reynods numbers ( 50 < Re < 100 ). As the bond number increases urther ( 100 < Bo < 200 ), skirt bubbes are ormed at higher Reynods numbers ( 20 < Re < 100 ). When the Bond number is increased to the range o 100 < Bo < 200 and Reynods number to the range100 < Re < 200, toroida bubbes are observed. From the simuation resuts, presented here, it can be concuded the current modeing method is robust enough to reasonaby predict the various bubbe shapes under wider ow regimes. Figure 1: Bubbe shapes under dierence ow regimes (extracted rom [8]). Parameters Bo Re Figure 2: The predicted bubbe shapes under various Reynods and Bond numbers. WIT Transactions on Engineering Sciences, Vo 52, ISSN (on-ine)

7 Advances in Fuid Mechanics VI 85 Test Case TS1 TS2 TS3 TS4 Test condition E = 8.67 M = 711 Re= E = 32.2 M = Re= 55.3 E = 243 M = 266 Re= 7.77 E = 115 M = Re= 94.0 Experiments Observed bubbe termina shape Predicted bubbe Termina Shape Simuations Modeing conditions Bo = 8.67 Re= U = Re,c= Bo = 32.2 Re= U = Re,c=52.96 Bo = 243 Re= U = Re,c=8.39 Bo = 115 Re= U = Re,c=88.70 TS5 E=641 M=43.1 Re=30.3 Bo = 641 Re= U = Re,c=29.93 Figure 3: Comparison o termina bubbe shapes and rising speeds predicted by simuation and observed experiments. 3.3 Comparison o termina bubbe shapes Detai comparison o the bubbe shapes and termina veocity is aso necessary to understand the accuracy o the numerica predictions. Figure 3 compares the termina bubbe shapes and rising speeds obtained rom experiments and simuations under various ow conditions. Typica bubbe shapes (spherica, eipsoid, eipsoid-cap, spherica-cap and skirt bubbes) ormed in dierent regimes can be predicted accordingy through simuations. The simiarity o bubbe shapes predicted in the simuations and observed in experiments is quite reasonabe. For exampe, the simuation resuts on predicted bubbe shape presented in Figure 3 shows that the bubbe base is dimpes or indented at the intermediate Reynods number ( 10 < Re < 100 ) and reativey high Bond number ( 20 < Bo < 50 ). The indentation is aso ceary visibe in bubbe photographs shown in cases TS3 and TS4, where the upper indentation may be seen near the axis o bubbes. At the rim o bubbe, the dierent reractive indices o the gas and the iquid prevent us rom seeing how the indentation joins the outer surace o the bubbe. I taken this eect into consideration, the simuation predicted termina bubbe shapes agree we with experimenta observation or the most study cases. WIT Transactions on Engineering Sciences, Vo 52, ISSN (on-ine)

8 86 Advances in Fuid Mechanics VI The termina bubbe veocity is another important indicator to quantitativey evauate the dierence between the experimenta and simuations. Based on the parameters given in the experiments, we can derive out the dimensioness parameters ( Re, Bo, ρ / ρ, b µ µ ) to perorm simuations. The simuation can b predict the termina bubbe rising veocity ( U ). Actuay, based on this dimensioness veocity, the dimensiona termina bubbe veocity U can be 1/ 2 cacuated asu gd, and the dimensiona termina bubbe veocity based Reynods number ( Re, c ) predicted in simuation can be cacuated as Re, c = Re U. The comparison o the Reynods numbers or the experiment (Re) and simuation cases ( Re, c ) is aso shown in Figure 3. The resuts rom simuation prediction agree with those o experiment with very we within 10% dierence. Test Case TW1 Test conditions E=96.2 M=0.962 Re=18.2 Experiments Observed termina bubbe wake Predicted termina bubbe wake Simuations Modeing conditions Bo=96.2 Re=31.0 TW2 E=94.3 M= Re=77.9 Bo=94.3 Re=116.3 TW3 E=114 M= Re=91.6 Bo=114 Re= Re,c= TW4 E=292 M=26.7 Re=22.1 Bo=292 Re=31.07 Figure 4: Comparison o bubbe wake predicted by simuation and observed in experiments. 3.4 Comparison o bubbe wake ow patterns The existence o a cosed toroida wake has been observed in experiments [5] through the ow visuaisation with the H 2 tracers. Wake ow circuation patterns predicted by the simuation agree we with the observations in experimenta as shown in Figure 4. The wake circuation within the bubbe base WIT Transactions on Engineering Sciences, Vo 52, ISSN (on-ine)

