Skip to main content Accessibility help
×
Home

Forced synchronization of quasiperiodic oscillations in a thermoacoustic system

  • Yu Guan (a1), Vikrant Gupta (a2), Minping Wan (a2) and Larry K. B. Li (a1)

Abstract

In self-excited combustion systems, the application of open-loop forcing is known to be an effective strategy for controlling periodic thermoacoustic oscillations, but it is not known whether and under what conditions such a strategy would work on thermoacoustic oscillations that are not simply periodic. In this study, we experimentally examine the effect of periodic acoustic forcing on a prototypical thermoacoustic system consisting of a ducted laminar premixed flame oscillating quasiperiodically on an ergodic $\mathbb{T}^{2}$ torus at two incommensurate natural frequencies, $f_{1}$ and  $f_{2}$ . Compared with that of a classical period-1 system, complete synchronization of this $\mathbb{T}_{1,2}^{2}$ system is found to occur via a more intricate route involving three sequential steps: as the forcing amplitude, $\unicode[STIX]{x1D716}_{f}$ , increases at a fixed forcing frequency, $f_{f}$ , the system transitions first (i) to ergodic $\mathbb{T}_{1,2,f}^{3}$ quasiperiodicity; then (ii) to resonant $\mathbb{T}_{1,f}^{2}$ quasiperiodicity as the weaker of the two natural modes, $f_{2}$ , synchronizes first, leading to partial synchronization; and finally (iii) to a $P1_{f}$ limit cycle as the remaining natural mode, $f_{1}$ , also synchronizes, leading to complete synchronization. The minimum $\unicode[STIX]{x1D716}_{f}$ required for partial and complete synchronization decreases as $f_{f}$ approaches either $f_{1}$ or $f_{2}$ , resulting in two primary Arnold tongues. However, when forced at an amplitude above that required for complete synchronization, the system can transition out of $P1_{f}$ and into $\mathbb{T}_{1,2,f}^{3}$ or $\mathbb{T}_{2,f}^{2}$ . The optimal control strategy is to apply off-resonance forcing at a frequency around the weaker natural mode ( $f_{2}$ ) and at an amplitude just sufficient to cause $P1_{f}$ , because this produces the largest reduction in thermoacoustic amplitude via asynchronous quenching. Analysis of the Rayleigh index shows that this reduction is physically caused by a disruption of the positive coupling between the unsteady heat release rate of the flame and the $f_{1}$ and $f_{2}$ acoustic modes. If the forcing is applied near the stronger natural mode ( $f_{1}$ ), however, resonant amplification can occur. We then phenomenologically model this $\mathbb{T}_{1,2}^{2}$ thermoacoustic system as two reactively coupled van der Pol oscillators subjected to external sinusoidal forcing, and find that many of its synchronization features – such as the three-step route to $P1_{f}$ , the double Arnold tongues, asynchronous quenching and resonant amplification – can be qualitatively reproduced. This shows that these features are not limited to our particular system, but are universal features of forced self-excited oscillators. This study extends the applicability of open-loop control from classical period-1 systems with just a single time scale to ergodic $\mathbb{T}^{2}$ quasiperiodic systems with two incommensurate time scales.

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Forced synchronization of quasiperiodic oscillations in a thermoacoustic system
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Forced synchronization of quasiperiodic oscillations in a thermoacoustic system
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Forced synchronization of quasiperiodic oscillations in a thermoacoustic system
      Available formats
      ×

