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The trajectory of the spray is studied theoretically and experimentally when a round liquid jet is injected into a supersonic crossflow vertically. A solid model of continuous liquid column is established in three-dimensional space. The cross-section deformation equation of the continuous liquid column along the injection direction is established using a method of micro-element analysis. The stress analysis of cross section is simplified into a two-dimensional droplet. The shape of the cross section is considered to continuously change from circular to elliptical shape. And the bow shock wave in front of the jet column is simplified into an oblique shock wave with a known shock angle. Based on this, the calculation of aerodynamic force is greatly simplified. A dimensionless parameter named effective deformation time of liquid column (the logogram is
${t_{\rm valid}}$ ) is defined and used to judge the end point of the liquid column quantitatively. The liquid jet trajectory and cross-section deformation can be calculated using MATLAB software. The instantaneous images of continuous liquid columns in supersonic crossflow are captured using high-spatial-resolution microscopic imaging methods. The microscopic imaging system is composed of a double pulse solid-state laser, computer, CCD camera, synchronous controller, microscope lens and laser diffuser. After passing through the laser diffuser, a plane background light with uniform distribution is formed on the scattering plate. The mean filtering method is used to filter the original image. After filtering, the range of gray distribution in the background area is obviously reduced. The distribution of gray value is more concentrated, and the background of the image is more uniform. Then the image edge detection function is used to obtain the near-field jet trajectory. The parameter variables studied include liquid injection pressure drop (1–2 MPa), liquid nozzle diameter (0.5 mm/1.0 mm), and liquid gas momentum ratio (3.32–7.27). The results show that the continuous liquid column model can better predict the jet trajectory on the center plane and the shape of the liquid column in three-dimensional space. It is indicated that the predictive result matches well with the experimental result. This study is of great significance for establishing the solid-particle coupling model of liquid jet in supersonic crossflows.[1] Xiao F, Dianat M, Mcguirk J J 2013 AIAA J. 51 2878
Google Scholar
[2] 夏同军, 董永强, 曹义刚 2013 62 214702
Google Scholar
Xia T J, Dong Y Q, Cao Y G 2013 Acta Phys. Sin. 62 214702
Google Scholar
[3] Heister S D 2011 Handbook of Atomization and Sprays: Theory and Applications (New York: Springer) pp65−660
[4] Li C Y, Li C, Xiao F, Li Q L, Zhu Y H 2019 Aerosp. Sci. Technol. 95 105426
Google Scholar
[5] Dixon D R, Gruber M R, Jackson T A, Lin K C 2005 43th AIAA Aerospace Sciences Meeting & Exhibit Reno, America, January 10−13, 2005 p733
[6] Wu L Y, Wang Z G, Li Q L, Li C 2016 J. Visualization 3 337
Google Scholar
[7] Hu R S, Li Q, Li C Y, Li C 2019 Acta Astronaut. 159 440
Google Scholar
[8] Lin K C, Kennedy P J 2002 40 th AIAA Aerospace Sciences Meeting & Exhibit Reno, America, January 14−17, 2002 p873
[9] Lin K C, Kennedy P J, Kennedy P J, Jackson T A 2004 42th AIAA Aerospace Sciences Meeting & Exhibit Reno, America, January 5−8, 2004 p971
[10] 刘静, 王辽, 张佳, 韦宝禧, 徐旭 2008 航空动力学报 4 146
Google Scholar
Liu J, Wang L, Zhang J, Wei B X, Xu X 2008 J. Aerosp. Power 4 146
Google Scholar
[11] 仝毅恒, 李清廉, 吴里银 2012 国防科技大学学报 2 73
Google Scholar
Tong Y H, Li Q L, Wu L Y 2012 J. Natl. Univ. Def. Technol. 2 73
Google Scholar
[12] 李春 2014 硕士学位论文 (长沙: 国防科学技术大学)
Li C 2012 M. S. Thesis (Changsha: National University of Defense Technology) (in Chinese)
[13] 吴里银, 王振国, 李清廉, 李春 中国专利 zl201410800056.5 [2018-2-2]
