受限制空间内的气泡行为数值与实验研究
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摘要
气泡广泛存在于自然界的液体之中,从多方面影响着我们的生活。局部压力的降低会使液体发生空化而产生蒸气泡(又称空泡),并经历发展、溃灭和回弹等一系列动力学过程。外界气体也可通过多种方式进入液体之中形成气泡,同时液体中存在的微量气体作为“气核”对空化的发生有重要的促进作用。气泡的行为对自然界中的许多现象具有重要的意义,在流体机械、液压系统、水工结构中造成振动、噪声、空蚀等多种危害,在生物医学、MEMS和工程领域则有广泛的应用前景。长期以来,气泡行为的研究是流体动力学研究中的热点领域。但是,现有关于在受限制空间中气泡行为的研究较少,而这一问题对于气泡的科学认识和应用都有重要的基础性意义。
     本文结合理论分析、数值模拟和高速摄像观测等手段,对受限制空间中的气泡行为开展了较为系统和深入的研究。总结了约束液面形成冲击气泡的各种现象,得到了液滴冲击约束液面现象的无量纲参数域,分析了壁面约束和液滴振荡对冲击气泡形成过程的影响机理。定量分析了反气泡的溃灭过程,建立了溃灭速度模型。观察了管道内空泡的动力学行为和双空泡干涉现象,建立了管道内空泡动力学行为的BEM数值模型,分析了管道约束对空泡行为的影响规律,揭示了管道内单空泡/双空泡动力学行为的机理。
     论文的主要内容如下:
     第一章,绪论。阐述了本课题研究的背景与意义;综述了冲击气泡形成过程和空泡动力学行为的研究历史与现状;概括了本文的主要内容。
     第二章,空泡动力学数值模型研究。采用BEM方法建立了空泡动力学数值计算模型,阐述了其控制方程、边界积分方程、离散化方法、算法设置和计算流程,针对空泡发展过程中拓扑形状的变化给出了相应的数值处理算法。
     第三章,约束液面冲击气泡形成研究。总结了受约束液面形成冲击气泡的四种类型,给出了冲击现象的参数平面分布图;分析了壁面约束对延时气泡、大气泡和穹顶气泡等现象的影响,提出液滴形状振荡是大气泡形成区域间断分布现象的原因;发现反气泡开始溃灭后极短时间内可达到稳定的溃灭速度,建立反气泡溃灭速度的计算模型,发现溃灭速度与(б/pe)1/2成正比。
     第四章,管道内空泡动力学行为及双空泡干涉的研究。通过高速摄像观察了管道内单空泡的形态演化,采用数值模拟研究了特征参数对空泡体积和周期的影响;观察了管道内的双空泡干涉现象,得到了干涉现象的参数区域分布图,通过数值模拟分析了管道内双空泡干涉的机理。
     第五章,总结与展望。对论文研究工作和创新点进行了总结,对后续的研究工作进行了展望。
Bubble is a common phenomenon in liquid. Vapor bubbles are formed by cavitation due to the decrease of local pressure, which will develop, collapse and rebound in the liquid. Other bubbles consist of external gas entrapped in the liquid, which play an inportant role in cavitation process as cavitation nuclei. Bubble causes several serious problems in fluid systems, such as vibration, noise and cavitation erosion. On the other hand, bubble can be widely applied in the areas of medical treatment. MEMS and engineering, while it is also a pivotal element in many natural processes. Although bubbles have attracted attention of many reaseachers. investigation on the behavior of bubbles in restricted space is uncommon in literature.
     In this thesis, the behavior of bubbles in restricted space is studied systematically and comprehensively, by means of theoretical analysis, numerical simulation and high-speed camera observation. The phenomena of bubble entrainment during drop impacting on restricted liquid surface are summarized. The effects of the restircting wall and the vibration of drop are analysised. The regimes of various outcomes of drop impact are given on non-dimentional parameter plane. The collapse of antibubble is studied quantitatively, to derive a model of the collapse velocity. The dynamic behavior of single/double spark-induced bubble in tube is observed by high-speed camera. BEM numerical model for the bubbles in tube is presented, with which the effect of tube wall on the behavior of the bubble is presented, and mechanism of single/double spark-induced bubble in tube is analysised.
