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The European Physical Journal H

, Volume 40, Issue 2, pp 159–204 | Cite as

The classical Rankine-Hugoniot jump conditions, an important cornerstone of modern shock wave physics: ideal assumptions vs. reality

  • Peter O. K. KrehlEmail author
Article

Abstract

The purpose of this paper is to discuss from a historical point of view the classical Rankine-Hugoniot (RH) relations in more detail than usually done in standard textbooks. Particularly focusing on the last seventy years, this paper (i) reviews their validity and limitations as interpreted by numerous users; (ii) summarizes their enormous extension also to other branches of science and engineering; and (iii) discusses the nontrivial problem of error estimation. Originally, the RH relations were derived for a plane-parallel steadily propagating aerial shock with a step wave profile; i.e., a wave with zero rise-time and constant thermodynamic as well as kinematic parameter values behind the shock front. But real shock waves are in most cases three-dimensional, have finite rise-times, and in almost all cases are unsteady waves. These real properties must produce a systematic error when applying the RH relations and the cardinal question arises how large this error will be in comparison with a random error caused by the applied high-speed diagnostics. However, numerical procedures of studying systematic errors of unsteady wave propagation are difficult to carry out because of various reasons and still pending.

Keywords

Shock Wave Shock Front Shock Tube Blast Wave Shock Compression 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Rankine-Hugoniot jump conditions. Wikipedia; http://en.wikipedia.org/wiki/Rankine%E2%80%93Hugoniot_conditions.
  2. 2.
    Multivariable calculusRankine-Hugoniot jump conditions derivation. Mathematics Stack Exchange; http://math.stackexchange.com/questions/865142/rankine-hugoniot-jump-condition-derivation.
  3. 3.
    P.O.K. Krehl, Shock wave physics and detonation physics − a stimulus for the emergence of numerous new branches in science and engineering, Eur. Phys. J. H. 36, 85–152 (2011).ADSGoogle Scholar
  4. 4.
    A. Mazzia, Numerical Methods for the Solution of Hyperbolic Conservation Laws. Rapporto Tecnico No. 68, Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università de Padova, Italia, 1998.Google Scholar
  5. 5.
    W.C. Griffith, W. Bleakney, Shock waves in gases, Am. J. Phys. 22, 597–612 (1954).ADSGoogle Scholar
  6. 6.
    G. Ben-Dor, Shock Wave Reflection Phenomena (Springer, New York, 1992).Google Scholar
  7. 7.
    C. Chalons, P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling, Commun. Math. Sci. 5, 533–551 (2007).zbMATHMathSciNetGoogle Scholar
  8. 8.
    D.F. Cioffi, C.F. McKee, E. Bertschinger, Dynamics of radiative supernova remnants, Astrophys. J. (Part 1) 334, 252–265 (1988).ADSGoogle Scholar
  9. 9.
    G.W. Swan, D.E. Duvall, C.K. Thornhill, On steady wave profiles in solids, J. Mech. Phys. Solids 21, 215–227 (1973).zbMATHADSGoogle Scholar
  10. 10.
    Y.J. Horie, Classification of steady-profile shocks in liquids, J. Appl. Phys. 45, 759–764 (1974).ADSGoogle Scholar
  11. 11.
    L. Euler, Continuation des recherches sur la théorie du mouvement des fluides, Hist. Acad. Roy. Sci. Belles Lettres (Berlin) 11, 316–361 (1757).Google Scholar
  12. 12.
    G.G. Stokes, On a difficulty in the theory of sound, Phil. Mag. 33 [III], 349–356 (1848).Google Scholar
  13. 13.
    G.F.B. Riemann, Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite. Abhandl. Königl. Gesell. Wiss. Gött. 8 [Math. Physik. Kl.], 243–265 (1860).Google Scholar
  14. 14.
    W.J.M. Rankine, On the thermodynamic theory of waves of finite longitudinal disturbance [read Dec. 16, 1869], Phil. Trans. Roy. Soc. Lond. 160, 277–286 (1870); Supplement. Ibid. pp. 287–288 (1870).Google Scholar
  15. 15.
    P.H. Hugoniot, Mémoire sur la propagation du mouvement dans les corps et plus spécialement dans les gaz parfaits. 1e Partie, J. Ecole Polytech. (Paris) 57, 3–97 (1887); Mémoire sur la propagation du mouvement dans les corps et plus spécialement dans les gaz parfaits. 2e Partie, J. Ecole Polytech. (Paris) 58, 1–125 (1889).Google Scholar
  16. 16.
    R.J.E. Clausius, Über verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie, Ann. Phys. 125 [II], 353–400 (1865).Google Scholar
  17. 17.
    P.O.K. Krehl, History of Shock Waves, Explosions, and Impact – a Chronological and Biographical Reference (Springer-Verlag, Berlin, 2009), pp. 340–341, 387–389, 1075–1077, 1094–1095, 1148–1149, 1175–1176.Google Scholar
  18. 18.
    R. Chéret, The life and work of Pierre-Henri Hugoniot, Shock Waves 2, 1–4 (1992).ADSGoogle Scholar
  19. 19.
    J.N. Johnson, R. Chéret, Shock waves in solids: an evolutionary perspective, Shock Waves 9, 193–200 (1999).zbMATHADSGoogle Scholar
  20. 20.
    M. Salas, The curious events leading to the theory of shock waves, Shock Waves 16, 477–487 (2007); A Shock-Fitting Primer (CRC Press, Boca Raton, FL, 2010).zbMATHADSGoogle Scholar
  21. 21.
    R.H. Cole, Underwater Explosions (Princeton University Press, Princeton, NJ, 1948)Google Scholar
  22. 22.
    C.E. Needham, Blast Waves (Springer-Verlag Berlin, Heidelberg, 2010), pp. 11–15.Google Scholar
  23. 23.
    R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves. Springer-Verlag, New York (corrected 5th printing 1999), pp. 121–126, 336–338.Google Scholar
  24. 24.
