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In shock bubble interaction (SBI), the baroclinic vorticity generated by misalignment of pressure and density gradient will lead to flow instability which promotes the mixing between the bubbles and surrounding gas. A numerical study on the flow and mixing of shock-accelerated elliptic helium cylinder with the surrounding air is presented in this study. To well simulate the SBI, compressible multi-component two-dimensional Navier-Stokes equations are solved by combining with double-flux model and five-order weighted essentially non-oscillatory scheme. Both the wave system evolution and the interface deformation are clearly illustrated by using the present numerical method. Quantitatively, the length scales of distorted interface, compressibility of helium cylinder, circulation, and total mixing rates of helium are measured and compared to investigate the mixing mechanism and structure effect of the helium cylinder. It is found that the evolution of elliptic interface is closely related to its shape. In the case of elliptic gas cylinder shock-accelerated along major axis, the most remarkable feature is the air jet which grows constantly with time and penetrates the downstream interface boundary, forming two independent vortices. The penetration speed of the air jet is found to increase with ellipse eccentricity increasing. In addition, like the case of the circular helium cylinder, typical free-precursor irregular shock wave refraction occurs when incident shock wave passes through the interface. In the case of shock-accelerated elliptic gas cylinder along minor axis, a distinct flat structure appears due to the shock compression during the evolution of interface, and then vorticity concentrates at the two ends of the ellipses, which finally bends the interface severely. Simple regular shock wave refraction occurs in the large frontal area of the helium cylinder. These features also grow intensely with the eccentricity of the initial elliptic interface increasing. The distinct morphologies of these elliptic interfaces also lead to the different behaviors of the interface features including the length and height. The comprehensive analysis shows that for the elliptic helium cylinder, the structure effect not only affects the interface evolution in a length-scale manner but also plays a role in their mixing process. The mixing rate of helium cylinder shocked along the major axis is significantly superior to that along the minor axis.
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Keywords:
- compressible flow /
- shock-bubble interaction /
- Richtmyer-Meshkov instability /
- mixing
[1] Richtmyer R D 1960 Commun. Pure Appl. Math. 13 297
[2] Meshkov E E 1969 Fluid Dyn. 4 101
[3] Lindl J D, Mccrory R L, Campbell E M 1992 Phys. Today 45 32
[4] Lindl J D, Amendt P, Berger R L 2004 Phys. Plasmas 11 339
[5] Yang J, Kubota T, Zukoski E E 1993 AIAA J. 31 854
[6] Arnett W D, Bahcall J N, Kirshner R P, Woosley S E 1989 Annu. Rev. Astron. Astrophys. 27 629
[7] Haas J F, Sturtevant B 1987 J. Fluid Mech. 181 41
[8] Jacobs J W 1992 J. Fluid Mech. 234 629
[9] Giordano J, Burtschell Y 2006 Phys. Fluids 18 036102
[10] Ranjan D, Niederhaus J H J, Oakley J G, Anderson M H, Greenough J A, Bonazza R 2008 Phys. Scripta 132 014020
[11] Tomkins C, Kumar S, Orlicz G, Prestridge K 2008 J. Fluid Mech. 611 131
[12] Shankar S K, Kawai S, Lele S K 2011 Phys. Fluids 23 024102
[13] Sha S, Chen Z H, Xue D W 2013 Acta Phys. Sin. 62 144701 (in Chinese) [沙莎, 陈志华, 薛大文 2013 物理学报 62 144701]
[14] Sha S, Chen Z H, Zhang Q B 2015 Acta Phys. Sin. 64 015201 (in Chinese) [沙莎, 陈志华, 张庆兵 2015 物理学报 64 015201]
[15] Sha S, Chen Z H, Xue D W, Zhang H 2014 Acta Phys. Sin. 63 085205 (in Chinese) [沙莎, 陈志华, 薛大文, 张辉 2014 物理学报 63 085205]
[16] Bai J S, Zou L Y, Wang T, Liu K, Huang W B, Liu J H, Li P, Tang D W, Liu C L 2010 Phys. Rev. E 82 056318
[17] Liao S F, Zou L Y, Huang X L, Liu J H, Zhang K, Wang Y P 2016 Sci. Sin.: Phys. Mech. Astron. 46 034702 (in Chinese) [廖深飞, 邹立勇, 黄熙龙, 刘金宏, 张珂, 王彦平 2016 中国科学: 物理学 力学 天文学 46 034702]