     The outline of this thesis is as follows,
     In chapter1, the background and significance of the thesis are proposed. Literature on the bubble formation during impacting process and the dynamics of cavitation bubble are reviewed. The main content of the thesis is summarized.
     In chapter2, BEM numerical model for the dynamic behavior of bubbles is presented, which is able to deal with bubble coalescence and ring bubble. The governing equations, boundary integral equations, descretization. calculation programme and other algorithms are introduced.
     In chapter3, the phenomena of bubble entrainment during drop impacting on restricted liquid surface are summarized into four types. The regimes of various outcomes of drop impact are given on parameter plane. The effects of restricting wall on delayed bubble, large bubble and canopy bubble are analysised. The discrete Weber number regime of large bubbles is reported, which is caused by the shape oscillation of the drop. A model of the collapse velocity of antibubbles is derived, which shows that the collapse velocity is proportionate to (σ/pe)1/2.
     In chapter4, the development and collapse of single spark-induced bubble in tube is studied, whose volume and vibration period are analysised by numerical simulation. The interaction of two spark-induced bubbles is observed by high-speed camera. The mechanism of two bubble interaction is analysised, and the regimes of various phenomena are presented on parameter plane.
     In chapter5, the research work of the thesis is summarized with further prospect.
引文
1. Harrison, M. An experimental study of single bubble cavitation noise. The Journal of the Acoustical Society of America.1952,24:p.776.
    2. Li, S.C. Cavitation of hydraulic machinery. Vol.1.2000, Imperial College Press London.
    3. Karimi, A. and J.L. Martin. Cavitation erosion of materials. International Materials Reviews.1986,31(1):p.1-26.
    4. Flynn, H.G. and C.C. Church. Transient pulsations of small gas bubbles in water. The Journal of the Acoustical Society of America.1988,84:p.985.
    5. Margulis, M.A. and I.M. Margulis. Luminescence mechanism of acoustic and laser-induced cavitation. Acoustical Physics.2006,52(3):p.283-292.
    6. Blake, J.R. and D.C. Gibson. Cavitation bubbles near boundaries. Annual Review of Fluid Mechanics.1987,19(1):p.99-123.
    7.路甬祥.液压气动技术手册.2002,机械工业出版社.
    8.陆亮.液压节流阀中的空化流动与噪声.硕士论文,浙江大学,2012.
    9. Kawai, A. and K. Suzuki. Removal mechanism of nano-bubble with afm for immersion lithography. Microelectronic Engineering.2006,83(4-9):p.655-658.
    10. Meletis, E.I., et al. Electrolytic plasma processing for cleaning and metal-coating of steel surfaces. Surface and Coatings Technology.2002,150(2-3):p.246-256.
    11. Growcock, F., A. Khan and G. Simon. Application of water-based and oil-based aphrons in drilling fluids. International Symposium on Oilfield Chemistry.2003.
    12. Mazue, G., et al. Large-scale ultrasonic cleaning system:design of a multi-transducer device for boat cleaning (20kHz). Ultrasonics Sonochemistry.2011,18(4):p.895-900.
    13.胥震,欧阳清,易定和.海洋污损生物防除方法概述及发展趋势.腐蚀科学与防护技术.2012,24(3):p.192-198.
    14. Kuijpers, M., et al. Cavitation-induced reactions in high-pressure carbon dioxide. Science.2002,298(5600):p.1969-1971.
    15. Le. H.P. Progress and trends in ink-jet printing technology. Journal of Imaging Science and Technology.1998.42(1):p.49-62.
