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Structure and stability of shock waves in granular gases

Published online by Cambridge University Press:  25 June 2019

Nick Sirmas Open the ORCID record for Nick Sirmas [Opens in a new window]  and
Matei I. Radulescu Open the ORCID record for Matei I. Radulescu [Opens in a new window]
Nick Sirmas*
Affiliation:
Department of Mechanical Engineering, University of Ottawa, Ottawa, ON K1N 6N5, Canada
Matei I. Radulescu
Affiliation:
Department of Mechanical Engineering, University of Ottawa, Ottawa, ON K1N 6N5, Canada
*
†Email address for correspondence: nsirmas@gmail.com
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Abstract

Previous experiments have revealed that shock waves driven through dissipative media may become unstable, for example, in granular gases, and in molecular gases undergoing strong relaxation effects. The current paper addresses this problem of shock stability at the Euler and Navier–Stokes continuum levels in a system of disks (two-dimensional) undergoing activated inelastic collisions. The dynamics of shock formation and stability is found to be in very good agreement with earlier molecular dynamic simulations (Sirmas & Radulescu, Phys. Rev. E, vol. 91, 2015, 023003). It was found that the modelling of shock instability requires the introduction of molecular noise for its development and sustenance. This is confirmed in two stability problems. In the first, the evolution of shock formation dynamics is monitored without noise, with only initial noise and with continuous molecular noise. Only the latter reproduces the results of shock instability of molecular dynamics simulations. In the second problem, the steady travelling wave solution is obtained for the shock structure in the inviscid and viscous limits and its nonlinear stability is studied with and without molecular fluctuations, again showing that instability can be sustained only in the presence of fluctuations. The continuum results show that instability takes the form of a rippled front of a wavelength comparable with the relaxation thickness of the steady shock wave, at scales at which molecular fluctuations become important, in excellent agreement with the molecular dynamic simulations.

JFM classification

Rarefied Gas Flow: Molecular dynamics Compressible Flows: Shock waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 873 , 25 August 2019 , pp. 568 - 607
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Copyright
© 2019 Cambridge University Press 

