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超声速气流中激波诱导液滴变形/破碎过程及机理综述

邓光洲,  杨顺华,  王艳华,  谢迅,  刘二伟

邓光洲, 杨顺华, 王艳华, 等. 超声速气流中激波诱导液滴变形/破碎过程及机理综述[J]. 航空动力学报, 2026, 42(X):20250512 doi: 10.13224/j.cnki.jasp.20250512
引用本文: 邓光洲, 杨顺华, 王艳华, 等. 超声速气流中激波诱导液滴变形/破碎过程及机理综述[J]. 航空动力学报, 2026, 42(X):20250512 doi: 10.13224/j.cnki.jasp.20250512
Deng Guangzhou, Yang Shunhua, Wang Yanhua, et al. Review of droplet deformation and breakup in supersonic flow induced by shock waves[J]. Journal of Aerospace Power, 2026, 42(X):20250512 doi: 10.13224/j.cnki.jasp.20250512
Citation: Deng Guangzhou, Yang Shunhua, Wang Yanhua, et al. Review of droplet deformation and breakup in supersonic flow induced by shock waves[J]. Journal of Aerospace Power, 2026, 42(X):20250512 doi: 10.13224/j.cnki.jasp.20250512

超声速气流中激波诱导液滴变形/破碎过程及机理综述

doi: 10.13224/j.cnki.jasp.20250512
基金项目: 北京流体动力科学研究中心资助
详细信息
    作者简介:

    邓光洲(1997-),男,博士生,主要从事超燃冲压发动机燃料雾化及燃烧研究。E-mail:572159721@qq.com

    通讯作者:

    杨顺华(1976-),男,研究员、博士生导师,博士,主要从事超燃冲压发动机总体研究。E-mail:ysh_cardc @126.com

  • 中图分类号: V231.3

Review of droplet deformation and breakup in supersonic flow induced by shock waves

  • 摘要:

    从液滴破碎动力学及两相物性影响、液滴近场激波系特征、湍流结构演化、液滴内部流动以及数值模拟方法研究等方面,综述了近年来超声速气流中激波诱导液滴变形/破碎的主要研究进展。相关研究主要集中在两相物性参数对破碎模式的影响规律、激波与液滴相互作用形成的复杂波系及其作用特征、湍流演化对液滴界面不稳定及破碎过程的调控作用等方面。认为激波诱导液滴破碎是气动力、界面不稳定性及湍流共同作用的复杂多尺度过程,但当前关于激波、湍流等因素对液滴破碎影响的研究仍停留在定性阶段。未来研究应以跨尺度高精度数值方法为基础,进一步加强激波与湍流对液滴破碎影响的定量表征,并实现激波与湍流对液滴破碎影响的解耦分析。

     

  • 图 1  雾化过程[12]

    Figure 1.  Atomization process[12]

    图 2  液滴破碎模式 [20]

    Figure 2.  Droplet breakup regimes[20]

    图 3  液滴破碎中的RT穿孔破碎模拟图像[18]

    Figure 3.  Simulated images of RT piercing breakup during droplet breakup[18]

    图 4  基于尺寸比的破碎转变[32]

    Figure 4.  Breakup transition based on size ratio[32]

    图 5  不稳定波实验图像[32]

    Figure 5.  Experimental image of instability waves[32]

    图 6  液滴破碎阈值随Oh变化[41]

    Figure 6.  Droplet breakup threshold as a function of Oh[41]

    图 7  正激波进化[32]

    Figure 7.  Normal shock wave evolution[32]

    图 8  激波影响的液滴变形过程

    Figure 8.  Droplet deformation process affected by shock waves

    图 9  液滴变形的近场压力云图[19]

    Figure 9.  Near-field pressure contour of droplet deformation[19]

    图 10  层流/湍流状态下液滴变形对比[94]

    Figure 10.  Comparison of droplet deformation under laminar/turbulent flow conditions[94]

    图 11  湍流大小对液滴变形的影响对比[94]

    Figure 11.  Comparison of the effects of turbulence scale on droplet deformation[94]

    图 12  非对称气团扫掠速度云图 [94]

    Figure 12.  Velocity contour of asymmetric airflow sweeping [94]

    图 13  激波冲击后液滴尾流区涡量云图[19]

    Figure 13.  Vorticity contour of the droplet wake region after shock wave impact

    图 14  尾流涡旋[60]

    Figure 14.  Wake vortices[60]

    图 15  液滴表面应力分布[119]

    Figure 15.  Droplet surface stress distribution[119]

    图 16  尾流卡门涡街[22]

    Figure 16.  Wake Kármán vortex street[22]

    图 17  液滴变形时间随Oh变化曲线[36]

    Figure 17.  Droplet deformation time curve as a function of Oh

    图 18  液滴总破碎时间随We和Oh变化曲线[40]

    Figure 18.  Total droplet breakup time curve as a function of We and Oh[40]

    图 19  液滴前后缘变形流动统计[62]

    图 20  第二奇异位置[62]

    图 21  模拟与势流理论对比[62]

    Figure 21.  Deformation flow statistics of droplet leading and trailing edges[62] Fig.20 Second singular position[62] Fig.21 Comparison between simulation and potential flow theory[62]

    表  1  自适应加密方法对比

    Table  1.   Comparison of adaptive mesh refinement methods

    对比项目数值误差估计流场特征捕捉几何结构检测
    驱动因素计算误差最小化流场特征识别界面追踪
    适用对象通用关键变量较多的流动仅关注几何界面
    计算难度较高较低适中
    代表性方法Richardson外推误差[148]、伴随误差[149]、
    小波系数误差[150]等
    密度梯度[31]、Q准则[151]、
    界面不稳定[152]等
    相场梯度[153]、细长结构[154]、
    各向异性[155]等
    下载: 导出CSV

    表  2  控制方程对比

    Table  2.   Comparison of governing equations

    对比项目 传统N-S方程 五方程扩散界面模型 七方程模型
    基本假设 瞬态平衡假设 瞬态平衡假设 趋向平衡假设
    控制方程 连续性方程、动量方程、能量方程、界面输运方程 两相连续性方程、混合动量方程、混合物能量方程、相分数输运方程 两相连续性、动量方程、能量方程、相分数演化方程
    计算效率 适中 较高 较低
    优势 界面几何精度高、
    通用性强
    效率与精度兼具、
    易于实现高阶格式
    物理解释完备、
    通用性强
    下载: 导出CSV

