Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T20:43:02.309Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear and weakly nonlinear instability of a premixed curved flame under the influence of its spontaneous acoustic field

Published online by Cambridge University Press:  07 October 2014

Raphaël C. Assier  and
Xuesong Wu
Raphaël C. Assier*
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
Xuesong Wu
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
*
†Present address: School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK. Email address for correspondence: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The stability of premixed flames in a duct is investigated using an asymptotic formulation, which is derived from first principles and based on high-activation-energy and low-Mach-number assumptions (Wu et al., J. Fluid Mech., vol. 497, 2003, pp. 23–53). The present approach takes into account the dynamic coupling between the flame and its spontaneous acoustic field, as well as the interactions between the hydrodynamic field and the flame. The focus is on the fundamental mechanisms of combustion instability. To this end, a linear stability analysis of some steady curved flames is undertaken. These steady flames are known to be stable when the spontaneous acoustic perturbations are ignored. However, we demonstrate that they are actually unstable when the latter effect is included. In order to corroborate this result, and also to provide a relatively simple model guiding active control, we derived an extended Michelson–Sivashinsky equation, which governs the linear and weakly nonlinear evolution of a perturbed flame under the influence of its spontaneous sound. Numerical solutions to the initial-value problem confirm the linear instability result, and show how the flame evolves nonlinearly with time. They also indicate that in certain parameter regimes the spontaneous sound can induce a strong secondary subharmonic parametric instability. This behaviour is explained and justified mathematically by resorting to Floquet theory. Finally we compare our theoretical results with experimental observations, showing that our model captures some of the observed behaviour of propagating flames.

JFM classification

Acoustics: Acoustics Reacting Flows: Combustion Instability: Instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 758 , 10 November 2014 , pp. 180 - 220
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© 2014 Cambridge University Press