Wu L Y, Wang Z G, Li Q L, Li C Chinese Patent zl201410800056.5 [20218-2-2] (in Chinese)
[14] 胡润生, 朱元昊, 张翔宇, 李清廉 2019 工程热 7 1659
Google Scholar
Hu R S, Zhu Y H, Zhang X Y, Li Q L 2019 J. Eng. Therm. 7 1659
Google Scholar
[15] 吴里银, 王振国, 李清廉, 李春 2016 68 094701
Google Scholar
Wu L Y, Wang Z G, Li Q L, Li C 2016 Acta Phys. Sin. 68 094701
Google Scholar
[16] 周曜智, 李清廉, 李晨阳 2020 推进技术 41 7
Google Scholar
Zhou Y Z, Li Q L, Li C Y 2020 J. Propul. Technol. 41 7
Google Scholar
[17] Mashayek A, Behzad M, Ashgriz N 2011 AIAA J. 49 2407
Google Scholar
[18] 李春 2020 博士学位论文 (长沙: 国防科技大学)
Li C 2020 Ph. D. Dissertation (Changsha: National University of Defense Technology) (in Chinese)
[19] Xiao F, Wang Z G, Sun M B, Liang J H, Liu N 2016 Int. J. Multiphase Flow 87 229
Google Scholar
[20] 李佩波, 王振国, 孙明波, 汪洪波 2016 宇航学报 37 79
Google Scholar
Li P B, Wang Z G, Sun M B, Wang H B 2016 J. Astronautics 37 79
Google Scholar
[21] Zhu Y X, Xiao F, Li Q L, Li C Y, Lin S 2018 Acta Astronaut. 154 119
Google Scholar
[22] Liu N, Wang Z G, Sun M B, Deiterding R, Wang H B 20191 Aerosp. Sci. Technol. 91 456
Google Scholar
[23] Douglas H F, Vitor V, Lucas S M, Francisco J S 2019 Int. J. Multiphase Flow 114 98
Google Scholar
[24] Li P B, Li C Y, Wang H B, Sun M B, Liu C Y, Wang Z G, Huang Y H 2019 Aerosp. Sci. Technol. 94 105401
Google Scholar
[25] Nguyen T, Karagozian A R 1992 J. Propul. Power 8 21
Google Scholar
[26] Clark M M 1988 Chem. Eng. Sci. 43 671
Google Scholar
[27] Inamura T 2000 J. Propul. Power 16 155
Google Scholar
[28] Sallam K A, Aalburg C, Faeth G M 2004 AIAA J. 42 2529
Google Scholar
[29] Wu P K, Kirkendall K A, Fuller R P, Nejad A S 1997 J. Propul. Power 13 64
Google Scholar
[30] 吴里银 2016 博士学位论文 (长沙: 国防科学技术大学
Wu L Y 2016 Ph. D. Dissertation (Changsha: National University of Defense Technology) (in Chinese)
[31] Lee K, Aalburg C, Diez F J, Faeth G M, Sallam K A 2007 AIAA J. 45 1907
Google Scholar
[32] Ranger A A, Nicholls J A 1969 AIAA J. 7 285
Google Scholar
[33] Chryssakis C A, Assanis, D N 2005 18th Annual Conference on Liquid and Atomization and Spray Systems, Irvine, America, May 15−18, 2005 p283
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图 2 显微成像系统原理图
Figure 2. Schematic diagram of microscopic imaging system
图 3 显微成像系统原理图 (a) QM1镜头; (b) QM1镜头工作特性曲线
Figure 3. Schematic diagram of microscopic imaging system: (a) QM1 camera lens; (b) QM1 microscopy camera lens working characteristic curve
图 4 液体微元的受力分析示意图
Figure 4. Aerodynamic drag coefficient calculation domain
图 5 二维液滴附近网格
Figure 5. Grids near the 2D droplet
图 6 对称面流向速度分布
Figure 6. Symmetrical flow velocity distributions
图 7 二维液滴表面静压分布
Figure 7. Surface static pressure distributions of 2D droplet
图 8 连续液柱变形分区示意图
Figure 8. Schematic diagram of continuous liquid column deformation partition
图 10 液体微元横截面变形示意图
Figure 10. Schematic diagram of cross-section deformation of liquid microelements
图 11 液体微元的受力分析示意图
Figure 11. Schematic diagram of force analysis of liquid microelements
图 12 二维液滴附近流向速度分布
Figure 12. Flow velocity distribution of 2D droplets
图 13 二维液滴气动阻力系数CD
Figure 13. Aerodynamic drag coefficient of 2D droplet CD
图 14 斜激波简化示意图
Figure 14. Simplified schematic diagram of oblique shock
图 15 斜激波波前/波后速度分解示意图
Figure 15. Schematic diagram of velocity decomposition of oblique shock wave front/back wave
图 16 液体微元横截面椭圆率计算结果
Figure 16. Calculation results of eccentricity of liquid microelement cross section
图 17 液体微元当地韦伯数计算结果
Figure 17. Schematic diagram of cross-section deformation of liquid microelements
图 18 剥离液滴质量计算结果
Figure 18. Schematic diagram of cross-section deformation of liquid microelements
图 19 CLC模型预测得到的连续液柱结构 (a) 侧视图; (b) 正视图
Figure 19. Continuous liquid column structure predicted by CLC model: (a) Side view; (b) front view
图 20 射流轨迹叠加结果
Figure 20. Superposition result of jet trajectory
图 21 喷嘴出口位置射流显微成像结果(基准工况)
Figure 21. Jet microscopic imaging results at the nozzle outlet (basic condition).
