PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Numerical investigation of indirect combustion noise mechanisms
in a nozzle
Thomas DECONINCK1; Antoine VALLON2; Alexis APPEL DE GARDANE 1 ; Yves DETANDT1 ;César LEGENDRE1 and Grégory LIELENS1
1Free Field Technologies, MSC Software Belgium
2

Safran Helicopter Engines

ABSTRACTThe entropy and vorticity disturbances generated in the combustion chamber of an aero-engine can beconverted into pressure fluctuations while accelerated in the turbine stages: this phenomenon is known asIndirect Noise. The mean flow plays a significant role in this process and should be considered in themodeling. As a first step, the present paper focus on modeling the interaction of entropy and acoustic waves
in a nozzle. First, the mean flow inside the nozzle is estimated by means of a steady-state CFD simulation.
Second, the wave propagation including the mean flow effects is computed by means of a high-order adaptive
Discontinuous Galerkin (DG) scheme in time domain solving the linearized Euler equations (LEE). This
scheme is chosen due to its ability to accurately represent the phenomena involved and its high parallel
scalability. The studied cases consist of a convergent/divergent nozzle configuration where entropy waves
are injected upstream by means of strong heat pulses. Three representative flow conditions with different
mass flow rates are considered. Finally, the numerical solutions are compared with available experimental
data and analytical models showing good agreement.
Keywords: Sound, Combustion Noise, Indirect Noise

1. INTRODUCTION
Combustion noise in aero-engines can be decomposed into direct combustion noise and indirect
combustion noise. The strong unsteadiness of the combustion chamber itself is responsible for thedirect combustion noise, whose acoustic waves traverse the turbine stages. Indirect combustion noise
is caused by the generation of acoustic waves within the turbine stages when temperature
inhomogeneities (associated with entropy spots) are accelerated through those stages.
Assuming a quasi-one-dimensional nozzle, Marble & Candle [1] derived wave transfer functions
linking entropy and acoustic waves for both subsonic and chocked nozzles. Their model has been welldemonstrated for compact cases, i.e. when one considers the axial length of the nozzle to be small
compared to the wavelength of the acoustic or entropy perturbations limiting the results to low
frequencies. In the same manner, the Cumpsty & Marble model [2] applies the actuator disk theory tomimic the acoustic behavior of turbine stages taking into account also the azimuthal modes, as wellas the mean flow deviation with respect to the axis. These two above mentioned models constitute acomputationally inexpensive modeling technique to assess the indirect combustion noise in a nozzle
using the compact assumption.
Full numerical simulations of the Linearized Euler Equations (LEE) offer an alternative to model
the combustion noise through the turbine stages. The LEE allow evaluating the propagation of any
kind of small perturbations (acoustic or not) within a given mean flow assuming isentropic andinviscid conditions. Actran DGM (from the commercial software FFT) solves the LEE in the time
domain using a quadrature-free Runge-Kutta discontinuous Galerkin method (RK-DGM) [3]. It is
implemented for unstructured grids in 2D, 2D axisymmetric and 3D and employs a fourth order
Runge-Kutta temporal scheme and supports variable order of the elements (p-adaptivity).
In the present work, we aim at assessing the mentioned modeling techniques for indirect noise5434

through the experimental Entropy Wave Generator (EWG) from the DLR [4,5], which has been
simulated by many groups. Section 2 describes the modeling technique approaches, with one
subsection dedicated to the Marble & Candel model with the compact assumption and the second
subsection dedicated to the LEE model in Actran DGM. A description of the EWG experimental setup
is presented in section 3. Numerical and experimental data are analyzed in section 4 and conclusions
are finally discussed in section 5.

2. INDIRECT COMBUSTION NOISE
2.1 The compact model
As wavelengths of the disturbances considered here are much larger than the axial length of thenozzle, acoustic and entropy waves propagate quasi-steadily through the flow. Based on firstprinciples conservation laws (for the mass flow rate, the total temperature and the entropy), matchingconditions are imposed between the region upstream of the nozzle and the region downstream of thenozzle. Marble & Candel [1] derived expressions for the up- and downstream propagating acousticpressure perturbations (traveling from the inlet to the outlet or conversely and indicated by thesuperscripts “+” and “-”, respectively) generated by both impinging entropy disturbances andimpinging acoustic pressure waves. The responses to acoustic wave and entropy disturbances when
considering a subsonic flow, are presented in Table 1.

