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基于虚功原理, 在Hu X Y等和Grenier N等的研究结果基础上推导了多相流光滑粒子流体动力学(smoothed particle hydrodynamics, SPH)控制方程, 采用精度较高的黏性力和表面张力模型, 发展了一套适用于具有大密度比和大黏性比界面的多相流SPH方法. 首先, 通过施加人工位移修正, 适当背景压力和异相界面力, 使得计算全程粒子分布相对均匀, 改善了界面处的失稳现象, 防止了异相界面处粒子的非物理性穿透; 在此基础上, 利用方形流体团振荡模型对表面张力模型进行了验证, 数值结果与解析解甚为吻合; 然后采用上浮气泡经典数值算例对比研究了不同黏性力计算方法、不同核函数的适用性以及人工位移修正的效果; 最后, 对单个气泡的上浮、变形、撕裂以及垂向两个气泡的追赶、融合等现象进行了模拟, 初步揭示了气泡上浮过程中各种有趣物理现象的细节过程和动力学机理.Based on the principle of virtual works, a multiphase smoothed particle hydrodynamics (SPH) model is further developed from the foundation of Hu X Y et al. (2006) and Grenier N et al. (2009). In the present model, the surface tension force implementation suitable for the multiphase flows with a large density ratio is applied, and this allows a good continuity at the multiphase interface. Artificial displacement correction is applied to keep the particles distributing uniformly in the whole flow field, and therefore any artificial viscous term is never needed; this is very important in the numerical simulation of viscous flows since the introduction of artificial viscosity changes the Reynolds number. Background pressure and interface sharpness force are added in the equation of state and the equation of momentum respectively to ensure the multiphase interface stability and smoothness; this is essential in the simulation of multiphase flows with large density difference at the multiphase interface. Two types of viscosity expressions suitable for multiphase flows are introduced and analyzed; the conclusion is that the formula proposed by Morris et al. (1997) and its similarly derived forms can give more accurate results. In the numerical validations, an oscillating droplet test is applied first to confirm the accuracy of the surface tension model and good results are achieved. This demonstrates that the artificial displacement and the interface sharp force will make negligible effects to the surface tension implementation. After that, two classic quantitative benchmarks of rising bubbles are simulated and the results of SPH agree well with the reference data. Moreover, in the two numerical benchmarks, the effect of the artificial displacement, the choice of the viscosity expression, and the type of the kernel function are compared and finally an optimal combination of these numerical aspects is recommended. Based on the above numerical investigations, the splitting process of an initially circular bubble is simulated and the numerical results agree well with the experimental data. In the last numerical case, the process of chasing and merging between two rising bubbles in vertical direction is simulated, based on which the mechanisms of these interesting interactions between two rising bubbles are analyzed. It is demonstrated in the present work that further improved multiphase SPH model may provide a potential method for the research of bubble dynamics.
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Keywords:
- multiphase flow /
- smoothed particle hydrodynamics /
- surface tension /
- rising bubble
[1] Shew W L, Pinton J F 2006 J. Stat. Mech. -Theory E 2006 01
[2] Zhang A M, Cui P, Cui J, Wang Q X 2015 J. Fluid Mech. 776 137
