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Based on Navier-Stokes equations, combining the fifth-order weighted essentially non-oscillatory scheme with the adaptive structured grid refinement technique, the interactions between converging shock and annular SF6 layers with different initial perturbation amplitudes and thickness are numerically investigated. The evolution mechanism of shock structure and interface are revealed in detail, and the variations of the circulation, mixing rate and turbulent kinetic energy are quantitatively analyzed. The dynamic mode decomposition method is used to analyze the dynamic characteristics of the vorticity. The results show that in the case with large initial perturbation amplitude, the transmitted shock wave forms Mach reflection structures both inside and outside of the inner interface of SF6 layer, and multiple shock focusing phenomena occur in the center. After the transmitted shock wave penetrates the outer interface, the circulation increases faster, and the “spike” and “bubble” structure on inner interface develop faster, so that the amplitude of the inner and outer interfaces and the gas mixing rate increase. As for the case with larger thickness of the gas layer, the phase of the transmitted shock wave changes inside the layer, which forms “bubble” at the crest of the inner interface and “spike” at the trough. When the thickness of the gas layer decreases, the crest of the inner interface does not move inside after being impacted, and “spike” and “bubble” are generated in the late stage. The dynamic modes show that the main structure of vorticity and the exchange of positive and negative vorticity on the main structure are determined by the modes with weak growth and low frequency, but the modes with weak growth and high frequency determine rapid exchange of positive and negative vorticity at the interface in the cases with weak coupling effect.
[1] Yang J, Kubota T, Zukoski E E 1993 AIAA J. 31 854
Google Scholar
[2] Cao L, Fei W L, Grosshans H, Cao N 2017 Appl. Sci. 7 880
Google Scholar
[3] Lindl J D, McCrory R L, Campbell E M 1992 Phys. Today 45 32
Google Scholar
[4] Richtmyer R D 1960 Commun. Pure Appl. Math. 13 297
Google Scholar
[5] Fraley G 1986 Phys. Fluids 29 376
Google Scholar
[6] Haehn N, Ranjan D, Weber C, Oakley J G, Anderson M H, Bonazza R 2010 Phys. Scr. T142 014067
Google Scholar
[7] Haehn N, Weber C, Oakley J, Anderson M, Ranjan D, Bonazza R 2012 Shock Waves 22 47
Google Scholar
[8] Luo X S, Wang M H, Si T, Zhai Z G 2015 J. Fluid Mech. 773 366
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[9] 沙莎, 陈志华, 薛大文, 张辉 2014 物理学报 63 085205
Google Scholar
Sha S, Chen Z H, Xue D W, Zhang H 2014 Acta Phys. Sin. 63 085205
Google Scholar
[10] Mikaelian K O 1990 Phys. Rev. A 42 3400
Google Scholar
[11] Lombardini M, Pullin D I 2009 Phys. Fluids 21 114103
Google Scholar
[12] Si T, Long T, Zhai Z G, Luo X S 2015 J. Fluid Mech. 784 225
Google Scholar
[13] Ding J C, Si T, Yang J M, Lu X Y, Zhai Z G, Luo X S 2017 Phys. Rev. Lett. 119 014501
Google Scholar
[14] Ding J C, Li J M, Sun R, Zhai Z G, Luo X S 2019 J. Fluid Mech. 878 277
Google Scholar
[15] Mikaelian K O 1995 Phys. Fluids 7 888
Google Scholar
[16] Sun R, Ding J C, Zhai Z G, Si T, Luo X S 2020 J. Fluid Mech. 902 A3
Google Scholar
[17] Li J M, Ding J C, Si T, Luo X S 2020 J. Fluid Mech. 884 R2
Google Scholar
[18] 徐建于, 黄生洪 2019 力学学报 51 998
Google Scholar
Xu J Y, Huang S H 2019 Chin. J. Theor. Appl. Mech. 51 998
Google Scholar
[19] 梁煜, 关奔, 翟志刚, 罗喜胜 2017 物理学报 66 064701
Google Scholar
Liang Y, Guan B, Zhai Z G, Luo X S 2017 Acta Phys. Sin. 66 064701
Google Scholar
[20] Zhou Z B, Ding J C, Zhai Z G, Cheng W, Luo X S 2020 Acta Mech. Sin. 36 356
Google Scholar
[21] Tang J G, Zhang F, Luo X S, Zhai Z G 2020 Acta Mech. Sin. 37 434
Google Scholar
[22] 何惠琴, 翟志刚, 司廷, 罗喜胜 2016 计算物理 33 66
Google Scholar
He H Q, Zhai Z G, Si T, Luo X S 2016 Chin. J. Comput. Phys. 33 66
Google Scholar
[23] Fu Y W, Yu C P, Li X L 2020 AIP Adv. 10 105302
Google Scholar
[24] Lombardini M, Hill D J, Pullin D I, Meiron D I 2011 J. Fluid Mech. 670 439
Google Scholar
[25] Hill D J, Pullin D I 2004 J. Comput. Phys. 194 435
Google Scholar
[26] Pantano C, Deiterding R, Hill D J, Pullin D I 2007 J. Comput. Phys. 221 63
Google Scholar
[27] Henry D, Movahed P, Johnsen E 2015 Shock Waves 25 329
Google Scholar
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图 2 网格无关性检验
Figure 2. Verification of the mesh resolution.
