Skip to main content Accessibility help
×
Home

Symmetry breaking of azimuthal thermoacoustic modes: the UQ perspective

  • M. Bauerheim (a1), A. Ndiaye (a1), P. Constantine (a2), S. Moreau (a3) and F. Nicoud (a4)...

Abstract

Since its introduction in the late 19th century, symmetry breaking has been found to play a crucial role in physics. In particular, it appears as one key phenomenon controlling hydrodynamic and acoustic instabilities in problems with rotational symmetries. A previous paper investigated its desired potential application to the control of circumferential thermoacoustic modes in one annular cavity coupled with multiple flames (Bauerheim et al., J. Fluid Mech., vol. 760, 2014, pp. 431–465). The present paper focuses on a similar problem when symmetry breaking appears unintentionally, for example when uncertainties due to tolerances are taken into account. It yields a large uncertainty quantification (UQ) problem containing numerous uncertain parameters. To tackle this well-known ‘curse of dimensionality’, a novel UQ methodology is used. It relies on the active subspace approach to construct a reduced set of input variables. This strategy is applied on two annular cavities coupled by 19 flames to determine its modal risk factor, i.e. the probability of an azimuthal acoustic mode being unstable. Since each flame is modelled by two uncertain parameters, it leads to a large UQ problem involving 38 parameters. An acoustic network model is then derived, which yields a nonlinear dispersion relation for azimuthal modes. This nonlinear problem, subject to bifurcations, is solved quasi-analytically. Results show that the dimension of the probabilistic problem can be drastically reduced, from 38 uncertain parameters to only 3. Moreover, it is found that the three active variables are related to physical quantities, which unveils underlying phenomena controlling the stability of the two coupled cavities. The first active variable is associated with a coupling strength controlling the bifurcation of the system, while the two others correspond to a symmetry-breaking effect induced by the uncertainties. Thus, an additional destabilization effect appear caused by the non-uniform pattern of the uncertainty distribution, which breaks the initial rotating symmetry of the annular cavities. Finally, the active subspace is exploited by fitting the response surface with polynomials (linear, quadratic and cubic). By comparing accuracy and cost, results prove that 5 % error can be achieved with only 30 simulations on the reduced space, whereas 2000 are required on the complete initial space. It exemplifies that this novel UQ technique can accurately predict the risk factor of an annular configuration at low cost as well as unveil key parameters controlling the stability.

