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Non-normality and internal flame dynamics in premixed flame–acoustic interaction

Published online by Cambridge University Press:  13 May 2011

Department of Aerospace Engineering, Indian Institute of Technology Madras, Chennai 600036, India
Department of Aerospace Engineering, Indian Institute of Technology Madras, Chennai 600036, India
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This paper investigates the non-normal nature of premixed flame–acoustic interaction. The thermoacoustic system is modelled using the acoustic equations for momentum and energy, together with the equation for the evolution of the flame front obtained from the kinematic G-equation. As the unsteady heat addition acts as a volumetric source, the flame front is modelled as a distribution of monopole sources. Evolutions of the system are characterized with a measure of energy due to fluctuations. In addition to the acoustic energy, the energy due to fluctuations considered in the present paper accounts for the energy of the monopole sources. The linearized operator for this thermoacoustic system is non-normal, leading to non-orthogonality of its eigenvectors. Non-orthogonal eigenvectors can cause transient growth even when all the eigenvectors are decaying. Therefore, classical linear stability theory cannot predict the finite-time transient growth observed in non-normal systems. In the present model, the state space variables include the monopole source strengths in addition to the acoustic variables. Inclusion of these variables in the state space is essential to account for the transient growth due to non-normality. A parametric study of the variation in transient growth due to change in parameters such as flame location and flame angle is performed. In addition to projections along the acoustic variables of velocity and pressure, the optimal initial condition for the self-evolving system has significant projections along the strength of the monopole distribution. Comparison of linear and corresponding nonlinear evolutions highlights the role of transient growth in subcritical transition to instability. The notion of phase between acoustic pressure and heat release rate as an indicator of stability is examined.

Copyright © Cambridge University Press 2011

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Annaswamy, A. M., Fleifil, M., Hathout, J. P. & Ghoneim, A. F. 1997 Impact of linear coupling on the design of active controllers for the thermoacoustic instability. Combust. Sci. Technol. 128, 131180.CrossRefGoogle Scholar
Bakas, N. A. 2009 Mechanisms underlying transient growth of planar perturbations in unbounded compressible shear flow. J. Fluid Mech. 639, 479507.CrossRefGoogle Scholar
Balasubramanian, K. & Sujith, R. I. 2008 a Non-normality and nonlinearity in combustion acoustic interaction in diffusion flames. J. Fluid Mech. 594, 2957.CrossRefGoogle Scholar
Balasubramanian, K. & Sujith, R. I. 2008 b Thermoacoustic instability in a Rijke tube: nonnormality and nonlinearity. Phys. Fluids 20, 044103.CrossRefGoogle Scholar
Bloomshield, F. S., Crump, J. E., Mathes, H. B., Stalnaker, R. A. & Beckstead, M. W. 1997 Nonlinear stability testing of full scale tactical motors. J. Propul. Power 13 (3), 356366.CrossRefGoogle Scholar
Boyer, L. & Quinard, J. 1990 On the dynamics of anchored flames. Combust. Flame 82, 5165.CrossRefGoogle Scholar
Candel, S. 2002 Combustion dynamics and control: progress and challenges. Proc. Combust. Inst. 29, 128.CrossRefGoogle Scholar
Chagelishvili, G. D., Rogava, A. D. & Segal, I. N. 1994 Hydrodynamic stability of compressible plane Couette flow. Phys. Rev. E 50, 42834285.Google ScholarPubMed
Chu, B. T. 1964 On the energy transfer to small disturbances in fluid flow (Part I). Acta Mechanica 1 (3), 215234.CrossRefGoogle Scholar