Copyright

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

Corresponding author

Email address for correspondence: larryli@ust.hk

References

Hide All
Acharya, V. S., Bothien, M. R. & Lieuwen, T. C. 2018 Non-linear dynamics of thermoacoustic eigen–mode interactions. Combust. Flame 194, 309321.
Afraimovich, V. S. & Shilnikov, L. P. 1991 Invariant two-dimensional tori, their breakdown and stochasticity. Am. Math. Soc. Transl. 149 (2), 201212.
Anishchenko, V., Nikolaev, S. & Kurths, J. 2007 Peculiarities of synchronization of a resonant limit cycle on a two-dimensional torus. Phys. Rev. E 76 (4), 046216.
Anishchenko, V., Nikolaev, S. & Kurths, J. 2008 Bifurcational mechanisms of synchronization of a resonant limit cycle on a two-dimensional torus. Chaos 18 (3), 037123.
Balanov, A., Janson, N., Postnov, D. & Sosnovtseva, O. 2008 Synchronization: From Simple to Complex. Springer Science and Business Media.
Balusamy, S., Li, L. K. B., Han, Z., Juniper, M. P. & Hochgreb, S. 2015 Nonlinear dynamics of a self-excited thermoacoustic system subjected to acoustic forcing. Proc. Combust. Inst. 35 (3), 32293236.
Battelino, P. M. 1988 Persistence of three-frequency quasiperiodicity under large perturbations. Phys. Rev. A 38 (3), 14951502.
Bellows, B., Hreiz, A. & Lieuwen, T. 2008 Nonlinear interactions between forced and self-excited acoustic oscillations in premixed combustor. J. Propul. Power 24 (3), 628631.
Biwa, T., Tozuka, S. & Yazaki, T. 2015 Amplitude death in coupled thermoacoustic oscillators. Phys. Rev. Appl. 3 (3), 034006.
Boashash, B. 1992 Estimating and interpreting the instantaneous frequency of a signal. Proc. IEEE 80 (4), 520538.
Boccaletti, S., Allaria, E., Meucci, R. & Arecchi, F. T. 2002 Experimental characterization of the transition to phase synchronization of chaotic CO2 laser systems. Phys. Rev. Lett. 89 (19), 194101.
Bonciolini, G. & Noiray, N. 2019 Bifurcation dodge: avoidance of a thermoacoustic instability under transient operation. Nonlinear Dyn. 1, 114.
Borkowski, L., Perlikowski, P., Kapitaniak, T. & Stefanski, A. 2015 Experimental observation of three-frequency quasiperiodic solution in a ring of unidirectionally coupled oscillators. Phys. Rev. E 91 (6), 062906.
Bothien, M. R., Moeck, J. P. & Paschereit, C. O. 2008 Active control of the acoustic boundary conditions of combustion test rigs. J. Sound Vib. 318 (4), 678701.
Bourehla, A. & Baillot, F. 1998 Appearance and stability of a laminar conical premixed flame subjected to an acoustic perturbation. Combust. Flame 114 (3), 303318.
Candel, S. 2002 Combustion dynamics and control: progress and challenges. Proc. Combust. Inst. 29 (1), 128.
Cao, L. 1997 Practical method for determining the minimum embedding dimension of a scalar time series. Physica D 110 (1), 4350.
Culick, F. E. C. 1971 Non-linear growth and limiting amplitude of acoustic oscillations in combustion chambers. Combust. Sci. Technol. 3 (1), 116.
Culick, F. E. C.2006 Unsteady motions in combustion chambers for propulsion systems. North Atlantic Treaty Organisation, AGARDograph AG-AVT-039.
Davitian, J., Getsinger, D., Hendrickson, C. & Karagozian, A. R. 2010 Transition to global instability in transverse-jet shear layers. J. Fluid Mech. 661, 294315.
Dewan, E. 1972 Harmonic entrainment of van der Pol oscillations: phaselocking and asynchronous quenching. IEEE Trans. Autom. Control 17 (5), 655663.
Dowling, A. P. & Morgans, A. S. 2005 Feedback control of combustion oscillations. Annu. Rev. Fluid Mech. 37, 151182.
