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The effect of diffusion on the dynamics of unsteady detonations

  • C. M. Romick (a1), T. D. Aslam (a2) and J. M. Powers (a1)


The dynamics of a one-dimensional detonation predicted by a one-step irreversible Arrhenius kinetic model are investigated in the presence of mass, momentum and energy diffusion. A study is performed in which the activation energy is varied and the length scales of diffusion and reaction are held constant. As the activation energy is increased, the system goes through a series of period-doubling events and eventually undergoes a transition to chaos. The rate at which these bifurcation points converge is calculated and shown to be in agreement with the Feigenbaum constant. Within the chaotic regime, there exist regions in which there are limit cycles consisting of a small number of oscillatory modes. When an appropriately fine grid is used to capture mass, momentum and energy diffusion, predictions are independent of the differencing scheme. Diffusion affects the behaviour of the system by delaying the onset of instability and strongly influencing the dynamics in the unstable regime. The use of the reactive Euler equations to predict detonation dynamics in the unstable and marginally stable regimes is called into question as the selected reactive and diffusive length scales are representative of real physical systems; reactive Navier–Stokes is a more appropriate model in such regimes.


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1. Al-Khateeb, A. N., Powers, J. M. & Paolucci, S. 2010 On the necessary grid resolution for verified calculation of premixed laminar flames. Commun. Comput. Phys. 8 (2), 304326.
2. Bourlioux, A., Majda, A. J. & Roytburd, V. 1991 Theoretical and numerical structure for unstable one-dimensional detonations. SIAM J. Appl. Maths 51 (2), 303343.
3. Clarke, J. F., Kassoy, D. R., Meharzi, N. E., Riley, N. & Vasantha, R. 1990 On the evolution of plane detonations. Proc. R. Soc. Lond. Ser. A. Math. Phys. Eng. Sci. 429 (1877), 259283.
4. Clarke, J. F., Kassoy, D. R. & Riley, N. 1986 On the direct initiation of a plane detonation wave. Proc. R. Soc. Lond. Ser. A. Math. Phys. Eng. Sci. 408 (1834), 129148.
5. Deiterding, R. 2009 A parallel adaptive method for simulating shock-induced combustion with detailed chemical kinetics in complex domains. Comput. Struct. 87, 769783.
6. Erpenbeck, J. J. 1964 Stability of idealized one-reaction detonations. Phys. Fluids 7 (5), 684696.
7. Fedkiw, R. P., Merriman, B. & Osher, S. 1997 High accuracy numerical methods for thermally perfect gas flows with chemistry. J. Comput. Phys. 132 (2), 175190.
8. Feigenbaum, M. J. 1979 The universal metric properties of nonlinear transformations. J. Stat. Phys. 21 (6), 669706.
9. Fickett, W. & Davis, W. C. 1979 Detonation. University of California Press.
10. Gasser, I. & Szmolyan, P. 1993 A geometric singular perturbation analysis of detonation and deflagration waves. SIAM J. Math. Anal. 24 (4), 968986.
11. He, X. & Karagozian, A. R. 2006 Pulse-detonation-engine simulations with alternative geometries and reaction kinetics. J. Propul. Power 22 (4), 852861.
12. 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 (1), 311329.
13. Hirschfelder, J. O. & Curtiss, C. F. 1958 Theory of detonations. I. Irreversible unimolecular reaction. J. Chem. Phys. 28 (6), 11301147.
14. Hu, X. Y., Khoo, B. C., Zhang, D. L. & Jiang, Z. L. 2004 The cellular structure of a two-dimensional detonation wave. Combust. Theor. Model. 8 (2), 339359.
15. 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.
16. Lee, H. I. & Stewart, D. S. 1990 Calculation of linear detonation instability: one-dimensional instability of planar detonations. J. Fluid Mech. 216, 103132.
17. Lyng, G. & Zumbrun, K. 2004 One-dimensional stability of viscous strong detonation waves. Arch. Rat. Mech. Anal. 173, 213277.
18. May, R. M. 1976 Simple mathematical models with very complicated dynamics. Nature 261, 459467.
19. 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 (1), 159170.
20. Oran, E. S., Weber, J. W., Stefaniw, E. I., Lefebvre, M. H. & Anderson, J. D. 1998 A numerical study of a two-dimensional detonation using a detailed chemical reaction model. Combust. Flame 113 (1–2), 147163.
21. Powers, J. M. 2006 Review of multiscale modelling of detonation. J. Propul. Power 22 (6), 12171229.
22. Powers, J. M. & Paolucci, S. 2005 Accurate spatial resolution estimates for reactive supersonic flow with detailed chemistry. AIAA J. 43 (5), 10881099.
23. Quirk, J. J. 1994 A contribution to the great Riemann solver debate. Intl J. Numer. Meth. Fluids 18, 555574.
24. 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.
25. Roache, P. J. 2002 Code verification by the method of manufactured solutions. Trans. ASME: J. Fluids Engng 124 (1), 410.
26. Seitenzahl, I. R., Meakin, C. A., Townsley, D. M., Lamb, D. Q. & Truran, J. W. 2009 Spontaneous initiation of detonations in white dwarf environments: determination of critical sizes. Astrophys. J. 696, 515527.
27. Sharpe, G. J. 1997 Linear stability of idealized detonations. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 453 (1967), 26032625.
28. Sharpe, G. J. & Falle, S. A. E. G. 2000 Numerical simulations of pulsating detonations: I. Nonlinear stability of steady detonations. Combust. Theor. Model. 4, 557574.
29. Shepherd, J. E. 2009 Detonation in gases. Proc. Combust. Inst. 32, 8398.
30. Singh, S., Rastigejev, Y., Paolucci, S. & Powers, J. M. 2001 Viscous detonation in using intrinsic low-dimensional manifolds and wavelet adaptive multilevel representation. Combust. Theor. Model. 5 (2), 163184.
31. Texier, B. & Zumbrun, K. 2011 Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions. Commun. Math. Phys. 302, 151.
32. Tsuboi, N., Eto, K. & Hayashi, A. K. 2007 Detailed structure of spinning detonation in a circular tube. Combust. Flame 149 (1–2), 144161.
33. Walter, M. A. T. & da Silva, L. F. F. 2006 Numerical study of detonation stabilization by finite length wedges. AIAA J. 44 (2), 353361.
34. Wang, B., He, H. & Yu, S. T. J. 2005 Direct calculation of wave implosion for detonation initiation. AIAA J. 43 (10), 21572169.
35. Watt, S. D. & Sharpe, G. J. 2005 Linear and nonlinear dynamics of cylindrically and spherically expanding detonation waves. J. Fluid Mech. 522, 329356.
36. Wood, W. W. 1963 Existence of detonations for large values of the rate parameter. Phys. Fluids 6 (8), 10811090.
37. Xu, S., Aslam, T. D. & Stewart, D. S. 1997 High resolution numerical simulation of ideal and non-ideal compressible reacting flows with embedded internal boundaries. Combust. Theor. Model. 1, 113142.
38. Ziegler, J. L., Deiterding, R., Shepherd, J. E. & Pullin, D. I. 2011 An adaptive high-order hybrid scheme for compressive, viscous flows with detailed chemistry. J. Comput. Phys. 230 (20), 75987630.
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