Research Papers

Thermoacoustics of Can-Annular Combustors

[+] Author and Article Information
G. Ghirardo

Ansaldo Energia Switzerland,
Römerstrasse 36,
Baden 5401, Switzerland
e-mail: giulio.ghirardo@ansaldoenergia.com

C. Di Giovine

Ansaldo Energia Switzerland,
Römerstrasse 36,
Baden 5401, Switzerland

J. 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
Trondheim 7491, Norway

M. R. Bothien

Ansaldo Energia Switzerland,
Römerstrasse 36,
Baden 5401, Switzerland

1Corresponding author.

Manuscript received June 22, 2018; final manuscript received June 28, 2018; published online September 14, 2018. Editor: Jerzy T. Sawicki.

J. Eng. Gas Turbines Power 141(1), 011007 (Sep 14, 2018) (10 pages) Paper No: GTP-18-1287; doi: 10.1115/1.4040743 History: Received June 22, 2018; Revised June 28, 2018

Can-annular combustors consist of a set of independent cans, connected on the upstream side to the combustor plenum and on the downstream side to the turbine inlet, where a transition duct links the round geometry of each can with the annular segment of the turbine inlet. Each transition duct is open on the sides toward the adjacent transition ducts, so that neighboring cans are acoustically connected through a so-called cross-talk open area. This theoretical, numerical, and experimental work discusses the effect that this communication has on the thermoacoustic frequencies of the combustor. We show how this communication gives rise to axial and azimuthal modes, and that these correspond to particularly synchronized states of axial thermoacoustic oscillations in each individual can. We show that these combustors typically show clusters of thermoacoustic modes with very close frequencies and that a slight loss of rotational symmetry, e.g., a different acoustic response of certain cans, can lead to mode localization. We corroborate the predictions of azimuthal modes, clusters of eigenmodes, and mode localization with experimental evidence.

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Fig. 1

Geometry of a typical transition duct and the simplifications leading to the two-dimensional (2D) model, appearing ultimately in Fig. 2(a). The communicating gap between two adjacent transition ducts is colored in gray, with an axial length Lgap. (a) three-dimensional (3D) sketch of a transition duct, (b) front view of the air volume, and (c) side view of the air volume.

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Fig. 2

(a) Geometry of the 2D model of the transition duct, simplified from the original geometry of Fig. 1. (b) Legend, common to the other four figures. ((c) and (d)) First nontrivial eigenmode of the studied geometry for an axial mode, i.e., m = 0 (λ = 2L). Absolute value of the acoustic pressure on the left and absolute value of the acoustic velocity on the right. The solutions of the axial case match the solutions of the problem with Neumann conditions applied over the whole boundary. ((e) and (f)) same of (c) and (d) but for a push–pull mode, i.e., m = N/2 = 7, for a set of N = 14 cans (λ = 4.26L). In (f), the acoustic velocity has a strong increase where the gap starts and is inhomogeneous over the gap length.

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Fig. 3

First eigenmode with azimuthal order m = 2, for a set of N = 14 communicating cans. Because the solution is complex-valued, we use different colormaps for the absolute value and the phase angle, presented in the two legends in (a). In (b), we present the solution in one can only: we observe that the phase is approximately constant on the cross section at the upstream end of the transition duct, on the left, far from the cross-talk area. The amplitude decreases slightly from left to right. In (c), we present the ensemble of all the 14 cans pulsating together for the mode in (b), reconstructed using Eq. (1). The phase changes mostly at the cross-talk area and changes two times (since the azimuthal order m is 2) the quantity 2π along the annulus. At the upstream end of each transition duct, the phase is approximately constant and the mode presents a certain azimuthal phase pattern. The phase between two cans is the difference of the phase value between two locations. Refer to Fig. 4(a) for experimental evidence of a second-order azimuthal mode like this one in an engine. (a) Legend for (b) and (c). (b) First azimuthal m = 2 mode in one can, pressure field. Legend in (a). (c) Phase in the 14 cans of the same mode of (a). The first can from the left is the same appearing in (b).

