Interaction of a pair of horizontally aligned bubbles in gravity field

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1 Interacton of a par of horzontally algned bubbles n gravty feld Han JIAO 1 ; Dongyan SHI 2 ; Zhka Wang 3 ; Hongqun LI 4 1 Harbn Engneerng Unversty, Chna 2 Harbn Engneerng Unversty, Chna 3 Harbn Engneerng Unversty, Chna 4 Harbn Engneerng Unversty, Chna ABSTRACT Both large deformaton and strong dscontnuty problems exst n the process of bubble-bubble nteracton. The lattce Boltzmann method whch s based on the mcroscopc level has naturally comparatve advantage to descrbe tny changes n multphase flow system. A three-dmensonal lattce Boltzmann (LB) model of two horzontally algned bubbles s presented to explore the bubble movement, topologcal transformaton and bubbles nteracton characterstcs. In ths model, the vscous effect s easly consdered, and surface tenson effect s realzed by ntroducng a dscrete LB body force term. To provde a reference, the solated bubble dynamcs drven by gravty s smulated frstly, and the surface tenson effect s studed. Then the effect of gap dstance and nterface wdth on the coalescence of two stable bubbles s dscussed. Fnally, the nteracton of two horzontal bubbles wth dfferent sze rato n gravty feld s explored carefully. The results not only show the detaled couplng and coalescence process of two rsng bubbles, but expose that the relatve sze rato plays an mportant role n the nteracton. The opposte phenomenon can be seen. When bubbles are of dfferent sze, the coalescence phenomenon wll happen, but when they are n the same sze, the rejecton phenomenon wll replace t. Keywords: Horzontal bubbles, Interacton, Lattce Boltzmann method I-INCE Classfcaton of Subjects Number(s): INTRODUCTION Bubble group plays an mportant role n the hydrodynamc characterstcs of the surroundng flud flow, especally affects some sophstcated underwater equpment, such as propeller. Ther collectve collapse phenomenon s one of the man sources of structure vbraton and underwater nose (1-3). Thus, t s qute necessary to study the nteracton mechansm of bubbles. At present, the studes of sngle-bubble movement n gravty feld, ncludng shapes and floatng velocty, already have relatvely adequate outcomes (4-6). By contrast, cases about mult-bubble moton characterstcs whch are closer to the actual engneerng are relatvely rare. When there s an another bubble n the vcnty of a bubble, nteractons can cause remarkable changes of the flow feld propertes and the movement of bubbles, ncludng bubble breakup and fuson (7-9). In 1996, Katz et al. (1) found that bubbles n a ppe wll produce collsons under the acton of the vortex at the Reynolds number between.2 and 35. In 1999, Krshna et al. (11) consdered the couplng effect between small and large bubbles. In 29, Sanada et al. (12) observed the nteracton between a par of horzontally bubbles by usng the hgh-speed photography, and found the fuson and reboundng phenomena. In 212, Zhang A M. et al. (13) studed the fuson of two horzontal bubbles near the free surface. Compared wth expermental research, numercal smulaton owns advantages of controllable envronment, accurate settng parameter and observable results. In 23, the dstrbuton of velocty and vortex around the bubble surface were successfully obtaned by Legendre et al. wth the body-ftted grd 1 jaohan19923@gmal.com 2 shdongyan@hrbeu.edu.cn 3 zhka.wa@gmal.com 4 lhongqun@hrbeu.edu.cn Inter-nose 214 Page 1 of 8

