Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-17T21:28:06.008Z Has data issue: false hasContentIssue false
  • English
  • Français

D’Yakov–Kontorovich instability in planar reactive shocks

Published online by Cambridge University Press:  19 September 2019

César Huete Open the ORCID record for César Huete [Opens in a new window]  and
Marcos Vera Open the ORCID record for Marcos Vera [Opens in a new window]
César Huete*
Affiliation:
Grupo de Mecánica de Fluidos, Universidad Carlos III, Av. Universidad 30, 28911 Leganés, Spain
Marcos Vera
Affiliation:
Grupo de Mecánica de Fluidos, Universidad Carlos III, Av. Universidad 30, 28911 Leganés, Spain
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The standard D’Yakov and Kontorovich (DK) instability occurs when a planar shock wave is perturbed and then oscillates with constant amplitude in the long-time regime. As a direct result, pressure perturbations generated directly behind the shock propagate downstream as non-evanescent sound waves, an effect known as spontaneous acoustic emission (SAE). To reach the DK regime, the slope of the Rankine–Hugoniot curve in the post-shock state must satisfy certain conditions, which have usually been related to non-ideal equations of state. This study reports that the DK instability and SAE can also occur in shocks moving in perfect gases when exothermic effects occur. In particular, a planar detonation, initially perturbed with a wavelength much larger than the detonation thickness, may exhibit constant-amplitude oscillations when the amount of heat released is positively correlated with the shock strength, a phenomenon that resembles the Rayleigh thermoacoustic instability. The opposite strongly damped oscillation regime is reached when the shock strength and the change in the heat released are negatively correlated. This study employs a linear perturbation model to describe the long-time and transient evolution of the detonation front, which is assumed to be infinitely thin, and the sound and entropy–vorticity fields generated downstream.

JFM classification

Compressible Flows: Shock waves Compressible Flows: Detonation waves Instability: Parametric instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 879 , 25 November 2019 , pp. 54 - 84
Check for updates
Copyright
© 2019 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

Ami Glasner, S., Livne, E., Steinberg, E., Yalinewich, A. & Truran, J. W. 2018 Ignition of detonation in accreted helium envelopes. Mon. Not. R. Astron. Soc. 476, 2238–2248.Google Scholar
Bates, J. W. 2004 Initial-value-problem solution for isolated rippled shock fronts in arbitrary fluid media. Phys. Rev. E 69, 056313.10.1103/PhysRevE.69.056313Google Scholar
Bates, J. W. 2007 Instability of isolated planar shock waves. Phys. Fluids 19, 094102.10.1063/1.2757706Google Scholar
Bates, J. W. 2012 On the theory of a shock wave driven by a corrugated piston in a non-ideal fluid. J. Fluid Mech. 691, 146–164.10.1017/jfm.2011.463Google Scholar
Bates, J. W. 2015 Theory of the corrugation instability of a piston-driven shock wave. Phys. Rev. E 91, 013014.Google Scholar
Bates, J. W. & Montgomery, D. C. 2000 The D’yakov–Kontorovich instability of shock waves in real gases. Phys. Rev. Lett. 84, 1180.10.1103/PhysRevLett.84.1180Google Scholar
Bourlioux, A., Majda, A. J. & Royburd, V. 1991 Theoretical and numerical structure for unstable one-dimensional detonations. SIAM J. Appl. Maths 51, 303–343.Google Scholar