Acoustic wave upstream (𝑃 = 𝜀)Acoustic wave downstream (𝑃 = 𝜀)

𝑃 = 𝜀, 𝑃 = 0

𝑃 = 0, 𝑃 = 𝜀11 − (𝛾 − 1)𝑀 𝑀
𝑀 −𝑀
1+𝑀
2
𝑃 =
𝜀1−𝑀
𝑀 + 𝑀 1 + 1 (𝛾 − 1)𝑀 𝑀
211 + 2 (𝛾 − 1)𝑀
2𝑀
1+𝑀
𝑃 =𝜀1+𝑀
𝑀 + 𝑀 1 + 1 (𝛾 − 1)𝑀 𝑀
22𝑀
𝑃 =𝑀 +𝑀

𝑃 =−

𝑀 −𝑀
1+𝑀

1
1 + (𝛾 − 1)𝑀2
𝜀1
1 + (𝛾 − 1)𝑀 𝑀
2
1
1 − 2 (𝛾 − 1)𝑀 𝑀
1−𝑀𝜀
𝑀 + 𝑀 1 + 1 (𝛾 − 1)𝑀 𝑀
2

1−𝑀
1−𝑀

Convected entropy wave ( s /𝑐 = 𝜎)
𝑃 = 0, 𝑃 = 0
𝑀 −𝑀
𝑃 =−1−𝑀

𝑃 =𝑀 −𝑀
1+𝑀

1𝑀𝜎
211 + (𝛾 − 1)𝑀 𝑀
21𝑀 𝜎
211 + 2 (𝛾 − 1)𝑀 𝑀

Table 1. Compact acoustic and entropy transfer functions in a subsonic nozzle
The acoustic processes in a choked nozzle differ since the acoustic waves in the supersonicdivergent cannot propagate to the convergent. For such supercritical nozzle, Marble & Candel [1]derived expressions of the transfer functions as shown in Table 2. A configuration in which a normal
shock is located downstream of the nozzle, as illustrated in Figure 1, had also been considered in their
original model.

Acoustic wave upstream (𝑃 = 𝜀)
𝑃 = 𝜀,𝑃 =𝑃 =11 − (𝛾 − 1)𝑀2𝜀
11 + 2 (𝛾 − 1)𝑀

11 − (𝛾 − 1)𝑀2
𝜀
11 + (𝛾 − 1)𝑀25435

11 + (𝛾 − 1)𝑀2𝑃 =11 + (𝛾 − 1)𝑀2Convected entropy wave ( s /𝑐 = 𝜎)
𝑃 = 0,

𝑃 =−

𝑀 +𝑀
2

𝜀

Convected entropy wave with shock (s /𝑐 = 𝜎)

1
𝜎
2

𝑃 =0

1
1 + (𝛾 − 1)𝑀
2

1𝑀𝜎
2
1
1 + 2 (𝛾 − 1)𝑀
1𝜎𝑀 −𝑀
2
𝑃 =1
21 + (𝛾 − 1)𝑀2
𝑃 =−𝑃 =−1𝑀𝜎21
1 + 2 (𝛾 − 1)𝑀−(1 + 𝑀 )𝑀 + 2𝑀 𝑀
𝑃 =1 + 𝑀 + 2𝑀 𝑀

1𝜎2
11 + (𝛾 − 1)𝑀
2Table 2. Compact acoustic and entropy transfer functions in a choked nozzle

Figure 1. Supersonic nozzle with a normal shock downstream
2.2 The LEE model in Actran DGM
Actran DGM solves the LEE in the time domain by means of a Discontinuous Galerkin Methods.
As the LEE are derived from the compressible Navier-Stokes equations, the LEE isentropic set ofequations is made of five partial differential equations. The isentropic equations propagate not only
the acoustic fluctuations (the upstream propagating acoustic mode characterized by the propagation
speed v0 -c and the downstream propagating acoustic mode characterized by the propagating speedv0 +c), but also two vortical modes and one entropy mode (characterized by the propagating speed v0 ).
The system of equations supports the acoustic propagation in supersonic mean flows and can be solved
for the acoustic propagation through a shock.
The element boundary integrals in the DGM scheme are computed by considering the solution in
the two adjacent elements. The scheme is designed to enforce a stable solution during the propagationof waves from one element to the other. For free field propagation, a non-reflecting boundary enforcesthat all the incoming modes are set to zero, the outgoing modes being free to leave the domain. At theinlet, a Thompson condition has been introduced to avoid this reflective character of the traditionalboundary conditions. For our case of interest, the incoming entropy wave is simulated with an applied
density perturbation at the boundary, while the pressure and velocity fluctuations are set to zero.

3. Experimental configuration
The Entropy Wave Generator (EWG) is a nozzle with the capability of inducing entropy waves via
a heating module [5]. The experimental setup is shown in Figure 2 while the nozzle itself is sketchedin Figure 3. In order to create an entropy wave, a temperature pulse is generated through a heatgenerator. The temperature pulse is a step with a duration of 100 ms, which occurs every second.