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Fig. 4

Experimental evidence of a second azimuthal and of a push–pull mode. The timeseries of 14 pressure sensors in a 14-can can-annular combustor are processed at the frequency of one thermoacoustic instability. We present the amplitude of pulsation at that frequency on the left axis (in arbitrary units) and the phase difference on the right vertical axis at this frequency in each can. The phase difference is between the first can and the nth can, so that the value is zero in the first can. (a) and (b) refer to a different frequency of oscillation and a different operating condition. (a) The phase pattern corresponds to an azimuthal mode of order m = 2 that is rotating, because the phase changes with an approximately constant slope twice the amount of 2π moving along the annulus. This matches the theoretical prediction of Fig. 3(c). (b) The phase difference between most adjacent cans is very close to ±π (m = N/2 = 7). We observe some variation of the amplitude between cans, characterized as mode localization in The Effect of Asymmetry section of the paper.

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Fig. 5

Phase of the equivalent reflection coefficient for the azimuthal modes for N = 14 cans, calculated with reference cross section at the red inlet of the transition duct of Fig. 2(a). The Helmholtz number is defined as He = /c.

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Fig. 6

Validation of the geometrical simplifications for the 2D model for N = 12. The dashed lines are obtained from a fixed complex 3D geometry, while the continuous lines are obtained with the bidimensional model proposed in this paper, with the nondimensional values estimated from the 3D geometry without applying further corrections. The good agreement validates the geometrical approximations made to map the complex 3D geometry to the simpler 2D model.

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Fig. 7

Sensitivity of the first two eigenfrequencies of a can system as a function of the geometry aspect ratio L/H, for all azimuthal wavenumber m, for N = 14. We plot the Helmholtz number of all the eigenmodes with He < 2π for each azimuthal mode m, as a function of the two nondimensional parameters governing can to can acoustic communication. As L/H increases, the Helmholtz number of the azimuthal modes converges slowly to π/2 and 3π/2 for the first two modes. For a fixed value of L/H, the eigenfrequencies for m = 5, 6, 7 are rather close, forming a cluster. These modes are closer together the larger is L/H.

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Fig. 8

Spectrogram of the timeseries of the azimuthal modes reconstructed from an engine. We can observe that more than one azimuthal mode is active in a small range of frequencies, forming a cluster. Within the cluster, each mode peaks at a frequency that increases with the azim. order m. The highest amplitude is not always in the same mode, and the system's energy appears to move randomly between the modes, with some statistical preference for the m = 3, 4, 5 modes.

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Fig. 9

(a) Sketch of the reflection/transmission transfer functions T(d) for an incoming traveling wave f. Each function T(d)(ω) is the transmitted Riemann invariant from one can to a can that is d cans apart. For example, T(0) is the reflection of a wave propagating downstream of one can and being reflected to the same can, and T(3) can be interpreted for example as the transmission of a wave traveling downstream can 4 and propagating upstream in can 7, or any other two cans that are 3 cans apart. ((b) and (c)) Gain and phase of the transfer functions T(d) between all cans.

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Fig. 10

Sketch of the nth can of the can-annular model. The can communicates with the others via the blue cross-talk area where the f, g waves are sketched.

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Fig. 11

Complex eigenvalues for the three cases considered. The eigenvalues have been normalized with the angular eigenfrequency of the quarter-wave mode of an isolated can. The vertical axis is the normalized growth rate of oscillation of each mode. The red numbers refer to the asymmetric case and describe the mth azimuthal order of the respective symmetric eigenmode. The amplitude and phase of each asymmetric mode is presented in Fig. 12 in the plot with the same red number in the top left corner.

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Fig. 12

Visualization of the azimuthal mode shapes associated with the eigenvalues of the asymmetric case (Fig. 11). Ordered from left to right and top to bottom with increasing oscillation frequency. The bar height indicates the pressure amplitude and the bar color the phase (same color legend as in Fig. 3), at the interface between each can and each transition duct. The number in the top left corner is the azimuthal order m of the corresponding mode in the symmetric case. Couple of modes with the same m differ very little in frequency and can result in a slowly modulated linear combination with approximately constant amplitude along the annulus. This is not the case for the perturbed push–pull mode in the bottom right corner, where cans 1, 2, 3, 10, 11, and 12 present much larger amplitude than the others.



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