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Research Papers: Gas Turbines: Heat Transfer

Uncertainty Quantification of Growth Rates of Thermoacoustic Instability by an Adjoint Helmholtz Solver

[+] Author and Article Information
Camilo F. Silva

Professur für Thermofluiddynamik,
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;
Engineering Department,
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

Wolfgang Polifke

Professur für Thermofluiddynamik,
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 139(1), 011901 (Aug 30, 2016) (11 pages) Paper No: GTP-16-1224; doi: 10.1115/1.4034203 History: Received June 17, 2016; Revised June 29, 2016

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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References

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Figures

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Fig. 1

Turbulent swirled combustor configuration under investigation. Adapted from Palies et al. [18].

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Fig. 8

Same as Fig. 7 for configuration C8 (l1=0.160 m)

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Fig. 6

Monte Carlo simulations around the unperturbed parameters, η0, containing 10,000 realizations. Top frames: configuration C4 (l1=0.096 m) where ω0=(2π·148+i110) s–1. Bottom frames: configuration C8 (l1=0.160 m) where ω0=(2π·130+i110) s–1. The benchmark solution is given by method 0 shown in black in column (a). The adjoint solutions are shown in column (b) method 1 (blue), (c) method 2 (orange), and (d) Method 3 (red). A summary of the methods is reported in Table 3. Growth rates on the right of the vertical line (damping rate DR) are unstable.

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Fig. 7

Distribution of the growth rates obtained from 10,000 Monte Carlo realizations for configuration C4 (l1=0.096 m). Top frames: distribution of the adjoint solution, ω̃i, with respect to the benchmark solution, ωi, given by the straight line. The higher the scattering, the lower the accuracy of the adjoint solution. Bottom frames: PDF of the perturbed growth rate. In all the panels, the benchmark solution is given by method 0 in black. The solutions obtained by the three adjoint-based methods (see Table 3) are reported in color in column (a) method 1 (blue), (b) method 2 (orange), and (c) method 3 (red). Growth rates to the right of the vertical line (damping rate DR) correspond to unstable modes.

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Fig. 2

First thermoacoustic mode at ω0=(2π·148 + i110) s–1. (a) Direct mode, p̂0 and (b) adjoint mode, p̂0†.

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Fig. 3

The geometrical parameter η0Ge is the plenum cone angle, α

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Fig. 9

Structure of the matrices L,A,H, and B. Blue dots represent nonzero elements.

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Fig. 4

Frequency drift around the reference angular frequency, ω0r=2π·148 s–1, due to perturbations to the system's parameters, η0, reported in Table 1. Top frames: Real{(1/2)(σ2/ω0)}/2π, where σ2=T2ord(1) □, σ2=T2ord(2) *, σ2=T2ord(3) △, σ2=T2ord(4) °. Bottom frames: method 0 (continuous black line), method 1 (dashed blue line), method 2 (dashed orange line), method 3 (continuous red line), explained in Table 3.

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Fig. 5

Growth-rate drift around the reference growth rate, ω0r=2π·148, s–1, due to perturbations to the system's parameters, η0, reported in Table 1. Top frames: Imag{(1/2)(σ2/ω0)}, where σ2=T2ord(1) □, σ2=T2ord(2) *, σ2=T2ord(3) △, σ2=T2ord(4) °. Bottom frames: method 0 (continuous black line), method 1 (dashed blue line), method 2 (dashed orange line), method 3 (continuous red line), explained in Table 3.

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Fig. 10

Growth rate trajectories as a function of velocity perturbation ratio |û|/u¯b at the flame base (flame B of Palies et al. [19]). The dashed–dotted lines surrounded by the gray band indicate the region where the growth rate is balanced by damping. Crosses indicate the amplitude at which limit cycles are expected. AVSP data taken from Silva et al. [4].

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