Improved thin layer model of classical Rayleigh-Taylor instability for the deformation of interface

Zhao Kai-Ge; Xue Chuang; Wang Li-Feng; Ye Wen-Hua; Wu Jun-Feng; Ding Yong-Kun; Zhang Wei-Yan; He Xian-Tu

doi:10.7498/aps.67.20172613

Abstract

The thin shell (layer) configuration is adopted in inertial-confinement fusion (ICF) implosions. The weakly nonlinear deformation of the thin shell significantly influences the performances of implosion acceleration and fusion ignition, which is an important issue for the study of ICF physics. Based on the thin layer model of Ott (Ott E 1972 Phys. Rev. Lett. 29 1429), an improved thin layer model is proposed to describe the deformation and nonlinear evolution of the perturbed interface induced by the Rayleigh-Taylor instability (RTI). Differential equations describing motion are obtained by analyzing the forces of fluid elements (i.e., Newton's second law), which are then solved by numerical method. Then the position of the perturbed interface with an initial perturbation can be obtained. The linear growth rate obtained from our thin layer approximation agrees with that from the classical RTI. For fixed Atwood number (wave number), the total amplitudes of the bubble and spike obtained from the improved thin layer model agree with those from the three-order weakly nonlinear model. In addition, we compare the deformation and evolution of the layer from our model with results of the numerical simulation. In the linear regime, the amplitudes of the bubble and spike obtained from our model agree with those from the numerical simulation. And the evolution of the perturbed interface obtained from the improved thin layer model is consistent with that from the numerical simulation. In the nonlinear regime, the evolution trends of the total amplitude of the bubble and spike for both the improved thin layer model and numerical results are the same. However, the amplitude of the bubble is obviously greater than that of the spike in the later stage of the perturbation. This is because of some shortcomings in the improved thin layer model. The first shortcoming is that ignoring the dynamical pressure in the pressure difference. In fact, the shear velocity of the fluids plays an important role in the nonlinear regime of the perturbation. The second shortcoming is that the surface area of the upper interface equals the lower interface in the whole perturbation process of the present model. Thus, the present model can be used to describe the nonlinear evolution of the perturbed interface before the mushroom structure. Finally, it is worth noting that the improved thin layer model can be used to describe the deformation and nonlinear evolution of a thin layer for arbitrary Atwood number with a perturbation of large initial amplitude and arbitrary distribution. The initial perturbations of the triangular and rectangular waves are also discussed.

Keywords:

Authors and contacts

1.
Graduate School, China Academy of Engineering Physics, Beijing 100088, China;
2.
Institute of Applied Physics and Computational Mathematics, Beijing 100094, China;
3.
Center for Applied Physics and Technology, HEDPS, Peking University, Beijing 100871, China

Corresponding author: Xue Chuang, xue_chuang@iapcm.ac.cn;wang_lifeng@iapcm.ac.cn,lif_wang@pku.edu.cn ; Wang Li-Feng, xue_chuang@iapcm.ac.cn;wang_lifeng@iapcm.ac.cn,lif_wang@pku.edu.cn ;

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11675026, 11475034, 11575033), the Foundation of President of Chinese Academy of Engineering Physics (Grant No. 2014-1-040), and the National Basic Research Program of China (Grant No. 2013CB834100).