图 22 不同喷注压降射流显微图像 (a)
$\Delta p$ = 0.97 MPa; (b)$\Delta p$ = 1.49 MPa; (c)$\Delta p$ = 2.05 MPaFigure 22. Microscopic images of jets with different injection pressure drops: (a)
$\Delta p$ = 0.97 MPa; (b)$\Delta p$ = 1.49 MPa; (c)$\Delta p$ = 2.05 MPa.
图 23 CLC模型预测得到的射流轨迹结果与实验结果(基准工况)
Figure 23. Jet trajectory results and experimental results predicted by the CLC model (basic condition)
图 24 超声速横流中液柱破碎位置与文献结果对比
Figure 24. Comparison of the broken position of liquid column in supersonic cross-flow with literature results
图 25 不同动量比和不同喷嘴直径下射流轨迹计算结果与本文实验结果比较 (a) d = 0.5 mm; (b) q = 3.32
Figure 25. Calculations for different momentum ratios and diffent nozzle diameters in comparison with experiments: (a) d = 0.5 mm; (b) q = 3.32
表 1 超声速横流参数
Table 1. Supersonic cross flow parameters
Ma = 2.1 Ma = 2.85 总温T0/K 300 300 总压P0/kPa 891 1410 静温T/K 159.4 114.3 静压P/kPa 97.7 48.1 密度/kg·m–3 2.13 1.47 声速/m·s–1 253 214 速度/m·s–1 531 611 
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表 2 不同网格计算得到的气动阻力系数
Table 2. Aerodynamic drag coefficients calculated from different grids.
类别 最小网格尺寸/d 网格量 气动阻力系数 Coarse 0.020 50840 1.372 Middle 0.010 225448 1.350 Fine 0.005 488752 1.349
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表 3 理论计算参数
Table 3. Theoretical calculation parameters.
编号 横向气流 (T0 = 300 K) 水射流 (密度: 998 kg·m–3) Ma P0/kPa ρg/kg·m–3 uj/m·s–1 d/mm $\Delta P$/MPa q Case 1 2.85 1410 1.47 44.7 0.5/1.0 1.0 3.323 Case 2 2.85 1410 1.47 54.8 0.5/1.0 1.5 4.985 Case 3 2.85 1410 1.47 63.2 0.5/1.0 2.0 6.647 Case 4 2.1 891 2.13 44.7 0.5/1.0 1.0 3.637 Case 5 2.1 891 2.13 54.8 0.5/1.0 1.5 5.456 Case 6 2.1 891 2.13 63.2 0.5/1.0 2.0 7.274
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表 4 三维连续液柱模型参数
Table 4. 3D continuous liquid column model parameters.
