Abstract
The paper is devoted to the numerical investigation of the stability of propagation of pulsating gas detonation waves. For various values of the mixture activation energy, detailed propagation patterns of the stable, weakly unstable, irregular, and strongly unstable detonation are obtained. The mathematical model is based on the Euler system of equations and the one-stage model of chemical reaction kinetics. The distinctive feature of the paper is the use of a specially developed computational algorithm of the second approximation order for simulating detonation wave in the shock-attached frame. In distinction from shock capturing schemes, the statement used in the paper is free of computational artifacts caused by the numerical smearing of the leading wave front. The key point of the computational algorithm is the solution of the equation for the evolution of the leading wave velocity using the second-order grid-characteristic method. The regimes of the pulsating detonation wave propagation thus obtained qualitatively match the computational data obtained in other studies and their numerical quality is superior when compared with known analytical solutions due to the use of a highly accurate computational algorithm.
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References
A. A. Vasiliev, V. V. Mitrofanov, and M. E. Topchiyan, “Detonation waves in gases”, Combust. Expl. Shock Waves, 23, 605–623 (1987).
V. A. Levin, V. V. Markov, T. A. Zhuravskaya, and S. F. Osinkin, “Nonlinear wave processes that occur during the initiation and propagation of gaseous detonation,” Proc. Steklov Inst. Math. 251, 192–205 (2005).
L. K. Cole, A. R. Karagozian, and J.-L. Cambier, “Stability of flame-shock coupling in detonation waves: ID dynamics,” Combustion Sci. Technol. 184, 1502–1525 (2012).
L. M. Faria, A. R. Kasimov, and R. R. Rosales, “Study of a model equation in detonation theory,” SIAM J. Appl. Math. 74 (2), 547–570 (2014).
A. I. Lopato and P. S. Utkin, “Pulsating detonation wave investigation using shock-capturing methods and calculations in shock-attached frame” Gorenie vzryv 8 (1), 145–150 (2015).
L. I. Sedov, V. P. Korobeinikov, and V. V. Markov, “The theory of propagation of blast waves”, Proc. Steklov Inst. Math. 2, 187–228 (1988).
L. He and J. H. S. Lee, “The dynamical limit of one-dimensional detonations,” Phys. Fluids 7, 1151–1158 (1995).
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 (2), 311–329 (2006).
A. R. Kasimov and D. S. Stewart, “On the dynamics of the self-sustained one-dimensional detonations: A numerical study in the shock-attached frame,” Phys. Fluids 16 (10), 3566–3578 (2004).
T. D. Aslam and J. M. Powers, “The dynamics of unsteady detonation in ozone,” Proc. of the 47th AIAA Aerospace Sci. Meeting and Exhibition, Orlando, Florida, 2009, Paper No. 2009-0632.
C. M. Romick, T. D. Aslam, and J. M. Powers, “The dynamics of unsteady detonation with diffusion,” Proc. of the 49th AIAA Aerospace Sci. Meeting including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, 2011, Paper No. 2011-799.
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).
Y. Daimon and A. Matsuo, “Detailed features of one-dimensional detonations,” Phys. Fluids 15 (1), 112–122 (2003).
J. H. S. Lee, The Detonation Phenomenon (Cambridge University Press, 2008).
A. S. Kholodov, “Numerical methods for the solution of hyperbolic equations and systems of equations,” in Encyclopedia of Low-Temperature Plasma, Part 2 (Nauka, Moscow, 2008), pp. 220–235 [in Russian].
A. G. Kulikovskii, N. G. Pogorelov, and A. Yu. Semenov, Mathematical Problems of Numerical Solution of Hyperbolic Systems of Equations (Fizmatlit, Moscow, 2001) [in Russian].
C.-W. Shu, “Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws,” ICASE Report No 97-65, NASA/CR-97-206253, 1997.
C.-W. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes,” J. Comput. Phys. 77, 439–471 (1988).
I. V. Semenov, P. S. Utkin, I. F. Akhmed’yanov, and I. S. Men’shov, “Application of high performance computing to the solution of interior ballistics problems,” Vychisl. Metody Programm. 12, 183–193 (2011).
A. I. Lopato and P. S. Utkin, “Mathematical modeling of pulsating detonation wave using ENO-schemes of different approximation orders,” Komp’yut. Issledovaniya Modelir. 6 (5), 643–653 (2014).
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Original Russian Text © A.I. Lopato, P.S. Utkin, 2016, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2016, Vol. 56, No. 5, pp. 856–868.
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Lopato, A.I., Utkin, P.S. Detailed simulation of the pulsating detonation wave in the shock-attached frame. Comput. Math. and Math. Phys. 56, 841–853 (2016). https://doi.org/10.1134/S0965542516050134
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DOI: https://doi.org/10.1134/S0965542516050134