2 Page 2 of 8 Inter-nose 214 technology. In 21, Rabha and Buwa et al. (14) dscussed the nteracton of bubbles n shear flow by usng the VOF method. Lattce Boltzmann Method (LBM) s another relatvely new developed numercal method. Compared wth tradtonal numercal methods, ths method, whch s of the mcroscopc propertes and ease of programmng, can acheve the second-order accuracy requrement n smulatng sngle-phase flows (15, 16), and multphase flows (9,17). At present, ths method manly ncludes the color model, the potental feld model and the free energy model. Dfferent from the other two models, the free energy model can ensure the thermodynamc consstency (18). In ths paper, the horzontal two bubble nteracton model n free feld s establshed based on ths model. The process to verfy the two bubble nteracton model s dvded nto two parts: an solated bubble n gravty feld and two bubble nteracton crossly affected by the bubble wall thckness and the gap dstance (18) unconsderng gravty. Fnally, the couplng characterstcs of a par of bubbles lad sde by sde were explored on the terms of the relatve poston and the relatve sze, respectvely. The mplemented of lattce Boltzmann method n smulatng bubble dynamcs s brefly descrbed n Secton 2,whle the smulaton results are presented n Secton 3. Fnally, the concluson s provded n Secton NUMERICAL METHOD 2.1 Governng Equaton The governng equatons to descrbe bubble dynamcs n non-newtonan vscous flow feld can be wrtten as (4, 19), ρ/ t + ( ρu) = 2 ( ρu)/ t + ( ρ uu) = P + µ u+f. (1) 2 / t + ( u) = ϒ M µ Where, the frst two equatons are the Naver-Stokes equatons and the thrd one s the Cahn-Hllard equaton. ρ, u and P are densty, velocty and pressure. Intally, to descrbe two-phase partcles movement unformly, ρ s defned as ρ = ( ρh + ρl)/2, where ρ H and ρldenote the densty of lqud and gas respectvely. F s the body force. denotes an order parameter to capture the two-phase nterface, = ( ρh ρl)/2. ϒ M s the moblty rate. µ s defned for calculatng the surface tenson (18) µ = 12 σ( )/ W 3 σw/2. (2) Whereσ s the surface tenson coeffcent, W s the thckness of the bubble wall. 2.2 Lattce Boltzmann Method To recover the Naver-Stokes equatons and the Cahn-Hllard equaton, two groups of lattce Boltzmann equatons are constructed (4, 2), ( ) f(, ) (, ) [ eq r + cδt t + δt f rt = f ( r, t) f( r, t)]/ τ ρ. (3) ( eq) [ ] g( r + cδt, t + δt) g( r, t) = (1 q) g( r + cδt, t) g( r, t) + [ g ( r, t) g( r, t)]/ τ. (4) f( r, t) and g( r, t) are number densty of the partcle dstrbuton at a local poston n the certan tme. By mult-scale expanson, Eq. (2) can obtan the Naver-Stokes equatons whle Eq. (3) can get the 1 Cahn-Hllard equaton wth second-order accuracy. q s the control coeffcent, q =. c τ +.5 denotes dscrete veloctes dependng on the lattce structure. To mprove the computatonal effcency, D3Q19 and D3Q7 models are separately adopted n Eq. (3) and (4). c = (,,) dx/ dt, = c = ( ± 1,,) dx/ dt,(, ± 1,) dx/ dt,(,, ± 1) dx/ dt, = 1~6. (5) c = ( ± 1, ± 1,) dx/ dt,( ± 1,, ± 1) dx/ dt,(, ± 1, ± 1) dx/ dt, = 7~18 The dstrbuton of nneteen dscrete veloctes n D3Q19 s shown n Eq. (5), and the dscrete velocty dstrbuton n D3Q7 s just the frst seven dscrete veloctes n Eq. (5). τ ρ andτ are the relaxaton coeffcents, and τ ρ s assocated wth µ. cs s lattce sound velocty, c = 1/ 3. s Page 2 of 8 Inter-nose 214