    D.H. Weinberg, Astronomy 825: Radiative Gas Dynamics, Chap. 7: Shocks (Winter 2003). Dept. of Astronomy, Mathematical and Physical Sciences, Ohio State University; http://www.astronomy.ohio-state.edu/_dhw/A825/notes7.pdf.
  25. 25.
    A. Bressan, Hyperbolic conservation laws – an illustrated tutorial (Dept. of Mathematics, Penn State University, University Park, PA (2009); http://www.math.psu.edu/bressan/PSPDF/clawtut09.pdf.
  26. 26.
    J.W. Forbes, Shock Wave Compression of Condensed Matter – a Primer (Springer-Verlag, Berlin, 2012), pp. 179–185.Google Scholar
  27. 27.
    G.W. Sutton, A. Sherman, Engineering Magnetohydrodynamics (McGraw-Hill Book Co., New York, 1965).Google Scholar
  28. 28.
    J. von Neumann, The point source solution. In Blast Wave, edited by K. Fuchs, J.O. Hirschfelder, J.L. Magee, R. Peierls, J. von Neumann (Rept. La-2000, LASL, 1947), pp. 27–55.Google Scholar
  29. 29.
    L.M. Barker, R.E. Hollenbach, Shock-wave studies of PMMA, fused silica, and sapphire, J. Appl. Phys. 41, 4208–4226 (1970).ADSGoogle Scholar
  30. 30.
    C.E. Ragan III, M.G. Silbert, B.C. Diven, Shock compression of molybdenum to 2.0 TPa by means of a nuclear explosion, J. Appl. Phys. 48, 2860–2870 (1977).ADSGoogle Scholar
  31. 31.
    B. Hayes, Particle-velocity gauges system for nanosecond sampling rate of shock and detonation waves, Rev. Sci. Instrum. 52, 594–603 (1981).ADSGoogle Scholar
  32. 32.
    S. Minshall, Properties of elastic and plastic waves determined by pin contactors and crystals, J. Appl. Phys. 26, 463–469 (1955).ADSGoogle Scholar
  33. 33.
    S.R. BrinkleyJr., J.G. Kirkwood, Theory of the propagation of shock waves, Phys. Rev. 71, 606–611 (1947).zbMATHMathSciNetADSGoogle Scholar
  34. 34.
    D. Benson, An efficient, accurate, simple ALE method for nonlinear finite element programs, Computat. Meth. Appl. Mech. Eng. 72, 305–350 (1989).zbMATHADSGoogle Scholar
  35. 35.
    Arbitrary Lagrangian Eulerian and Fluid-Structure Interaction: Numerical Simulation, edited by M. Souli, D.J. Benson (Wiley, London, UK and Hoboken, NJ, 2010).Google Scholar
  36. 36.
    W. Bleakney, D.K. Weimer, C.H. Fletcher, The shock tube: a facility for investigations in fluid dynamics, Rev. Sci. Instrum. 20, 807–815 (1949).ADSGoogle Scholar
  37. 37.
    R. LeVeque, Numerical Methods for Conservation Laws (Birkhäuser, Berlin, 1992), Chap. 2.1: Integral and differential forms, pp. 14–16.Google Scholar
  38. 38.
    R.K. Tsou, Conservation equations, in Dynamic Response of Materials to Intense Impulsive Loading, edited by P.C. Chou, A.K. Hopkins. Rept. AD-768-416, Air Force Materials Laboratory, Wright Patterson Air Force Base, OH (1972), Chap. 2, pp. 7–42.Google Scholar
  39. 39.
    V.N. Kukudzhanov, Numerical Continuum Mechanics (De Gruyter GmbH, Berlin/Boston, 2013), pp. 9–13.Google Scholar
  40. 40.
    Ames Research Staff, Equations, Tables, and Charts for Compressible Flow, Rept. NACA 1135 (1953); Normal shock waves, pp. 6–8.Google Scholar
  41. 41.
    Rankine-Hugoniot spreadsheets. Dewey McMillin & Associates Ltd., Victoria, BC V8N 2A4, Canada; see www.blastanalysis.com.
  42. 42.
    J.M. Walsh, M.H. Rice, R.G. McQueen, F.L. Yarger, Shock-wave compressions of twenty-seven metals. Equations of state of metals, Phys. Rev. 108 [II], 196–216 (1957).Google Scholar
  43. 43.
    J.M. Walsh, M.H. Rice, Dynamic compression of liquids from measurements on strong shock waves, J. Chem. Phys. 26, 815–823 (1957).ADSGoogle Scholar
  44. 44.
    M. van Thiel, A.S. Kusubov, A.C. Mitchell, Compendium of Shock Wave Data. Rept. UCRL-50108, Lawrence Radiation Laboratory (LRL), Livermore, CA (1977).Google Scholar
  45. 45.
    LASL Shock Hugoniot Data Bank, edited by S.P. Marsh (University of California Press, Berkeley, 1980).Google Scholar
  46. 46.
    SESAME: The Los Alamos National Laboratory Equations-of-State Database., edited by S.P. Lyon, J.D. Johnson. Rept. LA-UR-92-3407 (1992).Google Scholar
  47. 47.
    J.D. Johnson, G.I. Kerley, G.T. Rood, Recent Developments in the SESAME Equation-of-State Library, edited by B.I. Bennett. Rept. LA-7130, LASL, Los Alamos, NM (1978); Equations of state – theoretical formalism. Los Alamos Science No. 26, 192 (2000).Google Scholar
  48. 48.
    I.V. Lomonosov, K.V. Khishchenko, P.R. Levashov, D.V. Minakov, A.S. Zakharenkov, J.B. Aidun, International shock-wave data base. IPCP and JIHT, RAS and SNL, USA; see http://www.ihed.ras.ru/elbrus12/program/restore.php?id=820.
  49. 49.
