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Symmetry breaking of azimuthal thermo-acoustic modes in annular cavities: a theoretical study

  • M. Bauerheim (a1) (a2), P. Salas (a3), F. Nicoud (a4) and T. Poinsot (a5)

Abstract

Many physical problems containing rotating symmetry exhibit azimuthal waves, from electromagnetic waves in nanophotonic crystals to seismic waves in giant stars. When this symmetry is broken, clockwise (CW) and counter-clockwise (CCW) waves are split into two distinct modes which can become unstable. This paper focuses on a theoretical study of symmetry breaking in annular cavities containing a number $N$ of flames prone to azimuthal thermo-acoustic instabilities. A general dispersion relation for non-perfectly-axisymmetric cavities is obtained and analytically solved to provide an explicit expression for the frequencies and growth rates of all azimuthal modes of the configuration. This analytical study unveils two parameters affecting the stability of the mode: (i) a coupling strength corresponding to the cumulative effects of the $N$ flames and (ii) a splitting strength due to the symmetry breaking when the flames are different. This theory has been validated using a 3D Helmholtz solver and good agreement is found. When only two types of flames are introduced into the annular cavity, the splitting strength is found to depend on two parameters: the difference between the two burner types and the pattern used to distribute the flames along the azimuthal direction. To first order, this theory suggests that the most stable configuration is obtained for a perfectly axisymmetric configuration. Therefore, breaking the symmetry by mixing different flames cannot improve the stability of an annular combustor independently of the flame distribution pattern.

