Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T20:06:20.694Z Has data issue: false hasContentIssue false
  • English
  • Français

Theory of weakly nonlinear self-sustained detonations

Published online by Cambridge University Press:  03 November 2015

Luiz M. Faria ,
Aslan R. Kasimov  and
Rodolfo R. Rosales
Luiz M. Faria
Affiliation:
Applied Mathematics and Computational Science, King Abdullah University of Science and Technology, Room 4-2226, 4700 KAUST, Thuwal 23955-6900, Saudi Arabia Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Aslan R. Kasimov*
Affiliation:
Applied Mathematics and Computational Science, King Abdullah University of Science and Technology, Room 4-2226, 4700 KAUST, Thuwal 23955-6900, Saudi Arabia
Rodolfo R. Rosales
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose a theory of weakly nonlinear multidimensional self-sustained detonations based on asymptotic analysis of the reactive compressible Navier–Stokes equations. We show that these equations can be reduced to a model consisting of a forced unsteady small-disturbance transonic equation and a rate equation for the heat release. In one spatial dimension, the model simplifies to a forced Burgers equation. Through analysis, numerical calculations and comparison with the reactive Euler equations, the model is demonstrated to capture such essential dynamical characteristics of detonations as the steady-state structure, the linear stability spectrum, the period-doubling sequence of bifurcations and chaos in one-dimensional detonations and cellular structures in multidimensional detonations.

JFM classification

Nonlinear Dynamical Systems: Chaos Compressible Flows: Detonation waves Instability: Nonlinear instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 784 , 10 December 2015 , pp. 163 - 198
Check for updates
Copyright
© 2015 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

Barker, B., Humpherys, J., Lyng, G. & Zumbrun, K. 2015 Viscous hyperstabilization of detonation waves in one space dimension. SIAM J. Appl. Maths 75 (3), 885–906.Google Scholar
Bdzil, J. B. & Stewart, D. S. 2012 Theory of detonation shock dynamics. In Shock Waves Science and Technology Library, vol. 6, pp. 373–453. Springer.CrossRefGoogle Scholar
Blythe, P. A. & Crighton, D. G. 1989 Shock-generated ignition: the induction zone. Proc. R. Soc. Lond. A 426 (1870), 189–209.Google Scholar
Bourlioux, A., Majda, A. J. & Roytburd, V. 1991 Theoretical and numerical structure for unstable one-dimensional detonations. SIAM J. Appl. Maths 51, 303–343.Google Scholar
Browne, S. & Shepherd, J. 2007 Linear stability of detonations with reversible chemical reactions. In Proceedings of the Western States Section/Combustion Institute, Sandia National Laboratories, Livermore, pp. 1669–1676.Google Scholar
Burgers, J. M. 1948 A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. 1, 171–199.CrossRefGoogle Scholar
Choquet-Bruhat, Y. 1968 Ondes asymptotiques et approchées pour des systèmes d’équations aux dérivées partielles non linéaires. Institut Henri Poincaré.Google Scholar
Clavin, P. & Denet, B. 2002 Diamond patterns in the cellular front of an overdriven detonation. Phys. Rev. Lett. 88 (4), 044502.Google Scholar
Clavin, P. & Williams, F. A. 2002 Dynamics of planar gaseous detonations near Chapman–Jouguet conditions for small heat release. Combust. Theor. Model. 6 (1), 127–139.Google Scholar
Clavin, P. & Williams, F. A. 2009 Multidimensional stability analysis of gaseous detonations near Chapman–Jouguet conditions for small heat release. J. Fluid Mech. 624, 125–150.Google Scholar
Clavin, P. & Williams, F. A. 2012 Analytical studies of the dynamics of gaseous detonations. Phil. Trans. R. Soc. Lond. A 370 (1960), 597–624.Google ScholarPubMed
Erpenbeck, J. J. 1964 Stability of idealized one-reaction detonations. Phys. Fluids 7, 684–696.Google Scholar
Faria, L. M.2014 Qualitative and asymptotic theory of detonations. PhD thesis, King Abdullah University of Science and Technology.Google Scholar
Faria, L. M., Kasimov, A. R. & Rosales, R. R. 2014 Study of a model equation in detonation theory. SIAM J. Appl. Maths 74 (2), 547–570.CrossRefGoogle Scholar
Fickett, W. 1979 Detonation in miniature. Am. J. Phys. 47 (12), 1050–1059.CrossRefGoogle Scholar
Fickett, W. & Davis, W. C. 2011 Detonation: Theory and Experiment. Dover.Google Scholar
Glaz, H. M., Colella, P., Glass, I. I. & Deschambault, R. L. 1985 A numerical study of oblique shock-wave reflections with experimental comparisons. Proc. R. Soc. Lond. A 398 (1814), 117–140.Google Scholar
Hayes, W. D. & Probstein, R. F. 1959 Hypersonic Flow Theory. Academic.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 (1), 311–329.Google Scholar
Humpherys, J., Lyng, G. & Zumbrun, K. 2013 Stability of viscous detonations for Majda’s model. Physica D 259, 63–80.CrossRefGoogle Scholar
Hunter, J. K. 1995 Asymptotic equations for nonlinear hyperbolic waves. In Surveys in Applied Mathematics, vol. 2, pp. 167–276. Springer.Google Scholar
Hunter, J. K. & Brio, M. 2000 Weak shock reflection. J. Fluid Mech. 410, 235–261.Google Scholar
Kasimov, A. R., Faria, L. M. & Rosales, R. R. 2013 Model for shock wave chaos. Phys. Rev. Lett. 110 (10), 104104.Google 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, 3566.Google Scholar
Keller, J. B. 1978 Rays, waves and asymptotics. Bull. Am. Math. Soc. 84 (5), 727–750.CrossRefGoogle Scholar
Ketcheson, D. I., Mandli, K. T., Ahmadia, A. J., Alghamdi, A., Quezada de Luna, M., Parsani, M., Knepley, M. G. & Emmett, M. 2012 PyClaw: accessible, extensible, scalable tools for wave propagation problems. SIAM J. Sci. Comput. 34 (4), C210–C231.CrossRefGoogle Scholar
Korteweg, D. J. & de Vries, G. 1895 On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil. Mag. 39 (240), 422–443.Google Scholar
Lee, H. I. & Stewart, D. S. 1990 Calculation of linear detonation instability: one-dimensional instability of plane detonation. J. Fluid Mech. 212, 103–132.Google Scholar
Lee, J. H. S. 2008 The Detonation Phenomenon. Cambridge University Press.Google Scholar
LeVeque, R. J. 2002 Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.Google Scholar
Lin, C. C., Reissner, E. & Tsien, H. S. 1948 On two-dimensional non-steady motion of a slender body in a compressible fluid. J. Math. Phys. 27 (3), 220–231.CrossRefGoogle Scholar
Majda, A. 1980 A qualitative model for dynamic combustion. SIAM J. Appl. Maths 41 (1), 70–93.Google Scholar
Mazaheri, K., Mahmoudi, Y. & Radulescu, M. I. 2012 Diffusion and hydrodynamic instabilities in gaseous detonations. Combust. Flame 159 (6), 2138–2154.CrossRefGoogle Scholar
von Neumann, J. 1963 Oblique reflection of shocks. In Collected Works, vol. VI, pp. 238–299. Pergamon.Google Scholar
Ng, H., Higgins, A., Kiyanda, C., Radulescu, M., Lee, J., Bates, K. & Nikiforakis, N. 2005 Nonlinear dynamics and chaos analysis of one-dimensional pulsating detonations. Combust. Theor. Model. 9 (1), 159–170.CrossRefGoogle Scholar
Oran, E. & Boris, J. P. 2001 Numerical Simulation of Reactive Flow. Cambridge University Press.Google Scholar
Radulescu, M. I. & Tang, J. 2011 Nonlinear dynamics of self-sustained supersonic reaction waves: Fickett’s detonation analogue. Phys. Rev. Lett. 107 (16), 164503.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, 453–464.Google Scholar
Romick, C. M., Aslam, T. D. & Powers, J. M. 2015 Verified and validated calculation of unsteady dynamics of viscous hydrogen–air detonations. J. Fluid Mech. 769, 154–181.Google Scholar
Rosales, R. R. 1989 Diffraction effects in weakly nonlinear detonation waves. In Nonlinear Hyperbolic Problems, Lecture Notes in Mathematics, vol. 1402, pp. 227–239. Springer.CrossRefGoogle Scholar
Rosales, R. R. 1991 An introduction to weakly nonlinear geometrical optics. In Multidimensional Hyperbolic Problems and Computations, The IMA Volumes in Mathematics and Its Applications, vol. 29, pp. 281–310. Springer.Google Scholar
Rosales, R. R. & Majda, A. J. 1983 Weakly nonlinear detonation waves. SIAM J. Appl. Maths 43 (5), 1086–1118.Google Scholar
Short, M. 1997 Multidimensional linear stability of a detonation wave at high activation energy. SIAM J. Appl. Maths 57 (2), 307–326.CrossRefGoogle 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 (2024), 1795–1807.Google 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.CrossRefGoogle Scholar
Stewart, D. S., Aslam, T. D. & Yao, J. 1996 On the evolution of cellular detonation. In Proceedings of the Combustion Institute, vol. 26, pp. 2981–2989. The Combustion Institute.Google Scholar
Strogatz, S. H. 1994 Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Westview.Google Scholar
Sturtevant, B. & Kulkarny, V. A. 1976 The focusing of weak shock waves. J. Fluid Mech. 73 (4), 651–671.Google Scholar
Tabak, E. G. & Rosales, R. R. 1994 Focusing of weak shock waves and the von Neumann paradox of oblique shock reflection. Phys. Fluids 6 (5), 1874–1892.CrossRefGoogle Scholar
Taylor, B. D., Kasimov, A. R. & Stewart, D. S. 2009 Mode selection in weakly unstable two-dimensional detonations. Combust. Theor. Model. 13 (6), 973–992.CrossRefGoogle Scholar
Voitsekhovskii, B. V., Mitrofanov, V. V. & Topchian, M. Y.The structure of detonation front in gases. Report FTD-MTD-64-527. Foreign Technology Division, Wright Patterson Air Force Base, OH (AD 633-821).Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. John Wiley and Sons.Google Scholar
Williams, F. A. 1985 Combustion Theory. Westview.Google Scholar
Yao, J. & Stewart, D. S. 1996 On the dynamics of multi-dimensional detonation. J. Fluid Mech. 309, 225–275.Google Scholar
Zabolotskaya, E. A. & Khokhlov, R. V. 1969 Quasi-plane waves in the nonlinear acoustics of confined beams. Sov. Phys. Acoust. 15 (1), 35–40.Google Scholar
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), 7598–7630.Google Scholar