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Shock Propagation Following an Intense Explosion: Comparison Between Hydrodynamics and Simulations

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The solution for the radial distribution of pressure, density, temperature and flow velocity fields in a blast wave propagating through a medium at rest, following an intense explosion, starting from hydrodynamic equations, is one of the classic problems in gas dynamics. However, there is very little direct verification of the theory and its assumptions from simulations of microscopic models. In this paper, we compare the results and assumptions of the hydrodynamic theory with results from large scale event driven molecular dynamics simulations of a hard sphere gas in three dimensions. We find that the predictions for the radial distribution of the thermodynamic quantities do not match well with the numerical data. We improve the theory by replacing the ideal gas law with a more realistic virial equation of state for the hard sphere gas. While this improves the theoretical predictions, we show that they still fail to describe the data well. To understand the reasons for this discrepancy, the different assumptions of the hydrodynamic theory are tested within the simulations. A key assumption of the theory is the existence of a local equation of state. We validate this assumption by showing that the local pressure, temperature and density obey the equation of state for a hard sphere gas. However, the probability distribution of the velocity fluctuations has non-gaussian tails, especially away from the shock front, showing that the assumption of local equilibrium is violated. This, along with neglect of heat conduction, could be the possible reasons for the mismatch between theory and simulations.

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References

  1. Abdel-Raouf, A., Gretler, W.: Quasi-similar solutions for blast waves with internal heat transfer effects. Fluid Dyn. Res. 8, 273–285 (1991)

    Article  ADS  Google Scholar 

  2. Anand, R.: Shock dynamics of strong imploding cylindrical and spherical waves with non-ideal gas effects. Wave Motion 50, 1003–1015 (2013)

    Article  MathSciNet  Google Scholar 

  3. Antal, T., Krapivsky, P.L., Redner, S.: Exciting hard spheres. Phys. Rev. E 78, 030301 (2008). https://doi.org/10.1103/PhysRevE.78.030301

    Article  ADS  Google Scholar 

  4. Barbier, M., Villamaina, D., Trizac, E.: Blast dynamics in a dissipative gas. Phys. Rev. Lett. 115, 214301 (2015). https://doi.org/10.1103/PhysRevLett.115.214301

    Article  ADS  Google Scholar 

  5. Barbier, M., Villamaina, D., Trizac, E.: Microscopic origin of self-similarity in granular blast waves. Phys. Fluids 28, 083302 (2016). https://doi.org/10.1063/1.4961047

    Article  ADS  Google Scholar 

  6. Barenblatt, G.: Scaling, Self-similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics. Cambridge University Press, Cambridge (1987)

    MATH  Google Scholar 

  7. Berezinsky, V., Dokuchaev, V.: Hidden source of high-energy neutrinos in collapsing galactic nucleus. Astropart. Phys. 15, 87–96 (2001)

    Article  ADS  Google Scholar 

  8. Boudet, J.F., Cassagne, J., Kellay, H.: Blast shocks in quasi-two-dimensional supersonic granular flows. Phys. Rev. Lett. 103, 224501 (2009). https://doi.org/10.1103/PhysRevLett.103.224501

    Article  ADS  Google Scholar 

  9. Brode, H.L.: Numerical solutions of spherical blast waves. J. Appl. Phys. 26, 766 (1955). https://doi.org/10.1063/1.1722085

    Article  MathSciNet  MATH  ADS  Google Scholar 

  10. Castor, J., McCray, R., Weaver, R.: Interstellar bubbles. Astrophys. J. 200, L107–L110 (1975)

    Article  ADS  Google Scholar 

  11. Cheng, X., Xu, L., Patterson, A., Jaeger, H.M., Nagel, S.R.: Towards the zero-surface-tension limit in granular fingering instability. Nat. Phys. 4, 234 (2008)

    Article  Google Scholar 

  12. Cioffi, D.F., Mckee, C.F., Bertschinger, E.: Dynamics of radiative supernova remnants. Astrophys. J. 334, 252–265 (1988)

    Article  ADS  Google Scholar 

  13. Dokuchaev, V.I.: Self-similar shock solution with sustained energy injection. Astronomy Astrophys. 395, 1023–1029 (2002). https://doi.org/10.1051/0004-6361:20021305

    Article  MATH  ADS  Google Scholar 

  14. Edens, A., Ditmire, T., Hansen, J., Edwards, M., Adams, R., Rambo, P., Ruggles, L., Smith, I., Porter, J.: Study of high mach number laser driven blast waves. Phys. Plasmas 11, 4968–4972 (2004). https://doi.org/10.1063/1.1773553

