This article revisits the problem of indirect combustion noise in nozzles of finite length. The analytical model proposed by Moase for indirect combustion noise is rederived and applied to subcritical nozzles having shapes of increasing complexity. This model is based on the equations formulated by Marble and Candel for which an explicit solution is obtained in the subsonic framework. The discretization of the nozzle into n elementary units of finite length implies the determination of 2n integration constants for which a set of linear equations is provided in this article. The analytical method is applied to configurations of increasing complexity. Analytical solutions are compared to numerical results obtained using SUNDAY (a 1D nonlinear Euler solver in temporal space) and CEDRE (3D Navier–Stokes flow solver). Excellent agreement is found for all configurations thereby showing that acceleration discontinuities at the boundaries between adjacent elements do not influence the actual acoustic transfer functions. The issue of nozzle compactness is addressed. It is found that in the subcritical domain, spectral results should be nondimensionalized using the flow-through-time of the entire nozzle. Doing so, transfer functions of nozzles of different lengths are successfully compared and a compactness criterion is proposed that writes ω*0Ldζ/u(ζ)<1 where L is the axial length of the nozzle. Finally, the entropy wave generator (EWG) experimental setup is considered. Analytical results are compared to the results reported by Howe. Both models give similar trends and show the important role of the rising time of the fluctuating temperature front on the amplitude of the indirect acoustic emission. The experimental temperature profile and the impedance coefficients at the inlet and outlet are introduced into the analytical formulation. Results show that the indirect combustion noise mechanism is not alone responsible for the acoustic emission in the subcritical case.

References

1.
Candel
,
S. M.
, 1972, “Analytical Studies of Some Acoustic Problems of Jet Engines,” PhD thesis, California Institute of Technology, Pasadena, CA.
2.
Marble
,
F. E.
, 1973,, “
Acoustic Disturbance From Gas Non-Uniformities Convecting Through a Nozzle
,”
Symposium on Transportation Noise
,
Stanford University
,
Stanford, CA
.
3.
Morfey
,
C. L.
, 1973, “
Amplification of Aerodynamic Noise by Convected Flow Inhomogeneities
,”
J. Sound Vib.
,
31
, pp.
391
397
.
4.
Marble
,
F. E.
, and
Candel
,
S. M.
, 1977, “
Acoustic Disturbance From Gas Non-Uniformities Convecting Through a Nozzle
,”
J. Sound Vib.
,
55
(
2
), pp.
225
243
.
5.
Cumpsty
,
N. A.
, and
Marble
,
F. E.
, 1977, “
Core Noise From Gas Turbine Exhausts
,”
J. Sound Vib.
,
54
, pp.
297
309
.
6.
Poinsot
,
T.
, and
Veynante
,
D.
, 2012.
Theoretical and Numerical Combustion
, 3rd ed.
R. T.
Edwards
,
Flourtown
,
PA
.
7.
Culick
,
F. C. E.
, 2006, “
Unsteady Motions in Combustion Chambers for Propulsion Systems
,”
North Atlantic Treaty Organization,
Paper No. AG-AVT-039, available at: http://ftp.rta.nato.int/public//PubFullText/RTO/AG/RTO-AG-AVT-039///$$AG-AVT-039-TOC.pdf
8.
Blacodon
,
D.
, 2009, “
Combustion-Noise Characterization of a Turbofan Engine With Spectral Estimation Method
,”
J. Propul. Power
,
25
(
2
), pp.
374
379
.
9.
Howe
,
M. S.
, 2010, “
Indirect Combustion Noise
,”
J. Fluid Mech.
,
659
, pp.
267
288
.
10.
Bake
,
F.
,
Richter
,
C.
,
Mühlbauer
,
B.
,
Kings
,
N.
,
Rohle
,
I.
,
Thiele
,
F.
, and
Noll
,
B.
, 2009, “
The Entropy Wave Generator (EWG): A Reference Case on Entropy Noise
,”
J. Sound Vib.
,
326
, pp.
574
598
.
11.
Mühlbauer
,
B.
,
Noll
,
B.
, and
Aigner
,
M.
, 2009, “
Numerical Investigation of the Fundamental Mechanism of Entropy Noise Generation in Aero-Engines
,”
Acta Acust. United Ac.
,
95
(
3
), pp.
470
478
.
12.
Leyko
,
M.
,
Nicoud
,
F.
, and
Poinsot
,
T.
, 2009, “
Numerical and Analytical Investigation of the Indirect Noise in a Nozzle
,”
C. R. Mécanique
,
337
, pp.
415
425
.
13.
Goh
,
C. S.
, and
Morgans
,
A. S.
, 2011, “
Phase Prediction of the Response of Choked Nozzles to Entropy and Acoustic Disturbances
,”
J. Sound Vib.
,
330
(
21
), pp.
5184
5198
.
14.
Moase
,
W.
,
Brear
,
M. J.
, and
Manzie
,
C.
, 2007, “
The Forced Response of Choked Nozzles and Supersonic Diffusers
,”
J. Fluid Mech.
,
585
, pp.
281
304
.
15.
Abramowitz
,
M.
, and
Stegun
,
I.
, 1967.
Handbook of Mathematical Functions
,
National Bureau of Standards
,
Washington, DC
.
16.
Bogey
,
C.
, and
Bailly
,
C.
, 2004, “
A Family of Low Dispersive and Low Dissipative Explicit Schemes for Flow and Noise Computations
,”
J. Comput. Phys.
,
194
(
1
), pp.
194
214
.
17.
Cacqueray
,
N. D.
, 2010, “Méthodes numériques pour les écoulements supersoniques avec application au calcul du bruit rayonné par un jet sur-détendu,” PhD Thesis, École Centrale de Lyon, Lyon, France.
18.
Thompson
,
K. W.
, 1987, “
Time Dependent Boundary Conditions for Hyperbolic Systems
,”
J. Comput. Phys.
,
68
(
1
), pp.
1
24
.
19.
Poinsot
,
T. J.
, and
Lele
,
S. K.
, 1992, “
Boundary Conditions for Direct Simulations of Compressible Viscous Flows
,”
J. Comput. Phys.
,
101
(
1
), pp.
104
129
.
20.
Kaufmann
,
A.
,
Nicoud
,
F.
, and
Poinsot
,
T.
, 2002, “
Flow Forcing Techniques for Numerical Simulation of Combustion Instabilities
,”
Combust. Flame
,
131
(
4
), pp.
371
385
.
21.
Sigman
,
R. K.
, and
Zinn
,
B. T.
, 1983, “
A Finite Element Approach for Predicting Nozzle Admittances
,”
J. Sound Vib.
,
88
(
1
), pp.
117
131
.
22.
Vuillot
,
F.
,
Scherrer
,
D.
, and
Habiballah
,
M.
, 2003, “
CFD Code Validation for Space Propulsion Applications
,” Fifth International Conference on Liquid Space Propulsion, Chattanooga, TN, October 28–30.
23.
Durán
,
I.
, and
Moreau
,
S.
, 2011, “
Analytical and Numerical Study of the Entropy Wave Generator Experiment on Indirect Combustion Noise
,” 17th American Institute of Aeronautics and Astronautics/Council of European Aerospace Societies (AIAA/CEAS) Aeroacoustics Conference, Portland, OR, June 5–8, Paper No. AIAA-2011-2829.
24.
Leyko
,
M.
,
Moreau
,
S.
,
Nicoud
,
F.
, and
Poinsot
,
T.
, 2011, “
Numerical and Analytical Modeling of Entropy Noise in a Supersonic Nozzle With a Shock
,”
J. Sound Vib.
,
330
(
16
), pp.
3944
3958
.
25.
Holman
,
J. P.
, 1986,
Heat Transfer
,
McGraw-Hill Book Company
, New York.
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