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A unified framework for nonlinear combustion instability analysis based on the flame describing function

  • N. NOIRAY (a1), D. DUROX (a1), T. SCHULLER (a1) and S. CANDEL (a1)

Abstract

Analysis of combustion instabilities relies in most cases on linear analysis but most observations of these processes are carried out in the nonlinear regime where the system oscillates at a limit cycle. The objective of this paper is to deal with these two manifestations of combustion instabilities in a unified framework. The flame is recognized as the main nonlinear element in the system and its response to perturbations is characterized in terms of generalized transfer functions which assume that the gain and phase depend on the amplitude level of the input. This ‘describing function’ framework implies that the fundamental frequency is predominant and that the higher harmonics generated in the nonlinear element are weak because the higher frequencies are filtered out by the other components of the system. Based on this idea, a methodology is proposed to investigate the nonlinear stability of burners by associating the flame describing function with a frequency-domain analysis of the burner acoustics. These elements yield a nonlinear dispersion relation which can be solved, yielding growth rates and eigenfrequencies, which depend on the amplitude level of perturbations impinging on the flame. This method is used to investigate the regimes of oscillation of a well-controlled experiment. The system includes a resonant upstream manifold formed by a duct having a continuously adjustable length and a combustion region comprising a large number of flames stabilized on a multipoint injection system. The growth rates and eigenfrequencies are determined for a wide range of duct lengths. For certain values of this parameter we find a positive growth rate for vanishingly small amplitude levels, indicating that the system is linearly unstable. The growth rate then changes as the amplitude is increased and eventually vanishes for a finite amplitude, indicating the existence of a limit cycle. For other values of the length, the growth rate is initially negative, becomes positive for a finite amplitude and drops to zero for a higher value. This indicates that the system is linearly stable but nonlinearly unstable. Using calculated growth rates it is possible to predict amplitudes of oscillation when the system operates on a limit cycle. Mode switching and instability triggering may also be anticipated by comparing the growth rate curves. Theoretical results are found to be in excellent agreement with measurements, indicating that the flame describing function (FDF) methodology constitutes a suitable framework for nonlinear instability analysis.

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Abugov, D. & Obrezgov, O. 1978 Acoustic noise in turbulent flames. Combust., Explosions Shock Waves 14, 606612.
Ananthkrishnan, N., Deo, S. & Culick, F. E. C. 2005 Reduced-order modeling and dynamics of nonlinear acoustic waves in a combustion chamber. Combust. Sci. Technol. 177, 221247.
Armitage, C. A., Balachandran, R., Mastorakos, E. & Cant, R. S. 2006 Investigation of the nonlinear response of turbulent premixed flames to imposed inlet velocity oscillations. Combust. Flame 146, 419436.
Balachandran, R., Ayoola, B. O., Kaminski, C. F., Dowling, A. P. & Mastorakos, E. 2005 Experimental investigation of the nonlinear response of turbulent premixed flames to imposed inlet velocity oscillations. Combust. Flame 143, 3755.
Balasubramanian, K. & Sujith, R. I. 2008 Non-normality and nonlinearity in combustion-acoustic interaction in diffusion flames. J. Fluid Mech. 594, 2957.
Bellows, B. D., Bobba, M. K., Forte, A., Seitzman, J. M. & Lieuwen, T. 2007 Flame transfer function saturation mechanisms in a swirl-stabilized combustor. Proc. Combust. Inst. 31, 31813188.
Birbaud, A. L., Durox, D., Ducruix, S. & Candel, S. 2007 Dynamics of confined premixed flames submitted to upstream acoustic modulations. Proc. Combust. Inst. 31, 12571265.
Burnley, V. S. & Culick, F. E. C. 1999 On the energy transfer between transverse acoustic modes in a cylindrical combustion chamber. Combust. Sci. Technol. 144, 119.
Clavin, P. & Siggia, E. D. 1991 Turbulent premixed flames and sound generation. Combust. Sci. Technol. 78, 147155.
Crocco, L. 1951 Aspects of combustion instability in liquid propellant rocket motors. J. Am. Rocket Soc. 21, 163178.