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Figure 2. Sketch of the Entropy Wave Generator (EWG)

Figure 3. Sketch of the convergent-divergent nozzle
Three representative flow conditions with different mass flow rates are considered. The flow iseither supersonic (EWG1), subsonic (EWG2), or almost supersonic (EWG3). These flow conditions
had already been used for the European project RECORD [6] and are summarized in Table 3. All data
come from measurements except for the Mach numbers upstream and downstream of the nozzle andthe Mach number before shock that come a 1D flow code (which had been developed in the frameworkof the RECORD project).
Testcase

Massflowrate[kg/h]

Pressure setting Pressure nozzle
Machchamber against
throat against
numberambient pressure ambient pressure upstream
[kPa]
[kPa]
the nozzle

Machnumberdownstream
the nozzle

Machnumber atnozzle
throatMachnumberbefore
shock

EWG 1

42

11.17

-52.71

0.029

0.023

1.0

1.59

EWG 2

37

4.34

-32.65

0.027

0.019

0.71

-

EWG 3

40

7.34-45.49

0.028

0.020

0.87-

Table 3. Experimental flow conditions of the EWG

4. Aeroacoustic simulations
4.1 Numerical model
In Actran DGM, Euler’s equations are linearized around a stationary mean flow. Therefore, in orderto steer the LEE model, a simulation of the nozzle has been performed with the CFD software
OpenFOAM. The boundary layer is not modelled (wall slip boundary conditions at the nozzle walls)
and the density-based version of the solver has been considered without any turbulence model. An
axi-symmetric mesh with 32 000 cells has been built with the ANSA software. Details regarding theboundary conditions for the supersonic case (EWG1) are shown in Figure 4.

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Figure 4. CFD boundary conditions
As shown in Figure 5, the Actran DGM model uses a similar domain as the CFD domain, butevidently with different types of boundary conditions. The excitation is injected in the model througha Thompson boundary condition, while the semi-infinite duct is modelled with a buffer zone and a
non-reflecting boundary condition. An acoustic mesh has been built for the LEE model, comprising
about 9000 elements, capturing not only the acoustics at 200 Hz, but also the geometrical features of
the nozzle.

Figure 5. Components and boundary conditions of the Actran DGM model
For the simulations performed with the compact model, the nozzle domain is simply bypassedthrough a transfer admittance and with non-reflecting boundary conditions at the inlet and outlet. At
the inlet, the entropy signal is evaluated with the expression given in equation (1), while the density
signal is evaluated with the expression given in equation (2) in the Thompson boundary condition of
Actran DGM.
𝑠 = ∗ 𝑇′
(1)
𝜌=−

∗𝑠 =−

(2)

∗ 𝑇′

Looking at Figure 6, one can see that the experimental temperature signal includes notable noise.
Instead, an equivalent analytical signal has been considered in this study, whose expressions are givenin equations (3)-(4)-(5). The numerical values of t min, tmax, τ min , and τmax have been obtained by a leastsquare fit from the original experimental signal. The differences between the experimental andanalytical signal are the strongest at high frequencies, as shown in Figure 6.
(3)

𝑇(𝑡) = 0 𝑖𝑓 𝑡 < 𝑡5438

𝑇(𝑡) = 𝐴 ∙ 1 − 𝑒𝑖𝑓 𝑡<𝑡<𝑡(4)

𝑇(𝑡) = 𝐴 ∙ 𝑒

𝑖𝑓 𝑡<𝑡(5)Figure 6. Experimental vs analytical temperature signal in the time (left) and frequency (right) domains
It has been observed that Kelvin-Helmholtz instabilities occur in the Actran DGM model at low
frequencies and in the presence of high velocity gradients in the divergent section. These instabilities
prevent the computation convergence, and so decrease the results accuracy. To avoid this, the flow
variations in the radial direction have been removed to keep only the flow variations in the axial
direction, as illustrated in Figure 7 for the test case EWG1. By considering an equivalent 1D flow
(with a solver that had been developed in the framework of the RECORD project), the hydro-dynamic
instabilities are strongly reduced and allows the computation to converge at very low frequencies.
This simplification does not reduce the accuracy of the acoustic simulations with Actran DGM.

Figure 7.Original mean velocity flow from axi-symmetric model (left) vs equivalent 1D velocity flow (right)
4.2 Numerical results
In this section a comparison of the EWG data with numerical results obtained with the compact
model and the LEE model is presented. For brevity purposes, we will show results only for microphone
1 and 4 (see Figure 2) even though, from the test case description with the anechoic boundary
conditions, the results should not differ in between those microphones. We start considering the firsttest case, EWG1, in which a shock occurs in the downstream section of the nozzle. In the compact
model, the Mach numbers upstream and downstream of the nozzle, as well as the Mach number before
the shock are considered as inputs (see Table 3).
We observe differences in the experimental results at higher frequencies in between microphone 1and 4 that may indicate some type of reflections at the exit. As shown in Figure 8, the comparison in
between the compact and LEE model shows a very good match and, globally, there is also a good
agreement with the experimental results.
For the EWG2 test case, the flow remains subsonic through the nozzle, with a maximum Mach
number of 0.71 at the throat. For this test case and using the compact model of Marble and Candel[1], only the Mach values at the inlet and outlet of the nozzle should in principle be taken into account.
Nevertheless, the model does barely transform entropy into acoustics when both Mach numbers values
are comparable. As proposed by Bake et al. [4], the model has been distinctly split into an upstream
nozzle and a downstream diffuser considering the Mach number at the throat as extra input. The
impinging entropy disturbances enter the nozzle generating acoustic disturbances, and those acoustic
disturbances are solely transmitted in the diffuser (as for the supersonic cases when acoustic5439

