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Lock-in and quasiperiodicity in a forced hydrodynamically self-excited jet

Published online by Cambridge University Press:  11 June 2013

Larry K. B. Li  and
Matthew P. Juniper
Larry K. B. Li*
Affiliation:
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
Matthew P. Juniper
Affiliation:
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
*
†Email address for correspondence: l.li@gatescambridge.org
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Abstract

The ability of hydrodynamically self-excited jets to lock into strong external forcing is well known. Their dynamics before lock-in and the specific bifurcations through which they lock in, however, are less well known. In this experimental study, we acoustically force a low-density jet around its natural global frequency. We examine its response leading up to lock-in and compare this to that of a forced van der Pol oscillator. We find that, when forced at increasing amplitudes, the jet undergoes a sequence of two nonlinear transitions: (i) from periodicity to ${ \mathbb{T} }^{2} $ quasiperiodicity via a torus-birth bifurcation; and then (ii) from ${ \mathbb{T} }^{2} $ quasiperiodicity to 1:1 lock-in via either a saddle-node bifurcation with frequency pulling, if the forcing and natural frequencies are close together, or a torus-death bifurcation without frequency pulling, but with a gradual suppression of the natural mode, if the two frequencies are far apart. We also find that the jet locks in most readily when forced close to its natural frequency, but that the details contain two asymmetries: the jet (i) locks in more readily and (ii) oscillates more strongly when it is forced below its natural frequency than when it is forced above it. Except for the second asymmetry, all of these transitions, bifurcations and dynamics are accurately reproduced by the forced van der Pol oscillator. This shows that this complex (infinite-dimensional) forced self-excited jet can be modelled reasonably well as a simple (three-dimensional) forced self-excited oscillator. This result adds to the growing evidence that open self-excited flows behave essentially like low-dimensional nonlinear dynamical systems. It also strengthens the universality of such flows, raising the possibility that more of them, including some industrially relevant flames, can be similarly modelled.

JFM classification

Instability: Absolute/convective instability Flow Control: Flow Control Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
Papers
Information
Journal of Fluid Mechanics , Volume 726 , 10 July 2013 , pp. 624 - 655
Copyright
©2013 Cambridge University Press 