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References

Alder, B. J. & Wainwright, T. E. 1959 Studies in molecular dynamics. 1. General method. J. Chem. Phys. 31 (2), 459–466.10.1063/1.1730376Google Scholar
Ben-Naim, E., Chen, S. Y., Doolen, G. D. & Redner, S. 1999 Shocklike dynamics of inelastic gases. Phys. Rev. Lett. 83, 4069–4072.10.1103/PhysRevLett.83.4069Google Scholar
Bird, G. A. 1999 Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford University Press.Google Scholar
Bizon, C., Shattuck, M. D., Swift, J. B., McCormick, W. D. & Swinney, H. L. 1998 Patterns in 3D vertically oscillated granular layers: simulation and experiment. Phys. Rev. Lett. 80 (1), 57–60.10.1103/PhysRevLett.80.57Google Scholar
Boudet, J. F. & Kellay, H. 2013 Unstable blast shocks in dilute granular flows. Phys. Rev. E 87, 052202.10.1103/PhysRevE.87.052202Google Scholar
Brey, J. J., Maynar, P. & De Soria, M. I. G. 2009 Fluctuating hydrodynamics for dilute granular gases. Phys. Rev. E 79 (5), 051305.10.1103/PhysRevE.79.051305Google Scholar
Brey, J. J., Maynar, P. & de Soria, M. I. G. 2011 Fluctuating Navier–Stokes equations for inelastic hard spheres or disks. Phys. Rev. E 83 (4), 041303.10.1103/PhysRevE.83.041303Google Scholar
Bridges, F. G., Hatzes, A. & Lin, D. N. C. 1984 Structure, stability and evolution of Saturn’s rings. Nature 309 (5966), 333–335.10.1038/309333a0Google Scholar
Brilliantov, N. & Pöschel, T. 2004 Kinetic Theory of Granular Gases. Oxford University Press.10.1093/acprof:oso/9780198530381.001.0001Google Scholar
Campbell, C. S. 1990 Rapid granular flows. Annu. Rev. Fluid Mech. 22 (1), 57–90.10.1146/annurev.fl.22.010190.000421Google Scholar
Carrillo, J. A., Pöschel, T. & Salueña, C. 2008 Granular hydrodynamics and pattern formation in vertically oscillated granular disk layers. J. Fluid Mech. 591, 199–144.Google Scholar
Falle, S. A. E. G. 1991 Self-similar jets. Mon. Not. R. Astron. Soc. 250 (3), 581–596.10.1093/mnras/250.3.581Google Scholar
Falle, S. A. E. G. & Komissarov, S. S. 1996 An upwind numerical scheme for relativistic hydrodynamics with a general equation of state. Mon. Not. R. Astron. Soc. 278 (2), 586–602.10.1093/mnras/278.2.586Google Scholar
Fickett, W. & Davis, W. C. 2000 Detonation: Theory and Experiment. Dover.Google Scholar
Frost, D. L., Gregoire, Y., Petel, O., Goroshin, S. & Zhang, F. 2012 Particle jet formation during explosive dispersal of solid particles. Phys. Fluids 29 (9), 091109.Google Scholar
Glass, I. I. & Liu, W. S. 1978 Effects of hydrogen impurities on shock structure and stability in ionizing monatomic gases. 1. Argon. J. Fluid Mech. 84, 55–77.10.1017/S002211207800004XGoogle Scholar
Goldhirsch, I. 2003 Rapid granular flows. Annu. Rev. Fluid Mech. 35, 267–293.10.1146/annurev.fluid.35.101101.161114Google Scholar
Goldshtein, A. & Shapiro, M. 1995 Mechanics of collisional motion of granular materials. Part 1. General hydrodynamic equations. J. Fluid Mech. 282, 75–114.10.1017/S0022112095000048Google Scholar
Goldshtein, A., Shapiro, M. & Gutfinger, C. 1996 Mechanics of collisional motion of granular materials 3. Self-similar shock wave propagation. J. Fluid Mech. 316, 29–51.10.1017/S0022112096000432Google Scholar
Griffiths, R. W., Sandeman, R. J. & Hornung, H. G. 1976 Stability of shock waves in ionizing and dissociating gases. J. Phys. D Appl. Phys. 9 (12), 1681–1691.10.1088/0022-3727/9/12/006Google Scholar
Grun, J., Stamper, J., Manka, C., Resnick, J., Burris, R., Crawford, J. & Ripin, B. H. 1991 Instability of Taylor–Sedov blast waves propagating through a uniform gas. Phys. Rev. Lett. 66 (21), 2738–2741.10.1103/PhysRevLett.66.2738Google Scholar
Gad-el Hak, M. 2001 The MEMS Handbook. CRC press.10.1201/9781420050905Google Scholar
Helfand, E., Frisch, H. L. & Lebowitz, J. L. 1961 Theory of two- and one-dimensional rigid sphere fluids. J. Chem. Phys. 34 (3), 1037–1042.10.1063/1.1731629Google Scholar
Hornung, H. G. & Lemieux, P. 2001 Shock layer instability near the Newtonian limit of hypervelocity flows. Phys. Fluids 13 (8), 2394–2402.10.1063/1.1383591Google Scholar