Fraser, A. M. & Swinney, H. L. 1986 Independent coordinates for strange attractors from mutual information. Phys. Rev. A 33 (2), 11341140.
Gabor, D. 1946 Theory of communication. J. Inst. Electr. Engng (London) 3, 429457.
Gaydon, A. G. 1974 Spectroscopy of Flames. Chapman and Hall.
Glass, L. 2001 Synchronization and rhythmic processes in physiology. Nature 410 (6825), 277284.
Gollub, J. P. & Benson, S. V. 1980 Many routes to turbulent convection. J. Fluid Mech. 100 (3), 449470.
Gopalakrishnan, E. A., Tony, J., Sreelekha, E. & Sujith, R. I. 2016 Stochastic bifurcations in a prototypical thermoacoustic system. Phys. Rev. E 94 (2), 022203.
Gotoda, H., Okuno, Y., Hayashi, K. & Tachibana, S. 2015 Characterization of degeneration process in combustion instability based on dynamical systems theory. Phys. Rev. E 92 (5), 052906.
Gottwald, G. A. & Melbourne, I. 2004 A new test for chaos in deterministic systems. Proc. R. Soc. Lond. A 460 (2042), 603611.
Grassberger, P. & Procaccia, I. 1983 Characterization of strange attractors. Phys. Rev. Lett. 50 (5), 346349.
Guan, Y., Gupta, V., Kashinath, K. & Li, L. K. B. 2019a Open-loop control of periodic thermoacoustic oscillations: experiments and low-order modelling in a synchronization framework. Proc. Combust. Inst. 37, 53155323.
Guan, Y., He, W., Murugesan, M., Li, Q., Liu, P. & Li, L. K. B. 2019b Control of self-excited thermoacoustic oscillations using transient forcing, hysteresis and mode switching. Combust. Flame 202, 262275.
Guan, Y., Murugesan, M. & Li, L. K. B. 2018 Strange nonchaotic and chaotic attractors in a self-excited thermoacoustic oscillator subjected to external periodic forcing. Chaos 28 (9), 093109.
Gustafsson, F. 1996 Determining the initial states in forward-backward filtering. IEEE Trans. Signal Process. 44 (4), 988992.
Heckl, M. A. 1988 Active control of the noise from a Rijke tube. J. Sound Vib. 124 (1), 117133.
Hilborn, R. C. 2000 Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers. Oxford University Press.
Hong, S., Shanbhogue, S. J., Speth, R. L. & Ghoniem, A. F. 2013 On the phase between pressure and heat release fluctuations for propane/hydrogen flames and its role in mode transitions. Combust. Flame 160 (12), 28272842.
Hovel, P. 2010 Control of Complex Nonlinear Systems with Delay. Springer.
Jahnke, C. C. & Culick, F. E. C. 1994 Application of dynamical systems theory to nonlinear combustion instabilities. J. Propul. Power 10 (4), 508517.
Johnson, A., Uddin, M. & Pollard, A. 2005 Calibration of hot-wire probes using non-uniform mean velocity profiles. Exp. Fluids 39 (3), 525532.
Juniper, M. P. & Sujith, R. I. 2018 Sensitivity and nonlinearity of thermoacoustic oscillations. Annu. Rev. Fluid Mech. 50, 661689.
Kabiraj, L., Saurabh, A., Wahi, P. & Sujith, R. I. 2012a Route to chaos for combustion instability in ducted laminar premixed flames. Chaos 22 (2), 023129.
Kabiraj, L., Steinert, R., Saurabh, A. & Paschereit, C. O. 2015 Coherence resonance in a thermoacoustic system. Phys. Rev. E 92 (4), 042909.
Kabiraj, L. & Sujith, R. I. 2012 Nonlinear self-excited thermoacoustic oscillations: intermittency and flame blowout. J. Fluid Mech. 713, 376397.
Kabiraj, L., Sujith, R. I. & Wahi, P. 2012b Bifurcations of self-excited ducted laminar premixed flames. Trans. ASME: J. Engng Gas Turbines Power 134 (3), 031502.
Kantz, H. & Schreiber, T. 2003 Nonlinear Time Series Analysis, 2nd edn. Cambridge University Press.
Kashinath, K., Li, L. K. B. & Juniper, M. P. 2018 Forced synchronization of periodic and aperiodic thermoacoustic oscillations: lock-in, bifurcations and open-loop control. J. Fluid Mech. 838, 690714.
Kashinath, K., Waugh, I. C. & Juniper, M. P. 2014 Nonlinear self-excited thermoacoustic oscillations of a ducted premixed flame: bifurcations and routes to chaos. J. Fluid Mech. 761, 399430.
Keen, B. E. & Fletcher, W. H. W. 1970 Suppression of a plasma instability by the method of ‘asynchronous quenching’. Phys. Rev. Lett. 24 (4), 130.
Khalak, A. & Williamson, C. H. K. 1999 Motions, forces and mode transitions in vortex-induced vibrations at low mass-damping. J. Fluid Struct. 13, 813851.
Kulp, C. W. & Zunino, L. 2014 Discriminating chaotic and stochastic dynamics through the permutation spectrum test. Chaos 24 (3), 033116.
Li, L. K. B. & Juniper, M. P. 2013a Lock-in and quasiperiodicity in a forced hydrodynamically self-excited jet. J. Fluid Mech. 726, 624655.
Li, L. K. B. & Juniper, M. P. 2013b Lock-in and quasiperiodicity in hydrodynamically self-excited flames: experiments and modelling. Proc. Combust. Inst. 34 (1), 947954.
Li, L. K. B. & Juniper, M. P. 2013c Phase trapping and slipping in a forced hydrodynamically self-excited jet. J. Fluid Mech. 735, R5.
Libchaber, A. & Maurer, J. 1982 A Rayleigh Bénard experiment: helium in a small box. In Nonlinear Phenomena at Phase Transitions and Instabilities, pp. 259286. Springer.
Lieuwen, T. C. 2003 Combustion driven oscillations in gas turbines. In Turbomachinery International, pp. 1618. Turbomachinery International Publications.
Lieuwen, T. C. & Yang, V. 2005 Combustion Instabilities in Gas Turbine Engines: Operational Experience, Fundamental Mechanisms, and Modeling. American Institute of Aeronautics and Astronautics.
Loose, A., Wünsche, H. J. & Henneberger, F. 2010 Synchronization of quasiperiodic oscillations to a periodic force studied with semiconductor lasers. Phys. Rev. E 82 (3), 035201.
Lubarsky, E., Shcherbik, D., Bibik, A. & Zinn, B. T. 2003 Active control of combustion oscillations by non-coherent fuel flow modulation. In Ninth AIAA/CEAS Aeroacoustics Conference and Exhibit, AIAA Paper, p. 3180.
Martin, S., Leber, H. & Martienssen, W. 1984 Oscillatory and chaotic states of the electrical conduction in barium sodium niobate crystals. Phys. Rev. Lett. 53 (4), 303.
McManus, K. R., Vandsburger, U. & Bowman, C. T. 1990 Combustor performance enhancement through direct shear layer excitation. Combust. Flame 82 (1), 7592.
Minorsky, N. 1967 Comments on asynchronous quenching. IEEE Trans. Autom. Control 12 (2), 225227.
Moeck, J. P. & Paschereit, C. O. 2012 Nonlinear interactions of multiple linearly unstable thermoacoustic modes. Intl J. Spray Combust. 4 (1), 127.
Mondal, S., Pawar, S. A. & Sujith, R. I. 2017 Synchronous behaviour of two interacting oscillatory systems undergoing quasiperiodic route to chaos. Chaos 27 (10), 103119.
Mondal, S., Pawar, S. A. & Sujith, R. I. 2019 Forced synchronization and asynchronous quenching of periodic oscillations in a thermoacoustic system. J. Fluid Mech. 864, 7396.
Mongia, H. C., Held, T. J., Hsiao, G. C. & Pandalai, R. P. 2003 Challenges and progress in controlling dynamics in gas turbine combustors. J. Propul. Power 19 (5), 822829.
Newhouse, S., Ruelle, D. & Takens, F. 1978 Occurrence of strange Axiom A attractors near quasiperiodic flows on t m , m⩾3. Commun. Math. Phys. 64 (1), 3540.
