瑞利-泰勒不稳定问题的光滑粒子法模拟研究

杨秀峰; 刘谋斌

doi:10.7498/aps.66.164701

摘要

提出了一种适用于模拟多相流的光滑粒子法，该方法对密度方程在交界面处的离散格式进行了修正以适应多相流所涉及的大密度比问题，在不同相粒子之间施加了很小的排斥力以防止粒子穿透交界面，并采用了最新发展的双曲型光滑函数以消除应力不稳定问题.应用该多相流光滑粒子法模拟研究了单模态和多模态瑞利-泰勒不稳定问题.通过与文献中结果的对比研究表明：在模拟瑞利-泰勒不稳定问题时，本文方法的结果明显优于文献中的大部分光滑粒子法模拟结果，与Grenier等（2009 J.Comput.Phys.228 8380）的结果相当，但本文方法比Grenier等的方法简单方便.对于单模态瑞利-泰勒不稳定问题，研究了交界面的形态，涡结构的演化过程以及贯穿深度随时间的变化关系.对于多模态瑞利-泰勒不稳定问题，研究了交界面演化过程中小尺度结构合并成大尺度结构的过程，水平方向的平均密度随高度的变化关系，以及贯穿深度随时间的变化关系.

关键词:

Abstract

In this paper, we present a smoothed particle hydrodynamics (SPH) method for modeling multiphase flows. The multiphase SPH method includes a corrective discretization scheme for density approximation around the fluid interface to treat large density ratio, a small repulsive force between particles from different phases to prevent particles from unphysically penetrating fluid interface, and a newly-developed hyperbolic-shaped kernel function to remove possible stress instability. This multiphase SPH method is then used to study the single-and multi-mode Rayleigh-Taylor instability problems. A comparison between our results with the results from existing literature shows that our results are obviously better than most available results from other SPH simulations. The present results are close to those by Grenier et al. while the present multiphase SPH method is simpler and easier to implement than that in the work by Grenier et al. (Grenier, et al. 2009 J. Comput. Phys. 228 8380). For the single-mode Rayleigh-Taylor instability, the evolutions of the interface pattern and vortex structures, and the penetration depth each as a function of time are investigated. For the multi-mode Rayleigh-Taylor instability, the merging of small structures into a large structure during the evolution of the interface is studied. The horizontal average density and the penetration each as a function of height are also studied.

Keywords:

作者及机构信息

杨秀峰¹,
刘谋斌²

1.
Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA;

2.
北京大学工学院, 北京 100187

通信作者: 刘谋斌, mbliu@pku.edu.cn

基金项目: 国家自然科学基金（批准号：11302237，U1530110）资助的课题.

Authors and contacts

Yang Xiu-Feng¹,
Liu Mou-Bin²

1.
Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA;

2.
BIC-ESAT, College of Engineering, Peking University, Beijing 100187, China

Corresponding author: Liu Mou-Bin, mbliu@pku.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11302237, U1530110).

参考文献

[1]	Rayleigh L 1883 Proc. Lond. Math. Soc. 14 170
[2]	Taylor G 1950 Proc. Roy. Soc. A:Math. Phys. 201 192
[3]	Lewis D J 1950 Proc. Roy. Soc. A:Math. Phys. 202 81
[4]	Tryggvason G 1988 J. Comput. Phys. 75 253
[5]	Banerjee R, Kanjilal S 2015 J. Pure Appl. Ind. Phys. 5 73
[6]	Sharp D H 1984 Physica D 12 3
[7]	Kilkenny J, Glendinning S, Haan S, Hammel B, Lindl J, Munro D, Remington B, Weber S, Knauer J, Verdon C 1994 Phys. Plasmas 1 1379
[8]	Huang C S, Kelley M, Hysell D 1993 J. Geophys. Res.-Space Physics 98 15631
[9]	Alon U, Hecht J, Ofer D, Shvarts D 1995 Phys. Rev. Lett. 74 534
[10]	Dimonte G 2000 Phys. Plasmas 7 2255
[11]	Ramshaw J D 1998 Phys. Rev. E 58 5834
[12]	Glimm J, Saltz D, Sharp D H 1998 Phys. Rev. Lett. 80 712
[13]	Cheng B, Glimm J, Sharp D 2002 Phys. Rev. E 66 036312
[14]	Zhang Y S, He Z W, Gao F J, Li X L, Tian B L 2016 Phys. Rev. E 93 063102
[15]	He X, Chen S, Zhang R 1999 J. Comput. Phys. 152 642
[16]	Kadau K, Barber J L, Germann T C, Holian B L, Alder B J 2010 Phil. Trans. R. Soc. A 368 1547
[17]	Ramaprabhu P, Karkhanis V, Banerjee R, Varshochi H, Khan M, Lawrie A 2016 Phys. Rev. E 93 013118
[18]	Sagert I, Howell J, Staber A, Strother T, Colbry D, Bauer W 2015 Phys. Rev. E 92 013009
[19]	Liang H, Li Q, Shi B, Chai Z 2016 Phys. Rev. E 93 033113
[20]	Lucy L B 1977 Astron. J. 82 1013
[21]	Gingold R A, Monaghan J J 1977 Mon. Not. R. Astron. Soc. 181 375
[22]	Yang X, Peng S, Liu M, Shao J 2012 Int. J. Comp. Meth.-Sing 9 1240002
[23]	Yang X F, Peng S L, Liu M B 2014 Appl. Math. Model 38 3822
[24]	Yang X, Dai L, Kong S C 2017 Proc. Combust. Inst. 36 2393
[25]	Yang X F, Liu M B 2012 Acta Phys. Sin. 61 224701 (in Chinese)[杨秀峰, 刘谋斌2012 61 224701]
[26]	Yang X F, Liu M B, Peng S 2014 Comput. Fluids 92 199
[27]	Monaghan J J 1992 Ann. Rev. Astron. Astrophys. 30 543
[28]	Monaghan J J 2000 J. Comput. Phys. 159 290
[29]	Grenier N, Antuono M, Colagrossi A, Le Touzé D, Alessandrini B 2009 J. Comput. Phys. 228 8380
[30]	Yang X, Liu M 2013 Sci. China:Phys. Mech. Astron. 56 315
[31]	Bonet J, Lok T S 1999 Comput. Methods Appl. Mech. Engrg. 180 97
[32]	Colagrossi A, Landrini M 2003 J. Comput. Phys. 191 448
[33]	Chen Z, Zong Z, Liu M, Zou L, Li H, Shu C 2015 J. Comput. Phys. 283 169
[34]	Monaghan J, Rafiee A 2013 Int. J. Numerical Mech. Fluids 71 537
[35]	Hu X Y, Adams N A 2009 J. Comput. Phys. 228 2082
[36]	Yang X F, Liu M B 2016 Chin. J. Comput. Mech. 33 594(in Chinese)[杨秀峰, 刘谋斌2016计算力学学报33 594]
[37]	Layzer D 1955 Astrophys. J. 122 1