[3] Zhang A M, Sun P, Ming F 2015 Comput. Method Appl. M. 294 189
[4] Zhang A M, Li S, Cui J 2015 Phys. Fluids 27 062102
[5] Yu Z, Yang H, Fan L S 2011 Chem. Eng. Sci. 66 3441
[6] Hua J S, Stene J F, Lin P 2008 J. Comput. Phys. 227 3358
[7] Wang H, Zhang Z Y, Yang Y M, Zhang H S 2010 Chinese Phys. B 19 026801
[8] Annaland M, Deen N G, Kuipers J A M 2005 Chem. Eng. Sci. 60 2999
[9] Croce R, Griebel M, Schweitzer M A 2010 Int. J. Numer. Meth. Fl. 62 963
[10] Mahdi D, Mohammad T R, Hamidreza M 2015 Chinese Phys. B 24 024303
[11] Zhang A M, Wang S P, Wu G X 2013 Eng. Anal. Bound. Elem. 37 1179
[12] Zhang A M, Liu Y L 2015 J. Comput. Phys. 294 208
[13] Zhang A M, Wang S P, Huang C, Wang B 2013 Eur. J. Mech. B-Fluid 42 69
[14] Li S, Sun L Q, Zhang A M 2014 Acta Phys. Sin. 63 184701 (in Chinese) [李帅, 孙龙泉, 张阿漫 2014 物理学报 63 184701]
[15] Colagrossi A, Landrini M 2003 J. Comput. Phy. 191 448
[16] Chen Z, Zong Z, Li H T, Li J 2013 Ocean Eng. 59 129
[17] Sun P, Ming F, Zhang A 2015 Ocean Eng. 98 32
[18] Liu G R, Liu M B 2003 Smoothed Particle Hydrodynamics: A Meshfree Particle Method (Singapore: World Scientific)
[19] Hu X Y, Adams N A 2006 J. Comput. Phys. 213 844
[20] Grenier N, Antuono M, Colagrossi A, Le Touzé D, Alessandrini B 2009 J. Comput. Phys. 228 8380
[21] Sun P, Ming F, Zhang A, Yao X 2014 Proceedings of the 33rd International Conference on Ocean, Offshore and Arctic Engineering San Francisco June 8-14 2014
[22] Szewc K, Pozorski J, Minier J P 2013 Int. J. Multiphas. Flow 50 98
[23] Ji B, Luo X W, Wu Y L, Peng X X, Duan Y L 2013 Int. J. Multiphas. Flow 51 33
[24] Grenier N, Le Touzé D, Colagrossi A, Antuono M, Colicchio G 2013 Ocean Eng. 69 88
[25] Zainali A, Tofighi N, Shadloo M S, Yildiz M 2013 Comput. Method Appl. M. 254 99
[26] Hysing S, Turek S, Kuzmin D, Parolini N, Burman E, Ganesan S, Tobiska L 2009 Int. J. Numer. Meth. Fl. 60 1259
[27] Colagrossi A, Antuono M, Souto-Iglesias A, Le Touzé D 2011 Phys. Rev. E 84 026705
[28] Brackbill J U, Kothe D B, Zemach C 1992 J. Comput. Phys. 100 335
[29] Monaghan J J 1994 J. Comput. Phys. 110 399
[30] Colagrossi A, Bouscasse B, Antuono M, Marrone S 2012 Comput. Phys. Commun. 183 1641
[31] Marrone S, Colagrossi A, Antuono M, Colicchio G, Graziani G 2013 J. Comput. Phys. 245 456
[32] Chen Z, Zong Z, Liu M B, Zou L, Li H T, Shu C 2015 J. Comput. Phys. 283 169
[33] Yang X F, Liu M B 2012 Acta Phys. Sin. 61 224701 (in Chinese) [杨秀峰, 刘谋斌 2012 物理学报 61 224701]
[34] Jin H B, Ding X 2005 J. Comput. Phys. 202 699
[35] Molteni D, Colagrossi A 2009 Comput. Phys. Commun. 180 861
[36] Shepard D 1968 Proceedings of the 23rd ACM national conference: ACM 517
[37] Monaghan J, Gingold R 1983 J. Comput. Phys. 52 374
[38] Morris J P, Fox P J, Zhu Y 1997 J. Comput. Phys. 136 214
[39] Adami S, Hu X Y, Adams N A 2010 J. Comput. Phys. 229 5011
[40] Grenier N, Le Touzé D, Colagrossi A, Antuono M, Colicchio G 2013 Ocean Eng. 69 88
[41] Zhang A M, Cao X Y, Ming F R, Zhang Z F 2013 Appl. Ocean Res. 42 24
[42] Chen R, Tian W, Su G, Qiu S, Ishiwatari Y, Oka Y 2011 Chem. Eng. Sci. 66 5055
[43] Marrone S 2012 Ph. D. Dissertation (Rome: University Of Rome)
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[1] Shew W L, Pinton J F 2006 J. Stat. Mech. -Theory E 2006 01