    表  3  界面追踪方法对比

    Table  3.   Comparison of interface tracking methods

    对比项目 显式界面追踪 隐式界面追踪
    基本原理 用离散标记点直接追踪界面,标记点随流体做拉格朗日运动 在固定欧拉网格上定义标量函数(如体积分数α),界面由函数等值面隐式确定
    主要优势 界面锐利无数值扩散、几何信息精确 自动处理界面拓扑变化、网格规则便于并行
    主要劣势 拓扑变化处理困难、大变形时标记点存在畸变、对激波诱导的流动精度较差 存在数值扩散/几何界面精度相对较低
    计算效率 中小规模效率高、大规模并行困难 计算效率高、网格结构规则、易于并行优化
    代表性方法 front-tracking、SPH VOF、level-set、phase-field
    下载: 导出CSV
  • [1] Sathiyamoorthy K, Danish T H, Iyengar V S, et al. Penetration and combustion studies of tandem liquid jets in supersonic crossflow[J]. Journal of Propulsion and Power, 2020, 36(6): 920-930. doi: 10.2514/1.B38047
    [2] 吴里银. 超声速气流中液体横向射流破碎与雾化机理研究[D]. 长沙: 国防科学技术大学, 2016. Wu Liyin. Breakup and atomization mechanism of liquid jet in supersonic crossflows[D]. Changsha: National University of Defense Technology, 2016. (in Chinese

    Wu Liyin. Breakup and atomization mechanism of liquid jet in supersonic crossflows[D]. Changsha: National University of Defense Technology, 2016. (in Chinese)
    [3] Lin Kuocheng, Kastengren A L, Hammack S, et al. Exploration of water jets in supersonic crossflow using X-ray diagnostics[J]. Atomization and Sprays, 2020, 30(5): 331-350. doi: 10.1615/AtomizSpr.2020034448
    [4] Donggyu Y, Sung H G. Numerical simulations on a liquid jet in supersonic crossflow using a homogeneous mixture model with AMR[R]. AIAA-2023-3041, 2023.
    [5] Wen Jian, Hu Yong, Nakanishi A, et al. Atomization and evaporation process of liquid fuel jets in crossflows: a numerical study using Eulerian/Lagrangian method[J]. International Journal of Multiphase Flow, 2020, 129: 103331. doi: 10.1016/j.ijmultiphaseflow.2020.103331
    [6] Fuster D, Bagué A, Boeck T, et al. Simulation of primary atomization with an octree adaptive mesh refinement and VOF method[J]. International Journal of Multiphase Flow, 2009, 35(6): 550-565. doi: 10.1016/j.ijmultiphaseflow.2009.02.014
    [7] Vadivukkarasan M, Panchagnula M V. Combined Rayleigh–Taylor and Kelvin–Helmholtz instabilities on an annular liquid sheet[J]. Journal of Fluid Mechanics, 2017, 812: 152-177. doi: 10.1017/jfm.2016.784
    [8] Isaac K. Liquid jet atomization in supersonic flows[R]. AIAA-1990-2189, 1990.
    [9] Missoum A, Isaac K, Drallmeier J. Atomization of liquid fuels in supersonic flow[R]. AIAA-1992-3235, 1992.
    [10] 李佩波. 超声速气流中横向喷雾的混合及燃烧过程数值模拟[D]. 长沙: 国防科技大学, 2019. Li Peibo. Numerical simulation of mixing and combustion process of transverse spray in supersonic flows[D]. Changsha: National University of Defense Technology, 2019. (in Chinese

    Li Peibo. Numerical simulation of mixing and combustion process of transverse spray in supersonic flows[D]. Changsha: National University of Defense Technology, 2019. (in Chinese)
    [11] 张军龙, 常军涛, 王瑄, 等. 基于支板稳燃的超声速火焰特性研究进展[J]. 空气动力学学报, 2020, 38(3): 577-592. Zhang Junlong, Chang Juntao, Wang Xuan, et al. Recent research progress on flame characteristics in strut-equipped scramjet combustor[J]. Acta Aerodynamica Sinica, 2020, 38(3): 577-592. (in Chinese doi: 10.7638/kqdlxxb-2019.0168

    Zhang Junlong, Chang Juntao, Wang Xuan, et al. Recent research progress on flame characteristics in strut-equipped scramjet combustor[J]. Acta Aerodynamica Sinica, 2020, 38(3): 577-592. (in Chinese) doi: 10.7638/kqdlxxb-2019.0168
    [12] Bhatia B, Johny T, De A. Understanding the liquid jet break-up in various regimes at elevated pressure using a compressible VOF-LPT coupled framework[J]. International Journal of Multiphase Flow, 2023, 159: 104303. doi: 10.1016/j.ijmultiphaseflow.2022.104303
    [13] Huang Zhiwei, Zhu Ruixuan, Davy M. The aerodynamic breakup and interactions of evaporating water droplets with a propagating shock wave[J]. Flow, Turbulence and Combustion, 2025, 114(1): 243-273. doi: 10.1007/s10494-024-00581-z
    [14] Meng J C, Colonius T. Numerical simulations of the early stages of high-speed droplet breakup[J]. Shock Waves, 2015, 25(4): 399-414. doi: 10.1007/s00193-014-0546-z
    [15] Gobyzov O A, Ryabov M N, Bilsky A V. Study of deformation and breakup of submillimeter droplets’ spray in a supersonic nozzle flow[J]. Applied Sciences, 2020, 10(18): 6149. doi: 10.3390/app10186149
    [16] Xu Zhikun, Wang Tianyou, Che Zhizhao. Droplet deformation and breakup in shear flow of air[J]. Physics of Fluids, 2020, 32(5): 052109. doi: 10.1063/5.0006236
    [17] Zhao Hui, Nguyen D, Duke D J, et al. Effect of turbulence on drop breakup in counter air flow[J]. International Journal of Multiphase Flow, 2019, 120: 103108. doi: 10.1016/j.ijmultiphaseflow.2019.103108
    [18] Dorschner B, Biasiori-poulanges L, Schmidmayer K, et al. On the formation and recurrent shedding of ligaments in droplet aerobreakup[J]. Journal of Fluid Mechanics, 2020, 904: A20. doi: 10.1017/jfm.2020.699
    [19] 宋家喜, 潘书诚. 高马赫数下激波液滴相互作用的数值模拟研究[J]. 力学学报, 2022, 54(9): 2419-2434. Song Jiaxi, Pan Shucheng. Numerical investigation of shock-droplet interaction with high-Mach numbers[J]. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(9): 2419-2434. (in Chinese doi: 10.6052/0459-1879-22-191

