Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-06-03T00:27:09.427Z Has data issue: false hasContentIssue false
  • English
  • Français

Thermoacoustic interplay between intrinsic thermoacoustic and acoustic modes: non-normality and high sensitivities

Published online by Cambridge University Press:  06 September 2019

Francesca M. Sogaro Open the ORCID record for Francesca M. Sogaro [Opens in a new window] ,
Peter J. Schmid Open the ORCID record for Peter J. Schmid [Opens in a new window]  and
Aimee S. Morgans Open the ORCID record for Aimee S. Morgans [Opens in a new window]
Francesca M. Sogaro*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Peter J. Schmid
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Aimee S. Morgans
Affiliation:
Department of Mechanical Engineering, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: francesca.sogaro14@imperial.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

This study analyses the interplay between classical acoustic modes and intrinsic thermoacoustic (ITA) modes in a simple thermoacoustic system. The analysis is performed using a frequency-domain low-order network model as well as a time-domain spatially discretised model. Anti-correlated modal sensitivities are found to arise due to a pairwise interplay between acoustic and ITA modes. The magnitude of the sensitivities increases as the interplay between the modes grows stronger. The results show a global behaviour of the modes linked to the presence of exceptional points in the spectrum. The time-domain analysis results in a delay-differential equation and allows the investigation of non-normal behaviour and its consequences. Pseudospectral analysis reveals that energy amplification is crucially linked to an interplay between acoustic and ITA modes. While higher non-orthogonality between two modes is correlated with peaks in modal sensitivity, transient energy growth does not necessarily involve the most sensitive modes. In particular, growth estimates based on the Kreiss constant demonstrate that transient amplification relies critically on the proximity of the non-normal modes to the imaginary axis. The time scale for transient amplification is identified as the flame time delay, which is further corroborated by determining the optimal initial conditions responsible for the bulk of the non-modal energy growth. The flame is identified as an active and dominant contributor to energy gain. The frequency of the optimal perturbation matches the acoustic time scale, once more confirming an interplay between acoustic and ITA structures. Flame-based amplification factors of two to five are found, which are significant when feeding into the acoustic dynamics and eventually triggering nonlinear limit-cycle behaviour.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Reacting Flows: Flames
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 878 , 10 November 2019 , pp. 190 - 220
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aguilar, J. G. & Juniper, M. P. 2018 Adjoint methods for elimination of thermoacoustic oscillations in a model annular combustor via small geometry modifications. In Volume 4A: Combustion, Fuels, and Emissions, V04AT04A054. ASME.Google Scholar
Aguilar, J. G., Magri, L. & Juniper, M. P. 2017 Adjoint-based sensitivity analysis of low order thermoacoustic networks using a wave-based approach. J. Comput. Phys. 341, 163–181.Google Scholar
Annaswamy, A. M. & Ghoniem, A. F. 1995 Active control in combustion systems. IEEE Control Syst. Magazine 15 (6), 49–63.Google Scholar
Balasubramanian, K. & Sujith, R. I. 2008a Non-normality and nonlinearity in combustion-acoustic interaction in diffusion flames. J. Fluid Mech. 594 (2008), 29–57.Google Scholar
