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

# Comparison of Linear Stability Analysis With Experiments by Actively Tuning the Acoustic Boundary Conditions of a Premixed Combustor

[+] Author and Article Information
Mirko R. Bothien1

Institut für Strömungsmechanik und Technische Akustik  Technische Universität Berlin, 10623 Berlin, Germanymirko.bothien@tu-berlin.de

Jonas P. Moeck, Christian Oliver Paschereit

Institut für Strömungsmechanik und Technische Akustik  Technische Universität Berlin, 10623 Berlin, Germany

The time dependence for a certain mode reads $eiωeigt=e−ωgrt(cos(ωosct)+i sin(ωosct))$, where the prefactor determines whether the harmonic wave is attenuated or amplified.

1

Corresponding author.

J. Eng. Gas Turbines Power 132(12), 121502 (Aug 20, 2010) (10 pages) doi:10.1115/1.4000806 History: Received May 06, 2009; Revised July 22, 2009; Published August 20, 2010; Online August 20, 2010

## Abstract

Linear stability analysis by means of low-order network models is widely spread in industry and academia to predict the thermoacoustic characteristics of combustion systems. Even though a vast amount of publications on this topic exist, much less is reported on the predictive capabilities of such stability analyses with respect to real system behavior. In this sense, little effort has been made on investigating if predicted critical parameter values, for which the combustion system switches from stability to instability, agree with experimental observations. Here, this lack of a comprehensive experimental validation is addressed by using a model-based control scheme. This scheme is able to actively manipulate the acoustic field of a combustion test rig by imposing quasi-arbitrary reflection coefficients. It is employed to continuously vary the downstream reflection coefficient of an atmospheric swirl-stabilized combustion test rig from fully reflecting to anechoic. By doing so, the transient behavior of the system can be studied. In addition to that, an extension of the common procedure, where the stability of an operating point is classified solely based on the presence of high amplitude pressure pulsations and their frequency, is given. Generally, the predicted growth rates are only compared with measurements with respect to their sign, which obviously lacks a quantitative component. In contrast to that, in this paper, validation of linear stability analysis is conducted by comparing calculated and experimentally determined linear growth rates of unstable modes. Besides this, experimental results and model predictions are also compared in terms of frequency of the least stable mode. Excellent agreement between computations from the model and experiments is found. The concept is also used for active control of combustion instabilities. By tuning the downstream reflectivity of the combustion test rig, thermoacoustic instabilities can be suppressed. The underlying mechanism is an increase in the acoustic energy losses across the system boundary.

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## Figures

Figure 3

Schematical setup of the control concept for manipulation of the acoustic boundary condition of a duct

Figure 1

Schematic setup of the atmospheric test rig

Figure 2

Representation of the test rig as a network model consisting of up- and downstream reflection coefficients

Figure 4

Sketch of an actuated end element for derivation of the control law K

Figure 5

Magnitude (top) and phase (bottom) of downstream reflection coefficient. Solid: baseline case without control; dashed: controlled case for a control gain of Kg=0.4.

Figure 6

Spectra of acoustic pressure. Solid: baseline case without control; dashed: controlled case for a control gain of Kg=0.4.

Figure 7

Magnitude (top) and phase (bottom) of reflection coefficient for different control gains. Solid ◻: Kg=0 (w/o control), dotted ×: Kg=0.15, dashed-dotted ◇: Kg=0.325, and dashed ○: Kg=0.4.

Figure 17

Growth rates −I{ωeig} in s−1 determined from experiments (gray ○) and predicted by the model (black ×) as a function of the downstream reflection coefficient |Rcl| at the instability frequency

Figure 8

Spectra of acoustic pressure for different control gains. Solid ◻: Kg=0 (w/o control), dotted ×: Kg=0.15, dashed-dotted ◇: Kg=0.325, and dashed ○: Kg=0.4.

Figure 9

Spectral peak amplitude of acoustic pressure for increasing (black ×) and decreasing (gray ○) the control gain Kg

Figure 10

Spectral peak amplitude of OH∗-fluctuation for increasing (black ×) and decreasing (gray ○) the control gain Kg

Figure 11

|Rcl| at the frequency f(p̂max) at which the acoustic pressure is maximal versus the control gain Kg

Figure 12

Growth rate −I{ωeig} (top) and frequency (bottom) of the least stable mode as a function of |Rcl|

Figure 13

Probability density functions of acoustic pressure for different |Rcl| specified in Fig. 1

Figure 14

Time traces of acoustic pressure (black) and trigger signal (gray) during transition from controlled to uncontrolled state

Figure 15

Time traces of acoustic pressure (black dotted) during transition from controlled to uncontrolled state and identified exponential wave (gray)

Figure 16

Histogram of linear growth rates −I{ωeig} from controlled to uncontrolled state calculated with Eq. 7

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