Numerical Investigation of a Shock Accelerated Heavy Gas Cylinder in the Self-Similar Regime

Affiliation(s)

^{1}
Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang, China.

^{2}
National Key Laboratory of Shock Wave and Detonation Physics, Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang, China.

ABSTRACT

A detailed numerical simulation of a shock accelerated heavy gas (SF_{6})
cylinder surrounded by air gas is presented. It is a simplified configuration
of the more general shock-accelerated inhomogeneous flows which occur in a wide
variety of astrophysical systems. From the snapshots of the time evolution of
the gas cylinder, we find that the evolution of the shock accelerated gas
cylinder is in some ways similar to the roll-ups of a vortex sheet for both
roll up into a spiral and fall into a self-similar behavior. The systemic and
meaningful analyses of the negative circulation, the center of vorticity and
the vortex spacing are in a good agreement with results obtained from the prediction
of vorticity dynamics. Unlike the mixing zone width in single-mode or
multi-mode Richtmyer-Meshkov instability which doesn’t exist, a single power
law of time owing to the bubble and spike fronts follow a power law of *t*^{θ} with different power
exponents, the normalized length of the shock accelerated gas cylinder follows
a single power law with *θ* = 0.43 in its self-similar regime obtained from the
numerical results.

A detailed numerical simulation of a shock accelerated heavy gas (SF

Cite this paper

Wang, B. , Bai, J. and Wang, T. (2015) Numerical Investigation of a Shock Accelerated Heavy Gas Cylinder in the Self-Similar Regime.*International Journal of Astronomy and Astrophysics*, **5**, 38-46. doi: 10.4236/ijaa.2015.51006.

Wang, B. , Bai, J. and Wang, T. (2015) Numerical Investigation of a Shock Accelerated Heavy Gas Cylinder in the Self-Similar Regime.

References

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http://dx.doi.org/10.1002/cpa.3160130207

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http://dx.doi.org/10.1007/BF01015969

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http://dx.doi.org/10.1017/S0022112094003307

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http://dx.doi.org/10.1086/313364

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http://dx.doi.org/10.1063/1.2031347

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http://dx.doi.org/10.1017/S0022112078002189

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http://dx.doi.org/10.1017/S0022112086002732

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http://dx.doi.org/10.1086/173554

[11] Hwang, U., Flanagan, K.A. and Petre, R. (2005) Chandra X-Ray Observation of a Mature Cloud-Shock Interaction in the Bright Eastern Knot Region of Puppis A. Astrophysical Journal, 635, 355-364.

http://dx.doi.org/10.1086/497298

[12] Haas, J.F. and Sturtevant, B. (1987) Interaction of Weak Shock Waves with Cylindrical and Spherical Gas Inhomogeneities. Journal of Fluid Mechanics, 181, 41-76.

http://dx.doi.org/10.1017/S0022112087002003

[13] Jacobs, J.W. (1993) The Dynamics of Shock Accelerated Light and Heavy Gas Cylinders. Physics of Fluids A, 5, 2239-2247.

http://dx.doi.org/10.1063/1.858562

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http://dx.doi.org/10.1017/S0022112008002723

[15] Picone, J.M. and Boris, J.P. (1983) Vorticity Generation by Asymmetric Energy Deposition in a Gaseous Medium. Physics of Fluids, 26, 365-382.

http://dx.doi.org/10.1063/1.864173

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http://dx.doi.org/10.1063/1.2884787

[18] Shankar, S.K., Kawai, S. and Lele, S.K. (2011) Two-Dimensional Viscous Flow Simulation of a Shock Accelerated Heavy Gas Cylinder. Physics of Fluids, 23, Article ID: 024102.

http://dx.doi.org/10.1063/1.3553282

[19] Dimonte, G. and Schneider, M. (2000) Density Ratio Dependence of Rayleigh-Taylor Mixing for Sustained and Impulsive Acceleration Histories. Physics of Fluids, 12, 304-321.

http://dx.doi.org/10.1063/1.870309

[20] Alon, U., Hecht, J., Ofer, D. and Shvarts, D. (1995) Power Laws and Similarity of Rayleigh-Taylor and Richtmyer-Meshkov Mixing Fronts at All Density Ratios. Physical Review Letters, 74, 534-537.

http://dx.doi.org/10.1103/PhysRevLett.74.534

[21] Sohn, S.-I. (2008) Quantitative Modeling of Bubble Competition in Richtmyer-Meshkov Instability. Physical Review E, 78, Article ID: 017302.

http://dx.doi.org/10.1103/PhysRevE.78.017302

[22] Bai, J.S., Wang, T., Li, P., Zou, L.Y, and Liu, C.L. (2009) Numerical Simulation of the Hydrodynamic Instability Experiments and Flow Mixing. Science in China Series G, 52, 2017-2040.

http://dx.doi.org/10.1007/s11433-009-0277-9

[23] Colella, P. and Woodward, P.R., (1984) The Piecewise Parabolic Method (PPM) for Gas-Dynamical Simulations. Journal of Computational Physics, 54, 174-201.