Bourlioux, A., Majda, A. J. & Royburd, V. 1992 Theoretical and numerical structure for unstable two-dimensional detonations. Combust. Flame 90, 211–229.10.1016/0010-2180(92)90084-3Google Scholar
Browne, S., Ziegler, J. & Shepherd, J. E. 2008 Shock and Detonation Toolbox. GALCIT-Explosion Dynamics Laboratory.Google Scholar
Buckmaster, J. D. & Ludford, G. S. S. 1988 The effect of structure on stability of detonations. Role of the induction zone. Proc. Combust. Inst. 21, 1669–1676.10.1016/S0082-0784(88)80400-4Google Scholar
Campos, F. C. & Wouchuk, J. G. 2014 Analytical asymptotic velocities in linear Richtmyer–Meshkov-like flows. Phys. Rev. E 053007.Google Scholar
Clavin, P. & Denet, B. 2002 Diamond patterns in the cellular front of an overdriven detonation. Phys. Rev. Lett. 88, 044502.Google Scholar
Clavin, P. & Denet, B. 2018 Decay of plane detonation waves to the self-propagating Chapman–Jouguet regime. C. R. Méc. 845, 170–202.Google Scholar
Clavin, P. & He, L. 1996 Acoustic effects in the nonlinear oscillations of planar detonations. Phys. Rev. E 53, 4778.Google Scholar
Clavin, P., He, L. & Williams, F. A. 1997 Multidimensional stability analysis of overdriven gaseous detonations. Phys. Fluids 9, 3746–3785.10.1063/1.869520Google Scholar
Clavin, P. & Searby, G. 2016 Combustion Waves and Fronts in Flows: Flames, Shocks, Detonations, Ablation Fronts and Explosion of Stars, pp. 261–265. Cambridge University Press.10.1017/CBO9781316162453Google Scholar
Clavin, P. & Williams, F. A. 2009 Multidimensional stability analysis of gaseous detonations near Chapman–Jouguet conditions for small heat release. J. Fluid Mech. 324, 125–150.10.1017/S0022112008005387Google Scholar
Clavin, P. & Williams, F. A. 2012 Analytical studies of the dynamics of gaseous detonations. Phil. Trans. R. Soc. Lond. A 13, 597–624.Google Scholar
Coughlin, E. R., Ro, S. & Quataert, E. 2019 Weak shock propagation with accretion. II. Stability of self-similar solutions to radial perturbations. Astrophys. J. 874, 58.10.3847/1538-4357/ab09ecGoogle Scholar
Daou, R. & Clavin, P. 2003 Instability threshold of gaseous detonations. J. Fluid Mech. 482, 181–206.10.1017/S0022112003004038Google Scholar
D’Yakov, S. P. 1954 The stability of shockwaves: investigation of the problem of stability of shock waves in arbitrary media. Zh. Eksp. Teor. Fiz. 27, 288–295.Google Scholar
Erpenbeck, J. J. 1962 Stability of step shocks. Phys. Fluids 5, 1181.10.1063/1.1706503Google Scholar
Erpenbeck, J. J. 1964 Stability of idealized one-reaction detonations. Phys. Fluids 7, 684–696.10.1063/1.1711269Google Scholar
Faria, L. M., Kasimov, A. R. & Rosales, R. R. 2015 Theory of weakly nonlinear self-sustained detonations. J. Fluid Mech. 784, 163–198.Google Scholar
Fickett, W. & Davis, W. C. 2000 Detonation: Theory and Experiment, pp. 230–236. Courier Corporation.Google Scholar
Fowles, G. R. 1976 Conditional stability of shock waves–a criterion for detonation. Phys. Fluids 19, 227–238.Google Scholar
Fowles, G. R. 1981 Stimulated and spontaneous emission of acoustic waves from shock fronts. Phys. Fluids 24, 222–227.10.1063/1.863369Google Scholar
Fowles, G. R. & Swan, G. W. 1973 Stability of plane shock waves. Phys. Rev. Lett. 30, 1023–1025.10.1103/PhysRevLett.30.1023Google Scholar
Fraley, G. 1986 Rayleigh–Taylor stability for a normal shock wave-density discontinuity interaction. Phys. Fluids 29, 376–386.Google Scholar
Freeman, N. C. 1955 A theory of the stability of plane shock waves. Phil. Trans. R. Soc. Lond. A 228, 341–362.Google Scholar
Gamezo, V. N., Khokhlov, A. M. & Oran, E. S. 2004 Deflagrations and detonations in thermonuclear supernovae. Phys. Rev. Lett. 92, 211102.10.1103/PhysRevLett.92.211102Google Scholar
Gao, Y., Ng, H. D. & Lee, J. H. S. 1990 Experimental characterization of galloping detonations in unstable mixtures. Combust. Flame 162, 2405–2413.10.1016/j.combustflame.2015.02.007Google Scholar
Guha, A. 1994 A unified theory of aerodynamic and condensation shock waves in vapor–droplet flows with or without a carrier gas. Phys. Fluids 6, 1893–1913.Google Scholar
Huete, C., Jin, T., Martínez, R. D. & Luo, K. 2017 Interaction of a planar reacting shock wave with an isotropic turbulent vorticity field. Phys. Rev. E 96, 053104.Google Scholar
Huete, C., Sánchez, A. L. & Williams, F. A. 2013 Theory of interactions of thin strong detonations with turbulent gases. Phys. Fluids 25, 076105.Google Scholar
Huete, C., Wouchuk, J. G. & Velikovich, A. L. 2011 Analytical linear theory for the interaction of a planar shock wave with a 2D/3D random isotropic density field. Phys. Rev. E 83, 056320.Google Scholar
Huete, C., Wouchuk, J. G. & Velikovich, A. L. 2012 Analytical linear theory for the interaction of a planar shock wave with a 2D/3D random isotropic acoustic field. Phys. Rev. E 85, 026312.Google Scholar
Kabanov, D. I. & Kasimov, A. R. 2018 Linear stability analysis of detonations via numerical computation and dynamic mode decomposition. Phys. Fluids 30, 036103.10.1063/1.5020558Google 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, 3566–3578.10.1063/1.1776531Google Scholar
Khokhlov, A. M. 1989 The structure of detonation waves in supernovae. Mon. Not. R. Astron. Soc. 239, 785–808.Google Scholar
Khokhlov, A. M., Oran, E. S. & Wheeler, J. V. 1997 Deflagration-to-detonation transition in thermonuclear supernova. Astrophys. J. 239, 678–688.Google Scholar
Kontorovich, V. M. 1957 On the stability of shock waves. Zh. Eksp. Teor. Fiz. 33, 1525–1526.Google Scholar
Konyukhov, A. V., Likhachev, A. P., Fortov, V. E., Khishchenko, K. V., Anisimov, S. I., Lomonosov, I. V. & Konyukhov, A. V. 2009 On the neutral stability of a shock wave in real medias. J. Expl Theor. Phys. Lett. 90, 18–24.Google Scholar
Lapworth, K. C. 1959 An experimental investigation of the stability of plane shock waves. J. Fluid Mech. 6, 469–480.Google Scholar
Lee, H. I. & Stewart, D. S. 1990 Calculation of linear detonation instability: one-dimensional instability of plane detonation. J. Fluid Mech. 216, 103–132.10.1017/S0022112090000362Google Scholar
Levin, V. A. & Chernyi, G. G. 1967 Asymptotic laws of detonation wave behavior. Prikl. Mat. Mekh. 31, 393–405.Google Scholar
Liñan, A., Kurdyumov, V. N. & Sánchez, A. L. 2012 Initiation of reactive blast waves by external energy sources. C. R. Méc. 340, 829–844.10.1016/j.crme.2012.10.033Google Scholar
Majda, A. J. & Rosales, R. R. 1983 A theory for spontaneous Mach stem formation in reacting shock fronts, I. The basic perturbation analysis. SIAM J. Appl. Maths 43, 1310–1334.10.1137/0143088Google Scholar
Martínez-Ferrer, P. J., Buttay, R., Lehnasch, G. & Mura, A. 2014 A detailed verification procedure for compressible reactive multicomponent Navier–Stokes solver. J. Comput. Fluids 89, 88–110.Google Scholar
Martínez-Ruiz, D., Urzay, J., Sánchez, A. L., Liñan, A. & Williams, F. A. 2013 Dynamics of thermal ignition of spray flames in mixing layers. J. Fluid Mech. 734, 387–423.Google Scholar
McBride, J., Zehe, M. J. & Gordon, S.2002 NASA Glenn coefficients for calculating thermodynamic properties of individual species. Tech. Rep. NASA/TP-2002-211556. NASA, Glenn Research Center.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, 31–81.10.1017/S0022112007005046Google Scholar