Crocco, L., Grey, J. & T., Harrje D. 1960 Theory of liquid propellant rocket combustion instability and its experimental verification. J. Am. Rocket Soc. 30, 159168.
Culick, F. E. C. 1994 Some recent results for nonlinear acoustics in combustion-chambers. AIAA J. 32, 146169.
Culick, F. E. C. 2006 Unsteady motions in combustion chambers for propulsion systems. AGARDograph, NATO/RTO-AG-AVT-039.
Culick, F. E. C., Burnley, V. & Swenson, G. 1995 Pulsed instabilities in solid-propellant rockets. J. Propuls. Power 11, 657665.
Dowling, A. P. 1997 Nonlinear self-excited oscillations of ducted flame. J. Fluid Mech. 346, 271290.
Dowling, A. P. 1999 A kinematic model of ducted flame. J. Fluid Mech. 394, 5172.
Ducruix, S., Durox, D. & Candel, S. 2000 Theoretical and experimental determination of the transfer function of a laminar premixed flame. Proc. Combust. Inst. 28, 765773.
Fleifil, M., Annaswamy, A., Ghoneim, Z. & Ghoniem, A. 1996 Response of a laminar premixed flame to flow oscillations: a kinematic model and thermoacoustic instability results. Combust. Flame 106, 487510.
Howe, M. S. 1979 On the theory of unsteady high reynolds number flow through a circular aperture. Proc. R. Soc. Lond. A 366, 205223.
Hurle, I. R., Price, R. B., Sudgen, T. M. & Thomas, A. 1968 Sound emission from open turbulent premixed flames. Proc. R. Soc. Lond. A 303, 409427.
Jahnke, C. C. & Culick, F. E. C. 1994 Application of dynamical systems theory to nonlinear combustion instabilities. J. Propuls. Power 10, 508517.
Joulin, G. & Sivashinsky, G. I. 1991 Pockets in premixed flames and combustion rate. Combust. Sci. Technol. 77, 329335.
Kang, D. M., Culick, F. E. C. & Ratner, A. 2007 Combustion dynamics of a low-swirl combustor. Combust. Flame 151, 412425.
Keller, J. O. & Saito, K. 1987 Measurements of the combustion flow in a pulse combustor. Combust. Sci. Technol. 53, 137163.
Krebs, W., Bethke, S., Lepers, J., Flohr, P., Prade, B., Johnson, C. & Sattinger, S. 2005 Thermoacoustic design tools and passive control: Siemens power generation approaches. In Combustion Instabilities in Gas Turbines, Operational Experience, Fundamental Mechanisms, and Modeling (ed. Lieuwen, T. C. & Yang, V.). AIAA.
Lawn, C. J., Evesque, S. & Polifke, W. 2004 A model for the thermoacoustic response of a premixed swirl burner, part 1: acoustic aspects. Combust. Sci. Technol. 176, 13311358.
Lee, J. G. & Santavicca, D. A. 2003 Experimental diagnostics for the study of combustion instabilities in lean premixed combustors. J. Propuls. Power 19, 735750.
Lieuwen, T. 2002 Experimental investigation of limit-cycle oscillations in an unstable gas turbine combustor. J. Propuls. Power 18, 6167.
Lieuwen, T. 2005 Nonlinear kinematic response of premixed flames to harmonic velocity disturbances. Proc. Combust. Inst. 30, 17251732.
Lieuwen, T. & Neumeier, Y. 2002 Nonlinear pressure-heat release transfer function measurements in a premixed combustor. Proc. Combust. Inst. 29, 99105.
Lieuwen, T. C. & Yang, V. (Ed.) 2005 Combustion Instabilities in Gas Turbines, Operational Experience, Fundamental Mechanisms, and Modeling. AIAA.
Martin, C. E., Benoit, L., Sommerer, Y., Nicoud, F. & Poinsot, T. 2006 Large-eddy simulation and acoustic analysis of a swirled staged turbulent combustor. AIAA J. 44, 741750.
Matsui, Y. 1981 An experimental study on pyro-acoustic amplification of premixed laminar flames. Combust. Flame 43, 199209.