    Song Jiaxi, Pan Shucheng. Numerical investigation of shock-droplet interaction with high-Mach numbers[J]. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(9): 2419-2434. (in Chinese) doi: 10.6052/0459-1879-22-191
    [20] Jain S S, Tyagi N, Prakash R S, et al. Secondary breakup of drops at moderate Weber numbers: Effect of Density ratio and Reynolds number[J]. International Journal of Multiphase Flow, 2019, 117: 25-41. doi: 10.1016/j.ijmultiphaseflow.2019.04.026
    [21] Pilch M, Erdman C A. Use of breakup time data and velocity history data to predict the maximum size of stable fragments for acceleration-induced breakup of a liquid drop[J]. International Journal of Multiphase Flow, 1987, 13(6): 741-757. doi: 10.1016/0301-9322(87)90063-2
    [22] Song Xianzhao, Jiang Lin, Xie Lifeng, et al. Observation of von Kármán vortex street in a droplet breakup[J]. Physics of Fluids, 2021, 33(9): 093301. doi: 10.1063/5.0064873
    [23] Joseph D D, Belanger J, Beavers G S. Breakup of a liquid drop suddenly exposed to a high-speed airstream[J]. International Journal of Multiphase Flow, 1999, 25(6/7): 1263-1303. doi: 10.1016/s0301-9322(99)00043-9
    [24] Hwang S S, Liu Z, Reitz R D. Breakup mechanisms and drag coefficients of high-speed vaporizing liquid drops[J]. Atomization and Sprays, 1996, 6(3): 353-376. doi: 10.1615/AtomizSpr.v6.i3.60
    [25] Theofanous T G. Aerobreakup of Newtonian and viscoelastic liquids[J]. Annual Review of Fluid Mechanics, 2011, 43: 661-690. doi: 10.1146/annurev-fluid-122109-160638
    [26] Theofanous T G, Mitkin V V, Ng C L, et al. The physics of aerobreakup II: viscous liquids[J]. Physics of Fluids, 2012, 24(2): 022104. doi: 10.1063/1.3680867
    [27] Theofanous T G, Li G J, Dinh T N. Aerobreakup in rarefied supersonic gas flows[J]. Journal of Fluids Engineering, 2004, 126(4): 516-527. doi: 10.1115/1.1777234
    [28] Theofanous T G, Li G J, Dinh T N, et al. Aerobreakup in disturbed subsonic and supersonic flow fields[J]. Journal of Fluid Mechanics, 2007, 593: 131-170. doi: 10.1017/S0022112007008853
    [29] Theofanous T G, Mitkin V V, Ng C L. The physics of aerobreakup III: viscoelastic liquids[J]. Physics of Fluids, 2013, 25(3): 032101. doi: 10.1063/1.4792712
    [30] Theofanous T G, Li G J. On the physics of aerobreakup[J]. Physics of Fluids, 2008, 20(5): 052103. doi: 10.1063/1.2907989
    [31] Meng J C, Colonius T. Numerical simulation of the aerobreakup of a water droplet[J]. Journal of Fluid Mechanics, 2018, 835: 1108-1135. doi: 10.1017/jfm.2017.804
    [32] Sharma S, Pratap Singh A, Rao S S, et al. Shock induced aerobreakup of a droplet[J]. Journal of Fluid Mechanics, 2021, 929: A27. doi: 10.1017/jfm.2021.860
    [33] Wang Zhaoguang, Hopfes T, Giglmaier M, et al. Effect of Mach number on droplet aerobreakup in shear stripping regime[J]. Experiments in Fluids, 2020, 61(9): 193. doi: 10.1007/s00348-020-03026-1
    [34] Mizuno K, Yada T, Kamiya T, et al. Deformation behavior of liquid droplet in shock-induced atomization[J]. International Journal of Multiphase Flow, 2022, 155: 104141. doi: 10.1016/j.ijmultiphaseflow.2022.104141
    [35] Lee C H, Reitz R D. An experimental study of the effect of gas density on the distortion and breakup mechanism of drops in high speed gas stream[J]. International Journal of Multiphase Flow, 2000, 26(2): 229-244. doi: 10.1016/S0301-9322(99)00020-8
    [36] Guildenbecher D R, López-rivera C, Sojka P E. Secondary atomization[J]. Experiments in Fluids, 2009, 46(3): 371-402. doi: 10.1007/s00348-008-0593-2
    [37] Aalburg C, Van Leer B, Faeth G M. Deformation and drag properties of round drops subjected to shock-wave disturbances[J]. AIAA Journal, 2003, 41(12): 2371-2378. doi: 10.2514/2.6862
    [38] Liu Z, Reitz R D. An analysis of the distortion and breakup mechanisms of high speed liquid drops[J]. International Journal of Multiphase Flow, 1997, 23(4): 631-650. doi: 10.1016/S0301-9322(96)00086-9
