The frequency response of a confined premixed swirled flame is explored experimentally through the use of describing functions that depend on both the forcing frequency and the forcing level. In these experiments, the flame is forced by a loudspeaker connected to the bottom of the burner in the fresh gas region or by a set of loudspeakers connected to the combustion chamber exhaust tube in the burnt gas region. The experimental setup is equipped with a hot-wire (HW) probe and a microphone, both of which located in front of each other below the swirler. The forcing level is varied between $|v′0|/v¯0=0.10$ and 0.72 RMS, where $v¯0$ and $v′0$ are, respectively, the mean and the fluctuating velocity at the HW probe. An additional microphone is placed on a water-cooled waveguide connected to the combustion chamber backplate. A photomultiplier equipped with an OH* filter is used to measure the heat release rate fluctuations. The describing functions between the photomultiplier signal and the different pressure and velocity reference signals are then analyzed in the case of upstream and downstream forcing. The describing function measured for a given reference signal is shown to vary depending on the type of forcing. The impedance of the injector at the HW location is also determined for both upstream and downstream forcing. For all describing functions investigated, it is found that their phase lags do not depend on the forcing level, whereas their gains strongly depend on $|v′0|/v¯0$ for certain frequency ranges. It is furthermore shown that the flame describing function (FDF) measured with respect to the HW signal can be retrieved from the specific impedance at the HW location and the describing function determined with respect to the signal of the microphone located in front of the HW. This relationship is not valid when the signal from the microphone located at the combustion chamber backplate is considered. It is then shown that a one-dimensional (1D) acoustic model allows to reproduce the describing function computed with respect to the microphone signal inside the injector from the microphone signal located at the combustion chamber backplate in the case of downstream forcing. This relation does not hold for upstream forcing because of the acoustic dissipation across the swirler which is much larger compared to downstream forcing for a given forcing level set at the HW location. This study sheds light on the differences between upstream and downstream acoustic forcing when measuring describing functions. It is also shown that the upstream and downstream forcing techniques are equivalent only if the reference signal used to determine the FDF is the acoustic velocity in the fresh gases just before the flame.

## References

References
1.
Keller
,
J. J.
,
1995
, “
Thermoacoustic Oscillations in Combustion Chambers of Gas Turbines
,”
AIAA J.
,
33
(
12
), pp.
2280
2287
.
2.
Dowling
,
A. P.
, and
Stow
,
S. R.
,
2003
, “
Acoustic Analysis of Gas Turbine Combustors
,”
J. Propul. Power
,
19
(
5
), pp.
751
764
.
3.
Sattelmayer
,
T.
, and
Polifke
,
W.
,
2003
, “
Assessment of Methods for the Computation of Linear Stability of Combustors
,”
Combust. Sci. Technol.
,
175
(
3
), pp.
453
476
.
4.
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
.
5.
Camporeale
,
S. M.
,
Fortunato
,
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.
011506
.
6.
Laera
,
D.
,
Schuller
,
T.
,
Prieur
,
K.
,
Durox
,
D.
, and
Camporeale
,
S. M.
,
2017
, “
Flame Describing Function Analysis of Spinning and Standing Modes in an Annular Combustor and Comparison With Experiments
,”
Combust. Flame
,
184
, pp.
136
152
.
7.
Candel
,
S.
,
2002
, “
Combustion Dynamics and Control: Progress and Challenges
,”
Proc. Combust. Inst.
,
29
(
1
), pp.
1
28
.
8.
Dowling
,
A. P.
,
1997
, “
Nonlinear Self-Excited Oscillations of a Ducted Flame
,”
J. Fluid Mech.
,
346
, pp.
271
290
.
9.
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
, p.
139
.
10.
Palies
,
P.
,
Durox
,
D.
,
Schuller
,
T.
, and
Candel
,
S.
,
2011
, “
Nonlinear Combustion Instability Analysis Based on the Flame Describing Function Applied to Turbulent Premixed Swirling Flames
,”
Combust. Flame
,
158
(
10
), pp.
1980
1991
.
11.
Ćosić
,
B.
,
Moeck
,
J. P.
, and
Paschereit
,
C. O.
,
2014
, “
Nonlinear Instability Analysis for Partially Premixed Swirl Flames
,”
Combust. Sci. Technol.
,
186
(
6
), pp.
713
736
.
12.
Boudy
,
F.
,
Durox
,
D.
,
Schuller
,
T.
,
Jomaas
,
G.
, and
Candel
,
S.
,
2011
, “
Describing Function Analysis of Limit Cycles in a Multiple Flame Combustor
,”
ASME J. Eng. Gas Turbines Power
,
133
(
6
), p.
061502
.
13.
Mirat
,
C.
,
Durox
,
D.
, and
Schuller
,
T.
,
2015
, “
Stability Analysis of a Swirl Spray Combustor Based on Flame Describing Function
,”
Proc. Combust. Inst.
,
35
(
3
), pp.