    P.R. Levashov, K.V. Khishchenko, I.V. Lomonosov, V.E. Fortov, Database on shock-wave experiments and equations of state available via Internet, in APS Topical Conference on Shock Compression of Condensed Matter2003, edited by M.D. Furnish, Y.M. Gupta, J.W. Forbes. AIP Conf. Proc. 706, 87–90 (2004).ADSGoogle Scholar
  50. 50.
    J.M. Dewey, The Rankine-Hugoniot equations: their extensions and inversions related to blast waves, in Proc. 19th International Symposium on Military Aspects of Blast and Shock (MABS) [Calgary, Alberta, Canada; Oct. 1−6, 2006]. CD-ROM, publ. by Defence R&D, Suffield, Alberta, Canada.Google Scholar
  51. 51.
    E. Jouguet, La théorie thermodynamique de la propagation des explosions, in Verhandlungen des 2. Int. Kongresses für Technische Mechanik, edited by E. Meissner [Zurich, Switzerland; Sept. 12−17, 1926]. Füssli, Zurich (1927), pp. 12–22.Google Scholar
  52. 52.
    R. Becker, Stoßwelle und Detonation, Z. Phys. 8, 321–362 (1922).ADSGoogle Scholar
  53. 53.
    Encyclopaedic Dictionary of Physics, edited by J. Thewlis (Pergamon Press, Oxford, 1962), p. 718.Google Scholar
  54. 54.
    S.D. Poisson, Sur la chaleur des gaz et des vapeurs, Ann. Chem. Phys. 23 [II], 337–353 (1823).Google Scholar
  55. 55.
    A.F. Viñas, J.D. Scudder, Fast and optimal solution to the Rankine-Hugoniot problem. NASA Memorandum 86214 (May 1985).Google Scholar
  56. 56.
    S.K. Chakrabarti, Theory of Transonic Astrophysical Flows (World Scientific Publishing Co., Singapore, 1990), pp. 48–53.Google Scholar
  57. 57.
    A. Siegenthaler, J. Madhani, Outline of a theory of non-Rankine-Hugoniot shock wave in weak Mach reflection. 14th Australasian Fluid Mechanics Conference (Adelaide Univ., Adelaide, Austr.; 10–14 Dec. 2001).Google Scholar
  58. 58.
    A.H. Bepp, Underwater explosion measurements from small charges at short ranges, Phil. Trans. Roy. Soc. Lond. 244, 153–175 (1951).ADSGoogle Scholar
  59. 59.
    J.M. Walsh, R.H. Christian, Equation of state of metals from shock wave measurements, Phys. Rev. 97, 1544–1556 (1955).ADSGoogle Scholar
  60. 60.
    M.H. Rice, R.G. McQueen, J.M. Walsh, Compression of solids by strong shock waves, in Solid State Physics. Advances in Research and Applications, edited by F. Seitz, D. Turnbull (Academic Press, New York and London, 1958), Vol. 6, pp. 1–63.Google Scholar
  61. 61.
    N. Curle, Rankine-Hugoniot law, in Encyclopaedic Dictionary of Physics, edited by J. Thewlis (Pergamon Press, Oxford, 1962), pp. 194–195.Google Scholar
  62. 62.
    The Effects of Nuclear Weapons, edited by S. Glasstone. Prepared by the US Dept. of Defense, published by the US Atomic Energy Commission (Feb. 1964), Chap. 3.72.Google Scholar
  63. 63.
    J.M. Dewey, The properties of a blast wave obtained from an analysis of the particle trajectories, Proc. Roy. Soc. Lond. A 324, 275–299 (1971).ADSGoogle Scholar
  64. 64.
    Prof. John M. Dewey, private communication on June 8, 2012.Google Scholar
  65. 65.
    B.K. Godwal, S.K. Sikka, R. Chidambaram, Equation of state theories of condensed matter up to about 10 TPa, Phys. Rep. (Rev. Ser. Phys. Lett.) 102, 121–197 (1983).ADSGoogle Scholar
  66. 66.
    E. Murr, K.P. Staudhammer, Shock wave fundamentals: effects on the structure and behavior of engineering materials, in Shock Waves for Industrial Applications, edited by E. Murr (Noyes Publs., Park Ridge, NJ, 1988), pp. 13–15.Google Scholar
  67. 67.
    M.A. Barrios, D.G. Hicks, T.R. Boehly, D.E. Fratanduono, D.D. Meyerhofer, J.H. Eggert, P.M. Celliers, G.W. Collins, High-precision measurements of the EOS of hydrocarbons at 1−10 Mbar using laser-driven shock waves. LLE (Laboratory for Laser Energetics) Rev. 121, 6–21 (2009).Google Scholar
  68. 68.
    J.W. Forbes, Shock Wave Compression of Condensed Matter – a Primer (Springer-Verlag, Berlin, 2012), pp. 31–57.Google Scholar
  69. 69.
    Dr. Charles E. Needham, private communication on Dec. 17, 2013.Google Scholar
  70. 70.
    J. von Neumann, R.D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys. 21, 232–237 (1950).zbMATHMathSciNetADSGoogle Scholar
  71. 71.
    H.L. Brode, Blast wave from a spherical charge, Phys. Fluids 2, 217–229 (1959).zbMATHADSGoogle Scholar
  72. 72.
    W. Band, G.E. Duvall, Physical nature of shock propagation, Am. J. Phys. 29, 780–785 (1961).zbMATHADSGoogle Scholar
  73. 73.
    Lord Rayleigh, J.W. Strutt, Aerial plane waves of finite amplitude, Proc. Roy. Soc. Lond. A 84, 247–284 (1910).zbMATHADSGoogle Scholar
  74. 74.
    W. Band, Introduction to Mathematical Physics (Van Nostrand Company, Inc., Princeton, NJ, 1959).Google Scholar
  75. 75.
    G.E. Duvall, Shock waves in condensed media. Int. School of Physics Enrico Fermi (Lago di Como, Italy; July 14–26, 1969). Proc. publ. in Physics of High Energy Density, edited by P. Caldirola (Academic Press, New York, 1971), Vol.48, pp. 7–50.Google Scholar
  76. 76.