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Corresponding author

Email address for correspondence: bauerheim@cerfacs.fr

References

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Barman, A., Barman, S., Kimura, T., Fukuma, Y. & Otani, Y. 2010 Gyration mode splitting in magnetostatically coupled magnetic vortices in an array. J. Phys. D: Appl. Phys. 43, 422001.
Bauerheim, M., Cazalens, M. & Poinsot, T.2014a A theoretical study of mean azimuthal flow and asymmetry effects on thermo-acoustic modes in annular combustors. Proceedings of the 35th Combustion Institute (in press); doi:10.1016/j.proci.2014.05.053.
Bauerheim, M., Nicoud, F. & Poinsot, T. 2014b Theoretical analysis of the mass balance equation through a flame at zero and non-zero Mach numbers. Combust. Flame (in press); doi:10.1016/j.combustflame.2014.06.017.
Bauerheim, M., Parmentier, J. F., Salas, P., Nicoud, F. & Poinsot, T. 2014c An analytical model for azimuthal thermoacoustic modes in an annular chamber fed by an annular plenum. Combust. Flame 161, 13741389.
Berenbrink, P. & Hoffmann, S.2001 Suppression of dynamic combustion instabilities by passive and active means, GT2001-42.
Blimbaum, J., Zanchetta, M., Akin, T., Acharya, V., O’Connor, J., Noble, D. R. & Lieuwen, T. 2012 Transverse to longitudinal acoustic coupling processes in annular combustion chambers. Intl J. Spray Combust. Dyn. 4 (4), 275298.
Borisnika, S. V. 2006 Symmetry, degeneracy and optical confinement of modes in coupled microdisk resonators and photonic crystal cavities. IEEE J. Sel. Top. Quant. Electron. 12 (6), 11751182.
Bourgouin, J.-F., Durox, D., Moeck, J. P., Schuller, T. & Candel, S.2013 Self-sustained instabilities in an annular combustor coupled by azimuthal and longitudinal acoustic modes, GT2013-95010.
Busse, F. H. 1984 Oscillations of a rotating liquid drop. J. Fluid Mech. 142, 18.
Creighton, J. A. 1982 Splitting of degenerate vibrational modes due to symmetry perturbations in tetrahedral m4 and octahedral m6 clusters. Inorg. Chem. 21 (1), 14.
Crocco, L. 1951 Aspects of combustion instability in liquid propellant rocket motors. Part I. J. Am. Rocket Soc. 21, 163178.
Culick, F. E. C. & Kuentzmann, P.2006 Unsteady motions in combustion chambers for propulsion systems. NATO Research and Technology Organization.
Cummings, D. L. & Blackburn, D. A. 1991 Oscillations of magnetically levitated aspherical droplets. J. Fluid Mech. 224, 395416.
Davey, A. & Salwen, H. 1994 On the stability in an elliptic pipe which is nearly circular. J. Fluid Mech. 281, 357369.
Davies, P. O. A. L. 1988 Practical flow duct acoustics. J. Sound Vib. 124 (1), 91115.
Dowling, A. P. 1995 The calculation of thermoacoustic oscillations. J. Sound Vib. 180 (4), 557581.
Evesque, S. & Polifke, W.2002 Low-order acoustic modelling for annular combustors: validation and inclusion of modal coupling, GT2002-30064.
Evesque, S., Polifke, W. & Pankiewitz, C.2003 Spinning and azimuthally standing acoustic modes in annular combustors, AIAA Paper 2003-3182.
Feng, Z. C. & Sethna, P. R. 1989 Symmetry-breaking bifurcation in resonant surface waves. J. Fluid Mech. 199, 495518.
Gelbert, G., Moeck, J. P., Paschereit, C. O. & King, R. 2012 Feedback control of unstable thermoacoustic modes in an annular Rijke tube. Control Engng Practice 20, 770782.
Guckenheimer, J. & Mahalov, A. 1992 Instability induced by symmetry reduction. Phys. Rev. Lett. 68, 2257.
Guslienko, K. Y., Slavin, A. N., Tiberkevich, V. & Kim, S. K. 2008 Dynamic origin of azimuthal modes splitting in vortex-state magnetic dots. Phys. Rev. Lett. 24, 247203.
Hoffmann, F., Woltersdorf, G., Perzlmaier, K., Slavin, A. N., Tiberkevich, V. S., Bischof, A., Weiss, D. & Back, C. H. 2007 Mode degeneracy due to vortex core removal in magnetic disks. Phys. Rev. B 76, 014416.
Kammerer, M., Weigand, M., Curcic, M., Sproll, M., Vansteenkiste, A., Waeyenberge, B. V., Stoll, H., Woltersdorf, G., Back, C. H. & Schuetz, G. 2011 Magnetic vortex core reversal by excitation of spin waves. Nat. Commun. 2, 279; doi:10.1038/ncomms1277.
Kippenberg, T. J. 2010 Microresonators: particle sizing by mode splitting. Nat. Photon. 4, 910.
Kopitz, J., Huber, A., Sattelmayer, T. & Polifke, W.2005 Thermoacoustic stability analysis of an annular combustion chamber with acoustic low order modeling and validation against experiment, GT2005-68797. Reno, NV, USA.
Kosovichev, A. G. 1999 Inversion methods in helioseismology and solar tomography. J. Comput. Appl. Maths 109, 139.
Krebs, W., Flohr, P., Prade, B. & Hoffmann, S. 2002 Thermoacoustic stability chart for high intense gas turbine combustion systems. Combust. Sci. Technol. 174, 99128.
Krueger, U., Hueren, J., Hoffmann, S., Krebs, W., Flohr, P. & Bohn, D. 2000 Prediction and measurement of thermoacoustic improvements in gas turbines with annular combustion systems. Trans. ASME J. Engng Gas Turbines Power 123 (3), 557566.
Kumar, A. & Krousgrill, C. M. 2012 Mode-splitting and quasi-degeneracies in circular plate vibration problems: the example of free vibrations of the stator of a travelling wave ultrasonic motor. J. Sound Vib. 331 (26), 57885802.
Lavely, E. M.1983 Theoretical investigations in helioseismology. PhD thesis, Columbia University.
Lieuwen, T. & Yang, V. 2005 Combustion Instabilities in Gas Turbine Engines. Operational Experience, Fundamental Mechanisms and Modeling, Progress in Astronautics and Aeronautics, vol. 210. AIAA.
Lin, J. & Parker, R. G. 2000a Mesh stiffness variation instabilities in two-stage gear systems. J. Vib. Acoust. 124, 6876.
Lin, J. & Parker, R. G. 2000b Structured vibration characteristics of planetary gears with unequally spaced planets. J. Sound Vib. 235 (5), 921928.
Marble, F. E. & Candel, S. 1977 Acoustic disturbances from gas nonuniformities convected through a nozzle. J. Sound Vib. 55, 225243.
