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Research Papers

Effects of Nonlinear Modal Interactions on the Thermoacoustic Stability of Annular Combustors

[+] Author and Article Information
Alessandro Orchini

Institut für Strömungsmechanik und
Technische Akustik,
Technische Universität Berlin,
Berlin 10623, Germany
e-mail: a.orchini@campus.tu-berlin.de

Georg A. Mensah

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

Jonas P. Moeck

Institut für Strömungsmechanik und
Technische Akustik,
Technische Universität Berlin,
Berlin 10623, Germany;
Department of Energy and Process Engineering,
Norwegian University of
Science and Technology,
Trondheim 7491, Norway

1Corresponding author.

2Throughout this paper, the adjective (un)stable refers to the behavior of a particular solution of the thermoacoustic dynamical system in phase space in a Lyapunov sense. It can refer to two types of solutions: fixed points (zero amplitude) or limit cycles (finite amplitude). The shorthand “linearly (un)stable” will be used in place of “the fixed point solution is (un)stable.”

Manuscript received June 26, 2018; final manuscript received June 29, 2018; published online September 19, 2018. Editor: Jerzy T. Sawicki.

J. Eng. Gas Turbines Power 141(2), 021002 (Sep 19, 2018) (10 pages) Paper No: GTP-18-1356; doi: 10.1115/1.4040768 History: Received June 26, 2018; Revised June 29, 2018

In this theoretical and numerical analysis, a low-order network model is used to investigate a thermoacoustic system with discrete rotational symmetry. Its geometry resembles that of the MICCA combustor (Laboratoire EM2C, CentraleSupelec); the flame describing function (FDF) employed in the analysis is that of a single-burner configuration and is taken from experimental data reported in the literature. We show how most of the dynamical features observed in the MICCA experiment, including the so-called slanted mode, can be predicted within this framework, when the interaction between a longitudinal and an azimuthal thermoacoustic mode is considered. We show how these solutions relate to the symmetries contained in the equations that model the system. We also discuss how considering situations in which two modes are linearly unstable compromises the applicability of stability criteria available in the literature.

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References

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Bourgouin, J. , Durox, D. , Moeck, J. , Schuller, T. , and Candel, S. , 2014, “ Characterization and Modelling of a Spinning Thermoacoustic Instability in an Annular Combustor Equipped With Multiple Matrix Injectors,” ASME J. Eng. Gas Turbines Power, 137(2), p. 021503. [CrossRef]
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Figures

Grahic Jump Location
Fig. 1

Azimuthal view of the configuration under consideration, with N burners and featuring discrete rotational symmetry. The periodicity implies that index N + 1 is mapped into index 1.

Grahic Jump Location
Fig. 2

Single burner FDF gain for the MICCA experiment. Dots are values extracted from Ref. [7]. Extrapolation ensures that the gain drops to 0 at high amplitudes (A ≈ 1.5) and high frequencies, and remains 1 at zero frequency. The surface color indicates the FDF phase in rad.

Grahic Jump Location
Fig. 3

Acoustic modeshapes of the configuration under investigation. Color indicates the magnitude of the acoustic pressure. The mode structure reveals which parts of the combustor are most sensitive to a certain type of oscillations. When the FTF is included, thermoacoustic instabilities associated with the two lowest frequency modes are predicted.

Grahic Jump Location
Fig. 4

Evolution of growth rates and frequencies of the linearly unstable thermoacoustic modes (squares) when the oscillation amplitude (colours) is increased. The longitudinal (spinning) mode saturates to a limit cycle, σ = 0, at A ≈ 1.2 (A ≈ 0.75). The thermoacoustic modes converge to the corresponding acoustic modes (circles) at amplitudes A≳1.5, for which the gain of the FDF vanishes.

Grahic Jump Location
Fig. 5

Evolution of growth rates and frequencies of the spinning and standing mode structures when the amplitude is increased. For the standing modes, the amplitude refers to that at a velocity antinode. The spinning mode saturates faster because the amplitude increases uniformly at every burner, reducing the overall FDF gain.

Grahic Jump Location
Fig. 6

N2 stability criterion for the standing mode pattern. The real part of the pressure-based FDF evaluated at the positions of the burners and its interpolant are shown with red solid lines. The dashed line is sin(2θ). For this standing mode N2 < 0, meaning the limit cycle is unstable.

Grahic Jump Location
Fig. 7

Two-dimensional phase-spaces showing two possible scenarios for the amplitude evolution of a longitudinal (Alon) and an azimuthal (Aazi) mode, constrained to the FDF analysis. Stable (unstable) solutions of the dynamical system are represented with filled (empty) circles. Dashed red lines represent the system nullclines. Solid blue lines track some trajectories.

Grahic Jump Location
Fig. 8

Homotopy between a standing and a slanted mode, obtained solving Eq. (36) for ϵ ∈ [0, 1]. The slanted solution (yellow) is obtained via a small perturbation of the operator N around the standing solution (blue). These perturbations have a minimal effect on the system eigenfrequencies (see top-left inset), but a significant effect on the eigenvectors. The location of the 16 burners is highlighted in black to emphasize the shift in the nodal line (red-dashed).

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