3 Inter-nose 214 Page 3 of 8 ( eq) ( eq) The equlbrum densty functons, f, g, come from the Max-well dstrbuton (9), [ (1 3 /2) 6 ], ( eq) w ρ u µ = f =, (6) 2 2 w[ ρ (1 + 3c u + 9( cu ) /2 3 u /2) + 3 µ ], = 1~18 2 ( eq) 3( Γ µ + cs ), = g =. (7) [2 Γ µ + (2τ + 1)( c u) ]/4, = 1~6 where w s the weght coeffcent n the each dscrete drecton. w = 4/9, w 1~6 = 1/9, w 7~18 = 1/36. Γ s used to control the moblty rato and s related to the parameter ϒ n Eq. (1)(18) ϒ = (2τ 1)/(2τ + 1) Γ. (8) In order to accurately ntroduce the body force F n Eq. (1) as a lattce form, the dscrete scheme proposed by Guo Z. L. et al. (21) s adopted n ths paper, F = 3 w(1 1/2 τ ρ)[( c u) + 3( c uc ) ] F. (9) Meanwhle, combnng wth the sub-step operaton of Lee T. et al. (22), va ntroducng an ntermedate md varableg, the tradtonal sngle-step collson operaton s transformed nto two steps, makng the model more flexblty. The expresson s as follows, md ( eq) g ( r, t) = g( r, t) + ( g ( r, t) g( r, t))/ τ. (1) ( eq) md g( r + cδt, t + δt) = (1 1/( τ +.5)) g ( r + cδt, t) + 1/( τ +.5) g ( r, t) The macroscopc flud parameters, lke the densty ρ and the velocty u and the order parameter, can be calculated from 18 ρ = f = 18 u = [ fc + ( µ + F )/2]/ ρ. (11) = 6 = g = 3. RESULTS AND DISCUSSION 3.1 Isolated Rsng Bubble To explore the bubble behavor and the nteracton characterstcs of two horzontally arranged bubbles n gravty feld, the solated bubble was frstly smulated to confrm the ablty of the model to capture the multphase nterface. The flud propertes can be dstrcted by the latter three parameters (9), ( ρh ρl) g 4 ( ρh ρl) g 2 ρud Mo = µ,,re 3 2 H Eo = D =. (12) σ ρ σ 2 H Here, g stands for the gravty, µ H s the vscosty parameter of the flud component, D s the dameter of the bubble, and the other parameters are the same wth those defned before. Inter-nose 214 Page 3 of 8

4 Page 4 of 8 Inter-nose 214 (a) (b) (c) (d) (e) Fgure 1 The rsng process of a bubble wth dfferent surface tenson coeffcent n gravty feld. (a) σ =.1, (b) σ =.2, (c) σ =.5, (d) σ = 1., (e) σ = 2. In the smulaton, ρ H = 1, ρ L = 1, τ ρ =.6, τ δ =.7, Γ = 4, D = 2R = 4. Mo and Eo are changed accordng to the modfed surface tenson coeffcent σ. Fgure 1 shows the rsng process and the termnal shape wth dfferent surface tenson coeffcent. The smulaton results can verfy the newly bult model n the paper n two aspects: (1) the large deformaton of the multphase surface; (2) the termnal shape agrees well wth the observaton by Bhaga D. (5), and the trend of termnal shape changng along the value of Mo s same wth that n the lterature (23) where the termnal shape becomes more platter wth smaller Mo. 3.2 Two Horzontally Steady Bubbles To dentfy the model n dealng wth mult-bubble nteracton, the crossng effect of bubble wall wdth and gap dstance on the nteracton characterstcs s studed by smulatng two bubble n statc flow. For a comparatve analyss,the parameters are valued accordng to the lterature (18). The two bubbles are wth the same radus R = 2 δ l, and the gap dstance, d = 5. δ l. The parameters are chosen as, σ = 1., τ =.6, τ =.7, Γ = 4, and the bubble wall wdth, W s modfed. The perodc ρ δ boundary condton s employed at all boundares. (a) (b) (c) (d) (e) Fgure 2 Results of two statonary bubbles (W = 2.4δ l ).(a) t = ; (b) t = 2.21; (c) t = 2.37; (d) t = 2.69; (e) t = 3. (a) (b) (c) (d) (e) Fgure 3 Results of two statonary bubbles (W = 2.6 l δ ). (a) t = ; (b) t = 2.21; (c) t = 2.37; (d) t = 2.69; (e) t = 3. At frst, W s set as 2.4. That s, the gap dstance s greater than 2W. Numercal results are shown n Fgure 2. It can be easly observed that the two bubbles do not merge or even have the trend. Then, W s set as 2.6, whch means that the gap dstance s less than 2W. Other parameters reman unchanged. Numercal results are shown n Fg. 3. The coalescence process can be easly observed. The results are consstent wth Page 4 of 8 Inter-nose 214

