Research Papers

Quantification and Propagation of Uncertainties in Identification of Flame Impulse Response for Thermoacoustic Stability Analysis

[+] Author and Article Information
Shuai Guo

Professur für Thermofluiddynamik,
Technische Universität München,
Boltzmannstr. 15,
Garching D-85748, Germany
e-mail: guo@tfd.mw.tum.de

Camilo F. Silva

Professur für Thermofluiddynamik,
Technische Universität München,
Boltzmannstr. 15,
Garching D-85748, Germany
e-mail: silva@tfd.mw.tum.de

Abdulla Ghani

Professur für Thermofluiddynamik,
Technische Universität München,
Boltzmannstr. 15,
Garching D-85748, Germany
e-mail: ghani@tfd.mw.tum.de

Wolfgang Polifke

Professor Professur für Thermofluiddynamik,
Technische Universität München,
Boltzmannstr. 15,
Garching D-85748, Germany
e-mail: polifke@tfd.mw.tum.de

1Corresponding author.

Manuscript received August 28, 2018; final manuscript received September 14, 2018; published online October 23, 2018. Editor: Jerzy T. Sawicki.

J. Eng. Gas Turbines Power 141(2), 021032 (Oct 23, 2018) (10 pages) Paper No: GTP-18-1585; doi: 10.1115/1.4041652 History: Received August 28, 2018; Revised September 14, 2018

The thermoacoustic behavior of a combustion system can be determined numerically via acoustic tools such as Helmholtz solvers or network models coupled with a model for the flame dynamic response. Within such a framework, the flame response to flow perturbations can be described by a finite impulse response (FIR) model, which can be derived from large eddy simulation (LES) time series via system identification. However, the estimated FIR model will inevitably contain uncertainties due to, e.g., the statistical nature of the identification process, low signal-to-noise ratio, or finite length of time series. Thus, a necessary step toward reliable thermoacoustic stability analysis is to quantify the impact of uncertainties in FIR model on the growth rate of thermoacoustic modes. There are two practical considerations involved in this topic. First, how to efficiently propagate uncertainties from the FIR model to the modal growth rate of the system, considering it is a high dimensional uncertainty quantification (UQ) problem? Second, since longer computational fluid dynamics (CFD) simulation time generally leads to less uncertain FIR model identification, how to determine the length of the CFD simulation required to obtain satisfactory confidence? To address the two issues, a dimensional reduction UQ methodology called “Active subspace approach (ASA)” is employed in the present study. For the first question, ASA is applied to exploit a low-dimensional approximation of the original system, which allows accelerated UQ analysis. Good agreement with Monte Carlo analysis demonstrates the accuracy of the method. For the second question, a procedure based on ASA is proposed, which can serve as an indicator for terminating CFD simulation. The effectiveness of the procedure is verified in the paper.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.


