[american institute of aeronautics and astronautics 14th aiaa/ceas aeroacoustics conference (29th...

Turbofan Aft Noise Radiation: Progress Towards a

Realistic CAA Simulation

Damiano Casalino∗ Mattia Barbarino†

CIRA, Italian Aerospace Research Center, Capua, I-81043, Italy

The frequency-domain CAA code GFD is used to compute the sound radiation from aturbofan exhaust. Results obtained by solving a second-order wave equation for the velocitypotential are compared with the results obtained by solving a third-order wave equationfor the pressure perturbation. This second wave model has been verified against analyticalsolutions for an idealized bypass configuration. Similar simulations are carried out in thepresent paper for a buried nozzle configuration, but with the opposite intent to generatereference results for the verification of analytical models. Since the potential wave modelis valid only in the high-frequency limit, whereas the pressure wave model is valid only forparallel mean flows, a residual analysis is carried out in order to investigate the differencesbetween the two models for a realistic turbofan configuration, when both these methodsare not rigorously valid. It is pointed out that, even though the difference of the SPL far-field peak values does not exceed 4dB, further analysis are required in order to understandwhich model is best suited for this kind of computations. An additional outcome of thepresent activity is the preliminary investigation of an active control concept for the fannoise radiated from the bypass duct opening.

Nomenclature

A Governing matrixM Mach numberR Magnitude of the local relative residueT TemperatureU Mean-flow velocityVn Wall normal velocityi Imaginary unit (i =

�√−1

�)

c Speed of soundf Acoustic frequencyk Acoustic wavenumber (k = ω/ca)

m, n Circumferential and radial duct mode ordersp Pressurer1, r2 Inner and outer duct radiiu Velocity

Subscripts0 Mean-flow quantitiesa Atmospheric ambient quantities

Conventions

3D Three-dimensionalCAA Computational Aero-AcousticsFW-H Ffowcs-Williams and HawkingsGFD Green’s Function DiscretizationPML Perfectly Matched LayerRANS Reynolds-Averaged Navier-StokesSPL Sound Pressure Level

Symbolsγ Ratio of specific heats of the gasλ Sound wavelengthω Radian frequencyφ Velocity potential (u = ∇φ)

Π Logarithmic pressure (Π = γ−1 ln p)

ρ Density

Superscripts′ Perturbation quantities· Fourier component in frequency space· Dimensionless mean-flow quantities

∗Senior Research engineer, Rotorcraft Aerodynamics and Aeroacoustics Laboratory, CIRA, Member AIAA.†Junior Research engineer, Rotorcraft Aerodynamics and Aeroacoustics Laboratory, CIRA.

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American Institute of Aeronautics and Astronautics

14th AIAA/CEAS Aeroacoustics Conference (29th AIAA Aeroacoustics Conference)5 - 7 May 2008, Vancouver, British Columbia Canada

AIAA 2008-2882

Copyright © 2008 by Italian Aerospace Research Center. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

I. Introduction

The problem of CAA prediction of noise transmitted through and radiated from a realistic turbofanbypass duct in the presence of a sheared mean flow, plays a primary role in the aero-engine duct acousticsresearch domain. The emphasis recently given to this problem is due, on one hand, to the several analyticalworks showing the importance of a proper Kutta condition at the nozzle edge;1,2 on the other hand, to theoccurrence of instabilities in the shear layer when, in order to satisfy the proper edge condition, linearizedEuler’s perturbation models are used instead of acoustic potential models.3,4

More recently, some authors have contributed to the demystification of the bypass duct noise predictionproblem. On one side, alternative wave models, either based on perturbation equations deprived by theunstable vortical mode5 or derived from a Pridmore-Brown wave operator,6,7 have been shown to constitutevalid alternatives to the solution of linearized Euler’s equations, both for realistic and idealized exhaustmodels; on the other side, some authors solved the Euler’s equations in time domain without observing anyinstability in the shear layer, both for realistic8,9 and idealized configurations.10

In the present paper we continue to apply the third-order pressure perturbation model to idealized exhaustconfigurations, by using the same numerical methodology described in a previous and strictly related paper.7

