Nonlinear analysis of thermoacoustic instability is essential for the prediction of the frequencies, amplitudes, and stability of limit cycles. Limit cycles in thermoacoustic systems are reached when the energy input from driving processes and energy losses from damping processes balance each other over a cycle of the oscillation. In this paper, an integral relation for the rate of change of energy of a thermoacoustic system is derived. This relation is analogous to the well-known Rayleigh criterion in thermoacoustics, however, it can be used to calculate the amplitudes of limit cycles and their stability. The relation is applied to a thermoacoustic system of a ducted slot-stabilized 2-D premixed flame. The flame is modeled using a nonlinear kinematic model based on the G-equation, while the acoustics of planar waves in the tube are governed by linearized momentum and energy equations. Using open-loop forced simulations, the flame describing function (FDF) is calculated. The gain and phase information from the FDF is used with the integral relation to construct a cyclic integral rate of change of energy (CIRCE) diagram that indicates the amplitude and stability of limit cycles. This diagram is also used to identify the types of bifurcation the system exhibits and to find the minimum amplitude of excitation needed to reach a stable limit cycle from another linearly stable state for single-mode thermoacoustic systems. Furthermore, this diagram shows precisely how the choice of velocity model and the amplitude-dependence of the gain and the phase of the FDF influence the nonlinear dynamics of the system. Time domain simulations of the coupled thermoacoustic system are performed with a Galerkin discretization for acoustic pressure and velocity. Limit cycle calculations using a single mode, along with twenty modes, are compared against predictions from the CIRCE diagram. For the single mode system, the time domain calculations agree well with the frequency domain predictions. The heat release rate is highly nonlinear but, because there is only a single acoustic mode, this does not affect the limit cycle amplitude. For the twenty-mode system, however, the higher harmonics of the heat release rate and acoustic velocity interact, resulting in a larger limit cycle amplitude. Multimode simulations show that, in some situations, the contribution from higher harmonics to the nonlinear dynamics can be significant and must be considered for an accurate and comprehensive analysis of thermoacoustic systems.

References

References
1.
Lieuwen
,
T. C.
and
Yang
,
V.
,
2005
, “
Combustion Instabilities in Gas Turbine Engines
,”
Progress in Astronautics and Aeronautics
, Vol.
210
,
AIAA
,
Reston, Va
.
2.
Zinn
,
B. T.
and
Lores
,
M. E.
,
1972
, “
Application of the Galerkin Methods in the Solution of Nonlinear Axial Combustion Instability Problems in Liquid Rockets
,”
Combust. Sci. Technol.
,
4
, pp.
269
278
.10.1080/00102207108952493
3.
Culick
,
F. E. C.
,
1976
, “
Nonlinear Behaviour of Acoustic Waves in Combustion Chambers—Part 1
,”
Acta Astronaut.
,
3
, pp.
715
734
.10.1016/0094-5765(76)90107-7
4.
Culick
,
F. E. C.
,
1976
, “
Nonlinear Behaviour of Acoustic Waves in Combustion Chambers—Part 2
,”
Acta Astronaut.
,
3
, pp.
735
757
.10.1016/0094-5765(76)90108-9
5.
Yang
,
V.
and
Culick
,
F. E. C.
,
1990
, “
On the Existence and Stability of Limit Cycles for Transverse Acoustic Oscillations in a Cylindrical Combustion Chamber. I: Standing Modes
,”
Combust. Sci. Technol.
,
72
(
1
), pp.
37
65
.10.1080/00102209008951639
6.
Yang
,
V.
,
Kim
,
S. I.
, and
Culick
,
F. E. C.
,
1990
, “
Triggering of Longitudinal Pressure Oscillations in Combustion Chambers. I: Nonlinear Gas Dynamics
,”
Combust. Sci. Technol.
,
72
(
4
), pp.
183
214
.10.1080/00102209008951647
7.
Baum
,
J. D.
,
Levine
,
J. N.
, and
Lovine
,
R. L.
,
1988
, “
Pulsed Instability in Rocket Motors: A Comparison Between Predictions and Experiment
,”
J. Propul. Power
,
4
(
4
), pp.
308
316
.10.2514/3.23068
8.
Wicker
,
J. M.
,
Greene
,
W. D.
,
Kim
,
S-I.
, and
Yang
,
V.
,
1996
, “
Triggering of Longitudinal Pressure Oscillations in Combustion Chambers. I: Nonlinear Combustion Response
,”
J. Propul. Power
,
12
(
6
), pp.
1148
1158
.10.2514/3.24155
9.
Ananthkrishnan
,
N.
,
Deo
,
S.
