Abstract
Gas turbine combustors are prone to undesirable combustion dynamics in the form of thermoacoustic oscillations. Analysis of the stability of thermoacoustic systems in the frequency domain leads to nonlinear eigenvalue problems (NLEVP); here, “nonlinear” refers to the fact that the eigenvalue, the complex oscillation frequency, appears in a nonlinear fashion. In this paper, we employ a noniterative strategy based on contour integration in the complex eigenvalue plane, which returns all eigenvalues inside the contour. An introduction to the technique is given, and is complemented with guidelines for the specific application to thermoacoustic problems. Two prototypical nonlinear eigenvalue problems are considered: a network model of the classical Rijke tube with an analytic flame response model and a finite element discretization of an annular model combustor with an experimental flame transfer function (FTF). Computation of all eigenvalues in a domain of interest is vital to assess stability of these systems. We demonstrate that this is generally challenging for iterative strategies. An eigenvalue solver based on contour integration, in contrast, provides a reliable, noniterative method to achieve this goal.
Issue Section:
Research Papers
References
1.
Schuller
,
T.
,
Durox
,
D.
, and
Candel
,
S.
, 2003
, “
A Unified Model for the Prediction of Laminar Flame Transfer Functions: Comparisons Between Conical and V-Flame Dynamics
,” Combust. Flame
,
134
(1–2
), pp. 21
–34
.10.1016/S0010-2180(03)00042-72.
Lieuwen
,
T.
, 2003
, “
Modeling Premixed Combustion–Acoustic Wave Interactions: A Review
,” J. Propul. Power
,
19
(5
), pp. 765
–781
.10.2514/2.61933.
Lieuwen
,
T. C.
, and
Yang
,
V.
, eds., 2005
, Combustion Instabilities in Gas Turbine Engines—Operational Experience, Fundamental Mechanisms, and Modeling
, Vol.
210
,
AIAA
, Reston, VA
.4.
Staffelbach
,
G.
,
Gicquel
,
L. Y. M.
,
Boudier
,
G.
, and
Poinsot
,
T.
, 2009
, “
Large Eddy Simulation of Self-Excited Azimuthal Modes in Annular Combustors
,” Proc. Combust. Inst.
,
32
(2
), pp. 2909
–2916
.10.1016/j.proci.2008.05.0335.
Wolf
,
P.
,
Staffelbach
,
G.
,
Gicquel
,
L.
,
Müller
,
D.
, and
Poinsot
,
T.
, 2012
, “
Acoustic and Large Eddy Simulation Studies of Azimuthal Modes in Annular Combustion Chambers
,” Combust. Flame
,
159
(11
), pp. 3398
–3413
.10.1016/j.combustflame.2012.06.0166.
Poinsot
,
T.
, 2013
, “
Simulation Methodologies and Open Questions for Acoustic Combustion Instability Studies
,” CTR Annual Research Briefs
, Stanford, CA
, pp. 179
–188
.7.
Evesque
,
S.
, and
Polifke
,
W.
, 2002
, “
Low-Order Acoustic Modelling for Annular Combustors: Validation and Inclusion of Modal Coupling
,” ASME Paper No. GT-2002-30064
. 10.1115/GT-2002-300648.
Schuermans
,
B.
,
Bellucci
,
V.
, and
Paschereit
,
C. O.
, 2003
, “
Thermoacoustic Modeling and Control of Multi Burner Combustion Systems
,” ASME Paper No. GT-2003-38688
. 10.1115/GT-2003-386889.
Sattelmayer
,
T.
, and
Polifke
,
W.
, 2003
, “
Assessment of Methods for the Computation of the Linear Stability of Combustors
,” Combust. Sci. Technol.
,
175
(3
), pp. 453
–476
.10.1080/0010220030238210.
Mehrmann
,
V.
, and
Voss
,
H.
, 2014
, “
Nonlinear Eigenvalue Problems: A Challenge for Modern Eigenvalue Methods
,” GAMM Mitt. Ges. Angew. Math. Mech.
,
27
(2
), pp. 121
–152
. 11.
Güttel
,
S.
, and
Tisseur
,
F.
, 2017
, “
The Nonlinear Eigenvalue Problem
,” Acta Numer.
,
26
, pp. 1
–94
.10.1017/S096249291700003412.
Schuermans
,
B.
, 2003
, “
Modeling and Control of Thermoacoustic Instabilities
,” Doctoral thesis, EPFL, Lausanne, Switzerland.13.
Emmert
,
T.
,
Meindl
,
M.
,
Jaensch
,
S.
, and
Polifke
,
W.
, 2016
, “
Linear State Space Interconnect Modeling of Acoustic Systems
,” Acta Acust. Acust.
,
102
(5
), pp. 824
–833
.10.3813/AAA.91899714.
Bothien
,
M. R.
,
Moeck
,
J. P.
,
Lacarelle
,
A.
