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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.

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References

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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).
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Figures

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Fig. 1

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

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Fig. 2

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

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Fig. 3

Sketch of acoustic network model, flow from left to right

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Fig. 4

Workflow of UQ based on active subspace approach

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Fig. 5

Active subspace approach and DMC simulation

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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.

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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.

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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.

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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

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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.

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Fig. 11

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

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Fig. 12

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

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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.

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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.

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Fig. 15

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

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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.

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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

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Fig. 18

The proposed procedure for estimating CFD time length for FIR identification

Tables

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