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Geometrical shock dynamics applied to condensed phase materials

  • Brandon Lieberthal (a1) (a2), D. Scott Stewart (a2) and Alberto Hernández (a2)


Taylor blast wave (TBW) theory and geometrical shock dynamics (GSD) theory describe a radially expanding shock wave front through an inert material, typically an ideal gas, in the strong blast wave limit and weak acoustic limit respectively. We simulate a radially expanding blast shock in air using a hydrodynamic simulation code and numerically describe the intermediate region between these two limits. We test our description of the intermediate shock phase through a two-dimensional simulation of the Bryson and Gross experiment. We then apply the principles of GSD to materials that follow the Mie–Gruneisen equation of state, such as plastics and metals, and derive an equation that accurately relates the acceleration, velocity and curvature of the shock through these materials. Along with detonation shock dynamics (DSD), which describes detonation shock propagation through high explosive fluids, we develop a hybrid DSD/GSD model for the simulation of heterogeneous explosives. This model enables computationally efficient simulation of the shock front in high explosive/inert mixtures consisting of simple or complex geometric configurations. We simulate an infinite two-dimensional slab consisting of one half explosive, PBXN-9, and one half aluminium and model the boundary angle conditions using shock polar analysis. We also simulate a series of high explosive unit cells embedded with aluminium spherical particles, and we compare the propagation of the detonation shock front with a direct numerical simulation performed with the ALE3D code.


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Arienti, M., Morano, E. & Shepherd, J. E. 2004 Shock and detonation modeling with the Mie–Gruneisen equation of state. In Graduate Aeronautical Laboratories Report FM99-8, California Institute of Technology, Pasadena, CA.
Aslam, T.1996 Investigations on detonation shock dynamics. Thesis, University of Illinois at Urbana-Champaign.
Aslam, T. & Stewart, D. S. 1999 Detonation shock dynamics and comparisons with direct numerical simulation. Combust. Theor. Model. 3 (1), 77101.
Bdzil, J. B., Lieberthal, B. & Stewart, D. S.2010 Mesoscale modeling of metal-loaded high explosives. Report. Los Alamos National Laboratory (LANL).
Bdzil, J. B. & Stewart, D. S. 2012 Theory of Detonation Shock Dynamics, Shock Wave Science and Technology Reference Library, vol. 6, pp. 373453. Springer, book section 7.
Bdzil, J. B., Stewart, D. S. & Jackson, T. L. 2001 Program burn algorithms based on detonation shock dynamics: discrete approximations of detonation flows with discontinuous front models. J. Comput. Phys. 174 (2), 870902.
Brown, J. L. & Ravichandran, G. 2013 Analysis of oblique shock waves in solids using shock polars. Shock Waves 24 (4), 403413.
Bryson, A. E. & Gross, R. W. F. 1961 Diffraction of strong shocks by cones, cylinders, and spheres. J. Fluid Mech. 10 (01), 116.
Cates, J. E. & Sturtevant, B. 1997 Shock wave focusing using geometrical shock dynamics. Phys. Fluids 9 (10), 30583068.
Chisnell, R. F. 1955 The normal motion of a shock wave through a non-uniform one-dimensional medium. Proc. R. Soc. Lond. A 232, 350370.
Drikakis, D., Ofengeim, D., Timofeev, E. & Voionovich, P. 1997 Computation of non-stationary shock-wave/cylinder interaction using adaptive-grid methods. J. Fluids Struct. 11 (6), 665692.
Hernández, A., Bdzil, J. B. & Stewart, D. S. 2013 An mpi parallel level-set algorithm for propagating front curvature dependent detonation shock fronts in complex geometries. Combust. Theor. Model. 17 (1), 109141.
Hernández, A., Lieberthal, B. & Stewart, D. S. 2017 An explicit algorithm for imbedding solid boundaries in cartesian grids for the reactive Euler equation. Combust. Theory Model.; (submitted).
Hernández, A. & Stewart, D. S.2013 Newcode-1d [computer software]. University of Illinois at Urbana-Champaign.
Holman, S.2010 On the calibration of some ideal and non-ideal explosives. Thesis, University of Illinois at Urbana-Champaign.
Holm, D. D. & Logan, J. D. 1983 Self-similar detonation waves. J. Phys. A: Math. Gen. 16 (9), 20352047.
Hu, X. Y., Adams, N. A. & Shu, C.-W. 2013 Positivity-preserving method for high-order conservative schemes solving compressible euler equations. J. Comput. Phys. 242, 169180.
Hull, L. M.1997 Detonation propagation and mach stem formation in pbxn-9. Report. Los Alamos National Laboratory.
Jiang, G.-S. & Shu, C.-W.1995 Efficient implementation of weighted eno schemes. Report. DTIC Document.
Kapila, A. K., Bdzil, J. B. & Stewart, D. S. 2006 On the structure and accuracy of programmed burn. Combust. Theor. Model. 10 (2), 289321.
LeVeque, R. J. 2007 Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-state and Time-dependent Problems. SIAM.
Lieberthal, B. A., Bdzil, J. B. & Stewart, D. S. 2014 Modelling detonation of heterogeneous explosives with embedded inert particles using detonation shock dynamics: Normal and divergent propagation in regular and simplified microstructure. Combust. Theor. Model. 18 (2), 204241.
Matalon, M., Cui, C. & Bechtold, J. K. 2003 Hydrodynamic theory of premixed flames: effects of stoichiometry, variable transport coefficients and arbitrary reaction orders. J. Fluid Mech. 487, 179210.
Miller, G. H. & Puckett, E. G. 1996 A high-order godunov method for multiple condensed phases. J. Comput. Phys. 128 (1), 134164.
Mitchell, A. C. 1981 Shock compression of aluminum, copper, and tantalum. J. Appl. Phys. 52 (5), 3363.
Ofengeim, D. K. & Drikakis, D. 1997 Simulation of blast wave propagation over a cylinder. Shock Waves 7 (5), 305317.
Saenz, J. A., Taylor, B. D. & Stewart, D. S. 2012 Asymptotic calculation of the dynamics of self-sustained detonations in condensed phase explosives. J. Fluid Mech. 710, 166194.
Steinberg, D. 1996 Equation of State and Strength Properties of Selected Materials. Lawrence Livermore National Laboratory.
Tarver, C. M. & Urtiew, P. A. 2010 Theory and modeling of liquid explosive detonation. J. Energetic Materials 28 (4), 299317.
Taylor, G. 1950 The formation of a blast wave by a very intense explosion. i. Theoretical discussion. Proc. R. Soc. Lond. A 201, 159174.
Whitham, G. B. 1956 On the propagation of weak shock waves. J. Fluid Mech. 1 (03), 290318.
Whitham, G. B. 1957 A new approach to problems of shock dynamics. J. Fluid Mech. 2 (02), 145171.
Whitham, G. B. 2011 Linear and Nonlinear Waves. Wiley.
Wilkins, M. L., University of California, Berkeley & Lawrence Livermore, Laboratory 1963 Calculation of Elastic-plastic Flow. University of California Lawrence Radiation Laboratory.
Xu, S., Aslam, T. & Stewart, D. S.1997 High resolution numerical simulation of ideal and non-ideal compressible reacting flows with embedded internal boundaries.
Yao, J. & Stewart, D. S. 1996 On the dynamics of multi-dimensional detonation. J. Fluid Mech. 309, 225275.
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