    G.R. Fowles, R.F. Williams, Plane stress wave propagation in solids, J. Appl. Phys. 41, 360–363 (1970).ADSGoogle Scholar
  77. 77.
    R. Fowles, Determination of constitutive relations from plane wave experiments (US Defense Technical Information Center, Fort Belvoir, VA, 1970).Google Scholar
  78. 78.
    G.R. Fowles, Conservation relations for spherical and cylindrical stress waves, J. Appl. Phys. 41, 2740–2741 (1970).ADSGoogle Scholar
  79. 79.
    M. Cowperthwaite, R.F. Williams, Determination of constitutive relationships with multiple gauges in non-divergent waves, J. Appl. Phys. 42, 456–462 (1971).ADSGoogle Scholar
  80. 80.
    L. Seaman, Lagrangian analysis for multiple stress or velocity gages in alternating waves, J. Appl. Phys. 45, 4303–4314 (1974).ADSGoogle Scholar
  81. 81.
    J.B. Aidun, Y.M. Gupta, Analysis of Lagrangian gauge measurements of simple and nonsimple plane waves, J. Appl. Phys. 69, 6998–7014 (1991).ADSGoogle Scholar
  82. 82.
    W.J. Murri, D.R. Curran, C.F. Peterson, R.C. Crewdson, Response of Solids to Shock Waves. Tech. Rept. No. 001-71, Poulter Laboratory of SRI, Menlo Park, CA (1971). Later published in Advances in High Pressure Research, edited by R.H. Wentorf Jr. (Academic Press, London and New York, 1974), Vol. 4, pp. 1–163.Google Scholar
  83. 83.
    G.I. Taylor, The conditions necessary for discontinuous motion in gases, Proc. Roy. Soc. Lond. A 84, 371–377 (1910).zbMATHADSGoogle Scholar
  84. 84.
    J.M. Kelly, P.P. Gillis, Shock thickness in viscoplastic solids, J. Appl. Mech. 37, 163–170 (1970).ADSGoogle Scholar
  85. 85.
    R.T. Walsh, Finite difference methods, in Dynamic Response of Materials to Intense Impulsive Loading, edited by P.C. Chou, A.K. Hopkins. Rept. AD-768-416, Air Force Materials Laboratory, Wright Patterson Air Force Base, OH (Aug. 1972), Chap. 7, pp. 363–403.Google Scholar
  86. 86.
    G.E. Duvall, R.A. Graham, Phase transitions under shock wave loading, Rev. Mod. Phys. 49, 523–579 (1977).ADSGoogle Scholar
  87. 87.
    L. Barker, α-phase Hugoniot of iron, J. Appl. Phys. 46, 2544–2546 (1975).ADSGoogle Scholar
  88. 88.
    R.A. Graham, Measurement of wave profiles in shock-loaded solids, in High-Pressure Science and Technology, edited by K.D. Timmerhaus, M.S. Barber (Plenum Publ. Corp., New York, 1979), Vol. 2, pp. 854–869.Google Scholar
  89. 89.
    D.C. Wallace, Equation of state from weak shocks in solids, Phys. Rev. B20, 1495–1502 (1980).ADSGoogle Scholar
  90. 90.
    R. Blandford, D. Eichler, Particle acceleration at astrophysical shocks: a theory of cosmic ray origin, Phys. Rep. 154, 1–75 (1987).ADSGoogle Scholar
  91. 91.
    T.J. Ahrens, Equation of state, in High Pressure Shock Compression of Solids, edited by J.R. Asay, M. Shahinpoor (Springer-Verlag, New York, 1993), pp. 75–114.Google Scholar
  92. 92.
    Y. Sano, Shock jump equations for unsteady wave fronts, J. Appl. Phys. 82, 5382–5390 (1997).ADSGoogle Scholar
  93. 93.
    Y. Sano, Shock jump equations for unsteady wave fronts of finite rise time, J. Appl. Phys. 84, 6606–6613 (1998).ADSGoogle Scholar
  94. 94.
    Y. Sano, I. Miyamoto, Generalized smooth and weak-discontinuous unsteady waves, J. Math. Phys. 41, 6233–6247 (2000).zbMATHMathSciNetADSGoogle Scholar
  95. 95.
    Y. Sano, T. Sano, Unsteady state Rankine-Hugoniot jump conditions, in 15th APS Topical Conference on Shock Compression of Condensed Matter2007, edited by M. Elert, M.D. Furnish, R. Chau, N.C. Holmes, J. Nguyen, AIP Conf. Proc. 955, 267–270 (2007).Google Scholar
  96. 96.
    Y. Sano, T. Sano, Jump across an outgoing spherical shock wave front, in 15th APS Topical Conference on Shock Compression of Condensed Matter2007, edited by M. Elert, M.D. Furnish, R. Chau, N.C. Holmes, J. Nguyen, AIP Conf. Proc. 955, 271–274 (2007).Google Scholar
  97. 97.
    W.W. Anderson, Jump conditions for nonsteady waves, in 14th APS Topical Conference on Shock Compression of Condensed Matter2005, edited by M.D. Furnish, M.L. Elert, T.P. Russell, C.T. White, AIP Conf. Proc. 845, 1303–1306 (2006).Google Scholar
  98. 98.
    A. Balogh, R.A. Treumann, Physics of Collisionless Shocks (Springer-Verlag, New York, 2013), p. 30.Google Scholar
  99. 99.
    P.L. Sachdev, Shock Waves and Explosions. Monographs and Surveys in Pure & Applied Mathematics, No. 132 (Chapman & Hall/CRC, Boca Ration, FL, 2004), pp. 38–39.Google Scholar
  100. 100.
    F. de Hoffmann, E. Teller, Magneto-hydrodynamic shocks, Phys. Rev. 80, 692–703 (1950).zbMATHADSGoogle Scholar
  101. 101.
    Shocks and discontinuities (magnetohydrodynamics). Wikipedia;http://en.wikipedia.org/wiki/Shocks_and_discontinuities_(magnetohydrodynamics).