9 Advances in Fuid Mechanics VI 87 indentation is ceary shown in the case TW1 by the photograph, where the trace track disappears behind the rim o the bubbe. The simiar wake circuation pattern is reveaed in the simuation. Actuay, the simuation show that a second wake circuation occurs just behind the bubbe rim. This may be the reason that the bubbe photo o TW1 has some bright spots at outside o bubbe rim. The second circuation in the skirt bubbe wake becomes much cear rom the simuation. The photo or the skirt bubbe wake (TW4) does have much arge bright spot just underneath the bubbe. As the bubbe size increase (Reynods number increases), the wake voume increase as we, the wake seems to be torn away rom the bubbe (TW2 and TW3). In this case, the second bubbe wake circuation disappears, and the bubbe base indentation becomes smaer. Compared to the experimenta studies [5] on the rising bubbe shapes in iquid within dierent ow regimes, the bubbe shapes predicted using the present modeing approach are in reasonabe agreement. Most o the previous numerica studies on bubbe rising in iquid are imited to certain regimes where the bubbe shape deormation is minima. For exampe, in the work o [1], the bubbe rise was simuated in the regimes o ow Reynods or ow Bond numbers. The present work extends the capabiity to simuate the bubbe rise and deormation or a wider ow regime. 4 Concusions A ront tracking method or modeing two-phase uid systems has been examined, improved and vaidated or much wider ow regimes. The new agorithm adopted the treatment o the interace as inite thickness as proposed by Tryggvason et a. The uid properties (density, viscosity and surace tension) were varied smoothy over the interace, and updated with the new interace position. The interace is advected using the ront marker veocities that are interpoated rom the veocity ied in the ixed grid. The ront mesh size is adapted to match the background mesh size due to it variation in ront moving and deorming, and the ront mesh position is corrected to conserve the ront inner voume. The veocity ied has been soved impicity with the inite voume method over the ixed grid using the improved SIMPLE agorithm, which keeps the conservation o voume ux. The newy proposed agorithm is appied to simuate the rise o singe bubbe in a viscous iquid. The bubbe shapes and veocity in a wider ow regime are studied as a unction o the non dimensiona parameters such as Reynods number, Bond number, density ratio and viscosity ratio. The comparison o simuations with the avaiabe experiments shows satisactory agreements in termina bubbe shape, veocity and wake ow pattern. Reerences [1] Unverdi, S. O. & Tryggvason, G. A ront-tracking method or viscous, incompressibe, muti-uid ows. J. Comput. Phys. 100, 25-37, WIT Transactions on Engineering Sciences, Vo 52, ISSN (on-ine)

10 88 Advances in Fuid Mechanics VI [2] Bunner, B. & Tryggvason, G. Dynamics o homogenous bubby ows. Part 1. Rise veocity and microstructure o the bubbes. J. Fuid Mech. 466, 17-52, [3] Chen, L., Garimea, S. V., Reizes, J. A. & Leonardi, E., The deveopment o a bubbe rising in a viscous iquid. J. o Fuid Mech. 387, 61-96, [4] Ohta, M., Imura, T., Yoshida, Y. & Sussman M. A computationa study o the eect o initia bubbe conditions on the motion o a gas bubbe rising in viscous iquids. Int. J. o Mutiphase ow 31, , [5] Bhaga, D. & Weber, M. E. Bubbes in viscous iquid: shapes, wakes and veocities. J. Fuid Mech. 105, 61-85, [6] Peskin, C.S. & Printz, B.F. Improved voume conservation in the computation o ows with immersed boundaries, J. Comput. Phys. 105, 33-46, [7] Patankar, S.V. Numerica heat transer and uid ow. Hemisphere, [8] Cit, R., Grace, J. R. & Weber, M. E. Bubbes, Drops, and Partices. Academic Press, WIT Transactions on Engineering Sciences, Vo 52, ISSN (on-ine)