    Article  MATH  ADS  Google Scholar 

  15. Edwards, M.J., MacKinnon, A.J., Zweiback, J., Shigemori, K., Ryutov, D., Rubenchik, A.M., Keilty, K.A., Liang, E., Remington, B.A., Ditmire, T.: Investigation of ultrafast laser-driven radiative blast waves. Phys. Rev. Lett. 87, 085004 (2001). https://doi.org/10.1103/PhysRevLett.87.085004

    Article  ADS  Google Scholar 

  16. Falle, S.: A numerical calculation of the effect of stellar winds on the interstellar medium. Astonomy Astrophys. 43, 323–336 (1975)

    ADS  Google Scholar 

  17. Ghoniem, A., Kamel, M., Berger, S., Oppenheim, A.: Effect of internal heat transfer on the structure of self-similar blast waves. J. Fluid Mech. 117, 473–491 (1982). https://doi.org/10.1017/S0022112082001724

    Article  MATH  ADS  Google Scholar 

  18. Grasselli, Y., Herrmann, H.J.: Crater formation on a three dimensional granular heap. Granul. Matter 3(4), 201–204 (2001). https://doi.org/10.1007/s100350100095

    Article  Google Scholar 

  19. Guderley, G.: Powerful spherical and cylindrical compression shocks in the neighbourhood of the centre and of the cylinder axis. Lufifahrtforschung 19, 302–312 (1942)

    MathSciNet  Google Scholar 

  20. Gull, S.: A numerical model of the structure and evolution of young supernova remnants. Monthly Not. R. Astronomical Soc. 161, 47–69 (1973)

    Article  ADS  Google Scholar 

  21. Hirschler, T., Steiner, H.: A self-similar solution for the implosion problem in a dusty gas. Fluid Dyn. Res. 32, 61–67 (2003)

    Article  ADS  Google Scholar 

  22. Huang, H., Zhang, F., Callahan, P.: Granular fingering in fluid injection into dense granular media in a hele-shaw cell. Phys. Rev. Lett. 108, 258001 (2012). https://doi.org/10.1103/PhysRevLett.108.258001

    Article  ADS  Google Scholar 

  23. Isobe, M.: Hard sphere simulation in statistical physics—methodologies and applications. Mol. Simul. 42, 1317–1329 (2016). https://doi.org/10.1080/08927022.2016.1139106

    Article  Google Scholar 

  24. Jabeen, Z., Rajesh, R., Ray, P.: Universal scaling dynamics in a perturbed granular gas. Eur. Phys. Lett. 89(3), 34001 (2010)

    Article  ADS  Google Scholar 

  25. Johnsen, O., Toussaint, R., Måløy, K.J., Flekkøy, E.G.: Pattern formation during air injection into granular materials confined in a circular hele-shaw cell. Phys. Rev. E 74, 011301 (2006). https://doi.org/10.1103/PhysRevE.74.011301

    Article  ADS  Google Scholar 

  26. Joy, J.P., Pathak, S.N., Das, D., Rajesh, R.: Shock propagation in locally driven granular systems. Phys. Rev. E 96, 032908 (2017). https://doi.org/10.1103/PhysRevE.96.032908

    Article  ADS  Google Scholar 

  27. Landau, L., Lifshitz, E.: Course of Theoretical Physics—Fluid Mechanics. Butterwörth-Heinemann, Oxford (1987)

    Google Scholar 

  28. Latter, R.: Similarity solution for spherical shock wave. J. Appl. Phys. 26, 954 (1955). https://doi.org/10.1063/1.1722144

    Article  MathSciNet  MATH  ADS  Google Scholar 

  29. Lazarus, R.B.: Self-similar solutions for converging shocks and collapsing cavities. SIAM J. Numer. Anal. 18, 316–371 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  30. McCoy, B.M.: Advanced Statistical Mechanics. Oxford Science Publications, Oxford (2009)

    Book  Google Scholar 

  31. Metzger, P.T., Schuler, R.C.L., Immer, J.M.: Craters formed in granular beds by impinging jets of gas. AIP Conf. Proc. 1145, 767–770 (2009). https://doi.org/10.1063/1.3180041

    Article  ADS  Google Scholar 

  32. Moore, A.S., Symes, D.R., Smith, R.A.: Tailored blast wave formation: developing experiments pertinent to laboratory astrophysics. Phys. Plasmas 12, 052707-1–052707-7 (2005). https://doi.org/10.1063/1.1909199

    Article  ADS  Google Scholar 

  33. Ostriker, J.P., McKee, C.F.: Astrophysical blastwaves. Rev. Mod. Phys. 60, 1–68 (1988). https://doi.org/10.1103/RevModPhys.60.1

    Article  ADS  Google Scholar 

  34. Pathak, S.N., Jabeen, Z., Ray, P., Rajesh, R.: Shock propagation in granular flow subjected to an external impact. Phys. Rev. E 85, 061301 (2012). https://doi.org/10.1103/PhysRevE.85.061301