    16. Awad. S.B. Ultrasonic cavitations and precision cleaning. Precision Cleaning.1996. 4:p.12.
    17. Jyoti, K.K. and A.B. Pandit. Water disinfection by acoustic and hydrodynamic cavitation. Biochemical Engineering Journal.2001,7(3):p.201-212.
    18. Harris, G.R. Progress in medical ultrasound exposimetry. Ultrasonics, Ferroelectrics and Frequency Control. IEEE Transactions.2005,52(5):p.717-736.
    19. Carnell, M.T., R.D. Alcock and D.C. Emmony. Optical imaging of shock waves produced by a high-energy electromagnetic transducer. Physics in Medicine and Biology. 1993.38(11):p.1575.
    20. Paliwal, S. and S. Mitragotri. Ultrasound-induced cavitation:applications in drug and gene delivery. Expert Opinion on Drug Delivery.2006,3(6):p.713-726.
    21. Dadvand, A., et al. Boundary element analysis of the droplet dynamics induced by spark-generated bubble. Engineering Analysis with Boundary Elements.2012,36(11):p. 1595-1603.
    22. Pavard, D., et al. Removal of particles from holes in submerged plates with oscillating bubbles. Physics of Fluids.2009,21(8):p.0833048.
    23. Ohl, C., et al. Surface cleaning from laser-induced cavitation bubbles. Applied Physics Letters.2006,89(7):p.074102.
    24. Thorpe, S.A. Dynamical processes of transfer at the sea surface. Progress in Oceanography.1995,35(4):p.315-352.
    25. Longuet-Higgins, M.S. Bubble noise spectra The Journal of the Acoustical Society of America.1990,87:p.652-661.
    26. Lauterborn, W. and T. Kurz. Physics of bubble oscillations. Reports on Progress in Physics.2010,73(10):p.106501.
    27. Plesset. M.S. and A. Prosperetti. Bubble dynamics and cavitation. Annual Review of Fluid Mechanics.1977.9(1):p.145-185.
    28. Feng, Z.C. and L.G. Leal. Nonlinear bubble dynamics. Annual Review of Fluid Mechanics.1997.29(1):p.201-243.
    29. Worthington, A.M. and R.S. Cole. Impact with a liquid surface, studied by the aid of instantaneous photography. Philosophical Transactions of the Royal Society of London. Series a, Containing Papers of a Mathematical or Physical Character.1897,189:p.137-148.
    30. Edgerton, H.E. and J.R. Killian. Flash! Seeing the unseen by ultra high-speed photography.1954, CT Branford Co.
    31. Thoroddsen, S.T., T.G. Etoh and K. Takehara. High-speed imaging of drops and bubbles. Annual Review Fluid Mechanics.2008,40:p.257-285.
    32. Adrian, R.J. Particle-imaging techniques for experimental fluid mechanics. Annual Review of Fluid Mechanics.1991,23(1):p.261-304.
    33. Sigler, J. and R. Mesler. The behavior of the gas film formed upon drop impact with a liquid surface. Journal of Colloid and Interface Science.1990,134(2):p.459-474.
    34. Rein, M. Phenomena of liquid-drop impact on solid and liquid surfaces. Fluid Dynamics Research.1993,12(2):p.61-93.
    35. Oguz, H.N. and A. Prosperetti. Bubble entrainment by the impact of drops on liquid surfaces. Journal of Fluid Mechanics.1990,219:p.143-179.
    36. Prosperetti, A. and H.N. Oguz. The impact of drops on liquid surfaces and the underwater noise of rain. Annual Review of Fluid Mechanics.1993,25:p.577-602.
    37. Morton, D., M. Rudman and J.L. Liow. An investigation of the flow regimes resulting from splashing drops. Physics of Fluids.2000,12(4):p.747-763.
    38. Pumphrey, H.C. and P.A. Elmore. The entrainment of bubbles by drop impacts. Journal of Fluid Mechanics.1990,220:p.539-567.