References

Aldredge, R. C. 2005 Saffman–Taylor influence on flame propagation in thermoacoustically excited flow. Combust. Sci. Technol. 177, 53–73.Google Scholar
Al-Shahrany, A. S., Bradley, D., Lawes, M., Liu, K. & Woolley, R. 2006 Darrieus–Landau and thermo-acoustic instabilities in closed vessel explosions. Combust. Sci. Technol. 178, 1771–1802.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
Bloxsidge, G. J., Dowling, A. P. & Langhorne, P. J. 1988 Reheat buzz: an acoustically coupled combustion. Part 2. Theory. J. Fluid Mech. 193, 445–473.Google Scholar
Bychkov, V. V. 1998 Nonlinear equation for a curved stationary flame and the flame velocity. Phys. Fluids 10 (8), 2091–2098.Google Scholar
Bychkov, V. V. 1999 Analytical scalings for flame interaction with sound waves. Phys. Fluids 11 (10), 3168–3173.Google Scholar
Bychkov, V. V. & Liberman, M. A. 2000 Dynamics and stability of premixed flames. Phys. Rep. 325 (4–5), 115–237.Google Scholar
Cambray, P. & Joulin, G. 1994 Length-scales of wrinkling of weakly-forced, unstable premixed flames. Combust. Sci. Technol. 97, 405–428.Google Scholar
Candel, S. 2002 Combustion dynamics and control: progress and challenges. Proc. Combust. Inst. 29, 1–28.CrossRefGoogle Scholar
Clanet, C. & Searby, G. 1998 First experimental study of the Darrieus–Landau instability. Phys. Rev. Lett. 80 (17), 3867–3870.Google Scholar
Clanet, C., Searby, G. & Clavin, P. 1999 Primary acoustic instability of flames propagating in tubes: cases of spray and premixed gas combustion. J. Fluid Mech. 385, 157–197.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
Clavin, P. 1994 Premixed combustion and gasdynamics. Annu. Rev. Fluid Mech. 26, 321–352.CrossRefGoogle Scholar
Clavin, P. & Graña Otero, J. C. 2011 Curved and stretched flames: the two Markstein numbers. J. Fluid Mech. 686, 187–217.CrossRefGoogle Scholar
Clavin, P. & Williams, F. A. 1982 Effects of molecular diffusion and of thermal expansion on the structure and dynamics of premixed flames in turbulent flows of large scale and low intensity. J. Fluid Mech. 116, 251–282.Google Scholar
Creta, F. & Matalon, M. 2011 Propagation of wrinkled turbulent flames in the context of hydrodynamic theory. J. Fluid Mech. 680, 225–264.Google Scholar
Darrieus, G.1938 Propagation d’un front de flamme. Essai de théorie des vitesses anormale de déflagration par développement spontané de la turbulence. Unpublished works presented at La Technique Moderne.Google Scholar
Dowling, A. P. 1995 The calculation of thermoacoustic oscillations. J. Sound Vib. 180 (4), 557–581.CrossRefGoogle Scholar
Dowling, A. P. 1999 A kinematic model of a ducted flame. J. Fluid Mech. 394, 51–72.Google Scholar
Dowling, A. P. & Morgans, A. S. 2005 Feedback control of combustion oscillations. Annu. Rev. Fluid Mech. 37 (1), 151–182.CrossRefGoogle Scholar
Ducruix, S., Candel, S., Durox, D. & Schuller, T. 2003 Combustion dynamics and instabilities: elementary coupling and driving mechanisms. J. Propul. Power 19 (5), 722–734.Google Scholar
Ducruix, S., Durox, D. & Candel, S. 2000 Theoretical and experimental determinations of the transfer function of a laminar premixed flame. Proc. Combust. Inst. 28, 765–773.CrossRefGoogle Scholar
Durox, D., Schuller, T., Noiray, N., Birbaud, A. L. & Candel, S. 2009 Rayleigh criterion and acoustic energy balance in unconfined self-sustained oscillating flames. Combust. Flame 156 (1), 106–119.CrossRefGoogle Scholar
Fogla, N., Creta, F. & Matalon, M. 2013 Influence of the Darrieus–Landau instability on the propagation of planar turbulent flames. Proc. Combust. Inst. 34 (1), 1509–1517.Google Scholar
Gonzalez, M. 1996 Acoustic instability of a premixed flame propagating in a tube. Combust. Flame 107 (3), 245–259.CrossRefGoogle Scholar
Harrje, D. T. & Reardon, F. H.1972 Liquid propellant rocket combustion instability. NASA Rep. SP-194.Google Scholar
Helenbrook, B. T. & Law, C. K. 1999 The role of Landau–Darrieus instability in large scale flows. Combust. Flame 117 (98), 155–169.CrossRefGoogle Scholar
Kerstein, A. R., Ashurst, W. T. & Williams, F. A. 1988 Field equation for interface propagation in an unsteady homogeneous flow field. Phys. Rev. A 37 (7), 2728–2731.Google Scholar
Landau, L. 1944 On the theory of slow combustion. Acta Physicochim. USSR 19, 77–85.Google Scholar
Lieuwen, T. 2003 Modeling premixed combustion–acoustic wave interactions: a review. J. Propul. Power 19 (5), 765–781.CrossRefGoogle Scholar
Lieuwen, T. 2005 Nonlinear kinematic response of premixed flames to harmonic velocity disturbances. Proc. Combust. Inst. 30 (2), 1725–1732.Google Scholar
Lieuwen, T. & Yang, V. 2005 Combustion instabilities in gas turbine engines: operational experience, fundamental mechanisms, and modeling. Prog. Astronaut. Aeronaut. AIAA Paper 210.Google Scholar
Luzzato, C. M., Assier, R. C., Morgans, A. S. & Wu, X.2013 Modelling thermo-acoustic instabilities of an anchored laminar flame in a simple lean premixed combustor: including hydrodynamic effects. In Proceedings of the 19th AIAA/CEAS Aeroacoustics Conference, 2013-2003, pp. 1–15.Google Scholar
Mallard, F. E. & Le Châtelier, H. L. 1882 Étude sur la combustion des mélanges gazeux explosifs. J. Phys. Theor. Appl. 1 (1), 173–183.CrossRefGoogle Scholar
Marble, F. E. & Candel, S. M. 1979 An analytical study of the non-steady behavior of large combustors. Symp. (Intl) Combust. 17 (1), 761–769.CrossRefGoogle Scholar
