Parametric Investigation of Thermoacoustic Instability (TAI) in a Rijke Tube: A Time-Delay Perspective

This paper presents a novel deployment of a recent mathematical paradigm for predicting the thermo-acoustic instability (TAI) of a Rijke tube in the relevant parametric space. This benchmark problem in combustion science has been studied for over 1½ centuries with phenomenal achievements both in theoretical and practical fronts. The new paradigm is called the Cluster Treatment of Characteristic Roots (CTCR), which is originally developed to assess the asymptotic stability of Linear Time Invariant (LTI) Time-delayed Systems (TDS). A notorious subcategory within LTI-TDS is called “Neutral TDS”, which matches the characteristics of the linearized dynamics of thermo-acoustic instability. The CTCR is shown to reveal a non-conservative and exhaustive linear stability map of the Rijke tube within the space of its geometric and operational parameters. We present a review of this paradigm as well as several case studies to demonstrate its capabilities and some encouraging comparison with the earlier literature. This paper is a concept document and it is prepared with the intent of providing a breeding ground for studies beyond its present coverage.

[1]  S. Niculescu.  Delay Effects on Stability: A Robust Control Approach , 2001 .

[2]  P. Wahi,et al.  Subcritical bifurcation and bistability in thermoacoustic systems , 2013, Journal of Fluid Mechanics.

[3]  Maria A. Heckl,et al.  Active Control of the Noise from a Rijke Tube , 1988 .

[4]  P. Moin,et al.  Combustion instability due to the nonlinear interaction between sound and flame , 2003, Journal of Fluid Mechanics.

[5]  Tomás Vyhlídal,et al.  Mapping Based Algorithm for Large-Scale Computation of Quasi-Polynomial Zeros , 2009, IEEE Transactions on Automatic Control.

[6]  P. L. Rijke,et al.  LXXI. Notice of a new method of causing a vibration of the air contained in a tube open at both ends , 1859 .

[7]  Paul Kuentzmann,et al.  Unsteady Motions in Combustion Chambers for Propulsion Systems , 2006 .

[8]  Merrill W. Beckstead,et al.  A REVIEW OF RIJKE TUBES, RIJKE BURNERS AND RELATED DEVICES , 1993 .

[9]  R. Bellman,et al.  Differential-Difference Equations , 1967 .

[10]  M. Lighthill,et al.  The response of laminar skin friction and heat transfer to fluctuations in the stream velocity , 1954, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[11]  M. Juniper.  Triggering in the horizontal Rijke tube: non-normality, transient growth and bypass transition , 2010, Journal of Fluid Mechanics.

[12]  Wolfgang Polifke,et al.  Identification of heat transfer dynamics for non-modal analysis of thermoacoustic stability , 2011, Appl. Math. Comput..

[13]  Nejat Olgaç,et al.  A practical method for analyzing the stability of neutral type LTI-time delayed systems , 2004, Autom..

[14]  S. Candel,et al.  A review of active control of combustion instabilities , 1993 .

[15]  Christian Oliver Paschereit,et al.  Feedback control of unstable thermoacoustic modes in an annular Rijke tube , 2012 .

[16]  Carlos J. Moreno,et al.  The zeros of exponential polynomials (I) , 1973 .

[17]  A. Dowling.  Nonlinear self-excited oscillations of a ducted flame , 1997, Journal of Fluid Mechanics.

[18]  Hassan Fazelinia.  A novel stability analysis of systems with multiple time delays and its application to high speed milling chatter , 2007 .

[19]  Nejat Olgaç,et al.  Complete stability robustness of third-order LTI multiple time-delay systems , 2005, Autom..

[20]  Pankaj Wahi,et al.  Bifurcation Analysis of Thermoacoustic Instability in a Horizontal Rijke Tube , 2010 .

[21]  Ann P. Dowling,et al.  ADAPTIVE CONTROL OF COMBUSTION OSCILLATIONS , 1998 .

[22]  Z. Rekasius,et al.  A stability test for systems with delays , 1980 .

[23]  Jack K. Hale,et al.  Strong stabilization of neutral functional differential equations , 2002 .

[24]  Anuradha M. Annaswamy,et al.  Self-tuning regulators for combustion oscillations , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[25]  Richard Bellman,et al.  Differential-Difference Equations , 1967 .

[26]  Nejat Olgaç,et al.  A unique methodology for the stability robustness of multiple time delay systems , 2006, Syst. Control. Lett..

[27]  Nejat Olgac,et al.  The Cluster Treatment of Characteristic Roots and the Neutral Type Time-Delayed Systems , 2005 .

[28]  F. Nicoud,et al.  Acoustic modes in combustors with complex impedances and multidimensional active flames , 2007 .

[29]  A. Annaswamy,et al.  Response of a laminar premixed flame to flow oscillations: A kinematic model and thermoacoustic instability results , 1996 .

[30]  Wolfgang Polifke,et al.  CFD-based application of the Nyquist criterion to thermo-acoustic instabilities , 2008, J. Comput. Phys..

[31]  Naresh K. Sinha,et al.  Modern Control Systems , 1981, IEEE Transactions on Systems, Man, and Cybernetics.

[32]  T. Lieuwen,et al.  Experimental investigation of limit cycle oscillations in an unstable gas turbine combustor , 2000 .

[33]  Konstantin I. Matveev,et al.  Thermoacoustic instabilities in the Rijke tube: experiments and modeling , 2003 .

[34]  Sergio Bittanti,et al.  Identification of a model for thermoacoustic instabilities in a Rijke tube , 2002, IEEE Trans. Control. Syst. Technol..

[35]  Hadi Widjaya. Tjioe.  Feedback control of combustion oscillations , 2010 .

[36]  Raman Sujith,et al.  Thermoacoustic instability in a Rijke tube: non-normality and nonlinearity , 2007 .

[37]  Claude Sensiau,et al.  Combustion Instability Problems Analysis for High-Pressure Jet Engine Cores , 2008 .

[38]  Nejat Olgaç,et al.  Extended Kronecker Summation for Cluster Treatment of LTI Systems with Multiple Delays , 2007, SIAM J. Control. Optim..

[39]  W. Polifke,et al.  Combustion Stability Analysis of Rocket Engines with Resonators Based on Nyquist Plots , 2014 .

[40]  Bruno Schuermans,et al.  Theoretical and experimental investigations on damper performance for suppression of thermoacoustic oscillations , 2010 .

[41]  Jack K. Hale,et al.  On the zeros of exponential polynomials , 1980 .

[42]  S. Barnett.  Polynomials and linear control systems , 1983 .

[43]  Raman Sujith,et al.  Characterizing energy growth during combustion instabilities: Singularvalues or eigenvalues? , 2009 .