Abstract
Many important explosives and energetics applications involve multiphase formulations employing dispersed particles. While considerable progress has been made toward developing mathematical models and computational methodologies for these flows, significant challenges remain. In this work, we apply a mathematical model for compressible multiphase flows with dispersed particles to existing shock and explosive dispersal problems from the literature. The model is cast in an Eulerian framework, treats all phases as compressible, is hyperbolic, and satisfies the second law of thermodynamics. It directly applies the continuous-phase pressure gradient as a forcing function for particle acceleration and thereby retains relaxed characteristics for the dispersed particle phase that remove the constituent material sound velocity from the eigenvalues. This is consistent with the expected characteristics of dispersed particle phases and can significantly improve the stable time-step size for explicit methods. The model is applied to test cases involving the shock and explosive dispersal of solid particles and compared to data from the literature. Computed results compare well with experimental measurements, providing confidence in the model and computational methods applied.
Similar content being viewed by others
References
McGrath, T., St. Clair, J., Balachandar, S.: A compressible two-phase model for dispersed particle flows with application from dense to dilute regimes. J. Appl. Phys. 119, 174903 (2016). doi:10.1063/1.4948301
Baer, M., Nunziato, J.: A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiph. Flow 12, 861 (1986). doi:10.1016/0301-9322(86)90033-9
Bdzil, J., Menikoff, R., Son, S., Kapila, A., Stewart, D.: Two-phase modeling of deflagration-to-detonation transition in granular materials: a critical examination of modeling issues. Phys. Fluids 11(2), 378 (1999). doi:10.1063/1.869887
Saurel, R., Lemetayer, O.: A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation. J. Fluid Mech. 431, 239 (2001). doi:10.1017/S0022112000003098
Saurel, R., Favrie, N., Peptitpas, F., Lallemand, M.H., Gavrilyuk, S.: Modelling dynamic and irreversible powder compaction. J. Fluid Mech. 664, 348 (2010). doi:10.1017/S0022112010003794
Saurel, R., Le Martelot, S., Tosello, R., Lapebie, E.: Symmetric model of compressible granular mixtures with permeable interfaces. Phys. Fluids 26, 123304 (2014). doi:10.1063/1.4903259
Rogue, X., Rodriguez, G., Haas, J., Saurel, R.: Experimental and numerical investigation of the shock-induced fluidization of a particles bed. Shock Waves 8, 29 (1998). doi:10.1007/s001930050096
Sommerfeld, M.: The unsteadiness of shock waves propagating through gas-particle mixtures. Exp. Fluids. 3, 197 (1985). doi:10.1007/BF00265101
Zhang, F., Frost, D., Thibault, P., Murray, S.: Explosive dispersal of solid particles. Shock Waves 10, 431 (2001). doi:10.1007/PL00004050
Kapila, A., Menikoff, R., Bdzil, J., Son, S., Stewart, D.: Two-phase modeling of deflagration-to-detonation transition in granular materials: Reduced equations. Phys. Fluids 13(10), 3002 (2001). doi:10.1063/1.1398042
Godunov, S.K.: A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. (N.S.) 47(89), 271–306 (1959)
Colella, P.: A direct Eulerian MUSCL scheme for gas dynamics. SIAM J. Sci. Stat. Comput. (1985). doi:10.1137/0906009
Toro, E.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 2nd edn. Springer, Berlin (1999)
Davis, S.: Simplified second-order Godunov-type methods. SIAM J Sci. Stat. Comput. (1988). doi:10.1137/0909030
Leveque, R.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)
Wilkins, M.: Computer Simulation of Dynamic Phenomena. Springer, Berlin (1999)
Forbes, J.: Shock Wave Compression of Condensed Matter: A Primer. Springer, Berlin (2012)
Crowe, C., Sommerfield, M., Tsuji, Y.: Multiphase Flows with Droplets and Particles. CRC Press, New York (1998)
Ergun, S.: Fluid flow through packed columns. Chem. Eng. Prog. 48, 89–94 (1952)
Di Felice, R.: The voidage function for fluid-particle interaction systems. Int. J. Multiph. Flow 20, 153 (1994). doi:10.1016/0301-9322(94)90011-6
Grady, D.E., Chhabildas, L.C.: Shock wave properties of soda lime glass. In: Iyer, K.R., Chou, S.C. (eds.) Proceedings of 14th US Army Symposium on Solid Mechanics. Battelle Press, Myrtle Beach (1996)
Omang, M.G., Trulsen, J.K.: Multi-phase shock simulations with smoothed particle hydrodynamics (SPH). Shock Waves 24, 521 (2014). doi:10.1007/s00193-014-0506-7
Chang, E., Kailasanath, K.: Shock wave interactions with particles and liquid fuel droplets. Shock Waves 12, 333 (2003). doi:10.1007/s00193-002-0170-1
Dobratz, C.: LLNL Handbook of Explosives, UCRL-52997. Lawrence Livermore National Laboratory, Livermore (1985)
Acknowledgements
This work was supported by the Office of Naval Research under the In-House Laboratory and Independent Research (ILIR) Program and the Naval Undersea Research Program (NURP). The authors wish to thank the ILIR and NURP program management and review boards for their support and guidance of this research. Additionally, this work benefited from the U.S. Department of Energy, National Nuclear Security Administration, Advanced Simulation and Computing Program, as a Cooperative Agreement to the University of Florida under the Predictive Science Academic Alliance Program, under Contract No. DE-NA0002378 and from the Defense Threat Reduction Agency Basic Research Award No. HDTRA1-14-1-0028 to the University of Florida.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by D. Frost.
Rights and permissions
About this article
Cite this article
McGrath, T., St. Clair, J. & Balachandar, S. Modeling compressible multiphase flows with dispersed particles in both dense and dilute regimes. Shock Waves 28, 533–544 (2018). https://doi.org/10.1007/s00193-017-0726-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00193-017-0726-8