    [39] Soni S K, Kirar P K, Kolhe P, et al. Deformation and breakup of droplets in an oblique continuous air stream[J]. International Journal of Multiphase Flow, 2020, 122: 103141. doi: 10.1016/j.ijmultiphaseflow.2019.103141
    [40] Suryaprakash R, Tomar G. Secondary breakup of drops[J]. Journal of the Indian Institute of Science, 2019, 99(1): 77-91. doi: 10.1007/s41745-018-0094-0
    [41] Hsiang L P, Faeth G M. Drop deformation and breakup due to shock wave and steady disturbances[J]. International Journal of Multiphase Flow, 1995, 21(4): 545-560. doi: 10.1016/0301-9322(94)00095-2
    [42] Radhakrishna V, Shang Weixiao, Yao Longchao, et al. Experimental characterization of secondary atomization at high Ohnesorge numbers[J]. International Journal of Multiphase Flow, 2021, 138: 103591. doi: 10.1016/j.ijmultiphaseflow.2021.103591
    [43] Gelfand B E. Droplet breakup phenomena in flows with velocity lag[J]. Progress in Energy and Combustion Science, 1996, 22(3): 201-265. doi: 10.1016/S0360-1285(96)00005-6
    [44] Kumar R, Kuloor N K. The formation of bubbles and drops[M]//Advances in Chemical Engineering: Volume 8. Amsterdam: Elsevier, 1970: 255-368.
    [45] Cohen R D. Effect of viscosity on drop breakup[J]. International Journal of Multiphase Flow, 1994, 20(1): 211-216. doi: 10.1016/0301-9322(94)90016-7
    [46] Kékesi T, Amberg G, Wittberg L P. Drop deformation and breakup[J]. International Journal of Multiphase Flow, 2014, 66: 1-10.
    [47] Osher S, Sethian J A. Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations[J]. Journal of Computational Physics, 1988, 79(1): 12-49. doi: 10.1016/0021-9991(88)90002-2
    [48] Lin Kuocheng, Kennedy P, Jackson T. Structures of water jets in a Mach 1.94 supersonic crossflow[R]. AIAA-2004-971, 2004.
    [49] Peng Shuting, Chen Fuzhen, Yan Hong, et al. Three-dimensional numerical simulation of tandem droplets accelerated by continuous uniform airflow[J]. Physical Review Fluids, 2025, 10(2): 024304. doi: 10.1103/PhysRevFluids.10.024304
    [50] Jackiw I M, Ashgriz N. On aerodynamic droplet breakup[J]. Journal of Fluid Mechanics, 2021, 913: A33. doi: 10.1017/jfm.2021.7
    [51] Zhao Hui, Wu Zhaowei, Li Weifeng, et al. Transition Weber number between surfactant-laden drop bag breakup and shear breakup of secondary atomization[J]. Fuel, 2018, 221: 138-143. doi: 10.1016/j.fuel.2018.02.119
    [52] Kumar Chandra N, Sharma S, Basu S, et al. Shock-induced aerobreakup of a polymeric droplet[J]. Journal of Fluid Mechanics, 2023, 965: A1. doi: 10.1017/jfm.2023.377
    [53] Utheza F, Saurel R, Daniel E, et al. Droplet break-up through an oblique shock wave[J]. Shock Waves, 1996, 5(5): 265-273. doi: 10.1007/BF02425219
    [54] Ranjan D, Oakley J, Bonazza R. Shock-bubble interactions[J]. Annual Review of Fluid Mechanics, 2011, 43: 117-140. doi: 10.1146/annurev-fluid-122109-160744
    [55] Henderson L F. On the refraction of shock waves[J]. Journal of Fluid Mechanics, 1989, 198: 365-386. doi: 10.1017/S0022112089000170
    [56] Sembian S, Liverts M, Tillmark N, et al. Plane shock wave interaction with a cylindrical water column[J]. Physics of Fluids, 2016, 28(5): 056102. doi: 10.1063/1.4948274
    [57] Ben-Dor G. Shock wave reflection phenomena[M]. Berlin, Heidelberg: Springer, 2007.
    [58] Bryson A E, Gross R W F. Diffraction of strong shocks by cones, cylinders, and spheres[J]. Journal of Fluid Mechanics, 1961, 10(1): 1-16. doi: 10.1017/s0022112061000019
    [59] Heilig W H. Diffraction of a shock wave by a cylinder[J]. The Physics of Fluids, 1969, 12(5): I-154-I-157.
    [60] 易翔宇. 激波诱导高速气流中液滴的变形与破碎实验研究[D]. 合肥: 中国科学技术大学, 2017. Yi Xiangyu. Experimental study of the deformation and breakup of a liquid drop in shock induced gas flow[D]. Hefei: University of Science and Technology of China, 2017. (in Chinese

    Yi Xiangyu. Experimental study of the deformation and breakup of a liquid drop in shock induced gas flow[D]. Hefei: University of Science and Technology of China, 2017. (in Chinese)
    [61] Das P, Udaykumar H S. A sharp-interface method for the simulation of shock-induced vaporization of droplets[J]. Journal of Computational Physics, 2020, 405: 109005. doi: 10.1016/j.jcp.2019.109005
    [62] Guan Ben, Liu Yao, Wen C Y, et al. Numerical study on liquid droplet internal flow under shock impact[J]. AIAA Journal, 2018, 56(9): 3382-3387. doi: 10.2514/1.J057134
    [63] Chen H, Liang S M. Flow visualization of shock/water column interactions[J]. Shock Waves, 2008, 17(5): 309-321. doi: 10.1007/s00193-007-0115-9
    [64] 吴润龙, 李祝军, 丁航. 平面激波冲击并排液滴的三维数值模拟研究[J]. 力学学报, 2022, 54(11): 2958-2969. Wu Runlong, Li Zhujun, Ding Hang. Impact of a planar shock onto side-by-side droplets: a 3d numerical study[J]. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(11): 2958-2969. (in Chinese doi: 10.6052/0459-1879-22-358

    Wu Runlong, Li Zhujun, Ding Hang. Impact of a planar shock onto side-by-side droplets: a 3d numerical study[J]. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(11): 2958-2969. (in Chinese) doi: 10.6052/0459-1879-22-358
    [65] Igra D, Takayama K. Experimental investigation of two cylindrical water columns subjected to planar shock wave loading[J]. Journal of Fluids Engineering, 2003, 125(2): 325-331. doi: 10.1115/1.1538628
    [66] Mehta Y, Jackson T L, Zhang J, et al. Numerical investigation of shock interaction with one-dimensional transverse array of particles in air[J]. Journal of Applied Physics, 2016, 119(10): 104901. doi: 10.1063/1.4943616
    [67] Sridharan P, Jackson T L, Zhang J, et al. Shock interaction with deformable particles using a constrained interface reinitialization scheme[J]. Journal of Applied Physics, 2016, 119(6): 064904. doi: 10.1063/1.4941687
    [68] Xu Sheng, Fan Wenqi, Wu Wangxia, et al. Analysis of wave converging phenomena inside the shocked two-dimensional cylindrical water column[J]. Journal of Fluid Mechanics, 2023, 964: A12. doi: 10.1017/jfm.2023.239
    [69] Pettersen M S, Balibar S, Maris H J. Experimental investigation of cavitation in superfluid 4He[J]. Physical Review B, Condensed Matter, 1994, 49(17): 12062-12070. doi: 10.1103/PhysRevB.49.12062
    [70] Caupin F. Liquid-vapor interface, cavitation, and the phase diagram of water[J]. Physical Review E, 2005, 71(5): 051605. doi: 10.1103/PhysRevE.71.051605
    [71] Herbert E, Caupin F. The limit of metastability of water under tension: theories and experiments[J]. Journal of Physics: Condensed Matter, 2005, 17(45): S3597-S3602. doi: 10.1088/0953-8984/17/45/053
    [72] Zheng Q, Durben D J, Wolf G H, et al. Liquids at large negative pressures: water at the homogeneous nucleation limit[J]. Science, 1991, 254(5033): 829-832. doi: 10.1126/science.254.5033.829
    [73] Caupin F, Herbert E. Cavitation in water: a review[J]. Comptes Rendus Physique, 2006, 7(9/10): 1000-1017.
    [74] Herbert E, Balibar S, Caupin F. Cavitation pressure in water[J]. Physical Review E, 2006, 74(4): 041603.
    [75] El Mekki Azouzi M, Ramboz C, Lenain J F, et al. A coherent picture of water at extreme negative pressure[J]. Nature Physics, 2013, 9(1): 38-41. doi: 10.1038/nphys2475
    [76] Biasiori-poulanges L, Schmidmayer K. A phenomenological analysis of droplet shock-induced cavitation using a multiphase modeling approach[J]. Physics of Fluids, 2023, 35(1): 013312 doi: 10.1063/5.0127105
    [77] Tang Mao, Zhang Hongchao, Lu Jian. Pressure dynamics of an internal shock wave emission inside a water droplet and potential cavitation[J]. New Journal of Physics, 2023, 25(12): 123037. doi: 10.1088/1367-2630/ad1419
    [78] Sebastian D S, Muruganandam T M. Influence of oblique shock interaction on the liquid jet in supersonic crossflow[J]. Acta Astronautica, 2025, 235: 302-319. doi: 10.1016/j.actaastro.2025.05.040
    [79] Arefyev K Y, Guskov O V, Prokhorov A N, et al. Experimental research of gasdynamic liquid drops breakup in the supersonic flow with an oblique shock wave[J]. High Temperature, 2020, 58(6): 884-892. doi: 10.1134/S0018151X20060036
    [80] Minko A, Guskov O, Arefyev K, et al. Physical and mathematical modeling of the interaction of water droplets and high-speed gas flow[J]. Applied Sciences, 2021, 11(23): 11146. doi: 10.3390/app112311146
    [81] Tripathi M, Ganti H, Khare P. Interactions between shock waves and liquid droplet clusters: interfacial physics[J]. Journal of Fluids Engineering, 2022, 144(10): 101401. doi: 10.1115/1.4054181
    [82] 赵家兴, 岳亚军, 刘凇含, 等. 激波冲击下液滴的雾化特性研究[J]. 计算力学学报, 2024, 41(4): 781-789. Zhao Jiaxing, Yue Yajun, Liu Songhan, et al. Atomization characteristics of droplet under shock wave[J]. Chinese Journal of Computational Mechanics, 2024, 41(4): 781-789. (in Chinese doi: 10.7511/jslx20230214001

    Zhao Jiaxing, Yue Yajun, Liu Songhan, et al. Atomization characteristics of droplet under shock wave[J]. Chinese Journal of Computational Mechanics, 2024, 41(4): 781-789. (in Chinese) doi: 10.7511/jslx20230214001
    [83] 吴龙, 王江峰, 李龙飞, 等. 前置翼片式涡流发生器对燃烧室氢气燃料掺混特性的影响[J]. 空气动力学学报, 2023, 41(3): 33-43. Wu Long, Wang Jiangfeng, Li Longfei, et al. Effect of front vane vortex generators on hydrogen fuel mixing characteristics in combustor[J]. Acta Aerodynamica Sinica, 2023, 41(3): 33-43. (in Chinese doi: 10.7638/kqdlxxb-2021.0414