References

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[28]	Haan S W 1991 Phys. Fluids B 3 2349
[29]	Youngs D L 1984 Physica D 12 32
[30]	Zhang Y, Drake R P, Glimm J 2007 Phys. Plasmas 14 062703
[31]	Jacobs J W, Catton I 1988 J. Fluid Mech. 187 329
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[33]	Wilkinson J P, Jacobs J W 2007 Phys. Fluids 19 124102
[34]	Olson D H, Jacobs J W 2009 Phys. Fluids 21 034103
[35]	Wang L F, Ye W H, Li Y J 2010 Chin. Phys. Lett. 27 025203
[36]	Wang L F, Wu J F, Ye W H, Zhang W Y, He X T 2013 Phys. Plasmas 20 042708
[37]	Davies R M, Taylor G I 1950 Proc. Roy. Soc. A 200 375
[38]	Layzer D 1955 Astrophys. J. 122 1
[39]	Zhang Q 1998 Phys. Rev. Lett. 81 3391
[40]	Goncharov V N 2002 Phys. Rev. Lett. 88 134502
[41]	Sohn S 2003 Phys. Rev. E 67 026301
[42]	Abarzhi S I, Nishihara K, Glimm J 2003 Phys. Lett. A 317 470
[43]	Mikaelian K O 2003 Phys. Rev. E 67 026319
[44]	Tao Y S, Wang L F, Ye W H, Zhang G C, Zhang J C, Li Y J 2012 Acta Phys. Sin. 61 075207 (in Chinese) [陶烨晟, 王立锋, 叶文华, 张广财, 张建成, 李英骏 2012 物理学报 61 075207]
[45]	Ott E 1972 Phys. Rev. Lett. 29 1429
[46]	Manheimer W, Colombant D, Ott E 1984 Phys. Fluids 27 2164
[47]	Colombant D, Manheimer W, Ott E 1984 Phys. Rev. Lett. 53 446
[48]	Wang L F, Guo H Y, Wu J F, Ye W H, Liu J, Zhang W Y, He X T 2014 Phys. Plasmas 21 122710