Nicoud, F. , Benoit, L. , Sensiau, C. , and Poinsot, T. , 2007, “ Acoustic Modes in Combustors With Complex Impedances and Multidimensional Active Flames,” AIAA J., 45(2), pp. 426–441. [CrossRef]
Silva, C. F. , Emmert, T. , Jaensch, S. , and Polifke, W. , 2015, “ Numerical Study on Intrinsic Thermoacoustic Instability of a Laminar Premixed Flame,” Combust. Flame, 162(9), pp. 3370–3378. [CrossRef]
Magri, L. , and Juniper, M. P. , 2013, “ Sensitivity Analysis of a Time-Delayed Thermo-Acoustic System Via an Adjoint-Based Approach,” J. Fluid Mech., 719, pp. 183–202. [CrossRef]
Ndiaye, A. , Bauerheim, M. , and Nicoud, F. , 2015, “ Uncertainty Quantification of Thermoacoustic Instabilities on a Swirled Stabilized Combustor,” ASME Paper No. GT2015-44133.
Tay-Wo-Chong, L. , Bomberg, S. , Ulhaq, A. , Komarek, T. , and Polifke, W. , 2012, “ Comparative Validation Study on Identification of Premixed Flame Transfer Function,” ASME J. Eng. Gas Turbines Power, 134(2), p. 021502. [CrossRef]
Bauerheim, M. , Ndiaye, A. , Constantine, P. , Moreau, S. , and Nicoud, F. , 2016, “ Symmetry Breaking of Azimuthal Thermoacoustic Modes: The UQ Perspective,” J. Fluid Mech., 789, pp. 534–566. [CrossRef]
Magri, L. , Bauerheim, M. , Nicoud, F. , and Juniper, M. P. , 2016, “ Stability Analysis of Thermo-Acoustic Nonlinear Eigenproblems in Annular Combustors—Part II: Uncertainty Quantification,” J. Comput. Phys., 325, pp. 411–421. [CrossRef]
Silva, C. , Magri, L. , Runte, T. , and Polifke, W. , 2017, “ Uncertainty Quantification of Growth Rates of Thermoacoustic Instability by an Adjoint Helmholtz Solver,” ASME J. Eng. Gas Turbines Power, 139(1), p. 011901. [CrossRef]
Blumenthal, R. S. , Subramanian, P. , Sujith, R. , and Polifke, W. , 2013, “ Novel Perspectives on the Dynamics of Premixed Flames,” Combust. Flame, 160(7), pp. 1215–1224. [CrossRef]
Polifke, W. , 2014, “ Black-Box System Identification for Reduced Order Model Construction,” Ann. Nucl. Energy, 67, pp. 109–128. [CrossRef]
Sovardi, C. , Jaensch, S. , and Polifke, W. , 2016, “ Concurrent Identification of Aero-Acoustic Scattering and Noise Sources at a Flow Duct Singularity in Low Mach Number Flow,” J. Sound Vib., 377, pp. 90–105. [CrossRef]
Guo, S. , Silva, C. F. , Bauerheim, M. , Ghani, A. , and Polifke, W. , 2018, “ Evaluating the Impact of Uncertainty in Flame Impulse Response Model on Thermoacoustic Instability Prediction: A Dimensionality Reduction Approach,” Proc. Combust. Inst. (in press).
Constantine, P. , Dow, E. , and Wang, Q. , 2014, “ Active Subspace Methods in Theory and Practice: Applications to Kriging Surfaces,” SIAM J. Sci. Comput., 36(4), pp. A1500–A1524. [CrossRef]
Constantine, P. G. , 2015, Active Subspaces: Emerging Ideas in Dimension Reduction for Parameter Studies, Vol. 2, SIAM, Philadelphia, PA.
Lukaczyk, T. W. , Constantine, P. G. , Palacios, F. , and Alonso, J. J. , 2014, “ Active Subspaces for Shape Optimization,” AIAA Paper No. 2014-1171.
Jefferson, J. L. , Gilbert, J. M. , Constantine, P. G. , and Maxwell, R. M. , 2015, “ Active Subspaces for Sensitivity Analysis and Dimension Reduction of an Integrated Hydrologic Model,” Comput. Geosci., 83, pp. 127–138. [CrossRef]
Constantine, P. G. , and Diaz, P. , 2017, “ Global Sensitivity Metrics From Active Subspaces,” Reliab. Eng. Syst. Saf., 162, pp. 1–13. [CrossRef]
Bodén, H. , and Polifke, W. , 2015, “ Uncertainty Quantification Applied to Aeroacoustic Predictions,” Progress in Simulation, Control and Reduction of Ventilation Noise (VKI Lecture Series 2015), C. Schram , ed., von Karman Institute for Fluid Dynamics, Sint-Genesius-Rode, Belgium.
Keesman, K. J. , 2011, “ Time-Invariant System Identification,” System Identification (Advanced Textbooks in Control and Signal Processing), Springer, London, pp. 59–167.
Tay-Wo-Chong, L. , Komarek, T. , Kaess, R. , Föller, S. , and Polifke, W. , 2010, “ Identification of Flame Transfer Functions From LES of a Premixed Swirl Burner,” ASME Paper No. GT2010-22769.
Komarek, T. , and Polifke, W. , 2010, “ Impact of Swirl Fluctuations on the Flame Response of a Perfectly Premixed Swirl Burner,” ASME J. Eng. Gas Turbines Power, 132(6), p. 061503. [CrossRef]
Emmert, T. , Bomberg, S. , Jaensch, S. , and Polifke, W. , 2017, “ Acoustic and Intrinsic Thermoacoustic Modes of a Premixed Combustor,” Proc. Combust. Inst., 36(3), pp. 3835–3842. [CrossRef]
Cowan, G. , 1998, Statistical Data Analysis, 1st ed., Clarendon Press, Gloucestershire, UK.
Jaensch, S. , Merk, M. , Emmert, T. , and Polifke, W. , 2018, “ Identification of Flame Transfer Functions in the Presence of Intrinsic Thermoacoustic Feedback and Noise,” Combust. Theory Modell., 22(3), pp. 613–634. [CrossRef]
Tangirala, A. K. , 2014, Principles of System Identification: Theory and Practice, CRC Press, Boca Raton, FL.
Avdonin, A. , and Polifke, W. , 2018, “ Quantification of the Impact of Uncertainties in Operating Conditions on the Flame Transfer Function With Non-Intrusive Polynomial Chaos Expansion,” ASME Paper No. GT2018-75476.