http://dx.doi.org/10.1016/0021-9991(84)90143-8

[24] Vreman, W. (2004) An Eddy-Viscosity Subgrid-Scale Model for Turbulent Shear Flow: Algebraic Theory and Applications. Physics of Fluids, 16, 3670-3681.

http://dx.doi.org/10.1063/1.1785131

[25] Miles, J.W. (1958) On the Disturbed Motion of a Plane Vortex Sheet. Journal of Fluid Mechanics, 4, 538-552.

http://dx.doi.org/10.1017/S0022112058000653

[26] Samtaney, R. and Pullin, D.I. (1996) On Initial-Value and Self-Similar Solutions of the Compressible Euler Equations. Physics of Fluids, 8, 2650-2655.

http://dx.doi.org/10.1063/1.869050

[27] Jones, B.D. and Jacobs, J.W. (1997) A Membraneless Experimental for the Study of Richtmyer-Meshkov Instability of a Shock-Accelerated Gas Interface. Physics of Fluids, 9, 3078-3085.

http://dx.doi.org/10.1063/1.869416

[28] Zhang, S., Zabusky, N.J., Peng, G. and Gupta, S. (2004) Shock Gaseous Cylinder Interactions: Dynamically Validated Initial Conditions Provide Excellent Agreement between Experiments and Numerical Simulations to Late-Intermediate Time. Physics of Fluids, 16, 1203-1216.

http://dx.doi.org/10.1063/1.1651483

[29] Picone, J.M. and Boris, J.P. (1988) Vorticity Generation by Shock Propagation through Bubbles in a Gas. Journal of Fluid Mechanics, 189, 23-51.

http://dx.doi.org/10.1017/S0022112088000904

[30] Tong, B.G., Yin, X.Y. and Zhu, K.Q. (2009) Theory of Vortex Motion. 2nd Edition, University of Science and Technology of China Press, Hefei.

[1] Richtmyer, R.D. (1960) Taylor Instability in a Shock Acceleration of Compressible Fluids. Communications on Pure and Applied Mathematics, 13, 297-319.

http://dx.doi.org/10.1002/cpa.3160130207

[2] Meshkov, E.E. (1968) Instability of the Interface of Two Gases Accelerated by a Shock Wave. Soviet Fluid Dynamics, 4, 101-104.

http://dx.doi.org/10.1007/BF01015969

[3] Lindl, J.D., McCropy, R.L. and Campbell, E.M. (1992) Progress toward Ignition and Propagating Burn in Inertial Confinement Fusion. Physics Today, 45, 32-40.

http://dx.doi.org/10.1063/1.881318

[4] Yang, J., Kubota, T. and Zukoski, E.E. (1994) A Model for Characterization of a Vortex Pair Formed by Shock Passage over a Light-Gas Inhomogeneity. Journal of Fluid Mechanics, 258, 217-244.

http://dx.doi.org/10.1017/S0022112094003307

[5] Arnett, D. (2000) The Role of Mixing in Astrophysics. The Astrophysical Journal Supplement Series, 127, 213-217.

http://dx.doi.org/10.1086/313364

[6] Kumar, S., Orlicz, G., Tomkins, C., Goodenough, C., Prestridge, K., Vorobieff, P. and Benjamin, R. (2005) Stretching of Material Lines in Shock-Accelerated Gaseous Flows. Physics of Fluids, 17, Article ID: 082107.

http://dx.doi.org/10.1063/1.2031347

[7] Moore, D.W. (1975) The Rolling Up of a Semi-Infinite Vortex Sheet. Proceedings of the Royal Society of London A, 345, 417-430.

http://dx.doi.org/10.1098/rspa.1975.0147

[8] Pullin, D.I. (1978) The Large-Scale Structure of Unsteady Self-Similar Rolled-Up Vortex Sheets. Journal of Fluid Mechanics, 88, 401-430.

http://dx.doi.org/10.1017/S0022112078002189

[9] Krasny, R. (1986) A Study of Singularity Formation in a Vortex Sheet by the Point-Vortex Approximation. Journal of Fluid Mechanics, 167, 65-93.

http://dx.doi.org/10.1017/S0022112086002732

[10] Klein, R.I., McKee, C.F. and Colella, P. (1994) On the Hydrodynamic Interaction of Shock Waves with Interstellar Clouds. 1: Nonradiative Shocks in Small Clouds. The Astrophysical Journal, 420, 213-236.

http://dx.doi.org/10.1086/173554

[11] Hwang, U., Flanagan, K.A. and Petre, R. (2005) Chandra X-Ray Observation of a Mature Cloud-Shock Interaction in the Bright Eastern Knot Region of Puppis A. Astrophysical Journal, 635, 355-364.