    39. Liow, J.L. Splash formation by spherical drops. Journal of Fluid Mechanics.2001, 427:p.73-105.
    40. Liow, J.L. and D.E. Cole. High framing rate piv studies of an impinging water drop. 28th International Congress on High-Speed Imaging and Photonics. Canberra, Australia, 2008
    41. Weiss, D.A. and A.L. Yarin. Single drop impact onto liquid films:neck distortion, jetting, tiny bubble entrainment, and crown formation. Journal of Fluid Mechanics.1999, 385:p.229-254.
    42. Thoroddsen, S.T. The making of a splash. Journal of Fluid Mechanics.2012,690:p. 1-4.
    43. Thoroddsen, S.T. and K. Takehara. The coalescence cascade of a drop. Physics of Fluids.2000,12(6):p.1265.
    44. Yarin, A.L. Drop impact dynamics:splashing, spreading, receding, bouncing....2006. p.159-192.
    45. Zou, J., et al. Experimental study of a drop bouncing on a liquid surface. Physics of Fluids.2011,23(4):044101.
    46. Rein, M. The transitional regime between coalescing and splashing drops. Journal of Fluid Mechanics.1996,306:p.145-165.
    47. Rodgiguez, F. and R. Mesler. The penetration of drop-formed vortex rings into pools of liquid. Journal of Colloid and Interface Science.1988,121(1):p.121-129.
    48. Honey, E.M. and H.P. Kavehpour. Astonishing life of a coalescing drop on a free surface. Physical Review E.2006,73(2):p.027301.
    49. Cole, D. The splashing morphology of liquid-liquid impacts. PhD Thesis, James Cook University,2007.
    50. Harlow, F.H. and J.E. Welch. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Physics of Fluids.1965,8:p.2182.
    51. Morton, D., M. Rudman and J.L. Liow. An investigation of the flow regimes resulting from splashing drops. Physics of Fluids.2000,12(4):p.747-763.
    52. Watanabe, Y., A. Sarusatari and D.M. Ingram. Free-surface flows under impacting droplets. Journal of Computational Physics.2008,227(4):p.2344-2365.
    53. Berberovic, E. Investigation of free-surface flow associated with drop impact: numerical simulations and theoretical modeling. PhD Thesis, TU Darmstadt,2010.
    54.马理强等.液滴溅落问题的光滑粒子动力学模拟.物理学报.2012,61(5):054701.
    55. Franz, G.J. Splashes as sources of sound in liquids. The Journal of the Acoustical Society of America.1959,31:p.1080.
    56. Esmailizadeh, L. and R. Mesler. Bubble entrainment with drops. Journal of Colloid and Interface Science.1986,110(2):p.561-574.
    57. Thoroddsen, S.T., T.G. Etoh and K. Takehara. Air entrapment under an impacting drop. Journal of Fluid Mechanics.2003,478:p.125-134.
    58. MEDWIN, H., et al. The anatomy of underwater rain noise. Journal of the Acoustical Society of America.1992,92(3):p.1613-1623.
    59. Rein, M. The transitional regime between coalescing and splashing drops. Journal of Fluid Mechanics.1996,306:p.145-165.
    60. Elmore, P.A., G.L. Chahine and H.N. Oguz. Cavity and flow measurements of reproducible bubble entrainment following drop impacts. Experiments in Fluids.2001, 31(6):p.664-673.
    61. Postema, M. and O.H. Gilja. Ultrasound-directed drug delivery. Current Pharmaceutical Biotechnology.2007,8(6):p.355-361.
    62. Vandewalle, N., et al. Antibubbles, liquid onions and bouncing droplets. Colloids and Surfaces a:Physicochemical and Engineering Aspects.2009,344(1):p.42-47.
    63. Kim, P. and J. Vogel. Antibubbles:factors that affect their stability. Colloids and Surfaces a:Physicochemical and Engineering Aspects.2006,289(1):p.237-244.