Copyright

Corresponding author

Email address for correspondence: bauerheim@cerfacs.fr

References

Hide All
Bauerheim, M., Cazalens, M. & Poinsot, T. 2014a A theoretical study of mean azimuthal flow and asymmetry effects on thermo-acoustic modes in annular combustors. Proc. Combust. Inst. 35 (3), 32193227.
Bauerheim, M., Ndiaye, A., Constantine, P., Iaccarino, G., Moreau, S. & Nicoud, F. 2014b Uncertainty quantification of thermo-acoustic instabilities in annular combustors. In Proceedings of the Summer Program, pp. 209218.
Bauerheim, M., Nicoud, F. & Poinsot, T. 2014c Theoretical analysis of the mass balance equation through a flame at zero and non-zero Mach numbers. Combust. Flame 162 (1), 6067.
Bauerheim, M., Parmentier, J. F., Salas, P., Nicoud, F. & Poinsot, T. 2014d An analytical model for azimuthal thermoacoustic modes in an annular chamber fed by an annular plenum. Combust. Flame 161, 13741389.
Bauerheim, M., Salas, P., Nicoud, F. & Poinsot, T. 2014e Symmetry breaking of azimuthal thermoacoustic modes in annular cavities: a theoretical study. J. Fluid Mech. 760, 431465.
Bauerheim, M., Staffelbach, G., Worth, N. A., Dawson, J. R., Gicquel, L. Y. M & Poinsot, T. 2014f Sensitivity of LES-based harmonic flame response model for turbulent swirled flames and impact on the stability of azimuthal modes. Proc. Combust. Inst. 3, 33553363.
Bourgouin, J.-F.2014 Dynamique de flamme dans les foyeres annulaires comportant des injecteurs multiples. PhD thesis, Ecole Centrale de Paris (EM2C).
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.
Bourgouin, J.-F., Durox, D., Moeck, J. P., Schuller, T. & Candel, S.2014 Characterization and modeling of a spinning thermoacoustic instability in an annular combustor equipped with multiple matrix injectors. ASME Paper 2014-GT-25067.
Bourguet, R. & Jacono, D. Lo 2013 Flow-induced vibrations of a rotating cylinder. J. Fluid Mech. 740, 342380.
Busse, F. H. 1984 Oscillations of a rotating liquid drop. J. Fluid Mech. 142, 18.
Camarri, S. & Giannetti, F. 2010 Effect of confinement on three-dimensional stability in the wake of a circular cylinder. J. Fluid Mech. 642, 477487.
Campa, G., Camporeale, S. M., Guaus, A., Favier, J., Bargiacchi, M., Bottaro, A., Cosatto, E. & Mori, G.2011 A quantitative comparison between a low order model and a 3d FEM code for the study of thermoacoustic combustion instabilities. ASME Paper 2011-GT-45969.
Chantrasmi, T. & Iaccarino, G. 2012 Forward and backward uncertainty propagation for discontinuous system response using the Padé–Legendre method. Intl J. Uncertainty Quantification 2 (2), 125143.
Clavin, P., Kim, J. S. & Williams, F. A. 1994 Turbulence induced noise effects on high-frequency combustion instabilities. Combust. Sci. Technol. 96 (61–84).
Constantine, P. G. 2015 Active Subspaces: Emerging Ideas for Dimension Reduction in Parameter Studies. SIAM.
Constantine, P. G., Dow, E. & Wang, Q. 2014 Active Subspace methods in theory and pratice: applications to kriging surfaces. SIAM J. Sci. Comput. 36 (4), 15001524.
Crocco, L. 1952 Aspects of combustion instability in liquid propellant rocket motors. Part II. J. Am. Rocket Soc. 22, 716.
Crocco, L. & Cheng, S. I. 1956 Theory of Combustion Instability in Liquid Propellant Rocket Motors, Agardograph, vol. 8. Butterworths Science.
Cummings, D. L. & Blackburn, D. A. 1991 Oscillations of magnetically levitated aspherical droplets. J. Fluid Mech. 224, 395416.
Curie, P. 1894 Sur la symétrie dans les phénomènes physiques, symétrie d’un champ électrique et d’un champ magnétique. J. Phys. Theor. Appl. 3 (1), 393415.
Davey, A. & Salwen, H. 1994 On the stability in an elliptic pipe which is nearly circular. J. Fluid Mech. 281, 357369.
Dawson, J. R. & Worth, N. A. 2014 The effect of baffles on self-excited azimuthal modes in an annular combustor. Proc. Combust. Inst. 35 (3), 32833290.
Dempster, A. P. & Laird, N. M. 1977 Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. B 39 (1), 138.
Dowling, A. P. 1995 The calculation of thermoacoustic oscillations. J. Sound Vib. 180 (4), 557581.
Duchaine, F., Boudy, F., Durox, D. & Poinsot, T. 2011 Sensitivity analysis of transfer functions of laminar flames. Combust. Flame 158 (12), 23842394.
Feng, Z. C. & Sethna, P. R. 1989 Symmetry-breaking bifurcation in resonant surface waves. J. Fluid Mech. 199, 495518.
Ghirardo, G. & Juniper, M. 2013 Azimuthal instabilities in annular combustors: standing and spinning modes. Proc. R. Soc. Lond. A 469, 20130232.
