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Energy for Propulsion pp 125–150Cite as

Synchronization Transition in a Thermoacoustic System: Temporal and Spatiotemporal Analyses

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Abstract

The occurrence of thermoacoustic instability has been a major concern in the combustors used in power plants and propulsive systems such as gas turbine engines, rocket motors. A positive feedback between the inherent processes such as the acoustic field and the unsteady heat release rate of the combustor can result in the occurrence of large-amplitude, self-sustained pressure oscillations. Prior to the state of thermoacoustic instability, intermittent oscillations are observed in turbulent combustors. Such intermittent oscillations are characterized by an apparently random appearance of bursts of large-amplitude periodic oscillations interspersed between epochs of low-amplitude aperiodic oscillations. In most of the earlier studies, the pressure oscillations alone have been analyzed to explore the dynamical transition to thermoacoustic instability. The present chapter focuses on the instantaneous interaction between the acoustic field and the unsteady heat release rate observed during such a transition in a bluff-body-stabilized turbulent combustor. The instantaneous interaction of these oscillations will be discussed using the concepts of synchronization theory. First, we give a brief introduction to the synchronization theory so as to summarize the concepts of locking of phase and frequency of the oscillations. Then, the temporal and spatiotemporal aspects of the interaction will be presented in detail. We find that, during stable operation, aperiodic oscillations of the pressure and the heat release rate remain desynchronized, whereas synchronized periodic oscillations are noticed during the occurrence of thermoacoustic instability. Such a transition happens through intermittent phase-synchronized oscillations, wherein synchronization and desynchronization of the oscillators are observed during the periodic and the aperiodic epochs of the intermittent oscillations, respectively. Further, the spatiotemporal analysis reveals a very interesting pattern in the reaction zone. Phase asynchrony among the local heat release rate oscillators is observed during the stable operation, while they become phase-synchronized during the onset of thermoacoustic instability. Interestingly, the state of intermittent oscillations corresponds to a simultaneous existence of synchronized periodic and desynchronized aperiodic patterns in the reaction zone. Such a coexistence of synchrony and asynchrony in the reactive flow field mimics a chimera state.

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Abbreviations

\(R_a\) :

Rayleigh integral

\(\varDelta f\) :

Frequency detuning

\(\omega _0\) :

Natural angular frequency

E :

Embedding dimension

T :

Optimum time delay

V :

Volume of the combustion zone

\(\varTheta \) :

Heaviside step function

\(\mathbb {R}_{i,j}\) :

Recurrence Matrix

\(\varepsilon \) :

Cutoff threshold

\(P(\tau )\) :

Probability of recurrence

\(\tau \) :

Time lag

\(\bar{u}\) :

Mean flow velocity

\(p^\prime \) :

Pressure oscillations

\(\dot{q}^\prime \) :

Heat release rate oscillations

R(t):

Kuramoto order parameter

\(\bar{R}\) :

Time-averaged Kuramoto order parameter

\(\zeta (t)\) :

Analytic signal

RP:

Recurrence Plot

PMT:

Photomultiplier tube

PSD:

Power spectral density

PS:

Phase synchronization

GS:

Generalized synchronization

IPS:

Intermittent phase synchronization

FWHM:

Full width at half maximum

References

  1. D.M. Abrams, R. Mirollo, S.H. Strogatz, D.A. Wiley, Solvable model for chimera states of coupled oscillators. Phys. Rev. Lett. 101(8), 084–103 (2008)

    Google Scholar 

  2. S. Ahn, C. Park, L.L. Rubchinsky, Detecting the temporal structure of intermittent phase locking. Phys. Rev. E 84(1), 016–201 (2011)

    Google Scholar 

  3. N. Ananthkrishnan, S. Deo, F.E. Culick, Reduced-order modeling and dynamics of nonlinear acoustic waves in a combustion chamber. Combust. Sci. Technol. 177(2), 221–248 (2005)