Noiray, N., Durox, D., Schuller, T. & Candel, S. 2009 Dynamic phase converter for passive control of combustion instabilities. Proc. Combust. Inst. 32 (2), 31633170.
Noiray, N. & Schuermans, B. 2013 Deterministic quantities characterizing noise driven Hopf bifurcations in gas turbine combustors. Intl J. Nonlinear Mech. 50, 152163.
Odajima, K., Nishida, Y. & Hatta, Y. 1974 Synchronous quenching of drift-wave instability. Phys. Fluids 17 (8), 16311633.
Orchini, A. & Juniper, M. P. 2016 Flame double input describing function analysis. Combust. Flame 171, 87102.
Pawar, S. A., Sujith, R. I., Emerson, B. & Lieuwen, T. 2018 Characterization of forced response of density stratified reacting wake. Chaos 28 (2), 023108.
Pikovsky, A., Rosenblum, M. & Kurths, J. 2003 Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press.
Pikovsky, A., Rosenblum, M., Osipov, G. & Kurths, J. 1997 Phase synchronization of chaotic oscillators by external driving. Physica D 104 (3–4), 219238.
Poinsot, T. 2017 Prediction and control of combustion instabilities in real engines. Proc. Combust. Inst. 36 (1), 128.
van der Pol, B. 1927 Forced oscillations in a circuit with non-linear resistance. Phil. Mag. 3 (13), 6580.
van der Pol, B. & van der Mark, J. 1927 Frequency demultiplication. Nature 120 (3019), 363364.
Provansal, M., Mathis, C. & Boyer, L. 1987 Bénard–von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 122.
Rayleigh, Lord 1878 The explanation of certain acoustical phenomena. Nature 18, 319321.
Ruelle, D. & Takens, F. 1971 On the nature of turbulence. Commun. Math. Phys. 20 (3), 167192.
Schmid, P. J. & Henningson, D. S. 2012 Stability and Transition in Shear Flows. Springer Science and Business Media.
Shampine, L. F. & Reichelt, M. W. 1997 The Matlab ODE suite. SIAM J. Sci. Comput. 18 (1), 122.
Small, M. 2005 Applied Nonlinear Time Series Analysis: Applications in Physics, Physiology and Finance. World Scientific.
Stankevich, N. V., Kurths, J. & Kuznetsov, A. P. 2015 Forced synchronization of quasiperiodic oscillations. Commun. Nonlinear Sci. 20 (1), 316323.
Staubli, T. 1987 Entrainment of self-sustained flow oscillations: phaselocking or asynchronous quenching? J. Appl. Mech. 54, 707.
Takens, F. 1981 Detecting strange attractors in turbulence. In Dynamical Systems and Turbulence (ed. Rand, D. A. & Young, L. S.), Lecture Notes in Mathematics, pp. 366381. Springer.
Thévenin, J., Romanelli, M., Vallet, M., Brunel, M. & Erneux, T. 2011 Resonance assisted synchronization of coupled oscillators: frequency locking without phase locking. Phys. Rev. Lett. 107 (10), 104101.
Thompson, J. M. T. & Stewart, H. B. 2002 Nonlinear Dynamics and Chaos. John Wiley.
Van Buskirk, R. & Jeffries, C. 1985 Observation of chaotic dynamics of coupled nonlinear oscillators. Phys. Rev. A 31 (5), 3332.
Vishnu, R., Sujith, R. I. & Aghalayam, P. 2015 Role of flame dynamics on the bifurcation characteristics of a ducted V-flame. Combust. Sci. Technol. 187 (6), 894905.
Walden, R. W., Kolodner, P., Passner, A. & Surko, C. M. 1984 Nonchaotic Rayleigh–Bénard convection with four and five incommensurate frequencies. Phys. Rev. Lett. 53 (3), 242.
Welch, P. D. 1967 The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short modified periodograms. IEEE Trans. Audio Electroacoust. 15, 7073.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Forced synchronization of quasiperiodic oscillations in a thermoacoustic system

  • Yu Guan (a1), Vikrant Gupta (a2), Minping Wan (a2) and Larry K. B. Li (a1)

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