perturbations from the diffuser cannot propagate back to the nozzle). Some questions remain over thevalidity of this model and other authors have proposed alternatives (see Duran & Moreau [7] forexample).

Figure 8. Acoustic pressure spectrum at microphones 1 and 4 (EWG1)The DGM method had convergence issues for microphone 1 at low frequencies, therefore theresults below 45 Hz are not presented. Apart from this, the overall agreement in between the proposedmodels and the experimental results is quite good, as shown in Figure 9.

Figure 9. Acoustic pressure spectrum at microphones 1 and 4 (EWG2)
For the EWG3 test case, the flow becomes almost sonic at the throat (M = 0.88 on the axis). The
same methodology as for the EWG2 test case has been applied in the numerical model. For this test
case, the DGM method had convergence issues for microphone 1 at low frequencies, therefore theresults below 65 Hz are not presented. As for the other test cases, the overall agreement in between
the proposed models and the experimental results is quite good (see Figure 10).

Figure 10. Acoustic pressure spectrum at microphones 1 and 4 (EWG3)

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5. Conclusions
This study focused on the application of the compact model from Marble and Candel [1] and a LEE
model from Actran DGM to the EWG test case, which is a convergent-divergent nozzle.
Three representative flow conditions have been considered and have been computed with
OpenFOAM using an axi-symmetric model:
 EWG1: Supersonic flow at the nozzle throat followed by a shock in the diffuser
 EWG2: Subsonic flow at the nozzle throat
 EWG3: Almost sonic flow at the nozzle throat
Few assumptions have been made in the numerical set-up. A clean equivalent analytical signal has
been considered as input for the entropy disturbances and the radial variations of the CFD results
feeding the LEE model have been suppressed. Also, the Marble and Candel model has been adapted
for subsonic cases with distinct treatment for the upstream nozzle and the diffuser. Comparison of theEWG data with numerical results have been presented and show a good agreement between theimplemented models and the experimental data. As expected, the Cumpsty and Marble model [2]produces exactly the same results as the Marble and Candel model for this test case.
Future work will focus on the validation of the numerical models with realistic turbine cases (seefor example [8],[9]). The LEE model from Actran DGM should turn out to be more accurate for thoseindustrial test cases as less restrictions with respect to the compact model are applied.

ACKNOWLEDGEMENTS
Part of this work and the experimental results on EWG have been funded by the European
Commission with the 7 th Framework Program (Project RECORD grant 312444). The authors
acknowledge their financial support.

REFERENCES
1. F.E. Marble and S. Candel, Acoustic disturbances from gas non-uniformities convected through a nozzle,
J. Sound Vib. 55 (1977) 225–243.
2. N.A. Cumpsty and F.E. Marble, The interaction of entropy fluctuations with turbine blade rows; amechanism of turbojet engine noise, Proc. R. Soc. Lond. A357 (1977) 323–344.
3. R. Leneveu, B. Schiltz, J. Manera, and S. Caro, Parallel DGM scheme for LEE applied to axhaust andbypass problems, 13th AIAA/CEAS Aeroacoustics Conference (2007), AIAA Paper 2007-3510.
4. F. Bake, U. Michel and I. Röhle, Experimental investigation of the fundamental entropy noise mechanismin aero-engines, 13th AIAA/CEAS Aeroacoustics Conference (2007), AIAA Paper 2007-3694.
5. F. Bake, C. Richter, B. Mühlbauer, N. Kings, I. Röhle, F. Thiele and B. Noll, The Entropy Wave Generator(EWG): A reference case on entropy noise, J. Sound Vib. 326 (2009) 574–598.
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Researchon
CoreNoiseReduction,
https://www.xnoise.eu/about-xnoise/projects/generation-2-projects/record/
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invariants and the Magnus expansion, J. Fluid Mech. 723 (2013) 190-231.
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and an actuator disk theory to predict combustion noise in a helicopter engine, Combust. Flame 165 (2016)272-287.
9. M. Lemoult, A. Vallon, M. Roger, Exhaust noise from helicopter turboshaft engines, Proceedings of the8th International Symposium on Experimental and Computational Aerothermodynamics of Internal Flows
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