Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-27T02:20:56.102Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of detonations for an idealized condensed-phase model

Published online by Cambridge University Press:  08 January 2008

M. SHORT ,
I. I. ANGUELOVA ,
T. D. ASLAM ,
J. B. BDZIL ,
A. K. HENRICK  and
G. J. SHARPE
M. SHORT
Affiliation:
Theoretical and Applied Mechanics, University of Illinois, Urbana, IL 61801, USA
I. I. ANGUELOVA
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA
T. D. ASLAM
Affiliation:
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
J. B. BDZIL
Affiliation:
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
A. K. HENRICK
Affiliation:
Los Alamos National Laboratory, Los Alamos, NM 87545, USA Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
G. J. SHARPE
Affiliation:
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of travelling wave Chapman–Jouguet and moderately overdriven detonations of Zeldovich–von Neumann–Döring type is formulated for a general system that incorporates the idealized gas and condensed-phase (liquid or solid) detonation models. The general model consists of a two-component mixture with a one-step irreversible reaction between reactant and product. The reaction rate has both temperature and pressure sensitivities and has a variable reaction order. The idealized condensed-phase model assumes a pressure-sensitive reaction rate, a constant-γ caloric equation of state for an ideal fluid, with the isentropic derivative γ=3, and invokes the strong shock limit. A linear stability analysis of the steady, planar, ZND detonation wave for the general model is conducted using a normal-mode approach. An asymptotic analysis of the eigenmode structure at the end of the reaction zone is conducted, and spatial boundedness (closure) conditions formally derived, whose precise form depends on the magnitude of the detonation overdrive and reaction order. A scaling analysis of the transonic flow region for Chapman–Jouguet detonations is also studied to illustrate the validity of the linearization for Chapman–Jouguet detonations. Neutral stability boundaries are calculated for the idealized condensed-phase model for one- and two-dimensional perturbations. Comparisons of the growth rates and frequencies predicted by the normal-mode analysis for an unstable detonation are made with a numerical solution of the reactive Euler equations. The numerical calculations are conducted using a new, high-order algorithm that employs a shock-fitting strategy, an approach that has significant advantages over standard shock-capturing methods for calculating unstable detonations. For the idealized condensed-phase model, nonlinear numerical solutions are also obtained to study the long-time behaviour of one- and two-dimensional unstable Chapman–Jouguet ZND waves.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 595 , 25 January 2008 , pp. 45 - 82
Copyright
Copyright © Cambridge University Press 2008