[2] Zhang A M, Cui P, Cui J, Wang Q X 2015 J. Fluid Mech. 776 137
[3] Zhang A M, Sun P, Ming F 2015 Comput. Method Appl. M. 294 189
[4] Zhang A M, Li S, Cui J 2015 Phys. Fluids 27 062102
[5] Yu Z, Yang H, Fan L S 2011 Chem. Eng. Sci. 66 3441
[6] Hua J S, Stene J F, Lin P 2008 J. Comput. Phys. 227 3358
[7] Wang H, Zhang Z Y, Yang Y M, Zhang H S 2010 Chinese Phys. B 19 026801
[8] Annaland M, Deen N G, Kuipers J A M 2005 Chem. Eng. Sci. 60 2999
[9] Croce R, Griebel M, Schweitzer M A 2010 Int. J. Numer. Meth. Fl. 62 963
[10] Mahdi D, Mohammad T R, Hamidreza M 2015 Chinese Phys. B 24 024303
[11] Zhang A M, Wang S P, Wu G X 2013 Eng. Anal. Bound. Elem. 37 1179
[12] Zhang A M, Liu Y L 2015 J. Comput. Phys. 294 208
[13] Zhang A M, Wang S P, Huang C, Wang B 2013 Eur. J. Mech. B-Fluid 42 69
[14] Li S, Sun L Q, Zhang A M 2014 Acta Phys. Sin. 63 184701 (in Chinese) [李帅, 孙龙泉, 张阿漫 2014 物理学报 63 184701]
[15] Colagrossi A, Landrini M 2003 J. Comput. Phy. 191 448
[16] Chen Z, Zong Z, Li H T, Li J 2013 Ocean Eng. 59 129
[17] Sun P, Ming F, Zhang A 2015 Ocean Eng. 98 32
[18] Liu G R, Liu M B 2003 Smoothed Particle Hydrodynamics: A Meshfree Particle Method (Singapore: World Scientific)
[19] Hu X Y, Adams N A 2006 J. Comput. Phys. 213 844
[20] Grenier N, Antuono M, Colagrossi A, Le Touzé D, Alessandrini B 2009 J. Comput. Phys. 228 8380
[21] Sun P, Ming F, Zhang A, Yao X 2014 Proceedings of the 33rd International Conference on Ocean, Offshore and Arctic Engineering San Francisco June 8-14 2014
[22] Szewc K, Pozorski J, Minier J P 2013 Int. J. Multiphas. Flow 50 98
[23] Ji B, Luo X W, Wu Y L, Peng X X, Duan Y L 2013 Int. J. Multiphas. Flow 51 33
[24] Grenier N, Le Touzé D, Colagrossi A, Antuono M, Colicchio G 2013 Ocean Eng. 69 88
[25] Zainali A, Tofighi N, Shadloo M S, Yildiz M 2013 Comput. Method Appl. M. 254 99
[26] Hysing S, Turek S, Kuzmin D, Parolini N, Burman E, Ganesan S, Tobiska L 2009 Int. J. Numer. Meth. Fl. 60 1259
[27] Colagrossi A, Antuono M, Souto-Iglesias A, Le Touzé D 2011 Phys. Rev. E 84 026705
[28] Brackbill J U, Kothe D B, Zemach C 1992 J. Comput. Phys. 100 335
[29] Monaghan J J 1994 J. Comput. Phys. 110 399
[30] Colagrossi A, Bouscasse B, Antuono M, Marrone S 2012 Comput. Phys. Commun. 183 1641
[31] Marrone S, Colagrossi A, Antuono M, Colicchio G, Graziani G 2013 J. Comput. Phys. 245 456
[32] Chen Z, Zong Z, Liu M B, Zou L, Li H T, Shu C 2015 J. Comput. Phys. 283 169
[33] Yang X F, Liu M B 2012 Acta Phys. Sin. 61 224701 (in Chinese) [杨秀峰, 刘谋斌 2012 物理学报 61 224701]
[34] Jin H B, Ding X 2005 J. Comput. Phys. 202 699
[35] Molteni D, Colagrossi A 2009 Comput. Phys. Commun. 180 861
[36] Shepard D 1968 Proceedings of the 23rd ACM national conference: ACM 517
[37] Monaghan J, Gingold R 1983 J. Comput. Phys. 52 374
[38] Morris J P, Fox P J, Zhu Y 1997 J. Comput. Phys. 136 214
[39] Adami S, Hu X Y, Adams N A 2010 J. Comput. Phys. 229 5011
[40] Grenier N, Le Touzé D, Colagrossi A, Antuono M, Colicchio G 2013 Ocean Eng. 69 88
[41] Zhang A M, Cao X Y, Ming F R, Zhang Z F 2013 Appl. Ocean Res. 42 24
[42] Chen R, Tian W, Su G, Qiu S, Ishiwatari Y, Oka Y 2011 Chem. Eng. Sci. 66 5055
[43] Marrone S 2012 Ph. D. Dissertation (Rome: University Of Rome)
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