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Sensitivity analysis of a time-delayed thermo-acoustic system via an adjoint-based approach

Published online by Cambridge University Press:  19 February 2013

Luca Magri  and
Matthew P. Juniper
Luca Magri*
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: [email protected]
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Abstract

We apply adjoint-based sensitivity analysis to a time-delayed thermo-acoustic system: a Rijke tube containing a hot wire. We calculate how the growth rate and frequency of small oscillations about a base state are affected either by a generic passive control element in the system (the structural sensitivity analysis) or by a generic change to its base state (the base-state sensitivity analysis). We illustrate the structural sensitivity by calculating the effect of a second hot wire with a small heat-release parameter. In a single calculation, this shows how the second hot wire changes the growth rate and frequency of the small oscillations, as a function of its position in the tube. We then examine the components of the structural sensitivity in order to determine the passive control mechanism that has the strongest influence on the growth rate. We find that a force applied to the acoustic momentum equation in the opposite direction to the instantaneous velocity is the most stabilizing feedback mechanism. We also find that its effect is maximized when it is placed at the downstream end of the tube. This feedback mechanism could be supplied, for example, by an adiabatic mesh. We illustrate the base-state sensitivity by calculating the effects of small variations in the damping factor, the heat-release time-delay coefficient, the heat-release parameter, and the hot-wire location. The successful application of sensitivity analysis to thermo-acoustics opens up new possibilities for the passive control of thermo-acoustic oscillations by providing gradient information that can be combined with constrained optimization algorithms in order to reduce linear growth rates.

JFM classification

Acoustics: Acoustics Flow Control: Flow Control Flow Control: Instability control
Type
Papers
Information
Journal of Fluid Mechanics , Volume 719 , 25 March 2013 , pp. 183 - 202
Copyright
©2013 Cambridge University Press

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References

Balasubramanian, K. & Sujith, R. I. 2008a Thermoacoustic instability in a Rijke tube: non-normality and nonlinearity. Phys. Fluids 20, 044103.Google Scholar
Balasubramanian, K. & Sujith, R. I. 2008b Non-normality and nonlinearity in combustion-acoustic interaction in diffusion flames. J. Fluid Mech. 594, 29–57.CrossRefGoogle Scholar
Belegundu, A. D. & Arora, J. S. 1985 A sensitivity interpretation of adjoint variables in optimal design. Comput. Meth. Appl. Mech. Engng 48, 81–89.Google Scholar
Chandler, G. J., Juniper, M. P., Nichols, J. W. & Schmid, P. J. 2012 Adjoint algorithms for the Navier–Stokes equations in the low Mach number limit. J. Comput. Phys. 231, 1900–1916.Google Scholar
Chu, B. T. 1963 Analysis of a self-sustained thermally driven nonlinear vibration. Phys. Fluids 6 (11), 1638–1644.Google Scholar
Culick, F. E. C. 1971 Nonlinear growth and limiting amplitude of acoustic oscillations in combution chambers. Combust. Sci. Technol. 3, 1–16.CrossRefGoogle Scholar
Dennery, P. & Krzywicki, A. 1996 Mathematics for Physicists. Dover.Google Scholar
Giannetti, F., Camarri, S. & Luchini, P. 2010 Structural sensitivity of the secondary instability in the wake of a circular cylinder. J. Fluid Mech. 651, 319–337.Google Scholar
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167–197.Google Scholar
Gibbs, J. W. 1898 Fourier’s series. Lett. Nature 59, 200.CrossRefGoogle Scholar
Heckl, M. 1990 Nonlinear acoustic effects in the Rijke tube. Acustica 72, 63.Google Scholar
Hill, D. C. 1992 A theoretical approach for analysing the restabilization of wakes, NASA TM 103858.Google Scholar
Juniper, M. P. 2011 Triggering in the horizontal Rijke tube: non-normality, transient growth and bypass transition. J. Fluid Mech. 667, 272–308.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon.Google Scholar
Marquet, O., Sipp, D. & Jacquin, L. 2008 Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221–252.Google Scholar
Matveev, I. 2003 Thermo-acoustic instabilities in the Rijke tube: experiments and modelling. PhD thesis, CalTech.Google Scholar
Rayleigh, Lord 1878 The explanation of certain acoustical phenomena. Nature 18, 319–321.CrossRefGoogle Scholar
Salwen, H. & Grosch, C. E. 1981 The continuous spectrum of the Orr–Sommerfeld equation. Part 2. Eigenfuction expansions. J. Fluid Mech. 104, 445–465.Google Scholar
Sipp, D., Marquet, O., Meliga, P. & Barbagallo, A. 2010 Dynamics and control of global instabilities in open-flows: a linearized approach. Appl. Mech. Rev. 63 (3), 030801.CrossRefGoogle Scholar