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该文结合了Ott提出的修正连续性方程和Adami改进的动量方程, 对空气中的液滴碰撞问题进行了二维数值模拟. 为有效提高计算精度, 推导了适用于大密度差多相流的人工黏性和人工应力方程. 通过表面张力作用下方形液滴自然变化和空气中两液滴互溶的算例, 验证了算法的有效性; 对不同韦伯数 (8.8, 19.8)、不同碰撞参数 (0, 0.5)下的液滴碰撞过程进行了数值模拟, 并与VOF方法对比,取得了较为一致的结果; 进一步计算多个韦伯数、多个碰撞参数下的液滴碰撞, 得到了空气中二维液滴碰撞结果分布图,与实验结果相符合. 结果表明, 该算法对于求解涉及大密度差多相流的液滴碰撞破碎问题十分有效,而且该方法容易拓展到三维, 从而为进一步模拟火箭发动机的二次雾化过程奠定了基础.In this paper, the corrective continuity equation proposed by Ott and the momentum equation improved by Adami combine to solve two-dimensional simulation problems of droplet collisions in air. To effectively improve the calculation accuracy, the artificial viscosity equation and the artificial stress equation are derived which are suited for multi-phase flows with large density differences. This method is validated to be effective via examples of an initially square droplet under surface tension and in evolution process of two droplets in air. Droplet collisions for different Weber numbers (8.8, 19.8) and different impact parameters (0, 0.5) are simulated, all of which are compared with the results of VOF simulation. Through further calculation, distribution map of the two-dimensional droplet collision outcomes in air is obtained, which is in agreement with the experimental results. It is demonstrated that this method can be effective for solving problems of droplet collisions which are involved in multi-phase flows with large density differences, and is easily extended to three-dimensional simulation; thus it lays a foundation for further simulation of the secondary atomization in liquid rocket engines.
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Keywords:
- smoothed particle hydrodynamics /
- large density differences /
- multi-phase flow /
- droplet collision
[1] Park R W 1970 Ph. D. Dissertation (Madison: University of Wisconsin)
[2] Ashgriz N, Poo J Y 1990 J. Fluid Mech. 221 183
[3] Qian J, Law C K 1997 J. Fluid Mech. 331 59
[4] Willis K, Orme M 2003 Exp. Fluids 34 28
[5] Chen R H, Chen C T 2006 Exp. Fluids 41 453
[6] Chen R H 2007 Appl. Therm. Eng. 27 604
[7] Foote G B 1973 J. Comp. Phys. 11 507
[8] Mashayek F, Ashgriz N, Minkowycz W J, Shotorban B 2003 Int. J. Heat Mass Transfer 46 77
[9] Nobari M R, Jan Y J 1996 Phys. Fluids 8 29
[10] Poo J Y, Ashgriz N 1992 Proceeding of the 5th Annual Conference on Liquid Atomization and Spray System San Ramon, California, May 18–20, 1992 p110
[11] Sun Z H, Han R J 2008 Chin. Phys. B 17 3185
[12] Zhang A M, Yao X L 2008 Acta Phys. Sin. 57 339 (in Chinese) [张阿漫, 姚熊亮 2008 57 339]
[13] Liu M B, Chang J Z 2010 Acta Phys. Sin. 59 3654 (in Chinese) [刘谋斌, 常建忠 2010 59 3654]
[14] Ma L Q, Chang J Z, Liu H T, Liu M B 2012 Acta Phys. Sin. 61 054701 (in Chinese) [马理强, 常建忠, 刘汉涛, 刘谋斌 2012 61 054701]
[15] Meleán Y, Sigalotti L D G 2005 Int. J. Heat Mass Transfer 48 4041
[16] Qiang H F, Chen F Z, Gao W R 2012 Eng. Mech. 29 21 (in Chinese) [强洪夫, 陈福振, 高巍然 2012 工程力学 29 21]
[17] Malavé A A 2012 AIP Adv. 2 042106
[18] Frank Ott, Erik Schnetter 2003 arXiv: physics/0303112v3 [physics.comp-ph]
[19] Adami S, Hu X Y, Adams N A 2010 J. Comp. Phys. 229 5011
[20] Monaghan J J 1994 J. Comp. Phys. 110 399
[21] Hu X Y, Adams N A 2007 J. Comp. Phys. 227 264
[22] Morris J P 2000 Int. J. Numer. Met. Fluids 33 333
[23] Monaghan J J 1992 Annu. Rev. Astro. Astrophys. 30 543
[24] Monaghan J J 2000 J. Comp. Phys. 159 290
[25] Gray J P, Monaghan J J, Swift R P 2001 Comput. Methods. Appl. Mech. Eng. 190 6641
[26] Morris J P, Fox P J, Zhu Y 1997 J. Comp. Phys. 136 214
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[1] Park R W 1970 Ph. D. Dissertation (Madison: University of Wisconsin)
[2] Ashgriz N, Poo J Y 1990 J. Fluid Mech. 221 183
[3] Qian J, Law C K 1997 J. Fluid Mech. 331 59
[4] Willis K, Orme M 2003 Exp. Fluids 34 28
[5] Chen R H, Chen C T 2006 Exp. Fluids 41 453
[6] Chen R H 2007 Appl. Therm. Eng. 27 604
[7] Foote G B 1973 J. Comp. Phys. 11 507
[8] Mashayek F, Ashgriz N, Minkowycz W J, Shotorban B 2003 Int. J. Heat Mass Transfer 46 77
[9] Nobari M R, Jan Y J 1996 Phys. Fluids 8 29
[10] Poo J Y, Ashgriz N 1992 Proceeding of the 5th Annual Conference on Liquid Atomization and Spray System San Ramon, California, May 18–20, 1992 p110
[11] Sun Z H, Han R J 2008 Chin. Phys. B 17 3185
[12] Zhang A M, Yao X L 2008 Acta Phys. Sin. 57 339 (in Chinese) [张阿漫, 姚熊亮 2008 57 339]
[13] Liu M B, Chang J Z 2010 Acta Phys. Sin. 59 3654 (in Chinese) [刘谋斌, 常建忠 2010 59 3654]
[14] Ma L Q, Chang J Z, Liu H T, Liu M B 2012 Acta Phys. Sin. 61 054701 (in Chinese) [马理强, 常建忠, 刘汉涛, 刘谋斌 2012 61 054701]
[15] Meleán Y, Sigalotti L D G 2005 Int. J. Heat Mass Transfer 48 4041
[16] Qiang H F, Chen F Z, Gao W R 2012 Eng. Mech. 29 21 (in Chinese) [强洪夫, 陈福振, 高巍然 2012 工程力学 29 21]
[17] Malavé A A 2012 AIP Adv. 2 042106
[18] Frank Ott, Erik Schnetter 2003 arXiv: physics/0303112v3 [physics.comp-ph]
[19] Adami S, Hu X Y, Adams N A 2010 J. Comp. Phys. 229 5011
[20] Monaghan J J 1994 J. Comp. Phys. 110 399
[21] Hu X Y, Adams N A 2007 J. Comp. Phys. 227 264
[22] Morris J P 2000 Int. J. Numer. Met. Fluids 33 333
[23] Monaghan J J 1992 Annu. Rev. Astro. Astrophys. 30 543
[24] Monaghan J J 2000 J. Comp. Phys. 159 290
[25] Gray J P, Monaghan J J, Swift R P 2001 Comput. Methods. Appl. Mech. Eng. 190 6641
[26] Morris J P, Fox P J, Zhu Y 1997 J. Comp. Phys. 136 214
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