Balasubramanian, K. & Sujith, R. I. 2008b Thermoacoustic instability in a Rijke tube: non-normality and nonlinearity. Phys. Fluids 20 (4), 044103.Google Scholar
Blumenthal, R. S., Tangirala, A. K., Sujith, R. I. & Polifke, W. 2017 A systems perspective on non-normality in low-order thermoacoustic models: full norms, semi-norms and transient growth. Intl J. Spray Combust. Dyn. 9 (1), 19–43.Google Scholar
Candel, S., Durox, D., Ducruix, S., Birbaud, A.-L., Noiray, N. & Schuller, T. 2009 Flame dynamics and combustion noise: progress and challenges. Intl J. Aeroacoust. 8 (1), 1–56.Google Scholar
Candel, S., Durox, D. & Schuller, T. 2004 Flame interactions as a source of noise and combustion instabilities. In 10th AIAA/CEAS Aeroacoustics Conf. - AIAA 2004-2928, pp. 2004–2928.Google Scholar
Courtine, E., Selle, L., Nicoud, F., Polifke, W., Silva, C., Bauerheim, M. & Poinsot, T. 2014 Causality and intrinsic thermoacoustic instability modes. In CTR Proceedings of the Summer Program, pp. 169–178.Google Scholar
Crocco, L. 1969 Research on combustion instability in liquid propellant rockets. Symp. (Int.) Combust. 12 (1), 85–99.Google Scholar
Dowling, A. P. 1997 Nonlinear self-excited oscillations of a ducted flame. J. Fluid Mech. 346, 271–290.Google Scholar
Dowling, A. P. & Ffowcs Williams, J. E. 1983 Sound and Sources of Sound. E. Horwood.Google Scholar
Dowling, A. P. & Morgans, A. S. 2005 Feedback control of combustion oscillations. Annu. Rev. Fluid Mech. 37, 151–182.Google Scholar
Dowling, A. P. & Stow, S. R. 2003 Acoustic analysis of gas turbine combustors. J. Propul. Power 19 (5), 751–764.Google Scholar
Ducruix, S., Schuller, T., Durox, D. & Candel, S. 2003 Combustion dynamics and instabilities: elementary coupling and driving mechanisms. J. Propul. Power 19 (5), 722–734.Google Scholar
Emmert, T., Bomberg, S., Jaensch, S. & Polifke, W. 2017 Acoustic and intrinsic thermoacoustic modes of a premixed combustor. Proc. Combust. Inst. 36 (3), 3835–3842.Google Scholar
Emmert, T., Bomberg, S. & Polifke, W. 2015 Intrinsic thermoacoustic instability of premixed flames. Combust. Flame 162 (1), 75–85.Google Scholar
Emmert, T., Meindl, M., Jaensch, S. & Polifke, W. 2016 Linear state space interconnect modeling of acoustic systems. Acta Acust. 102 (5), 824–833.Google Scholar
Fedkiw, R. P., Aslam, T., Merriman, B. & Osher, S. 1999 A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the Ghost Fluid Method). J. Comput. Phys. 152 (2), 457–492.Google Scholar
Foures, D. P. G., Caulfield, C. P. & Schmid, P. J. 2012 Variational framework for flow optimization using seminorm constraints. Phys. Rev. E 86, 026306.Google Scholar
Goh, C. S. & Morgans, A. S. 2013 The influence of entropy waves on the thermoacoustic stability of a model combustor. Combust. Sci. Technol. 185 (2), 249–268.Google Scholar
Hoeijmakers, M., Kornilov, V., Lopez Arteaga, I., de Goey, P. & Nijmeijer, H. 2014 Intrinsic instability of flame-acoustic coupling. Combust. Flame 161 (11), 2860–2867.Google Scholar
Hoeijmakers, M., Kornilov, V., Lopez Arteaga, I., de Goey, P. & Nijmeijer, H. 2016 Flame dominated thermoacoustic instabilities in a system with high acoustic losses. Combust. Flame 169, 209–215.Google Scholar
Horn, R. A. & Johnson, C. R. 1991 Topics in Matrix Analysis. Cambridge University Press.Google Scholar
Hosseini, N., Kornilov, V. N., Lopez Arteaga, I., Polifke, W., Teerling, O. J. & de Goey, L. P. H. 2018 Intrinsic thermoacoustic modes and their interplay with acoustic modes in a Rijke burner. Intl J. Spray Combust. Dyn. 10 (4), 315–325.Google Scholar
