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Stochastic dynamics of vortex-acoustic lock-in: effects of coloured, multiplicative noise

Published online by Cambridge University Press:  08 October 2025

Sathesh Mariappan Open the ORCID record for Sathesh Mariappan [Opens in a new window]
Sathesh Mariappan*
Affiliation:
Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, Uttar Pradesh, India
*
Corresponding author: Sathesh Mariappan, sathesh@iitk.ac.in
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Abstract

The paper discusses the stochastic dynamics of the vortex shedding process in the presence of external harmonic excitation and coloured multiplicative noise. The situation is encountered in a turbulent practical combustor experiencing combustion instability. Acoustic feedback and turbulent flow are imitated by the harmonic and stochastic excitations, respectively. The Ornstein–Uhlenbeck process is used to generate the noise. A low-order model for vortex shedding is used. The Fokker–Planck framework is used to obtain the evolution of the probability density function of the shedding time period. Stochastic lock-in and resonance characteristics are studied for various parameters associated with the harmonic (amplitude, frequency) and noise (amplitude, correlation time, multiplicative noise factor) excitations. We observed that: (i) the stochastic lock-in (s-lock-in) boundary strongly depends on the noise correlation time; (ii) the parameter sites for s-lock-in can be approximately identified from the noise-induced shedding statistics; and (iii) stochastic resonance is significant for some intermediate correlation times. The effects of the above-mentioned observations are discussed in the context of combustion instability.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Vortex Flows: Vortex dynamics Nonlinear Dynamical Systems: Low-dimensional models

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JFM Papers
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Journal of Fluid Mechanics , Volume 1020 , 10 October 2025 , A55
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© The Author(s), 2025. Published by Cambridge University Press

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References

Achenbach, E. & Heinecke, E. 1981 On vortex shedding from smooth and rough cylinders in the range of Reynolds numbers 6 $\times$ 103 to 5 $\times$ 106. J. Fluid Mech. 109, 239–251.10.1017/S002211208100102XCrossRefGoogle Scholar
Anishchenko, V.S., Astakhov, V., Neiman, A., Vadivasova, T. & Schimansky-Geier, L. 2007 Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments. vol. 111. Springer.Google Scholar
Bagheri, G., Hosseini, S.E. & Wahid, M.A. 2014 Effects of bluff body shape on the flame stability in premixed micro-combustion of hydrogen–air mixture. Appl. Therm. Engng 67 (1–2), 266–272.10.1016/j.applthermaleng.2014.03.040CrossRefGoogle Scholar
Balanov, A., Janson, N., Postnov, D. & Sosnovtseva, O. 2008 Synchronization: From Simple to Complex. Springer Science & Business Media.Google Scholar
Bonciolini, G., Boujo, E. & Noiray, N. 2016 Effects of turbulence-induced colored noise on thermoacoustic instabilities in combustion chambers. In International Symposium: Thermoacoustic Instabilities in Gas Turbines and Rocket Engines, ETH Zürich.Google Scholar
Bonciolini, G., Boujo, E. & Noiray, N. 2017 Output-only parameter identification of a colored-noise-driven Van-der-Pol oscillator: Thermoacoustic instabilities as an example. Phys. Rev. E 95 (6), 062217.10.1103/PhysRevE.95.062217CrossRefGoogle ScholarPubMed
Bonciolini, G., Faure-Beaulieu, A., Bourquard, C. & Noiray, N. 2021 Low order modelling of thermoacoustic instabilities and intermittency: flame response delay and nonlinearity. Combust. Flame 226, 396–411.10.1016/j.combustflame.2020.12.034CrossRefGoogle Scholar
Boulal, S., Genot, A., Klein, J.-M., Fabignon, Y., Vincent-Randonnier, A. & Sabelnikov, V. 2023 On the hydro-acoustic coupling responsible for the flashback limit-cycle of a premixed flame at a backward-facing step. Combust. Flame 257, 112999.10.1016/j.combustflame.2023.112999CrossRefGoogle Scholar
Britto, A.B. & Mariappan, S. 2019 Lock-in phenomenon of vortex shedding in oscillatory flows: an analytical investigation pertaining to combustors. J. Fluid Mech. 872, 115–146.10.1017/jfm.2019.353CrossRefGoogle Scholar
Britto, A.B. & Mariappan, S. 2021 Lock-in phenomenon of vortex shedding in flows excited with two commensurate frequencies: a theoretical investigation pertaining to combustion instability. J. Fluid Mech. 925, A9.10.1017/jfm.2021.624CrossRefGoogle Scholar
Britto, A.B. & Mariappan, S. 2023 Self-excited vortex-acoustic lock-in in a bluff body combustor. J. Fluid Mech. 975, A16.10.1017/jfm.2023.845CrossRefGoogle Scholar
Burkitt, A.N. 2006 A review of the integrate-and-fire neuron model: i. homogeneous synaptic input. Biol. Cybern. 95, 1–19.10.1007/s00422-006-0068-6CrossRefGoogle ScholarPubMed
Chakravarthy, S.R., Sivakumar, R. & Shreenivasan, O.J. 2007 Vortex-acoustic lock-on in bluff-body and backward-facing step combustors. Sadhana 32 (1), 145–154.10.1007/s12046-007-0013-yCrossRefGoogle Scholar
Clements, R.R. 1973 An inviscid model of two-dimensional vortex shedding. J. Fluid Mech. 57 (2), 321–336.10.1017/S0022112073001187CrossRefGoogle Scholar
Culick, F.E.C. 2006 Unsteady motions in combustion chambers for propulsion systems. Tech. Rep. AG-AVT-039,RTO AGARDorgraph.Google Scholar
Emerson, B. & Lieuwen, T. 2015 Dynamics of harmonically excited, reacting bluff body wakes near the global hydrodynamic stability boundary. J. Fluid Mech. 779, 716–750.10.1017/jfm.2015.450CrossRefGoogle Scholar
Fage, A. & Johansen, F.C. 1927 On the flow of air behind an inclined flat plate of infinite span. Proc. R. Soc. Lond. A 116 (773), 170–197.Google Scholar
Faure-Beaulieu, A., Indlekofer, T., Dawson, J.R. & Noiray, N. 2021 Experiments and low-order modelling of intermittent transitions between clockwise and anticlockwise spinning thermoacoustic modes in annular combustors. Proc. Combust. Inst. 38 (4), 5943–5951.10.1016/j.proci.2020.05.008CrossRefGoogle Scholar
Frisch, H.L. & Nowakowski, B. 1993 The thermalized Fokker–Planck equation. J. Chem. Phys. 98 (11), 8963–8969.10.1063/1.464455CrossRefGoogle Scholar
Gao, N., Zhang, Z., Deng, J., Guo, X., Cheng, B. & Hou, H. 2022 Acoustic metamaterials for noise reduction: a review. Adv. Materials Technol. 7 (6), 2100698.Google Scholar
Goldberg, D.H., Cauwenberghs, G. & Andreou, A.G. 2001 Probabilistic synaptic weighting in a reconfigurable network of VLSI integrate-and-fire neurons. Neural Networks 14 (6–7), 781–793.10.1016/S0893-6080(01)00057-0CrossRefGoogle Scholar
Goldstein, R.J., Lau, K.Y. & Leung, C.C. 1983 Velocity and turbulence measurements in combustion systems. Exp. Fluids 1 (2), 93–99.10.1007/BF00266261CrossRefGoogle Scholar
Gottwald, G.A. & Harlim, J. 2013 The role of additive and multiplicative noise in filtering complex dynamical systems.Proc. R. Soc. A: Math., Phys. Engng Sci. 2155, 20130096.10.1098/rspa.2013.0096CrossRefGoogle Scholar
Graham, R. & Schenzle, A. 1982 Stabilization by multiplicative noise. Phys. Rev. A 26 (3), 1676.Google Scholar
Griffin, O.M. 1985 Vortex shedding from bluff bodies in a shear flow-a review. ASME J. Fluids Engng 107, 298–306.10.1115/1.3242481CrossRefGoogle Scholar
Guk, S., Seo, S. & Lee, M. 2024 An image-based spatiotemporal approach for detecting coherence resonance in annular model gas-turbine combustor. Phys. Fluids 36 (5), 054121.10.1063/5.0208950CrossRefGoogle Scholar
Han, M., Han, X., Wang, X. & Li, L. 2022 Thermoacoustic instabilities with varying geometries of the main-stage exit in a centrally staged burner. Phys. Fluids 34 (12), 125112.Google Scholar
Han, X., Li, J. & Morgans, A.S. 2015 Prediction of combustion instability limit cycle oscillations by combining flame describing function simulations with a thermoacoustic network model. Combust. Flame 162 (10), 3632–3647.Google Scholar
Hill, P.G. & Peterson, C.R. 1992 Mechanics and Thermodynamics of Propulsion. 2ndedn. India, Dorling Kindersley India Pvt. Ltd.Google Scholar
Hosseinalipour, S.M., Fattahi, A., Khalili, H., Tootoonchian, F. & Karimi, N. 2020 Experimental investigation of entropy waves’ evolution for understanding of indirect combustion noise in gas turbine combustors. Energy 195, 116978.Google Scholar
Jegadeesan, V. & Sujith, R.I. 2013 Experimental investigation of noise induced triggering in thermoacoustic systems. Proc. Combust. Inst. 34 (2), 3175–3183.Google Scholar
Kabiraj, L., Steinert, R., Saurabh, A. & Paschereit, C.O. 2015 Coherence resonance in a thermoacoustic system. Phys. Rev. E 92 (4), 042909.Google Scholar
Kobayashi, T., Murayama, S., Hachijo, T. & Gotoda, H. 2019 Early detection of thermoacoustic combustion instability using a methodology combining complex networks and machine learning. Phys. Rev. Appl. 11 (6), 064034.10.1103/PhysRevApplied.11.064034CrossRefGoogle Scholar
Lee, M., Kim, K.T., Gupta, V. & Li, L.K.B. 2021 System identification and early warning detection of thermoacoustic oscillations in a turbulent combustor using its noise-induced dynamics. Proc. Combust. Inst. 38 (4), 6025–6033.10.1016/j.proci.2020.06.057CrossRefGoogle Scholar
Li, L.K.B. & Juniper, M.P. 2013 Lock-in and quasiperiodicity in hydrodynamically self-excited flames: experiments and modelling. Proc. Combust. Inst. 34 (1), 947–954.10.1016/j.proci.2012.06.022CrossRefGoogle Scholar
Li, X., Xu, B., Li, X., Pang, K., Li, X. & Zhang, H. 2023 Effects of multiplicative and additive colored noises on the stability of a simplified thermoacoustic combustor. Combust. Flame 249, 112413.10.1016/j.combustflame.2022.112413CrossRefGoogle Scholar
Li, X. & Zhao, D. 2017 Coherence resonance in a premixed combustor driven by a turbulence-induced color noise. In 24th International Congress On Sound and Vibration, ICSV 2017.Google Scholar
Lieuwen, T. 2012 Unsteady Combustor Physics, 1st edn. Cambridge University Press.Google Scholar
Lieuwen, T.C. 2002 Experimental investigation of limit-cycle oscillations in an unstable gas turbine combustor. J. Propul. Power 18 (1), 61–67.Google 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
Löcher, M., Cigna, D. & Hunt, E.R. 1998 Noise sustained propagation of a signal in coupled bistable electronic elements. Phys. Rev. Lett. 80 (23), 5212.10.1103/PhysRevLett.80.5212CrossRefGoogle Scholar
Lyu, Z., Fang, Y. & Wang, G. 2023 Precursor detection of thermoacoustic instability using statistical complexity and artificial neural network. Phys. Fluids 35 (6), 064101.Google Scholar
Majda, A.J., Timofeyev, I. & Vanden Eijnden, E. 1999 Models for stochastic climate prediction. Proc. Natl Acad. Sci. 96 (26), 14687–14691.10.1073/pnas.96.26.14687CrossRefGoogle ScholarPubMed
Mariappan, S. 2022 Stochastic dynamics of vortex-acoustic lock-in: stochastic bifurcation and resonance. J. Fluid Mech. 941, A38.10.1017/jfm.2022.303CrossRefGoogle Scholar
Mathur, T., Lin, K.-C., Kennedy, P., Gruber, M., Donbar, J., Jackson, T. & Billig, F. 2000 Liquid jp-7 combustion in a scramjet combustor. In 36th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, pp. 3581.Google Scholar
Matveev, K.I. & Culick, F.E.C. 2003 A model for combustion instability involving vortex shedding. Combust. Sci. Technol. 175 (6), 1059–1083.10.1080/00102200302349CrossRefGoogle Scholar
Mcmanus, K., Poinsot, T. & Candel, S. 1993 A review of active control of combustion instabilities. Prog. Energ. Combust. 19, 1–29.Google Scholar
Middleton, J.W., Chacron, M.J., Lindner, B. & Longtin, A. 2003 Firing statistics of a neuron model driven by long-range correlated noise. Phys. Rev. E 68 (2), 021920.Google ScholarPubMed
Nair, V. & Suijith, R.I. 2013 Identifying homoclinic orbits in the dynamics of intermittent signals through recurrence quantification. Chaos 756, 1–7.Google Scholar
Nair, V. & Sujith, R.I. 2015 A reduced-order model for the onset of combustion instability: physical mechanisms for intermittency and precursors. Proc. Combust. Inst. 35 (3), 3193–3200.10.1016/j.proci.2014.07.007CrossRefGoogle Scholar
Nair, V. & Sujith, R.I. 2014 Multifractality in combustion noise: predicting an impending combustion instability. J. Fluid Mech. 747 (x), 635–655.10.1017/jfm.2014.171CrossRefGoogle Scholar
Noiray, N. & Schuermans, B. 2013 Deterministic quantities characterizing noise driven Hopf bifurcations in gas turbine combustors. Intl J. Non-Linear Mech. 50, 152–163.10.1016/j.ijnonlinmec.2012.11.008CrossRefGoogle Scholar
Nozaki, D., Collins, J.J. & Yamamoto, Y. 1999 Mechanism of stochastic resonance enhancement in neuronal models driven by 1/f noise. Phys. Rev. E 60 (4), 4637.Google ScholarPubMed
Oefelein, J.C. & Yang, V. 1993 Comprehensive review of liquid-propellant combustion instabilities in F-1 engines. J. Propul. Power 9 (5), 657–677.Google Scholar
Palleschi, V. & De Rosa, M. 1992 Numerical solution of the Fokker–Planck equation. II. Multidimensional case. Phys. Lett. A 163 (5–6), 381–391.10.1016/0375-9601(92)90843-BCrossRefGoogle Scholar
Pawar, S.A., Seshadri, A., Unni, V.R. & Sujith, R.I. 2017 Thermoacoustic instability as mutual synchronization between the acoustic field of the confinement and turbulent reactive flow. J. Fluid Mech. 827, 664–693.10.1017/jfm.2017.438CrossRefGoogle Scholar
Plesser, H.E. & Geisel, T. 1999 Markov analysis of stochastic resonance in a periodically driven integrate-and-fire neuron. Phys. Rev. E 59 (6), 7008.10.1103/PhysRevE.59.7008CrossRefGoogle Scholar
Plesser, H.E. & Tanaka, S. 1997 Stochastic resonance in a model neuron with reset. Phys. Lett. A 225 (4–6), 228–234.Google Scholar