Chu, B. T. & Kovasznay, L. S. G. 1957 Non-linear interactions in a viscous heat-conducting compressible gas. J. Fluid Mech. 3, 494514.CrossRefGoogle Scholar
Coats, C. M. 1996 Coherent structures in combustion. Prog. Energy Combust. Sci. 22, 427509.CrossRefGoogle Scholar
Dowling, A. P. 1997 Nonlinear self-excited oscillations of a ducted flame. J. Fluid Mech. 346, 271290.CrossRefGoogle Scholar
Dowling, A. P. 1999 A kinematic model of a ducted flame. J. Fluid Mech. 394, 5172.CrossRefGoogle Scholar
Dowling, A. P. & Morgens, A. S. 2005 Feedback control of combustion oscillations. Annu. Rev. Fluid Mech. 37, 151182.CrossRefGoogle Scholar
Dowling, A. P. & Williams, F. W. 1983 Sound and Sources of Sound. Ellis Horwood.Google Scholar
Farrell, B. F. & Ioannou, P. J. 2000 Transient and asymptotic growth of two-dimensional perturbations in viscous compressible shear flow. Phys. Fluids 12, 30213028.CrossRefGoogle Scholar
Fleifil, M., Annaswamy, A., Ghoneim, Z. & Ghoneim, A. 1996 Response of a laminar premixed flame to flow oscillations: a kinematic model and thermoacoustic instability results. Combust. Flame 106, 487510.CrossRefGoogle Scholar
Giauque, A., Poinsot, T., Brear, M. J. & Nicoud, F. 2006 Budget of disturbance energy in gaseous reacting flows. In Proc. 2006 Summer Program, Center for Turbulence Research, Stanford University.Google Scholar
Hirsch, M. W., Smale, S. & Devaney, R. L. 2004 Differential Equations, Dynamical Systems and an Introduction to Chaos, 2nd edn. Academic Press/Elsevier.Google Scholar
Howe, M. S. 2003 Theory of Vortex Sound. Cambridge University Press.Google Scholar
Jiang, G. S. & Shu, C. W. 1996 Efficient implementation of weighted WENO schemes. J. Comput. Phys. 126, 202228.CrossRefGoogle Scholar
Juniper, M. P. 2010 Triggering in the horizontal Rijke tube: non-normality, transient growth and bypass transition. J. Fluid Mech. 667, 272308.CrossRefGoogle Scholar
van Kampen, J. F. 2006 Acoustic pressure oscillations induced by confined turbulent premixed natural gas flames. PhD thesis, University of Twente.Google Scholar
Karimi, N., Brear, M., Jin, S.-H. & Monty, J. P. 2009 Linear and nonlinear forced response of conical, ducted, laminar, premixed flames. Combust. Flame 156, 22012212.CrossRefGoogle Scholar
Kerstein, A. R., Ashurst, W. T. & Williams, F. A. 1988 Field equation for interface propagation in an unsteady homogenous flow field. Phys. Rev. A 37 (7), 27282731.CrossRefGoogle Scholar
Lieuwen, T. 2003 Modeling premixed combustion–acoustic wave interactions: a review. J. Propul. Power 195, 765781.CrossRefGoogle Scholar
Lieuwen, T. 2005 Nonlinear kinematic response of premixed flames to harmonic velocity disturbances. Proc. Combust. Inst. 30, 17251732.CrossRefGoogle Scholar
Lieuwen, T. & Zinn, B. T. 1998 The role of equivalence ratio oscillations in driving combustion instabilities in low NOX gas turbines. In 27th Intl Symp. on Combustion, pp. 1809–1816.Google Scholar
Lighthill, M. J. 1952 On sound generated aerodynamically. I. General theory. Proc. R. Soc. Lond. A 211, 564587.CrossRefGoogle Scholar
Mack, L. M. 1969 Boundary-layer stability theory. Tech. Rep. Doc. 900–277. JPL.Google Scholar
Mariappan, S. & Sujith, R. I. 2010 Thermoacoustic instability in a solid rocket motor: non-normality and nonlinear instabilities. J. Fluid Mech. 653, 133.CrossRefGoogle Scholar
Markstein, G. H. 1964 Nonsteady Flame Propagation. Pergamon.Google Scholar
Matveev, K. 2003 a A model for combustion instability involving vortex shedding. Combust. Sci. Technol. 175 (6), 10591083.CrossRefGoogle Scholar
Matveev, K. 2003 b Thermoacoustic instabilities in the Rijke tube: experiments and modeling. PhD thesis, California Institute of Technology, Pasadena.Google Scholar
McManus, K., Poinsot, T. & Candel, S. 1993 A review of active control of combustion instabilities. Prog. Energy Combust. Sci. 19, 129.CrossRefGoogle Scholar
Meirovitch, L. 1967 Analytical Methods in Vibration. Macmillan.Google Scholar