Mazzei, A., Gotzinger, S., Menezes, L. de. S., Zumofen, G., Benson, O. & Sandoghdar, V. 2007 Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light. Phys. Rev. Lett. 99, 173603.
Moeck, J. P., Paul, M. & Paschereit, C.2010 Thermoacoustic instabilities in an annular flat Rijke tube, GT2010-23577.
Nicoud, F., Benoit, L., Sensiau, C. & Poinsot, T. 2007 Acoustic modes in combustors with complex impedances and multidimensional active flames. AIAA J. 45, 426441.
Noiray, N., Bothien, M. & Schuermans, B. 2011 Analytical and numerical analysis of staging concepts in annular gas turbines. Combust. Theor. Model. 15 (5), 585606.
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.
Noiray, N. & Schuermans, B. 2013 On the dynamic nature of azimuthal thermoacoustic modes in annular gas turbine combustion chambers. Proc. R. Soc. Lond. A 469 (2151).
O’Connor, J. & Lieuwen, T.2012a Influence of transverse acoustic modal structure on the forced response of a swirling nozzle flow, GT2012-70053.
O’Connor, J. & Lieuwen, T. 2012b Recirculation zone dynamics of a transversely excited swirl flow and flame. Phys. Fluids 24, 075107.
O’Connor, J. & Lieuwen, T. 2012c Further characterization of the disturbance field in a transversely excited swirl-stabilized flame. Trans. ASME J. Engng Gas Turbines Power 134 (1), 011501.
O’Connor, J. & Lieuwen, T. 2014 Transverse combustion instabilities: acoustic, fluid mechanics and flame processes. Prog. Energy Combust. Sci. (in press).
Oefelein, J. C. & Yang, V. 1993 Comprehensive review of liquid-propellant combustion instabilities in F-1 engines. J. Propul. Power 9 (5), 657677.
Palies, P.2010 Dynamique et instabilités de combustion de flammes swirlées. PhD thesis, Ecole Centrale Paris.
Pang, L., Tetz, K. A. & Fainman, Y. 2007 Observation of the splitting of degenerate surface plasmon polariton modes in a two-dimensional metallic nanohole array. Appl. Phys. Lett. 90 (11), 111103.
Pankiewitz, C. & Sattelmayer, T. 2003 Time domain simulation of combustion instabilities in annular combustors. Trans. ASME J. Engng Gas Turbines Power 125 (3), 677685.
Parmentier, J. F., Salas, P., Wolf, P., Staffelbach, G., Nicoud, F. & Poinsot, T. 2012 A simple analytical model to study and control azimuthal instabilities in annular combustion chamber. Combust. Flame 159, 23742387.
Perrin, R. & Charnley, T. 1973 Group theory and the bell. J. Sound Vib. 31 (4), 411418.
Pierce, A. D. 1981 Acoustics: an Introduction to its Physical Principles and Applications. McGraw-Hill.
Poinsot, T. & Veynante, D. 2011 Theoretical and Numerical Combustion, 3rd edn. www.cerfacs.fr/elearning.
Polifke, W., Poncet, A., Paschereit, C. O. & Doebbeling, K. 2001 Reconstruction of acoustic transfer matrices by instationary computational fluid dynamics. J. Sound Vib. 245 (3), 483510.
Schuermans, B., Bellucci, V. & Paschereit, C.2003 Thermoacoustic modeling and control of multiburner combustion systems, GT2003-38688.
Schuermans, B., Paschereit, C. & Monkewitz, P.2006 Non-linear combustion instabilities in annular gas-turbine combustors, AIAA paper 2006-0549.
Schuller, T., Durox, D., Palies, P. & Candel, S. 2012 Acoustic decoupling of longitudinal modes in generic combustion systems. Combust. Flame 159, 19211931.
Selle, L., Benoit, L., Poinsot, T., Nicoud, F. & Krebs, W. 2006 Joint use of compressible large-eddy simulation and Helmholtz solvers for the analysis of rotating modes in an industrial swirled burner. Combust. Flame 145 (1–2), 194205.
Sensiau, C., Nicoud, F. & Poinsot, T. 2009 A tool to study azimuthal and spinning modes in annular combustors. Intl J. Aeroacoust. 8 (1), 5768.
Silva, C. F., Nicoud, F., Schuller, T., Durox, D. & Candel, S. 2013 Combining a Helmholtz solver with the flame describing function to assess combustion instability in a premixed swirled combustor. Combust. Flame 160, 17431754.
Silva, F., Guillemain, Ph., Kergomard, J., Mallaroni, B. & Norris, A. N. 2009 Approximation formulae for the acoustic radiation impedance of a cylindrical pipe. J. Sound Vib. 322, 255263.
Simonelli, F. & Gollub, J. P. 1989 Surface wave mode interactions: effects of symmetry and degeneracy. J. Fluid Mech. 199, 471494.
Staffelbach, G., Gicquel, L. Y. M., Boudier, G. & Poinsot, T. 2009 Large eddy simulation of self-excited azimuthal modes in annular combustors. Proc. Combust. Inst. 32, 29092916.
Stow, S. R. & Dowling, A. P.2001 Thermoacoustic oscillations in an annular combustor, GT2001-0037, New Orleans, Louisiana.
Stow, S. R. & Dowling, A. P.2003 Modelling of circumferential modal coupling due to Helmholtz resonators, GT2003-38168, Atlanta, Georgia, USA.
Strahle, W. C. 1972 Some results in combustion generated noise. J. Sound Vib. 23 (1), 113125.
Tripathy, S. C., Jain, K. & Bhatnagar, A. 2000 Helioseismic solar cycle changes and splitting coefficients. J. Astrophys. Astron. 21, 349352.
Wolf, P., Staffelbach, G., Gicquel, L. Y. M., Muller, J. D. & Poinsot, T. 2012 Acoustic and large eddy simulation studies of azimuthal modes in annular combustion chambers. Combust. Flame 159, 33983413.
Wolf, P., Staffelbach, G., Roux, A., Gicquel, L., Poinsot, T. & Moureau, V. 2009 Massively parallel LES of azimuthal thermo-acoustic instabilities in annular gas turbines. C. R. Acad. Sci. Méc. 337 (6–7), 385394.
Worth, N. A. & Dawson, J. R. 2013a Modal dynamics of self-excited azimuthal instabilities in an annular combustion chamber. Combust. Flame 160 (11), 24762489.
Worth, N. A. & Dawson, J. R. 2013b Self-excited cricumferential instabilities in a model annular gas turbine combustor: global flame dynamics. Proc. Combust. Inst. 34, 31273134.
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Symmetry breaking of azimuthal thermo-acoustic modes in annular cavities: a theoretical study

  • M. Bauerheim (a1) (a2), P. Salas (a3), F. Nicoud (a4) and T. Poinsot (a5)

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