An annular exhaust with a buried inner nozzle is considered and benchmark results are generated for theverification of Wiener-Hopf analytical models or other numerical results. This concludes CIRA researchactivity on idealized exhaust configurations and code verification. The core of the paper is focused onturbofan aft noise computation for a realistic exhaust configuration, the same used in the European projectTURNEX and provided by the European project CoJEN.11 For this configuration, a RANS simulation iscarried out by using the commercial CFD software Fluenttm . Then frequency-domain CAA computations arecarried out for two values of the reduced frequency, and for different duct acoustic modes. Two wave modelsare used. The first one is a third-order pressure perturbation model derived from Lilley’s equation, butretaining all the mean-flow gradient terms,6,12 the second one is a second-order acoustic velocity potentialmodel derived by Pierce13 for sound propagation in rotational mean flows. Both these models are notrigorously valid for the addressed case: the pressure perturbation equation fails as acoustic analogy model inthe presence of axial mean-flow gradients, whereas the acoustic potential equation can be rigorously appliedonly for mean flows that vary slowly over the length and time scale of the sound waves (high-frequencylimit), and this is not the case in the presence of thin shear layers close to the nozzle edge. The pressure fieldcomputed by using the potential solution of the Pierce’s equation is used to compute the local residue withrespect to the third-order wave equation. The residual field is finally correlated with macroscopic propertiesof the mean flow.

One possible application of the code GFD is the feasibility study of passive/active noise mitigation devicesfor aero-engine nacelles. Some preparatory activities for future activities whose goal is the design of an activecontrol system for the bypass duct have been carried out and reported in this paper: the demonstration ofa control concept based on vibrating actuators close to the bypass duct exit, and the computational costassessment for axi-symmetric and equivalent 3D computations.

The paper is organized as follows. In section (II) the pressure perturbation and the acoustic potentialwave models are presented. Benchmark results obtained by using the pressure perturbation model for anidealized exhaust with an inner buried nozzle are collected in section (III). Then results, for a realistic exhaustgeometry are presented in section (IV) with emphasis given to the comparison between the two wave models.Finally the main outcomes of the present work are drawn in section (V).

II. Wave propagation models

II.A. Third-order pressure perturbation model

Considering a compressible inviscid perfect isentropic gas mean flow, with superimposed linear disturbances,it is possible to rearrange the Euler’s governing equations into the form of an acoustic analogy equation inwhich a third-order wave operator acts on the quantity Π′ = (1/γ) p′/p0 and the remaining terms, shiftedat the right-hand side, are treated as aeroacoustic sources, i.e.:

D0

Dt

[D2

0Π′

Dt2− ∂

∂xj

(c20

∂Π′

∂xj

)]+ 2

∂Uk

∂xj

∂

∂xk

(c20

∂Π′

∂xj

)= S, (1)

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where p0, T0, c0, Ui denote mean-flow quantities, D0/Dt is the mean-flow Lagrangian derivative, and theright-hand acts as a pure source term only in the case of a parallel mean shear flow. In the case of a genericmean flow with non-zero mean-flow gradients in more than one direction, the right-hand side includes termsthat are linear in the perturbation quantities and do not vanish outside the source region. Therefore, for ageneric mean flow, Eq. (1) constitutes an ”abuse” of Lilley’s acoustic analogy model. By considering harmonicperturbations of radian frequency ω and making use of the mean-flow Lagrangian derivative D0/Dt=−i ω +Ui ∂/∂xi, Eq. (1) takes the following form:

i k3Π′ − 3 k2Ui∂Π′

∂xi− 3 i k UiUj

∂2Π′

∂xi∂xj− 3 i k

(Uj

∂Ui

∂xj

)∂Π′

∂xi

+Uk∂

∂xk

(UiUj

∂2Π′

∂xi∂xj

)+ Uk

∂

∂xk

[(Uj

∂Ui

∂xj

)∂Π′

∂xi

]+ i k

∂T

∂xj

∂Π′

∂xj

+i k T∂2Π′

∂x2j

− Uk∂

∂xk

(∂T

∂xj

∂Π′

∂xj+ T

∂2Π′

∂x2j

)+ 2

∂Uk

∂xj

[∂T

∂xk

∂Π′

∂xj+ T

∂2Π′

∂xk∂xj

]= 0, (2)

where k is the acoustic wavenumber in ambient conditions, and Ui = Ui/ca and T = T0/Ta are mean-flow quantities made dimensionless by ambient reference conditions. Eq. (2) is the third-order pressureperturbation equation used in the present work, but solved for the quantity p′pa/p0.

