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Shock waves induced by a planar obstacle in a vibrated granular gas

Published online by Cambridge University Press:  07 March 2018

Alexandre Vilquin ,
Hamid Kellay  and
Jean-François Boudet Open the ORCID record for Jean-François Boudet [Opens in a new window]
Alexandre Vilquin
Affiliation:
Laboratoire Onde et Matière d’Aquitaine (UMR CNRS 5798), Université de Bordeaux, 351 cours de la Libération, 33405 Talence, France
Hamid Kellay
Affiliation:
Laboratoire Onde et Matière d’Aquitaine (UMR CNRS 5798), Université de Bordeaux, 351 cours de la Libération, 33405 Talence, France
Jean-François Boudet*
Affiliation:
Laboratoire Onde et Matière d’Aquitaine (UMR CNRS 5798), Université de Bordeaux, 351 cours de la Libération, 33405 Talence, France
*
†Email address for correspondence: [email protected]
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Abstract

The low value of the speed of sound in dilute granular media permits the study of the properties of supersonic flows for a wide range of Mach numbers. In this paper, we report the experimental observation of a subsonic–supersonic transition in a vibrated granular gas. The shock fronts studied are obtained by simply pushing a rectangular obstacle into the granular gas for different obstacle velocities. The supersonic regime is characterized by the formation of normal shock waves whose width increases when the Mach number decreases to values close to 1. The bimodal model proposed by Mott-Smith in the 1950s provides a good description for the velocity distributions as well as the macroscopic quantities for shock waves in molecular gases but remains inadequate for dissipative media like granular gases and plasmas. Here by examining the shock front structure for a wide range of Mach numbers, we adapt the Mott-Smith bimodal description to a dissipative medium. By using balance equations from granular kinetic theory and taking into account different dissipation sources, the proposed model allows us to understand how this dissipation modifies temperature, mean velocity and volume fraction profiles through the shock front.

JFM classification

Granular media: Granular media Rarefied Gas Flow: Kinetic theory Compressible Flows: Shock waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 842 , 10 May 2018 , pp. 163 - 187
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Copyright
© 2018 Cambridge University Press 