  102. 102.
    A.H. Taub, Relativistic Rankine-Hugoniot equations, Phys. Rev. 74 [II], 328–334 (1948).zbMATHMathSciNetADSGoogle Scholar
  103. 103.
    S.M. Carioli, Solutions of the Rankine-Hugoniot relations in relativistic magnetohydrodynamics, Phys. Fluids 29, 672–675 (1986).zbMATHMathSciNetADSGoogle Scholar
  104. 104.
    X.-B. Lin, Generalized Rankine-Hugoniot condition and shock solutions for quasi-linear hyperbolic systems. Dept. of Mathematics, North Carolina State University, Raleigh, NC (June 2, 2000); see http://www4.ncsu.edu/xblin/preprint/shock.pdf.
  105. 105.
    V.M. Shelkovich, Delta-shocks in the Navier-Stokes system of granular hydrodynamics. Poster presented at the 14th Int. Conference on Hyperbolic Problems: Their Theory, Numerics, Applications (HYP 2012) [Università di Padova, Italy; June 25–29, 2012].Google Scholar
  106. 106.
    V.M. Shelkovich, Concept of delta-shock type solutions to systems of conservation laws and the Rankine-Hugoniot conditions, Operator Theory: Advances & Applications 231, 297–305 (2013).MathSciNetGoogle Scholar
  107. 107.
    S.K. Lele, Shock-jump relations in a turbulent flow, Phys. Fluids A 4, 2900–2905 (1992).zbMATHMathSciNetADSGoogle Scholar
  108. 108.
    M.A. Liberman, Introduction to Physics and Chemistry of Combustion: Explosion, Flame, Detonation (Springer-Verlag, Berlin, 2008).Google Scholar
  109. 109.
    C.M. Tarver, Chemical energy release in one-dimensional detonation waves in gaseous explosives, Combust. Flame 46, 111–133 (1982).Google Scholar
  110. 110.
    H. Nieuwenhuijzen, C. de Jager, M. Cuntz, A. Lobel, L. Achmad, A generalized version of the Rankine-Hugoniot relations including ionization, dissociation, radiation and related phenomena, A&A 280, 195–200 (1993).ADSGoogle Scholar
  111. 111.
    H. Ockendon, J.R. Ockendon, Waves and Compressible Flow (Springer-Verlag, New York, 2004).Google Scholar
  112. 112.
    Y. He, X. Hu, Y. Hu, Z. Jiang, J. Lü, Rankine-Hugoniot relations of an axial shock in cylindrical non-neutral plasma, Phys. Plasmas 13, 092116 (2006).MathSciNetADSGoogle Scholar
  113. 113.
    Y. Zheng, Systems of Conservation Laws: Two-Dimensional Riemann Problems (Birkhäuser, Boston, 2001), pp. 86–88.Google Scholar
  114. 114.
    K.C. Hall, A linearized Euler analysis of unsteady flows in turbomachinery. Ph.D. thesis, Dept. of Aeronautics & Astronautics, MIT, Cambridge, USA (May 1987); later partly publ. with W.S. Clark and C.B. Lorence in J. Turbomach. 116, 477–488 (1994).Google Scholar
  115. 115.
    R.F. Chisnell, The normal motion of a shock wave through a non-uniform one dimensional medium, Proc. Roy. Soc. Lond. A232, 350–370 (1955).MathSciNetADSGoogle Scholar
  116. 116.
    R.F. Chisnell, The motion of a shock wave in a channel, with applications to cylindrical and spherical shock waves, J. Fluid Mech. 2, 286–298 (1957).zbMATHMathSciNetADSGoogle Scholar
  117. 117.
    G.B. Whitham, On the propagation of shock waves through regions of non-uniform area or flow, J. Fluid Mech. 4, 337–368 (1958).zbMATHMathSciNetADSGoogle Scholar
  118. 118.
    G.P. Zank, Y. Zhou, W.H. Matthaeus, W.K.M. Rice, The interaction of turbulence with shock waves: a basic model, Phys. Fluids 14, 3766–3774 (2002).MathSciNetADSGoogle Scholar
  119. 119.
    Yu.P. Raizer, Heating of a gas by a powerful light pulse, Sov. Phys. J. Exp. Theor. Phys. 21, 1009–1017 (1965).ADSGoogle Scholar
  120. 120.
    E. Daniel, J. Massoni, Jump relations across shock waves in condensed multiphase flows: a comparison between numerical and analytical solutions. 18ème Congrès Français de Mécanique (Grenoble, France; August 27–31, 2007).Google Scholar
  121. 121.
    W. Fickett, W.C. Davis, Detonation (University of California Press, Berkeley and Los Angeles, 1979), pp. 16–20, 98–102.Google Scholar
  122. 122.
    S.L. Gavrilyuk, R. Saurel, Rankine-Hugoniot relations for shocks in heterogeneous mixtures, J. Fluid Mech. 575, 495–507 (2007).zbMATHMathSciNetADSGoogle Scholar
  123. 123.
    O. Thual, Modeling rollers for shallow water flows, J. Fluid Mech. 728, 1–4 (2013).zbMATHADSGoogle Scholar
  124. 124.
    Y.C. Huang, F.G. Hammitt, T.M. Mitchell, Note on shock wave velocity in high-speed liquid-solid impact, J. Appl. Phys. 44, 1868–1869 (1973).ADSGoogle Scholar
  125. 125.
    R. Ghoshal, N. Mitra, Non-contact near-field underwater explosion induced shock-loading of submerged rigid structures: nonlinear compressibility effects in fluid structure interaction, J. Appl. Phys. 112, 024911 (2012).ADSGoogle Scholar
  126. 126.
    H.G. David, S.D. Hamann, Some properties of compressional waves in Lennard-Jones-and-Devonshire liquids, Austral. J. Chem. 14, 372–386 (1961).Google Scholar
  127. 127.