    Article  Google Scholar 

  4. S. Candel, Combustion dynamics and control: progress and challenges. Proc. Combust. Inst. 29(1), 1–28 (2002)

    Article  Google Scholar 

  5. S.R. Chakravarthy, O.J. Shreenivasan, B. Boehm, A. Dreizler, J. Janicka, Experimental characterization of onset of acoustic instability in a nonpremixed half-dump combustor. J. Acoust. Soc. Am 122(1) 120–127 (2007)

    Article  Google Scholar 

  6. H. Chaté, P. Manneville, Transition to turbulence via spatio-temporal intermittency. Phys. Rev. Lett. 58(2), 112 (1987)

    Article  Google Scholar 

  7. L. Crocco, S.I. Cheng, Theory of combustion instability in liquid propellant rocket motors, Technical report. Princeton Univ, NJ, 1956

    Google Scholar 

  8. F. Culick, Nonlinear behavior of acoustic waves in combustion chambersi. Acta Astronaut. 3(9–10), 715–734 (1976)

    Article  MATH  Google Scholar 

  9. F. Culick, Some recent results for nonlinear acoustics in combustion chambers. AIAA J. 32(1), 146–169 (1994)

    Article  MATH  Google Scholar 

  10. F. Culick, P. Kuentzmann, Unsteady motions in combustion chambers for propulsion systems, Technical report. Nato Research and Technology Organization Neuilly-Sur-Seine (France), 2006

    Google Scholar 

  11. S. Datta, S. Mondal, A. Mukhopadhyay, D. Sanyal, S. Sen, An investigation of nonlinear dynamics of a thermal pulse combustor. Combust. Theory Model. 13(1), 17–38 (2009)

    Article  MATH  Google Scholar 

  12. S. Domen, H. Gotoda, T. Kuriyama, Y. Okuno, S. Tachibana, Detection and prevention of blowout in a lean premixed gas-turbine model combustor using the concept of dynamical system theory. Proc. Combust. Inst. 35(3), 3245–3253 (2015)

    Article  Google Scholar 

  13. A.P. Dowling, Nonlinear self-excited oscillations of a ducted flame. J. Fluid Mech. 346, 271–290 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. J.P. Eckmann, S.O. Kamphorst, D. Ruelle, Recurrence plots of dynamical systems. EPL (Europhysics Letters) 4(9), 973 (1987)

    Article  Google Scholar 

  15. E.M. Essaki Arumugam, M.L. Spano, A chimeric path to neuronal synchronization. Chaos: Interdiscip. J. Nonlinear Sci. 25(1), 013–107 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. H. Fujisaka, T. Yamada, Stability theory of synchronized motion in coupled-oscillator systems. Prog. Theor. Phys. 69(1), 32–47 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Gleick, Chaos: Making a New Science, Enhanced edn. (Open Road Media, 2011)

    Google Scholar 

  18. H. Gotoda, H. Nikimoto, T. Miyano, S. Tachibana, Dynamic properties of combustion instability in a lean premixed gas-turbine combustor. Chaos: Interdiscip. J. Nonlinear Sci. 21(1), 013–124 (2011)

    Article  Google Scholar 

  19. H. Gotoda, Y. Shinoda, M. Kobayashi, Y. Okuno, S. Tachibana, Detection and control of combustion instability based on the concept of dynamical system theory. Phys. Rev. E 89(2), 022–910 (2014)

    Google Scholar 

  20. F. Guethe, D. Guyot, G. Singla, N. Noiray, B. Schuermans, Chemiluminescence as diagnostic tool in the development of gas turbines. Appl. Phys. B 107(3), 619–636 (2012)

    Article  Google Scholar 

  21. U. Hegde, D. Reuter, B. Daniel, B. Zinn, Flame driving of longitudinal instabilities in dump type ramjet combustors. Combust. Sci. Technol. 55(4–6), 125–138 (1987)