    Article  ADS  Google Scholar 

  35. Pinto, S.F., Couto, M.S., Atman, A.P.F., Alves, S.G., Bernardes, A.T., de Resende, H.F.V., Souza, E.C.: Granular fingers on jammed systems: new fluidlike patterns arising in grain-grain invasion experiments. Phys. Rev. Lett. 99, 068001 (2007). https://doi.org/10.1103/PhysRevLett.99.068001

    Article  ADS  Google Scholar 

  36. Plooster, M.N.: Shock waves from line sources: numerical solutions and experimental measurements. Phys. Fluids 13, 2665 (1970). https://doi.org/10.1063/1.1692848

    Article  ADS  Google Scholar 

  37. Ponchaut, N.F., Hornung, H., Pullin, D., Mouton, C.: On imploding cylindrical and spherical shock waves in a perfect gas. J. Fluid Mech. 560, 103–122 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  38. Rapaport, D.C.: The Art of Molecular Dynamics Simulations. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  39. Sandnes, B., Knudsen, H.A., Måløy, K.J., Flekkøy, E.G.: Labyrinth patterns in confined granular-fluid systems. Phys. Rev. Lett. 99, 038001 (2007). https://doi.org/10.1103/PhysRevLett.99.038001

    Article  ADS  Google Scholar 

  40. Sedov, L.: Propagation of strong shock waves. J. Appl. Math. Mech. 10, 241–250 (1946)

    Google Scholar 

  41. Sedov, L.: Similarity and Dimensional Methods in Mechanics, 10th edn. CRC Press, Florida (1993)

    Google Scholar 

  42. Steiner, H., Gretler, W.: The propagation of spherical and cylindrical shock waves in real gases. Phys. Fluids 6, 2154 (1994). https://doi.org/10.1063/1.868218

    Article  MATH  ADS  Google Scholar 

  43. Taylor, G.: The formation of a blast wave by a very intense explosion. I. Theoretical discussion. Proc. R. Soc. Lond. Ser. A 201(1065), 159–174 (1950)

    Article  ADS  Google Scholar 

  44. Taylor, G.: The formation of a blast wave by a very intense explosion. II. The atomic explosion of 1945. Proc. R. Soc. Lond. A 201(1065), 175–186 (1950)

    Article  ADS  Google Scholar 

  45. von Neumann, J.. : In: Collected Works. Pergamon Press, Oxford, p. 219 (1963)

  46. VonNeumann, J., Richtmyer, R.: A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys. 21, 232 (1950). https://doi.org/10.1063/1.1699639

    Article  MathSciNet  MATH  ADS  Google Scholar 

  47. Walsh, A.M., Holloway, K.E., Habdas, P., de Bruyn, J.R.: Morphology and scaling of impact craters in granular media. Phys. Rev. Lett. 91, 104301 (2003). https://doi.org/10.1103/PhysRevLett.91.104301

    Article  ADS  Google Scholar 

  48. Weaver, R., McCray, R., Castor, J.: Interstellar bubbles. II. Structure and evolution. Astrophysica J. 218, 377–395 (1977)

    Article  ADS  Google Scholar 

  49. Whitham, G.: Linear and Nonlinear Waves. Wiley, New York (1974)

    MATH  Google Scholar 

  50. Woltjer, L.: Supernova remnants. Ann. Rev. Astron. Astrophys. 10(1), 129–158 (1972). https://doi.org/10.1146/annurev.aa.10.090172.001021

    Article  ADS  Google Scholar 

  51. Zel’dovich, Y.B., Raizer, Y.P.: Physics of Shock Waves and High Temperature Hydrodynamic Phenomena. Dover Publications Inc, New York (2002)

    Google Scholar 

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Acknowledgements

The simulations were carried out on the supercomputer Nandadevi at The Institute of Mathematical Sciences.

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Authors and Affiliations

  1. The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai, 600113, India

    Jilmy P. Joy & R. Rajesh

  2. Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai, 400094, India

    Jilmy P. Joy & R. Rajesh

  3. Department of Physics, St. Thomas College (Autonomous), Thrissur, Kerala, 680001, India

    Jilmy P. Joy

  4. Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore

    Sudhir N. Pathak

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  1. Jilmy P. Joy
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Correspondence to Jilmy P. Joy.

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Communicated by Deepak Dhar.

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Joy, J.P., Pathak, S.N. & Rajesh, R. Shock Propagation Following an Intense Explosion: Comparison Between Hydrodynamics and Simulations. J Stat Phys 182, 34 (2021). https://doi.org/10.1007/s10955-021-02715-3

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