Skip to main content
SpringerLink
Log in

Qualitative Properties of the Collisional Model for Describing Shock-Wave Dynamics in Gas Suspensions

  • Published:
Mathematical Models and Computer Simulations Aims and scope

Abstract

A The paper presents theoretical analysis of a two-phase medium model for describing shock-wave processes in dense gas particle mixtures, taking into account the chaotic motion and collisions of particles. Hyperbolicity regions and composite-type domains are identified in the governing system of equations. It is shown that the hyperbolicity region expands beyond the collisionless model. An approximate hyperbolized model is presented. Numerical solutions of the problem on the development of various shock-wave structure types are compared. The convergence properties in numerical simulations of non-conservative composite-type equations were established using the Harten and Gentry–Martin–Daly schemes. The conditions of the hyperbolized model applicability to different types of flows are obtained. It is shown that, in the general case, shock-wave processes in gas suspensions should preferably be analyzed within a complete model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.

Similar content being viewed by others

REFERENCES

  1. Kh. A. Rakhamulin, “Basics of gas dynamics for interpenetrating movements of compressible media,” Prikl. Mat. Mekh. 20, 184–195 (1956).

    Google Scholar 

  2. R. I. Nigmatulin, Dynamics of Multiphase Media, Part 1 (Nauka, Moscow, 1987; CRC, Boca Raton, FL, 1990).

  3. V. N. Nikolaevskii, “Hydrodynamic analysis of shock adiabats of heterogeneous mixtures of substances,” J. Appl. Mech. Tech. Phys., No. 3, 407–411 (1969).

  4. N. N. Yanenko, R. I. Soloukhin, A. N. Papyrin, and V. M. Fomin, Supersonic Two-Phase Flows under Conditions of Particle Velocity Nonequilibrium (Nauka, Moscow, 1980) [in Russian].

    Google Scholar 

  5. A. E. Medvedev, A. V. Fedorov, and V. M. Fomin, “Description of ignition and combustion of gas mixtures with solid particles by methods of the mechanics of continuous media,” Combust., Explos. Shock Waves 20, 127–133 (1984).

    Article  Google Scholar 

  6. M. R. Baer and J. W. Nunziato, “A two-phase mixture theory for the deflagration to detonation transition (DDT) in reactive granular materials,” Int. J. Multiphase Flow 12, 861–889 (1986).

    Article  Google Scholar 

  7. A. V. Fedorov, “Mathematical description of the flow of a mixture of condensed materials at high pressures,” in Physical Gas Dynamics of Reacting Media, Collection of Articles (Nauka, Novosibirsk, 1990), pp. 119–128 [in Russian].

    Google Scholar 

  8. J. M. Powers, D. S. Stewart, and H. Krier, “Theory of two-phase detonation, Part I: Modelling,” Combust. Flame 80, 264–279 (1990).

    Article  Google Scholar 

  9. K. A. Gonthier and J. M. Powers, “A numerical investigation of transient detonation in granulated material,” Shock Waves 6, 183–195 (1996).

    Article  Google Scholar 

  10. S. Xu and D. S. Stewart, “Deflagration-to-detonation transition in porous energetic materials: a comparative model study,” J. Eng. Math. 31, 143–172 (1997).

    Article  Google Scholar 

  11. J. B. Bdzil, R. Menikoff, S. F. Son, A. K. Kapila, and D. S. Stewart, “Two-phase modeling of DDT in granular materials: a critical examination of modeling issues,” Phys. Fluids, 378–402 (1999).

  12. A. V. Fedorov, “The structure of a shock wave in a mixture of two solids (hydrodynamic approximation),” Model. Mekh. 5 (4), 135–158 (1991).

    MathSciNet  MATH  Google Scholar 

  13. E. V. Varlamov and A. V. Fedorov, “Traveling wave in a non-isothermal mixture of solids,” Model. Mekh. 5 (3), 14–26 (1991).

    MathSciNet  Google Scholar 

  14. A. V. Fedorov and N. N. Fedorova, “Structure, propagation, and reflection of shock waves in a mixture of solids (the hydrodynamic approximation),” J. Appl. Mech. Tech. Phys. 33, 487–494 (1992).

    Article  MathSciNet  Google Scholar 

  15. A. V. Fedorov and A. A. Zhilin, “The shock-wave structure in a two-velocity mixture of compressible media with different pressures,” J. Appl. Mech. Tech. Phys. 39, 166–174 (1998).

    Article  MathSciNet  Google Scholar 

  16. A. A. Zhilin and A. V. Fedorov, “Propagation of shock waves in a two-phase mixture with different pressures of the components,” J. Appl. Mech. Tech. Phys. 40, 46–53 (1999).

    Article  MathSciNet  Google Scholar 

  17. A. G. Kutushev and D. A. Rudakov, “Numerical investigation of the parameters of the air shocks associated with the expansion of a powder layer,” Combust., Explos. Shock Waves 28, 670–676 (1992).

    Article  Google Scholar 

  18. A. G. Kutushev, Mathematical Modeling of Wave Processes in Aero-Dispersed and Powdered Media (Nedra, St. Petersburg, 2003) [in Russian].

    Google Scholar 

  19. A. A. Gubaidullin, A. Britain, and D. N. Dudko, “Air shock wave interaction with an obstacle covered by porous material,” Shock Waves 13, 41–48 (2003).

    Article  Google Scholar 

  20. M. A. Gol’dshtik, “Elementary theory of the boiling layer,” J. Appl. Mech. Tech. Phys., No. 6, 851–856 (1972).

    Article  Google Scholar 

  21. M. A. Gol’dshtik and B. N. Kozlov, “Elementary theory of concentrated dispersed systems,” J. Appl. Mech. Tech. Phys., No. 4, 851–856 (1973).

