Skip to main content Accessibility help

Propagation of a strong shock over a random bed of spherical particles

  • Y. Mehta (a1), C. Neal (a1), K. Salari (a2), T. L. Jackson (a1), S. Balachandar (a1) and S. Thakur (a1)...


Propagation of a strong incident shock through a bed of particles results in complex wave dynamics such as a reflected shock, a transmitted shock, and highly unsteady flow inside the particle bed. In this paper we present three-dimensional numerical simulations of shock propagation in air over a random bed of particles. We assume the flow is inviscid and governed by the Euler equations of gas dynamics. Simulations are carried out by varying the volume fraction of the particle bed at a fixed shock Mach number. We compute the unsteady inviscid streamwise and transverse drag coefficients as a function of time for each particle in the random bed for different volume fractions. We show that (i) there are significant variations in the peak drag for the particles in the bed, (ii) the mean peak drag as a function of streamwise distance through the bed decreases with a slope that increases as the volume fraction increases, and (iii) the deviation from the mean peak drag does not correlate with local volume fraction. We also present the local Mach number and pressure contours for the different volume fractions to explain the various observed complex physical mechanisms occurring during the shock–particle interactions. Since the shock interaction with the random bed of particles leads to transmitted and reflected waves, we compute the average flow properties to characterize the strength of the transmitted and reflected shock waves and quantify the energy dissipation inside the particle bed. Finally, to better understand the complex wave dynamics in a random bed, we consider a simpler approximation of a planar shock propagating in a duct with a sudden area change. We obtain Riemann solutions to this problem, which are used to compare with fully resolved numerical simulations.


Corresponding author

Email address for correspondence:


Hide All
Akiki, G., Jackson, T. L. & Balachandar, S. 2016 Force variation within arrays of monodisperse spherical particles. Phys. Rev. Fluids 1, 044202.
Akiki, G., Jackson, T. L. & Balachandar, S. 2017 Pairwise interaction extended point-particle model for a random array of monodisperse spheres. J. Fluid Mech. 813, 882928.
Ashok S. Sangani, D. Z. Z. & Prosperetti, A. 1991 The added mass, basset, and viscous drag coefficients in nondilute bubbly liquids undergoing small-amplitude oscillatory motion. Phys. Fluids A 3 (12), 29552970.
Auton, T. R., Hunt, J. C. R. & Prud’Homme, M. 1988 The force exerted on a body in inviscid unsteady non-uniform rotational flow. J. Fluid Mech. 197, 241257.
Boiko, V. M., Klinkov, K. V. & Poplavskii, S. V. 2004 Collective bow shock ahead of a transverse system of spheres in a supersonic flow behind a moving shock wave. Fluid Dyn. 39 (2), 330338.
Britan, A., Elperin, T., Igra, O. & Jiang, J. P. 1995 Acceleration of a sphere behind planar shock waves. Exp. Fluids 20 (2), 8490.
Bulat, P., Ilyina, T., Volkov, K., Silnikov, M. & Chernyshov, M. 2016 Interaction of a shock wave with an array of particles and effect of particles on the shock wave weakening. Acta Astron. 135, 131138.
Collins, T. J. B., Poludnenko, A., Cunningham, A. & Frank, A. 2005 Shock propagation in deuterium-tritium-saturated foam. Phys. Plasmas 12 (6), 062705.
Eames, I. & Hunt, J. C. R. 1997 Inviscid flow around bodies moving in weak density gradients without buoyancy effects. J. Fluid Mech. 353, 331355.
Han, E., Hantke, M. & Warnecke, G. 2012 Exact Riemann solutions to compressible Euler equations in ducts with discontinuous cross-section. J. Hyperbolic Differ. Equ. 09 (03), 403449.
Haselbacher, A. 2005 A WENO reconstruction algorithim for unstructured grids based on explicit stencil construction. In 43rd AIAA Aerospace Sciences Meeting and Exhibit, AIAA.
Igra, O. & Takayama, K. 1993 Shock tube study of the drag coefficient of a sphere in a non-stationary flow. Proc. R. Soc. Lond. A 442 (1915), 231247.
Kröner, D. & Thanh, M. D. 2005 Numerical solutions to compressible flows in a nozzle with variable cross-section. SIAM J. Numer. Anal. 43 (2), 796824.
Lefloch, P. G. & Thanh, M. D. 2003 The Riemann problem for fluid flows in a nozzle with discontinuous cross-section. Commun. Math. Sci. 1 (4), 763797.
Ling, Y., Haselbacher, A. & Balachandar, S. 2009 Transient phenomena in one-dimensional compressible gas–particle flows. Shock Waves 19 (1), 6781.
Ling, Y., Haselbacher, A., Balachandar, S., Najjar, F. M. & Stewart, D. S. 2013 Shock interaction with a deformable particle: direct numerical simulation and point-particle modeling. J. Appl. Phys. 113 (1), 013504.
Ling, Y., Wagner, J. L., Beresh, S. J., Kearney, S. P. & Balachandar, S. 2012 Interaction of a planar shock wave with a dense particle curtain: modeling and experiments. Phys. Fluids 24 (11), 113301.
Liou, M.-S. 1996 A sequel to AUSM: AUSM+. J. Comput. Phys. 129 (2), 364382.
Liu, Y. & Kendall, M. A. F. 2006 Numerical analysis of gas and micro-particle interactions in a hand-held shock-tube device. Biomed. Microdevices 8 (4), 341351.
Loth, E. 2008 Compressibility and rarefaction effects on drag of a spherical particle. AIAA J. 46 (9), 22192228.
Lu, C., Sambasivan, S., Kapahi, A. & Udaykumar, H. 2012 Multi-scale modeling of shock interaction with a cloud of particles using an artificial neural network for model representation. Proc. IUTAM 3, 2552.
Matsumura, Y. & Jackson, T. L. 2014 Numerical simulation of fluid flow through random packs of polydisperse cylinders. Phys. Fluids 26 (12), 123302.
Mehta, Y., Jackson, T. L., Zhang, J. & Balachandar, S. 2016a Numerical investigation of shock interaction with one-dimensional transverse array of particles in air. J. Appl. Phys. 119 (10), 104901.
Mehta, Y., Neal, C., Jackson, T., Balachandar, S. & Thakur, S. 2016b Shock interaction with three-dimensional face centered cubic array of particles. Phys. Rev. Fluids 1, 054202.
Naiman, H. & Knight, D. D. 2007 The effect of porosity on shock interaction with a rigid, porous barrier. Shock Waves 16 (4), 321337.
Parmar, M., Haselbacher, A. & Balachandar, S. 2009 Modeling of the unsteady force for shock–particle interaction. Shock Waves 19 (4), 317329.
Parmar, M., Haselbacher, A. & Balachandar, S. 2011 Generalized Basset–Boussinesq–Oseen equation for unsteady forces on a sphere in a compressible flow. Phys. Rev. Lett. 106, 084501.
Parmar, M., Haselbacher, A. & Balachandar, S. 2012 Equation of motion for a sphere in non-uniform compressible flows. J. Fluid Mech. 699, 352375.
Persson, P.-O. & Strang, G. 2004 A simple mesh generator in matlab. SIAM Rev. 46 (2), 329345.
Quintanilla, J. & Torquato, S. 1997 Local volume fraction fluctuations in random media. J. Chem. Phys. 106 (7), 27412751.
Regele, J., Rabinovitch, J., Colonius, T. & Blanquart, G. 2014 Unsteady effects in dense, high speed, particle laden flows. Intl J. Multiphase Flow 61, 113.
Rycroft, C. H. 2009 Voro++: a three-dimensional Voronoi cell library in C++. Chaos: Interdiscip. J. Nonlinear Sci. 19 (4), 041111.
Si, H. 2015 TetGen, a delaunay-based quality tetrahedral mesh generator. ACM Trans. Math. Softw. 41 (2), 11:1–11:36.
Sridharan, P., Jackson, T. L., Zhang, J. & Balachandar, S. 2015 Shock interaction with one-dimensional array of particles in air. J. Appl. Phys. 117 (7), 075902.
Sun, M., Saito, T., Takayama, K. & Tanno, H. 2005 Unsteady drag on a sphere by shock wave loading. Shock Waves 14 (1), 39.
Suzuki, K., Himeki, H., Watanuki, T. & Abe, T. 2000 Experimental studies on characteristics of shock wave propagation through cylinder array. Inst. Space Astron. Sci. Rep. 3, 676.
Tanno, H., Itoh, K., Saito, T., Abe, A. & Takayama, K. 2003 Interaction of a shock with a sphere suspended in a vertical shock tube. Shock Waves 13 (3), 191200.
Tanno, H., Komuro, T., Takahashi, M., Takayama, K., Ojima, H. & Onaya, S. 2004 Unsteady force measurement technique in shock tubes. Rev. Sci. Instrum. 75 (2), 532536.
Torquato, S. 2002 Random Heterogeneous Materials – Microstructure and Macroscopic Properties. Springer.
Wagner, J. L., Beresh, S. J., Kearney, S. P., Pruett, B. O. M. & Wright, E. K. 2012a Shock tube investigation of quasi-steady drag in shock–particle interactions. Phys. Fluids 24 (12), 123301.
Wagner, J. L., Beresh, S. J., Kearney, S. P., Trott, W. M., Castaneda, J. N., Pruett, B. O. & Baer, M. R. 2012b A multiphase shock tube for shock wave interactions with dense particle fields. Exp. Fluids 52 (6), 15071517.
Yazdchi, K., Srivastava, S. & Luding, S. 2012 Micro–macro relations for flow through random arrays of cylinders. Composites A 43 (11), 20072020.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Propagation of a strong shock over a random bed of spherical particles

  • Y. Mehta (a1), C. Neal (a1), K. Salari (a2), T. L. Jackson (a1), S. Balachandar (a1) and S. Thakur (a1)...


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed