Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-05-16T22:43:04.036Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear dynamics of premixed flames: from deterministic stages to stochastic influence

Published online by Cambridge University Press:  25 September 2020

B. Radisson Open the ORCID record for B. Radisson [Opens in a new window] ,
B. Denet Open the ORCID record for B. Denet [Opens in a new window]  and
C. Almarcha Open the ORCID record for C. Almarcha [Opens in a new window]
B. Radisson
Affiliation:
Aix-Marseille Université, CNRS, École Centrale Marseille, Institut de Recherche sur les Phénomènes Hors Équilibre, UMR 7342, 49 rue F. Joliot Curie, 13013Marseille, France Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA90089-1191, USA
B. Denet
Affiliation:
Aix-Marseille Université, CNRS, École Centrale Marseille, Institut de Recherche sur les Phénomènes Hors Équilibre, UMR 7342, 49 rue F. Joliot Curie, 13013Marseille, France
C. Almarcha*
Affiliation:
Aix-Marseille Université, CNRS, École Centrale Marseille, Institut de Recherche sur les Phénomènes Hors Équilibre, UMR 7342, 49 rue F. Joliot Curie, 13013Marseille, France
*
†Email address for correspondence: almarcha@irphe.univ-mrs.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Premixed flame propagation is a rich interface problem. Instabilities and nonlinearities lead to the formation of cusps pointing toward the burnt gas. These cusps, which undergo complex dynamics, enhance the reaction rate by increasing the flame surface. These crests can be interpreted as pole solutions of the Michelson–Sivashinsky equation that evolve according to ordinary differential equations. Thanks to a quasi-bidimensional experimental facility (a Hele-Shaw burner) we evaluate the accuracy of the description of flame dynamics by elementary interaction between cusps. In particular, we address the time for which a direct comparison between experiments and numerical integration is feasible. The sensitivity to initial conditions and noise is discussed. We demonstrate that at any time of evolution, interesting features can be recovered by describing the flame surface evolution as pole dynamics.

JFM classification

Nonlinear Dynamical Systems: Pattern formation Reacting Flows: Flames
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 903 , 25 November 2020 , A17
Check for updates
Copyright
© The Author(s), 2020. Published by 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

REFERENCES

Al Sarraf, E., Almarcha, C., Quinard, J., Radisson, B., Denet, B. 2018 a Quantitative analysis of flame instabilities in a Hele-Shaw burner. Flow Turbul. Combust. 101 (3), 851–868.CrossRefGoogle Scholar
Al Sarraf, E., Almarcha, C., Quinard, J., Radisson, B., Denet, B., Garcia-Ybarra, P. 2018 b Darrieus–Landau instability and Markstein numbers of premixed flames in a Hele-Shaw cell. Proc. Combust. Inst. 37 (2), 1783–1789.CrossRefGoogle Scholar
Almarcha, C., Denet, B. & Quinard, J. 2015 Premixed flames propagating freely in tubes. Combust. Flame 162 (4), 1225–1233.CrossRefGoogle Scholar
Almarcha, C., Radisson, B., Al Sarraf, E., Villermaux, E., Denet, B. & Quinard, J. 2018 Interface dynamics, pole trajectories, and cell size statistics. Phys. Rev. E 98 (3), 030202.CrossRefGoogle Scholar
Altantzis, C., Frouzakis, C. E., Tomboulides, A. G., Matalon, M. & Boulouchos, K. 2012 Hydrodynamic and thermodiffusive instability effects on the evolution of laminar planar lean premixed hydrogen flames. J. Fluid Mech. 700, 329–361.CrossRefGoogle Scholar
Assier, R. C. & Wu, X. 2014 Linear and weakly nonlinear instability of a premixed curved flame under the influence of its spontaneous acoustic field. J. Fluid Mech. 758, 180–220.CrossRefGoogle Scholar
Bayes, T., Price, R. & Canton, J. 1763 An essay towards solving a problem in the doctrine of chances.Google Scholar
Bessis, D. & Fournier, J. D. 1984 Pole condensation and the Riemann surface associated with a shock in Burgers’ equation. J. Phys. Lett. 45 (17), 833–841.CrossRefGoogle Scholar
Bradley, D. 1999 Instabilities and flame speeds in large-scale premixed gaseous explosions. Phil. Trans. R. Soc. Lond. A 357 (1764), 3567–3581.CrossRefGoogle Scholar
Bychkov, V. & Kleev, A. 1999 The nonlinear equation for curved flames applied to the problem of flames in cylindrical tubes. Phys. Fluids 11 (7), 1890–1895.CrossRefGoogle Scholar
Bychkov, V. V. 1998 Nonlinear equation for a curved stationary flame and the flame velocity. Phys. Fluids 10 (8), 2091–2098.CrossRefGoogle Scholar
Bychkov, V. V., Kovalev, K. A. & Liberman, M. A. 1999 Nonlinear equation for curved nonstationary flames and flame stability. Phys. Rev. E 60 (3), 2897–2911.CrossRefGoogle ScholarPubMed
Cambray, P., Joulain, K. & Joulin, G. 1994 Mean evolution of wrinkle wavelengths in a model of weakly-turbulent premixed flame. Combust. Sci. Technol. 103 (1–6), 265–282.CrossRefGoogle Scholar
Cambray, P. & Joulin, G. 1992 On moderately-forced premixed flames. Symp. Combust. 24 (1), 61–67.CrossRefGoogle Scholar
Clanet, C. & Searby, G. 1998 First experimental study of the Darrieus–Landau instability. Phys. Rev. Lett. 80 (17), 3867–3870.CrossRefGoogle Scholar
Creta, F., Fogla, N. & Matalon, M. 2011 Turbulent propagation of premixed flames in the presence of Darrieus–Landau instability. Combust. Theor. Model. 15 (2), 267–298.CrossRefGoogle Scholar
Creta, F. & Matalon, M. 2011 Strain rate effects on the nonlinear development of hydrodynamically unstable flames. Proc. Combust. Inst. 33 (1), 1087–1094.CrossRefGoogle Scholar
Darrieus, G. 1938 Propagation d'un front de flamme. La Technique Moderne 30, 18.Google Scholar
Denet, B. 2006 Stationary solutions and Neumann boundary conditions in the Sivashinsky equation. Phys. Rev. E 74 (3), 036303.CrossRefGoogle ScholarPubMed
Denet, B. 2007 Sivashinsky equation in a rectangular domain. Phys. Rev. E 75 (4), 046310.CrossRefGoogle Scholar
Du, P., Kibbe, W. A. & Lin, S. M. 2006 Improved peak detection in mass spectrum by incorporating continuous wavelet transform-based pattern matching. Bioinformatics 22 (17), 2059–2065.CrossRefGoogle ScholarPubMed
El-Rabii, H., Joulin, G. & Kazakov, K. A. 2010 Stability analysis of confined V-shaped flames in high-velocity streams. Phys. Rev. E 81 (6), 066312.CrossRefGoogle ScholarPubMed
Escobedo, M., Mischler, S. & Ricard, M. R. 2005 On self-similarity and stationary problem for fragmentation and coagulation models. Ann. Inst. Henri Poincaré 22, 99–125.CrossRefGoogle Scholar
Filyand, L., Sivashinsky, G. I. & Frankel, M. L. 1994 On self-acceleration of outward propagating wrinkled flames. Physica D 72 (1–2), 110–118.CrossRefGoogle Scholar
Fournier, N. & Laurençot, P. 2005 Existence of self-similar solutions to Smoluchowski's coagulation equation. Commun. Math. Phys. 256 (3), 589–609.CrossRefGoogle Scholar
Frankel, M. L. & Sivashinsky, G. I. 1982 The effect of viscosity on hydrodynamic stability of a plane flame front. Combust. Sci. Technol. 29 (3–6), 207–224.CrossRefGoogle Scholar
Friedlander, S. K. & Wang, C. S. 1966 The self-preserving particle size distribution for coagulation by Brownian motion. J. Colloid Interface Sci. 22 (2), 126–132.CrossRefGoogle Scholar
Giada, L., Giacometti, A. & Rossi, M. 2002 Pseudospectral method for the Kardar-Parisi-Zhang equation. Phys. Rev. E 65 (3), 036134.CrossRefGoogle ScholarPubMed
Gostintsev, Y. A., Istratov, A. G. & Shulenin, Y. V. 1989 Self-similar propagation of a free turbulent flame in mixed gas mixtures. Combust. Explos. Shock Waves 24 (5), 563–569.CrossRefGoogle Scholar
Groff, E. G. 1982 The cellular nature of confined spherical propane-air flames. Combust. Flame 48, 51–62.CrossRefGoogle Scholar
Gutkowski, A. & Jarosiński, J. 2009 Flame propagation in narrow channels and mechanism of its quenching. In Combustion Phenomena, Selected Mechanisms of Flame Formation, Propagation and Extinction (ed. Jarosinski, J. & Veyssiere, B.), pp. 102–110. CRC Press.Google Scholar
Gutman, S. & Sivashinsky, G. I. 1990 The cellular nature of hydrodynamic flame instability. Physica D 43 (1), 129–139.CrossRefGoogle Scholar
Hidy, G. M. 1965 On the theory of the coagulation of noninteracting particles in Brownian motion. J. Colloid Sci. 20 (2), 123–144.CrossRefGoogle Scholar
Istratov, A. G. & Librovich, V. V. 1969 Stability of flames. Tech. Rep. Army Foreign Science and Technology Center.Google Scholar
Jang, H. J., Jang, G. M. & Kim, N. I. 2018 Unsteady propagation of premixed methane/propane flames in a mesoscale disk burner of variable-gaps. Proc. Combust. Inst. 37 (2), 1861–1868.CrossRefGoogle Scholar
Jomaas, G., Law, C. K. & Bechtold, J. K. 2007 On transition to cellularity in expanding spherical flames. J. Fluid Mech. 583, 1–26.CrossRefGoogle Scholar
Joulin, G. 1988 On a model for the response of unstable premixed flames to turbulence. Combust. Sci. Technol. 60 (1–3), 1–5.CrossRefGoogle Scholar
Joulin, G. 1989 On the hydrodynamic stability of curved premixed flames. J. Phys. 50 (9), 1069–1082.CrossRefGoogle Scholar
Joulin, G. & Cambray, P. 1992 On a tentative, approximate evolution equation for markedly wrinkled premixed flames. Combust. Sci. Technol. 81 (4–6), 243–256.CrossRefGoogle Scholar
Joulin, G., El-Rabii, H. & Kazakov, K. A. 2008 On-shell description of unsteady flames. J. Fluid Mech. 608, 217–242.CrossRefGoogle Scholar
Joulin, G. & Sivashinsky, G. 1994 Influence of momentum and heat losses on the large-scale stability of Quasi-2D premixed flames. Combust. Sci. Technol. 98 (1–3), 11–23.CrossRefGoogle Scholar
Karlin, V. 2002 Cellular flames may exhibit a non-modal transient instability. Proc. Combust. Inst. 29 (2), 1537–1542.CrossRefGoogle Scholar
Karlin, V. 2004 Estimation of the linear transient growth of perturbations of cellular flames. Math. Models Meth. Appl. Sci. 14 (8), 1191–1210.CrossRefGoogle Scholar
Karlovitz, B., Denniston, D. W. & Wells, F. E. 1951 Investigation of turbulent flames. J. Chem. Phys. 19 (5), 541–547.CrossRefGoogle Scholar
Kazakov, K. A. 2005 a Exact equation for curved stationary flames with arbitrary gas expansion. Phys. Rev. Lett. 94 (9), 094501.CrossRefGoogle ScholarPubMed
Kazakov, K. A. 2005 b On-shell description of stationary flames. Phys. Fluids 17 (3), 032107.CrossRefGoogle Scholar
Kazakov, K. A. & Liberman, M. A. 2002 a Effect of vorticity production on the structure and velocity of curved flames. Phys. Rev. Lett. 88 (6), 064502.CrossRefGoogle ScholarPubMed
Kazakov, K. A. & Liberman, M. A. 2002 b Nonlinear equation for curved stationary flames. Phys. Fluids 14 (3), 1166–1181.CrossRefGoogle Scholar
Kupervasser, O., Olami, Z. & Procaccia, I. 1996 Geometry of developing flame fronts: analysis with pole decomposition. Phys. Rev. Lett. 76 (1), 146.CrossRefGoogle ScholarPubMed
Landau, L. D. 1944 On the theory of slow combustion. Acta Phys. 19, 77–85.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon.Google Scholar
Laplace, P.-S. 1774 Mémoire sur la probabilité des causes par les évènements. Memoires de Mathematiques et Physique, Presentés à l'Académie Royale des Sciences 66, 621–656.Google Scholar
Layzer, D. 1955 On the instability of superposed fluids in a gravitational field. Astrophys. J. 122, 1.CrossRefGoogle Scholar
Lee, Y. C. & Chen, H. H. 1982 Nonlinear dynamical models of plasma turbulence. Phys. Scr. T2A, 41–47.CrossRefGoogle Scholar
Mallard, E. & Le Chatelier, H. 1883 Recherches Expérimentales et Théoriques Sur La Combustion Des Mélanges Gazeux Explosives. H. Dunod et E. Pinat.Google Scholar
Manton, J., von Elbe, G. & Lewis, B. 1952 Nonisotropic propagation of combustion waves in explosive gas mixtures and the development of cellular flames. J. Chem. Phys. 20 (1), 153–157.CrossRefGoogle Scholar
Markstein, G. H. 1949 Cell structure of propane flames burning in tubes. J. Chem. Phys. 17 (4), 428–429.CrossRefGoogle Scholar
Markstein, G. H. 1951 Experimental and theoretical studies of flame-front stability. J. Aeronaut. Sci. 18 (3), 199–209.CrossRefGoogle Scholar
Matalon, M. & Matkowsky, B. J. 1982 Flames as gasdynamic discontinuities. J. Fluid Mech. 124 (1), 239–259.CrossRefGoogle Scholar
Meunier, P. & Leweke, T. 2003 Analysis and treatment of errors due to high velocity gradients in particle image velocimetry. Exp. Fluids 35 (5), 408–421.CrossRefGoogle Scholar
Michelson, D. M. & Sivashinsky, G. I. 1977 Nonlinear analysis of hydrodynamic instability in laminar flames–II. Numerical experiments. Acta Astron. 4 (11–12), 1207–1221.CrossRefGoogle Scholar
Michelson, D. M. & Sivashinsky, G. I. 1982 Thermal-expansion induced cellular flames. Combust. Flame 48, 211–217.CrossRefGoogle Scholar
Olami, Z., Galanti, B., Kupervasser, O. & Procaccia, I. 1997 Random noise and pole dynamics in unstable front propagation. Phys. Rev. E 55 (3), 2649.CrossRefGoogle Scholar
Patyal, A. & Matalon, M. 2018 Nonlinear development of hydrodynamically-unstable flames in three-dimensional laminar flows. Combust. Flame 195, 128–139.CrossRefGoogle Scholar
Pelce, P. & Clavin, P. 1982 Influence of hydrodynamics and diffusion upon the stability limits of laminar premixed flames. J. Fluid Mech. 124 (1), 219–237.CrossRefGoogle Scholar
Pelce, P. & Clavin, P. 1987 The stability of curved fronts. Europhys. Lett. 3 (8), 907–913.CrossRefGoogle Scholar
Pelcé-Savornin, C., Quinard, J. & Searby, G. 1988 The flow field of a curved flame propagating freely upwards. Combust. Sci. Technol. 58 (4–6), 337–346.CrossRefGoogle Scholar
Quinard, J. 1984 Limite de stabilité et structures cellulaires dans les flammes de prémélange: étude expérimentale. PhD thesis.Google Scholar
Rabaud, M., Couder, Y. & Gerard, N. 1988 Dynamics and stability of anomalous Saffman-Taylor fingers. Phys. Rev. A 37 (3), 935–947.CrossRefGoogle ScholarPubMed
Radisson, B., Piketty-Moine, J. & Almarcha, C. 2019 Coupling of vibro-acoustic waves with premixed flame. Phys. Rev. Fluids 4 (12), 121201.CrossRefGoogle Scholar