5 Inter-nose 214 Page 5 of 8 the concluson that the coalescence phenomenon only happens when the gap dstance s less than 2W, whch s frst drew by Zheng H. W. (18) by usng a two dmensonal lattce Boltzmann model. 3.3 Two Horzontally Rsng Bubbles Keepng the gap dstance and the bubble wall wdth stable, the coalescence processes wth the radus rato of 1:1, 4:3, 2:1 are studed respectvely. The perodc boundares are used for all boundares, and to remove the nfluence of the boundares, the ntal dstance between the boundary and the bubble center s set as four tmes as the bubble radus. The other parameters are set as Eo=39.96,σ=1., τ ρ=.6, τ φ=.7, Γ=4 and W=2.6δ l. (a) (b) (c) (d) (e) (f) Fgure 4 Movement trajectory of two horzontal bubbles wth the same sze. (a) t =.79; (b) t = 1.58; (c) t = 2.21; (d) t = 3.79; (e) t = 4.74; (f) t = The movement trajectory of two horzontal bubbles wth the same sze, R = 2, s shown n Fgure 4. And the pressure pattern and velocty vectors are shown n t. The repulson phenomenon can be clearly found, but the attracton appears at the ntal stage. As the bubbles rse, the surround fluds begn to move and format two vortex rngs. The parts of the two vortex rngs n the gap seem to be canceled, and repulsve force n the horzontal drecton appears. The gap dstance becomes bgger and bgger, untl the two bubbles can merely nfluence each other. And each bubble expresses as the dynamcs of an solated bubble rsng. (a) (b) (c) (d) (e) (f) (g) (h) () Fgure 5 Coalescence process of two horzontal bubbles wth R 1 :R 2 = 2:1. (a) t =.79; (b) t = 1.58; (c) t = 2.37; (d) t = 4.74; (e) t = 7.12; (f) t = 7.91; (g) t = 8.7; (h) t = 1.28; () t = Fgure 5 shows the coalescence process of two horzontal bubbles wth R 1 :R 2 = 2:1. After the bubbles are released, the bgger bubble goes faster, and the gap dstance ncreases at frst. As the vortex appears, the smaller bubble movement s nfluenced by both gravty and the vortex. When t Inter-nose 214 Page 5 of 8

6 Page 6 of 8 Inter-nose 214 comes nto the wake of the bgger bubble, the gap dstance decreases quckly. At about t = 7.12, a slght bulge on the button surface of the bgger bubble appears duo to the approach of the smaller bubble. The wave moton of the button surface ncludng the surface breakng and rebult s shown n Fgure 5(f)-(g). After the wave moton comes smooth under the effect of surface tenson, the merged bubble keeps rsng Case1: Smaller bubble(r = 1) Case1: Bgger bubble(r = 2) Case2: Smaller bubble(r = 15) Case2: Bgger bubble(r=2) Case3: Two bubbles wth same sze v t Fgure 6 Comparson of the rsng velocty of the two horzontal bubbles wth dfferent relatve sze n the coalescence process. The rsng veloctes of the two horzontal bubbles wth dfferent relatve sze n the coalescence process are shown n Fgure 6. Accordng to the dfferent radus rato, the workng condtons are classfed nto three cases: case 1 (R 1 :R 2 = 2:1), case 2 (R 1 :R 2 = 4:3), and case 3 (R 1 :R 2 = 1:1). t s the dmensonless tme, and v s the vertcal velocty. All curves show that the velocty ncreases at a faster rate under the effect of gravty n the begnnng. Subsequently, the vortex appears, and the velocty of the smaller bubble s suppressed, but that of the bgger bubble s merely nfluenced. For the bubbles wth the same sze, ther curves completely concde. It can be found that the varaton of velocty matches well wth the phenomenon of deformaton shown n Fgures 4-5. The results also show that the relatve sze plays an mportant role n the coalescence process of two horzontal bubbles. The larger the sze dfference s, the larger the effects of the bgger bubble on the smaller one wll be. However, the bgger bubble s barely nfluenced by the smaller one. And the completely adverse phenomena of coalescence and repulson appear when two horzontal bubbles have the same sze. To dentfy the dscovery mentoned above and test the ntal gap dstance effectng on t, more cases are smulated wth the workng condtons of A-B-C where A, B and C stand for the frst bubble radus, the second bubble radus and the ntal gap dstance respectvely. Extraneous three cases are carred out for each radus rato condton. The more detaled nformaton about the ntal condton s lsted n table 1. Table 1 Informaton about the key velocty and tme ponts Cases (A-B-C) t 1 t 2 t 3 t 4 v 1 v 2 v Page 6 of 8 Inter-nose 214