    64. Dorbolo, S.E.P., H.E. Caps and N. Vandewalle. Fluid instabilities in the birth and death of antibubbles. New Journal of Physics.2003,5(1):p.161.
    65. Kim, P.G. and H.A. Stone. Dynamics of the formation of antibubbles. Epl (Europhysics Letters).2008,83(5):p.54001.
    66. Postema, M., et al. Creating antibubbles with ultrasound.2005.
    67. Silpe, J.E. and D.W. McGrail. Magnetic antibubbles:formation and control of magnetic macroemulsions for fluid transport applications. Journal of Applied Physics. 2013,113(17):p.17B304.
    68. Kim, P. and J. Vogel. Antibubbles:factors that affect their stability. Colloids and Surfaces a:Physicochemical and Engineering Aspects.2006,289(1):p.237-244.
    69. Dorbolo, S.E.P., et al. Antibubble lifetime:influence of the bulk viscosity and of the surface modulus of the mixture. Colloids and Surfaces a:Physicochemical and Engineering Aspects.2010,365(1):p.43-45.
    70. Poortinga, A.T. Micron-sized antibubbles with tunable stability. Colloids and Surfaces a:Physicochemical and Engineering Aspects.2012,419(1):p.15-20.
    71. Dorbolo. S.E.P., et al. Aging of an antibubble. Europhysics Letters.2005.69(6):p. 966.
    72. Eri. A. and K. Okumura. Lifetime of a two-dimensional air bubble. Physical Review E.2007,76(6):p.060601.
    73. Poortinga. A.T. Long-lived antibubbles:stable antibubbles through pickering stabilization. Langmuir.2011.27(6):p.2138-2141.
    74. Scheid. B., et al. Antibubble dynamics:the drainage of an air film with viscous interfaces. Physical Review Letters.2012,109(26):p.264502.
    75. Rayleigh, L. On the pressure developed in a liquid during the collapse of a spherical cavity. Philosophical Magazine Series 6.1917,34(200):p.94-98.
    76. Brennen, C.E. Cavitation and bubble dynamics.1995:Oxford University Press.
    77. Besant, W.H. Hydrostatics and hydromechanics.1859, London:G.Bell.
    78. Plesset. M.S. The dynamics of cavitation bubbles. Journal of Applied Mechanics. 1949,16(3):p.227-282.
    79. Gilmore. F.R. The growth or collapse of a spherical bubble in a viscous compressible liquid.1952. (unpublished. from internet)
    80. Noltingk, B.E. and E.A. Neppiras. Cavitation produced by ultrasonics. Proceedings of the Physical Society. Section B.1950.63(9):p.674.
    81. Prosperetti, A. and A. Lezzi. Bubble dynamics in a compressible liquid. Part 1. First-order theory. Journal of Fluid Mechanics.1986,168:p.457-478.
    82. Akhatov. I., et al. Collapse and rebound of a laser-induced cavitation bubble. Physics of Fluids.2001.13:p.2805.
    83. Benjamin. T.B. and A.T. Ellis. Collapse of cavitation bubbles and pressures thereby produced against solid boundaries. Philosophical Transactions of the Royal Society of London Series a-Mathematical and Physical Sciences.1966,260(1110):p.221.
    84. Blake. J.R., B.B. Taib and G. Doherty. Transient cavities near boundaries. Part 1. Rigid boundary. Journal of Fluid Mechanics.1986.170:p.479-497.
    85. Brujan. E.A., et al. Dynamics of laser-induced cavitation bubbles near an elastic boundary. Journal of Fluid Mechanics.2001.433:p.251-281.
    86. Kling, C.L. and F.G. Hammitt. A photographic study of spark-induced cavitation bubble collapse. Journal of Basic Engineering.1972,94:p.825.
    87. Shima, A., et al. Mechanism of impact pressure generation from spark-generated bubble collapse near a wall. Aiaa Journal.1983,21(1):p.55-59.
    88. Chahine, G.L. Experimental and asymptotic study of nonspherical bubble collapse. Applied Scientific Research.1982,38(1):p.187-197.