    A. Satoh, Rankine-Hugoniot relations for Lennard-Jones liquid, J. Fluid Eng. 116, 625–630 (1994).Google Scholar
  128. 128.
    R.J. Seeger, H. Polachek, On shock-wave phenomena: waterlike substances, J. Appl. Phys. 22, 640–654 (1951).zbMATHMathSciNetADSGoogle Scholar
  129. 129.
    L. Davison, R.A. Graham, Shock compression of solids, Phys. Rep. 55, 255–379 (1979).ADSGoogle Scholar
  130. 130.
    L. Davison, Shockwave structure in porous solids, J. Appl. Phys. 42, 5503–5512 (1971).ADSGoogle Scholar
  131. 131.
    L.G. Bolkhovitinov, Yu.B. Khvostov, The Rankine-Hugoniot relation for shock waves in very porous media, Nature 274, 882–883 (1978).ADSGoogle Scholar
  132. 132.
    P.P. Gillis, Elastic precursor decay in tantalum, J. Appl. Phys. 42, 2145–2146 (1971).ADSGoogle Scholar
  133. 133.
    G.E. Duvall, Shock waves and equations of state, in Dynamic Response of Materials to Intense Impulsive Loading, edited by P.C. Chou, A.K. Hopkins. Rept. AD-768-416, Air Force Materials Laboratory, Wright Patterson Air Force Base, OH (1972), Chap. 4, pp. 89–118.Google Scholar
  134. 134.
    J.K. Chao, B. Goldstein, Modification of the Rankine-Hugoniot relations for shocks in space, J. Geophys. Res. 77, 5455–5466 (1972).ADSGoogle Scholar
  135. 135.
    J.J. Sanderson, R.A. Uhrig Jr., Extended Rankine-Hugoniot relations for collisionless shocks, J. Geophys. Res.: Space Phys. 83, 1395–1400 (1978).ADSGoogle Scholar
  136. 136.
    D. Winterhalter, M.G. Kivelson, R.J. Walker, C.T. Russell, The MHD Rankine-Hugoniot jump conditions and the terrestrial bow shock: a statistical comparison, Adv. Space Res. 4, 287–292 (1984).ADSGoogle Scholar
  137. 137.
    E.C. Roelof, S.M. Krimigis, D.G. Mitchell, R.B. Decker, J.D. Richardson, M. Gruntsman, H. Funsten, D. McComas, Implications of generalized Rankine-Hugoniot conditions for the PUI population at the Voyager 2 termination shock, in Proc. 9th Annual International Astrophysics Conference, edited by J. Le Roux, G.P. Zank, A.J. Coates, V. Florinski. AIP Conf. Proc. 1302, 133–141 (2010).Google Scholar
  138. 138.
    R.E. Lee, S.C. Chapman, R.O. Dendy, Numerical simulations of local shock reformation and ion acceleration in supernova remnants. 31st EPS Conference on Plasma Physics [London, UK; June 26–July, 2, 2004]. ECA 28G, Paper O-4.15 (2004).Google Scholar
  139. 139.
    G. Pallocchia, A.A. Samsonov, M.B. Bavassano Cattaneo, M.F. Marcucci, H. Rème, C.M. Carr, J.B. Cao, Interplanetary shock transmitted into the Earth’s magnetosheath: cluster and double star observations, Ann. Geophys. 28, 1141–1156 (2010).ADSGoogle Scholar
  140. 140.
    R.D. Blandford, C.F. McKee, Fluid dynamics of relativistic blast waves, Phys. Fluids 19, 1130–1138 (1976).zbMATHADSGoogle Scholar
  141. 141.
    Y. Gao, C.K. Law, Rankine-Hugoniot relations in relativistic combustion waves. arXiv:1210.3455 [astro-ph.CO] (2012).Google Scholar
  142. 142.
    P. Jenny, B. Müller, Rankine-Hugoniot-Riemann solver with considering source terms and multi-dimensional effects, J. Comput. Phys. 145, 575–610 (1997).ADSGoogle Scholar
  143. 143.
    S. Jaisankar, S.V.R. Rao, A central Rankine-Hugoniot solver for hyperbolic conservation laws, J. Comput. Phys. 228, 770–798 (2009).zbMATHMathSciNetADSGoogle Scholar
  144. 144.
    A. Konyukhov, A. Likhachev, V. Fortov, S. Anisimov, Nonlinear analysis of stability of plane shock waves in media with arbitrary thermodynamic properties, in 28th Int. Symposium on Shock Waves, edited by K. Kontis (Springer-Verlag, Heidelberg & Berlin, 2012), Vol. 2, pp. 531–536.Google Scholar
  145. 145.
    R.F. Chisnell, The motion of a shock wave through a nonuniform one-dimensional medium, Proc. Roy. Soc. Lond. A 232, 350–370 (1957).MathSciNetADSGoogle Scholar
  146. 146.
    L. Crussard, Ondes de choc et onde explosive, Bull. Soc. Industrie Minérale 6 [IV], 257–364 (1907); Propriété de l’onde explosive, C. R. Acad. Sci. Paris 144, 417–420 (1907).Google Scholar
  147. 147.
    G.E. Duvall, Semiannual report, 1 February 1973 to 31 July 1973. Contract No.DAAG-46-C-0104, AMMRC, Watertown, MA (1973).Google Scholar
  148. 148.
    S.D. Hamann, Effects of intense shock waves, in Advances in High Pressure Research, edited by R.S. Bradley (Academic Press, London & New York, 1966), Vol. 1, pp. 85–141.Google Scholar
  149. 149.
    D.P. Dandekar, Behavior of porous tungsten under shock compression at room temperature, J. Appl. Phys. 48, 2871–2879 (1977).ADSGoogle Scholar
  150. 150.
    C.F. Petersen, W.J. Murri, M. Cowperthwaite, Hugoniot and release-adiabat measurements for selected geological materials, J. Geophys. Res. 75, 2063–2072 (1970).ADSGoogle Scholar
  151. 151.
