Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-26T19:36:11.644Z Has data issue: false hasContentIssue false
  • English
  • Français

Forced synchronization of periodic and aperiodic thermoacoustic oscillations: lock-in, bifurcations and open-loop control

Published online by Cambridge University Press:  22 January 2018

Karthik Kashinath Open the ORCID record for Karthik Kashinath [Opens in a new window] ,
Larry K. B. Li Open the ORCID record for Larry K. B. Li [Opens in a new window]  and
Matthew P. Juniper
Karthik Kashinath*
Affiliation:
Lawrence Berkeley National Laboratory, Climate Science Department – Earth and Environmental Sciences Area, 1 Cyclotron Road, Berkeley, CA 94720, USA
Larry K. B. Li
Affiliation:
Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
Matthew P. Juniper
Affiliation:
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
*
†Email address for correspondence: karthikkashinath@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Synchronization is a universal concept in nonlinear science but has received little attention in thermoacoustics. In this numerical study, we take a dynamical systems approach to investigating the influence of harmonic acoustic forcing on three different types of self-excited thermoacoustic oscillations: periodic, quasi-periodic and chaotic. When the periodic system is forced, we find that: (i) at low forcing amplitudes, it responds at both the forcing frequency and the natural (self-excited) frequency, as well as at their linear combinations, indicating quasi-periodicity; (ii) above a critical forcing amplitude, the system locks in to the forcing; (iii) the bifurcations leading up to lock-in and the critical forcing amplitude required for lock-in depend on the proximity of the forcing frequency to the natural frequency; (iv) the response amplitude at lock-in may be larger or smaller than that of the unforced system and the system can exhibit hysteresis and the jump phenomenon owing to a cusp catastrophe; and (v) at forcing amplitudes above lock-in, the oscillations can become unstable and transition to chaos, or switch between different stable attractors depending on the forcing amplitude. When the quasi-periodic system is forced at a frequency equal to one of the two characteristic frequencies of the torus attractor, we find that lock-in occurs via a saddle-node bifurcation with frequency pulling. When the chaotic system is forced at a frequency close to the dominant frequency of its strange attractor, we find that it is possible to destroy chaos and establish stable periodic oscillations. These results show that the open-loop application of harmonic acoustic forcing can be an effective strategy for controlling periodic or aperiodic thermoacoustic oscillations. In some cases, we find that such forcing can reduce the response amplitude by up to 90 %, making it a viable way to weaken thermoacoustic oscillations.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Nonlinear Dynamical Systems: Chaos Acoustics: Noise control
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 838 , 10 March 2018 , pp. 690 - 714
Check for updates
Copyright
© 2018 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

Abel, M., Ahnert, K. & Bergweiler, S. 2009 Synchronization of sound sources. Phys. Rev. Lett. 103 (11), 114301.CrossRefGoogle ScholarPubMed
Afraimovich, V. S. & Shilnikov, L. P. 1983 On invariant two-dimensional tori, their breakdown and stochasticity. In Methods of the Qualitative Theory of Differential Equations, pp. 3–26. Gor’kov. Gos. University.Google Scholar
Annaswamy, A. M. 2006 Nonlinear modeling and control of combustion dynamics. In Control of Fluid Flow, pp. 95–121. Springer.CrossRefGoogle Scholar
Aronson, D. G., Ermentrout, G. B. & Kopell, N. 1990 Amplitude response of coupled oscillators. Physica D 41 (3), 403–449.Google Scholar
Balanov, A., Janson, N., Postnov, D. & Sosnovtseva, O. 2009 1 : 1 forced synchronization of periodic oscillations. In Synchronization: From Simple to Complex, chap. 3, Springer.Google Scholar