    Article  Google Scholar 

  22. A. V. Fedorov, “Structure of a combination discontinuity in gas suspensions in the presence of random pressure from particles,” J. Appl. Mech. Tech. Phys., No. 5, 851–856 (1992).

  23. A. V. Fedorov and T. A. Khmel, “Description of shock wave processes in gas suspensions using the molecular-kinetic collisional model,” Heat Transfer Res. 43, 95–107 (2012).

    Article  Google Scholar 

  24. T. A. Khmel’ and A. V. Fedorov, “Description of dynamic processes in two-phase colliding media with the use of molecular-kinetic approaches,” Combust., Explos. Shock Waves 38, 196–207 (2014).

    Article  Google Scholar 

  25. D. Gidaspow, Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions (Academic, Boston, 1994).

    MATH  Google Scholar 

  26. A. Goldshtein and M. Shapiro, “Mechanics of collisional motion of granular materials. Part I. General hydrodynamics equations,” J. Fluid Mech. 282, 75–114 (1995).

    Article  MathSciNet  Google Scholar 

  27. A. Goldshtein, M. Shapiro, and C. Gutfinger, “Mechanics of collisional motion of granular materials. Part 3. Self-similar shock wave propagation,” J. Fluid Mech. 316, 29–51 (1996).

    Article  Google Scholar 

  28. B. E. Gel’fand, S. P. Medvedev, A. N. Polenov, E. I. Timofeev, S. M. Frolov, and S. A. Tsyganov, “Measurement of the velocity of weak disturbances of bulk density in porous media,” J. Appl. Mech. Tech. Phys. 27 (5), 127–130 (1986).

    Article  Google Scholar 

  29. T. A. Khmel’ and A. V. Fedorov, “Modeling of propagation of shock and detonation waves in dusty media with allowance for particle collisions,” Combust., Explos. Shock Waves 50 (5), 196–207 (2014).

    Article  Google Scholar 

  30. T. A. Khmel and A. V. Fedorov, “Numerical simulation of dust dispersion using molecular-kinetic model for description of particle-to-particle collisions,” J. Loss Prevent. Process Ind. 36, 223–229 (2015).

    Article  Google Scholar 

  31. T. A. Khmel and A. V. Fedorov, “Role of particle collisions in shock wave interaction with a dense spherical layer of a gas suspension,” Combust., Explos. Shock Waves 53, 444–452 (2017).

    Article  Google Scholar 

  32. T. A. Khmel and A. V. Fedorov, “Numerical study of dispersion of a rough dense layer of particles under the action of an expanding shock wave,” Combust., Explos. Shock Waves 53, 696–704 (2017).

    Article  Google Scholar 

  33. T. A. Khmel and A. V. Fedorov, “Numerical technologies for investigations of heterogeneous detonations of gas particle suspensions,” Mat. Model. 18 (8), 49–63 (2006).

    MathSciNet  MATH  Google Scholar 

  34. V. F. Kuropatenko, “Momentum and energy exchange in nonequilibrium multicomponent media,” J. Appl. Mech. Tech. Phys. 46, 1–8 (2005).

    Article  Google Scholar 

  35. V. F. Kuropatenko, “Mesomechanics of one-component and multicomponent materials,” Fiz. Mezomekh. 4 (3), 49–55 (2001).

    Google Scholar 

  36. A. I. Ivandaev, A. G. Kutushev, and D. A. Rudakov, “Numerical investigation of throwing a powder layer by a compressed gas,” Combust., Explos. Shock Waves 31, 459–465 (1995).

    Article  Google Scholar 

  37. A. N. Kraiko and L. E. Sternin, “Theory of flows of a two-velocity continuous medium containing solid or liquid particles,” J. Appl. Math. Mech. 29, 482–496 (1965).

    Article  Google Scholar 

  38. S. P. Kiselev, G. A. Ruev, A. I. Trunev, V. M. Fomin, and M. Sh. Shavaliev, Shock-Wave Processes in Two-Component and Two-Phase Media (Nauka, Novosibirsk, 1992) [in Russian].

    Google Scholar 

  39. Yu. V. Kazakov, “Numerical simulation of the propagation of rarefaction and detonation waves in gas mixtures,” Cand. Sci. (Phys. Math.) Dissertation (Novosibirsk, Krasnoyarsk, 1989).

  40. I. A. Bedarev and A. V. Fedorov, “Structure and stability of shock waves in a gas-particle mixture with two pressure,” Vychisl. Tekhnol. 20 (2), 3–19 (2015).

    MATH  Google Scholar 

  41. A. V. Fedorov and I. A. Bedarev, “The shock-wave strukture in a gas-mixture with chaotic pressure,” Math. Models Comput. Simul. 10 (1), 1–14 (2018).

    Article  MathSciNet  Google Scholar 

Download references

Funding

The work was supported by the Russian Science Foundation (project no. 16-19-00010).

Author information

Authors and Affiliations

  1. Institute of Theoretical and Applied Mechanics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia

    A. V. Fedorov & T. A. Khmel

Authors
  1. A. V. Fedorov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. A. Khmel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Fedorov.

Additional information

Translated by I. Pertsovskaya

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fedorov, A.V., Khmel, T.A. Qualitative Properties of the Collisional Model for Describing Shock-Wave Dynamics in Gas Suspensions. Math Models Comput Simul 11, 818–830 (2019). https://doi.org/10.1134/S2070048219050077

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S2070048219050077

Keywords:

Search

Navigation