    Wu Long, Wang Jiangfeng, Li Longfei, et al. Effect of front vane vortex generators on hydrogen fuel mixing characteristics in combustor[J]. Acta Aerodynamica Sinica, 2023, 41(3): 33-43. (in Chinese) doi: 10.7638/kqdlxxb-2021.0414
    [84] Sevik M, Park S H. The splitting of drops and bubbles by turbulent fluid flow[J]. Journal of Fluids Engineering, 1973, 95(1): 53-60. doi: 10.1115/1.3446958
    [85] Baldyga J, Podgórska W. Drop break-up in intermittent turbulence: Maximum stable and transient sizes of drops[J]. The Canadian Journal of Chemical Engineering, 1998, 76(3): 456-470. doi: 10.1002/cjce.5450760316
    [86] Prevish T. Secondary droplet breakup in highly turbulent flowfields[D]. Ann Arbor: University of Michigan, 1998.
    [87] Andersson R, Andersson B. On the breakup of fluid particles in turbulent flows[J]. AIChE Journal, 2006, 52(6): 2020-2030. doi: 10.1002/aic.10831
    [88] Andersson R, Andersson B. Modeling the breakup of fluid particles in turbulent flows[J]. AIChE Journal, 2006, 52(6): 2031-2038. doi: 10.1002/aic.10832
    [89] Hinze J O. Fundamentals of the hydrodynamic mechanism of splitting in dispersion processes[J]. AIChE Journal, 1955, 1(3): 289-295. doi: 10.1002/aic.690010303
    [90] Kolmogorov A N. On the disintegration of drops in a turbulent flow[J]. Doklady Akademii Nauk SSSR, 1949, 66(5): 825-828.
    [91] Rivière A, Mostert W, Perrard S, et al. Sub-Hinze scale bubble production in turbulent bubble break-up[J]. Journal of Fluid Mechanics, 2021, 917: A40. doi: 10.1017/jfm.2021.243
    [92] Institute for Liquid Atomization and Spray Systems. 11th International Annual Conference on Liquid Atomization and Spray Systems 2009 (ICLASS 2009)[M]. Red Hook, US: Curran, 2011.
    [93] Rivière A, Ruth D J, Mostert W, et al. Capillary driven fragmentation of large gas bubbles in turbulence[J]. Physical Review Fluids, 2022, 7(8): 083602. doi: 10.1103/PhysRevFluids.7.083602
    [94] Tang Kaitao, Adcock T A A, Mostert W. Droplet bag formation in turbulent airflows[J]. Physical Review Fluids, 2025, 10(3): 033604. doi: 10.1103/PhysRevFluids.10.033604
    [95] Jiao Daokuan, Jiao Kui, Zhang Fan, et al. Direct numerical simulation of droplet deformation in turbulent flows with different velocity profiles[J]. Fuel, 2019, 247: 302-314. doi: 10.1016/j.fuel.2019.03.010
    [96] Jackiw I M, Ashgriz N. Prediction of the droplet size distribution in aerodynamic droplet breakup[J]. Journal of Fluid Mechanics, 2022, 940: A17. doi: 10.1017/jfm.2022.249
    [97] Kornev N, Hassel E. Method of random spots for generation of synthetic inhomogeneous turbulent fields with prescribed autocorrelation functions[J]. Communications in Numerical Methods in Engineering, 2007, 23(1): 35-43. doi: 10.1002/cnm.880
    [98] Cui Zhiwen, Qiu Jingran, Jiang Xinyu, et al. Effect of fluid inertial torque on the rotational and orientational dynamics of tiny spheroidal particles in turbulent channel flow[J]. Journal of Fluid Mechanics, 2023, 977: A20. doi: 10.1017/jfm.2023.942
    [99] Zhao Hui, Liu Haifeng, Cao Xiankui, et al. Breakup characteristics of liquid drops in bag regime by a continuous and uniform air jet flow[J]. International Journal of Multiphase Flow, 2011, 37(5): 530-534. doi: 10.1016/j.ijmultiphaseflow.2010.12.006
    [100] Zhao Hui, Liu Haifeng, Li Weifeng, et al. Morphological classification of low viscosity drop bag breakup in a continuous air jet stream[J]. Physics of Fluids, 2010, 22(11): 114103. doi: 10.1063/1.3490408
    [101] Chou W H, Faeth G M. Temporal properties of secondary drop breakup in the bag breakup regime[J]. International Journal of Multiphase Flow, 1998, 24(6): 889-912. doi: 10.1016/S0301-9322(98)00015-9
    [102] Liu A B, Mather D, Reitz R D. Modeling the effects of drop drag and breakup on fuel sprays[R]. Warrendale, PA: SAE International, 1993.
    [103] Vela-martín A, Avila M. Deformation of drops by outer eddies in turbulence[J]. Journal of Fluid Mechanics, 2021, 929: A38. doi: 10.1017/jfm.2021.879
    [104] Xu Xu, Qi Yinghe, Zhong Shijie, et al. Intermittency of bubble deformation in turbulence[J]. Physical Review Letters, 2024, 133(21): 214001. doi: 10.1103/PhysRevLett.133.214001
    [105] Wang Yayun, Sierakowski A, Prosperetti A. Rotational dynamics of a particle in a turbulent stream[J]. Physical Review Fluids, 2019, 4(6): 064304. doi: 10.1103/PhysRevFluids.4.064304
    [106] Sanadi D, Rajamanickam K, Basu S. Analysis of hollow-cone spray injected in an unconfined, isothermal, coannular swirling jet environment[J]. Atomization and Sprays, 2017, 27(1): 7-29. doi: 10.1615/AtomizSpr.2016016249
    [107] Hanson A R, Domich E G, Adams H S. Shock tube investigation of the breakup of drops by air blasts[J]. The Physics of Fluids, 1963, 6(8): 1070-1080. doi: 10.1063/1.1706864
    [108] Nicholls J A, Ranger A A. Aerodynamic shattering of liquid drops[J]. AIAA Journal, 1969, 7(2): 285-290. doi: 10.2514/3.5087
    [109] Lefebvre A H, McDonell V G. Atomization and Sprays[M]. 2 ed. | Boca Raton: Taylor & Francis, CRC Press, 2017.
    [110] Ranger A A, Nlcholls J A. Shape and surrounding flowfield of a drop in a high-speed gas stream[J]. AIAA Journal, 1970, 8(9): 1720-1722. doi: 10.2514/3.5980
    [111] Rind E, Castro I P. On the effects of free-stream turbulence on axisymmetric disc wakes[J]. Experiments in Fluids, 2012, 53(2): 301-318. doi: 10.1007/s00348-012-1288-2
    [112] Tang K, Adcock T A A, Mostert W. Bag film breakup of droplets in uniform airflows[J]. Journal of Fluid Mechanics, 2023, 970: A9. doi: 10.1017/jfm.2023.605
    [113] Tang K, Adcock T A A, Mostert W. Fragmentation of colliding liquid rims[J]. Journal of Fluid Mechanics, 2024, 987: A18. doi: 10.1017/jfm.2024.392
    [114] Vledouts A, Quinard J, Vandenberghe N, et al. Explosive fragmentation of liquid shells[J]. Journal of Fluid Mechanics, 2016, 788: 246-273. doi: 10.1017/jfm.2015.716
    [115] Néel B, Lhuissier H, Villermaux E. ‘Fines’ from the collision of liquid rims[J]. Journal of Fluid Mechanics, 2020, 893: A16. doi: 10.1017/jfm.2020.235
    [116] Kalita J C, Sen S. Unsteady separation leading to secondary and tertiary vortex dynamics: the sub-α- and sub-β-phenomena[J]. Journal of Fluid Mechanics, 2013, 730: 19-51. doi: 10.1017/jfm.2013.272
    [117] Bouard R, Coutanceau M. The early stage of development of the wake behind an impulsively started cylinder for 40<Re <104[J]. Journal of Fluid Mechanics, 1980, 101(3): 583-607. doi: 10.1017/S0022112080001814
    [118] 易翔宇, 朱雨建, 杨基明. 激波诱导高速气流中液滴的初期变形[J]. 爆炸与冲击, 2017, 37(5): 853-862. Yi Xiangyu, Zhu Yujian, Yang Jiming. Early-stage deformation of liquid drop in shock induced high-speed flow[J]. Explosion and Shock Waves, 2017, 37(5): 853-862. (in Chinese doi: 10.11883/1001-1455(2017)05-0853-10

    Yi Xiangyu, Zhu Yujian, Yang Jiming. Early-stage deformation of liquid drop in shock induced high-speed flow[J]. Explosion and Shock Waves, 2017, 37(5): 853-862. (in Chinese) doi: 10.11883/1001-1455(2017)05-0853-10
    [119] 易翔宇, 朱雨建, 杨基明. 相近韦伯数条件下激波波后液滴初期变形的影响机制[J]. 爆炸与冲击, 2018, 38(3): 525-533. Yi Xiangyu, Zhu Yujian, Yang Jiming. Mechanism of early-stage drop deformation in shock induced flow at limited Weber numbers[J]. Explosion and Shock Waves, 2018, 38(3): 525-533. (in Chinese doi: 10.11883/bzycj-2016-0269