    Article  Google Scholar 

  22. N.E. Huang, Z. Shen, S.R. Long, M.C. Wu, H.H. Shih, Q. Zheng, N.C. Yen, C.C. Tung, H.H. Liu, The empirical mode decomposition and the hilbert spectrum for nonlinear and non-stationary time series analysis, in Proceedings of the Royal Society of London A: mathematical, physical and engineering sciences, vol. 454 (The Royal Society, 1998), pp. 903–995

    Article  MathSciNet  MATH  Google Scholar 

  23. Y. Ikeda, J. Kojima, H. Hashimoto, Local chemiluminescence spectra measurements in a high-pressure laminar methane/air premixed flame. Proc. Combust. Inst. 29(2), 1495–1501 (2002)

    Article  Google Scholar 

  24. C.C. Jahnke, F.E. Culick, Application of dynamical systems theory to nonlinear combustion instabilities. J. Propuls. Power 10(4), 508–517 (1994)

    Article  Google Scholar 

  25. M.P. Juniper, R. Sujith, Sensitivity and nonlinearity of thermoacoustic oscillations. Ann. Rev. Fluid Mech. 50(1) (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. L. Kabiraj, R. Sujith, Nonlinear self-excited thermoacoustic oscillations: intermittency and flame blowout. J. Fluid Mech. 713, 376–397 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. L. Kabiraj, A. Saurabh, P. Wahi, R. Sujith, Route to chaos for combustion instability in ducted laminar premixed flames. Chaos: Interdiscip. J. Nonlinear Sci. 22(2), 023–129 (2012)

    Article  Google Scholar 

  28. L, Kabiraj, R. Sujith, P. Wahi, Investigating the dynamics of combustion-driven oscillations leading to lean blowout. Fluid Dyn. Res. 44(3), 031–408 (2012)

    Article  MATH  Google Scholar 

  29. K. Kashinath, I.C. Waugh, M.P. Juniper, Nonlinear self-excited thermoacoustic oscillations of a ducted premixed flame: bifurcations and routes to chaos. J. Fluid Mech. 761, 399–430 (2014)

    Article  Google Scholar 

  30. J.O. Keller, L. Vaneveld, D. Korschelt, G. Hubbard, A. Ghoniem, J. Daily, A. Oppenheim, Mechanism of instabilities in turbulent combustion leading to flashback. AIAA J. 20(2), 254–262 (1982)

    Article  Google Scholar 

  31. D.W. Kendrick, T.J. Anderson, W.A. Sowa, T.S. Snyder, Acoustic sensitivities of lean-premixed fuel injectors in a single nozzle rig, in ASME 1998 International Gas Turbine and Aeroengine Congress and Exhibition (American Society of Mechanical Engineers, 1998)

    Google Scholar 

  32. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, vol. 19 (Springer Science & Business Media, 2012)

    Google Scholar 

  33. M. Lakshmanan, D.V. Senthilkumar, Dynamics of Nonlinear Time-Delay Systems (Springer Science & Business Media, 2011)

    Google Scholar 

  34. S. Lei, A. Turan, Nonlinear/chaotic behaviour in thermo-acoustic instability. Combust. Theory Model. 13(3), 541–557 (2009)

    Article  MATH  Google Scholar 

  35. T. Lieuwen, Modeling premixed combustion-acoustic wave interactions: a review. J. Propuls. power 19(5), 765–781 (2003)

    Article  Google Scholar 

  36. T. Lieuwen, Online combustor stability margin assessment using dynamic pressure data. Trans. ASME-A-Eng. Gas Turbines Power 127(3), 478–482 (2005)

    Article  Google Scholar 

  37. T. Lieuwen, B.T. Zinn, Application of multipole expansions to sound generation from ducted unsteady combustion processes. J. Sound Vib. 235(3), 405–414 (2000)