Jarlebring, E.2008 The spectrum of delay-differential equations: numerical methods, stability and perturbation. PhD thesis, KTH Royal Institute of Technology.Google Scholar
Juniper, M. P. 2011 Triggering in the horizontal Rijke tube: non-normality, transient growth and bypass transition. J. Fluid Mech. 667, 272–308.Google Scholar
Juniper, M. P. & Sujith, R. I. 2018 Sensitivity and nonlinearity of thermocoustic oscillations. Annu. Rev. Fluid Mech. 50 (2017), 661–689.Google Scholar
Lele, S. K. 1992 Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103 (1), 16–42.Google Scholar
Lieuwen, T. C. & Yang, V. 2005 Combustion Instabilities in Gas Turbine Engines. Operational Experience, Fundamental Mechanisms and Modelling. The American Institute of Aeronautics and Astronautics.Google Scholar
Magri, L., Balasubramanian, K., Sujith, R. I. & Juniper, M. P. 2013 Non-normality in combustion–acoustic interaction in diffusion flames: a critical revision. J. Fluid Mech. 733, 681–683.Google Scholar
Magri, L., Bauerheim, M. & Juniper, M. P. 2016 Stability analysis of thermo-acoustic nonlinear eigenproblems in annular combustors. Part I. Sensitivity. J. Comput. Phys. 325, 395–410.Google Scholar
Magri, L. & Juniper, M. P. 2014 Adjoint-based linear analysis in reduced-order thermo-acoustic models. Intl J. Spray Combust. Dyn. 6 (3), 225–246.Google Scholar
Mangesius, H. & Polifke, W. 2011 A discrete-time, state-space approach for the investigation of non-normal effects in thermoacoustic systems. Intl J. Spray Combust. Dyn. 3 (4), 331–350.Google Scholar
Marble, F. E. & Candel, S. M. 1977 Acoustic disturbance from gas non-uniformities convected through a nozzle. J. Sound Vib. 55 (2), 225–243.Google Scholar
Mariappan, S. & Sujith, R. I. 2010 Thermoacoustic instability in a solid rocket motor: non-normality and nonlinear instabilities. J. Fluid Mech. 653, 1–33.Google Scholar
Mariappan, S., Sujith, R. I. & Schmid, P. J. 2011 Non-normality of thermoacoustic interactions: an experimental investigation. In 47th AIAA/ASME/SAE/ASEE Jt. Prop. Conf. & Exh., pp. 1–29. AIAA.Google Scholar
Mariappan, S., Sujith, R. I. & Schmid, P. J. 2015 Experimental investigation of non-normality of thermoacoustic interaction in an electrically heated Rijke tube. Int. J. Spray Combust. Dyn. 7 (4), 315–352.Google Scholar
Matveev, K. I. & Culick, F. E. C. 2003 A model for combustion instability involving vortex shedding. Combust. Sci. Technol. 175 (6), 1059–1083.Google Scholar
McManus, K. R., Poinsot, T. & Candel, S. M. 1993 A review of active control of combustion instabilities. Prog. Energy Combust. Sci. 19 (1), 1–29.Google Scholar
Mensah, G. A., Magri, L., Silva, C. F., Buschmann, P. E. & Moeck, J. P. 2018 Exceptional points in the thermoacoustic spectrum. J. Sound Vib. 433, 124–128.Google Scholar
Morgans, A. S. & Dowling, A. P. 2007 Model-based control of combustion instabilities. J. Sound Vib. 299 (1-2), 261–282.Google Scholar
Nagarajan, S., Lele, S. K. & Ferziger, J. H. 2003 A robust high-order compact method for large eddy simulation. J. Comput. Phys. 191 (2), 392–419.Google Scholar
Nicoud, F., Benoit, L., Sensiau, C. & Poinsot, T. 2007 Acoustic modes in combustors with complex impedances and multidimensional active flames. AIAA J. 45 (2), 426–441.Google Scholar
Poinsot, T. 2017 Prediction and control of combustion instabilities in real engines. Proc. Combust. Inst. 36 (1), 1–28.Google Scholar
Poinsot, T., Candel, S., Esposito, E., Lang, W. & Bourienne, F. 1989 Suppression of combustion instabilities by active control. J. Propul. Power 5 (1), 14–20.Google Scholar
Poinsot, T. & Veynante, D. 2005 Theoretical and Numerical Combustion. R.T. Edwards, Inc.Google Scholar