图 3 计算模型示意图 (is, 初始激波; R, 外界面位置; R0, 外界面平均半径; r0, 内界面半径; a0: 初始扰动幅值)
Figure 3. Illustration of computational model (is, initial shock; R, location of outer interface; R0, mean radius of outer interface; r0, initial radius of inner interface; a0, initial amplitude).
图 4 case 1的界面与激波结构演变过程示意图(ts, 透射激波; rs, 反射激波; rrw, 反射稀疏波; m, 马赫杆; T, 三波点; SF, 激波聚焦; srs, 二次反射激波; spike, “尖钉”结构; bubble, “气泡”结构; sts, 二次透射激波; trs, 三次反射激波; tts, 三次透射激波; 下文符号含义相同)
Figure 4. Evolution of the interface and shock wave structures of case 1 (ts, transmitted shock; rs, reflected shock; rrw, reflected rarefaction wave; m, Mach stem; T, triple point; SF, shock focusing; srs, the second reflected shock; spike, “spike” structure; bubble; “bubble” structure; sts, the second transmitted shock; trs, the third reflected shock; tts, the third transmitted shock. The meaning of these abbreviations is similar hereinafter).
图 5 case 1中不同时刻流场涡量分布图
Figure 5. Distribution of vorticity at different times of case 1.
图 6 case 2的界面与激波结构演变过程
Figure 6. Evolution of the interface and shock wave structures of case 2.
图 7 case 3的界面与激波结构演变过程(frs. 四次反射激波; fts, 四次透射激波)
Figure 7. Evolution of the interface and shock wave structures of case 3 (frs, the forth reflected shock; fts, the forth transmitted shock).
图 8 case 4的界面与激波结构演变过程
Figure 8. Evolution of the interface and shock wave structures of case 4.
图 9 case 5的界面与激波结构演变过程
Figure 9. Evolution of the interface and shock wave structures of case 5.
图 10 case 1中不同时刻压力沿径向分布图
Figure 10. Variations of pressure along the radial of case 1.
图 11 直角坐标下t = 0.1 ms时case 1的组分分布图
Figure 11. Illustration of the fraction of SF6 for case 1 in Cartesian coordinate system at t = 0.1 ms.
图 12 cases 1—5中(a)内界面与(b)外界面扰动幅值演化过程
Figure 12. Evolution of the amplitude of (a) inner interface and (b) outer interface of cases 1–5.
图 13 cases 1—5中(a) 环量绝对值
$ \left|{ \varGamma }\right| $ 与(b)混合率随时间变化情况Figure 13. Evolution of (a) absolute value of circulation
$ \left|{ \varGamma }\right| $ and (b) mixing rate of cases 1–5.
图 14 cases 1—5中湍动能随时间分布图 (a) case 1; (b) case 2; (c) case 3; (d) case 4; (e) case 5
Figure 14. Distributions of turbulent kinetic energy (TKE) of (a) case 1, (b) case 2, (c) case 3, (d) case 4 and (e) case 5.
图 15 cases 1—5中DMD频谱分布图 (a) case 1; (b) case 2; (c) case 3; (d) case 4; (e) case 5
Figure 15. Distributions of the frequency spectrum of the DMD modes of (a) case 1, (b) case 2, (c) case 3, (d) case 4, and (e) case 5.
图 16 (a)—(e) cases 1—5中涡量的DMD模态的实数部分 (a1)—(e1) DM1; (a2)—(e2) DM2; (a3)—(e3) DM3; (a4)—(e4) DM4
Figure 16. (a)–(e) Representation of DMD modes with their real parts using contours of vorticity of cases 1–5: (a1)–(e1) DM1; (a2)–(e2) DM2; (a3)–(e3) DM3; (a4)–(e4) DM4.