Poinsot, T. 2017 Prediction and control of combustion instabilities in real engines. Proc. Combust. Inst. 36 (1), 1–28.10.1016/j.proci.2016.05.007CrossRefGoogle Scholar
Poinsot, T.J., Trouve, A.C., Veynante, D.P., Candel, S.M. & Esposito, E.J. 1987 Vortex-driven acoustically coupled combustion instabilities. J. Fluid Mech. 177, 265–292.10.1017/S0022112087000958CrossRefGoogle Scholar
Priestley, M.B. 1981 Spectral analysis and time series: probability and mathematical statistics.Google Scholar
Rajaram, R. & Lieuwen, T. 2009 Acoustic radiation from turbulent premixed flames. J. Fluid Mech. 637, 357–385.10.1017/S0022112009990681CrossRefGoogle Scholar
Ramlow, L., Falcke, M. & Lindner, B. 2023 An integrate-and-fire approach to Ca2+ signaling. Part I: Renewal model. Biophys. J. 122 (4), 713–736.Google ScholarPubMed
Rayleigh, L. 1878 The explanation of certain acoustical phenomenon. Nature 18, 319–321.10.1038/018319a0CrossRefGoogle Scholar
Reuther, N. & Kähler, C.J. 2020 Effect of the intermittency dynamics on single and multipoint statistics of turbulent boundary layers. J. Fluid Mech. 897, A11.Google Scholar
Schadow, K. & Gutmark, E. 1992 Combustion instability related to vortex shedding in dump combustors and their passive control. Prog. Energ. Combust. 18 (2), 117–132.10.1016/0360-1285(92)90020-2CrossRefGoogle Scholar
Sengupta, U., Rasmussen, C.E. & Juniper, M.P. 2021 Bayesian machine learning for the prognosis of combustion instabilities from noise. J. Engng Gas Turbines Power 143 (7), 071001.Google Scholar
Shivashankara, B., Strahle, W. & Handley, J. 1973 Combustion noise radiation by open turbulent flames. In Aeroacoustics Conference, pp. 1025.Google Scholar
Singh, G. & Mariappan, S. 2021 Experimental investigation on the route to vortex-acoustic lock-in phenomenon in bluff body stabilized combustors. Combust. Sci. Technol. 193 (9), 1538–1566.10.1080/00102202.2019.1700961CrossRefGoogle Scholar
Tateno, T. & Jimbo, Y. 2000 Stochastic mode-locking for a noisy integrate-and-fire oscillator. Phys. Lett. A 271 (4), 227–236.Google Scholar
Vishnoi, N., Gupta, V., Saurabh, A. & Kabiraj, L. 2022 System parameter identification of a colored-noise-driven rijke tube simulator. J. Engng Gas Turbines Power 144 (9), 091017.10.1115/1.4055212CrossRefGoogle Scholar
Vishnoi, N. & Kabiraj, L. 2024 Effect of background noise characteristics on early warning indicators of thermoacoustic instability. Combust. Flame 269, 113687.Google Scholar
Waugh, I.C. & Juniper, M. 2011 a Triggering in a thermoacoustic system with stochastic noise. Intl J. Spray Combust. 3, 224–242.Google Scholar
Waugh, I.C. & Juniper, M.P. 2011 b Triggering in a thermoacoustic system with stochastic noise. Intl J. Spray Combust. 3 (3), 225–241.Google Scholar
Wilczek, M., Daitche, A. & Friedrich, R. 2011 On the velocity distribution in homogeneous isotropic turbulence: correlations and deviations from gaussianity. J. Fluid Mech. 676, 191–217.Google Scholar
Wilczek, M. & Friedrich, R. 2009 Dynamical origins for non-Gaussian vorticity distributions in turbulent flows. Phys. Rev. E 80 (1), 016316.10.1103/PhysRevE.80.016316CrossRefGoogle ScholarPubMed
Zaikin, A.A. & Schimansky-Geier, L. 1998 Spatial patterns induced by additive noise. Phys. Rev. E 58 (4), 4355.Google Scholar
Zare, A., Jovanović, M.R. & Georgiou, T.T. 2017 Colour of turbulence. J. Fluid Mech. 812, 636–680.Google Scholar
Zukoski, E.E. & Oates, G.C. 1985 Afterburners. aerothermodynamics of aircraft engine components .Google Scholar