Melling, T. H. 1973 The acoustic impedance of perforates at medium and high sound pressure levels. J. Sound Vib. 29, 165.
Minorsky, N. 1962 Nonlinear oscillations. D. Van Nostrand.
Morgans, A. S. & Stow, S. R. 2007 Model-based control of combustion instabilities in annular combustors. Combust. Flame 150, 380399.
Noiray, N., Durox, D., Schuller, T. & Candel, S. 2006 Self-induced instabilities of premixed flames in a multiple injection configuration. Combust. Flame 145, 435446.
Noiray, N., Durox, D., Schuller, T. & Candel, S. 2007 Passive control of combustion instabilities involving premixed flames anchored on perforated plates. Proc. Combust. Inst. 31, 12831290.
Paschereit, C. O., Schuermans, B., Polifke, W. & Mattson, O. 2002 Measurement of transfer matrices and source terms of premixed flames. Trans. ASME: J. Engng Gas Turbines Power, 124, 239247.
Peracchio, A. A. & Proscia, W. M. 1999 Nonlinear heat-release/acoustic model for thermoacoustic instability in lean premixed combustors. Trans. ASME: J. Engng Gas Turbines Power 121, 415421.
Poinsot, T. & Candel, S. 1988 A nonlinear model for ducted flame combustion instabilities. Combust. Sci. Technol. 61, 121153.
Poinsot, T., Veynante, D., Bourienne, F., Candel, S., Esposito, E. & Surjet, J. 1988 Initiation and suppression of combustion instabilities by active control. Proc. Combust. Inst. 22, 13631370.
Poinsot, T., Yip, B., Veynante, D., Trouve, A., Samaniengo, J. M. & Candel, S. 1992 Active control – an investigation method for combustion instabilities. J. Phys. Paris III 2, 13311357.
Price, R. B., Hurle, I. R. & Sudgen, T. M. 1968 Optical studies of generation of noise in turbulent flames. Proc. Combust. Inst. 12, 10931102.
Rienstra, S. W. & Hirschberg, A. 2005 An Introduction to Acoustics. Eindhoven University of Technology: Report IWDE 92-06.
Roux, S., Lartigue, G., Poinsot, T. & Bérat, T. 2005 Studies of mean and unsteady flow in a swirled combustor using experiments, acoustic analysis and large eddy simulations. Combust. Flame 141, 4054.
Sattelmayer, T. 2003 Influence of the combustor aerodynamics on combustion instabilities from equivalence ratio fluctuations. Trans. ASME. J. Engng for Gas Turbines and Power 125, 1119.
Schuller, T., Durox, D. & Candel, S. 2002 Dynamics of and noise radiated by a perturbed impinging premixed jet flame. Combust. Flame 128, 88110.
Schuller, T., Durox, D. & Candel, S. 2003 a Self-induced combustion oscillations of laminar premixed flames stabilized on annular burners. Combust. Flame 135, 525537.
Schuller, T., Durox, D. & Candel, S. 2003 b A unified model for the prediction of laminar flame transfer functions : comparisons between conical and v-flame dynamics. Combust. Flame 134, 2134.
Searby, G. & Rochwerger, D. 1991 A parametric acoustic instability in premixed flames. J. Fluid Mech. 231, 529543.
Strahle, W. C. 1978 Combustion noise. Prog. Energy Combust. Sci. 4, 157176.
Wicker, J. M., Greene, W. D., Kim, S. & Yang, V. 1996 Triggering of longitudinal combustion instabilities in rocket motors: Nonlinear combustion response. J. Propul. Power 12, 11481158.
Wu, X. S., Wang, M., Moin, P. & Peters, N. 2003 Combustion instability due to the nonlinear interaction between sound and flame. J. Fluid Mech. 497, 2353.
Yang, V., Kim, S. I. & Culick, F. E. C. 1990 Triggering of longitudinal pressure oscillations in combustion chambers. 1. nonlinear gas dynamics. Combust. Sci. Technol. 72, 183214.
Zinn, B. T. & Lores, M. E. 1972 Application of galerkin method in solution of nonlinear axial combustion instability problems in liquid rockets. Combust. Sci. Technol. 4, 269.
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