表 1 不同双层重气柱几何参数表
Table 1. Structural parameters of cylinder of different cases.
Case R0/mm r0/mm a0/mm n λ/mm a0/λ Case 1 20 10 1 6 20.94 0.048 Case 2 20 10 0.5 6 20.94 0.024 Case 3 20 10 2 6 20.94 0.096 Case 4 20 5 1 6 20.94 0.048 Case 5 20 15 1 6 20.94 0.048 
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表 2 气体参数表
Table 2. Parameters of gases.
Gas γ M/(g·mol–1) ρ/(kg·m–3) Air 1.399 28.967 1.23 SF6 1.103 128.491 5.45
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[1] Yang J, Kubota T, Zukoski E E 1993 AIAA J. 31 854
Google Scholar
[2] Cao L, Fei W L, Grosshans H, Cao N 2017 Appl. Sci. 7 880
Google Scholar
[3] Lindl J D, McCrory R L, Campbell E M 1992 Phys. Today 45 32
Google Scholar
[4] Richtmyer R D 1960 Commun. Pure Appl. Math. 13 297
Google Scholar
[5] Fraley G 1986 Phys. Fluids 29 376
Google Scholar
[6] Haehn N, Ranjan D, Weber C, Oakley J G, Anderson M H, Bonazza R 2010 Phys. Scr. T142 014067
Google Scholar
[7] Haehn N, Weber C, Oakley J, Anderson M, Ranjan D, Bonazza R 2012 Shock Waves 22 47
Google Scholar
[8] Luo X S, Wang M H, Si T, Zhai Z G 2015 J. Fluid Mech. 773 366
Google Scholar
[9] 沙莎, 陈志华, 薛大文, 张辉 2014 物理学报 63 085205
Google Scholar
Sha S, Chen Z H, Xue D W, Zhang H 2014 Acta Phys. Sin. 63 085205
Google Scholar
[10] Mikaelian K O 1990 Phys. Rev. A 42 3400
Google Scholar
[11] Lombardini M, Pullin D I 2009 Phys. Fluids 21 114103
Google Scholar
[12] Si T, Long T, Zhai Z G, Luo X S 2015 J. Fluid Mech. 784 225
Google Scholar
[13] Ding J C, Si T, Yang J M, Lu X Y, Zhai Z G, Luo X S 2017 Phys. Rev. Lett. 119 014501
Google Scholar
[14] Ding J C, Li J M, Sun R, Zhai Z G, Luo X S 2019 J. Fluid Mech. 878 277
Google Scholar
[15] Mikaelian K O 1995 Phys. Fluids 7 888
Google Scholar
[16] Sun R, Ding J C, Zhai Z G, Si T, Luo X S 2020 J. Fluid Mech. 902 A3
Google Scholar
[17] Li J M, Ding J C, Si T, Luo X S 2020 J. Fluid Mech. 884 R2
Google Scholar
[18] 徐建于, 黄生洪 2019 力学学报 51 998
Google Scholar
Xu J Y, Huang S H 2019 Chin. J. Theor. Appl. Mech. 51 998
Google Scholar
[19] 梁煜, 关奔, 翟志刚, 罗喜胜 2017 物理学报 66 064701
Google Scholar
Liang Y, Guan B, Zhai Z G, Luo X S 2017 Acta Phys. Sin. 66 064701
Google Scholar
[20] Zhou Z B, Ding J C, Zhai Z G, Cheng W, Luo X S 2020 Acta Mech. Sin. 36 356
Google Scholar
[21] Tang J G, Zhang F, Luo X S, Zhai Z G 2020 Acta Mech. Sin. 37 434
Google Scholar
[22] 何惠琴, 翟志刚, 司廷, 罗喜胜 2016 计算物理 33 66
Google Scholar
He H Q, Zhai Z G, Si T, Luo X S 2016 Chin. J. Comput. Phys. 33 66
Google Scholar
[23] Fu Y W, Yu C P, Li X L 2020 AIP Adv. 10 105302
Google Scholar
[24] Lombardini M, Hill D J, Pullin D I, Meiron D I 2011 J. Fluid Mech. 670 439
Google Scholar
[25] Hill D J, Pullin D I 2004 J. Comput. Phys. 194 435
Google Scholar
[26] Pantano C, Deiterding R, Hill D J, Pullin D I 2007 J. Comput. Phys. 221 63
Google Scholar
[27] Henry D, Movahed P, Johnsen E 2015 Shock Waves 25 329
Google Scholar
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