, and
Paschereit
,
C. O.
, 2007
, “
Time Domain Modelling and Stability Analysis of Complex Thermoacoustic Systems
,” Proc. Inst. Mech. Eng., Part A
,
221
(5
), pp. 657
–668
.10.1243/09576509JPE38415.
Kopitz
,
J.
, and
Polifke
,
W.
, 2008
, “
CFD-Based Application of the Nyquist Criterion to Thermo-Acoustic Instabilities
,” J. Comput. Phys.
,
227
(14
), pp. 6754
–6778
.10.1016/j.jcp.2008.03.02216.
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
.10.2514/1.2493317.
Camporeale
,
S. M.
,
Fortunator
,
B.
, and
Campa
,
G.
, 2011
, “
A Finite Element Method for Three-Dimensional Analysis of Thermo-Acoustic Combustion Instability
,” ASME J. Eng. Gas Turbines Power
,
133
(1
), p. 015506
. 10.1115/1.400060618.
Mensah
,
G. A.
,
Campa
,
G.
, and
Moeck
,
J. P.
, 2016
, “
Efficient Computation of Thermoacoustic Modes in Industrial Annular Combustion Chambers Based on Bloch Wave Theory
,” ASME J. Eng. Gas Turbines Power
,
138
(8
), p. 081502
.10.1115/1.403233519.
Stow
,
S. R.
, and
Dowling
,
A. P.
, 2001
, “
Thermoacoustic Oscillations in an Annular Combustor
,” ASME Paper No. 2001-GT-0037
. 20.
Hoeijmakers
,
M.
,
Kornilov
,
V.
,
Arteaga
,
I. L.
,
de Goey
,
P.
, and
Nijmeijer
,
H.
, 2014
, “
Intrinsic Instability of Flame–Acoustic Coupling
,” Combust. Flame
,
161
(11
), pp. 2860
–2867
.10.1016/j.combustflame.2014.05.00921.
Bomberg
,
S.
,
Emmert
,
T.
, and
Polifke
,
W.
, 2015
, “
Thermal Versus Acoustic Response of Velocity Sensitive Premixed Flames
,” Proc. Combust. Inst.
,
35
(3
), pp. 3185
–3192
.10.1016/j.proci.2014.07.03222.
Emmert
,
T.
,
Bomberg
,
S.
, and
Polifke
,
W.
, 2015
, “
Intrinsic Thermoacoustic Instability of Premixed Flames
,” Combust. Flame
,
162
(1
), pp. 75
–85
.10.1016/j.combustflame.2014.06.00823.
Dellnitz
,
M.
,
Schütze
,
O.
, and
Zheng
,
Q.
, 2002
, “
Locating All the Zeros of an Analytic Function in One Complex Variable
,” J. Comput. Appl. Math.
,
138
(2
), pp. 325
–333
.10.1016/S0377-0427(01)00371-524.
Brebion
,
M.
, 2017
, “
Joint Numerical and Experimental Study of Thermo-Acoustic Instabilities
,” Ph.D. thesis, Institut National Polytechnique de Toulouse, Université de Toulouse, Toulouse, France.25.
Beyn
,
W.-J.
, 2012
, “
An Integral Method for Solving Nonlinear Eigenvalue Problems
,” Linear Algebra Appl.
,
436
(10
), pp. 3839
–3863
.10.1016/j.laa.2011.03.03026.
Schulze
,
M.
,
Hummel
,
T.
,
Klarmann
,
N.
,
Berger
,
F.
,
Schuermans
,
B.
, and
Sattelmayer
,
T.
, 2016
, “
Linearized Euler Equations for the Prediction of Linear High-Frequency Stability in Gas Turbine Combustors
,” ASME J. Eng. Gas Turbines Power
,
139
(3
), p. 031510
.10.1115/1.403445327.
Gikadi
,
J.
, 2014
, “
Prediction of Acoustic Modes in Combustors Using Linearized Navier-Stokes Equations in Frequency Space
,” Ph.D. thesis, Technische Universität München, Munich, Germany.28.
Mensah
,
G. A.
,
Buschmann
,
P. E.
,
Orchini
,
A.
, and
Moeck
,
J. P.
, 2018
, “
Pyholtz
,” Elsevier
, Amsterdam, The Netherlands
, accessed Oct. 21, 2019, https://bitbucket.org/pyholtzdevelopers/public.git29.
Barel
,
M. V.
, and
Kravanja
,
P.
, 2016
, “
Nonlinear Eigenvalue Problems and Contour Integrals
,” J. Comput. Appl. Math.
,
292
, pp. 526
–540
.10.1016/j.cam.2015.07.01230.
Keldysh
,
M. V.
, 1971
, “
On the Completeness of the Eigenfunctions of Some Classes of Non-Self-adjoint Linear Operators
,” Russ. Math. Surv.
,
26
(4
), p. 15
.10.1070/RM1971v026n04ABEH00398531.