    Article  Google Scholar 

  38. T. Lieuwen, H. Torres, C. Johnson, B.T. Zinn, A mechanism of combustion instability in lean premixed gas turbine combustors. Trans.-Am. Soc. Mech. Eng.-J. Eng. Gas Turbines Power 123(1), 182–189 (2001)

    Article  Google Scholar 

  39. T.C. Lieuwen, Experimental investigation of limit-cycle oscillations in an unstable gas turbine combustor. J. Propuls. Power 18(1), 61–67 (2002)

    Article  Google Scholar 

  40. T.C. Lieuwen, V. Yang, in Combustion Instabilities in Gas Turbine Engines(operational experience, fundamental mechanisms and modeling). Progress in Astronautics and Aeronautics, 2005

    Google Scholar 

  41. E.N. Lorenz, Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)

    Article  MATH  Google Scholar 

  42. N. Marwan, M.C. Romano, M. Thiel, J. Kurths, Recurrence plots for the analysis of complex systems. Phys. Rep. 438(5), 237–329 (2007)

    Article  MathSciNet  Google Scholar 

  43. K. McManus, T. Poinsot, S. Candel, A review of active control of combustion instabilities. Prog. Energy Combust. Sci. 19(1), 1–29 (1993)

    Article  Google Scholar 

  44. S. Mondal, A. Mukhopadhyay, S. Sen, Effects of inlet conditions on dynamics of a thermal pulse combustor. Combust. Theory Model. 16(1), 59–74 (2012)

    Article  MATH  Google Scholar 

  45. S. Mondal, A. Mukhopadhyay, S. Sen, Dynamic characterization of a laboratory-scale pulse combustor. Combust. Sci. Technol. 186(2), 139–152 (2014)

    Article  Google Scholar 

  46. S. Mondal, V.R. Unni, R. Sujith, Chimera-like states observed during the transition to thermoacoustic instability in turbulent combustor, in Conference on Nonlinear Systems & Dynamics IISER Kolkata, vol. 16, p. 18, 2016

    Google Scholar 

  47. S. Mondal, A. Mukhopadhyay, S. Sen, Bifurcation analysis of steady states and limit cycles in a thermal pulse combustor model. Combust. Theory Model. 21(3), 487–502 (2017)

    Article  MathSciNet  Google Scholar 

  48. S. Mondal, S. Pawar, R. Sujith, Synchronous behaviour of two interacting oscillatory systems undergoing quasiperiodic route to chaos. Chaos: Interdiscip. J. Nonlinear Sci. 27(10), 103–119 (2017)

    Article  MathSciNet  Google Scholar 

  49. S. Mondal, V.R. Unni, R. Sujith, Onset of thermoacoustic instability in turbulent combustors: an emergence of synchronized periodicity through formation of chimera-like states. J. Fluid Mech. 811, 659–681 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  50. S. Mondal, A. Mukhopadhyay, S. Sen, Characterization of turbulent combustion systems using dynamical systems theory, in Modeling and Simulation of Turbulent Combustion (Springer, 2018), pp. 543–567

    Google Scholar 

  51. V. Nair, R. Sujith, Multifractality in combustion noise: predicting an impending combustion instability. J. Fluid Mech. 747, 635–655 (2014)

    Article  Google Scholar 

  52. V. Nair, G. Thampi, S. Karuppusamy, S. Gopalan, R. Sujith, Loss of chaos in combustion noise as a precursor of impending combustion instability. Int. J. Spray Combust. Dyn. 5(4), 273–290 (2013)

    Article  Google Scholar 

  53. V. Nair, G. Thampi, R. Sujith, Intermittency route to thermoacoustic instability in turbulent combustors. J. Fluid Mech. 756, 470–487 (2014)

    Article  Google Scholar 

  54. G.V. Osipov, B. Hu, C. Zhou, M.V. Ivanchenko, J. Kurths, Three types of transitions to phase synchronization in coupled chaotic oscillators. Phys. Rev. Lett. 91(2), 024–101 (2003)