[18] Zhai Z G, Si T, Zou L Y, Luo X S 2013 Acta Mech. Sin. 29 24
[19] Zhai Z G, Dong P, Luo X S 2017 Chin. J. High Pressure Phys. 31 718 (in Chinese) [翟志刚, 董平, 罗喜胜 2017 高压物理学报 31 718]
[20] Fan M R, Zhai Z G, Si T, Luo X S, Zou L Y, Tan D W 2012 Sci. China: Phys. Mech. Astron. 55 284
[21] Wang M, Si T, Luo X 2015 Shock Waves 25 347
[22] Huang X L, Liao S F, Zou L Y, Liu J H, Cao R Y 2017 Explo. Shock Wave 37 829 (in Chinese) [黄熙龙, 廖深飞, 邹立勇, 刘金宏, 曹仁义 2017 爆炸与冲击 37 829]
[23] Abgrall R, Karni S 2001 J. Comput. Phys. 169 594
[24] Ern A, Giovangigli V 1994 Multicomponent Transport Algorithms (Heidelberg: Springer-Verlag) pp329-389
[25] Kee R J, Coltrin M E, Glarborg P 2003 Chemically Reacting Flow Theory and Practice (Hoboken: John Wiley Sons) pp487-530
[26] Svehla R A 1962 Estimated Viscosities and Thermal Conductivities of Gases at High Temperatures (NASA Technical Report R-132) pp20-24
[27] Spiteri R J, Ruuth S J 2003 Siam J. Numer. Anal. 40 469
[28] Verwer J G, Sommeijer B P, Hundsdorfer W 2004 J. Comput. Phys. 201 61
[29] Houim R W, Kuo K K 2011 J. Comput. Phys. 230 8527
[30] Quirk J J, Karni S 1996 J. Fluid Mech. 318 129
[31] Bagabir A, Drikakis D 2001 Shock Waves 11 209
[32] Zhai Z G, Wang M H, Si T, Luo X S 2014 J. Fluid Mech. 757 800
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[1] Richtmyer R D 1960 Commun. Pure Appl. Math. 13 297
[2] Meshkov E E 1969 Fluid Dyn. 4 101
[3] Lindl J D, Mccrory R L, Campbell E M 1992 Phys. Today 45 32
[4] Lindl J D, Amendt P, Berger R L 2004 Phys. Plasmas 11 339
[5] Yang J, Kubota T, Zukoski E E 1993 AIAA J. 31 854
[6] Arnett W D, Bahcall J N, Kirshner R P, Woosley S E 1989 Annu. Rev. Astron. Astrophys. 27 629
[7] Haas J F, Sturtevant B 1987 J. Fluid Mech. 181 41
[8] Jacobs J W 1992 J. Fluid Mech. 234 629
[9] Giordano J, Burtschell Y 2006 Phys. Fluids 18 036102
[10] Ranjan D, Niederhaus J H J, Oakley J G, Anderson M H, Greenough J A, Bonazza R 2008 Phys. Scripta 132 014020
[11] Tomkins C, Kumar S, Orlicz G, Prestridge K 2008 J. Fluid Mech. 611 131
[12] Shankar S K, Kawai S, Lele S K 2011 Phys. Fluids 23 024102
[13] Sha S, Chen Z H, Xue D W 2013 Acta Phys. Sin. 62 144701 (in Chinese) [沙莎, 陈志华, 薛大文 2013 物理学报 62 144701]
[14] Sha S, Chen Z H, Zhang Q B 2015 Acta Phys. Sin. 64 015201 (in Chinese) [沙莎, 陈志华, 张庆兵 2015 物理学报 64 015201]
[15] Sha S, Chen Z H, Xue D W, Zhang H 2014 Acta Phys. Sin. 63 085205 (in Chinese) [沙莎, 陈志华, 薛大文, 张辉 2014 物理学报 63 085205]
[16] Bai J S, Zou L Y, Wang T, Liu K, Huang W B, Liu J H, Li P, Tang D W, Liu C L 2010 Phys. Rev. E 82 056318
[17] Liao S F, Zou L Y, Huang X L, Liu J H, Zhang K, Wang Y P 2016 Sci. Sin.: Phys. Mech. Astron. 46 034702 (in Chinese) [廖深飞, 邹立勇, 黄熙龙, 刘金宏, 张珂, 王彦平 2016 中国科学: 物理学 力学 天文学 46 034702]
[18] Zhai Z G, Si T, Zou L Y, Luo X S 2013 Acta Mech. Sin. 29 24
[19] Zhai Z G, Dong P, Luo X S 2017 Chin. J. High Pressure Phys. 31 718 (in Chinese) [翟志刚, 董平, 罗喜胜 2017 高压物理学报 31 718]
[20] Fan M R, Zhai Z G, Si T, Luo X S, Zou L Y, Tan D W 2012 Sci. China: Phys. Mech. Astron. 55 284
[21] Wang M, Si T, Luo X 2015 Shock Waves 25 347
[22] Huang X L, Liao S F, Zou L Y, Liu J H, Cao R Y 2017 Explo. Shock Wave 37 829 (in Chinese) [黄熙龙, 廖深飞, 邹立勇, 刘金宏, 曹仁义 2017 爆炸与冲击 37 829]
[23] Abgrall R, Karni S 2001 J. Comput. Phys. 169 594
[24] Ern A, Giovangigli V 1994 Multicomponent Transport Algorithms (Heidelberg: Springer-Verlag) pp329-389
[25] Kee R J, Coltrin M E, Glarborg P 2003 Chemically Reacting Flow Theory and Practice (Hoboken: John Wiley Sons) pp487-530
[26] Svehla R A 1962 Estimated Viscosities and Thermal Conductivities of Gases at High Temperatures (NASA Technical Report R-132) pp20-24
[27] Spiteri R J, Ruuth S J 2003 Siam J. Numer. Anal. 40 469
[28] Verwer J G, Sommeijer B P, Hundsdorfer W 2004 J. Comput. Phys. 201 61
[29] Houim R W, Kuo K K 2011 J. Comput. Phys. 230 8527
[30] Quirk J J, Karni S 1996 J. Fluid Mech. 318 129
[31] Bagabir A, Drikakis D 2001 Shock Waves 11 209
[32] Zhai Z G, Wang M H, Si T, Luo X S 2014 J. Fluid Mech. 757 800
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