基于FTM方法的双气泡融合特性模拟
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摘要
采用界面追踪法(FTM)对气泡融合现象进行数值模拟,将模拟结果与文献结果进行对比,验证了计算模型的准确性。结果表明,同轴双气泡上升速度均高于单独气泡的上升速度,且融合后气泡与等直径单气泡上升速度相同。气泡间距较小时,跟随气泡的上升速度更高。引导气泡的厄特沃什数Eo=0.36～9,Eo较大时两气泡上升阶段时间较短,但接触阶段时间较长,接触阶段气泡间的液膜在压力作用下逐渐变薄,最终破裂,气泡融合。Eo(27)4.16时,气泡融合所需时间随Eo增加而增加;Eo(29)4.16时,气泡融合所需时间不再变化。莫顿数Mo=0.57,Eo=5.04～18.72时,存在特定的双气泡初始角度θc,当0°≤θ≤θc时,双气泡相互排斥;当θc≤θ≤90°时,双气泡融合,且θc随Eo增加而降低.
The front tracking method(FTM), which can track the maker points and capture the changes of the interface accurately was used to simulate the phenomenon of bubble coalescence. All governing equations were solved by using a second-order accurate project method, using centered-differences on a fixed, staggered grid and considering the effect of surface tension at the interface. The numerical simulations were compared with experimental and computational results from other literatures which modified the accuracy of calculation model. In this work, the rising process of coaxial bubbles and the processafter fusion were analyzed in detail, and the existence of specific initial angle qc and the relationship between qc and Eotvos number(Eo) were analyzed. It was found that the rising velocity of both bubbles were higher than single bubble, and the coalesced bubble had the equal velocity with equivalent diameter single bubble in the rising process of coaxial double bubbles. The trailing bubble had higher velocity with the shorter distance of bubbles. In the range of Eo of leading bubble was 0.36～9, the time of rising stage was shorter and the time of contact stage was longer when the Eo of leading bubble increased. During contact stage, the thickness of liquid film between bubbles decreased because of the effect of pressure. Liquid film broke and coalescence of bubbles happened in the coalescence moment. The required time of coalescence increased as the distance of bubbles or Eo increased. However, the required time was stable as Eo was larger than 4.16. When the Morton number(Mo) was 0.57 and the range of Eo of leading bubble was 5.04～18.72, it was found that there was a specific initial angle θc. The two bubbles repelled each other for 0°≤θ≤θc but merge for θc≤θ≤90°, and qc decreased with the increase of Eo.
The front tracking method(FTM), which can track the maker points and capture the changes of the interface accurately was used to simulate the phenomenon of bubble coalescence. All governing equations were solved by using a second-order accurate project method, using centered-differences on a fixed, staggered grid and considering the effect of surface tension at the interface. The numerical simulations were compared with experimental and computational results from other literatures which modified the accuracy of calculation model. In this work, the rising process of coaxial bubbles and the processafter fusion were analyzed in detail, and the existence of specific initial angle qc and the relationship between qc and Eotvos number(Eo) were analyzed. It was found that the rising velocity of both bubbles were higher than single bubble, and the coalesced bubble had the equal velocity with equivalent diameter single bubble in the rising process of coaxial double bubbles. The trailing bubble had higher velocity with the shorter distance of bubbles. In the range of Eo of leading bubble was 0.36～9, the time of rising stage was shorter and the time of contact stage was longer when the Eo of leading bubble increased. During contact stage, the thickness of liquid film between bubbles decreased because of the effect of pressure. Liquid film broke and coalescence of bubbles happened in the coalescence moment. The required time of coalescence increased as the distance of bubbles or Eo increased. However, the required time was stable as Eo was larger than 4.16. When the Morton number(Mo) was 0.57 and the range of Eo of leading bubble was 5.04～18.72, it was found that there was a specific initial angle θc. The two bubbles repelled each other for 0°≤θ≤θc but merge for θc≤θ≤90°, and qc decreased with the increase of Eo.
引文
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[15]Unverdi S O,Tryggvason G.A front-tracking method for viscous,incompressible,multi-fluid flows[J].Journal of Computational Physics,1992,100(1):25-37.
[16]Bhaga D,Weber M E.Bubbles in viscous liquids:shapes,wakes and velocities[J].Journal of Fluid Mechanics,2006,105:61-85.
[17]Hysing S,Turek S,Kuzmin D,et al.Quantitative benchmark computations of two-dimensional bubble dynamics[J].International Journal for Numerical Methods in Fluids,2010,60(11):1259-1288.
[2]Yang L,Wang K,Tan J,et al.Experimental study of microbubble coalescence in a T-junction microfluidic device[J].Microfluidics&Nanofluidics,2012,12(5):715-722.
[3]Hasan B O.Breakage of drops and bubbles in a stirred tank:a review of experimental studies[J].Chinese Journal of Chemical Engineering,2017,25(6):698-711.
[4]Fan W Y,Yin X H.Numerical study on interaction between two bubbles rising side by side in CMC solution[J].Chinese Journal of Chemical Engineering,2013,21(7):705-713.
[5]Akker H E V D.Lattice boltzmann simulations for multi-scale chemical engineering[J].Current Opinion in Chemical Engineering,2018,21:67-75.
[6]Hassanzadeh A,Firouzi M,Albijanic B,et al.A review on determination of particle-bubble encounter using analytical,experimental and numerical methods[J].Minerals Engineering,2018,122(6):296-311.
[7]Ma M,Lu J C,Tryggvason G.Using statistical learning to close two-fluid multiphase flow equations for a simple bubbly system[J].Physics of Fluids,2015,27(9):092101.
[8]Cai Z Q,Gao Z M,Bao Y Y,et al.Formation and motion of conjunct bubbles in glycerol-water solutions[J].Industrial&Engineering Chemistry Research,2012,51(4):1990-1996.
[9]Anwar S.Lattice Boltzmann modeling of buoyant rise of single and multiple bubbles[J].Computers&Fluids,2013,88:430-439.
[10]Balcázar N,Lehmkuhl O,Jofre L,et al.Level-set simulations of buoyancy-driven motion of single and multiple bubbles[J].International Journal of Heat&Fluid Flow,2015,56:91-107.
[11]Abbassi W,Besbes S,Elhajem M,et al.Numerical simulation of free ascension and coaxial coalescence of air bubbles using the volume of fluid method(VOF)[J].Computers&Fluids,2017,161(1):47-59.
[12]Chakraborty I,Biswas G,Ghoshdastidar P S.A coupled level-set and volume-of-fluid method for the buoyant rise of gas bubbles in liquids[J].International Journal of Heat&Mass Transfer,2013,58(1/2):240-259.
[13]Pozorski S.Simulations of single bubbles rising through viscous liquids using smoothed particle hydrodynamics[J].International Journal of Multiphase Flow,2013,50:98-105.
[14]Rahmat A,Tofighi N,Yildiz M.Numerical simulation of the electrohydrodynamic effects on bubble rising using the SPHmethod[J].International Journal of Heat&Fluid Flow,2016,62:313-323.
[15]Unverdi S O,Tryggvason G.A front-tracking method for viscous,incompressible,multi-fluid flows[J].Journal of Computational Physics,1992,100(1):25-37.
[16]Bhaga D,Weber M E.Bubbles in viscous liquids:shapes,wakes and velocities[J].Journal of Fluid Mechanics,2006,105:61-85.
[17]Hysing S,Turek S,Kuzmin D,et al.Quantitative benchmark computations of two-dimensional bubble dynamics[J].International Journal for Numerical Methods in Fluids,2010,60(11):1259-1288.