    Yi Xiangyu, Zhu Yujian, Yang Jiming. Mechanism of early-stage drop deformation in shock induced flow at limited Weber numbers[J]. Explosion and Shock Waves, 2018, 38(3): 525-533. (in Chinese) doi: 10.11883/bzycj-2016-0269
    [120] Poplavski S V, Minakov A V, Shebeleva A A, et al. On the interaction of water droplet with a shock wave: Experiment and numerical simulation[J]. International Journal of Multiphase Flow, 2020, 127: 103273. doi: 10.1016/j.ijmultiphaseflow.2020.103273
    [121] Thoraval M J, Takehara K, Etoh T G, et al. Von Kármán vortex street within an impacting drop[J]. Physical Review Letters, 2012, 108(26): 264506. doi: 10.1103/PhysRevLett.108.264506
    [122] Williamson C K. Vortex dynamics in the cylinder wake[J]. Annual Review of Fluid Mechanics, 1996, 28: 477-539. doi: 10.1146/annurev.fl.28.010196.002401
    [123] Hu D L, Chan B, Bush J W M. The hydrodynamics of water strider locomotion[J]. Nature, 2003, 424(6949): 663-666. doi: 10.1038/nature01793
    [124] Baudet C, Ciliberto S, Pinton J F. Spectral analysis of the von Kármán flow using ultrasound scattering[J]. Physical Review Letters, 1991, 67(2): 193-195. doi: 10.1103/PhysRevLett.67.193
    [125] Samanta T, Tian Hui, Nakariakov V M. Evidence for vortex shedding in the Sun’s hot corona[J]. Physical Review Letters, 2019, 123(3): 035102. doi: 10.1103/PhysRevLett.123.035102
    [126] Dynnikova G Y, Dynnikov Y A, Guvernyuk S V, et al. Stability of a reverse Karman vortex street[J]. Physics of Fluids, 2021, 33(2): 024102. doi: 10.1063/5.0035575
    [127] Kim I. On the morphology of two-dimensional laminar vortex streets behind triangles[J]. Physics of Fluids, 2021, 33(3): 033601. doi: 10.1063/5.0037288
    [128] Zhu Jianxun, Jiang Fengjian, Holmedal L E. Three-dimensional wake transition behind an elliptic cylinder near a moving wall[J]. Physics of Fluids, 2021, 33(4): 043606. doi: 10.1063/5.0048219
    [129] Zhu Hongjun, Zhong Jiawen, Zhou Tongming. Wake structure characteristics of three tandem circular cylinders at a low Reynolds number of 160[J]. Physics of Fluids, 2021, 33(4): 044113. doi: 10.1063/5.0050385
    [130] Cuadra A, Di Renzo M, Hoste J O E, et al. Review of shock-turbulence interaction with a focus on hypersonic flow[J]. Physics of Fluids, 2025, 37(4): 045129. doi: 10.1063/5.0255816
    [131] Dai Z, Faeth G M. Temporal properties of secondary drop breakup in the multimode breakup regime[J]. International Journal of Multiphase Flow, 2001, 27(2): 217-236. doi: 10.1016/S0301-9322(00)00015-X
    [132] Boiko V M, Papyrin A N, Poplavskii S V. Dynamics of droplet breakup in shock waves[J]. Journal of Applied Mechanics and Technical Physics, 1987, 28(2): 263-269. doi: 10.1007/BF00918731
    [133] Engel O G. Fragmentation of waterdrops in the zone behind an air shock[J]. Journal of Research of the National Bureau of Standards, 1958, 60(3): 245. doi: 10.6028/jres.060.029
    [134] Simpkins P G, Bales E L. Water-drop response to sudden accelerations[J]. Journal of Fluid Mechanics, 1972, 55(4): 629-639. doi: 10.1017/S0022112072002058
    [135] Ranger A A, Nicholls J A. Atomization of liquid droplets in a convective gas stream[J]. International Journal of Heat and Mass Transfer, 1972, 15(6): 1203-1211. doi: 10.1016/0017-9310(72)90185-8
    [136] Simpkins P. On the distortion and breakup of suddenly accelerated droplets[R]. AIAA-1971-325, 1971.
    [137] 肖毅, 施红辉, 吴宇, 等. 激波与液滴作用的空气动力学现象的实验研究[J]. 浙江理工大学学报, 2013, 30(2): 203-207. Xiao Yi, Shi Honghui, Wu Yu, et al. Experimental study on aerodynamic phenomenon of interaction between shock wave and liquid drop[J]. Journal of Zhejiang Sci-Tech University, 2013, 30(2): 203-207. (in Chinese doi: 10.3969/j.issn.1673-3851.2013.02.015