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), 3229–3236.CrossRefGoogle Scholar
Belair, J. & Glass, L. 1983 Self-similarity in periodically forced oscillators. Phys. Lett. A 96 (3), 113–116.CrossRefGoogle Scholar
Bellows, B. D., Hreiz, A. & Lieuwen, T. 2008 Nonlinear interactions between forced and self-excited acoustic oscillations in premixed combustor. J. Propul. Power 24 (3), 628–631.CrossRefGoogle Scholar
Blevins, R. D. 1985 The effect of sound on vortex shedding from cylinders. J. Fluid Mech. 161, 217–237.CrossRefGoogle Scholar
Chen, S. C. 1990 Growth and frequency pushing effects in relativistic magnetron phase-locking. IEEE Trans. Plasma Sci. 18 (3), 570–576.CrossRefGoogle Scholar
Davitian, J., Getsinger, D., Hendrickson, C. & Karagozian, A. R. 2010 Transition to global instability in transverse-jet shear layers. J. Fluid Mech. 661, 294–315.CrossRefGoogle Scholar
Dowling, A. P. 1997 Nonlinear self-excited oscillations of a ducted flame. J. Fluid Mech. 346, 271–290.CrossRefGoogle Scholar
Dowling, A. P. & Morgans, A. S. 2005 Feedback control of combustion oscillations. Annu. Rev. Fluid Mech. 37, 151–182.CrossRefGoogle Scholar
Gabor, D. 1946 Theory of communication. J. Inst. Electr. Engrs (London) 3, 429–457.Google Scholar
Getsinger, D. R., Hendrickson, C. & Karagozian, A. R. 2012 Shear layer instabilities in low-density transverse jets. Exp. Fluids 53 (3), 783–801.CrossRefGoogle Scholar
Giannakopoulos, K. & Deliyannis, T. 2001 Jump phenomenon in an OTA simulated LC circuit. Intl J. Electronics 88 (1), 1–11.CrossRefGoogle Scholar
Glass, L., Guevara, M. R., Belair, J. & Shrier, A. 1984 Global bifurcations of a periodically forced biological oscillator. Phys. Lett. A 29 (3), 1348.Google Scholar
Gotoda, H., Amano, M., Miyano, T., Ikawa, T., Maki, K. & Tachibana, S. 2012 Characterization of complexities in combustion instability in a lean premixed gas-turbine model combustor. Chaos 22, 043128.CrossRefGoogle Scholar
Gotoda, H., Nikimoto, H., Miyano, T. & Tachibana, S. 2011 Dynamic properties of combustion instability in a lean premixed gas-turbine combustor. Chaos 21 (1), 013124.CrossRefGoogle Scholar
Gottlieb, S. & Shu, C. 1998 Total variation diminishing Runge–Kutta schemes. Math. Comput. 67 (221), 73–85.CrossRefGoogle Scholar
Hemchandra, S.2009 Dynamics of turbulent premixed flames in acoustic fields. PhD thesis, Georgia Institute of Technology, USA.Google Scholar
Holmes, P. J. & Rand, D. A. 1978 Bifurcations of the forced van der Pol oscillator. Q. Appl. Maths 35, 495–509.CrossRefGoogle Scholar
Hopfield, J. J. 1994 Neurons, dynamics and computation. Phys. Today 47 (2), 40–46.CrossRefGoogle Scholar
Hovel, P. 2010 Control of Complex Nonlinear Systems with Delay. Springer.CrossRefGoogle Scholar
Huygens, C. 1673 The pendulum clock. In Horologium Oscillatorium. Iowa State University Press; translated in 1986.Google Scholar
Illingworth, S. J. & Morgans, A. S. 2010 Adaptive feedback control of combustion instability in annular combustors. Combust. Sci. Technol. 182 (2), 143–164.CrossRefGoogle Scholar
Jiang, G. S. & Peng, D. 2000 Weighted ENO schemes for Hamilton–Jacobi equations. SIAM J. Sci. Comput. 21 (6), 2126–2143.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
Kabiraj, L. & Sujith, R. I. 2012 Nonlinear self-excited thermoacoustic oscillations: intermittency and flame blowout. J. Fluid Mech. 713, 376–397.CrossRefGoogle Scholar
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.Google Scholar
Kantz, H. & Schreiber, T. 2003 Nonlinear Time Series Analysis, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Karimi, N., Brear, M., Jin, S. H. & Monty, J. P. 2009 Linear and non-linear forced response of a conical ducted laminar premixed flame. Combust. Flame 156 (11), 2201–2212.CrossRefGoogle Scholar
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, 399–430.CrossRefGoogle Scholar
