Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-19T01:39:59.187Z Has data issue: false hasContentIssue false
  • English
  • Français

Diffusional-thermal instability of diffusion flames

Published online by Cambridge University Press:  26 April 2006

J. S. Kim ,
F. A. Williams  and
P. D. Ronney
J. S. Kim
Affiliation:
Center for Energy and Combustion Research, Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla, CA 92093-0411, USA Present address: Environment Research Center, Korea Institute of Science and Technology, PO Box 131, Cheongryang, Seoul, 130-650 Korea.
F. A. Williams
Affiliation:
Center for Energy and Combustion Research, Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla, CA 92093-0411, USA
P. D. Ronney
Affiliation:
Department of Mechanical Engineering, University of Southern California, Los Angeles, CA 90089-1453, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The diffusional–thermal instability, which gives rise to striped quenching patterns that have been observed for diffusion flames, is analysed by studying the model of a one-dimensional convective diffusion flame in the diffusion-flame regime of activation-energy asymptotics. Attention is focused principally on near-extinction conditions with Lewis numbers less than unity, in which the reactants with high diffusivity diffuse into the strong segments of the reaction sheet, so that the regions between the strong segments become deficient in reactant and subject to the local quenching that leads to the striped patterns. This analysis differs from other flame stability analyses in that the complete description of the dispersion relation is obtained from a composite expansion of the results of an analysis with the conventional convective-diffusive scaling and one with reaction-zone scaling. The results predict that striped patterns will occur, for flames sufficiently close to quasi-steady extinction, with a finite wavenumber that in convective–diffusive scaling is proportional to the cube root of the Zel'dovich number. The convective–diffusive response contributes to the stabilization of long-wavelength disturbances by through positive excess enthalpies by which the flame becomes more resistant to instability, while the reaction-zone response provides stabilization of short-wavelength disturbances by transverse diffusion, within the reactive inner layer, which relaxes the perturbed scalar fields towards their unperturbed states. As quasi-steady extinction is approached, marginal stability arises first at an intermediate range between these two scalings. Parametric results for this bifurcation point are obtained through numerical solutions of the associated generalized eigenvalue problems. Comparisons with measured pattern dimensions for different sets of reactants and diluents reveal excellent qualitative agreement.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 327 , 25 November 1996 , pp. 273 - 301
Copyright
© 1996 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

Barenblatt, G. I., Zel'dovich, Y. B. & Istratov, A. G. 1962 On diffusional-thermal stability of a laminar flame. Prikl. Mekh. Tekhn. Fiz. 4, 21–26 (in Russian).Google Scholar
Buckmaster, J. D. 1992 Flame instability. In Major Research Topics in Combustion (ed. M. Y. Hussaini, A. Kumar & R. G. Voigt). Springer.
Buckmaster, J. D. & Ludford, G. S. S. 1982 Theory of Laminar Flames. Cambridge University Press.
Buckmaster, J. D., Nachman, A. & Taliaferro, S. 1983 The fast time instability of diffusion flames. Physica 9D, 408–424.Google Scholar
Chen, R., Mitchell, G. B. & Ronney, P. D. 1992 Diffusive-thermal instability and flame extinction in nonpremixed combustion. In Twenty-Fourth Symp. (Intl) on Combustion, pp. 213–221. The Combustion Institute, Pittsburgh, PA.
Chung, S. H. & Law, C. K. 1983 Structure and extinction of convective diffusion flames with general Lewis numbers. Combust. Flame 52, 59–79.Google Scholar
Clavin, P. 1985 Dynamic behavior of premixed flame fronts in laminar and turbulent flows. Prog. Energy Combust. Sci. 11, 1–59.Google Scholar
Darrieus, G. 1938 Propagation d'un front de flamme. Unpublished works presented at La Technique Moderne (and at Congrés de Mécanique Appliquée Paris, 1945).
Dongworth, M. R. & Melvin, A. 1976 The transition to instability in a steady hydrogen-oxygen diffusion flame. Combust. Sci. Technol. 14, 177–182.Google Scholar
Garside, J. E. & Jackson, B. 1953 The formation and some properties of polyhedral burner flames. In Fourth Symp. (Intl) on Combustion, pp. 545–552. The Williams and Wilkins Co. Baltimore, MD.
Ishizuka, S. & Tsuji, H. 1981 An experimental study of effect of inert gases on extinction of laminar diffusion flames. In Eighteenth Symp. (Intl) on Combustion, pp. 213–221. The Combustion Institute, Pittsburgh, PA.
Joulin, G. 1986 Shapes, velocities and propagation limits of a non-adiabatic cellular flame. Combust. Sci. Technol. 47, 69–79.Google Scholar
Joulin, G. & Clavin, P. 1979 Linear stability analysis of nonadiabatic flames: diffusional-thermal model. Combust. Flame 35, 139–153.Google Scholar
Joulin, G. & Sivashinsky, G. I. 1983 On the dynamics of nearly-extinguished non-adiabatic cellular flames. Combust. Sci. Technol 31, 75–90.Google Scholar
Kirkby, L. L. & Schmitz, R. A. 1966 An analytical study of the stability of a laminar diffusion flame. Combust. Flame 10, 205–220.Google Scholar
Landau, L. 1944 On the theory of slow combustion. Acta Physicochimica URSS 19, 77–85.Google Scholar
Law, C. K. & Chung, S. H. 1982 Steady state diffusion flame structure with Lewis number variations. Combust. Sci. Technol. 29, 129–145.Google Scholar
Liñán, A. 1974 The asymptotic structure of counterflow diffusion flame for large activation energies. Acta Astronautica 1, 1007–1039.Google Scholar
Melvin, A., Moss, J. B. & Clarke, J. F. 1971 The structure of a reaction-broadened diffusion flame. Combust. Sci. Technol. 4, 17–30.Google Scholar
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics. pp. 766–769. McGraw-Hill.
Peters, N. 1978 On the stability of Liñán's “premixed flame regime”. Combust. Flame 33, 315–318.Google Scholar
Seshadri, K. & Treviño, C. 1989 The influence of the Lewis numbers of the reactants on the asymptotic structure of counterflow and stagnant diffusion flames. Combust. Sci. Technol. 64, 243–261.Google Scholar
Sivashinsky, G. I. 1977 Diffusional-thermal theory of cellular flames. Combust. Sci. Technol. 15, 137–146.Google Scholar
Stewart, D. S. 1986 On the stability of the reaction zone of the plane deflagration. Combust. Flame 64, 157–165.Google Scholar
Stewart, D. S. & Buckmaster, J. D. 1986 The stability of Liñán's “premixed flame regime” revisted. SIAM J. Appl. Maths 46, 582–587.Google Scholar
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I Proc. R. Soc. Lond. A 201, 192–196.Google Scholar
Titchmarsh, E. C. 1946 Eigenfunction Expansions Associated with Second-Order Differential Equations, pp. 107–128. Clarendon Press.
Turing, A. M. 1952 The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37–72.Google Scholar