http://dx.doi.org/10.1086/497298

[12] Haas, J.F. and Sturtevant, B. (1987) Interaction of Weak Shock Waves with Cylindrical and Spherical Gas Inhomogeneities. Journal of Fluid Mechanics, 181, 41-76.

http://dx.doi.org/10.1017/S0022112087002003

[13] Jacobs, J.W. (1993) The Dynamics of Shock Accelerated Light and Heavy Gas Cylinders. Physics of Fluids A, 5, 2239-2247.

http://dx.doi.org/10.1063/1.858562

[14] Tomkins, C.D., Kumar, S., Orlicz, G. and Prestridge, K.P. (2008) An Experimental Investigation of Mixing Mechanisms in Shock-Accelerated Flow. Journal of Fluid Mechanics, 611, 131-150.

http://dx.doi.org/10.1017/S0022112008002723

[15] Picone, J.M. and Boris, J.P. (1983) Vorticity Generation by Asymmetric Energy Deposition in a Gaseous Medium. Physics of Fluids, 26, 365-382.

http://dx.doi.org/10.1063/1.864173

[16] Quirk, J.J. and Karni, S. (1994) On the Dynamics of a Shock-Bubble Interaction. NASA CR 194978, ICASE Report No. 94-75.

[17] Weirs, V.G., Dupont, T. and Plewa, T. (2008) Three-Dimensional Effects in Shock-Cylinder Interactions. Physics of Fluids, 20, Article ID: 044102.

http://dx.doi.org/10.1063/1.2884787

[18] Shankar, S.K., Kawai, S. and Lele, S.K. (2011) Two-Dimensional Viscous Flow Simulation of a Shock Accelerated Heavy Gas Cylinder. Physics of Fluids, 23, Article ID: 024102.

http://dx.doi.org/10.1063/1.3553282

[19] Dimonte, G. and Schneider, M. (2000) Density Ratio Dependence of Rayleigh-Taylor Mixing for Sustained and Impulsive Acceleration Histories. Physics of Fluids, 12, 304-321.

http://dx.doi.org/10.1063/1.870309

[20] Alon, U., Hecht, J., Ofer, D. and Shvarts, D. (1995) Power Laws and Similarity of Rayleigh-Taylor and Richtmyer-Meshkov Mixing Fronts at All Density Ratios. Physical Review Letters, 74, 534-537.

http://dx.doi.org/10.1103/PhysRevLett.74.534

[21] Sohn, S.-I. (2008) Quantitative Modeling of Bubble Competition in Richtmyer-Meshkov Instability. Physical Review E, 78, Article ID: 017302.

http://dx.doi.org/10.1103/PhysRevE.78.017302

[22] Bai, J.S., Wang, T., Li, P., Zou, L.Y, and Liu, C.L. (2009) Numerical Simulation of the Hydrodynamic Instability Experiments and Flow Mixing. Science in China Series G, 52, 2017-2040.

http://dx.doi.org/10.1007/s11433-009-0277-9

[23] Colella, P. and Woodward, P.R., (1984) The Piecewise Parabolic Method (PPM) for Gas-Dynamical Simulations. Journal of Computational Physics, 54, 174-201.

http://dx.doi.org/10.1016/0021-9991(84)90143-8

[24] Vreman, W. (2004) An Eddy-Viscosity Subgrid-Scale Model for Turbulent Shear Flow: Algebraic Theory and Applications. Physics of Fluids, 16, 3670-3681.

http://dx.doi.org/10.1063/1.1785131

[25] Miles, J.W. (1958) On the Disturbed Motion of a Plane Vortex Sheet. Journal of Fluid Mechanics, 4, 538-552.

http://dx.doi.org/10.1017/S0022112058000653

[26] Samtaney, R. and Pullin, D.I. (1996) On Initial-Value and Self-Similar Solutions of the Compressible Euler Equations. Physics of Fluids, 8, 2650-2655.

http://dx.doi.org/10.1063/1.869050

[27] Jones, B.D. and Jacobs, J.W. (1997) A Membraneless Experimental for the Study of Richtmyer-Meshkov Instability of a Shock-Accelerated Gas Interface. Physics of Fluids, 9, 3078-3085.

http://dx.doi.org/10.1063/1.869416

[28] Zhang, S., Zabusky, N.J., Peng, G. and Gupta, S. (2004) Shock Gaseous Cylinder Interactions: Dynamically Validated Initial Conditions Provide Excellent Agreement between Experiments and Numerical Simulations to Late-Intermediate Time. Physics of Fluids, 16, 1203-1216.

http://dx.doi.org/10.1063/1.1651483

[29] Picone, J.M. and Boris, J.P. (1988) Vorticity Generation by Shock Propagation through Bubbles in a Gas. Journal of Fluid Mechanics, 189, 23-51.

http://dx.doi.org/10.1017/S0022112088000904

[30] Tong, B.G., Yin, X.Y. and Zhu, K.Q. (2009) Theory of Vortex Motion. 2nd Edition, University of Science and Technology of China Press, Hefei.