Skip to main content Accessibility help

Dynamics of detonations with a constant mean flow divergence

  • Matei I. Radulescu (a1) and Bijan Borzou (a1)


An exponential horn geometry is introduced in order to establish cellular detonations with a constant mean lateral mass divergence, propagating at quasi-steady speeds below the Chapman–Jouguet value. The experiments were conducted in $2\text{C}_{2}\text{H}_{2}+5\text{O}_{2}+21\text{Ar}$ and $\text{C}_{3}\text{H}_{8}+5\text{O}_{2}$ . Numerical simulations were also performed for weakly unstable cellular detonations to test the validity of the exponential horn geometry. The experiments and simulations demonstrated that such quasi-steady state detonations can be realized, hence permitting us to obtain the relations between the detonation speed and mean lateral flow divergence for cellular detonations in an unambiguous manner. The experimentally obtained speed ( $D$ ) dependencies on divergence ( $K$ ) were compared with the predictions for steady detonations with lateral flow divergence obtained with the real thermo-chemical data of the mixtures. For the $2\text{C}_{2}\text{H}_{2}+5\text{O}_{2}+21\text{Ar}$ system, reasonable agreement was found between the experiments and steady wave prediction, particularly for the critical divergence leading to failure. Observations of the reaction zone structure in these detonations indicated that all the gas reacted very close to the front, as the transverse waves were reactive. The experiments obtained in the much more unstable detonations in $\text{C}_{3}\text{H}_{8}+5\text{O}_{2}$ showed significant differences between the experimentally derived $D(K)$ curve and the prediction of steady wave propagation. The latter was found to significantly under-predict the detonability of cellular detonations. The transverse waves in this mixture were found to be non-reactive, hence permitting the shedding of non-reacted pockets, which burn via turbulent flames on their surface. It is believed that the large differences between experiment and the inviscid model in this class of cellular structures is due to the importance of diffusive processes in the burn-out of the non-reacted pockets. The empirical tuning of a global one-step chemical model to describe the macro-scale kinetics in cellular detonations revealed that the effective activation energy was lower by 14 % in $2\text{C}_{2}\text{H}_{2}+5\text{O}_{2}+21\text{Ar}$ and 54 % in the more unstable $\text{C}_{3}\text{H}_{8}+5\text{O}_{2}$ system. This confirms previous observations that diffusive processes in highly unstable detonations are responsible for reducing the thermal ignition character of the gases processed by the detonation front.


Corresponding author

Email address for correspondence:


Hide All
Austin, J.2003 The role of instability in gaseous detonation. PhD thesis, California Institute of Technology, Pasadena, California.
Bdzil, J. B. & Stewart, D. S. 2007 The dynamics of detonation in explosive systems. Annu. Rev. Fluid Mech. 39, 263292.
Bhattacharjee, R.2013 Experimental investigation of detonation re-initiation mechanisms following a Mach reflection of a quenched detonation. MSc thesis, Department of Mechanical Engineering, University of Ottawa, Ottawa, ON.
Blanquart, G.2018 CaltechMech. California institute of technology (
Borzou, B.2016 The influence of cellular structure on the dynamics of detonations with constant mass divergence. PhD thesis, University of Ottawa, Ottawa, ON.
Camargo, A., Ng, H. D., Chao, J. & Lee, J. H. S. 2010 Propagation of near-limit gaseous detonations in small diameter tubes. Shock Waves 20, 499508.
Chao, J., Ng, H. D. & Lee, J. H. S. 2009 Detonability limits in thin annular channels. Proc. Combust. Inst. 32, 23492354.
Chinnayya, A., Hadjadj, A. & Ngomo, D. 2013 Computational study of detonation wave propagation in narrow channels. Phys. Fluids 25, 036101.
Dennis, K., Maley, L., Liang, Z. & Radulescu, M. I. 2014 Implementation of large scale shadowgraphy in hydrogen explosion phenomena. Intl J. Hydrog. Energy 39 (21), 1134611353.
Dupré, G., Péraldi, O., Joannon, J., Lee, J. H. S. & Knystautas, R. 1991 Limit criterion of detonation in circular tubes. In Dynamics of Detonations and Explosions: Detonations (ed. Kuhl, A. et al. ), Progress in Astronautics and Aeronautics, vol. 133, pp. 156169. American Institute of Aeronautics and Astronautics.
Falle, S. A. E. G. & Giddings, J. R. 1993 Body capturing using adaptive Cartesian grids. In Numerical Methods for Fluid Dynamics IV, pp. 337343. Oxford University Press.
Falle, S. A. E. G. 1991 Self-similar jets. Mon. Not. R. Astron. Soc. 250, 581596.
Fay, J. A. 1959 Two-dimensional gaseous detonations:velocity deficit. Phys. Fluids 2, 283289.
Fickett, W. & Davis, W. C. 1979 Detonation Theory and Experiment. Dover.
Gamezo, V. N., Desbordes, D. & Oran, E. S. 1999 Formation and evolution of two-dimenional cellular detonations. Combust. Flame 116, 154165.
Gamezo, V. N., Vasil’ev, A. A., Khokhlov, A. M. & Oran, E. S. 2000 Fine cellular structures produced by marginal detonations. Proc. Combust. Inst. 28, 611617.
Gao, Y., Zhang, B., Ng, H. D. & Lee, J. H. S. 2016 An experimental investigation of detonation limits in hydrogen–oxygen–argon mixtures. Intl J. Hydrog. Energy 41 (14), 60766083.
Goodwin, D. G., Moffat, H. K. & Speth, R. L.2017 Cantera: an object-oriented software toolkit for chemical kinetics, thermodynamics, and transport processes., version 2.3.0.
Han, W. H., Huang, J., Du, N., Liu, Z. G., Kong, W. J. & Wang, C. 2017a Effect of cellular instability on the initiation of cylindrical detonations. Chin. Phys. Lett. 34 (5), 054701.
Han, W. H., Kong, W. J., Gao, Y. & Law, C. K. 2017b The role of global curvature on the structure and propagation of weakly unstable cylindrical detonations. J. Fluid Mech. 813 (5), 458481.
He, L. T. & Clavin, P. 1994 On the direct initiation of gaseous detonations by an energy source. J. Fluid Mech. 277, 227284.
Kao, S.2008 Detonation stability with reversible kinetics. PhD thesis, California Institute of Technology, Pasadena, California.
Kao, S. & Shepherd, J. E.2008 Numerical solution methods for control volume explosions and ZND detonation structure. GALCIT Rep. FM2006.007. California Institute of Technology, Pasadena, California.
Kasimov, A. R. & Stewart, D. S. 2005 Asymptotic theory of evolution and failure of self-sustained detonations. J. Fluid Mech. 525, 161192.
Klein, R., Krok, J. C. & Shepherd, J. E.1995 Curved quasi-steady detonations: asymptotic analysis and detailed chemical kinetics. GALCIT Rep. FM 95-04. California Institute of Technology, Pasadena, California.
Lee, J. H. S. 2008 The Detonation Phenomenon. Cambridge University Press.
Maxwell, B., Bhattacharjee, R., Lau-Chapdelaine, S., Falle, S., Sharpe, G. J. & Radulescu, M. I. 2017 Influence of turbulent fluctuations on detonation propagation. J. Fluid Mech. 818, 646696.
Mazaheri, K., Mahmoudi, Y., Sabzpooshani, M. & Radulescu, M. I. 2015 Experimental and numerical investigation of propagation mechanism of gaseous detonations in channels with porous walls. Combust. Flame 162, 26382659.
McBride, B. J. & Gordon, S.1996 Computer program for calculation of complex chemical equilibrium compositions. Tech. Rep. E-8017-1, National Aeronautics and Space Administration report, Washington DC.
Moen, I. O., Donato, M., Knystautas, R. & Lee, J. H. S. 1981 The influence of confinement on the propagation of detonations near the detonability limits. Proc. Combust. Inst. 18, 16151622.
Radulescu, M. I.2003 The propagation and failure mechanisms of gaseous detonations: experiments in porous-walled tubes. PhD thesis, Department of Mechanical Engineering, McGill University, Montreal, QC.
Radulescu, M. I. 2017 The usefulness of a 1D hydrodynamic model for the detonation structure for predicting detonation dynamic parameters. Extended abstract 1140. In Proceedings of the 26th International Colloquium on the Dynamics of Explosions and Reactive Systems, Boston, MA. Institute for Dynamics of Explosions and Reactive Systems.
Radulescu, M. I. & Lee, J. H. S. 2002 The failure mechanism of gaseous detonations: experiments in porous wall tubes. Combust. Flame 131, 2946.
Radulescu, M. I., Sharpe, G. J. & Law, C. K. 2007a Effect of cellular instabilities on the blast initiation of weakly unstable detonations. In Proceedings of the 21st International Colloquium on the Dynamics of Explosions and Reactive Systems, Poitiers, France. Institute for Dynamics of Explosions and Reactive Systems.
Radulescu, M. I., Sharpe, G. J., Law, C. K. & Lee, J. H. S. 2007b The hydrodynamic structure of unstable cellular detonations. J. Fluid Mech. 580, 3181.
Reynaud, M., Virot, F. & Chinnayya, A. 2017 A computational study of the interaction of gaseous detonations with a compressible layer. Phys. Fluids 29 (5), 056101.
Richtmyer, R. D. & Morton, K. W. 1967 Difference Methods for Initial-Value Problems. Interscience Publishers.
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.
Sharpe, G. J. & Quirk, J. J. 2008 Nonlinear cellular dynamics of the idealized detonation model: regular cells. Combust. Theor. Model. 12 (1), 121.
Shepherd, J. E. 2009 Detonation in gases. Proc. Combust. Inst. 32, 8398.
Short, M. & Bdzil, J. B. 2003 Propagation laws for steady curved detonations with chain-branching kinetics. J. Fluid Mech. 479, 3964.
Subbotin, V. A. 1976 Two kinds of transverse wave structures in multifront detonation. Combust. Explos. Shock Waves 11, 8388.
Van Leer, B. 1977 Towards the ultimate conservative difference scheme. Part III. Upstream-centered finite-difference schemes for ideal compressible flow. J. Comput. Phys. 23, 263275.
Wang, H., You, X., Joshi, A. V., Davis, S. G., Laskin, A., Egolfopoulos, F. & Law, C. K.2007 USC Mech version II: high-temperature combustion reation model of H2/CO/C1-C4 compounds (
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
Williams, F. A. 1985 Combustion Theory, 2nd edn. Benjamin/Cummings.
Williams, F. A.2014 Chemical-Kinetic Mechanisms for Combustion Applications. San Diego Mechanism web page, Mechanical and Aerospace Engineering (Combustion Research), University of California at San Diego (
Wood, W. W. & Kirkwood, J. G. 1954 Diameter effect in condensed explosives: the relation between velocity and radius of curvature of the detonation wave. J. Chem. Phys. 22 (11), 19201924.
Wood, W. W. & Kirkwood, J. G. 1957 Hydrodynamics of a reacting and relaxing fluid. J. Appl. Phys. 28, 395398.
Yao, J. & Stewart, D. S. 1995 On the normal detonation shock velocity–curvature relationship for materials with large activation energy. Combust. Flame 100, 519528.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Dynamics of detonations with a constant mean flow divergence

  • Matei I. Radulescu (a1) and Bijan Borzou (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed