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Sensitivity of the Rayleigh criterion in thermoacoustics

Published online by Cambridge University Press:  08 November 2019

Luca Magri Open the ORCID record for Luca Magri [Opens in a new window] ,
Matthew P. Juniper Open the ORCID record for Matthew P. Juniper [Opens in a new window]  and
Jonas P. Moeck
Luca Magri*
Affiliation:
Cambridge University Engineering Department, Trumpington Street, Cambridge CB2 1PZ, UK
Matthew P. Juniper
Affiliation:
Cambridge University Engineering Department, Trumpington Street, Cambridge CB2 1PZ, UK
Jonas P. Moeck
Affiliation:
Department of Energy and Process Engineering, NTNU, NO-7491 Trondheim, Norway
*
†Email address for correspondence: [email protected]
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Abstract

Thermoacoustic instabilities are one of the most challenging problems faced by gas turbine and rocket motor manufacturers. The key instability mechanism is described by the Rayleigh criterion. The Rayleigh criterion does not directly show how to alter a system to make it more stable. This is the objective of sensitivity analysis. Because thermoacoustic systems have many design parameters, adjoint sensitivity analysis has been proposed to obtain all the sensitivities with one extra calculation. Although adjoint sensitivity analysis can be carried out in both the time and the frequency domain, the frequency domain is more natural for a linear analysis. Perhaps surprisingly, the Rayleigh criterion has not yet been rigorously derived and comprehensively interpreted in the frequency domain. The contribution of this theoretical paper is threefold. First, the Rayleigh criterion is interpreted in the frequency domain with integral formulae for the complex eigenvalue. Second, the first variation of the Rayleigh criterion is calculated both in the time and frequency domain, both with and without Lagrange multipliers (adjoint variables). The Lagrange multipliers are physically related to the system’s observables. Third, an adjoint Rayleigh criterion is proposed. The paper also points out that the conclusions of Juniper (Phys. Rev. Fluids, vol. 3, 2018, 110509) apply to the first variation of the Rayleigh criterion, not to the Rayleigh criterion itself. The mathematical relations of this paper can be used to compute sensitivities directly from measurable quantities to enable optimal design.

JFM classification

Compressible Flows: Gas dynamics
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 882 , 10 January 2020 , R1
Copyright
© 2019 Cambridge University Press 

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References

Aguilar, J. G., Magri, L. & Juniper, M. P. 2017 Adjoint-based sensitivity analysis of low order thermoacoustic networks using a wave-based approach. J. Comput. Phys. 341, 163–181.Google Scholar
Brear, M. J., Nicoud, F., Talei, M., Giauque, A. & Hawkes, E. R. 2012 Disturbance energy transport and sound production in gaseous combustion. J. Fluid Mech. 707, 53–73.Google Scholar
Camarri, S. 2015 Flow control design inspired by linear stability analysis. Acta Mechanica 226 (4), 979–1010.Google Scholar
Candel, S. 2002 Combustion dynamics and control: progress and challenges. Proc. Combust. Inst. 29, 1–28.Google Scholar
Candel, S. M. 1975 Acoustic conservation principles and an application to plane and modal propagation in nozzles and diffusers. J. Sound Vib. 41 (2), 207–232.Google Scholar
Chu, B. T. 1965 On the energy transfer to small disturbances in fluid flow (Part I). Acta Mechanica 1 (3), 215–234.Google Scholar
Dowling, A. P. & Morgans, A. S. 2005 Feedback control of combustion oscillations. Annu. Rev. Fluid Mech. 37, 151–182.Google Scholar
George, K. J. & Sujith, R. I. 2011 On Chu’s disturbance energy. J. Sound Vib. 330 (22), 5280–5291.Google Scholar
George, K. J. & Sujith, R. I. 2012 Disturbance energy norms: a critical analysis. J. Sound Vib. 331 (7), 1552–1566.Google Scholar
Ghirardo, G., Juniper, M. P. & Moeck, J. P. 2016 Weakly nonlinear analysis of thermoacoustic instabilities in annular combustors. J. Fluid Mech. 805, 52–87.Google Scholar
Greiner, W. & Reinhardt, G. 1996 Field Quantization. Springer.Google Scholar
Juniper, M. P. 2018 Sensitivity analysis of thermoacoustic instability with adjoint Helmholtz solvers. Phys. Rev. Fluids 3, 110509.Google Scholar
Juniper, M. P. & Sujith, R. I. 2018 Sensitivity and nonlinearity of thermocoustic oscillations. Annu. Rev. Fluid Mech. 50 (1), 661–689.Google Scholar
Karimi, N., Brear, M. J. & Moase, W. H. 2008 Acoustic and disturbance energy analysis of a flow with heat communication. J. Fluid Mech. 597, 67–89.Google 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, Inc.Google Scholar
Lord Rayleigh 1878 The explanation of certain acoustical phenomena. Nature 18, 319–321.Google Scholar
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46 (1), 493–517.Google Scholar
Magri, L. 2019 Adjoint methods as design tools in thermoacoustics. Appl. Mech. Rev. 71 (2), 020801.Google Scholar
Magri, L. & Juniper, M. P. 2013 Sensitivity analysis of a time-delayed thermo-acoustic system via an adjoint-based approach. J. Fluid Mech. 719, 183–202.Google Scholar
Magri, L. & Juniper, M. P. 2014 Global modes, receptivity, and sensitivity analysis of diffusion flames coupled with duct acoustics. J. Fluid Mech. 752, 237–265.Google Scholar
Myers, M. K. 1991 Transport of energy by disturbances in arbitrary steady flows. J. Fluid Mech. 226, 383–400.Google Scholar
Nicoud, F. & Poinsot, T. 2005 Thermoacoustic instabilities: should the Rayleigh criterion be extended to include entropy changes? Combust. Flame 142 (1–2), 153–159.Google Scholar
Poinsot, T. 2017 Prediction and control of combustion instabilities in real engines. Proc. Combust. Inst. 36 (1), 1–28.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.Google Scholar
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