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References

Alsmeyer, H. 1976 Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam. J. Fluid Mech. 74 (03), 497–513.CrossRefGoogle Scholar
Amarouchene, Y. & Kellay, H. 2006 Speed of sound from shock fronts in granular flows. Phys. Fluids 18 (3), 031707.CrossRefGoogle Scholar
Anderson, J. D. 1990 Modern Compressible Flow: With Historical Perspective, vol. 12. McGraw-Hill.Google Scholar
Andreotti, B., Forterre, Y. & Pouliquen, O.2012 Les milieux granulaires-entre fluide et solide: Entre fluide et solide. EDP sciences. Collection Savoirs Actuels. http://laboutique.edpsciences.fr/produit/9782759800971.Google Scholar
Aumaître, S., Fauve, S., McNamara, S. & Poggi, P. 2001 Power injected in dissipative systems and the fluctuation theorem. Eur. Phys. J. B 19 (3), 449–460.CrossRefGoogle Scholar
Barbier, M., Villamaina, D. & Trizac, E. 2015 Blast dynamics in a dissipative gas. Phys. Rev. Lett. 115 (21), 214301.CrossRefGoogle Scholar
Barrat, A. & Trizac, E. 2002 Molecular dynamics simulations of vibrated granular gases. Phys. Rev. E 66 (5), 051303.CrossRefGoogle ScholarPubMed
Boudet, J. F., Amarouchene, Y. & Kellay, H. 2006 Dynamics of impact cratering in shallow sand layers. Phys. Rev. Lett. 96 (15), 158001.CrossRefGoogle ScholarPubMed
Boudet, J. F., Amarouchene, Y. & Kellay, H. 2007 The granular jump. J. Fluid Mech. 572, 413–432.CrossRefGoogle Scholar
Boudet, J. F., Amarouchene, Y. & Kellay, H. 2008 Shock front width and structure in supersonic granular flows. Phys. Rev. Lett. 101 (25), 254503.CrossRefGoogle ScholarPubMed
Boudet, J. F., Cassagne, J. & Kellay, H. 2009 Blast shocks in quasi-two-dimensional supersonic granular flows. Phys. Rev. Lett. 103 (22), 224501.CrossRefGoogle ScholarPubMed
Boudet, J. F. & Kellay, H. 2010 Drag coefficient for a circular obstacle in a quasi-two-dimensional dilute supersonic granular flow. Phys. Rev. Lett. 105 (10), 104501.CrossRefGoogle Scholar
Bougie, J., Moon, S. J., Swift, J. B. & Swinney, H. L. 2002 Shocks in vertically oscillated granular layers. Phys. Rev. E 66 (5), 051301.CrossRefGoogle ScholarPubMed
Brilliantov, N. & Pöschel, T. 2003 Hydrodynamics and transport coefficients for dilute granular gases. Phys. Rev. E 67 (6), 061304.CrossRefGoogle ScholarPubMed
Faug, T., Childs, P., Wyburn, E. & Einav, I. 2015 Standing jumps in shallow granular flows down smooth inclines. Phys. Fluids 27 (7), 073304.CrossRefGoogle Scholar
Goldhirsch, I. 2003 Rapid granular flows. Annu. Rev. Fluid Mech. 35 (1), 267–293.CrossRefGoogle Scholar
Goldshtein, A. & Shapiro, M. 1995 Mechanics of collisional motion of granular materials. Part 1. General hydrodynamic equations. J. Fluid Mech. 282, 75–114.CrossRefGoogle Scholar
Goldshtein, A., Shapiro, M. & Gutfinger, C. 1996 Mechanics of collisional motion of granular materials. Part 3. Self-similar shock wave propagation. J. Fluid Mech. 316, 29–51.CrossRefGoogle Scholar
Gray, J. M. N. T., Tai, Y. C. & Noelle, S. 2003 Shock waves, dead zones and particle-free regions in rapid granular free-surface flows. J. Fluid Mech. 491, 161–181.CrossRefGoogle Scholar
Gray, J. M. N. T. & Cui, X. 2007 Weak, strong and detached oblique shocks in gravity-driven granular free-surface flows. J. Fluid Mech. 579, 113–136.CrossRefGoogle Scholar
Gregori, G., Ravasio, A., Murphy, C. D., Schaar, K., Baird, A., Bell, A. R., Benuzzi-Mounaix, A., Bingham, R., Constantin, C., Drake, R. P. et al. 2012 Generation of scaled protogalactic seed magnetic fields in laser-produced shock waves. Nature 481 (7382), 480–483.CrossRefGoogle ScholarPubMed
Grossman, E. L., Zhou, T. & Ben-Naim, E 1997 Towards granular hydrodynamics in two dimensions. Phys. Rev. E 55 (4), 4200–4207.CrossRefGoogle Scholar
Haff, P. K. 1983 Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401–430.CrossRefGoogle Scholar
Hákonardóttir, K. M. & Hogg, A. J. 2005 Oblique shocks in rapid granular flows. Phys. Fluids 17 (7), 077101.CrossRefGoogle Scholar
Harnett, L. N. & Muntz, E. P. 1972 Experimental investigation of normal shock wave velocity distribution functions in mixtures of argon and helium. Phys. Fluids 15 (4), 565–572.CrossRefGoogle Scholar
Heil, P., Rericha, E. C., Goldman, D. I. & Swinney, H. L. 2004 Mach cone in a shallow granular fluid. Phys. Rev. E 70 (6), 060301.CrossRefGoogle Scholar
Holian, B. L., Patterson, C. W., Mareschal, M. & Salomons, E. 1993 Modeling shock waves in an ideal gas: going beyond the Navier–Stokes level. Phys. Rev. E 47 (1), R24.CrossRefGoogle Scholar
Holtz, T. & Muntz, E. P. 1983 Molecular velocity distribution functions in an argon normal shock wave at Mach number 7. Phys. Fluids 26 (9), 2425–2436.CrossRefGoogle Scholar
