Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-23T21:42:16.324Z Has data issue: false hasContentIssue false
  • English
  • Français

On-shell description of unsteady flames

Published online by Cambridge University Press:  11 July 2008

GUY JOULIN ,
HAZEM EL-RABII  and
KIRILI A. KAZAKOV
GUY JOULIN
Affiliation:
Laboratoire de Combustion et de Détonique, CNRS/ENSMA, 1 av. Clément Ader, 86961 Futuroscope, Poitiers, France
HAZEM EL-RABII
Affiliation:
Laboratoire de Combustion et de Détonique, CNRS/ENSMA, 1 av. Clément Ader, 86961 Futuroscope, Poitiers, France
KIRILI A. KAZAKOV
Affiliation:
Department of Theoretical Physics, Physics Faculty, Moscow State University, 119899, Moscow, Russian Federation
Get access Rights & Permissions [Opens in a new window]

Abstract

The problem of a non-perturbative description of unsteady premixed flames with arbitrary gas expansion is addressed in the two-dimensional case. Considering the flame as a surface of discontinuity with arbitrary local burning rate and gas velocity jumps, we show that the flame-front dynamics can be determined without having to solve the flow equations in the bulk. On the basis of the Thomson circulation theorem, an implicit integral representation of the downstream gas velocity is constructed. It is then simplified by a successive stripping of the potential contributions to obtain an explicit expression for the rotational component near the flame front. We prove that the unknown potential component is left bounded and divergence-free by this procedure, and hence can be eliminated using the dispersion relation for its on-shell value (i.e. the value along the flame front). The resulting system of integro-differential equations relates the on-shell fresh-gas velocity and the front position. As limiting cases, these equations contain all the theoretical results on flame dynamics established so far, including the linear equation describing the Darrieus–Landau instability of planar flames, and the nonlinear Sivashinsky–Clavin equation for flames with weak gas expansion.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 608 , 10 August 2008 , pp. 217 - 242
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

Bychkov, V. V. & Liberman, M. A. 2000 Dynamics and stability of premixed flames. Phys. Rep. 325, 115.CrossRefGoogle Scholar
Clavin, P., Masse, L. & Williams, F. A. 2005 Comparison of flame-front instabilities with instabilities of ablation fronts in inertial-confinement fusion. Combust. Sci. Tech. 177, 979.CrossRefGoogle Scholar
Darrieus, G. 1938 Unpublished work presented at La Technique Moderne, Paris.Google Scholar
Frankel, M. L. 1990 An equation of surface dynamics modeling flame fronts as density discontinuities in potential flows. Phys. Fluids A 2, 1879.CrossRefGoogle Scholar
Hayes, W. D. 1957 The vorticity jump across a gasdynamic discontinuity. J. Fluid Mech. 2, 595.CrossRefGoogle Scholar
Hillebrandt, W. & Niemeyer, J. C. 2000 Type Ia supernova explosion models. Annu. Rev. Astron. Astrophys. 38, 191.CrossRefGoogle Scholar
Joulin, G. & Cambray, P. 1992 On a tentative, approximate evolution equation for markedly wrinkled premixed flames. Combust. Sci. Tech. 81, 243.CrossRefGoogle Scholar
Kazakov, K. A. 2005 a Exact equation for curved stationary flames with arbitrary gas expansion. Phys. Rev. Lett. 94, 094501.CrossRefGoogle ScholarPubMed
Kazakov, K. A. 2005 b On-shell description of stationary flames. Phys. Fluids 17, 032107.CrossRefGoogle Scholar
Kazakov, K. A. & Liberman, M. A. 2002 Nonlinear equation for curved stationary flames. Phys. Fluids 14, 1166.CrossRefGoogle Scholar
Landau, L. D. 1944 On the theory of slow combustion. Acta Phys.-Chim USSR 19, 77.Google Scholar
Landau, L. D. & Lifschitz, E. 1987 Fluid Mechanics. Pergamon.Google Scholar
Mallard, E. & Le Chatelier, H. L. 1881 Sur la vitesse de propagation de l'inflammation dans les mélanges gazeux explosifs. C. R. Acad. Sci. (Paris) 93, 145.Google Scholar
Markstein, G. H. 1951 Experimental and theoretical studies of flame front stability. J. Aeron. Sci. 18, 199.CrossRefGoogle Scholar
Markstein, G. H. 1964 Nonsteady Flame Propagation. Pergamon.Google Scholar
Matalon, M. & Matkowsky, B. J. 1982 Flames as gasdynamic discontinuities. J. Fluid Mech. 124, 239.CrossRefGoogle Scholar
Pelce, P. & Clavin, P. 1982 Influences of hydrodynamics and diffusion upon the stability limits of laminar premixed flames. J. Fluid Mech. 124, 219.CrossRefGoogle Scholar
Searby, G. & Rochwerger, D. 1991 A parametric acoustic instability in premixed flames. J. Fluid Mech. 231, 529.CrossRefGoogle Scholar
Sivashinsky, G. I. 1977 Nonlinear analysis of hydrodynamic instability in laminar flames. -I. Derivation of basic equations. Acta. Astron. 4, 1177.CrossRefGoogle Scholar
Sivashinsky, G. I. & Clavin, P. 1987 On the nonlinear theory of hydrodynamic instability in flames. J. Phys. (Paris) 48, 193.CrossRefGoogle Scholar
Zel'dovich, Ya. B., Istratov, A. G., Kidin, N. I. & Librovich, V. B. 1980 Flame propagation in tubes: hydrodynamics and stability. Combust. Sci. Tech. 24, 1.CrossRefGoogle Scholar