Prosperetti
,
A.
, 2011
, Advanced Mathematics for Applications
,
Cambridge University Press
, Cambridge, UK
.32.
Crocco
,
L.
, and
Cheng
,
S.-I.
, 1956
, Theory of Combustion Instability in Liquid Propellant Rocket Motors
,
Butterworths Scientific Publications
, Oxford, UK
.33.
McManus
,
K. R.
,
Poinsot
,
T.
, and
Candel
,
S.
, 1993
, “
A Review of Active Control of Combustion Instabilities
,” Prog. Energy Combust. Sci.
,
19
(1
), pp. 1
–29
.10.1016/0360-1285(93)90020-F34.
Mensah
,
G. A.
,
Magri
,
L.
,
Silva
,
C. F.
,
Buschmann
,
P. E.
, and
Moeck
,
J. P.
, 2018
, “
Exceptional Points in the Thermoacoustic Spectrum
,” J. Sound Vib.
,
433
, pp. 124
–128
.10.1016/j.jsv.2018.06.06935.
Flohr
,
P.
,
Eroglu
,
A.
,
Brunner
,
P.
, and
Hellat
,
J.
, 2009
, “
Combustor Design for Low Emissions and Long Lifetime Requirements
,” ASME Paper No. GT2009-59540
. 10.1115/GT2009-5954036.
Poinsot
,
T.
, and
Veynante
,
D.
, 2012
, Theoretical and Numerical Combustion
, 3rd ed.,
RT Edwards
.37.
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
.10.1063/1.494003938.
Poinsot
,
T.
, 2017
, “
Prediction and Control of Combustion Instabilities in Real Engines
,” Proc. Combust. Inst.
,
36
(1
), pp. 1
–28
.10.1016/j.proci.2016.05.00739.
Bourgouin
,
J.-F.
,
Durox
,
D.
,
Moeck
,
J.
,
Schuller
,
T.
, and
Candel
,
S.
, 2015
, “
A New Pattern of Instability Observed in an Annular Combustor: The Slanted Mode
,” Proc. Combust. Inst.
,
35
(3
), pp. 3237
–3244
.10.1016/j.proci.2014.06.02940.
Laera
,
D.
,
Prieur
,
K.
,
Durox
,
D.
,
Schuller
,
T.
,
Camporeale
,
S.
, and
Candel
,
S.
, 2017
, “
Impact of Heat Release Distribution on the Spinning Modes of an Annular Combustor With Multiple Matrix Burners
,” ASME J. Eng. Gas Turbines Power
,
139
(5
), p. 051505
.10.1115/1.403520741.
Orchini
,
A.
,
Mensah
,
G. A.
, and
Moeck
,
J. P.
, 2018
, “
Effects of Nonlinear Modal Interactions on the Thermoacoustic Stability of Annular Combustors
,” ASME J. Eng. Gas Turbines Power
,
141
(2
), p. 021002
.10.1115/1.404076842.
Laera
,
D.
,
Prieur
,
K.
,
Durox
,
D.
,
Schuller
,
T.
,
Camporeale
,
S.
, and
Candel
,
S.
, 2016
, “
Flame Describing Function Analysis of Standing Modes in an Annular Combustor and Comparison With Experiments
,” Thermoacoustic Instabilities in Gas Turbines and Rocket Engines: Industry Meets Academia
, Munich, Germany
, Paper No. GTRE-044.43.
Gustavsen
,
B.
, and
Semlyen
,
A.
, 1999
, “
Rational Approximation of Frequency Domain Responses by Vector Fitting
,” IEEE Trans. Power Delivery
,
14
(3
), pp. 1052
–1061
.10.1109/61.77235344.
Durox
,
D.
,
Bourgouin
,
J.-F.
,
Moeck
,
J. P.
,
Philip
,
M.
,
Schuller
,
T.
, and
Candel
,
S.
, 2013
, “
Nonlinear Interactions in Combustion Instabilities Coupled by Azimuthal Acoustic Modes
,” International Summer School and Workshop on Non-Normal and Nonlinear Effects in Aero- and Thermoacoustics
, Munich, Germany
, June 18–21, p. 14
.45.
Mensah
,
G. A.
,
Magri
,
L.
,
Orchini
,
A.
, and
Moeck
,
J. P.
, 2018
, “
Effects of Asymmetry on Thermoacoustic Modes in Annular Combustors: A Higher-Order Perturbation Study
,” ASME J. Eng. Gas Turbines Power
, 141
(4
), p. 041030
.10.1115/1.404100746.
Jones
,
E.
,
Oliphant
,
T.
, and
Peterson
,
P.
, 2001
–2019, “
SciPy: Open Source Scientific Tools for Python
,” accessed Oct. 21, 2019, http://www.scipy.org/47.
Oliphant
,
T.
, 2006
–2019, NumPy: A Guide to NumPy
,
Trelgol Publishing
.Copyright © 2020 by ASME
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