Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T12:59:50.023Z Has data issue: false hasContentIssue false
  • English
  • Français

The Zeldovich spontaneous reaction wave propagation concept in the fast/modest heating limits

Published online by Cambridge University Press:  22 February 2016

D. R. Kassoy
D. R. Kassoy*
Affiliation:
Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309-0427, USA
*
†Email address for correspondence: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

Quantitative mathematical models describe planar, spontaneous, reaction wave propagation (Zeldovich, Combust. Flame, vol. 39, 1980, pp. 211–214) in a finite hot spot volume of reactive gas. The results describe the complete thermomechanical response of the gas to a one-step, high-activation-energy exothermic reaction initiated by a tiny initial temperature non-uniformity in a gas at rest with uniform pressure. Initially, the complete conservation equations, including all transport terms, are non-dimensionalized to identify parameters that quantify the impact of viscosity, conduction and diffusion. The results demonstrate unequivocally that transport terms are tiny relative to all other terms in the equations, given the relevant time and length scales. The asymptotic analyses, based on the reactive Euler equations, describe both induction and post-induction period models for a fast heat release rate (induction time scale short compared to the acoustic time of the spot), as well as a modest heat release rate (induction time scale equivalent to the acoustic time). Analytical results are obtained for the fast heating rate problem and emphasize the physics of near constant-volume heating during the induction period. Weak hot spot expansion is the source of fluid expelled from the original finite volume and is a ‘piston-effect’ source of acoustic mechanical disturbances beyond the spot. The post-induction period is characterized by the explosive appearance of an ephemeral, spatially uniform high-temperature, high-pressure spot embedded in a cold, low-pressure environment. In analogy with a shock tube the subsequent expansion process occurs on the acoustic time scale of the spot and will be the source of shocks propagating beyond the spot. The modest heating rate induction period is characterized by weakly compressible phenomena that can be described by a novel system of linear wave equations for the temperature, pressure and induced velocity perturbations driven by nonlinear chemical heating, which provides physical insights difficult to obtain from the more familiar ‘Clarke equation’. When the heating rate is modest, reaction terms in the post-induction period Euler equations exhibit a form of singular behaviour in the high-activation-energy limit, implying the need to use a nonlinear exponential scaling for time and space, developed originally to describe spatially uniform thermal explosions (Kassoy, Q. J. Mech. Appl. Maths, vol. 30, 1977, pp. 71–89). Here again the result will be the explosive appearance of an ephemeral spatially uniform high-temperature, high-pressure hot spot. These results demonstrate that an initially weak temperature non-uniformity in a finite hot spot can be the source of acoustic and shock wave mechanical disturbances in the gas beyond the spot that may be related to rocket engine instability and engine knock.

JFM classification

Reacting Flows: Combustion Compressible Flows: Gas dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 791 , 25 March 2016 , pp. 439 - 463
Check for updates
Copyright
© 2016 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