Hoover, W. G. & Hoover, C. G. 2010 Well-posed two-temperature constitutive equations for stable dense fluid shock waves using molecular dynamics and generalizations of Navier–Stokes–Fourier continuum mechanics. Phys. Rev. E 81 (4), 046302.CrossRefGoogle ScholarPubMed
Huang, K., Miao, G., Zhang, P., Yun, Y. & Wei, R. 2006 Shock wave propagation in vibrofluidized granular materials. Phys. Rev. E 73 (4), 041302.CrossRefGoogle ScholarPubMed
Hugoniot, H. 1887 Mémoire sur la propagation des mouvements dans les corps et spécialement dans les gaz parfaits (première partie). J. l’École Polytechnique 57, 397.Google Scholar
Johnson, C. G. & Gray, J. M. N. T. 2011 Granular jets and hydraulic jumps on an inclined plane. J. Fluid Mech. 675, 87–116.CrossRefGoogle Scholar
Landau, L. D. & Lifchitz, E. M. 1967 Physique Théorique. Tome VI, Physique Statistique, pp. 434–437. Mir, Moscou.Google Scholar
Landau, L. D. & Lifchitz, E. M. 1971 Physique Théorique. Tome VI, Mécanique des Fluides. Mir, Moscou.Google Scholar
Lifshitz, J. M. & Kolsky, H. 1964 Some experiments on anelastic rebound. J. Mech. Phys. Solids 12 (1), 35–43.CrossRefGoogle Scholar
Lumpkin, F. E. III & Chapman, D. R. 1992 Accuracy of the Burnett equations for hypersonic real gas flows. J. Thermophys. Heat Transfer 6 (3), 419–425.CrossRefGoogle Scholar
Lun, C. K. K., Savage, S. B., Jeffrey, D. J. & Chepurniy, N. 1984 Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flowfield. J. Fluid Mech. 140, 223–256.CrossRefGoogle Scholar
Mazouffre, S., Vankan, P., Engeln, R. & Schram, D. C. 2001 Behavior of the H atom velocity distribution function within the shock wave of a hydrogen plasma jet. Phys. Rev. E 64 (6), 066405.CrossRefGoogle Scholar
Mejean, S., Faug, T. & Einav, I. 2017 A general relation for standing normal jumps in both hydraulic and dry granular flows. J. Fluid Mech. 816, 331–351.CrossRefGoogle Scholar
Mott-Smith, H. M. 1951 The solution of the Boltzmann equation for a shock wave. Phys. Rev. 82 (6), 885.CrossRefGoogle Scholar
Muntz, E. P. & Harnett, L. N. 1969 Molecular velocity distribution function measurements in a normal shock wave. Phys. Fluids 12, 2027–2035.CrossRefGoogle Scholar
Ogawa, S. 1978 Multitemperature theory of granular materials. In Proceedings U.S.-Japan Seminar on Continuum-Mechanical and Statistical Approaches in the Mechanics of Granular Materials, Gakajutsu Bunken Fukyu-Kai, pp. 208–217.Google Scholar
Pham-Van-Diep, G., Erwin, D. & Muntz, E. P. 1989 Nonequilibrium molecular motion in a hypersonic shock wave. Science 245 (4918), 624–626.CrossRefGoogle Scholar
Rankine, W. M. 1870 On the thermodynamic theory of waves of finite longitudinal disturbance. Phil. Trans. R. Soc. Lond. 160, 277–288.Google Scholar
Rericha, E. C., Bizon, C., Shattuck, M. D. & Swinney, H. L. 2001 Shocks in supersonic sand. Phys. Rev. Lett. 88 (1), 014302.CrossRefGoogle ScholarPubMed
Roeller, K., Clewett, J. P., Bowley, R. M., Herminghaus, S. & Swift, M. R. 2011 Liquid-gas phase separation in confined vibrated dry granular matter. Phys. Rev. Lett. 107 (4), 048002.CrossRefGoogle ScholarPubMed
Rouyer, F. & Menon, N. 2000 Velocity fluctuations in a homogeneous 2D granular gas in steady state. Phys. Rev. Lett. 85 (17), 3676.CrossRefGoogle Scholar
Savage, S. B. 1988 Streaming motions in a bed of vibrationally fluidized dry granular material. J. Fluid Mech. 194, 457–478.CrossRefGoogle Scholar
Savage, S. B. & Jeffrey, D. J. 1981 The stress tensor in a granular flow at high shear rates. J. Fluid Mech. 110, 255–272.CrossRefGoogle 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 (2), 023003.CrossRefGoogle ScholarPubMed
Van Zon, J. S. & Mackintosh, F. C. 2004 Velocity distributions in dissipative granular gases. Phys. Rev. Lett. 93 (3), 038001.CrossRefGoogle ScholarPubMed
Vankan, P., Mazouffre, S., Engeln, R. & Schram, D. C. 2005 Inflow and shock formation in supersonic, rarefied plasma expansions. Phys. Plasmas 12 (10), 102303.CrossRefGoogle Scholar
Vilquin, A.2015 Structure des ondes de choc dans les gaz granulaires. Doctoral dissertation, Université de Bordeaux.Google Scholar
Vilquin, A., Boudet, J. F. & Kellay, H. 2016 Structure of velocity distributions in shock waves in granular gases with extension to molecular gases. Phys. Rev. E 94 (2), 022905.CrossRefGoogle ScholarPubMed
Windows-Yule, C. R. K. & Parker, D. J. 2013 Boltzmann statistics in a three-dimensional vibrofluidized granular bed: Idealizing the experimental system. Phys. Rev. E 87 (2), 022211.CrossRefGoogle Scholar
Windows-Yule, C. R. K., Rosato, A. D., Parker, D. J. & Thornton, A. R. 2015 Maximizing energy transfer in vibrofluidized granular systems. Phys. Rev. E 91 (5), 052203.CrossRefGoogle ScholarPubMed