7 Inter-nose 214 Page 7 of Note: t 1 and t 2 stand for the tme pont that the bgger and the smaller bubbles frstly get the peak velocty, and t 3 s the tme that the veloctes of the two bubbles frstly get equalty. t 4 s the tme that the bgger bubble frstly gets the valley velocty. v 1 and v 2 stand for the frst peak velocty of the bgger and the smaller bubbles respectvely. v 3 stands for the frst valley velocty of the bgger bubble. Table 1 also shows the nformaton about the key velocty and tme ponts. As shown n the second, sxth and eghth columns, the values do not fluctuate much n dfferent workng condtons, whch means that the gap dstance and the smaller bubble have lttle effect on the bgger bubble. Comparng wth t 2, t 3 and t 4, the results that the relatve sze has more obvous effect than the gap dstance can be ndcated. By decreasng the gap dstance and ncreasng the radus rato, the coalescence process can be accelerated. Combnng the thrd and the seventh columns, t can be clearly seen that as the sze dfference decreases, the tme wasted by the smaller bubble to reach v 2 and the value of v 2 ncreases. At the same tme, keepng the relatve sze stable, by ncreasng the ntal gap dstance can also ncrease the value of v 2. It can be concluded that both the ntal gap dstance and the relatve sze play an mportant role n the coalescence process, but the latter one may be more obvous. Wth the radus rato and the gap dstance ncreasng, the bgger bubble wll be less nfluenced whle the small one s nfluenced serously. 1.6 x Start touchng tme Steps Cases Fgure 7 Tme of two horzontal bubbles begnnng to touch at dfferent ntal condtons (A-B-C) A s the radus of the bgger bubble, B s the radus of the smaller bubble, C s the gap dstance between the two bubbles. Fgure 7 shows the effects of the relatve sze couplng wth the gap dstance on the bubble mergng tme. It can be found that the whole tendency of the curve s plotted as the shape of an nverse letter 'Z'. It ndcates that the dfference of ther startng touchng tme s almost proportonal to the dfference n the ntal gap dstance. However, the relatve sze has a greater effect on t. 4. CONCLUSION Based on the lattce Boltzmann method, a three-dmensonal bubble-bubble nteracton model consderng the surface tenson and the gravty s bult. By usng ths model, the nteracton of two gravty-drven bubbles horzontally algned n the vscous flow feld s explored. Frstly, two general cases that an solated bubble rsng n gravty feld and two stable bubbles nteracton unconsderng the gravty are studed to verfy the ablty of the model n smulatng the multphase nterface movement. Then, the effect of radus rato on the rsng process of two horzontally algned bubbles s researched. Two contrary phenomena, repulson and coeffcent are observed. When the two bubbles are the same sze, the repulson appears. But, the coalescence phenomenon wll take the place of t when the sze s dfferent. And wth the same gap dstance, the more the bubbles dffer from each other n sze, the more easly the two bubbles wll merge together. Fnally, the effects of the gap dstance couplng relatve sze are researched. It s found that both radus rato and gas dstance play an mportant role n the nteracton. Compared wth the radus rato, the effect of the radus rato seems weaker. Decreasng the gap dstance can do help for the two bubbles Inter-nose 214 Page 7 of 8