Cited By

[1]	Wang J H 1994 Nonstationary Flow and Shock for Two-Dimensional (Beijing: Science Press) p10 (in Chinese) [王继海 1994 二维非定常流和激波 (北京:科学出版社) 第10页]
[2]	Rayleigh L 1893 Proc. R. Math. Soc. 14 170
[3]	Taylor G I 1950 Proc. R. Soc. London: Ser. A 201 192
[4]	Chandrasekhar S 1961 Hydrodynamic and Hydromagnetic Stability (London: Oxford University Press) pp429-514
[5]	Nuckolls J H, Wood J, Thiessen A, Zimmerman G 1972 Nature 239 139
[6]	Lindl J D, Amendt P, Berger R L, Glendinning S G, Glenzer S H, Haan S W, Kauffman R L, Landen O L, Suter L J 2004 Phys. Plasmas 11 339
[7]	Atzeni S, Meyer-ter-Vehn J 2004 The physics of Inertial Fusion: Beam Plasma Interaction, Hydrodynamics, Hot Dense Matter (Oxford: Oxford University Press)
[8]	He X T, Zhang W Y 2007 Eur. Phys. J. D 44 227
[9]	Remington B A, Drake R P, Ryutov D D 2006 Rev. Mod. Phys. 78 755
[10]	Remington B A, Arnett D, Drake R P, Takabe H 1999 Science 284 1488
[11]	Committee on High Energy Density Plasma Physics Plasma Science Committee Board on Physics and Astronomy Division on Engineering and Physical Science 2001 Frontiers in High Density Physics (Washington, DC: Academic Press)
[12]	Vlemmings W H T, Diamond P J, Imai H 2006 Nature 440 58
[13]	Wang L F, Ye W H, Li Y J 2010 Phys. Plasmas 17 052305
[14]	Liu W H, Wang L F, Ye W H, He X T 2012 Phys. Plasmas 19 042705
[15]	Wang L F, Wu J F, Fan Z F, Ye W H, He X T, Zhang W Y, Dai Z S, Gu J F, Xue C 2012 Phys. Plasmas 19 112706
[16]	Wang L F, Ye W H, Sheng Z M, Don W S, Li Y J, He X T 2010 Phys. Plasmas 17 122706
[17]	Ye W H, Wang L F, He X T 2010 Phys. Plasmas 17 122704
[18]	Wang L F, Ye W H, He X T, Zhang W Y, Sheng Z M, Yu M Y 2012 Phys. Plasmas 19 100701
[19]	Wang L F, Ye W H, Wu J F, Liu J, Zhang W Y, He X T 2016 Phys. Plasmas 23 052713
[20]	Wang L F, Ye W H, Wu J F, Liu J, Zhang W Y, He X T 2016 Phys. Plasmas 23 122702
[21]	Wang L F, Ye W H, He X T, Wu J F, Fan Z F, Xue C, Guo H Y, Miao W Y, Yuan Y T, Dong J Q, Jia G, Zhang J, Li Y J, Liu J, Wang M, Ding Y K, Zhang W Y 2017 Sci. China: Phys. Mech. Astron. 60 055201
[22]	Zhang W Y, Ye W H, Wu J F, Miao W Y, Fan Z F, Wang L F, Gu J F, Dai Z S, Cao Z Y, Xu X W, Yuan Y T, Kang D G, Li Y S, Yu X J, Liu C L, Xue C, Zheng W D, Wang M, Pei W B, Zhu S P, Jiang S E, Liu S Y, Ding Y K, He X T 2014 Sci. Sin.: Phys. Mech. Astron. 44 1 (in Chinese) [张维岩, 叶文华, 吴俊峰, 缪文勇, 范征锋, 王立锋, 谷建法, 戴振生, 曹柱荣, 徐小文, 袁永腾, 康洞国, 李永升, 郁晓瑾, 刘长礼, 薛创, 郑无敌, 王敏, 裴文兵, 朱少平, 江少恩, 刘慎业, 丁永坤, 贺贤土 2014 中国科学: 物理学力学天文学 44 1]
[23]	Reipurth B, Bally J 2001 Annu. Rev. Astron. Astrophys. 39 403
[24]	Jacobs J W, Catton I 1988 J. Fluid Mech. 187 353
[25]	Kull H J 1991 Phys. Rep. 206 197
[26]	Sanz J 1994 Phys. Rev. Lett. 73 2700
[27]	Garnier J, Raviart P A, Cherfils-Clrouin C, Masse L 2003 Phys. Rev. Lett. 90 185003
[28]	Haan S W 1991 Phys. Fluids B 3 2349
[29]	Youngs D L 1984 Physica D 12 32
[30]	Zhang Y, Drake R P, Glimm J 2007 Phys. Plasmas 14 062703
[31]	Jacobs J W, Catton I 1988 J. Fluid Mech. 187 329
[32]	Waddell J T, Niederhaus C E, Jacobs J W 2001 Phys. Fluids 13 1263
[33]	Wilkinson J P, Jacobs J W 2007 Phys. Fluids 19 124102
[34]	Olson D H, Jacobs J W 2009 Phys. Fluids 21 034103
[35]	Wang L F, Ye W H, Li Y J 2010 Chin. Phys. Lett. 27 025203
[36]	Wang L F, Wu J F, Ye W H, Zhang W Y, He X T 2013 Phys. Plasmas 20 042708
[37]	Davies R M, Taylor G I 1950 Proc. Roy. Soc. A 200 375
[38]	Layzer D 1955 Astrophys. J. 122 1
[39]	Zhang Q 1998 Phys. Rev. Lett. 81 3391
[40]	Goncharov V N 2002 Phys. Rev. Lett. 88 134502
[41]	Sohn S 2003 Phys. Rev. E 67 026301
[42]	Abarzhi S I, Nishihara K, Glimm J 2003 Phys. Lett. A 317 470
[43]	Mikaelian K O 2003 Phys. Rev. E 67 026319
[44]	Tao Y S, Wang L F, Ye W H, Zhang G C, Zhang J C, Li Y J 2012 Acta Phys. Sin. 61 075207 (in Chinese) [陶烨晟, 王立锋, 叶文华, 张广财, 张建成, 李英骏 2012 物理学报 61 075207]
[45]	Ott E 1972 Phys. Rev. Lett. 29 1429
[46]	Manheimer W, Colombant D, Ott E 1984 Phys. Fluids 27 2164
[47]	Colombant D, Manheimer W, Ott E 1984 Phys. Rev. Lett. 53 446
[48]	Wang L F, Guo H Y, Wu J F, Ye W H, Liu J, Zhang W Y, He X T 2014 Phys. Plasmas 21 122710

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