Ribner, H. S. 1959 Acoustic energy flux from shock-turbulence interaction. J. Fluid Mech. 35, 299–310.Google Scholar
Röpke, F. K. & Sim, S. A. 2018 Models for Type Ia supernovae and related astrophysical transients. Space Sci. Rev. 214, 72.Google Scholar
Sánchez, 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
Sharpe, G. J. 2001 Transverse waves in numerical simulations of cellular detonations. J. Fluid Mech. 447, 31–51.Google Scholar
Sharpe, F. & Sharpe, S. A. 2001 Two-dimensional numerical simulations of idealized detonations. Proc. R. Soc. Lond. A 456, 2081–2100.10.1098/rspa.2000.0603Google Scholar
Shepherd, J. E. 2009 Detonation in gases. Proc. Combust. Inst. 32, 83–98.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.Google Scholar
Short, M. & Stewart, D. S. 1997 Low-frequency two-dimensional linear instability of plane detonation. J. Fluid Mech. 339, 249–295.Google Scholar
Short, M. & Stewart, D. S. 1999 The multi-dimensional stability of weak–heat–release detonations. J. Fluid Mech. 382, 109–135.10.1017/S0022112098003759Google Scholar
Smirnov, N. N., Betelin, V. B., Kushnirenko, A. G., Nikitin, V. F., Dushin, V. R. & Nerchenko, V. A. 2013 A unified theory of aerodynamic and condensation shock waves in vapor–droplet flows with or without a carrier gas. Acta Astronaut. 87, 114–129.Google Scholar
Smolders, H. J. & van Dongen, M. E. H. 1992 Shock wave structure in a mixture of gas, vapour and droplets. Shock Waves 2, 225–267.Google Scholar
Touber, E. & Alferez, N. 2019 Shock-induced energy conversion of entropy in non-ideal fluids. J. Fluid Mech. 864, 807–847.10.1017/jfm.2019.25Google Scholar
Tumin, A. 2007 Initial-value problem for small disturbances in an idealized one-dimensional detonation. Phys. Fluids 19, 106105.Google Scholar
Velikovich, A. L., Huete, C. & Wouchuk, J. G. 2012 Effect of shock-generated turbulence on the Hugoniot jump conditions. Phys. Rev. E 85, 016301.Google Scholar
Velikovich, A. L., Murakami, M., Taylor, B. D., Giuliani, J. L., Iwamoto, Y. & Wouchuk, J. G. 2016 Stability of stagnation via an expanding accretion shock wave. Phys. Plasmas 23, 052706.10.1063/1.4948492Google Scholar
Vimercati, D., Gori, G. & Guardone, A. 2018 Non-ideal oblique shock waves. J. Fluid Mech. 847, 266–285.Google Scholar
Vishniac, E. T. & Ryu, D. 1989 On the stability of decelerating shocks. Astrophys. J. 337, 917–926.Google Scholar
Wetta, N., Pain, J. C. & Heuzé, O. 2018 D’yakov–Kontorovitch instability of shock waves in hot plasmas. Phys. Rev. E 98, 033205.Google Scholar
Williams, F. A. 1961 Structure of detonations in dilute sprays. Phys. Fluids 4, 1434–1443.Google Scholar
Williams, F. A. 1985 Combustion Theory, pp. 20–33. Benjamin Cummings.Google Scholar
Wouchuk, J. G., Huete Ruiz de Lira, C. & Velikovich, A. L. 2009 Analytical linear theory for the interaction of a planar shock wave with an isotropic turbulent vorticity field. Phys. Rev. E 79, 066315.Google Scholar
Wouchuk, J. G. & López Cabada, J. 2004 Spontaneous acoustic emission of a corrugated shock wave in the presence of a reflecting surface. Phys. Rev. E 70, 046303.Google Scholar
Wright, W. P., Nagaraj, G., Kneller, J. P., Scholberg, K. & Seitenzahl, I. R. 2016 Neutrinos from type Ia supernovae: the deflagration-to-detonation transition scenario. Phys. Rev. D 94, 025026.Google Scholar
Zaidel, P. M. 1960 Shock wave from a slightly curved piston. Z. Angew. Math. Mech. J. Appl. Math. Mech. 24, 316–327.Google Scholar
Zaidel, R. M. 1961 Stability of detonation waves in gas mixtures. Dokl. Akad. Nauk SSSR 136, 1142.Google Scholar