Rahibe, M., Aubry, N., Sivashinsky, G. I. & Lima, R. 1995 Formation of wrinkles in outwardly propagating flames. Phys. Rev. E 52 (4), 3675–3686.CrossRefGoogle ScholarPubMed
Rastigejev, Y. & Matalon, M. 2006 a Nonlinear evolution of hydrodynamically unstable premixed flames. J. Fluid Mech. 554 (1), 371–392.CrossRefGoogle Scholar
Rastigejev, Y. & Matalon, M. 2006 b Numerical simulation of flames as gas-dynamic discontinuities. Combust. Theor. Model. 10 (3), 459–481.CrossRefGoogle Scholar
Saffman, P. G. & Taylor, G. 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245 (1242), 312–329.Google Scholar
Searby, G. & Quinard, J. 1990 Direct and indirect measurements of Markstein numbers of premixed flames. Combust. Flame 82 (3–4), 298–311.CrossRefGoogle Scholar
Searby, G., Truffaut, J.-M. & Joulin, G. 2001 Comparison of experiments and a nonlinear model equation for spatially developing flame instability. Phys. Fluids 13 (11), 3270–3276.CrossRefGoogle Scholar
Sharif, J., Abid, M. & Ronney, P. D. 1999 Premixed-gas flame propagation in Hele-Shaw cells. In Spring Technical Meeting, Joint US Sections (15–17 March 1999). Combustion Institute, Washington, DC.Google Scholar
Sivashinsky, G. I. 1977 Nonlinear analysis of hydrodynamic instability in laminar flames–I. Derivation of basic equations. Acta Astron. 4 (11–12), 1177–1206.CrossRefGoogle Scholar
Sivashinsky, G. I. & Clavin, P. 1987 On the nonlinear theory of hydrodynamic instability in flames. J. Phys. 48 (2), 193–198.CrossRefGoogle Scholar
Smithells, A. & Ingle, H. 1892 XV.–The structure and chemistry of flames. J. Chem. Soc. Trans. 61, 204–216.CrossRefGoogle Scholar
Swift, D. L. & Friedlander, S. K. 1964 The coagulation of hydrosols by Brownian motion and laminar shear flow. J. Colloid Sci. 19 (7), 621–647.CrossRefGoogle Scholar
Taylor, G. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A 201 (1065), 192–196.Google Scholar
Thual, O., Frisch, U. & Hénon, M. 1985 Application of pole decomposition to an equation governing the dynamics of wrinkled flame fronts. J. Phys. 46 (9), 1485–1494.CrossRefGoogle Scholar
Truffaut, J.-M. & Searby, G. 1999 Experimental study of the Darrieus–Landau instability on an inverted-‘V’ flame, and measurement of the Markstein number. Combust. Sci. Technol. 149 (1–6), 35–52.CrossRefGoogle Scholar
Uberoi, M. S. 1959 Flow field of a flame in a channel. Phys. Fluids 2 (1), 72–78.CrossRefGoogle Scholar
Vaynblat, D. & Matalon, M. 2000 a Stability of pole solutions for planar propagating flames. I. Exact eigenvalues and eigenfunctions. SIAM J. Appl. Maths 60 (2), 679–702.CrossRefGoogle Scholar
Vaynblat, D. & Matalon, M. 2000 b Stability of pole solutions for planar propagating flames. II. Properties of eigenvalues/eigenfunctions and implications to stability. SIAM J. Appl. Maths 60 (2), 703–728.CrossRefGoogle Scholar
Villermaux, E. & Duplat, J. 2003 Mixing is an aggregation process le mélange est un processus d'agrégation. C. R. Mecanique 7 (331), 515–523.CrossRefGoogle Scholar
Von Lavante, E. & Strehlow, R. A. 1983 The mechanism of lean limit flame extinction. Combust. Flame 49 (1–3), 123–140.CrossRefGoogle Scholar
Yu, R., Bai, X.-S. & Bychkov, V. 2015 Fractal flame structure due to the hydrodynamic Darrieus–Landau instability. Phys. Rev. E 92 (6), 063028.CrossRefGoogle ScholarPubMed
Zeldovich, Y. B., Istratov, A. G., Kidin, N. L. & Librovich, V. B. 1980 Flame propagation in tubes: hydrodynamics and stability. Combust. Sci. Technol. 24, 1–13.CrossRefGoogle Scholar