Jaeger, H. M., Nagel, S. R. & Behringer, R. P. 1996 Granular solids, liquids, and gases. Rev. Mod. Phys. 68 (94), 1259–1273.10.1103/RevModPhys.68.1259Google Scholar
Jenkins, J. & Richman, M. 1985 Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks. Phys. Fluids 28 (12), 3485–3494.10.1063/1.865302Google Scholar
Kamenetsky, V., Goldshtein, A., Shapiro, M. & Degani, D. 2000 Evolution of a shock wave in a granular gas. Phys. Fluids 12 (11), 3036–3049.10.1063/1.1287514Google Scholar
Kuwabara, G. & Kono, K. 1987 Restitution coefficient in a collision between two spheres. Japan. J. Appl. Phys. 26 (8R), 1230.10.1143/JJAP.26.1230Google Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics, 2nd edn. Butterworth-Heinemann.Google Scholar
Maxwell, B.M.N., Bhattacharjee, R.R., Lau-Chapdelaine, S.S.M., Falle, S.A.E.G., Sharpe, G. J. & Radulescu, M. I. 2017 Influence of turbulent fluctuations on detonation propagation. J. Fluid Mech. 818, 646–696.10.1017/jfm.2017.145Google Scholar
Meerson, B. & Puglisi, A. 2005 Towards a continuum theory of clustering in a freely cooling inelastic gas. Europhys. Lett. 70 (4), 478–484.10.1209/epl/i2004-10507-8Google Scholar
Mishin, G. I., Bedin, A. P., Yushchenkova, N. I., Skvortsov, G. E. & Ryazin, A. P. 1981 Anomalous relaxation and the instability effect of shock waves in gases. Zh. Tekh. Fiz. 51 (11), 2315–2324.Google Scholar
Pöschel, T., Brilliantiov, N. V. & Schwager, T. 2003 Long-time behavior of granular gases with impact-velocity dependent coefficient of restitution. Physica A 325, 274–283.10.1016/S0378-4371(03)00206-1Google Scholar
Pöschel, T. & Schwager, T. 2005 Computational Granular Dynamics: Models and Algorithms. Springer.Google Scholar
Pouliquen, O., Delour, J. & Savage, S. B. 1997 Fingering in granular flows. Nature 386 (6627), 816.10.1038/386816a0Google Scholar
Ramírez, R., Pöschel, T., Brilliantov, N. V. & Schwager, T. 1999 Coefficient of restitution of colliding viscoelastic spheres. Phys. Rev. E 60, 4465–4472.10.1103/PhysRevE.60.4465Google Scholar
Rericha, E. C., Bizon, C., Shattuck, M. D. & Swinney, H. L. 2002 Shocks in supersonic sand. Phys. Rev. Lett. 88 (1), 014302.Google Scholar
Rodriguez, V., Saurel, R., Jourdan, G. & Houas, L. 2013 Solid-particle jet formation under shock-wave acceleration. Phys. Rev. E 88, 063011.10.1103/PhysRevE.88.063011Google Scholar
Salueña, C., Almazán, L. & Brilliantov, N. V. 2011 Mechanisms of cluster formation in force-free granular gases. Math. Modelling Nat. Phenom. 6 (4), 175–190.10.1051/mmnp/20127108Google Scholar
Semenov, A., Berezkina, M. & Krassovskaya, I. 2012 Classification of pseudo-steady shock wave reflection types. Shock Waves 22, 307–316.10.1007/s00193-012-0373-zGoogle Scholar
Short, M. & Stewart, D. S. 1998 Cellular detonation stability. Part 1. A normal-mode linear analysis. J. Fluid Mech. 368, 229–262.10.1017/S0022112098001682Google Scholar
Sirmas, N.2017 Dynamics and stability of shock waves in granular gases undergoing activated inelastic collisions. PhD thesis, University of Ottawa.Google Scholar
Sirmas, N. & Radulescu, M. I. 2015 Evolution and stability of shock waves in dissipative gases characterized by activated inelastic collisions. Phys. Rev. E 91, 023003.10.1103/PhysRevE.91.023003Google Scholar
Sirmas, N., Tudorache, M., Barahona, J. & Radulescu, M. I. 2012 Shock waves in hard disk fluids. Shock Waves 22 (3), 237–247.10.1007/s00193-012-0354-2Google Scholar
Tan, M. L. & Goldhirsch, I. 1998 Rapid granular flows as mesoscopic systems. Phys. Rev. Lett. 81 (14), 3022.10.1103/PhysRevLett.81.3022Google Scholar
Torquato, S. 1995 Nearest-neighbor statistics for packings of hard spheres and disks. Phys. Rev. E 51, 3170–3182.Google Scholar
Vincenti, W. G. & Kruger, C. H. 1975 Introduction to Physical Gas Dynamics. Krieger.Google Scholar
Zeldovich, Y. B. & Raizer, Y. P. 1966 Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Academic Press.Google Scholar