    Google Scholar 

  55. S.A. Pawar, R. Vishnu, M. Vadivukkarasan, M. Panchagnula, R. Sujith, Intermittency route to combustion instability in a laboratory spray combustor. J. Eng. Gas Turbines Power 138(4), 041–505 (2016)

    Article  Google Scholar 

  56. S.A. Pawar, A. Seshadri, V.R. Unni, R.I. Sujith, Thermoacoustic instability as mutual synchronization between the acoustic field of the confinement and turbulent reactive flow. J. Fluid Mech. 827, 664–693 (2017). https://doi.org/10.1017/jfm.2017.438

    Article  MathSciNet  Google Scholar 

  57. S.A. Pawar, S. Mondal, N.B. George, R. Sujith, Synchronization behaviour during the dynamical transition in swirl-stabilized combustor: temporal and spatiotemporal analysis, in 2018 AIAA Aerospace Sciences Meeting, p. 0394, 2018

    Google Scholar 

  58. A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, vol. 12 (Cambridge University Press, 2003)

    Google Scholar 

  59. T.J. Poinsot, A.C. Trouve, D.P. Veynante, S.M. Candel, E.J. Esposito, Vortex-driven acoustically coupled combustion instabilities. J. Fluid Mech. 177, 265–292 (1987)

    Article  Google Scholar 

  60. A.A. Putnam, Combustion Driven Oscillations in Industry (Elsevier Publishing Company, 1971)

    Google Scholar 

  61. J.W.S. Rayleigh, The explanation of certain acoustical phenomena. Nature 18(455), 319–321 (1878)

    Article  Google Scholar 

  62. M.C. Romano, M. Thiel, J. Kurths, I.Z. Kiss, J. Hudson, Detection of synchronization for non-phase-coherent and non-stationary data. EPL (Europhysics Letters) 71(3), 466 (2005)

    Article  Google Scholar 

  63. M. Rosenblum, A. Pikovsky, Synchronization: from pendulum clocks to chaotic lasers and chemical oscillators. Contemp. Phys. 44(5), 401–416 (2003)

    Article  Google Scholar 

  64. M.G. Rosenblum, A.S. Pikovsky, J. Kurths, Phase synchronization of chaotic oscillators. Phys. Rev. Lett. 76(11), 1804 (1996)

    Article  MATH  Google Scholar 

  65. M.G. Rosenblum, A.S. Pikovsky, J. Kurths, From phase to lag synchronization in coupled chaotic oscillators. Phys. Rev. Lett. 78(22), 4193 (1997)

    Article  MATH  Google Scholar 

  66. N.F. Rulkov, M.M. Sushchik, L.S. Tsimring, H.D. Abarbanel, Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E 51(2), 980 (1995)

    Article  Google Scholar 

  67. K. Schadow, E. Gutmark, Combustion instability related to vortex shedding in dump combustors and their passive control. Prog. Energy Combust. Sci. 18(2), 117–132 (1992)

    Article  Google Scholar 

  68. S. Sen, S. Mondal, A. Mukhopadhyay, Dynamics of thermal pulse combustor, in Energy Combustion and Propulsion (New Perspectives, Athena Academic, 2015), pp. 269–312

    Google Scholar 

  69. B.I. Shraiman, Order, disorder, and phase turbulence. Phys. Rev. Lett. 57(3), 325 (1986)

    Article  MathSciNet  Google Scholar 

  70. D.A. Smith, E.E. Zukoski, Combustion instability sustained by unsteady vortex combustion, in AIAA Joint Propulsion Conference (1985)

    Google Scholar 

  71. J.D. Sterling, Nonlinear analysis and modelling of combustion instabilities in a laboratory combustor. Combust. Sci. Technol. 89(1–4), 167–179 (1993)