Putnam, A. A. 1971 Combustion-Driven Oscillations in Industry. Elsevier.Google Scholar
Rayleigh, J. W. S. 1945 The Theory of Sound. Dover.Google Scholar
Reddy, S. C., Schmid, P. J. & Henningson, D. S. 1993 Pseudospectra of the Orr–Sommerfeld operator. SIAM J. Appl. Math. 53 (1), 15–47.Google Scholar
Renard, P.-H., Thévenin, D., Rolon, J. C. & Candel, S. 2000 Dynamics of flame/vortex interactions. Prog. Energy Combust. Sci. 26 (3), 225–282.Google Scholar
Richards, G. A., Straub, D. L. & Robey, E. H. 2003 Passive control of combustion dynamics in stationary gas turbines. J. Propul. Power 19 (5), 795–810.Google Scholar
Sattelmayer, T. & Polifke, W. 2003 Assessment of methods for the computation of the linear stability of combustors. Combust. Sci. Technol. 175 (3), 453–476.Google Scholar
Sayadi, T., LeChenadec, V., Schmid, P. J., Richecoeur, F. & Massot, M. 2014 Thermoacoustic instability – a dynamical system and time domain analysis. J. Fluid Mech. 753, 448–471.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39 (1), 129–162.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Silva, C. F., Yong, K. J. & Magri, L. 2018 Thermoacoustic modes of quasi-one-dimensional combustors in the region of marginal stability. Trans. ASME; J. Engng Gas Turbines Power 141 (2), 21022–21028.Google Scholar
Skene, C. S. & Schmid, P. J. 2019 Adjoint-based parametric sensitivity analysis for swirling m-flames. J. Fluid Mech. 859, 516–542.Google Scholar
Sterling, J. D. & Zukoski, E. E. 1991 Nonlinear dynamics of laboratory combustor pressure oscillations. Combust. Sci. Technol. 77 (4-6), 225–238.Google Scholar
Subramanian, P., Sujith, R. I. & Wahi, P. 2013 Subcritical bifurcation and bistability in thermoacoustic systems. J. Fluid Mech. 715, 210–238.Google Scholar
Sujith, R. I., Juniper, M. P. & Schmid, P. J. 2016 Non-normality and nonlinearity in thermoacoustic instabilities. Int. J. Spray Combust. Dyn. 8 (2), 119–146.Google Scholar
Toh, K.-C. & Trefethen, L. N. 1996 Calculation of pseudospectra by the Arnoldi iteration. SIAM J. Sci. Comput. 17 (1), 1–15.Google Scholar
Trefethen, L. N. 1999 Computation of pseudospectra. Acta Numerica 8, 247–295.Google Scholar
Trefethen, L. N. & Embree, M. 2005 Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press.Google Scholar
Wieczorek, K., Sensiau, C., Polifke, W. & Nicoud, F. 2011 Assessing non-normal effects in thermoacoustic systems with mean flow. Phys. Fluids 23 (10), 107103.Google Scholar
Wright, T. G. & Trefethen, L. N. 2001 Large-scale computation of pseudospectra using ARPACK and eigs. SIAM J. Sci. Comput. 23 (2), 591–605.Google Scholar
Yang, D., Sogaro, F. M., Morgans, A. S. & Schmid, P. J. 2019 Optimising the acoustic damping of multiple Helmholtz resonators attached to a thin annular duct. J. Sound Vib. 444, 69–84.Google Scholar
Zhao, D. 2012 Transient growth of flow disturbances in triggering a Rijke tube combustion instability. Combust. Flame 159 (6), 2126–2137.Google Scholar
Zhao, D., Li, X. Y., Liu, P., Zhao, H., Lu, Z. & Wang, B. 2018 A review of active control approaches in stabilizing combustion systems in aerospace industry. Prog. Aerosp. Sci. 97, 35–60.Google Scholar

Sogaro Supplementary Movie

Animated version of figure 2 (a, b, c), xf= 0.48L, which allows to 'follow' the movement of the modes in the complex plane as the flame time delay varies.

Download Sogaro Supplementary Movie(Video)
Video 5.5 MB

Sogaro Supplementary Movie

Animated version of figure 3 (a, b, c), xf= 0.4L, which allows to 'follow' the movement of the modes in the complex plane as the flame time delay varies.

Download Sogaro Supplementary Movie(Video)
Video 6.6 MB