8 Page 8 of 8 Inter-nose 214 coalescence. Durng the nteracton process, the bgger bubble s merely nfluenced whle the smaller bubble s nfluenced serously. ACKNOWLEDGEMENTS Ths paper s funded by the Internatonal Exchange Program of Harbn Engneerng Unversty for Innovaton-orented Talents Cultvaton. And the authors are grateful to the Program of the Scentsts Fund for Outstandng Young Scholars of Chna (Grant No ) and the Program for New Century Excellent Talents n Unversty (Grant No. NCET154) for support. REFERENCES 1. Hsao CT, Chahne GL. Effect of a propeller and gas dffuson on bubble nucle dstrbuton n a lqud. J Hydrodyn. 212; 24(6): Zhu Z, Fang S. Numercal nvestgaton of cavtaton performance of shp propellers. J Hydrodyn. 212; 24(3): J B, Luo X, Peng X. Numercal analyss of cavtaton evoluton and excted pressure fluctuaton around a propeller n non-unform wake. Int J Multphas Flow. 212; 43: Amaya-Bower L, Lee T. Sngle bubble rsng dynamcs for moderate Reynolds number usng lattce Boltzmann method. Comput Fluds. 21; 39(7): Bhaga D, Weber ME. Bubbles n vscous lquds: shapes, wakes and veloctes. J Flud Mech. 1981; 15: Ohta M, Imura T, Yoshda Y. A computatonal study of the effect of ntal bubble condtons on the moton of a gas bubble rsng n vscous lquds. Int J Multphas Flow. 25; 31(2): Chen RH, Tan WX, Su GH. Numercal nvestgaton on coalescence of bubble pars rsng n a stagnant lqud. Chem Eng Sc. 211; 66(21): Yu Z, Yang H, Fan LS. Numercal smulaton of bubble nteractons usng an adaptve lattce Boltzmann method. Chem Eng Sc. 211; 66(14): Cheng M, Hua J, Lou J. Smulaton of bubble bubble nteracton usng a lattce Boltzmann method. Comput Fluds. 21; 39(2): Katz J, Meneveau C. Wake-nduced relatve moton of bubbles rsng n lne. Int J Multphas Flow. 1996; 22(2): Krshna R, Urseanu MI, Van Baten JM. Rse velocty of a swarm of large gas bubbles n lquds. Chem Eng Sc.1999; 54(2): Sanada T, Sato A, Shrota M. Moton and coalescence of a par of bubbles rsng sde by sde. Chem Eng Sc. 29; 64(11): Zhang AM, Yao XL. The nteracton between multple bubbles and the free surface. Chnese Phys B. 28; 17(3): Rabha SS, Buwa VV. Volume-of-flud (VOF) smulatons of rse of sngle/multple bubbles n sheared lquds. Chem Eng Sc. 21; 65(1): Åsén PO, Kress G, Rempfer D. Drect numercal smulatons of localzed dsturbances n ppe Poseulle flow. Comput Fluds. 21; 39(6): Hguera FJ, Succ S. Smulatng the flow around a crcular cylnder wth a lattce Boltzmann equaton. EPL (Europhyscs Lett). 1989; 8(6): Inamuro T, Ogata T, Tajma S. A lattce Boltzmann method for ncompressble two-phase flows wth large densty dfferences. J Comput Phys. 24; 198(2): Zheng HW, Shu C, Chew YT. A lattce Boltzmann model for multphase flows wth large densty rato. J Comput Phys. 26; 218(1): Shan X, Chen H. Lattce Boltzmann model for smulatng flows wth multple phases and components. Phys Rev E. 1993; 47(3): Qan Y H, d'humères D, Lallemand P. Lattce BGK models for Naver-Stokes equaton. EPL (Europhyscs Lett). 1992; 17(6): Guo Z, Zheng C, Sh B. Dscrete lattce effects on the forcng term n the lattce Boltzmann method. Phys Rev E. 22; 65(4): Lee T, Ln CL. A stable dscretzaton of the lattce Boltzmann equaton for smulaton of ncompressble two-phase flows at hgh densty rato. J Comput Phys. 25; 26(1): Anwar S. Lattce Boltzmann modelng of buoyant rse of sngle and multple bubbles. Comput Fluds. 213; 88: Page 8 of 8 Inter-nose 214