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.

References

References
1.
Bauerheim
,
M.
,
Nicoud
,
F.
, and
Poinsot
,
T.
,
2016
, “
Progress in Analytical Methods to Predict and Control Azimuthal Combustion Instability Modes in Annular Chambers
,”
Phys. Fluids
,
28
(
2
), p.
021303
.
2.
Stow
,
S.
, and
Dowling
,
A.
,
2009
, “
A Time-Domain Network Model for Nonlinear Thermoacoustic Oscillations
,”
ASME J. Eng. Gas Turbines Power
,
131
(
3
), p.
031502
.
3.
Noiray
,
N.
,
Bothien
,
M.
, and
Schuermans
,
B.
,
2011
, “
Investigation of Azimuthal Staging Concepts in Annular Gas Turbines
,”
Combust. Theory Model.
,
15
(
5
), pp.
585
606
.
4.
Ghirardo
,
G.
,
Juniper
,
M.
, and
Moeck
,
J.
,
2016
, “
Weakly Nonlinear Analysis of Thermoacoustic Instabilities in Annular Combustors
,”
J. Fluid Mech.
,
805
, pp.
52
87
.
5.
Worth
,
N. A.
, and
Dawson
,
J. R.
,
2013
, “
Self-Excited Circumferential Instabilities in a Model Annular Gas Turbine Combustor: Global Flame Dynamics
,”
Proc. Combust. Inst.
,
34
(
2
), pp.
3127
3134
.
6.
Worth
,
N. A.
, and
Dawson
,
J. R.
,
2013
, “
Modal Dynamics of Self-Excited Azimuthal Instabilities in an Annular Combustion Chamber
,”
Combust. Flame
,
160
(
11
), pp.
2476
2489
.
7.
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
.
8.
Bourgouin
,
J.-F.
,
Durox
,
D.
,
Moeck
,
J.
,
Schuller
,
T.
, and
Candel
,
S.
,
2014
, “
A New Pattern of Instability Observed in an Annular Combustor: The Slanted Mode
,”
Proc. Combust. Inst.
,
35
(
3
), pp.
3237
3244
.
9.
Prieur
,
K.
,
Durox
,
D.
, and
Schuller
,
T.
,
2017
, “
A Hysteresis Phenomenon Leading to Spinning or Standing Azimuthal Instabilities in an Annular Combustor
,”
Combust. Flame
,
175
, pp.
283
291
.
10.
Mensah
,
G.
, and
Moeck
,
J.
,
2017
, “
Limit Cycles of Spinning Thermoacoustic Modes in Annular Combustors: A Bloch-Wave and Adjoint-Perturbation Approach
,”
ASME
Paper No. GT2017-64817.
11.
Laera
,
D.
,
Schuller
,
T.
,
Prieur
,
K.
,
Durox
,
D.
,
Camporeale
,
S. M.
, and
Candel
,
S.
,
2017
, “
Flame Describing Function Analysis of Spinning and Standing Modes in an Annular Combustor and Comparison With Experiments
,”
Combust. Flame
,
184
, pp.
136
152
.
12.
Bauerheim
,
M.
,
Parmentier
,
J.
,
Salas
,
P.
,
Nicoud
,
F.
, and
Poinsot
,
T.
,
2014
, “
An Analytical Model for Azimuthal Thermoacoustic Modes in an Annular Chamber Fed by an Annular Plenum
,”
Combust. Flame
,
161
(
5
), pp.
1374
1389
.
13.
Pikovsky
,
A.
,
Rosenblum
,
M.
, and
Kurths
,
J.
,
2003
,
Synchronization: A Universal Concept in Nonlinear Sciences
,
Cambridge University Press
, Cambridge, UK.
14.
Moeck
,
J.
, and
Paschereit
,
C.
,
2012
, “
Nonlinear Interactions of Multiple Linearly Unstable Thermoacoustic Modes
,”
Int. J. Spray Combust. Dyn.
,
4
(
1
), pp.
1
28
.
15.
Orchini
,
A.
, and
Juniper
,
M.
,
2016
, “
Flame Double Input Describing Function Analysis
,”
Combust. Flame
,
171
, pp.
87
102
.
16.
Schuermans
,
B.
,
2003
, “
Modeling and Control of Thermoacoustic Instabilities
,”
Ph.D. thesis
,
EPF Lausanne
,
Switzerland
.https://infoscience.epfl.ch/record/33275/files/EPFL_TH2800.pdf
17.
Morse
,
P.
, and
Feshbach
,
H.
,
1953
,
Methods of Theoretical Physics—Part I
,
McGraw-Hill
,
New York
.
18.
Bellucci
,
V.
,
Schuermans
,
B.
,
Nowak
,
D.
,
Flohr
,
P.
, and
Paschereit
,
C.
,
2005
, “
Thermoacoustic Modeling of a Gas Turbine Combustor Equipped With Acoustic Dampers
,”
ASME J. Turbomach.
,
127
(
2
), pp. 372–379.
19.
Moeck
,
J.
,
Paul
,
M.
, and
Paschereit
,
C.
,
2010
, “
Thermoacoustic Instabilities in an Annular Rijke Tube
,”
ASME
Paper No. GT2010-23577.
20.
Dowling
,
A.
,
1997
, “
Nonlinear Self-Excited Oscillations of a Ducted Flame
,”
J. Fluid Mech.
,
346
, pp.
271
290
.
21.
Noiray
,
N.
,
Durox
,
D.
,
Schuller
,
T.
, and
Candel
,
S.
,
2008
, “
A Unified Framework for Nonlinear Combustion Instability Analysis Based on the Flame Describing Function
,”
J. Fluid Mech.
,
615
, pp.
139
167
.
22.
Orchini
,
A.
,
Illingworth
,
S.
, and
Juniper
,
M.
,
2015
, “
Frequency Domain and Time Domain Analysis of Thermoacoustic Oscillations With Wave-Based Acoustics
,”
J. Fluid Mech.
,
775
, pp.
387
414
.
23.
Nicoud
,
F.
,
Benoit
,
L.
,
Sensiau
,
C.
, and
Poinsot
,
T.
,
2007
, “
Acoustic Modes in Combustors With Complex Impedances and Multidimensional Active Flames
,”
AIAA J.
,
45
(
2
), pp.
426
441
.
24.
Gray
,
R.
,
2006
, “
Toeplitz and Circulant Matrices: A Review
,”
Foundations Trends Commun. Inf. Theory
,
2
(
3
), pp.
155
239
.
25.
Moeck
,
J.
,
Durox
,
D.
,
Schuller
,
T.
, and
Candel
,
S.
,
2018
, “
Nonlinear Thermoacoustic Mode Synchronization in Annular Combustors
,” Proc. Combust. Inst. (in press).
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