DOI: https://doi.org/10.26089/NumMet.v18r101

A modification of the CABARET scheme for numerical simulation of one-dimensional detonation flows using a one-stage irreversible model of chemical kinetics

Authors

  • A.V. Danilin
  • A.V. Solovjev
  • A.M. Zaitsev

Keywords:

one-velocity multicomponent medium
systems of hyperbolic equations
CABARET scheme
computational fluid dynamics (CFD)
onservative methods
detonation
Arrhenius reaction

Abstract

An algorithm for numerical simulation of one-dimensional detonation using a one-stage irreversible model of chemical kinetics is proposed. The discretization of the convective parts of governing equations is made in accordance with the balance-characteristic CABARET (Compact Accurately Boundary Adjusting-REsolution Technique) approach. The approximation of source terms is performed implicitly without splitting into physical processes with a regulated order of approximation. It is shown that the numerically obtained Chapman-Jouget detonation parameters are in exact agreement with the analytical solution. It is also shown that, in the case of unstable detonation, the numerical results are dependent on the order of approximation chosen for the right-hand sides of the governing equations.

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Published

2017-01-16

Issue

Vol. 18 (2017): Issue 1.

Section

Section 1. Numerical methods and applications

Author Biographies

A.V. Danilin

Nuclear Safety Institute (IBRAE) of RAS
• Junior Researcher

A.V. Solovjev

Nuclear Safety Institute (IBRAE) of RAS
• Leading Researcher

A.M. Zaitsev

Nuclear Safety Institute (IBRAE) of RAS
• Junior Researcher

References

  1. D. L. Chapman, “On the Rate of Explosion in Gases,” Philos. Mag. series 47}, 90-104 (1899).
  2. E. Jouguet, “On the Propagation of Chemical Reactions in Gases,” J. Math. Pures Appl. series 1}, 347-425 (1905).
  3. E. Jouguet, Mécanique des Explosifs (Octave Doin, Paris, 1917).
  4. Yu. P. Raizer, Introduction to Hydrogasdynamics and the Theory of Shock Waves for Physicists (Intellekt, Dolgoprudnyi, 2011) [in Russian].
  5. Ya. B. Zeldovich, “To the Theory of Detonation Propagation in Gaseous Systems,” Zh. Eksp. Teor. Fiz. 10 (5), 542-568 (1940).
  6. J. von Neumann, Collected Works , Vol. 6 (Pergamon, New York, 1963).
  7. W. Döring, “On Detonation Processes in Gases,” Ann. Phys. series 43}, 421-436 (1943).
  8. J. J. Erpenbeck, “Stability of Steady-State Equilibrium Detonations,” Phys. Fluids 5 (5), 604-614 (1962).
  9. K. I. Shchelkin, “Instability of Combustion and Detonation of Gases,” Usp. Fiz. Nauk 87 (2), 273-302 (1965) [Sov. Phys. Usp. 8 (5), 780-797 (1966)].
  10. W. Fickett and W. W. Wood, “Flow Calculations for Pulsating One-Dimensional Detonations,” Phys. Fluids series 9} (5), 903-916 (1966).
  11. A. Bourlioux, A. J. Majda, and V. Roytburd, “Theoretical and Numerical Structure for Unstable One-Dimensional Detonations,” SIAM J. Appl. Math. 51 (2), 303-343 (1991).
  12. H. I. Lee and D. S. Stewart, “Calculation of Linear Detonation Instability: One-Dimensional Instability of Plane Detonation,” J. Fluid Mech. 216, 103-132 (1990).
  13. P. Clavin and L. He, “Stability and Nonlinear Dynamics of One-Dimensional Overdriven Detonations in Gases.’’ J. Fluid Mech. 306, 353-378 (1996).
  14. A. K. Henrick, T. D. Aslam, and J. M. Powers, “Simulations of Pulsating One-Dimensional Detonations with True Fifth Order Accuracy,” J. Comput. Phys. 213 (1), 311-329 (2006).
  15. J. J. Quirk, “Godunov-Type Schemes Applied to Detonation Flows,” in Combustion in High-Speed Flows (Springer, Dordrecht, 1994), Vol. 1, pp. 575-596.
  16. M. V. Papalexandris, A. Leonard, and P. E. Dimotakis, “Unsplit Algorithms for Multidimensional Systems of Hyperbolic Conservation Laws with Source Terms,” Comput. Math. Appl. 44 (1-2), 25-49 (2002).
  17. P. Hwang, R. P. Fedkiw, B. Merriman, et al., “Numerical Resolution of Pulsating Detonation Waves,” Combust. Theory Model. 4 (3), 217-240 (2000).
  18. L. K. Cole, A. R. Karagozian, and J.-L. Cambier, “Stability of Flame-Shock Coupling in Detonation Waves: 1D Dynamics,” Combust. Sci. Technol. 184 (10-11), 1502-1525 (2012).
  19. A. I. Lopato and P. S. Utkin, “Mathematical Modeling of Pulsating Detonation Wave Using ENO-Schemes of Different Approximation Orders,” Komput. Issled. Model. 6 (5), 643-653 (2014).
  20. A. I. Lopato and P. S. Utkin, “Two Approaches to the Mathematical Modeling of Detonation Waves,” Mat. Model. 28 (2), 133-145 (2016) [Math. Models Comput. Simul. 8 (5), 585-594 (2016)].
  21. V. M. Goloviznin and A. A. Samarskii, “Some Characteristics of Finite Difference Scheme ’Cabaret’,” Mat. Model. 10 (1), 101-116 (1998).
  22. V. M. Goloviznin and S. A. Karabasov, “Nonlinear Correction of Cabaret Scheme,” Mat. Model. 10 (12), 107-123 (1998).
  23. V. M. Goloviznin, S. A. Karabasov, and I. M. Kobrinskii, “Balance-Characteristic Schemes with Separated Conservative and Flux Variables,” Mat. Model. 15 (9), 29-48 (2003).
  24. V. M. Goloviznin, “Balanced Characteristic Method for 1D Systems of Hyperbolic Conservation Laws in Eulerian Representation,” Mat. Model. 18 (11), 14-30 (2006).
  25. S. A. Karabasov and V. M. Goloviznin, “Compact Accurately Boundary-Adjusting High-Resolution Technique for Fluid Dynamics,” J. Comput. Phys. 228 (19), 7426-7451 (2009).
  26. V. G. Kondakov, A Generalization of the ’Cabaret’ Scheme to Multidimensional Equations of Gas Dynamics , Candidate’s Dissertation in Mathematics and Physics (Moscow State Univ., Moscow, 2014).
  27. A. V. Danilin and A. V. Solovjev, “A Modification of the CABARET Scheme for the Computation of Multicomponent Gaseous Flows,” Vychisl. Metody Programm. 16, 18-25 (2015).
  28. A. V. Danilin, A. V. Solovjev, and A. M. Zaitsev, “A Modification of the CABARET Scheme for Numerical Simulation of Multicomponent Gaseous Flows in Two-Dimensional Domains,” Vychisl. Metody Programm. 16, 436-445 (2015).
  29. V. M. Goloviznin, A. V. Solovjev, and V. A. Isakov, “An Approximation Algorithm for the Treatment of Sound Points in the CABARET Scheme,” Vychisl. Metody Programm. 17, 166-176 (2016).

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