II.B. Second-order acoustic potential model

Considering a compressible inviscid perfect isentropic gas mean flow, perturbed by an irrotational acousticfield, it is possible to rearrange the Euler’s governing equations into the form of a second-order wave equationfor the acoustic velocity potential φ′, i.e.:

1ρ0

∇ · (ρ0∇φ′) − D0

Dt

(1c20

D0φ′

Dt

)= 0. (3)

Again, by considering harmonic perturbations of radian frequency ω and making use of dimensionless mean-flow quantities ρ = ρ0/ρa, T = T0/Ta and U = U0/ca, the following convected wave equation can beobtained:

1ρ

∂ρ

∂xi

∂φ′

∂xi+

∂2φ′

∂x2i

+k2

Tφ′ +2i k Ui

∂φ′

∂xi+ UiUj

∂2φ′

∂xi∂xj+ Ui

∂Uj

∂xi

∂φ′

∂xj− i k

T 2Ui

∂T

∂xiφ′ +

1T 2

Ui∂T

∂xiUj

∂φ′

∂xj= 0. (4)

This equation is accompanied by a linear expression for the acoustic potential that reads:

p′ = ρaca

(i kφ′ − Ui

∂φ′

∂xi

). (5)

Eq. (4) is the second-order acoustic potential equation used in the present work. It describes the propagationof acoustic perturbation through a generic rotational flow and is rigorously valid only in the short wavelengthlimit, as discussed by Pierce.13

II.C. Boundary conditions

PML buffers are used to absorb outgoing acoustic waves, both in the axial and radial directions. Theformulation is the same used in Ref.,7 with a PML thickness equal to 2λa and a damping factor that isincreased quadratically across the buffer, from 0 to its maximum value of 2 at the far-field boundary. 3Dgrids are obtained by circumferential extrusion of a two-dimensional structured grid. In the resulting annularfar-field buffer, the cylindrical PML change of variables applied in Ref.7 is used. Axial PML buffers are alsoused to impose a duct inlet mode without affecting the outgoing waves.

Slip conditions are imposed on solid walls in the form of a vanishing normal derivative of p′ or φ′. In thecase of a vibrating wall with velocity Vn, the conditions ∂p′/∂n ' i kρacaVn and ∂φ′/∂n = Vn are used forthe third- and second-order wave models, respectively.

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III. Buried nozzle benchmark solutions

This section is devoted to the presentation of reference results for an idealized turbofan exhaust thatconsists of a semi-infinite outer duct and a semi-infinite inner duct with a buried nozzle, as sketched inFig. (1). A triple piecewise constant flow is considered, with subscripts a, b and j denoting external, bypassand core jet flow quantities, respectively. The employed numerical procedure is described in more details ina previous and strictly related paper.7

y

xr

1 2 lr

PM

Laxia

l buff

ers

PM

Laxia

l buffe

r

PML radial buffer

U Tr

U Tr

U Tr

aa a

bb b

j jj

Figure 1. Illustration of the bypass duct model with a jet flow issuing from a buried nozzle.

30

40

50

60

70

80

90

0 20 40 60 80 100 120 140 160 180

SPL

(dB)

Observation angle (deg)

30

40

50

60

70

80

90

0 20 40 60 80 100 120 140 160 180

SPL

(dB)


Figure 2. SPL contour plots for co-planer (left column) and buried (right column) exhaust systems. SPL directivity(middle column): co-planer (solid line) versus buried (dashed line) exhaust systems. Inlet mode m = 10, n = 1. Top:kr2 =20, bottom: kr2 =40.