    89. Turangan, C.K., et al. Experimental and numerical study of transient bubble-elastic membrane interaction. Journal of Applied Physics.2006,100(5):p.0549105.
    90. Lauterborn, W. and H. Bolle. Experimental investigations of cavitation-bubble collapse in the neighbourhood of a solid boundary. Journal of Fluid Mechanics.1975, 72(2):p.391-393.
    91. Vogel, A., W. Lauterborn and R. Timm. Optical and acoustic investigations of the dynamics of laser-produced cavitation bubbles near a solid boundary. Journal of Fluid Mechanics.1989,206(1):p.299-338.
    92. Giovanneschi, P. and D. Dufresne. Experimental study of laser-induced cavitation bubbles. Journal of Applied Physics.1985,58:p.651.
    93. Cole, R.H. and R. Weller. Underwater explosions. Physics Today.1948,1:p.35.
    94. Klaseboer, E., et al. Experimental and numerical investigation of the dynamics of an underwater explosion bubble near a resilient/rigid structure. Journal of Fluid Mechanics. 2005,537(6):p.387-413.
    95. Brett, J.M. and G. Yiannakopolous. A study of explosive effects in close proximity to a submerged cylinder. International Journal of Impact Engineering.2008,35(4):p.206-225.
    96. Hung, C.F., et al. Dynamic response of cylindrical shell structures subjected to underwater explosion. Ocean Engineering.2009,36(8):p.564-577.
    97.张阿漫.水下爆炸气泡三维动态特性研究.博士论文,哈尔滨工程大学,2006.
    98. Zhang, A.M. and X. Yao. Interaction of underwater explosion bubble with complex elastic-plastic structure. Applied Mathematics and Mechanics.2008,29(1):p.89-100.
    99. Blake, J.R., B.B. Taib and G. Doherty. Transient cavities near boundaries part 2. Free surface. Journal of Fluid Mechanics.1987,181:p.197-212.
    100. Taib, B.B. Boundary integral method applied to cavitation bubble dynamics. PhD Thesis, University of Wollongong,1985.
    101. Wang, Q.X., et al. Strong interaction between a buoyancy bubble and a free surface. Theoretical and Computational Fluid Dynamics.1996,8(1):p.73-88.
    102. Rungsiyaphornrat, S., et al. The merging of two gaseous bubbles with an application to underwater explosions. Computers & Fluids.2003,32(8):p.1049-1074.
    103. Klaseboer, E. and B.C. Khoo. An oscillating bubble near an elastic material. Journal of Applied Physics.2004,96(10):p.5808-5818.
    104. Klaseboer, E. and B.C. Khoo. Boundary integral equations as applied to an oscillating bubble near a fluid-fluid interface. Computational Mechanics.2004,33(2):p. 129-138.
    105. Lee, M., E. Klaseboer and B.C. Khoo. On the boundary integral method for the rebounding bubble. Journal of Fluid Mechanics.2007,570(1):p.407-429.
    106. Lundgren, T.S. and N.N. Mansour. Vortex ring bubbles. Journal of Fluid Mechanics. 1991,224(5):p.177-196.
    107. Best, J.P. The formation of toroidal bubbles upon the collapse of transient cavities. Journal of Fluid Mechanics.1993,251:p.79-107.
    108. Zhang, S., J.H. Duncan and G.L. Chahine. The final stage of the collapse of a cavitation bubble near a rigid wall. Journal of Fluid Mechanics.1993,257:p.147-181.
    109. Wang, Q.X., et al. Nonlinear interaction between gas bubble and free surface. Computers & Fluids.1996,25(7):p.607-628.
    110. Blake, J.R., J.M. Boulton-Stone and R.P. Tong. Boundary integral methods for rising, bursting and collapsing bubbles. BE Applications in Fluid Mechanics, Phower H. Southampton:Computational Mechanics Publications.1995:p.31-71.