    Article  Google Scholar 

  72. W.C. Strahle, Combustion noise. Prog. Energy Combust. Sci. 4(3), 157–176 (1978)

    Article  Google Scholar 

  73. P. Subramanian, Dynamical Systems Approach to the Investigation of Thermoacoustic Instabilities, 2011

    Google Scholar 

  74. P. Subramanian, S. Mariappan, R. Sujith, P. Wahi, Bifurcation analysis of thermoacoustic instability in a horizontal rijke tube. Int. J. Spray Combust. Dyn. 2(4), 325–355 (2010)

    Article  Google Scholar 

  75. R. Sujith, M. Juniper, P. Schmid, Non-normality and nonlinearity in thermoacoustic instabilities. Int. J. Spray Combust. Dyn. 8(2), 119–146 (2016)

    Article  Google Scholar 

  76. F. Takens et al., Detecting strange attractors in turbulence. Lect. Notes Math. 898(1), 366–381 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  77. J. Tony, E. Gopalakrishnan, E. Sreelekha, R. Sujith, Detecting deterministic nature of pressure measurements from a turbulent combustor. Phys. Rev. E 92(6), 062–902 (2015)

    Google Scholar 

  78. V.R. Unni, R. Sujith, Multifractal characteristics of combustor dynamics close to lean blowout. J. Fluid Mech. 784, 30–50 (2015)

    Article  Google Scholar 

  79. J.M. Wilhite, B.J. Dolan, L. Kabiraj, R.V. Gomez, E.J. Gutmark, C.O. Paschereit, Analysis of combustion oscillations in a staged mldi burner using decomposition methods and recurrence analysis, in 54th AIAA Aerospace Sciences Meeting, AIAA SciTech Forum, (AIAA 2016-1156) (American Institute of Aeronautics and Astronautics, 2016), pp. 1–17

    Google Scholar 

  80. T. Yalçınkaya, Y.C. Lai, Phase characterization of chaos. Phys. Rev. Lett. 79(20), 3885 (1997)

    Article  Google Scholar 

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Acknowledgements

The authors would like to thank Office of Naval Research Global (ONRG) for the funding (grant no. N62909-14-1-N 299); Dr. R. Kolar is the contract monitor from ONRG. SM gratefully acknowledges the institute postdoctoral fellowship from IIT Madras. The authors gratefully acknowledge the valuable discussions with Mr. A. Seshadri, Dr. V. R. Unni, Dr. D. V. Senthilkumar, Dr. V. K. Chandrasekar, and Prof. M. Lakshmanan. The authors would also like to acknowledge the help provided by Mr. N. Babu, Mr. Thilagraj, Mr. Manikandan, and Mr. Syam for conducting the experiments. We are grateful to Mr. Thilagraj for providing the schematic of the experimental setup and to Dr. T. Komarek and Prof. W. Polifke for providing the design, which was adapted to fabricate this setup.

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  1. Indian Institute of Technology Madras, Chennai, 600036, India

    Sirshendu Mondal, Samadhan A. Pawar & R. I. Sujith

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Editors and Affiliations

  1. ACRI, The CFD Innovators, Los Angeles, California, USA

    Akshai K. Runchal

  2. Department of Mechanical Engineering, University of Maryland, Department of Mechanical Engineering, College Park, Maryland, USA

    Ashwani K. Gupta

  3. Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, India

    Abhijit Kushari

  4. Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Department of Aerospace Engineering, Kanpur, Uttar Pradesh, India

    Ashoke De

  5. Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, Department of Mechanical Engineering, Chicago, Illinois, USA

    Suresh K. Aggarwal

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Mondal, S., Pawar, S.A., Sujith, R.I. (2018). Synchronization Transition in a Thermoacoustic System: Temporal and Spatiotemporal Analyses. In: Runchal, A., Gupta, A., Kushari, A., De, A., Aggarwal, S. (eds) Energy for Propulsion . Green Energy and Technology. Springer, Singapore. https://doi.org/10.1007/978-981-10-7473-8_6

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