Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-25T06:25:10.979Z Has data issue: false hasContentIssue false
  • English
  • Français

Phase-reduction analysis of periodic thermoacoustic oscillations in a Rijke tube

Published online by Cambridge University Press:  29 December 2021

Calum S. Skene Open the ORCID record for Calum S. Skene [Opens in a new window]  and
Kunihiko Taira Open the ORCID record for Kunihiko Taira [Opens in a new window]
Calum S. Skene*
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, CA 90095, USA
Kunihiko Taira
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, CA 90095, USA
*
†Email address for correspondence: c.s.skene@leeds.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Phase-reduction analysis captures the linear phase dynamics with respect to a limit cycle subjected to weak external forcing. We apply this technique to study the phase dynamics of the self-sustained oscillations produced by a Rijke tube undergoing thermoacoustic instability. Through the phase-reduction formulation, we are able to reduce these dynamics to a scalar equation for the phase, which allows us to efficiently determine the synchronisation properties of the system. For the thermoacoustic system, we find the conditions for which $m:n$ frequency locking occurs, which sheds light on the mechanisms behind asynchronous and synchronous quenching. We also reveal the optimal placement of pressure actuators that provide the most efficient route to synchronisation.

JFM classification

Acoustics: Acoustics Instability: Instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 933 , 25 February 2022 , A35
Check for updates
Copyright
© The Author(s), 2021. Published by 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.)

Footnotes

‡

Present address: Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK.

