Details

Fiammetta Conforto^[a] and Giorgio Martalò^[b]

Detonation waves in multi-temperature reacting gas mixtures

Pages: 107-142
Received: 24 August 2023
Accepted: 19 October 2023
Mathematics Subject Classification: 35L67, 35Q31, 80A17, 80A32.
Keywords: Reacting gas mixtures, hyperbolic equations, detonation wave, singularity analysis.
Authors address:
[a]: University of Messina, Department of Chemical, Biological, Pharmaceutical and Environmental Sciences, V.le F. Stagno d'Alcontres 31, Messina, 98166, Italy
[b]: University of Parma, Department of Mathematical, Physical and Computer Sciences, Parco Area delle Scienze 53/A, Parma, 43124, Italy

Abstract: The 1-D steady detonation problem is studied for a mixture of four gases, undergoing a reversible bimolecular reaction. The multi-velocity and multi-temperature governing system provides an Euler level description, which is derived as a principal subsystem of the hyperbolic balance laws obtained from the 13-moment Grad approximation of the reactive Boltzmann equations. Suitable jump conditions for the onset of detonation processes are analysed and the occurrence of compatible stable sub-shocks along the travelling wave solution in the transient is detected. The results confirm the clear drawbacks in dealing with multi-component flows and with implicit constraints induced by chemical equilibrium conditions. The paper aims to highlight the key points to be sorted out to ensure the well-posedness of the detonation problem and to manage sub-shock occurrence.

References

[1]: V. Artale, F. Conforto, G. Martalò and A. Ricciardello, Shock structure and multiple sub-shocks in Grad 10-moment binary mixtures of monoatomic gases, Ricerche Mat. 68 (2019), no. 2, 485-502. DOI
[2]: V. Artale, F. Conforto, G. Martalò and A. Ricciardello, Shock structure solutions of Grad 13-moment equations for binary gas mixtures, Wave Motion 115 (2022), 103055. DOI
[3]: M. Bisi, F. Conforto and G. Martalò, Sub-shock formation in Grad 10-moment equations for a binary gas mixture, Continuum Mech. Thermodyn. 28 (2016), no. 5, 1295-1324. DOI
[4]: M. Bisi, F. Conforto and G. Martalò, Sub-shock formation in reacting gas mixtures, In: "From Particle Systems to Partial Differential Equations III'', P. Gonçalves and A. J. Soares, eds., Springer Proceedings in Mathematics & Statistics, 162, Springer Cham, 2016, 51-72. DOI
[5]: M. Bisi, F. Conforto R. Monaco and A. Ricciardello, On the steady deflagration process for a gas mixture undergoing irreversible reactions, Ricerche Mat. 68 (2019), no. 1, 13-35. DOI
[6]: M. Bisi, M. Groppi and G. Spiga, Grad's distribution functions in the kinetic equations for a chemical reaction, Continuum Mech. Thermodyn. 14 (2002), no. 2, 207-222. DOI
[7]: M. Bisi, M. Groppi and G. Spiga, Flame structure from a kinetic model for chemical reactions, Kinetic and Related Models 3 (2010), 17-34. DOI
[8]: M. Bisi, G. Martalò and G. Spiga, Shock wave structure of multi-temperature Euler equations from kinetic theory for a binary mixture, Acta Appl. Math. 132 (2014), 95-105. DOI
[9]: G. Boillat and T. Ruggeri, Hyperbolic principal subsystems: entropy convexity and subcharacteristic conditions, Arch. Rational Mech. Anal. 137 (1997), 305-320. DOI
[10]: G. Boillat and T. Ruggeri, On the shock structure problem for hyperbolic system of balance laws and convex entropy, Continuum Mech. Thermodyn. 10 (1998), no. 5, 285-292. DOI
[11]: F. Conforto, M. Groppi, R. Monaco and G. Spiga, Steady detonation waves for gases undergoing dissociation/recombination and bimolecular reactions, Continuum Mech. Thermodyn. 16 (2004), no.1-2, 149-161. DOI
[12]: F. Conforto, M. Groppi, R. Monaco and G. Spiga, Steady detonation problem for slow and fast chemical reactions, In: "Modelling and Numerics of Kinetic Dissipative Systems'', L. Pareschi, G. Russo and G. Toscani, eds., Nova Science, Hauppauge, NY, 2006, 105-117. MR
[13]: F. Conforto, M. Groppi, R. Monaco and G. Spiga, Kinetic approach to deflagration processes in a recombination reaction, Kinetic and Related Models 4 (2011), no. 1, 259-276. DOI
[14]: F. Conforto, M. Groppi, R. Monaco and G. Spiga, Steady Combustion Waves Driven by a Recombination Reaction in a Gas Mixture, Acta Appl. Math. 122 (2012), no. 1, 127-140. DOI
[15]: F. Conforto, A. Mentrelli and T. Ruggeri, Shock structure and multiple sub-shocks in binary mixtures of Eulerian fluids, Ricerche Mat. 66 (2017), no. 1, 221-231. DOI
[16]: F. Conforto, R. Monaco and A. Ricciardello, Discontinuous Shock Structure in a Reacting Mixture Modelled by Grad 13 Moment Approximation, Acta Appl. Math. 132 (2014), 225-236. DOI
[17]: F. Conforto, R. Monaco and A. Ricciardello, Analysis of steady combustion processes in a recombination reaction, Continuum Mech. Thermodyn. 26 (2014), 503-519. DOI
[18]: F. Conforto, R. Monaco, F. Schürrer and I. Ziegler, Detonation wave structure arising from the kinetic theory of reacting gases, In: "WASCOM 2001 - 11th Conference on Waves and Stability in Continuous Media'', R. Monaco, M. Pandolfi Bianchi and S. Rionero, eds., World Scientific, River Edge, NJ, 2002, 161-168. MR
[19]: F. Conforto, R. Monaco, F. Schürrer and I. Ziegler, Steady detonation waves via the Boltzmann equation for a reacting mixture, J. Phys. A: Math. Gen. 36 (2003), 5381-5398. DOI
[20]: W. Fickett and W. C. Davis, Detonation, University of California Press, Berkeley, 1979.
[21]: I. Glassman, Combustion, Academic Press, San Diego, 1996. DOI
[22]: M. Groppi, A. Rossani and G. Spiga, Kinetic theory of a diatomic gas with reactions of dissociation and recombination through a transition state, J. Phys. A: Math. Gen. 33 (2000), 8819-8833. DOI
[23]: K. K. Kuo, Principles of combustion, John Wiley, New York, 1986. Article (2nd ed.)
[24]: A. Rossani and G. Spiga, A note on the kinetic theory of chemically reacting gases, Phys. A: Stat. Mech. Appl. 272 (1999), no.3-4, 563-573. DOI
[25]: T. Ruggeri and S. Taniguchi, A complete classification of sub-shocks in the shock structure of a binary mixture of Eulerian gases with different degrees of freedom, Phys. Fluids 34 (2022), no. 6, 066116. DOI