Morfey, C. L. 1971 Sound transmission and generation in ducts with flow. J. Sound Vib. 14 (1), 3755.CrossRefGoogle Scholar
Morse, P. M. & Ingard, K. U. 1968 Theoretical Acoustics. McGraw-Hill.Google Scholar
Myers, M. K. 1991 Transport of energy by disturbances in arbitrary steady flows. J. Fluid Mech. 226, 383400.CrossRefGoogle Scholar
Nicoud, F., Benoit, A., Sensiau, C. & Poinsot, T. 2007 Acoustic modes in combustors with complex impedances and multidimensional active flames. AIAA J. 45 (2), 426441.CrossRefGoogle Scholar
Nicoud, F. & Poinsot, T. 2005 Thermoacoustic instabilities: should the Rayleigh criterion be extended to include entropy changes? Combust. Flame 142, 153159.CrossRefGoogle Scholar
Nicoud, F. & Wieczorek, K. 2009 About the zero Mach number assumption in the calculation of thermoacoustic instabilities. Intl J. Spray Combust. Dyn. 1 (1), 67111.CrossRefGoogle Scholar
Noiray, N., Durox, D., Schuller, T. & Candel, S. 2006 Self-induced instabilities of premixed flames in a multiple injection configuration. Combust. Flame 145, 435446.CrossRefGoogle 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, 139167.CrossRefGoogle Scholar
Poinsot, T. J., Trouve, A. C., Veynante, D. P., Candel, S. M. & Esposito, E. J. 1987 Vortex-driven acoustically coupled combustion instabilities. J. Fluid Mech. 177, 265292.CrossRefGoogle Scholar
Rienstra, S. W. & Hirschberg, A. 2008 An introduction to acoustics. IWDE Rep. 92–06.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Schuermans, B., Bellucci, V., Guethe, F., Meili, F., Flohr, P. & Paschereit, O. 2004 A detailed analysis of thermoacoustic interaction mechanisms in a turbulent premixed flame. In Proc. ASME Turbo Expo 2004: Power for Land, Sea, and Air, 14–17 June, Vienna, Austria.Google Scholar
Schuller, T., Durox, D. & Candel, S. 2003 A unified model for the prediction of laminar flame transfer functions: comparisons between conical and v-flame dynamics. Combust. Flame 134, 2134.CrossRefGoogle Scholar
Sterling, J. D. & Zukoski, E. E. 1991 Nonlinear dynamics of laboratory combustor pressure oscillations. Combust. Sci. Technol. 77, 225238.CrossRefGoogle Scholar
Subramanian, P., Mariappan, S., Wahi, P. & Sujith, R. I. 2010 Bifurcation analysis of thermoacoustic instability in a horizontal Rijke tube. Intl J. Combust. Spray Dyn. 2 (4), 325356.CrossRefGoogle Scholar
Trefethen, L. N. & Embree, M. 2005 Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operator. Princeton University Press.Google Scholar
Wicker, J. M., Greene, W. D., Kin, S.-I. & Yang, V. 1996 Triggering of longitudinal combustion instabilities in rocket motors: nonlinear combustion response. J. Propul. Power 12, 11481158.CrossRefGoogle Scholar
Wolf, A., Swift, J. B., Swinney, H. L. & Vastano, J. A. 1985 Determining Lyapunov exponents from a time series. Physica D 16, 285317.CrossRefGoogle Scholar
Wu, X., Wang, M., Moin, P. & Peters, N. 2003 Combustion instability due to nonlinear interaction between sound and flame. J. Fluid Mech. 497, 2353.CrossRefGoogle Scholar
You, D., Huang, Y. & Yang, V. 2005 A generalized model of acoustic response of turbulent premixed flame and its application to gas-turbine combustion instability analysis. Combust. Sci. Technol. 177, 11091150.CrossRefGoogle Scholar
Zhang, S. & Shu, C. W. 2007 A new smoothness indicator for the WENO schemes and its effects on the convergence to steady state solutions. J. Sci. Comput. 31, 273305.CrossRefGoogle Scholar
Zhong, X. 1998 Upwind compact and explicit high order finite difference schemes for direct numerical simulation of high speed flows. J. Comput. Phys. 144 (2), 662709.CrossRefGoogle Scholar
Zinn, B. T. & Lores, M. E. 1971 Application of the Galerkin method in the solution of non-linear axial combustion instability problems in liquid rockets. Combust. Sci. Technol. 4 (1), 269278.CrossRefGoogle Scholar
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