Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-13T12:23:27.195Z Has data issue: false hasContentIssue false
  • English
  • Français

Lock-in phenomenon of vortex shedding in flows excited with two commensurate frequencies: a theoretical investigation pertaining to combustion instability

Published online by Cambridge University Press:  23 August 2021

Abraham Benjamin Britto  and
Sathesh Mariappan Open the ORCID record for Sathesh Mariappan [Opens in a new window]
Abraham Benjamin Britto
Affiliation:
Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur208016, India
Sathesh Mariappan*
Affiliation:
Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur208016, India
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

An analytical investigation is performed to study the dynamics of vortex shedding behaviour during two commensurate frequency velocity excitations, with an emphasis on the phenomenon of lock-in. We attempt to theoretically study the dynamical features of lock-in under two-frequency excitations and contrast the behaviour with single-frequency excitation. We employ an existing low-order model to characterise the vortex shedding process behind a step/bluff-body. The continuous-time domain model is transformed to a nonlinear dynamical map that relates time instances of successive vortex shedding. Further, these time instances are converted to phase instances, involving which criteria for a generic p : q phase lock-in is obtained. Four parameters are involved: amplitude and frequency of the two excitation components termed as primary and secondary. Bifurcations occurring are investigated using return maps. The inclusion of secondary excitation leads to the existence of two orders of lock-in within a single lock-in boundary. Furthermore, our results indicate that secondary excitation can be used as a control in order to tailor the 1:1 lock-in region formed by the primary excitation. Finally, analytical expressions are obtained to identify lock-in boundaries and their salient geometrical features. Interesting features such as the occurrence of bistability and change in the order of lock-in are observed, which can be explored further with future experiments.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Nonlinear Dynamical Systems: Low-dimensional models Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 925 , 25 October 2021 , A9
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.)