References

REFERENCES

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.CrossRefGoogle Scholar
Balasubramanian, K. & Sujith, R.I. 2008 Thermoacoustic instability in a Rijke tube: non-normality and nonlinearity. Phys. Fluids 20 (4), 044103.CrossRefGoogle Scholar
Barkley, D. & Henderson, R.D. 1996 Three-dimensional floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215–241.CrossRefGoogle Scholar
Boccaletti, S., Pisarchik, A.N., del Genio, C.I. & Amann, A. 2018 Synchronization: From Coupled Systems to Complex Networks. Cambridge University Press.CrossRefGoogle Scholar
Candel, S. 2002 Combustion dynamics and control: progress and challenges. Proc. Combust. Inst. 29 (1), 1–28.CrossRefGoogle Scholar
Correa, S.M. 1998 Power generation and aeropropulsion gas turbines: from combustion science to combustion technology. Symp. (Intl) Combust. 27 (2), 1793–1807.CrossRefGoogle Scholar
Culick, F.E.C. 1996 Combustion Instabilities in Propulsion Systems, Nato Science Series E, vol. 306, pp. 173–241. Springer.Google Scholar
Dowling, A.P. & Morgans, A.S. 2005 Feedback control of combustion oscillations. Annu. Rev. Fluid Mech. 37 (1), 151–182.CrossRefGoogle Scholar
Dupère, I.D.J. & Dowling, A.P. 2005 The use of helmholtz resonators in a practical combustor. Trans. ASME: J. Engng Gas Turbines Power 127 (2), 268–275.Google Scholar
Ermentrout, G.B. & Terman, D. 2010 Mathematical Foundations of Neuroscience, Interdisciplinary Applied Mathematics, vol. 35. Springer.CrossRefGoogle Scholar
Floquet, G. 1883 Sur les équations différentielles linéaires à coefficients périodiques. Ann. Sci. Éc. Norm. Supér. 12, 47–88.CrossRefGoogle Scholar
Hale, J.K. 1977 Theory of Functional Differential Equations, Applied Mathematical Sciences, vol. 3. Springer-Verlag.CrossRefGoogle Scholar
Heckl, M.A. 1990 Non-linear acoustic effects in the Rijke tube. Acta Acust. 72 (1), 63–71.Google Scholar
Iima, M. 2019 Jacobian-free algorithm to calculate the phase sensitivity function in the phase reduction theory and its applications to Kármán's vortex street. Phys. Rev. E 99, 062203.CrossRefGoogle ScholarPubMed
Juniper, M.P. 2011 Triggering in the horizontal Rijke tube: non-normality, transient growth and bypass transition. J. Fluid Mech. 667, 272–308.CrossRefGoogle Scholar
Juniper, M.P. & Sujith, R.I. 2018 Sensitivity and nonlinearity of thermoacoustic oscillations. Annu. Rev. Fluid Mech. 50 (1), 661–689.CrossRefGoogle Scholar
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, 690–714.CrossRefGoogle Scholar
Kawamura, Y. & Nakao, H. 2015 Phase description of oscillatory convection with a spatially translational mode. Physica D 295–296, 11–29.CrossRefGoogle Scholar
Khodkar, M.A., Klamo, J.T. & Taira, K. 2021 Phase-locking of laminar wake to periodic vibrations of a circular cylinder. Phys. Rev. Fluids 6, 034401.CrossRefGoogle Scholar
Khodkar, M.A. & Taira, K. 2020 Phase-synchronization properties of laminar cylinder wake for periodic external forcings. J. Fluid Mech. 904, R1.CrossRefGoogle Scholar
King, L.V. 1914 XII. On the convection of heat from small cylinders in a stream of fluid: determination of the convection constants of small platinum wires with applications to hot-wire anemometry. Philos. Trans. R. Soc. Lond. A 214 (509–522), 373–432.Google Scholar
Kotani, K., Yamaguchi, I., Ogawa, Y., Jimbo, Y., Nakao, H. & Ermentrout, G.B. 2012 Adjoint method provides phase response functions for delay-induced oscillations. Phys. Rev. Lett. 109, 044101.CrossRefGoogle ScholarPubMed
Kuramoto, Y. 1984 Chemical Oscillations, Waves, and Turbulence. Springer-Verlag.CrossRefGoogle Scholar
Lieuwen, T.C. & Yang, V. 2005 Combustion Instabilities in Gas Turbine Engines: Operational Experience, Fundamental Mechanisms, and Modeling, Progress in Astronautics and Aeronautics, vol. 210. AIAA.Google Scholar
Loe, I.A., Nakao, H., Jimbo, Y. & Kotani, K. 2021 Phase-reduction for synchronization of oscillating flow by perturbation on surrounding structure. J. Fluid Mech. 911, R2.CrossRefGoogle Scholar
Magri, L. 2019 Adjoint methods as design tools in thermoacoustics. Appl. Mech. Rev. 71 (2), 020801.CrossRefGoogle Scholar
Magri, L. & Juniper, M.P. 2013 Sensitivity analysis of a time-delayed thermo-acoustic system via an adjoint-based approach. J. Fluid Mech. 719, 183–202.CrossRefGoogle Scholar
Mahashabde, A., et al. 2011 Assessing the environmental impacts of aircraft noise and emissions. Prog. Aerosp. Sci. 47 (1), 15–52.CrossRefGoogle 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.CrossRefGoogle Scholar
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, 73–96.CrossRefGoogle Scholar
Nair, A.G., Taira, K., Brunton, B.W. & Brunton, S.L. 2021 Phase-based control of periodic fluid flows. J. Fluid Mech. 927, A30.CrossRefGoogle Scholar
Nakao, H. 2016 Phase reduction approach to synchronisation of nonlinear oscillators. Contemp. Phys. 57 (2), 188–214.CrossRefGoogle Scholar
Novičenko, V. & Pyragas, K. 2012 Phase reduction of weakly perturbed limit cycle oscillations in time-delay systems. Physica D 241 (12), 1090–1098.CrossRefGoogle Scholar
Pikovsky, A., Rosenblum, M.G. & Kurths, J. 2003 Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press.CrossRefGoogle Scholar
Rackauckas, C. & Nie, Q. 2017 Differentialequations.jl: a performant and feature-rich ecosystem for solving differential equations in Julia. J. Open Res. Softw. 5 (1), 15.CrossRefGoogle Scholar
Rand, R. 2012 Differential-Delay Equations, pp. 83–117. Springer Berlin Heidelberg.Google Scholar
Rayleigh, J.L. 1878 The explanation of certain acoustical phenomena. Nature 18, 319–321.CrossRefGoogle Scholar
Rijke, P.L. 1859 LXXI. Notice of a new method of causing a vibration of the air contained in a tube open at both ends. Lond. Edinb. Dublin Philos. Mag. J. Sci. 17 (116), 419–422.CrossRefGoogle Scholar
Roy, A., Mondal, S., Pawar, S.A. & Sujith, R.I. 2020 On the mechanism of open-loop control of thermoacoustic instability in a laminar premixed combustor. J. Fluid Mech. 884, A2.CrossRefGoogle Scholar
Sayadi, T., Le Chenadec, V., Schmid, P.J., Richecoeur, F. & Massot, M. 2014 Thermoacoustic instability: a dynamical system and time domain analysis. J. Fluid Mech. 753, 448–471.CrossRefGoogle Scholar
Simmendinger, C., Wunderlin, A. & Pelster, A. 1999 Analytical approach for the floquet theory of delay differential equations. Phys. Rev. E 59, 5344–5353.CrossRefGoogle ScholarPubMed
Skene, C.S. & Taira, K. 2021 csskene/phase-reduction_rijke-tube: Initial release. https://doi.org/10.5281/zenodo.4981470CrossRefGoogle Scholar
Taira, K. & Nakao, H. 2018 Phase-response analysis of synchronization for periodic flows. J. Fluid Mech. 846, R2.CrossRefGoogle Scholar
Verner, J.H. 2010 Numerically optimal Runge–Kutta pairs with interpolants. Numer. Algorithms 53 (2–3), 383–396.CrossRefGoogle Scholar
Wischert, W., Wunderlin, A., Pelster, A., Olivier, M. & Groslambert, J. 1994 Delay-induced instabilities in nonlinear feedback systems. Phys. Rev. E 49, 203–219.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Zhao, D. & Li, X.Y. 2015 A review of acoustic dampers applied to combustion chambers in aerospace industry. Prog. Aerosp. Sci. 74, 114–130.CrossRefGoogle Scholar
Zhao, D., Lu, Z., Zhao, H., Li, X.Y., Wang, B. & Liu, P. 2018 A review of active control approaches in stabilizing combustion systems in aerospace industry. Prog. Aerosp. Sci. 97, 35–60.CrossRefGoogle Scholar