, and
Culick
,
F. E. C.
,
2005
, “
Reduced-Order Modeling and Dynamics of Nonlinear Acoustic Waves in a Combustion Chamber
,”
Combust. Sci. Technol.
,
177
(
2
), pp.
221
248
.10.1080/00102200590900219
10.
Poinsot
,
T.
, and
Candel
,
S.
,
1988
, “
A Nonlinear Model for Ducted Flame Combustion Instabilities
,”
Combust. Sci. Technol.
,
61
, pp.
121
153
.10.1080/00102208808915760
11.
Dowling
,
A. P.
,
1997
, “
Nonlinear Self-Excited Oscillations of a Ducted Flame
,”
J. Fluid Mech.
,
346
, pp.
271
290
.10.1017/S0022112097006484
12.
Dowling
,
A. P.
,
1999
, “
A Kinematic Model of a Ducted Flame
,”
J. Fluid Mech.
,
394
, pp.
51
72
.10.1017/S0022112099005686
13.
Stow
,
S. R.
, and
Dowling
,
A. P.
,
2004
, “
Low-Order Modelling of Thermoacoustic Limit Cycles
,” ASME Turbo Expo, Vienna, Austria, June 14–17,
ASME
Paper No. GT2004-54245.10.1115/GT2004-54245
14.
Stow
,
S. R.
, and
Dowling
,
A. P.
,
2008
, “
A Time-Domain Network Model for Nonlinear Thermoacoustic Oscillations
,” ASME Turbo Expo, Berlin, June 9–13,
ASME
Paper No. GT2008-50770.10.1115/GT2008-50770
15.
Lieuwen
,
T.
,
2005
, “
Nonlinear Kinematic Response of Premixed Flames to Harmonic Velocity Disturbances
,”
Proc. Combust. Inst.
,
29
, pp.
99
105
.10.1016/S1540-7489(02)80017-7
16.
Noiray
,
N.
,
Durox
,
D.
,
Schuller
,
T.
, and
Candel
,
S. M.
,
2008
, “
A Unified Framework for Nonlinear Combustion Instability Analysis Based on the Flame Describing Function
,”
J. Fluid Mech.
,
615
, pp.
139
167
.10.1017/S0022112008003613
17.
Moeck
,
J. P.
,
Bothien
,
M. R.
,
Schimek
,
S.
,
Lacarelle
,
A.
, and
Paschereit
,
C. O.
,
2008
, “
Subcritical Thermoacoustic Instabilities in a Premixed Combustor
,”
14th AIAA/CEAS Aeroacoustics Conference
, Vancouver, Canada, May 5–7,
AIAA
Paper No. 2008-2946.10.2514/6.2008-2946
18.
Subramanian
,
P.
,
Gupta
,
V.
,
Tulsyan
,
B.
, and
Sujith
,
R. I.
,
2010
, “
Can Describing Function Technique Predict Bifurcations in Thermoacoustic Systems?
,”
16th AIAA/CEAS Aeroacoustics Conference
, Stockholm, Sweden, June 7–9,
AIAA
Paper No. 2010-3860.10.2514/6.2010-3860
19.
Matveev
,
I.
,
2003
, “
Thermo-Acoustic Instabilities in the Rijke Tube: Experiments and Modeling
,” Ph.D. thesis,
CalTech
,
Pasadena, CA
.
20.
Juniper
,
M. P.
,
2011
, “
Triggering in the Horizontal Rijke Tube: Non-Normality, Transient Growth and Bypass Transition
,”
J. Fluid Mech.
,
667
, pp.
272
308
.10.1017/S0022112010004453
21.
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
, pp.
21
34
.10.1016/S0010-2180(03)00042-7
22.
Preetham
,
Santosh
,
H.
, and
Lieuwen
,
T.
,
2008
, “
Dynamics of Premixed Flames Forced by Harmonic Velocity Disturbances
,”
J. Propul. Power
,
24
(
6
), pp.
1390
1402
.10.2514/1.35432
23.
Shreekrishna
,
Hemchandra
,
S.
, and
Lieuwen
,
T.
,
2010
, “
Premixed Flame Response to Equivalence Ratio Perturbations
,”
Combust. Theory Model.
,
14
, pp.
681
714
.10.1080/13647830.2010.502247
24.
Abu-Orf
,
G.
, and
Cant
,
R. S.
,
1996
, “
Reaction Rate Modelling for Premixed Turbulent Methane-Air Flames
,”
Proceedings of the Joint Meeting of Spanish
,
Portuguese
, Swedish and British Sections of the Combustion Institute, Madeira, Portugal, April 1–4.
25.