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

REFERENCES

Bdzil, J. B., Aslam, T. D., Henninger, R. & Quirk, J. J. 2003 High-explosive performance: Understanding the effects of finite-length reaction zone. Los Alamos Science 28, 96–110.Google Scholar
Bdzil, J. B., Short, M., Sharpe, G. J., Aslam, T. D. & Quirk, J. J. 2006 Higher-order DSD for detonation propagation: DSD for detonation driven by multi-step chemistry models with disparate rates. In Proc. Thirteenth Symposium (Intl) on Detonation. Office of Naval Research Rep. ONR 351-07-01, pp. 726–736.Google Scholar
Bourlioux, A. & Majda, A. J. 1992 Theoretical and numerical structure for unstable two-dimensional detonations. Combust. Flame 90, 211–229.Google Scholar
Bourlioux, A., Majda, A. J. & Roytburd, V. 1991 a Theoretical and numerical structure for unstable one-dimensional detonations. SIAM J. Appl. Maths 51, 303–343.CrossRefGoogle Scholar
Bourlioux, A., Majda, A. J. & Roytburd, V. 1991 b Nonlinear development of low frequency one-dimensional instabilities for reacting shock waves. In Dynamical Issues in Combustion Theory (ed. Fife, P. C., Linan, A. & Williams, F. A.). IMA Volumes in Mathematics and its Applications, vol. 35, pp. 63–82.Google Scholar
Clavin, P. & He, L. T. 1996 Stability and nonlinear dynamics of one-dimensional overdriven detonations in gases. J. Fluid Mech. 306, 353–378.CrossRefGoogle Scholar
Clavin, P., He, L. T., & Williams, F. A. 1997 Multidimensional stability analysis of overdriven gaseous detonations. Phys. Fluids 9, 3764–3785.CrossRefGoogle Scholar
Clavin, P. & Williams, F. A. 2002 Dynamics of planar gaseous detonations near Chapman–Jouguet conditions for small heat release. Combust. Theory Model. 6, 127–139.Google Scholar
Daou, R. & Clavin, P. 2003 Instability threshold of gaseous detonations. J. Fluid Mech. 482, 181–206.Google Scholar
Davis, W. C. 1981 Fine structure in nitromethane/acetone detonations. In Proc. Seventh Symposium (Intl) on Detonation, pp. 958–964.Google Scholar
D'yakov, S. 1954 The stability of shock waves: Investigation of the problem of the stability of shock waves in arbitrary media. Zh. Eksp. Teor. Fiz. 27, 288.Google Scholar
Engelke, R. & Bdzil, J. B. 1983 A study of the steady-state reaction-zone structure of a homogeneous and a heterogeneous explosive. Phys. Fluids 26, 1210–1221.Google Scholar
Erpenbeck, J. J. 1964 Stability of idealized one-reaction detonations. Phys. Fluids 7, 684–696.Google Scholar
Fickett, W. & Davis, W. C. 1979 Detonation. University of California Press.Google Scholar
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, 611–617.CrossRefGoogle Scholar
Gorchkov, V., Kiyanda, C. B., Short, M. & Quirk, J. J. 2007 A detonation stability formulation for arbitrary equations of state and multi-step reaction mechanisms. Proc. Combust. Inst. 31, 2397–2405.CrossRefGoogle Scholar
Gustavsen, R. L., Sheffield, S. A., & Alcon, R. R. 2000 Progress in measuring detonation wave profiles in PBX-9501. In Proc. Eleventh Symposium (Intl) on Detonation. Office of Naval Research Rep. ONR 3330000-5, pp. 821–827.Google Scholar
Henrick, A. K. 2007 Shock-fitted solutions for two-dimensional detonation. PhD Thesis, University of Notre Dame.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, 311–329.CrossRefGoogle Scholar
Hill, L. G., Bdzil, J. B & Aslam, T. D. 2002 Front curvature rate stick measurements and detonation shock dynamics calibration for PBX-9502 over a wide temperature range. In Proc. Eleventh Symposium (Intl) on Detonation. Office of Naval Research Rep. ONR 33300-5, pp. 1029–1037.Google Scholar
Klein, R. 1991. On the dynamics of weakly curved detonations. In Dynamical Issues in Combustion Theory (ed. Fife, P. C., Linan, A. & Williams, F. A.). IMA Volumes in Mathematics and its Applications, vol. 35, pp. 127–166.Google Scholar
Kontorovich, V. 1957 Concerning the stability of shock waves. Zh. Eksp. Teor. Fiz. 33, 1525.Google Scholar
Lee, E. L. & Tarver, C. M. 1980 Phenomenological model of shock initiation in heterogeneous explosives. Phys. Fluids 23, 2362–2372.Google Scholar
Lee, H. I. & Stewart, D. S. 1990 Calculation of linear instability: One-dimensional instability of plane detonation. J. Fluid Mech. 216, 103–132.CrossRefGoogle Scholar
Majda, A. & Roytburd, V. 1992 Low-frequency multidimensional instabilities for reacting shock-waves. Stud. Appl. Maths 87, 135–174.CrossRefGoogle 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 detonation. Combust. Theory Model. 9, 159–170.CrossRefGoogle Scholar
Radulescu, M. I., Ng, H. D., Lee, J. H. S. & Varatharajan, B. 2002 Effect of argon dilution on the stability of acetylene-oxygen detonations. Proc. Combust. Inst. 29, 2825–2831.Google Scholar
Roberts, A. E. 1945 Stability of a plane steady shock. Los Alamos National Rep. 299. Los Alamos National Laboratory.Google Scholar
Roe, P. 1998 Linear bicharacteristic schemes without dissipation. SIAM J. Sci. Comput. 19, 1405–1427.Google Scholar
Sanchez, A. L., Carretero, M., Clavin, P. & Williams, F. A. 2001 One-dimensional overdriven detonations with branched-chain kinetics. Phys. Fluids 13, 776–792.Google Scholar
Seitz, W. L., Stacy, H. L., Engelke, R., Tang, P. K. & Wackerle, J. 1989 Detonation reaction zone structure of PBX9502. In Proc. Ninth Symposium (Intl) on Detonation. Office of Naval Research Rep. ONR 113291-7, pp. 657–669.Google Scholar
Sharpe, G. J. 1997 Linear stability of idealized detonations. Proc. R. Soc. Lond. A 453, 2603–2625.Google Scholar
Sharpe, G. J. 1999 Linear stability of pathological detonations. J. Fluid Mech. 401, 311–338.Google Scholar
Sharpe, G. J. 2001 Transverse waves in numerical simulations of cellular detonations. J. Fluid Mech. 447, 31–51.Google Scholar
Sharpe, G. J. & Falle, S. A. E. G. 1999 One-dimensional numerical simulations of idealized detonations. Proc. R. Soc. Lond. A 455, 1203–1214.Google Scholar
Sheffield, S. A., Engelke, R., Alcon, R. R., Gustavsen, R. L., Robbins, D. L., Stahl, D. B., Stacy, H. L. & Whitehead, M. C. 2002 Particle velocity measurements of the reaction zone in nitromethane. In Proc. Twelfth Symposium (Intl) on Detonation. Office of Naval Research Rep. ONR 333-05-02, pp. 159–166.Google Scholar
Short, M. & Blythe, P. A. 2002 Structure and stability of weak-heat-release detonations for finite Mach numbers. Proc. R. Soc. Lond. A 458, 1795–1807.Google Scholar
Short, M. & Dold, J. W. 1996 Multi-dimensional linear stability of a detonation wave with a model three-step chain-branching reaction. Math. Comput. Model. 24, 115–123.Google Scholar
Short, M. & Quirk, J. J. 1997 On the nonlinear stability and detonability limit of a detonation wave for a model 3-step chain-branching reaction. J. Fluid. Mech. 339, 89–119.CrossRefGoogle Scholar
Short, M. & Sharpe, G. J. 2003 Pulsating instability of detonations with a two-step chain-branching reaction model: Theory and numerics. Combust. Theory Model. 7, 401–416.CrossRefGoogle Scholar
Short, M. & Stewart, D. S. 1998 Cellular detonation stability. Part 1. A normal-mode linear analysis. J. Fluid Mech. 368, 229–262.CrossRefGoogle Scholar
Short, M. & Stewart, D. S. 1999 The multi-dimensional stability of weak-heat-release detonations. J. Fluid Mech. 382, 109–135.Google Scholar
Tarver, C. M., Kury, J. W. & Breithaupf, R. D. 1997 Detonation waves in triaminotrinitrobenzene. J. Appl. Phys. 82, 3771–3782.Google Scholar
Wasow, W. 2002 Asymptotic Expansions for Ordinary Differential Equations. Dover.Google Scholar
Yao, J. & Stewart, D. S. 1996 On the dynamics of multi-dimensional detonation. J. Fluid Mech. 309, 225–275.Google Scholar