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Research Papers: Gas Turbines: Combustion, Fuels, and Emissions

Low-Order Modeling of Nonlinear High-Frequency Transversal Thermoacoustic Oscillations in Gas Turbine Combustors

[+] Author and Article Information
Tobias Hummel

Lehrstuhl für Thermodynamik,
Technische Universität München,
Garching 85748, Germany;
Institute for Advanced Study,
Technische Universität München,
Garching 85748, Germany
e-mail: hummel@td.mw.tum.de

Klaus Hammer

Lehrstuhl für Thermodynamik,
Technische Universität München,
Garching 85748, Germany
e-mail: klaus.hammer@tum.de

Pedro Romero

Lehrstuhl für Thermodynamik,
Technische Universität München,
Garching 85748, Germany
e-mail: romero@td.mw.tum.de

Bruno Schuermans

Institute for Advanced Study,
Technische Universität München,
Garching 85748, Germany;
GE Power,
Baden 5401, Switzerland
e-mail: bruno.schuermans@ge.com

Thomas Sattelmayer

Lehrstuhl für Thermodynamik,
Technische Universität München,
Garching 85748, Germany
e-mail: sattelmayer@td.mw.tum.de

1Corresponding author.

Contributed by the Combustion and Fuels Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received July 11, 2016; final manuscript received November 30, 2016; published online February 23, 2017. Editor: David Wisler.

J. Eng. Gas Turbines Power 139(7), 071503 (Feb 23, 2017) (11 pages) Paper No: GTP-16-1322; doi: 10.1115/1.4035529 History: Received July 11, 2016; Revised November 30, 2016

This paper analyzes transversal thermoacoustic oscillations in an experimental gas turbine combustor utilizing dynamical system theory. Limit-cycle acoustic motions related to the first linearly unstable transversal mode of a given 3D combustor configuration are modeled and reconstructed by means of a low-order dynamical system simulation. The source of nonlinearity is solely allocated to flame dynamics, saturating the growth of acoustic amplitudes, while the oscillation amplitudes are assumed to always remain within the linearity limit. First, a reduced order model (ROM) which reproduces the combustor's modal distribution and damping of acoustic oscillations is derived. The ROM is a low-order state-space system, which results from a projection of the linearized Euler equations (LEE) into their truncated eigenspace. Second, flame dynamics are modeled as a function of acoustic perturbations by means of a nonlinear transfer function. This function has a linear and a nonlinear contribution. The linear part is modeled analytically from first principles, while the nonlinear part is mathematically cast into a cubic saturation functional form. Additionally, the impact of stochastic forcing due to broadband combustion noise is included by additive white noise sources. Then, the acoustic and the flame system is interconnected, where thermoacoustic noncompactness due to the transversal modes' high frequency (HF) is accounted for by a distributed source term framework. The resulting nonlinear thermoacoustic system is solved in frequency and time domain. Linear growth rates predict linear stability, while envelope plots and probability density diagrams of the resulting pressure traces characterize the thermoacoustic performance of the combustor from a dynamical systems theory perspective. Comparisons against experimental data are conducted, which allow the rating of the flame modes in terms of their capability to reproduce the observed combustor dynamics. Ultimately, insight into the physics of high-frequency, transversal thermoacoustic systems is created.

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References

Figures

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Fig. 1

MIMO modeling concept with schematic of the thermoacoustic feedback loop for a representative compact subregion

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Fig. 2

Schematic of the experimental combustor

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Fig. 3

T1 pressure mode shape and probe locations

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Fig. 4

Mesh, excitation sources, and boundary conditions

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Fig. 5

Complex eigenfrequencies of closed loop for both operation points

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Fig. 6

Frequency responses of the unstable case's open loop systems: (a) amplitude and (b) phase

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Fig. 7

Stable case: temporal oscillations (indicated in gray boxes) of Fourier amplitudes. (a) Experiment—real g(t), (b) ROM—real g(t), (c) experiment—real f(t), and (d) ROM—real f(t).

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Fig. 8

Stable case: probability density functions. (a) Experiment and (b) ROM.

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Fig. 9

Stable case: spin ratio histograms. (a) Experiment and (b) ROM.

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Fig. 10

Unstable case: temporal oscillations (indicated in gray boxes) of Fourier amplitudes. (a) Experiment—real g(t), (b) ROM—real g(t), (c) experiment—real f(t), and (d) ROM—real f(t).

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Fig. 11

Unstable case: probability density distributions. (a) Experiment and (b) ROM.

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Fig. 12

Unstable case: spin ratio histograms. (a) Experiment and (b) ROM.

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Fig. 13

PDF of swapped growth rate simulations

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Fig. 14

Simulated envelope evolutions

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