Guckenheimer, J. & Mahalov, A. 1992 Instability induced by symmetry reduction. Phys. Rev. Lett. 68, 2257.
Hoeijmakers, M., Lopez Arteaga, I., Kornilov, V., Nijmeijer, H. & de Goey, P. 2013 Accuracy assessment of thermoacoustic instability models using binary classification. Intl J. Spray Combust. Dyn. 5 (3), 201224.
Juniper, M. P., Magri, L., Bauerheim, M. & Nicoud, F. 2015 Sensitivity analysis of thermo-acoustic eigenproblems with adjoint methods. In Proceedings of the Summer Program, pp. 189198.
Kaarnioja, V.2013 Smolyak quadrature. Masters thesis, University of Helsinki, Department of Mathematics and Statistics.
Kammerer, M., Weigand, M., Curcic, M., Sproll, M., Vansteenkiste, A., Waeyenberge, B., Van Stoll, H., Woltersdorf, G., Back, C. H. & Schuetz, G. 2011 Magnetic vortex core reversal by excitation of spin waves. Nat. Commun. 2, 279.
Kedia, K. S., Altay, H. M. & Ghoniem, A. F. 2011 Impact of flame-wall interaction on premixed flame dynamics and transfer function characteristics. Proc. Combust. Inst. 33, 11131120.
Kerschen, G., Golinval, J.-C., Vakakis, A. F. & Bergman, L. A. 2005 The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview. Nonlinear Dyn. 41, 147169.
Krebs, W., Flohr, P., Prade, B. & Hoffmann, S. 2002 Thermoacoustic stability chart for high intense gas turbine combustion systems. Combust. Sci. Technol. 174, 99128.
Levich, E. & Tsinober, A. 1983 On the role of helicity structures in three-dimensional turbulent flow. Phys. Lett. A 93 (6), 293297.
Lieuwen, T. & Banaszuk, A. 2005 Background noise effects on combustor stability. J. Propul. Power 21 (1), 2531.
Lieuwen, T. & Yang, V. 2005 Combustion Instabilities in Gas Turbine Engines. Operational Experience, Fundamental Mechanisms and Modeling. vol. 210. Progress in Astronautics and Aeronautics, AIAA.
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. ASME Paper 2010-GT-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.
Nobile, F., Tempone, R. & Webster, C. G. 2008 A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46 (5), 23092345.
Noiray, N., Bothien, M. & Schuermans, B. 2011 Investigation of azimuthal staging concepts in annular gas turbines. Combust. Theory Model. 15 (5), 585606.
O’Connor, J. & Lieuwen, T. 2014 Transverse combustion instabilities: acoustic, fluid mechanics and flame processes. Prog. Energy Combust. Sci. 49, 139.
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.
Poinsot, T. & Veynante, D. 2005 Theoretical and Numerical Combustion, 2nd edn (ed. R. T. Edwards).
Polifke, W.1990 Aspects of helicity in turbulent flows. PhD thesis, City University of New York.
Polifke, W., Poncet, A., Paschereit, C. O. & Doebbeling, K. 2001 Reconstruction of acoustic transfer matrices by instationnary computational fluid dynamics. J. Sound Vib. 245 (3), 483510.
Prasad, K., Agrawal, A. & Sharma, A. 2013 Poiseuille flow across an eccentricity confined stationary/rotating cylinder. Ocean Engng 73, 4154.
Rowley, C. W. 2005 Model reduction for fluids, using balanced proper orthogonal decomposition. Intl J. Bifurcation Chaos 15 (997).
Saint-Michel, B., Daviaud, F. & Dubrulle, B. 2014 A zero-mode mechanism for spontaneous symmetry breaking in a turbulent von Karman flow. New J. Phys. 16, 013055.
Schuermans, B., Bellucci, V. & Paschereit, C.2003 Thermoacoustic modeling and control of multiburner combustion systems. ASME Paper 2003-GT-38688.
Schuller, T., Durox, D., Palies, P. & Candel, S. 2012 Acoustic decoupling of longitudinal modes in generic combustion systems. Combust. Flame 159, 19211931.
Simonelli, F. & Gollub, J. P. 1989 Surface wave mode interactions: effects of symmetry and degeneracy. J. Fluid Mech. 199, 471494.
Wolf, P., Staffelbach, G., Balakrishnan, R., Roux, A. & Poinsot, T. 2010 Azimuthal instabilities in annular combustion chambers. In Proceedings of the Summer Program, pp. 259269.
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. 2013 Self-excited cricumferential instabilities in a model annular gas turbine combustor: global flame dynamics. Proc. Combust. Inst. 34, 31273134.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Related content

Powered by UNSILO

Symmetry breaking of azimuthal thermoacoustic modes: the UQ perspective

  • M. Bauerheim (a1), A. Ndiaye (a1), P. Constantine (a2), S. Moreau (a3) and F. Nicoud (a4)...

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.