References

REFERENCES

Acharya, V.S., Bothien, M.R. & Lieuwen, T.C. 2018 Non-linear dynamics of thermoacoustic eigen-mode interactions. Combust. Flame 194, 309–321.CrossRefGoogle Scholar
Anderson, W.E. & Yang, V. 1995 Liquid Rocket Engine Combustion Instability. American Institute of Aeronautics and Astronautics.CrossRefGoogle Scholar
Anishchenko, V.S., Safonova, M.A. & Chua, L.O. 1993 Confirmation of the afraimovich-shilnikov torus-breakdown theorem via a torus circuit. IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 40 (11), 792–800.CrossRefGoogle Scholar
Balachandran, R., Dowling, A.P. & Mastorakos, E. 2008 Non-linear response of turbulent premixed flames to imposed inlet velocity oscillations of two frequencies. Flow Turbul. Combust. 80 (4), 455.CrossRefGoogle 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
Britto, A.B. & Mariappan, S. 2019 a Lock-in phenomenon of vortex shedding in oscillatory flows: an analytical investigation pertaining to combustors. J. Fluid Mech. 872, 115–146.CrossRefGoogle Scholar
Britto, A.B. & Mariappan, S. 2019 b Understanding the route to lock-in phenomenon in vortex shedding combustors. In 26th International Congress on Sound and Vibration, vol. 294. IIAV.Google Scholar
Candel, S. 2002 Combustion dynamics and control: progress and challenges. Proc. Combust. Inst. 29 (1), 1–28.CrossRefGoogle Scholar
Candel, S.M. 1992 Combustion instabilities coupled by pressure waves and their active control. In Symposium (International) on Combustion, vol. 24, pp. 1277–1296. Elsevier.CrossRefGoogle Scholar
Chakravarthy, S.R., Shreenivasan, O.J., Boehm, B., Dreizler, A. & Janicka, J. 2007 Experimental characterization of onset of acoustic instability in a nonpremixed half-dump combustor. Acoust. Soc. Am. J. 122 (1), 120.CrossRefGoogle Scholar
Crocco, L. 1969 Research on combustion instability in liquid propellant rockets. In Symposium (International) on Combustion, vol. 12, pp. 85–99. Elsevier.CrossRefGoogle Scholar
Culick, F.E.C., Burnley, V. & Swenson, G. 1995 Pulsed instabilities in solid-propellant rockets. J. Propul. Power 11, 657–665.CrossRefGoogle Scholar
Daan, S., Beersma, D.G. & Borbély, A.A. 1984 Timing of human sleep: recovery process gated by a circadian pacemaker. Am. J. Physiol. 246 (2), R161–R183.Google ScholarPubMed
Dowling, A.P. & Morgans, A.S. 2005 Feedback control of combustion oscillations. Annu. Rev. Fluid Mech. 37, 151–182.CrossRefGoogle 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.CrossRefGoogle Scholar
Emerson, B., Murphy, K. & Lieuwen, T. 2013 Flame density ratio effects on vortex dynamics of harmonically excited bluff body stabilized flames. In Turbo Expo: Power for Land, Sea, and Air, vol. 55102, p. V01AT04A014. American Society of Mechanical Engineers.CrossRefGoogle Scholar
Emerson, B., O'Connor, J., Noble, D. & Lieuwen, T. 2012 Frequency locking and vortex dynamics of an acoustically excited bluff body stabilized flame. In 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, p. 451. AIAA.CrossRefGoogle Scholar
Franceschini, V. 1983 Bifurcations of tori and phase locking in a dissipative system of differential equations. Physica D 6 (3), 285–304.CrossRefGoogle Scholar
Gardini, L., Lupini, R., Mammana, C. & Messia, M.G. 1987 Bifurcations and transitions to chaos in the three-dimensional Lotka–Volterra map. SIAM J. Appl. Maths 47 (3), 455–482.CrossRefGoogle Scholar
Gollub, J.P. & Benson, S.V. 1980 Many routes to turbulent convection. J. Fluid Mech. 100 (3), 449–470.CrossRefGoogle Scholar
Graves, C., Glass, L., Laporta, D., Meloche, R. & Grassino, A. 1986 Respiratory phase locking during mechanical ventilation in anesthetized human subjects. Am. J. Physiol. 250 (5), R902–R909.Google ScholarPubMed
Guan, Y., Gupta, V., Wan, M. & Li, L.K.B. 2019 a Forced synchronization of quasiperiodic oscillations in a thermoacoustic system. J. Fluid Mech. 879, 390–421.CrossRefGoogle Scholar
Guan, Y., He, W., Murugesan, M., Li, Q., Liu, P. & Li, L.K.B. 2019 b Control of self-excited thermoacoustic oscillations using transient forcing, hysteresis and mode switching. Combust. Flame 202, 262–275.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Gyllenberg, M., Jiang, J. & Niu, L. 2020 Chaotic attractors in Atkinson–Allen model of four competing species. J. Biol. Dyn. 14 (1), 440–453.CrossRefGoogle ScholarPubMed
Haeringer, M., Merk, M. & Polifke, W. 2019 Inclusion of higher harmonics in the flame describing function for predicting limit cycles of self-excited combustion instabilities. Proc. Combust. Inst. 37 (4), 5255–5262.CrossRefGoogle Scholar