Grahic Jump Location
Fig. 1

Normalized velocity and global heat release rate fluctuations, the total length of the data is 350 ms

Grahic Jump Location
Fig. 2

Impulse response. Each discrete stem represents one coefficient hk, upper and lower dot lines constitute the 95% confidence interval.

Grahic Jump Location
Fig. 3

Sketch of acoustic network model, flow from left to right

Grahic Jump Location
Fig. 4

Workflow of UQ based on active subspace approach

Grahic Jump Location
Fig. 5

Active subspace approach and DMC simulation

Grahic Jump Location
Fig. 6

Eigenmodes from deterministic analysis. For case A, the dominant mode is quarter wave mode [22], with a frequency of 434.2 Hz and a growth rate of −4 rad/s. For case B, the dominant mode is intrinsic mode [22], with a frequency of 97.5 Hz and a growth rate of −4 rad/s.

Grahic Jump Location
Fig. 7

Contour plot of the joint PDF of the modal growth rate and frequency. The contours (from outside to inside) correspond to 10%, 30%, 50%, 70%, and 90% of the maximum probability. The triangle is the deterministic solution (same as Fig. 6). Statistics regarding the marginal distribution of the modal growth rate are presented in Table 2: (a) case A and (b) case B.

Grahic Jump Location
Fig. 8

Eigenvalues in λASA in descending order. The prominent gap between the first and second eigenvalues indicates that a one-dimensional subspace exists: (a) case A and (b) case B.

Grahic Jump Location
Fig. 9

Components of the first eigenvector in W1, which will be used as the linear combination coefficients to form the single active variable: (a) case A and (b) case B

Grahic Jump Location
Fig. 10

Sufficient summary plot of the modal growth rate against active variable for each sample. We fit a quadratic function to link active variable and modal growth rate for each case.

Grahic Jump Location
Fig. 11

Probability density function of thermoacoustic growth rate of dominant mode produced by ASA and DMC: (a) case A and (b) case B

Grahic Jump Location
Fig. 12

A feasible workflow for estimating appropriate CFD simulation time to achieve predefined confidence requirements

Grahic Jump Location
Fig. 13

Finite impulse response models identified from time series of different length. Here, confidence intervals (represented by ±3 standard deviations) of FIR model coefficients become narrower as length of time series increases.

Grahic Jump Location
Fig. 14

200 ms, 500 ms, and 1400 ms of synthetic series are used to identify “FIR-200,” “FIR-500,” and “FIR-1400” model, respectively. Active subspace approach will only be implemented once on “FIR-200” to derive the SM.

Grahic Jump Location
Fig. 15

(a) Eigenvalues in descending order and (b) components of the first eigenvector

Grahic Jump Location
Fig. 16

Sufficient summary plot of the modal growth rate against active variable. A quadratic regression model is fitted to describe the relation between active variable and modal growth rate.

Grahic Jump Location
Fig. 17

Comparison of PDF results produced by SM and DME, considering the uncertainty information of (a) “FIR-500” model and (b) “FIR-1400” model

Grahic Jump Location
Fig. 18

The proposed procedure for estimating CFD time length for FIR identification



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In