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Subcritical bifurcation and bistability in thermoacoustic systems

  • Priya Subramanian (a1), R. I. Sujith (a1) and P. Wahi (a2)


This paper analyses subcritical transition to instability, also known as triggering in thermoacoustic systems, with an example of a Rijke tube model with an explicit time delay. Linear stability analysis of the thermoacoustic system is performed to identify parameter values at the onset of linear instability via a Hopf bifurcation. We then use the method of multiple scales to recast the model of a general thermoacoustic system near the Hopf point into the Stuart–Landau equation. From the Stuart–Landau equation, the relation between the nonlinearity in the model and the criticality of the ensuing bifurcation is derived. The specific example of a model for a horizontal Rijke tube is shown to lose stability through a subcritical Hopf bifurcation as a consequence of the nonlinearity in the model for the unsteady heat release rate. Analytical estimates are obtained for the triggering amplitudes close to the critical values of the bifurcation parameter corresponding to loss of linear stability. The unstable limit cycles born from the subcritical Hopf bifurcation undergo a fold bifurcation to become stable and create a region of bistability or hysteresis. Estimates are obtained for the region of bistability by locating the fold points from a fully nonlinear analysis using the method of harmonic balance. These analytical estimates help to identify parameter regions where triggering is possible. Results obtained from analytical methods compare reasonably well with results obtained from both experiments and numerical continuation.


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Allgower, E. L. & Georg, K. 1990 Computational Solution of Nonlinear Systems of Equations, Lectures in Applied Mathematics Series, vol. 26. American Mathematical Society.
Ananthakrishnan, N., Deo, S. & Culick, F. E. C. 2005 Reduced-order modelling and dynamics of nonlinear acoustic waves in a combustion chamber. Combust. Sci. Technol. 177 (28), 221248.
Annaswamy, A. M., Fleifil, M., Hathout, J. P. & Ghoneim, A. F. 1997 Impact of linear coupling on the design of active controllers for the thermoacoustic instability. Combust. Sci. Technol. 128, 131180.
Balasubramanian, K. & Sujith, R. I. 2008 Thermoacoustic instability in a Rijke tube: non-normality and nonlinearity. Phys. Fluids 20, 044103.
Bloomshield, F. S., Crump, J. E., Mathes, H. B., Stalnaker, R. A. & Beckstead, M. W. 1997 Nonlinear stability testing of full scale tactical motors. J. Propul. Power 13 (3), 356366.
Chandrasekhar, S. 1953 The instability of a layer of fluid heated below and subject to Coriolis forces. Proc. R. Soc. Lond. A 217, 306327.
Cooke, K. L. & Grossman, Z. 1982 Discrete delay, distributed delay and stability switches. J. Math. Anal. Appl. 86, 592627.
Crocco, L. 1969 Research on combustion instability in liquid propellant rockets. Symp. (Intl) Combust. 12 (1), 8599.
Crocco, L. & Cheng, S. 1956 Theory of Combustion Instability in Liquid Propellant Rocket Motors. Butterworths Scientific Publications.
Cross, M. & Greenside, H. 2009 Pattern Formation and Dynamics in Nonequilibrium Systems. Cambridge University Press.
Culick, F. E. C. 1963 Stability of high frequency pressure oscillations in rocket combustion chambers. AIAA J. 1, 10971104.
Culick, F. E. C. 1976a Nonlinear behaviour of acoustic waves in combustion chambers. Part I. Acta Astron. 3, 715734.
Culick, F. E. C. 1976b Nonlinear behaviour of acoustic waves in combustion chambers. Part II. Acta Astron. 3, 735757.
Das, S. L. & Chatterjee, A. 2002 Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations. Nonlinear Dyn. 30, 323335.
Dessi, D., Mastroddi, F. & Morino, L. 2004 A fifth order multiple scale solution for Hopf bifurcations. Comput. Struct. 82, 27232731.
Dowling, A. P. 1997 Nonlinear self-excited oscillations of a ducted flame. J. Fluid Mech. 346, 271290.
Dowling, A. P. 1999 A kinematic model of a ducted flame. J. Fluid Mech. 394, 5172.
Engelborghs, K., Luzyanina, T. & Roose, D. 2002 Numerical bifurcation analysis of delay differential equations using dde-biftool. ACM Trans. Math. Softw. 28 (1), 121.
Govindarajan, R. & Narasimha, R. 1995 Stability of spatially developing boundary layers in pressure gradients. J. Fluid Mech. 300, 117147.
Haken, H. 1983 Synergetics: Introduction & Advanced Topics. Springer.
Heckl, M. A. 1990 Nonlinear acoustic effects in the Rijke tube. Acustica 72, 6371.
Hillborn, R. C. 1994 Chaos and Nonlinear Dynamics. Oxford University Press.
Jahnke, C. C. & Culick, F. E. C. 1994 Application of dynamical systems theory to nonlinear combustion instabilities. J. Propul. Power 10, 508517.
Juniper, M. P. 2011 Triggering in the horizontal Rijke tube: non-normality, transient growth and bypass transition. J. Fluid Mech. 667, 272308.
King, L. V. 1914 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. Proc. R. Soc. Lond. A 90, 271289.
Kuang, Y. 1993 Delay Differential Equations with Applications in Population Dynamics. Academic.
Kuramoto, Y. 2003 Chemical Oscillations, Waves and Turbulence. Courier Dover.
Landau, L. D. 1944 On the problem of turbulence. Dokl. Acad. 44, 339342.
Lei, S. & Turan, A. 2009 Nonlinear/chaotic behaviour in thermo-acoustic instability. Combust. Theor. Model. 13 (3), 541557.
Lieuwen, T. 2002 Experimental investigation of limit-cycle oscillations in an unstable gas turbine combustor. J. Propul. Power 18, 6167.
Lighthill, M. J. 1954 The response of laminar skin friction and heat transfer to fluctuations in the stream velocity. Proc. R. Soc. Lond. A 224, 123.
Mariappan, S., Sujith, R. I. & Schmid, P. T. 2011 Non-normality of thermoacoustic interactions: an experimental investigation. In Proceedings of 47th AIAA/ASME/SAE/ASEE Joint Propulsion Conference.
Matveev, K. I. 2003a A model for combustion instability involving vortex shedding. Combust. Sci. Technol. 175 (6), 10591083.
Matveev, K. I. 2003b Thermo-acoustic instabilities in the Rijke tube: experiments and modelling. PhD thesis, California Institute of Technology, Pasadena.
Michiles, W. & Niculescu, S. I. 2007 Stability and Stabilization of Time-delay Systems: An Eigenvalue-based Approach. SIAM.
Nayfeh, A. H. 1971 Third-harmonic resonance in the interaction of capillary and gravity waves. J. Fluid Mech. 48, 385395.
Nayfeh, A. H. & Balachandran, B. 1990 Motion near a Hopf bifurcation of a three-dimensional system. Mech. Res. Commun. 17, 191198.
Nayfeh, A. H. & Balachandran, B. 1995 Applied Nonlinear Dynamics. Wiley & Sons.
Newell, A. C. & Whitehead, J. A. 1969 Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38, 279303.
Nicoli, C. & Pelce, P. 1989 One-dimensional model for the Rijke tube. J. Fluid Mech. 202, 8396.
Nicoud, F., Benoit, L., Sensiau, C. & Poinsot, T. 2007 Acoustic modes in combustors with complex impedances and multidimensional active flames. AIAA J. 45, 426441.
Nicoud, F. & Wieczorek, K. 2009 About the zero Mach number assumption in the calculation of thermoacoustic instabilities. Intl J. Spray Combust. Dyn. 1, 67111.
Noiray, N., Durox, D., Schuller, T. & Candel, S. 2008 A unified framework for nonlinear combustion instability analysis based on the flame descrbing function. J. Fluid Mech. 615, 139167.
Provansal, M., Mathis, C. & Boyer, L. 1987 Benard-von Karman instability: transient and forced regimes. J. Fluid Mech. 182, 122.
Rosales, R. 2004 Hopf bifurcations: notes on nonlinear dynamics and chaos. MIT Open Courseware. 18.385j/2.036j. MIT.
Saha, A., Bhattacharya, B. & Wahi, P. 2009 A comparative study on the control of friction-driven oscillations by time-delayed feedback. Nonlinear Dyn. 60, 1537.
Schuermans, B., Belucci, V., Guethe, F., Meili, F., Flohr, P. & Paschereit, O. 2004A detailed analysis of thermoacoustic interaction mechanisms in a turbulent premixed flame. In Proceedings of ASME Turbo Expo 2004: Power for Land, Sea, and Air.
Selimefendigil, F. & Polifke, W. 2011 A nonlinear frequency domain model for limit cycles in thermoacoustic systems with modal coupling. Intl J. Spray Combust. Dyn. 3, 303330.
Shivamoggi, B. K. 2003 Perturbation Methods for Differential Equations. Birkhauser.
Song, W.-S., Lee, S., Shin, D.-S. & Na, Y. 2006 Thermo-acoustic instability in the horizontal Rijke tube. J. Meas. Sci. Technol. 20, 905913.
Sterling, J. D. & Zukowski, E. E. 1991 Nonlinear dynamics of laboratory combustor pressure oscillations. Combust. Sci. Technol. 77, 225238.
Stewartson, K. & Stuart, J. T. 1971 A nonlinear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48, 529545.
Strogatz, S. H. 2000 Nonlinear Dynamics and Chaos: with applications to Physics, Biology, Chemistry, and Engineering, 1st edn. Westview.
Subramanian, P., Mariappan, S., Sujith, R. I. & Wahi, P. 2010 Bifurcation analysis of thermoacoustic instability in a horizontal Rijke tube. Intl J. Spray Combust. Dyn. 2 (4), 325356.
Tam, K. K. 1968 On the asymptotic solution of the Orr–Sommerfeld equation by the method of multiple-scales. J. Fluid Mech. 34, 145158.
Vidyasagar, M. 1993 Nonlinear System Analysis. Prentice-Hall.
Wahi, P. & Chatterjee, A. 2004 Averaging oscillations with small fractional damping and delayed terms. Nonlinear Dyn. 38 (1–2), 322.
Wahi, P. & Chatterjee, A. 2005 Regenerative tool chatter near a codimension 2 Hopf point using multiple scales. Nonlinear Dyn. 40, 323338.
Wahi, P. & Chatterjee, A. 2008 Self-interrupted regenerative metal cutting in turning. Intl J. Nonlinear Mech. 43, 111123.
Wicker, J. M., Greene, W. D., Kim, S. I. & Yang, V. 1996 Triggering of longitudinal combustion instabilities in rocket motors: nonlinear combustion response. J. Propul. Power 12, 11481158.
Yang, V. & Anderson, W. 1995 Liquid Rocket Engine Combustion Instability. Progress in Aeronautics and Astronautics, AIAA.
Zeytounian, R. Kh. 2002 Asymptotic Modelling of Fluid Flow Phenomena. Springer.
Zinn, B. T. & Lores, M. E. 1971 Application of the Galerkin method in the solution of nonlinear axial combustion instability problems in liquid rockets. Combust. Sci. Technol. 4, 269278.
Zinn, B. T. & Powell, E. A. 1971 Nonlinear combustion instability in liquid-propellant rocket engines. Symp. (Intl) Combust. 13 (1), 491503.
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