    111. Zhang, Y.L., et al. Three-dimensional computation of bubbles near a free surface. Journal of Computational Physics.1998,146(1):p.105-123.
    112. Zhang, Y.L., et al.3d jet impact and toroidal bubbles. Journal of Computational Physics.2001,166(2):p.336-360.
    113. Wang, C., B.C. Khoo and K.S. Yeo. Elastic mesh technique for 3d bim simulation with an application to underwater explosion bubble dynamics. Computers & Fluids.2003, 32(9):p.1195-1212.
    114. Blake, J.R. and D.C. Gibson. Growth and collapse of a vapour cavity near a free surface. Journal of Fluid Mechanics.1981,111:p.123-140.
    115. Boulton-Stone, J.M. A comparison of boundary integral methods for studying the motion of a two-dimensional bubble in an infinite fluid Computer Methods in Applied Mechanics and Engineering.1993,102(2):p.213-234.
    116. Zhang, Y.L., et al. Simulation of three-dimensional bubbles using desingularized boundary integral method. International Journal for Numerical Methods in Fluids.1999, 31(8):p.1311-1320.
    117. Wang, C. and B.C. Khoo. An indirect boundary element method for three-dimensional explosion bubbles. Journal of Computational Physics.2004,194(2):p.451-480.
    118. Buogo, S. and G.B. Cannelli. Implosion of an underwater spark-generated bubble and acoustic energy evaluation using the Rayleigh model. The Journal of the Acoustical Society of America.2002,111(6):p.2594-2600.
    119. Cole, R.H. Underwater explosions.1948, Princeton University Press.
    120. Lauterborn, W. and T. Kurz. Physics of bubble oscillations. Reports on Progress in Physics.2010,73(10):p.10650110.
    121. Tong, R.P., et al. The role of 'splashing' in the collapse of a laser-generated cavity near a rigid boundary. Journal of Fluid Mechanics.1999,380:p.339-361.
    122. Klaseboer, E., B.C. Khoo and K.C. Hung. Dynamics of an oscillating bubble near a floating structure. Journal of Fluids and Structures.2005,21(4):p.395-412.
    123. Gibson, D.C. Cavitation adjacent to plane boundaries. Proceedings of 3rd Australasian Conference on Hydraulics and Fluid Mechanics. Sydney,1968.
    124. Chahine, G.L. Interaction between an oscillating bubble and a free surface. Journal of Fluids Engineering.1977,99:p.709.
    125. Chahine, G. and A. Bovis. Oscillation and collapse of a cavitation bubble in the vicinity of a two-liquid interface. Cavitation and Inhomogeneities in Underwater Acoustics.1980, Springer Berlin Heidelberg, p.23-29.
    126. Ohl, S.W., E. Klaseboer and B.C. Khoo. The dynamics of a non-equilibrium bubble near bio-materials. Physics in Medicine and Biology.2009,54(20):p.6313-6336.
    127. Bretherton, F.P. The motion of long bubbles in tubes. Journal of Fluid Mechanics. 1961,10:p.166-188.
    128. Shosho, C.E. and M.E. Ryan. An experimental study of the motion of long bubbles in inclined tubes. Chemical Engineering Science.2001,56(6):p.2191-2204.
    129. Clanet, C, G. Raud and G. Searby. On the motion of bubbles in vertical tubes of arbitrary cross-sections:some complements to the Dumitrescu-Taylor problem. Journal of Fluid Mechanics.2004,519:p.359-376.
    130. HAZEL, A.L. and M. HEIL. The steady propagation of a semi-infinite bubble into a tube of elliptical or rectangular cross-section. Journal of Fluid Mechanics.2002,470: p.91-114.
    131. Amaya-Bower, L. and T. Lee. Numerical simulation of single bubble rising in vertical and inclined square channel using lattice Boltzmann method Chemical Engineering Science.2011,66(5):p.935-952.
