Hostname: page-component-7c8c6479df-hgkh8 Total loading time: 0 Render date: 2024-03-19T04:42:56.127Z Has data issue: false hasContentIssue false

Mean structure of one-dimensional unstable detonations with friction

Published online by Cambridge University Press:  06 March 2014

Aliou Sow
Affiliation:
CORIA-UMR 6614, Normandie University, CNRS-University & INSA of Rouen, 76800 Saint-Etienne du Rouvray, France
Ashwin Chinnayya
Affiliation:
CORIA-UMR 6614, Normandie University, CNRS-University & INSA of Rouen, 76800 Saint-Etienne du Rouvray, France Institut PPrime, CNRS UPR 3346, ENSMA & University of Poitiers, 86961 Futuroscope-Chasseneuil, France
Abdellah Hadjadj*
Affiliation:
CORIA-UMR 6614, Normandie University, CNRS-University & INSA of Rouen, 76800 Saint-Etienne du Rouvray, France
*
Email address for correspondence: hadjadj@coria.fr

Abstract

This investigation deals with the study of the mean structure of a mildly unstable non-ideal detonation wave. The analysis is based on the integration of one-dimensional reactive Euler equations with friction forces using a third-order Runge–Kutta scheme and a fifth-order weighted essentially non-oscillatory (WENO5) spatial discretization. A one-step Arrhenius reaction mechanism is used for modelling the chemical reaction. When the frictional forces are active, the limit cycle based on the post-shock pressure reveals an enhanced pulsating behaviour of the downstream subsonic reaction zone compared to the ideal case. The results show that the detonation-velocity deficit increases as the mean reaction zone becomes thicker compared to the generalized ZND model. A new master equation, based on the Favre-averaged quantities, is derived and analysed along with new sonicity and thermicity conditions. The analysis of the species, momentum and energy balances reveals that the presence of mechanical fluctuations within the reaction zone constitutes another source of energy withdrawal, meaning that the detonation deviates from its laminar structure. Furthermore, the compressibility of the flow is analysed and the relationships between the fluctuations of temperature, velocity and reactive scalar are discussed in terms of strong Reynolds analogies.