Clarke, J. F. 1978 A progress report on the theoretical analysis of the interaction between a shock wave and an explosive gas mixture. College of Aeronautics Report 7801. Cranfield Institute of Technology, Cranfield, UK.Google Scholar
Clarke, J. F. 1985 Finite amplitude waves in compressible gases. In The Mathematics of Combustion (ed. Buckmaster, J. D.), pp. 183–245. SIAM.CrossRefGoogle Scholar
Clarke, J. F. & Kassoy, D. R. 1984 High Speed Deflagration and Compressibility Effects in Dynamics of Shock Waves, Explosions and Detonations (ed. Bowen, J. R., Manson, N., Oppenheim, A. K. & Soloukhin, R. I.), Progress in Astronautics and Aeronautics, vol. 94, pp. 175–185.Google Scholar
Clarke, J. F., Kassoy, D. R. & Riley, N. 1984a Shocks generated in a confined gas due to rapid heat addition at the boundary I. Weak shock waves. Proc. R. Soc. Lond. A 393, 309–329.Google Scholar
Clarke, J. F., Kassoy, D. R. & Riley, N. 1984b Shocks generated in a confined gas due to rapid heat addition at the boundary II. Strong shock waves. Proc. R. Soc. Lond. A 393, 331–351.Google Scholar
Eckett, C. A., Quirk, J. J. & Shepherd, J. 2000 The role of unsteadiness in direct initiation of gaseous detonations. J. Fluid Mech. 421, 147–183.Google Scholar
Friedman, A. & Herrero, M. A. 1990 A nonlinear, nonlocal wave-equation arising in combustion theory. Nonlinear Anal. 14, 93–106.Google Scholar
Gu, X. J., Emerson, D. R. & Bradley, D. 2003 Modes of reaction front propagation from hot spots. Combust. Flame 133, 63–74.Google Scholar
Jackson, T. L., Kapila, A. K. & Stewart, D. S. 1989 Evolution of a reaction center in an explosive material. SIAM J. Appl. Maths 49, 432–458.Google Scholar
Kapila, A. K. & Dold, J. W. 1989 A theoretical picture of shock-to-detonation transition in a homogeneous explosive. In Proceedings of the 9th International Symposium on Detonation OCNR 113291-7, Office of Naval Research, Washington, DC, pp. 219–227.Google Scholar
Kapila, A. K., Schwendeman, D. W., Quirk, J. J. & Hawa, T. 2002 Mechanisms of detonation formation due to a temperature gradient. Combust. Theor. Model. 6, 553–594.CrossRefGoogle Scholar
Kassoy, D. R. 1977 The supercritical spatially homogeneous thermal explosion: initiation to completion. Q. J. Mech. Appl. Maths 30, 71–89.Google Scholar
Kassoy, D. R. 2010 The response of a compressible gas to extremely rapid transient, spatially resolved energy addition: an asymptotic formulation. J. Engng Maths 68, 249–262.Google Scholar
Kassoy, D. R. 2014 a Non-diffusive ignition of a gaseous reactive mixture following time-resolved, spatially distributed energy addition. Combust. Theor. Model. 18, 101–116.Google Scholar
Kassoy, D. R. 2014b Mechanical disturbances arising from thermal power deposition in a gas. AIAA J. doi:10.2514/1.J052807 abstract content.Google Scholar
Kassoy, D. R. & Clarke, J. F. 1985 The structure of a high speed deflagration with a finite origin. J. Fluid Mech. 150, 253–280.Google Scholar
Kevorkian, J. & Cole, J. D. 1968 Perturbation Methods in Applied Mathematics, pp. 482–496. Springer.Google Scholar
Kulkarni, R., Zellhuber, M. & Polifke, W. 2013 LES based investigation of autoignition in turbulent co-flow configurations. Combust. Theor. Model. 17, 224–259.Google Scholar
Kurdyumov, V., Sanzhez, A. L. & Liñan, A. L. 2003 A heat propagation from a concentrated external energy source in a gas. J. Fluid Mech. 491, 379–410.Google Scholar
Lieuwen, T. C. 2012 Unsteady Combustor Physics. Cambridge University Press.Google Scholar
Makviladze, G. M. & Rogatykh, D. I. 1991 Non-uniformities in initial temperature and concentration as a cause of explosive chemical reaction in combustible gases. Combust. Flame 87, 347–356.Google Scholar
Peters, N., Kerschgens, B. & Paczko, G. 2013 Super-knock predictions using a refined theory of turbulence. SAE Intl J. Engines 6, 953–967.Google Scholar
Poludnenko, A. Y., Gardiner, T. A. & Oran, E. S. 2011 Spontaneous transition of turbulent flames to detonation in unconfined media. Phys. Rev. Lett. 107, 054501,1–4.Google Scholar
Radulescu, M. I., Sharpe, G. J. & Bradley, D. 2013 A universal parameter quantifying explosion hazards, detonability and hot spot formation: the number. In Proceedings of the Seventh International Seminar on Fire and Explosion Hazards (ed. Bradley, D., Makhviladze, G., Molikov, V., Sunderland, P. & Taminini, F.). University of Maryland/Publisher, Research Publishing.Google Scholar
Sankaran, R., Hong, G. I., Hawkes, E. R. & Chen, J. H. 2005 The effects of non-uniform temperature distribution on the ignition of a lean homogeneous hydrogen–air mixture. Proc. Combust. Inst. 30, 875–882.Google Scholar
Seitenzahl, I. R., Meakin, C. A., Townley, D. M., Lamb, D. Q. & Truran, J. W. 2009 Spontaneous initiation of detonations in white dwarf environments: determination of critical sizes. Astrophys. J. 696, 515–527.CrossRefGoogle Scholar
Short, M. 1995 The initiation of detonation from general non-uniformly distributed initial conditions. Phil. Trans. R. Soc. Lond. A 353, 173–203.Google Scholar
Short, M. 1996 Homogeneous thermal explosion in a compressible atmosphere. Proc. R. Soc. Lond. A 452, 1127–1138.Google Scholar
Short, M. 1997 On the critical conditions for the initiation of a detonation in a nonuniformly perturbed reactive fluid. SIAM J. Appl. Maths 57, 1242–1280.Google Scholar
Sileem, A., Kassoy, D. R. & Hayashi, A. K. 1991 Thermally initiated detonation through deflagration to detonation transition. Proc. R. Soc. Lond. A 435, 459–482.Google Scholar
Taylor, G. I. 1946 The air wave surrounding an expanding sphere. Proc. R. Soc. Lond. A 186, 273–292.Google Scholar
Taylor, G. I. 1950 The formation of a blast wave by a very intense explosion. Proc. R. Soc. Lond. A 201, 159–174.Google Scholar
Vasquez-Espi’, C. & Liñan, A. 2001 Fast-non-diffusive ignition of a gaseous reacting mixture subject to a point energy addition. Combust. Theor. Model. 5, 495–498.Google Scholar
Zeldovich, Ya. B. 1980 Regime classification of an exothermic reaction with nonuniform initial conditions. Combust. Flame 39, 211–214.CrossRefGoogle Scholar
Zeldovich, Ya. B., Gelfand, B. E., Tsyganov, S.A., Frolov, S. M. & Polenov, A. N. 1988 Concentration and temperature nonuniformities of combustible mixtures as reason for pressure wave generation. Progr. Astronaut. Aeronaut. 114, 99–123.Google Scholar
Zeldovich, Ya. B., Librovich, V. B., Makviladze, G. M. & Sivashinsky, G. J. 1970 On the develoopment of detonation in a non-uniformly preheated gas. Acta Astronaut. 15, 313–321.Google Scholar