    H. Eyring, R.E. Powell, G.H. Duffey, R.B. Parlin, The stability of detonation, Chem. Rev. 45, 69–181 (1949), Appendix A.Google Scholar
  152. 152.
    Dr. Jerry W. Forbes, private communications on Nov. 22, 2014.Google Scholar
  153. 153.
    M. Müller, Energy dissipated at the shock wave during its propagation in sea water. Colloquium Fluid Dynamics, Institute of Fluid Dynamics, Prague (2007).Google Scholar
  154. 154.
    A.J. Eggers Jr., One-Dimensional Flows of an Imperfect Diatomic Gas. NACA Rept. 959 (1950).Google Scholar
  155. 155.
    F. Marconi, M. Salas, L. Yaeger, Development of computer code for calculating the steady super/hypersonic inviscid flow around real configurations. Vol. 1: Computational technique. Rept. NASA CR-2675 (1976).Google Scholar
  156. 156.
    G.R. Fowles, Shock wave compression of hardened and annealed 2024 aluminum, J. Appl. Phys. 32, 1475–1487 (1961).ADSGoogle Scholar
  157. 157.
    D.C. Pack, Shock wave phenomena, in Research Frontiers in Fluid Dynamics, edited by R.J. Seeger, G. Temple (Interscience Publications, New York, 1965), Chap. 8, pp. 212–249.Google Scholar
  158. 158.
    G.E. Duvall, Concepts of shock wave propagation, Bull. Seismology Soc. Am. 52, No. 4, 869–893 (Oct. 1962).Google Scholar
  159. 159.
    T.-P. Liu, Hyperbolic conservation laws with relaxation, Commun. Math. Phys. 108, 153–175 (1987).zbMATHADSGoogle Scholar
  160. 160.
    J.W. Taylor, M.H. Rice, Elastic-plastic properties of iron, J. Appl. Phys. 34, 364–371 (1963).ADSGoogle Scholar
  161. 161.
    M. Sichel, Structure of weak non-Hugoniot shocks, Phys. Fluids 6, 653–662 (1963).zbMATHMathSciNetADSGoogle Scholar
  162. 162.
    J.M. Dewey, Spherical shock waves, in Handbook of Shock Waves, edited by G. Ben-Dor, O. Igra, T. Elperin (Academic Press, San Diego, 2001), Vol. 2, pp. 441–481.Google Scholar
  163. 163.
    H. Grad, The profile of a steady plane shock wave, Commun. Pure Appl. Math. 5, 257–300 (1952).zbMATHMathSciNetGoogle Scholar
  164. 164.
    Y.B. Zel’dovich, On the possibility of rarefaction shock waves, Zh. Eksp. Teor. Fiz. 16, 363–364 (1946).Google Scholar
  165. 165.
    W.E. Drummond, Multiple shock production, J. Appl. Phys. 28, 998–1001 (1957).MathSciNetADSGoogle Scholar
  166. 166.
    S.S. Kutateladze, Al. A. Borisov, A.A. Borisov, V.E. Nakoryakov, Experimental detection of a rarefaction shock wave near a liquid-vapor critical point, Sov. Phys. Dokl. 25, 392–393 (1980).ADSGoogle Scholar
  167. 167.
    G.I. Taylor, The dynamics of the combustion products behind plane and spherical detonation fronts in explosives, Proc. Roy. Soc. Lond. A 200, 235–247 (1949/1950).ADSGoogle Scholar
  168. 168.
    Y.M. Gupta, Shock waves in condensed media. In McGraw-Hill Encyclopedia of Science & Technology, 9th edn. (McGraw-Hill Book Co., New York, 2005), pp. 438–439.Google Scholar
  169. 169.
    W. Band, G.E. Duvall, Physical nature of shock propagation, Am. J. Phys. 29, 780–785 (1961).zbMATHADSGoogle Scholar
  170. 170.
    L. Barker, L. Hollenbach, Shock wave study of the alpha-epsilon phase transition in iron, J. Appl. Phys. 45, 4872–4887 (1974).ADSGoogle Scholar
  171. 171.
    H.A. Bethe, E. Teller, Deviations from Thermal Equilibrium in Shock Waves. BL Rept. X-117, BRL, Aberdeen Proving Ground, MD (1941).Google Scholar
  172. 172.
    G.E. Duvall, G.R. Fowles, Shock waves, in High Pressure Physics and Chemistry, edited by R.S. Bradley (Academic Press, New York, 1963), Vol. 2, p. 212.Google Scholar
  173. 173.
    G.A. Lyzenga, T.J. Ahrens, W.J. Nellis, A.C. Mitchell, The temperature of shock-compressed water, J. Chem. Phys. 76, 6282–6286 (1982).ADSGoogle Scholar
  174. 174.
    L.V. Al’tshuler, K.K. Krupnikov, B.B. Lebedev, V.I. Zhuchikin, M.I. Brazhnik, Dynamic compressibility and equation of state of iron under high pressure, Sov. Phys. 7, 606–614 (1958).Google Scholar
  175. 175.
    A.C. Mitchell, W.J. Nellis, Shock compression of aluminum, copper, and tantalum, J. Appl. Phys. 52, 3363–3374 (1981).ADSGoogle Scholar
  176. 176.
    R.G. Shreffler, W.E. Deal, Free surface properties of explosive-driven metal plates, J. Appl. Phys. 24, 44–48 (1953).ADSGoogle Scholar
  177. 177.
    W.J. Carter, S.P. Marsh, J.N. Fritz, R.G. McQueen, The equation of state of selected materials for high-pressure references, in Accurate Characterization of the High-Pressure Environment, edited by E.C. Lloyd. NBS Special Publication No. 326, US Government Printing Office, Washington, DC (1971), pp. 147–158.Google Scholar
  178. 178.
    F.G. Friedlander, The diffraction of sound pulses. I. Diffraction by a semi-infinite plate, Proc. Roy. Soc. Lond. A 186, 322–344 (1946).zbMATHADSGoogle Scholar
  179. 179.