    132. Yuan, H., H.N. Oguz and A. Prosperetti. Growth and collapse of a vapor bubble in a small tube. International Journal of Heat and Mass Transfer.1999,42(19):p.3643-3657.
    133. Ory, E., et al. Growth and collapse of a vapor bubble in a narrow tube. Physics of Fluids.2000,12(6):p.1268-1277.
    134. Miao, H.Y. and S.M. Gracewski. Coupled fem and bem code for simulating acoustically excited bubbles near deformable structures. Computational Mechanics.2008, 42(1):p.95-106.
    135. Ni, B.Y., et al. Experimental and numerical study on the growth and collapse of a bubble in a narrow tube. Acta Mechanica Sinica.2012,28(5):p.1248-1260.
    136. Lauterborn, W. and W. Hentschel. Cavitation bubble dynamics studied by high speed photography and holography:part one. Ultrasonics.1985,23(6):p.260-268.
    137. Lauterborn, W. Cavitation bubble dynamics-new tools for an intricate problem. Applied Scientific Research.1982,38(1):p.165-178.
    138. Pelekasis, N.A. and J.A. Tsamopoulos. Bjerknes forces between two bubbles. Part 2. Response to an oscillatory pressure field. Journal of Fluid Mechanics.1993,254:p. 501-527.
    139. Pelekasis, N.A. and J.A. Tsamopoulos. Bjerknes forces between two bubbles. Part 1. Response to a step change in pressure. Journal of Fluid Mechanics.1993,254:p.467-499.
    140. Tomita, Y., A. Shima and K. Sato. Dynamic behavior of two-laser-induced bubbles in water. Applied Physics Letters.1990,57(3):p.234-236.
    141. Tomita, Y., A. Shima and T. Ohno. Collapse of multiple gas bubbles by a shock wave and induced impulsive pressure. Journal of Applied Physics.1984,56(1):p.125-131.
    142. Bremond, N., et al. Controlled multibubble surface cavitation. Phys. Rev. Lett.2006, 96(22):p.224501.
    143. Chew, L.W., et al. Interaction of two oscillating bubbles near a rigid boundary. Experimental Thermal and Fluid Science.2013,44(1):p.108-113.
    144. Aghdam, A.H., et al. Characterization of the interaction of two oscillating bubbles near a thin elastic membrane. Experiments in Fluids.2012,53(6):p.1723-1735.
    145. Blake, J.R., et al. Interaction of 2 cavitation bubbles with a rigid boundary. Journal of Fluid Mechanics.1993,255:p.707-721.
    146. Robinson, P.B., et al. Interaction of cavitation bubbles with a free surface. Journal of Applied Physics.2001,89(12):p.8225-8237.
    147. Zhang, Z.Y. and H.S. Zhang. Surface tension effects on the behavior of two cavities near a rigid wall. Physical Review E.2005,71(6):p.063026.
    148. Li, Z.R., et al. A boundary element method for the simulation of non-spherical bubbles and their interactions near a free surface. Acta Mechanica Sinica.2012,28(1):p. 51-65.
    149. Fong, S.W., et al. Interactions of multiple spark-generated bubbles with phase differences. Experiments in Fluids.2009,46(4):p.705-724.
    150. Chew, L.W., et al. Interaction of two differently sized oscillating bubbles in a free field. Physical Review E.2011,84(6):p.0663076.
    151. Davis, P.J.等.数值积分法.1986,高等教育出版社.
    152. Longuet-Higgins, M.S. and E.D. Cokelet. The deformation of steep surface waves on water. I. A numerical method of computation. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.1976,350(1660):p.1-26.
    153.任玉亮.冲击气泡形成机理及实验研究.硕士论文,浙江大学,2013.
    154. Dingle, N. and Y. Lee. Terminal fallspeeds of raindrops. Journal of Applied Meteorology.1972,11(5):p.877-879.
    155.徐明.近壁面空泡的可视化实验研究.博士论文,浙江大学,2013.