Li, L. K. B.2012 Forcing of globally unstable jets and flames. PhD thesis, University of Cambridge, Department of Engineering, United Kingdom.Google Scholar
Li, L. K. B. & Juniper, M. P. 2013a Lock-in and quasiperiodicity in a forced hydrodynamically self-excited jet. J. Fluid Mech. 726, 624–655.CrossRefGoogle Scholar
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), 947–954.CrossRefGoogle Scholar
Li, L. K. B. & Juniper, M. P. 2013c Phase trapping and slipping in a forced hydrodynamically self-excited jet. J. Fluid Mech. 735 (R5), 1–11.CrossRefGoogle Scholar
Lieuwen, T. C. & Yang, V. 2005 Combustion Instabilities in Gas Turbine Engines: Operational Experience, Fundamental Mechanisms, and Modeling. American Institute of Aeronautics and Astronautics.Google Scholar
Lubarsky, E., Shcherbik, D., Bibik, A. & Zinn, B. T. 2003 Active control of combustion oscillations by non-coherent fuel flow modulation. AIAA J. 3180 (9), 730–745.Google Scholar
Miranda, J. M. G. 2004 Synchronization and Control of Chaos: An Introduction for Scientists and Engineers. Imperial College Press.CrossRefGoogle Scholar
Nayfeh, A. H. & Balachandran, B. 2004 Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. Wiley-VCH.Google Scholar
Nayfeh, A. H. & Mook, D. T. 1995 Nonlinear Oscillations. Wiley.CrossRefGoogle Scholar
Olinger, D. J. 1992 Lock-in states in the dripping mode of the capillary jet. Exp. Fluids 15 (2), 155–158.CrossRefGoogle Scholar
Orchini, A., Illingworth, S. J. & Juniper, M. P. 2015 Frequency domain and time domain analysis of thermoacoustic oscillations with wave-based acoustics. J. Fluid Mech. 775, 387–414.CrossRefGoogle Scholar
Petrov, V., Ouyang, Q. & Swinney, H. L. 1997 Resonant pattern formation in a chemical system. Nature 388, 655–657.CrossRefGoogle Scholar
Pikovsky, A. S., Rosenblum, M. G. & Kurths, J. 2003 Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press.Google Scholar
van der Pol, B. 1927 Forced oscillations in a circuit with non-linear resistance. Phil. Mag. 3 (13), 65–80.CrossRefGoogle Scholar
van der Pol, B. & van der Mark, J. 1927 Frequency demultiplication. Nature 120 (3019), 363–364.CrossRefGoogle Scholar
Preetham, Santosh, H. & Lieuwen, T. 2008 Dynamics of laminar premixed flames forced by harmonic velocity disturbances. J. Propul. Power 24 (6), 1390–1402.CrossRefGoogle Scholar
Provansal, M., Mathis, C. & Boyer, L. 1987 Bénard–von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 1–22.CrossRefGoogle Scholar
Rompala, K., Rand, R. & Howland, H. 2007 Dynamics of three coupled van der Pol oscillators with application to circadian rhythms. Commun. Nonlinear Sci. 12 (5), 794–803.CrossRefGoogle Scholar
Scheffer, M., Carpenter, S., Foley, J. A., Folke, C. & Walker, B. 2001 Catastrophic shifts in ecosystems. Nature 413, 591–596.CrossRefGoogle ScholarPubMed
Small, M. 2005 Applied Nonlinear Time Series Analysis: Applications in Physics, Physiology and Finance. World Scientific.Google Scholar
Strogatz, S. H. 1994 Nonlinear Dynamics and Chaos. Perseus Books.Google Scholar
Thompson, J. M. T. & Stewart, H. B. 2002 Nonlinear Dynamics and Chaos. Wiley.Google Scholar
Venkatesan, A. & Lakshmanan, M. 1997 Bifurcation and chaos in the double-well Duffing–van der Pol oscillator: numerical and analytical studies. Phys. Rev. E 56 (6), 6321.Google Scholar
Walsh, J. E., Johnston, G. L., Davidson, R. C. & Sullivan, D. J. 1989 Theory of phase-locked regenerative oscillators with nonlinear frequency-shift effects. In OE/LASE’89, pp. 161–169. International Society for Optics and Photonics.Google Scholar
Wang, J. & Lim, T. C. 2011 On the boundary between nonlinear jump phenomenon and linear response of hypoid gear dynamics. Adv. Acoust. Vib. 2011, 1–13.Google Scholar
Williams, F. A. 1994 Combustion Theory, 2nd edn. Westview Press.Google Scholar
Xia, M. & Sun, Q. P. 2015 Jump phenomena of rotational angle and temperature of NiTi wire in nonlinear torsional vibration. Intl J. Solids Struct. 56–57, 220–234.CrossRefGoogle Scholar