类别 $x{\rm{/}}d$ $y{\rm{/}}d$ $z/d$ 考虑质量损失 0—1.92 0—4.17 –1.03—1.03
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[1] Xiao F, Dianat M, Mcguirk J J 2013 AIAA J. 51 2878
Google Scholar
[2] 夏同军, 董永强, 曹义刚 2013 62 214702
Google Scholar
Xia T J, Dong Y Q, Cao Y G 2013 Acta Phys. Sin. 62 214702
Google Scholar
[3] Heister S D 2011 Handbook of Atomization and Sprays: Theory and Applications (New York: Springer) pp65−660
[4] Li C Y, Li C, Xiao F, Li Q L, Zhu Y H 2019 Aerosp. Sci. Technol. 95 105426
Google Scholar
[5] Dixon D R, Gruber M R, Jackson T A, Lin K C 2005 43th AIAA Aerospace Sciences Meeting & Exhibit Reno, America, January 10−13, 2005 p733
[6] Wu L Y, Wang Z G, Li Q L, Li C 2016 J. Visualization 3 337
Google Scholar
[7] Hu R S, Li Q, Li C Y, Li C 2019 Acta Astronaut. 159 440
Google Scholar
[8] Lin K C, Kennedy P J 2002 40 th AIAA Aerospace Sciences Meeting & Exhibit Reno, America, January 14−17, 2002 p873
[9] Lin K C, Kennedy P J, Kennedy P J, Jackson T A 2004 42th AIAA Aerospace Sciences Meeting & Exhibit Reno, America, January 5−8, 2004 p971
[10] 刘静, 王辽, 张佳, 韦宝禧, 徐旭 2008 航空动力学报 4 146
Google Scholar
Liu J, Wang L, Zhang J, Wei B X, Xu X 2008 J. Aerosp. Power 4 146
Google Scholar
[11] 仝毅恒, 李清廉, 吴里银 2012 国防科技大学学报 2 73
Google Scholar
Tong Y H, Li Q L, Wu L Y 2012 J. Natl. Univ. Def. Technol. 2 73
Google Scholar
[12] 李春 2014 硕士学位论文 (长沙: 国防科学技术大学)
Li C 2012 M. S. Thesis (Changsha: National University of Defense Technology) (in Chinese)
[13] 吴里银, 王振国, 李清廉, 李春 中国专利 zl201410800056.5 [2018-2-2]
Wu L Y, Wang Z G, Li Q L, Li C Chinese Patent zl201410800056.5 [20218-2-2] (in Chinese)
[14] 胡润生, 朱元昊, 张翔宇, 李清廉 2019 工程热 7 1659
Google Scholar
Hu R S, Zhu Y H, Zhang X Y, Li Q L 2019 J. Eng. Therm. 7 1659
Google Scholar
[15] 吴里银, 王振国, 李清廉, 李春 2016 68 094701
Google Scholar
Wu L Y, Wang Z G, Li Q L, Li C 2016 Acta Phys. Sin. 68 094701
Google Scholar
[16] 周曜智, 李清廉, 李晨阳 2020 推进技术 41 7
Google Scholar
Zhou Y Z, Li Q L, Li C Y 2020 J. Propul. Technol. 41 7
Google Scholar
[17] Mashayek A, Behzad M, Ashgriz N 2011 AIAA J. 49 2407
Google Scholar
[18] 李春 2020 博士学位论文 (长沙: 国防科技大学)
Li C 2020 Ph. D. Dissertation (Changsha: National University of Defense Technology) (in Chinese)
[19] Xiao F, Wang Z G, Sun M B, Liang J H, Liu N 2016 Int. J. Multiphase Flow 87 229
Google Scholar
[20] 李佩波, 王振国, 孙明波, 汪洪波 2016 宇航学报 37 79
Google Scholar
Li P B, Wang Z G, Sun M B, Wang H B 2016 J. Astronautics 37 79
Google Scholar
[21] Zhu Y X, Xiao F, Li Q L, Li C Y, Lin S 2018 Acta Astronaut. 154 119
Google Scholar
[22] Liu N, Wang Z G, Sun M B, Deiterding R, Wang H B 20191 Aerosp. Sci. Technol. 91 456
Google Scholar
[23] Douglas H F, Vitor V, Lucas S M, Francisco J S 2019 Int. J. Multiphase Flow 114 98
Google Scholar
[24] Li P B, Li C Y, Wang H B, Sun M B, Liu C Y, Wang Z G, Huang Y H 2019 Aerosp. Sci. Technol. 94 105401
Google Scholar
[25] Nguyen T, Karagozian A R 1992 J. Propul. Power 8 21
Google Scholar
[26] Clark M M 1988 Chem. Eng. Sci. 43 671
Google Scholar
[27] Inamura T 2000 J. Propul. Power 16 155
Google Scholar
[28] Sallam K A, Aalburg C, Faeth G M 2004 AIAA J. 42 2529
Google Scholar
[29] Wu P K, Kirkendall K A, Fuller R P, Nejad A S 1997 J. Propul. Power 13 64
Google Scholar
[30] 吴里银 2016 博士学位论文 (长沙: 国防科学技术大学
Wu L Y 2016 Ph. D. Dissertation (Changsha: National University of Defense Technology) (in Chinese)
[31] Lee K, Aalburg C, Diez F J, Faeth G M, Sallam K A 2007 AIAA J. 45 1907
Google Scholar
[32] Ranger A A, Nicholls J A 1969 AIAA J. 7 285
Google Scholar
[33] Chryssakis C A, Assanis, D N 2005 18th Annual Conference on Liquid and Atomization and Spray Systems, Irvine, America, May 15−18, 2005 p283
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