    W.E. Baker, Explosions in Air (University of Texas Press, Austin, 1973).Google Scholar
  180. 180.
    M. Larcher, Pressure-time functions for the description of air blast waves. JRC Technical Note 46829, Joint Research Centre, Ispra, Italy (2008).Google Scholar
  181. 181.
    H. Honma, I.I. Glass, C.H. Wong, O. Hoist-Jensen, D. Xu, Experimental and numerical studies of weak blast waves in air, Shock Waves 1, 111–119 (1991).ADSGoogle Scholar
  182. 182.
    G.R. Fowles, Experimental techniques and instrumentation, in Dynamic response of materials to intense impulsive loading, edited by P.C. Chou, A.K. Hopkins. Rept. AD-768-416, Air Force Materials Laboratory, Wright Patterson Air Force Base, OH (1972), Chap. 8, pp. 405–480.Google Scholar
  183. 183.
    T.J. Ahrens, Shock wave techniques for geophysics and planetary physics, Meth. Exp. Phys. 24, 185–210 (1987).Google Scholar
  184. 184.
    J.W. Forbes, Shock Wave Compression of Condensed Matter – a Primer (Springer-Verlag, Berlin, 2012); pp. 68–79.Google Scholar
  185. 185.
    L.M. Barker, M. Shahinpoor, L.C. Chhabildas, Experimental and diagnostic techniques, in High-pressure shock compression of solids, edited by J.R. Asay, M. Shahinpoor (Springer-Verlag, New York, 1993), pp. 43–73.Google Scholar
  186. 186.
    Y. Beers, Introduction to the Theory of Error (Addison-Wesley, London, 1957), p. 4.Google Scholar
  187. 187.
    R.W. Goranson, D. Bancroft, B.L. Burton, T. Blechar, E.E. Houston, E.F. Gittings, S.A. Landeen, Dynamic determination of the compressibility of metals, J. Appl. Phys. 26, 1472–1479 (1955).ADSGoogle Scholar
  188. 188.
    R.G. McQueen, S.P. Marsh, Equation of state for nineteen metallic elements from shock-wave measurements to two megabars, J. Appl. Phys. 31, 1253–1269 (1960).ADSGoogle Scholar
  189. 189.
    F.S. Minshall, The dynamic response of iron and iron alloys to shock waves, in Response of Metals to High-Velocity Deformation, edited by V.F. Zackay, P.G. Shewmon (Interscience, New York, 1961), pp. 249–272.Google Scholar
  190. 190.
    R.E. Duff, E. Houston, Measurement of the Chapman-Jouguet pressure and reaction zone length in a detonating high explosive, J. Chem. Phys. 23, 1268–1273 (1955).ADSGoogle Scholar
  191. 191.
    N.L. Coleburn, J.W. Forbes, Irreversible transformation of hexagonal boron nitride by shock compression, J. Chem. Phys. 48, 555–559 (1968).ADSGoogle Scholar
  192. 192.
    R. Schall, G. Thomer, Flash Radiographic Measurement of the Shock Compressibility of Magnesium Alloy, Lucite, and Polyethylene. Rept. AFSWC-TDR-62-134, Air Force Systems Command, Kirtland Air Force Base, NM (1962).Google Scholar
  193. 193.
    R. Schall, Die Zustandsgleichung des Wassers bei hohen Drucken nach Röntgenblitzaufnahmen intensiver Stoßwellen, Z. Angew. Phys. 2, 252–254 (1950).Google Scholar
  194. 194.
    R. Schall, G. Thomer, Röntgenblitzaufnahmen von Stoßwellen in festen, flüssigen und gasförmigen Medien, Z. Angew. Phys. 3, 41–44 (1951).Google Scholar
  195. 195.
    F. Jamet, G. Thomer, Flash Radiography (Elsevier, Amsterdam, 1976), pp. 120–122.Google Scholar
  196. 196.
    G.E. Duvall, Problems in shock wave research [invited paper], in Conference on Metallurgical Effects at High Strain Rates [Albuquerque, NM; Feb. 5−8, 1973], edited by R.W. Rohde, B.M. Butcher, J.R. Holland, C.H. Karnes (Plenum Press, New York, 1973), pp. 1–13.Google Scholar
  197. 197.
    M. Ross, W. Nellis, A. Mitchell, Shock-wave compression of liquid argon to 910 kbar, Chem. Phys. Lett. 68, 532–535 (1979).ADSGoogle Scholar
  198. 198.
    A.H. Jones, W.H. Isbell, C.J. Maiden, Measurements of the very-high-pressure properties of materials using a light-gas gun, J. Appl. Phys. 37, 3493–3499 (1966).ADSGoogle Scholar
  199. 199.
    C.E. Morris, Shock-wave equation-of-state studies, Shock Waves 1, 213–222 (1991).ADSGoogle Scholar
  200. 200.
    G.V. oriskov et al., Shock compression of liquid deuterium up to 109 GPa, Phys. Rev. B 71, 092104 (2005).ADSGoogle Scholar
  201. 201.
    G. Chabrier et al., Hydrogen and helium at high density and astrophysical implications, in High Energy Density Laboratory Astrophysics, edited by S.V. Lebedev (Springer-Verlag, Dordrecht, The Netherlands, 2007), pp. 257–261.Google Scholar
  202. 202.
    J.W. Forbes, Shock Wave Compression of Condensed Matter – a Primer (Springer-Verlag, Berlin, 2012), pp. 82–90.Google Scholar
  203. 203.
    R.F. Smith, J.H. Eggert, A. Jankowski, P.M. Celliers, M.J. Edwards, Y.M. Gupta, J.R. Asay, G.W. Collins, Stiff response of aluminum under shockless compression to 110 GPa, Phys. Rev. Lett. 98, 065701 (2007).ADSGoogle Scholar
  204. 204.
    P. Krehl, Measurement of low shock pressures with piezoresistive carbon gauges, Rev. Sci. Instrum. 49, 1477–1484 (1978).ADSGoogle Scholar

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  1. 1.NimburgGermany

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