3291
3298
.
14.
Hochgreb
,
S.
,
Dennis
,
D.
,
Ayranci
,
I.
,
Bainbridge
,
W.
, and
Cant
,
S.
,
2013
, “
Forced and Self-Excited Instabilities From Lean Premixed, Liquid-Fuelled Aeroengine Injectors at High Pressures and Temperatures
,”
ASME
Paper No. GT2013-95311.
15.
Truffin
,
K.
, and
Poinsot
,
T.
,
2005
, “
Comparison and Extension of Methods for Acoustic Identification of Burners
,”
Combust. Flame
,
142
(
4
), pp.
388
400
.
16.
Gatti
,
M.
,
Gaudron
,
R.
,
Mirat
,
C.
, and
Schuller
,
T.
,
2017
, “
Effects of the Injector Design on the Transfer Function of Premixed Swirling Flames
,”
ASME
Paper No. GT2017-638.
17.
Gaudron
,
R.
,
Gatti
,
M.
,
Mirat
,
C.
, and
Schuller
,
T.
,
2017
, “
Analysis of the Transfer Function of Large and Small Premixed Laminar Conical Flames
,”
ASME
Paper No. GT2017-64231.
18.
Schuller
,
T.
,
Durox
,
D.
, and
Candel
,
S.
,
2003
, “
Self-Induced Combustion Oscillations of Laminar Premixed Flames Stabilized on Annular Burners
,”
Combust. Flame
,
135
(
4
), pp.
525
537
.
19.
Birbaud
,
A. L.
,
Durox
,
D.
,
Ducruix
,
S.
, and
Candel
,
S.
,
2007
, “
Dynamics of Free Jets Submitted to Upstream Acoustic Modulations
,”
Phys. Fluids
,
19
, p.
013602
.
20.
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
.
21.
Schuermans
,
B.
,
Paschereit
,
C. O.
, and
Monkewitz
,
P.
, “
Non-Linear Combustion Instabilities in Annular Gas-Turbine Combustors
,”
AIAA
Paper No. AIAA 2006-549.
22.
Noiray
,
N.
, and
Schuermans
,
B.
,
2013
, “
On the Dynamic Nature of Azimuthal Thermoacoustic Modes in Annular Gas Turbine Combustion Chambers
,”
Proc. R. Soc. A
,
469
, p. 20120535.
23.
Ghirardo
,
G.
,
Juniper
,
M. P.
, and
Moeck
,
J. P.
,
2015
, “
Stability Criteria for Standing and Spinning Waves in Annular Combustors
,”
ASME
Paper No. GT2015-43127.
24.
Hurle
,
I.
,
Price
,
R.
,
Sugden
,
T.
, and
Thomas
,
A.
,
1968
, “
Sound Emission From Open Turbulent Premixed Flames
,”
Proc. R. Soc. A
,
303
(
1475
), pp.
409
427
.
25.
Paschereit
,
C. O.
, and
Polifke
,
W.
,
1998
, “
Investigation of the Thermoacoustic Characteristics of a Lean Premixed Gas Turbine Burner
,”
ASME
Paper No. 98-GT-582.
26.
Fischer
,
A.
,
Hirsch
,
C.
, and
Sattelmayer
,
T.
,
2006
, “
Comparison of Multi-Microphone Transfer Matrix Measurements With Acoustic Network Models of Swirl Burners
,”
J. Sound Vib.
,
298
(
1–2
), pp.
73
83
.
27.
Rienstra
,
S.
, and
Hirschberg
,
A.
,
2016
,
An Introduction to Acoustics
,
Eindhoven University of Technology
,
Eindhoven, The Netherlands
.
28.
Lieuwen
,
T.
,
2005
,
,
Cambridge University Press
,
Cambridge, UK
.
29.
Howe
,
M. S.
,
1998
,
Acoustics of Fluid-Structure Interactions
,
Cambridge University Press
,
Cambridge, UK
.
30.
Palies
,
P.
,
Durox
,
D.
,
Schuller
,
T.
, and
Candel
,
S.
,
2011
, “
Experimental Study on Effects of Swirler Geometry and Swirl Number on Flame Describing Functions
,”
Combust. Sci. Technol.
,
183
(
7
), pp.
704
717
.
31.
Lighthill
,
M. J.
,
1952
, “
On Sound Generated Aerodynamically
,”
Proc. R. Soc. London A
,
211
(
1107
), pp.
564
587
.
32.
Howe
,
M. S.
,
1979
, “
On the Theory of Unsteady High Reynolds Number Flow Through a Circular Aperture
,”
Proc. R. Soc. London A
,
366
(
1725
), pp.
205
223
.
33.
Ni
,
F.
,
Miguel-Brebion
,
M.
,
Nicoud
,
F.
, and
Poinsot
,
T.
,
2017
, “
Accounting for Acoustic Damping in a Helmholtz Solver
,”
AIAA J.
,
55
(
4
), pp.
1205
1220
.