    Xiao Yi, Shi Honghui, Wu Yu, et al. Experimental study on aerodynamic phenomenon of interaction between shock wave and liquid drop[J]. Journal of Zhejiang Sci-Tech University, 2013, 30(2): 203-207. (in Chinese) doi: 10.3969/j.issn.1673-3851.2013.02.015
    [138] Bian Qingyong, Zhu Chengxiang, Wang Jingxin, et al. Numerical investigation on the characteristics of single droplet deformation in the airflow at different temperatures[J]. Physics of Fluids, 2022, 34(7): 073307. doi: 10.1063/5.0094748
    [139] Ortiz C, Joseph D D, Beavers G S. Acceleration of a liquid drop suddenly exposed to a high-speed airstream[J]. International Journal of Multiphase Flow, 2004, 30(2): 217-224. doi: 10.1016/j.ijmultiphaseflow.2003.11.004
    [140] Boiko V M, Poplavski S V. On the dynamics of drop acceleration at the early stage of velocity relaxation in a shock wave[J]. Combustion, Explosion, and Shock Waves, 2009, 45(2): 198-204. doi: 10.1007/s10573-009-0026-4
    [141] Boiko V M, Poplavskii S V. Particle and drop dynamics in the flow behind a shock wave[J]. Fluid Dynamics, 2007, 42(3): 433-441. doi: 10.1134/S0015462807030118
    [142] Hsiang L P, Faeth G M. Near-limit drop deformation and secondary breakup[J]. International Journal of Multiphase Flow, 1992, 18(5): 635-652. doi: 10.1016/0301-9322(92)90036-G
    [143] Gel’fand B E, Gubin S A, Kogarko S M, et al. Singularities of the breakup of viscous liquid droplets in shock waves[J]. Journal of Engineering Physics, 1973, 25(3): 1140-1142. doi: 10.1007/BF00838143
    [144] Boiko V M, Poplavski S V. Experimental study of two types of stripping breakup of the drop in the flow behind the shock wave[J]. Combustion, Explosion, and Shock Waves, 2012, 48(4): 440-445. doi: 10.1134/S0010508212040107
    [145] Yoshida T, Takayama K. Interaction of liquid droplets with planar shock waves[J]. Journal of Fluids Engineering, 1990, 112(4): 481-486. doi: 10.1115/1.2909431
    [146] Wadhwa A R, Magi V, Abraham J. Transient deformation and drag of decelerating drops in axisymmetric flows[J]. Physics of Fluids, 2007, 19(11): 113301. doi: 10.1063/1.2800038
    [147] Popinet S. Numerical models of surface tension[J]. Annual Review of Fluid Mechanics, 2018, 50: 49-75. doi: 10.1146/annurev-fluid-122316-045034
    [148] Berger M J, Colella P. Local adaptive mesh refinement for shock hydrodynamics[J]. Journal of Computational Physics, 1989, 82(1): 64-84. doi: 10.1016/0021-9991(89)90035-1
    [149] Fidkowski K J, Darmofal D L. Review of output-based error estimation and mesh adaptation in computational fluid dynamics[J]. AIAA Journal, 2011, 49(4): 673-694. doi: 10.2514/1.J050073
    [150] Harten A. Multiresolution algorithms for the numerical solution of hyperbolic conservation laws[J]. Communications on Pure and Applied Mathematics, 1995, 48(12): 1305-1342. doi: 10.1002/cpa.3160481201
    [151] Hunt J C R, Wray A A, Moin P. Eddies, streams, and convergence zones in turbulent flows[C]//Studying turbulence using numerical simulation databases 2: proceedings of the 1988 Summer Program. Stanford: Center for Turbulence Research, Stanford University, 1988: 193-208.
    [152] Sharp D H. An overview of Rayleigh-Taylor instability[J]. Physica D: Nonlinear Phenomena, 1984, 12(1/2/3): 3-18.
    [153] Mirjalili B, Jain S, Dodd M. Interface-capturing methods for two-phase flows: an overview and recent developments[C]//Center for Turbulence Research Annual Research Briefs. Stanford: Center for Turbulence Research, Stanford University, 2017: 117-135.
    [154] Shinjo J, Umemura A. Simulation of liquid jet primary breakup: Dynamics of ligament and droplet formation[J]. International Journal of Multiphase Flow, 2010, 36(7): 513-532. doi: 10.1016/j.ijmultiphaseflow.2010.03.008
    [155] Alauzet F, Loseille A. A decade of progress on anisotropic mesh adaptation for computational fluid dynamics[J]. Computer-Aided Design, 2016, 72: 13-39. doi: 10.1016/j.cad.2015.09.005
    [156] Kaiser J W J, Winter J M, Adami S, et al. Investigation of interface deformation dynamics during high-Weber number cylindrical droplet breakup[J]. International Journal of Multiphase Flow, 2020, 132: 103409. doi: 10.1016/j.ijmultiphaseflow.2020.103409
    [157] Kapila A K, Menikoff R, Bdzil J B, et al. Two-phase modeling of deflagration-to-detonation transition in granular materials: Reduced equations[J]. Physics of Fluids, 2001, 13(10): 3002-3024. doi: 10.1063/1.1398042
    [158] Allaire G, Clerc S, Kokh S. A five-equation model for the simulation of interfaces between compressible fluids[J]. Journal of Computational Physics, 2002, 181(2): 577-616. doi: 10.1006/jcph.2002.7143
    [159] Baer M R, Nunziato J W. A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials[J]. International Journal of Multiphase Flow, 1986, 12(6): 861-889. doi: 10.1016/0301-9322(86)90033-9
    [160] Subrahmanyam N, Lal B. A Textbook of Sound[M]. 2nd ed. New Delhi: Vikas Publishing House, 1995.
    [161] Saurel R, Abgrall R. A multiphase Godunov method for compressible multifluid and multiphase flows[J]. Journal of Computational Physics, 1999, 150(2): 425-467. doi: 10.1006/jcph.1999.6187
    [162] Tryggvason G, Bunner B, Esmaeeli A, et al. A front-tracking method for the computations of multiphase flow[J]. Journal of Computational Physics, 2001, 169(2): 708-759. doi: 10.1006/jcph.2001.6726
    [163] Pozrikidis C. Boundary integral and singularity methods for linearized viscous flow[M]. Cambridge: Cambridge University Press, 1992.
    [164] Monaghan J J. Simulating free surface flows with SPH[J]. Journal of Computational Physics, 1994, 110(2): 399-406. doi: 10.1006/jcph.1994.1034
    [165] Liu G R, Liu M B, Li Shaofan. Smoothed particle hydrodynamics: a meshfree method[J]. Computational Mechanics, 2004, 33(6): 491. doi: 10.1142/9789812564405
    [166] Colagrossi A, Landrini M. Numerical simulation of interfacial flows by smoothed particle hydrodynamics[J]. Journal of Computational Physics, 2003, 191(2): 448-475. doi: 10.1016/S0021-9991(03)00324-3
    [167] Tryggvason G, Scardovelli R, Zaleski S. Direct numerical simulations of gas-liquid multiphase flows[M]. Cambridge: Cambridge University Press, 2011.
    [168] Hirt C W, Nichols B D. Volume of fluid (VOF) method for the dynamics of free boundaries[J]. Journal of Computational Physics, 1981, 39(1): 201-225. doi: 10.1016/0021-9991(81)90145-5
    [169] Jacqmin D. Calculation of two-phase navier–stokes flows using phase-field modeling[J]. Journal of Computational Physics, 1999, 155(1): 96-127. doi: 10.1006/jcph.1999.6332
    [170] Xu Zhikun, Wang Tianyou, Che Zhizhao. Droplet breakup in airflow with strong shear effect[J]. Journal of Fluid Mechanics, 2022, 941: A54. doi: 10.1017/jfm.2022.326
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