Hertzberg, J.R., Shepherd, I.G. & Talbot, L. 1991 Vortex shedding behind rod stabilized flames. Combust. Flame 86 (1–2), 1–11.CrossRefGoogle Scholar
Huang, Y., Sung, H.-G., Hsieh, S.-Y. & Yang, V. 2003 Large-eddy simulation of combustion dynamics of lean-premixed swirl-stabilized combustor. J. Propul. Power 19 (5), 782–794.CrossRefGoogle Scholar
Huang, Y. & Yang, V. 2009 Dynamics and stability of lean-premixed swirl-stabilized combustion. Prog. Energy Combust. Sci. 35 (4), 293–364.CrossRefGoogle Scholar
Humbert, S.C., Gensini, F., Andreini, A., Paschereit, C.O. & Orchini, A. 2021 Nonlinear analysis of self-sustained oscillations in an annular combustor model with electroacoustic feedback. Proc. Combust. Inst. 38 (4), 6085–6093.CrossRefGoogle Scholar
Joos, F. & Vortmeyer, D. 1986 Self-excited oscillations in combustion chambers with premixed flames and several frequencies. Combust. Flame 65 (3), 253–262.CrossRefGoogle Scholar
Kabiraj, L., Saurabh, A., Wahi, P. & Sujith, R.I. 2012 a 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. 2012 b Bifurcations of self-excited ducted laminar premixed flames. Trans. ASME J. Engng Gas Turbines Power 134 (3), 031502.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
Keller, J.J. 1995 Thermoacoustic oscillations in combustion chambers of gas turbines. AIAA J. 33 (12), 2280–2287.CrossRefGoogle Scholar
Kim, K.T. 2017 Nonlinear interactions between the fundamental and higher harmonics of self-excited combustion instabilities. Combust. Sci. Technol. 189 (7), 1091–1106.CrossRefGoogle Scholar
Li, L.K.B. & Juniper, M.P. 2013 a 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. 2013 b 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. 2013 c Phase trapping and slipping in a forced hydrodynamically self-excited jet. J. Fluid Mech. 735, R5.CrossRefGoogle Scholar
Lieuwen, T. 2003 Combustion driven oscillations in gas turbines. Turbomach. Intl 44 (1), 16–18.Google Scholar
Lieuwen, T.C. 2012 Unsteady Combustor Physics. Cambridge University Press.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
Matveev, K.I. & Culick, F.E.C. 2003 A model for combustion instability involving vortex shedding. Combust. Sci. Technol. 175 (6), 1059–1083.CrossRefGoogle Scholar
Moeck, J.P. & Paschereit, C.O. 2012 Nonlinear interactions of multiple linearly unstable thermoacoustic modes. Intl J. Spray Combust. Dyn. 4 (1), 1–27.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, 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.CrossRefGoogle Scholar
Orchini, A. & Juniper, M.P. 2016 Flame double input describing function analysis. Combust. Flame 171, 87–102.CrossRefGoogle Scholar
Pastrone, D., Casalino, L. & Carmicino, C. 2014 Analysis of acoustics and vortex shedding interactions in hybrid rocket motors. J. Propul. Power 30 (6), 1613–1619.CrossRefGoogle 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.CrossRefGoogle Scholar
Pikovsky, A. & Rosenblum, M. 2007 Synchronization. Scholarpedia 2 (12), 1459.CrossRefGoogle 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.CrossRefGoogle Scholar
Rayleigh, , 1878 The explanation of certain acoustical phenomena. Nature 18, 319–321.CrossRefGoogle Scholar
Schadow, K.C. & Gutmark, E. 1992 Combustion instability related to vortex shedding in dump combustors and their passive control. Prog. Energy Combust. Sci. 18 (2), 117–132.CrossRefGoogle Scholar
Singaravelu, B. & Mariappan, S. 2016 Stability analysis of thermoacoustic interactions in vortex shedding combustors using Poincaré map. J. Fluid Mech. 801, 597–622.CrossRefGoogle 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.CrossRefGoogle Scholar
Strogatz, S.H., Abrams, D.M., McRobie, A., Eckhardt, B. & Ott, E. 2005 Crowd synchrony on the millennium bridge. Nature 438 (7064), 43–44.CrossRefGoogle ScholarPubMed
Van Veen, L. 2005 The quasi-periodic doubling cascade in the transition to weak turbulence. Physica D 210 (3–4), 249–261.CrossRefGoogle Scholar
Zukoski, E. 1985 Combustion instability sustained by unsteady vortex combustion. In 21st Joint Propulsion Conference, p. 1248. AIAA.CrossRefGoogle Scholar

Britto and Mariappan supplementary movie 1

See pdf file for movie caption

Download Britto and Mariappan supplementary movie 1(Video)
Video 467.1 KB

Britto and Mariappan supplementary movie 2

See pdf file for movie caption

Download Britto and Mariappan supplementary movie 2(Video)
Video 509.2 KB
Supplementary material: PDF

Britto and Mariappan supplementary material

Captions for movies 1-2

Download Britto and Mariappan supplementary material(PDF)
PDF 16.1 KB