Thermoacoustic instabilities are often calculated with Helmholtz solvers combined with a low-order model for the flame dynamics. Typically, such a formulation leads to an eigenvalue problem in which the eigenvalue appears under nonlinear terms, such as exponentials related to the time delays that result from the flame model. The objective of the present paper is to quantify uncertainties in thermoacoustic stability analysis with a Helmholtz solver and its adjoint. This approach is applied to the model of a combustion test rig with a premixed swirl burner. The nonlinear eigenvalue problem and its adjoint are solved by an in-house adjoint Helmholtz solver, based on an axisymmetric finite-volume discretization. In addition to first-order correction terms of the adjoint formulation, as they are often used in the literature, second-order terms are also taken into account. It is found that one particular second-order term has significant impact on the accuracy of the predictions. Finally, the probability density function (PDF) of the growth rate in the presence of uncertainties in the input parameters is calculated with a Monte Carlo approach. The uncertainties considered concern the gain and phase of the flame response, the outlet acoustic reflection coefficient, and the plenum geometry. It is found that the second-order adjoint method gives quantitative agreement with results based on the full nonlinear eigenvalue problem, while requiring much fewer computations.
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January 2017
Research-Article
Uncertainty Quantification of Growth Rates of Thermoacoustic Instability by an Adjoint Helmholtz Solver
Camilo F. Silva,
Camilo F. Silva
Professur für Thermofluiddynamik,
Technische Universität München,
Garching b. München D-85747, Germany
e-mail: silva@tfd.mw.tum.de
Technische Universität München,
Garching b. München D-85747, Germany
e-mail: silva@tfd.mw.tum.de
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Luca Magri,
Luca Magri
Center for Turbulence Research,
Stanford University,
Stanford, CA 94305-3024;
Stanford University,
Stanford, CA 94305-3024;
Engineering Department,
Cambridge University,
Trumpington Street,
Cambridge CB2 1PZ, UK
e-mail: lmagri@stanford.edu
Cambridge University,
Trumpington Street,
Cambridge CB2 1PZ, UK
e-mail: lmagri@stanford.edu
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Thomas Runte,
Thomas Runte
Professur für Thermofluiddynamik,
Technische Universität München,
Garching b. München D-85747, Germany
e-mail: thomas_runte1@gmx.de
Technische Universität München,
Garching b. München D-85747, Germany
e-mail: thomas_runte1@gmx.de
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Wolfgang Polifke
Wolfgang Polifke
Professur für Thermofluiddynamik,
Technische Universität München,
Garching b. München D-85747, Germany
e-mail: polifke@tfd.mw.tum.de
Technische Universität München,
Garching b. München D-85747, Germany
e-mail: polifke@tfd.mw.tum.de
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Camilo F. Silva
Professur für Thermofluiddynamik,
Technische Universität München,
Garching b. München D-85747, Germany
e-mail: silva@tfd.mw.tum.de
Technische Universität München,
Garching b. München D-85747, Germany
e-mail: silva@tfd.mw.tum.de
Luca Magri
Center for Turbulence Research,
Stanford University,
Stanford, CA 94305-3024;
Stanford University,
Stanford, CA 94305-3024;
Engineering Department,
Cambridge University,
Trumpington Street,
Cambridge CB2 1PZ, UK
e-mail: lmagri@stanford.edu
Cambridge University,
Trumpington Street,
Cambridge CB2 1PZ, UK
e-mail: lmagri@stanford.edu
Thomas Runte
Professur für Thermofluiddynamik,
Technische Universität München,
Garching b. München D-85747, Germany
e-mail: thomas_runte1@gmx.de
Technische Universität München,
Garching b. München D-85747, Germany
e-mail: thomas_runte1@gmx.de
Wolfgang Polifke
Professur für Thermofluiddynamik,
Technische Universität München,
Garching b. München D-85747, Germany
e-mail: polifke@tfd.mw.tum.de
Technische Universität München,
Garching b. München D-85747, Germany
e-mail: polifke@tfd.mw.tum.de
1Correspondence author.
Contributed by the Heat Transfer Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received June 17, 2016; final manuscript received June 29, 2016; published online August 30, 2016. Editor: David Wisler.
J. Eng. Gas Turbines Power. Jan 2017, 139(1): 011901 (11 pages)
Published Online: August 30, 2016
Article history
Received:
June 17, 2016
Revised:
June 29, 2016
Citation
Silva, C. F., Magri, L., Runte, T., and Polifke, W. (August 30, 2016). "Uncertainty Quantification of Growth Rates of Thermoacoustic Instability by an Adjoint Helmholtz Solver." ASME. J. Eng. Gas Turbines Power. January 2017; 139(1): 011901. https://doi.org/10.1115/1.4034203
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