施引文献

[1]	Rayleigh L 1883 Proc. Lond. Math. Soc. 14 170
[2]	Taylor G 1950 Proc. Roy. Soc. A:Math. Phys. 201 192
[3]	Lewis D J 1950 Proc. Roy. Soc. A:Math. Phys. 202 81
[4]	Tryggvason G 1988 J. Comput. Phys. 75 253
[5]	Banerjee R, Kanjilal S 2015 J. Pure Appl. Ind. Phys. 5 73
[6]	Sharp D H 1984 Physica D 12 3
[7]	Kilkenny J, Glendinning S, Haan S, Hammel B, Lindl J, Munro D, Remington B, Weber S, Knauer J, Verdon C 1994 Phys. Plasmas 1 1379
[8]	Huang C S, Kelley M, Hysell D 1993 J. Geophys. Res.-Space Physics 98 15631
[9]	Alon U, Hecht J, Ofer D, Shvarts D 1995 Phys. Rev. Lett. 74 534
[10]	Dimonte G 2000 Phys. Plasmas 7 2255
[11]	Ramshaw J D 1998 Phys. Rev. E 58 5834
[12]	Glimm J, Saltz D, Sharp D H 1998 Phys. Rev. Lett. 80 712
[13]	Cheng B, Glimm J, Sharp D 2002 Phys. Rev. E 66 036312
[14]	Zhang Y S, He Z W, Gao F J, Li X L, Tian B L 2016 Phys. Rev. E 93 063102
[15]	He X, Chen S, Zhang R 1999 J. Comput. Phys. 152 642
[16]	Kadau K, Barber J L, Germann T C, Holian B L, Alder B J 2010 Phil. Trans. R. Soc. A 368 1547
[17]	Ramaprabhu P, Karkhanis V, Banerjee R, Varshochi H, Khan M, Lawrie A 2016 Phys. Rev. E 93 013118
[18]	Sagert I, Howell J, Staber A, Strother T, Colbry D, Bauer W 2015 Phys. Rev. E 92 013009
[19]	Liang H, Li Q, Shi B, Chai Z 2016 Phys. Rev. E 93 033113
[20]	Lucy L B 1977 Astron. J. 82 1013
[21]	Gingold R A, Monaghan J J 1977 Mon. Not. R. Astron. Soc. 181 375
[22]	Yang X, Peng S, Liu M, Shao J 2012 Int. J. Comp. Meth.-Sing 9 1240002
[23]	Yang X F, Peng S L, Liu M B 2014 Appl. Math. Model 38 3822
[24]	Yang X, Dai L, Kong S C 2017 Proc. Combust. Inst. 36 2393
[25]	Yang X F, Liu M B 2012 Acta Phys. Sin. 61 224701 (in Chinese)[杨秀峰, 刘谋斌2012 61 224701]
[26]	Yang X F, Liu M B, Peng S 2014 Comput. Fluids 92 199
[27]	Monaghan J J 1992 Ann. Rev. Astron. Astrophys. 30 543
[28]	Monaghan J J 2000 J. Comput. Phys. 159 290
[29]	Grenier N, Antuono M, Colagrossi A, Le Touzé D, Alessandrini B 2009 J. Comput. Phys. 228 8380
[30]	Yang X, Liu M 2013 Sci. China:Phys. Mech. Astron. 56 315
[31]	Bonet J, Lok T S 1999 Comput. Methods Appl. Mech. Engrg. 180 97
[32]	Colagrossi A, Landrini M 2003 J. Comput. Phys. 191 448
[33]	Chen Z, Zong Z, Liu M, Zou L, Li H, Shu C 2015 J. Comput. Phys. 283 169
[34]	Monaghan J, Rafiee A 2013 Int. J. Numerical Mech. Fluids 71 537
[35]	Hu X Y, Adams N A 2009 J. Comput. Phys. 228 2082
[36]	Yang X F, Liu M B 2016 Chin. J. Comput. Mech. 33 594(in Chinese)[杨秀峰, 刘谋斌2016计算力学学报33 594]
[37]	Layzer D 1955 Astrophys. J. 122 1

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