Investigation of Normal Shock Structure by Using Navier-Stokes Equations with the Second Viscosity

doi:10.19596/j.cnki.1001-246x.8121

Abstract

Abstract: To investigate influence mechanism of the second viscosity on internal flow of a normal shock wave, one-dimensional Navier-Stokes equations are numerically solved. It indicates that the second viscosity has a smoothing effect on density, heat flow and energy distribution in the shock wave, which results in a decrease of peak value of heat and entropy flows, and an increase of shock thickness. Due to the production of normal viscous dissipation, some lost mechanical energy is converted into internal energy. As considering the second viscosity, density distribution and shock thickness are greatly improved. They are in good agreement with experimental data. In addition, Knudsen number is obtained 0.12≤Kn≤0.4 within Mach number range from 1.2 to 10. It indicates that Navier-Stokes equations with the second viscosity simulate normal shock structure more accurately.

Key words: Navier-Stokes equations, Stokes assumption, the second viscosity, bulk viscosity, shock wave structure

CLC Number:

LI Xindong, ZHAO Yingkui, HU Zongmin, JIANG Zonglin. Investigation of Normal Shock Structure by Using Navier-Stokes Equations with the Second Viscosity[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2020, 37(5): 505-513.

References

[1] 武际可. 力学史[M]. 上海:上海辞书出版社, 2010.
[2] STOKE G G. On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids[J]. Transactions of the Cambridge Philosophical Society, 1845, 8(22):287-342.
[3] 吴望一. 流体力学[M]. 北京:北京大学出版社, 1982.
[4] 泽尔泽尔道维奇R B, 莱依捷尔IO P. 激波与高温流体动力学现象物理学[M]. 张树才,译. 北京:科学出版社, 1985.
[5] LANDAU L D, LIFSHITZ E M. Fluid mechanics[M]. Beijing:Pergamon, 1959.
[6] WANG C, UHLENBECK G E. Transport phenomena in polyatomic gases[R]. Report No CM-681, 1951.
[7] MONCHICK L, YUN K S, MASON E A. Formal kinetic theory of transport phenomena in polyatomic gas mixtures[J]. The Journal of Chemical Physics, 1963, 39(3):654-669.
[8] CHAPMAN S, COWLING T G. The mathematical theory of non-uniform gases[M]. Cambridge:Cambridge University Press, 1970.
[9] 李馨东, 胡宗民, 姜宗林. 可压缩流体体积黏性的连续介质理论[J]. 中国科学:物理学力学天文学, 2016, 46(3):034701.
[10] LI X D, HU Z M, JIANG Z. Continuum perspective of bulk viscosity in compressible fluids[J]. Journal of Fluid Mechanics, 2017, 812:966-990.
[11] SHERMAN D S. A low-density wind-tunnel study of shock-wave structure and relaxation phenomena in gases[R]. NACA TN-3298, 1955.
[12] PRANGSMA G J, ALBERGA A H, BEENAKKER J J M. Ultrasonic determination of the volume viscosity of N₂, CO, CH₄ and CD₄ between 77 and 300 K[J]. Physics, 1973, 64(2):278-288.
[13] ASH R L, ZUKERWAR A J, ZHENG Z Q. Second coefficient of viscosity in air[R]. NASA CR-187783, 1991.
[14] GRAVES R E, ARGROW B M. Bulk viscosity:Past to present[J]. Journal of Thermophysics and Heat Transfer, 1999, 13(3):337-342.
[15] ELIZAROVA T G, KHOKHLOV A A, MONTERO S. Numerical simulation of shock wave structure in nitrogen[J]. Physics of Fluids, 2007, 19, 068102.
[16] CHIKITKIN A V, ROGOV B V, RIRSKY G A, et al. Effect of bulk viscosity in supersonic flow past spacecraft[J]. Applied Numerical Mathematics, 2015, 93:47-60.
[17] EMANUEL G. Effect of bulk viscosity on a hypersonic boundary layer[J]. Physics of Fluids, 1992, 4:491-495.
[18] GONZALEZ H, EMANUEL G. Effect of bulk viscosity on Couette flow[J]. Physics of Fluids, 1993, 5:1267-1268.
[19] CRAMER M S, BAHMANI F. Effect of large bulk viscosity on large-Reynolds-number flows[J]. Journal of Fluid Mechanics, 2014, 751:142-163.
[20] BAHMANI F, CRAMER M S. Suppression of shock-induced separation in fluids having large bulk viscosities[J]. Journal of Fluid Mechanics, 2014, 756:1-10.
[21] BILLET G, GIOVANGIGLI V, GASSOWSKI G D. Impact of volume viscosity on a shock/hydrogen bubble interaction[R]. Ecole Polytechnique Centre de Mathématiques Appliquées, 2007, UMR CNRS-7641.
[22] FRU G, JANIGA G, THEVENIN D. Impact of volume viscosity on the structure of turbulent premixed flames in the thin reaction zone regime[J]. Flow Turbulence Combust, 2012, 88:451-478.
[23] PAN S W, JOHNSEN E. The role of bulk viscosity on the decay of compressible, homogeneous, isotropic turbulence[J]. Journal of Fluid Mechanics, 2017, 833:717-744.
[24] LI X D, ZHAO Y K, OUYANG B Y, et al. Numerical investigation of bulk viscosity effect on two-dimensional toroidal shock wave focusing[J]. Chinese Journal of Computational Physics, 2017, 34(4):394-402.
[25] 李志辉, 张涵信. 激波结构内流动问题的气体运动论描述[J]. 空气动力学学报, 2007, 25(4):411-436.
[26] 沈青, 胡振华,徐晓燕, 等. 正激波结构与反射的蒙特卡罗模拟[J]. 空气动力学学报, 1991, 9(4):435-441.
[27] KOGAN M N. Rarefied gas dynamics[M]. New York:Plenum, 1969.
[28] BIRD G A. Molecular gas dynamics and the direct simulation of gas flows[M]. Oxford:Clarendon, 1998.
[29] ALSMEYER H. Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam[J]. Journal of Fluid Mechanics, 1976, 74(3):497-513.
[30] 李维新. 一维不定常流与冲击波[M]. 北京:国防工业出版社, 2003.
[31] ANDERSON J D. Hypersonic and high-temperature gas dynamics[M]. AIAA, 2006.
[32] 卞阴贵, 徐立功. 气动热力学[M]. 合肥:中国科学技术大学出版社, 2011.