Fleifil
,
M.
,
Annaswamy
,
A.
,
Ghoneim
,
Z.
, and
Ghoneim
,
A.
,
1996
, “
Response of a Laminar Premixed Flame to Flow Oscillations: A Kinematic Model and Thermoacoustic Instability Results
,”
Combust. Flame
,
106
, pp.
487
510
.10.1016/0010-2180(96)00049-1
26.
Ducruix
,
S.
,
Durox
,
D.
, and
Candel
,
S.
,
2000
, “
Theoretical and Experimental Determination of the Transfer Function of a Laminar Premixed Flame
,”
Proc. Combust. Inst.
,
28
, pp.
765
773
.10.1016/S0082-0784(00)80279-9
27.
Baillot
,
F.
,
Durox
,
D.
, and
Prud'homme
,
R.
,
1992
, “
Experimental and Theoretical Study of a Premixed Vibrating Flame
,”
Combust. Flame
,
88
(
2
), pp.
149
168
.10.1016/0010-2180(92)90049-U
28.
Baillot
,
F.
,
Bourehla
,
A.
, and
Durox
,
D.
,
1996
, “
The Characteristics Method and Cusped Flame Fronts
,”
Combust. Sci. Technol.
,
112
(
1
), pp.
327
350
.10.1080/00102209608951963
29.
Cuquel
,
A.
,
Durox
,
D.
, and
Schuller
,
T.
,
2011
, “
Theoretical and Experimental Determination of the Flame Transfer Function of Confined Premixed Conical Flames
,”
7th Mediterranean Combustion Symposium
,
Cagliari, Sardinia, Italy
, September 11–15.
30.
Michalke
,
A.
,
1971
, “
Instability of a Compressible Circular Free Jet With Consideration of the Influence of the Jet Boundary Thickness
,” NASA Report No. TM 75190.
31.
Jiang
,
G.-S.
, and
Peng
,
D.
,
2000
, “
Weighted ENO Schemes for Hamilton-Jacobi Equations
,”
SIAM J. Sci. Comput. (USA)
,
6
, pp.
2126
2143
.10.1137/S106482759732455X
32.
Gottlieb
,
S.
, and
Shu
,
C.
,
1998
, “
Total Variation Diminishing Runge-Kutta Schemes
,”
Math. Comput.
,
67
, pp.
73
85
.10.1090/S0025-5718-98-00913-2
33.
Peng
,
D.
,
Merriman
,
B.
,
Osher
,
S.
,
Zhao
,
H.
, and
Kang
,
M.
,
1999
, “
A PDE-Based Fast Local Level Set Method
,”
J. Comput. Phys.
,
155
(
2
), pp.
410
438
.10.1006/jcph.1999.6345
34.
Hemchandra
,
S.
,
2009
, “
Dynamics of Turbulent Premixed Flames in Acoustic Fields
,” Ph.D. thesis,
Georgia Institute of Technology
,
Atlanta, GA
.
35.
Karimi
,
N.
,
Brear
,
M. J.
,
Jin
,
S.-H.
, and
Monty
,
J. P.
,
2009
, “
Linear and Non-Linear Forced Response of a Conical, Ducted, Laminar Premixed Flame
,”
Combust. Flame
,
156
, pp.
2201
2212
.10.1016/j.combustflame.2009.06.027
36.
Smereka
,
P.
,
2006
, “
The Numerical Approximation of a Delta Function With Application to Level Set Methods
,”
J. Comput. Phys.
,
211
, pp.
77
90
.10.1016/j.jcp.2005.05.005
37.
Dowling
,
A. P.
, and
Morgans
,
A. S.
,
2005
, “
Feedback Control of Combustion Oscillations
,”
Ann. Rev. Fluid Mech.
,
37
, pp.
151
182
.10.1146/annurev.fluid.36.050802.122038
38.
Rayleigh
,
J. W. S.
,
1945
,
The Theory of Sound
, Vol.
II
,
Dover
,
New York
.
39.
Durox
,
D.
,
Schuller
,
T.
,
Noiray
,
N.
, and
Candel
,
S.
,
2009
, “
Experimental Analysis of Nonlinear Flame Transfer Functions for Different Flame Geometries
,”
Proc. Combust. Inst.
,
32
(
1
), pp.
1391
1398
.10.1016/j.proci.2008.06.204
40.
Polifke
,
W.
, and
Lawn
,
C.
,
2007
, “
On the Low-Frequency Limit of Flame Transfer Functions
,”
Combust. Flame
,
151
(
3
), pp.
437
451
.10.1016/j.combustflame.2007.07.005
You do not currently have access to this content.