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References

Abarbanel, H. D. I. 1996 Analysis of Observed Chaotic Data. Springer.Google Scholar
Albarède, P. & Monkewitz, P. A. 1992 A model for the formation of oblique shedding and ‘chevron’ patterns in cylinder wakes. Phys. Fluids A 4 (4), 744–756.Google Scholar
Anderson, K. R., Hertzberg, J. & Mahalingam, S. 1996 Classification of absolute and convective instabilities in premixed bluff body stabilized flames. Combust. Sci. Technol. 112 (1), 257–269.Google Scholar
Baek, S. J. & Sung, H. J. 2000 Quasi-periodicity in the wake of a rotationally oscillating cylinder. J. Fluid Mech. 408, 275–300.Google Scholar
Balachandran, R., Ayoola, B. O., Kaminski, C. F., Dowling, A. P. & Mastorakos, E. 2005 Experimental investigation of the nonlinear response of turbulent premixed flames to imposed inlet velocity oscillations. Combust. Flame 143 (1–2), 37–55.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
Barbi, C., Favier, D. P., Maresca, C. A. & Telionis, D. P. 1986 Vortex shedding and lock-on of a circular cylinder in oscillatory flow. J. Fluid Mech. 170, 527–544.Google 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.Google Scholar
Blevins, R. D. 1985 The effect of sound on vortex shedding from cylinders. J. Fluid Mech. 161, 217–237.Google Scholar
Broze, G. & Hussain, F. 1994 Nonlinear dynamics of forced transitional jets: periodic and chaotic attractors. J. Fluid Mech. 263, 93–132.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. J. Acoust. Soc. Am. 122 (1), 120–127.Google Scholar
Chomaz, J. M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.Google Scholar
Craik, A. D. D. 1988 Wave Interactions and Fluid Flows. Cambridge University Press.Google 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.Google Scholar
Dewan, E. M. 1972 Harmonic entrainment of van der Pol oscillations: phaselocking and asynchronous quenching. IEEE Trans. Automat. Control 17 (5), 655–663.Google Scholar
Facchinetti, M. L., de Langre, E. & Biolley, F. 2002 Vortex shedding modelling using diffusive van der Pol oscillators. C. R. Mec. 330, 451–456.Google Scholar
Facchinetti, M. L., de Langre, E. & Biolley, F. 2004 Coupling of structure and wake oscillators in vortex-induced vibrations. J. Fluid Struct. 19 (2), 123–140.CrossRefGoogle Scholar
Fauve, S. 1998 Pattern forming instabilities. In Hydrodynamics and Nonlinear Instabilities (ed. Godr̀eche, C. & Manneville, P.). chap. 4, Cambridge University Press.Google Scholar
Fraser, A. M. & Swinney, H. L. 1986 Independent coordinates for strange attractors from mutual information. Phys. Rev. A 33 (2), 1134–1140.Google Scholar
Gaster, M. 1969 Vortex shedding from slender cones at low Reynolds numbers. J. Fluid Mech. 38 (3), 565–576.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.Google Scholar
Gotoda, H. & Ueda, T. 2002 Transition from periodic to non-periodic motion of a Bunsen-type premixed flame tip with burner rotation. Proc. Combust. Inst. 29 (1), 1503–1509.Google Scholar
Grassberger, P. & Procaccia, I. 1983 Characterization of strange attractors. Phys. Rev. Lett. 50 (5), 346–349.Google Scholar
Guicking, D. & Haars, K. 1991 On the natural frequency of the van der Pol oscillator. Acustica 73 (3), 158–161.Google Scholar
Hallberg, M. P., Srinivasan, V., Gorse, P. & Strykowski, P. J. 2007 Suppression of global modes in low-density axisymmetric jets using coflow. Phys. Fluids 19 (1) 014102.Google Scholar
Hallberg, M. P. & Strykowski, P. J. 2006 On the universality of global modes in low-density axisymmetric jets. J. Fluid Mech. 569, 493–507.Google Scholar
Hallberg, M. P. & Strykowski, P. J. 2008 Open-loop control of fully nonlinear self-excited oscillations. Phys. Fluids 20 (4) 041703.Google Scholar
Hartlen, R. T. & Currie, I. G. 1970 Lift-oscillator model of vortex-induced vibration. J. Engng Mech. ASCE 96 (5), 577–591.Google Scholar
Henry, B., Lovell, N. & Camacho, F. 2000 Nonlinear dynamics time series analysis. In Nonlinear Biomedical Signal Processing: Dynamic Analysis and Modelling (ed. Akay, M.). chap. 1, Wiley–IEEE.Google Scholar
Hilborn, R. C. 2000 Chaos and Nonlinear Dynamics, 2nd edn. Oxford University Press.Google Scholar
Ho, C. M. & Huerre, P. 1984 Perturbed free shear layers. Annu. Rev. Fluid Mech. 16, 365–424.CrossRefGoogle Scholar
Holmes, P. J. & Rand, D. A. 1978 Bifurcations of the forced van der Pol oscillator. Q. Appl. Maths 35, 495–509.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.Google Scholar
Iwan, W. D. & Blevins, R. D. 1974 A model for vortex induced oscillation of structures. J. Appl. Mech. 41 (3), 581–586.Google Scholar
Jendoubi, S. & Strykowski, P. J. 1994 Absolute and convective instability of axisymmetric jets with external flow. Phys. Fluids 6 (9), 3000–3009.CrossRefGoogle Scholar
Johnson, A., Uddin, M. & Pollard, A. 2005 Calibration of hot-wire probes using non-uniform mean velocity profiles. Exp. Fluids 39 (3), 525–532.Google Scholar
Juniper, M. P., Li, L. K. B. & Nichols, J. W. 2009 Forcing of self-excited round jet diffusion flames. Proc. Combust. Inst. 32 (1), 1191–1198.Google Scholar
Kantz, H. & Schreiber, T. 2003 Nonlinear Time Series Analysis, 2nd edn. Cambridge University Press.Google Scholar
Karniadakis, G. E. & Triantafyllou, G. S. 1989 Frequency selection and asymptotic states in laminar wakes. J. Fluid Mech. 199, 441–469.CrossRefGoogle Scholar
Kennel, M. B., Brown, R. & Abarbanel, H. D. I. 1992 Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys. Rev. A 45 (6), 3403–3411.CrossRefGoogle ScholarPubMed
Koepke, M. E. & Hartley, D. M. 1991 Experimental verification of periodic pulling in a nonlinear electronic oscillator. Phys. Rev. A 44 (10), 6877–6887.Google Scholar
Koopmann, G. H. 1967 The vortex wakes of vibrating cylinders at low Reynolds numbers. J. Fluid Mech. 28 (3), 501–512.CrossRefGoogle Scholar
Kuznetsov, Y. 2004 Elements of Applied Bifurcation Theory, 3rd edn. Springer.Google Scholar
Kyle, D. M. & Sreenivasan, K. R. 1993 The instability and breakdown of a round variable-density jet. J. Fluid Mech. 249, 619–664.Google Scholar
Lesshafft, L., Huerre, P., Sagaut, P. & Terracol, M. 2006 Nonlinear global modes in hot jets. J. Fluid Mech. 554, 393–409.Google Scholar
Lesshafft, L. & Huerre, P. 2007 Linear impulse response in hot round jets. Phys. Fluids 19 (2), 024102.Google Scholar
Li, L. K. B. 2011 Forcing of globally unstable jets and flames. PhD thesis, University of Cambridge, Department of Engineering.Google 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.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.CrossRefGoogle Scholar
Manneville, P. 2010 Instabilities, Chaos and Turbulence, 2nd edn. Imperial College Press.Google Scholar
Mehta, R. D. & Bradshaw, P. 1979 Design rules for small low speed wind tunnels. Aeronaut. J. 83 (827), 443–449.Google Scholar
Monkewitz, P. A., Bechert, D. W., Barsikow, B. & Lehmann, B. 1990 Self-excited oscillations and mixing in a heated round jet. J. Fluid Mech. 213, 611–639.Google Scholar
Monkewitz, P. A., Huerre, P. & Chomaz, J. M. 1993 Global linear stability analysis of weakly non-parallel shear flows. J. Fluid Mech. 251, 1–20.Google Scholar
Mullin, T. & Price, T. J. 1989 An experimental observation of chaos arising from the interaction of steady and time-dependent flows. Nature 340, 294–296.Google Scholar
Nayfeh, A. H. & Balachandran, B. 2004 Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. Wiley–VCH.Google Scholar
Nichols, J. W., Chomaz, J. M. & Schmid, P. J. 2007 Secondary instability in variable-density round jets. In Advances in Turbulence XI: Proceedings of the 11th EUROMECH European Turbulence Conference (ed. Palma, J. M. L. M. & Silva Lopes, A.). Springer Series in Physics, vol. 117. pp. 32–34. Springer.Google Scholar
Noack, B. R., Ohle, F. & Eckelmann, H. 1991 On cell formation in vortex streets. J. Fluid Mech. 227, 293–308.Google Scholar
Olinger, D. J. 1992 Lock-in states in the dripping mode of the capillary jet. Exp. Fluids 15 (2), 155–158.Google Scholar
Olinger, D. J. 1993 A low-dimensional model for chaos in open fluid flows. Phys. Fluids 5 (8), 1947–1951.Google Scholar
van der Pol, B. 1926 On relaxation oscillations. Phil. Mag. 2, 978–992.Google Scholar
van der Pol, B. & van der Mark, J. 1927 Frequency demultiplication. Nature 120, 363–364.Google Scholar
Provansal, M., Mathis, C. & Boyer, L. 1987 Bénard–von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 1–22.Google Scholar
Rogers, D. E. & Marble, F. E. 1956 A mechanism for high-frequency oscillations in ramjet combustors and afterburners. Jet Propul. 26 (6), 456–462.Google Scholar
Schadow, K. C. & Gutmark, E. 1992 Combustion instability related to vortex shedding in dump combustors and their passive control. Prog. Energ. Combust. 18 (2), 117–132.Google Scholar
Settles, G. S. 2001 Schlieren and Shadowgraph Techniques: Visualizing Phenomena in Transparent Media. Springer.Google Scholar
Shampine, L. F. & Reichelt, M. W. 1997 The MATLAB ODE suite. SIAM J. Sci. Comput. 18 (1), 1–22.Google Scholar
Small, M. 2005 Applied Nonlinear Time Series Analysis: Applications in Physics, Physiology and Finance. World Scientific Publishing Company.Google Scholar
Sreenivasan, K. R., Raghu, S. & Kyle, D. 1989 Absolute instability in variable density round jets. Exp. Fluids 7 (5), 309–317.Google Scholar
Sreenivasan, K. R., Strykowski, P. J. & Olinger, D. J. 1987 Hopf bifurcation, Landau equation, and vortex shedding behind circular cylinders. In Forum on Unsteady Flow Separation, vol. 52. pp. 1–13. ASME.Google Scholar
Srinivasan, V., Hallberg, M. P. & Strykowski, P. J. 2010 Viscous linear stability of axisymmetric low-density jets: parameters influencing absolute instability. Phys. Fluids 22 (2), 024103.CrossRefGoogle Scholar
Stansby, P. K. 1976 The locking-on of vortex shedding due to the cross-stream vibration of circular cylinders in uniform and shear flows. J. Fluid Mech. 74 (4), 641–665.Google Scholar
Strogatz, S. H. 1994 Nonlinear Dynamics and Chaos. Perseus Books.Google Scholar
Student, 1908 The probable error of a mean. Biometrika 6 (1), 1–25.Google Scholar
Subbarao, E. R. & Cantwell, B. J. 1992 Investigation of a co-flowing buoyant jet: experiments on the effect of Reynolds number and Richardson number. J. Fluid Mech. 245, 69–90.Google Scholar
Takens, F. 1981 Detecting strange attractors in turbulence. In Dynamical Systems and Turbulence (ed. Rand, D.A. & Young, L.S.). Lecture Notes in Mathematics, vol. 898. pp. 366–381. Springer.Google Scholar
Theiler, J. 1986 Spurious dimension from correlation algorithms applied to limited time-series data. Phys. Rev. A – Gen. Phys. 34 (3), 2427–2432.Google Scholar
Van Atta, C. W. & Gharib, M. 1987 Ordered and chaotic vortex streets behind circular cylinders at low Reynolds numbers. J. Fluid Mech. 174, 113–133.Google Scholar
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, 70–73.Google Scholar
Zinn, B. T. & Lieuwen, T. C. 2005 Combustion instabilities: basic concepts. In Combustion Instabilities in Gas Turbine Engines: Operational Experience, Fundamental Mechanisms, and Modelling (ed. Lieuwen, T.C. & Yang, V.). chap. 1, American Institute of Aeronautics and Astronautics.Google Scholar