Type
Papers
Copyright
© 2014 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abderrahmane, H. A., Paquet, F. & Ng, H. D. 2011 Applying nonlinear dynamic theory to one-dimensional pulsating detonations. Combust. Theor. Model. 15 (2), 205225.Google Scholar
Anderson, D. A., Tannehill, J. C. & Pletcher, R. H. 1984 Computational Fluid Mechanics and Heat Transfer. Hemisphere.Google Scholar
Austin, J. M., Pintgen, F. & Shepherd, J. E. 2005 Reaction zones in highly unstable detonations. Proc. Combust. Inst. 32, 18491857.Google Scholar
Chaudhuri, A., Hadjadj, A., Chinnayya, A. & Palerm, S. 2011 Numerical study of compressible mixing layers using high-order weno schemes. J. Sci. Comput. 47 (2), 170197.Google Scholar
Chinnayya, A., Hadjadj, A. & Ngomo, D. 2013 Computational study of detonation wave propagation in narrow channels. Phys. Fluids 25, 036101.Google Scholar
Dionne, J.-P., Ng, H. D. & Lee, J. H. S. 2000 Transient development of friction-induced low-velocity detonations. Proc. Combust. Inst. 28, 645651.Google Scholar
Erpenbeck, J. J. 1964 Stability of idealized one-reaction detonations. Phys. Fluids 7, 684696.Google Scholar
Fay, J. A. 1959 Two-dimensional gaseous detonations: velocity deficit. Phys. Fluids 2 (3), 283289.Google Scholar
Fickett, W. & Davis, W. C. 1979 Detonation, Theory and Experiment. Dover.Google Scholar
Frolov, S. M. 1997 Zel’dovich theory of detonability limits. In Advances in Combustion Science: In Honor of Ya. B. Zel’dovich (ed. Sirignano, W. A., Merzhanov, A. G. & De Luca, L.), Progress in Astronautics and Aeronautics, vol. 173, pp. 341347. AIAA.Google Scholar
Gamezo, V. N., Desbordes, D. & Oran, E. S. 1999 Formation and evolution of two-dimensional cellular detonations. Combust. Flame 116 (1–3), 154165.Google Scholar
Henrick, A. K., Aslam, T. D. & Powers, J. M. 2005 Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys. 207, 542567.Google Scholar
Henrick, A. K., Aslam, T. D. & Powers, J. M. 2006 Simulations of pulsating one-dimensional detonations with true fifth order accuracy. J. Comput. Phys. 213, 311329.Google Scholar
Higgins, A. 2012 Steady one-dimensional detonations. In Shock Wave Science and Technology Reference Library, Vol. 6: Detonation Dynamics, pp. 33105. Springer.Google Scholar
Jiang, G.-S. & Shu, C.-W. 1996 Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126 (1), 202228.Google Scholar
Kasimov, A., Faria, L. & Rosales, R. R. 2013 A model for shock wave chaos. Phys. Rev. Lett. 110, 104104.CrossRefGoogle Scholar
Kasimov, A. R. & Stewart, D. S. 2004 On the dynamics of self-sustained one-dimensional detonations: a numerical study in the shock-attached frame. Phys. Fluids 16 (10), 35663578.Google Scholar
Lee, H. I. & Stewart, D. Scott 1990 Calculation of linear detonation instability: one-dimensional instability of plane detonation. J. Fluid Mech. 216, 103132.Google Scholar
Lee, J. H. S. 2008 The Detonation Phenomenon. Cambridge University Press.Google Scholar
Lee, J. H. S. & Radulescu, M. I. 2005 On the hydrodynamic thickness of cellular detonations. Combust. Explosions Shock Waves 41 (6), 745765.Google Scholar
Leung, C., Radulescu, M. I. & Sharpe, G. J. 2010 Characteristics analysis of the one-dimensional pulsating dynamics of chain-branching detonations. Phys. Fluids 22 (12), 126101.Google Scholar
Manson, N. & Guénoche, H.1957 Effect of charge diameter on the velocity of detonation waves in gas mixtures. In Proceedings of the 6th Symposium (International) on Combustion, pp. 631–639. Reinhold.Google Scholar
Massa, L., Chauhan, M. & Lu, F. K. 2011 Detonation-turbulence interaction. Combust. Flame 158, 17881806.Google Scholar
Mazaheri, K., Mahmoudi, Y. & Radulescu, M. I. 2012 Diffusion and hydrodynamic instabilities in gaseous detonations. Combust. Flame 159, 21382154.Google Scholar
Menikoff, R. & Shaw, Milton S. 2010 Reactive burn models and ignition & growth concept. In EPJ Web of Conferences, New Models and Hydrocodes for Shock Wave Processes in Condensed Matter (ed. Soulard, L.), vol. 10, p. 00003. EDP Sciences.Google Scholar
Ng, H. D., Higgins, A. J., Kiyanda, C. B., Radulescu, M. I., Lee, J. H. S., Bates, K. R. & Nikiforakis, N. 2005 Nonlinear dynamics and chaos analysis of one-dimensional pulsating detonations. Combust. Theor. Model. 9, 159170.CrossRefGoogle Scholar
Ngomo, D., Chaudhuri, A., Chinnayya, A. & Hadjadj, A. 2010 Numerical study of shock propagation and attenuation in narrow tubes including friction and heat losses. Comput. Fluids 39 (9), 17111721.Google Scholar
Nikolaev, Y. A. & Gaponov, O. A. 1995 On gas detonation limits. Combust. Explosions Shock Waves 31 (3), 395400.Google Scholar
Nikolaev, Y. A. & Zak, D. V. 1989 Quasi-one-dimensional model of self-sustaining multifront gas detonation with losses and turbulence taken into account. Combust. Explosions Shock Waves 25 (2), 225233.Google Scholar
Oran, E. S. & Gamezo, V. N. 2007 Origins of the deflagration-to-detonation transition in gas-phase combustion. Combust. Flame 148, 447.Google Scholar
Petitpas, F., Saurel, R., Franquet, E. & Chinnayya, A. 2009 Modelling detonation waves in condensed energetic materials: multiphase CJ conditions and multidimensional computations. Shock Waves 19 (5), 377401.Google Scholar
Radulescu, M. I. & Lee, J. H. S. 2002 The failure mechanism of gaseous detonations: experiments in porous wall tubes. Combust. Flame 131 (1–2), 2946.Google Scholar
Radulescu, M. I., Sharpe, G. J., Law, C. K. & Lee, J. H. S. 2007 The hydrodynamic structure of unstable cellular detonations. J. Fluid Mech. 580, 3181.Google Scholar
Romick, C. M., Aslam, T. D. & Powers, J. M.2011a The dynamics of unsteady detonation with diffusion. In AIAA Paper 2011-0799. Available at: http://www.nd.edu/ powers/paper.list.Google Scholar
Romick, C. M., Aslam, T. D. & Powers, J. M.2011b Verified calculation of non-linear dynamics of viscous detonation. In 23rd ICDERS International Colloquium on the Dynamics of Explosions and Reactive Systems, Irvine, California.Google Scholar
Romick, C. M., Aslam, T. D. & Powers, J. M. 2012 The effect of diffusion on the dynamics of unsteady detonations. J. Fluid Mech. 699, 453464.Google Scholar
Saurel, R., Chinnayya, A. & Renaud, F. 2003 Thermodynamic analysis and numerical resolution of turbulent–fully ionized plasma flow model. Shock Waves 13 (4), 283297.Google Scholar
Sharpe, G. J. & Falle, S. A. E. G. 2000 Numerical simulations of pulsating detonations: I. Nonlinear stability of steady detonations. Combust. Theory Model. 4, 557574.Google Scholar
Shepherd, J. E. 2009 Detonation in gases. Proc. Combust. Inst. 32, 8398.Google Scholar
Smits, A. J. & Dussauge, J.-P. 2006 Turbulent Shear Layers in Supersonic Flow. Springer.Google Scholar
Watt, S. D. & Sharpe, G. J. 2004 One-dimensional linear stability of curved detonations. Proc. R. Soc. Lond. A 460 (2049), 25512568.Google Scholar
White, D. R. 1961 Turbulent structure of gaseous detonation. Phys. Fluids 4 (4), 465480.Google Scholar
Xu, S., Aslam, T. & Stewart, D. S. 1997 High resolution numerical simulation of ideal and non-ideal compressible reacting flows with embedded internal boundaries. Combust. Theory Model. 1 (1), 113142.CrossRefGoogle Scholar
Zel’dovich, Y. B. 1940 On the theory of detonation propagation in gaseous systems. Zh. Eksp. Teor. Fiz. 10, 542 (translated in NACA TM-1261, 1950).Google Scholar
Zel’dovich, Y. B. & Kompaneets, A. S. 1960 Theory of Detonation. Academic.Google Scholar
Zhang, F., Chue, R. S., Frost, D. L., Lee, J. H. S., Thibault, P. & Yee, C. 1995 Effects of area change and friction on detonation stability in supersonic ducts. Proc. R. Soc. Lond. A 449, 3149.Google Scholar
Zhang, F. & Lee, J. H. S. 1994 Friction-induced oscillatory behaviour of one-dimensional detonations. Proc. R. Soc. Lond. A 446, 87105.Google Scholar