Markstein, G. H. 1953 Instability phenomena in combustion waves. Proc. Combust. Inst. 4, 44–59.Google Scholar
Markstein, G. H. 1964 Non-steady Flame Propagation. Pergamon.Google Scholar
Markstein, G. H.1970 Flames as amplifiers of fluid mechanical disturbances. In Proceedings of the 6th National Congress for Applied Mechanics, Cambridge, MA, pp. 11–33.Google Scholar
Markstein, G. H. & Squire, W. 1955 On the stability of a plane flame front in oscillating flow. J. Acoust. Soc. Am. 27 (3), 416–424.Google Scholar
Matalon, M. & Matkowsky, B. J. 1982 Flames as gasdynamic discontinuities. J. Fluid Mech. 124, 239–259.Google Scholar
Matkowsky, B. J. & Sivashinsky, G. I. 1979 An asymptotic derivation of two models in flame theory associated with the constant density approximation. SIAM J. Appl. Maths 37 (3), 686–699.CrossRefGoogle Scholar
Michelson, D. M. & Sivashinsky, G. I. 1977 Nonlinear analysis of hydrodynamic instability in laminar flames – II. Numerical experiments. Acta Astron. 4, 1207–1221.Google Scholar
Noiray, N., Durox, D., Schuller, T. & Candel, S. 2008 A unified framework for nonlinear combustion instability analysis based on the flame describing function. J. Fluid Mech. 615, 139–167.Google Scholar
Orszag, S. A. 1971 On the elimination of aliasing in finite-difference schemes by filtering high-wavenumber components. J. Atmos. Sci. 28, 1074.Google Scholar
Pelcé, P. & Clavin, P. 1982 Influence of hydrodynamics and diffusion upon the stability limits of laminar premixed flames. J. Fluid Mech. 124, 219–237.Google Scholar
Pelcé, P. & Rochwerger, D. 1992 Vibratory instability of cellular flames propagating in tubes. J. Fluid Mech. 239, 293–307.CrossRefGoogle Scholar
Piraux, J. & Lombard, B. 2001 A new interface method for hyperbolic problems with discontinuous coefficients: one-dimensional acoustic example. J. Comput. Phys. 168 (1), 227–248.Google Scholar
Poinsot, T., Trouvé, A., Veynante, D., Candel, S. & Esposito, E. 1987 Vortex-driven acoustically coupled combustion instabilities. J. Fluid Mech. 177, 265–292.Google Scholar
Preetham, , Santosh, H. & Lieuwen, T. 2008 Dynamics of laminar premixed flames forced by harmonic velocity disturbances. J. Propul. Power 24 (6), 1390–1402.Google Scholar
Rastigejev, Y. & Matalon, M. 2006 Nonlinear evolution of hydrodynamically unstable premixed flames. J. Fluid Mech. 554, 371–392.Google Scholar
Raushenbakh, B. V. 1961 Vibrational Combustion. Fizmatgiz.Google Scholar
Rayleigh, Lord 1878 The Theory of Sound, vol. 2. Macmillan.Google Scholar
Schadow, K. C. & Gutmark, E. 1992 Combustion instability related to vortex shedding in dump combustors and their passive control. Prog. Energy Combust. Sci. 18, 117–132.Google Scholar
Schuller, T., Durox, D. & Candel, S. 2003 A unified model for the prediction of laminar flame transfer functions. Combust. Flame 134 (1–2), 21–34.Google Scholar
Searby, G. 1992 Acoustic instability in premixed flames. Combust. Sci. Technol. 81, 221–231.CrossRefGoogle Scholar
Searby, G. & Rochwerger, D. 1991 A parametric acoustic instability in premixed flames. J. Fluid Mech. 231, 529–543.Google Scholar
Searby, G. & Truffaut, J. M. 2001 Experimental studies of laminar flame instabilities. In Coherent Structures in Complex Systems (ed. Reguera, D., Bonilla, L. L. & Rubi, J. M.), pp. 159–181. Springer.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.Google Scholar
Shin, D.-H. & Lieuwen, T. 2013 Flame wrinkle destruction processes in harmonically forced, turbulent premixed flames. J. Fluid Mech. 721, 484–513.Google Scholar
Sivashinsky, G. I. 1977 Nonlinear analysis of hydrodynamic instability in laminar flames – I. Derivation of basic equations. Acta Astron. 4, 1177–1206.CrossRefGoogle Scholar
Steinberg, A. M., Boxx, I., Stöhr, M., Carter, C. D. & Meier, W. 2010 Flow–flame interactions causing acoustically coupled heat release fluctuations in a thermo-acoustically unstable gas turbine model combustor. Combust. Flame 157 (12), 2250–2266.Google Scholar
Trefethen, L. N. 2000 Spectral Methods in MATLAB. SIAM.Google Scholar
Vaynblat, D. & Matalon, M. 2000a Stability of pole solutions for planar propagating flames: I. Exact eigenvalues and eigenfunctions. SIAM J. Appl. Maths 60 (2), 679–702.Google Scholar
Vaynblat, D. & Matalon, M. 2000b 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.Google Scholar
Williams, F. A. 1985 Combustion Theory. Benjamin Cummings.Google Scholar
Wu, X. & Law, C. K. 2009 Flame–acoustic resonance initiated by vortical disturbances. J. Fluid Mech. 634, 321–357.Google Scholar
Wu, X. & Moin, P. 2010 Large-activation-energy theory for premixed combustion under the influence of enthalpy fluctuations. J. Fluid Mech. 655, 3–37.Google Scholar
Wu, X., Wang, M., Moin, P. & Peters, N. 2003 Combustion instability due to the nonlinear interaction between sound and flame. J. Fluid Mech. 497, 23–53.Google Scholar
Yang, V. & Culick, F. E. C. 1986 Analysis of low frequency combustion instabilities in a laboratory ramjet combustor. Combust. Sci. Technol. 45 (1–2), 1–25.Google Scholar
Yu, K. H., Trouvé, A. & Daily, J. W. 1991 Low-frequency pressure oscillations in a model ramjet combustor. J. Fluid Mech. 232, 47–72.Google Scholar
Zhang, C. & LeVeque, R. J. 1997 The immersed interface method for acoustic wave equations with discontinuous coefficients. Wave Motion 25 (3), 237–263.Google Scholar