Axi-symmetric computations are carried out by using a rectangular uniform grid of size λa/12. Thezero thickness solid walls are modeled by cloning the grid nodes and forcing a disconnection into the meshtopology. The nodes at the geometrical duct terminations are treated in the same way as the other nodes,the unit normal vector being parallel to the radial direction. The inner and the outer bypass duct radii arer1 = 0.8 m and r2 = 1.2 m, respectively. The origin of the reference system is located at the center of thebypass nozzle section. The computational domain extends from 0 to 4.8 m in the radial direction and from−3 m to 5 m in the axial direction. All the simulations are carried out by considering an inlet power levelcorresponding to 120 dB (reference power 1 · 10−12). The far-field directivity is computed at a distance of100 m from the origin by performing a Kirchhoff integration upon a cylindrical surface of radius 1.3 m. Themean-flow properties are listed in the table (1).

Two configurations are considered: a co-planar nozzle (l = 0m) and a buried nozzle (l = 1/r2). Compu-tations are carried out for the circumferential duct mode orders m = 10 and m = 20 and the radial modeorders n = 1 (first), n = 2 and n = 3 at two values of the frequency, say f = 876 Hz and f = 1752 Hz,corresponding to the Helmholtz numbers kr2 = 20 and kr2 = 40, respectively. Only results for the caseswith cut-on inlet modes are presented in Figs. (2) through (6). In each figure, the near-field SPL contour

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30

40

50

60

70

80

90

0 20 40 60 80 100 120 140 160 180

SPL

(dB)


30

40

50

60

70

80

90

0 20 40 60 80 100 120 140 160 180

SPL

(dB)


Figure 3. SPL contour plots for co-planer (left column) and buried (right column) exhaust systems. SPL directivity(middle column): co-planer (solid line) versus buried (dashed) exhaust systems. Inlet mode m=10, n=2. Top: kr2 =20,bottom: kr2 =40.

30

40

50

60

70

80

90

0 20 40 60 80 100 120 140 160 180

SPL

(dB)


30

40

50

60

70

80

90

0 20 40 60 80 100 120 140 160 180

SPL

(dB)



ca Ua ρa Ta Ub ρb Tb Uj ρj Tj

330 99 1.3 271 210 1.16 305 265 0.5 699

Table 1. Mean-flow properties for the idealized bypass configuration (SI units).

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20

30

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50

60

70

80

90

100

0 20 40 60 80 100 120 140 160 180

SPL

(dB)


20

30

40

50

60

70

80

90

100

0 20 40 60 80 100 120 140 160 180

SPL

(dB)



20

30

40

50

60

70

80

90

100

0 20 40 60 80 100 120 140 160 180

SPL

(dB)


20

30

40

50

60

70

80

90

100

0 20 40 60 80 100 120 140 160 180

SPL

(dB)


Figure 6. SPL contour plots for co-planer (left column) and buried (right column) exhaust systems. SPL directivity(middle column): co-planer (solid line) versus buried (dashed line) exhaust systems. Inlet mode m=20, kr2 =40. Top:n=2, bottom: n=3.

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plots and the far-field SPL directivity patterns are shown. These results may be useful for the verificationof both analytical models and other numerical results.

IV. Radiation from a turbofan exhaust

IV.A. Case study description

In this section axi-symmetric results of sound radiation from a realistic bypass exhaust are presented. Thesame configuration has been recently used by Leneveu et al.4 to verify the commercial code Actran-DGM ,and corresponds to an exhaust configuration, known as ”short cowl nozzle”, used in the EU project CoJEN.11

For this case, a RANS k − ε mean-flow computation has been carried out by using the commercial softwareFluenttm . The external Mach number is 0, whereas inflow Mach number at both the bypass and jet inletsections is 0.338. Contour plots of the dimensionless mean-flow velocity U are shown in Fig. (7). Standardthermodynamic inflow conditions have been considered at the bypass and jet sections. The inner and outerradii of the bypass inlet section are r1 = 0.788 m and r2 = 1.405 m, respectively.

——

——

——

——

——

——

——

——

–

——

——

——

——

——

——

——

——

–

Figure 7. Dimensionless mean-flow velocity. Solution obtained by using the commercial CFD software Fluenttm .

CAA computations have been carried out for two frequencies, i.e. f = 2162.59 Hz and f = 4325.18 Hz,corresponding to the Helmholtz numbers kr2 = 55 and kr2 = 110, respectively. The circumferential modem = 52 and the radial modes n = 1 (first), n = 2 and n = 3 are considered. The radial modes n = 2and n = 3 are cut-on only at the second frequency. The computational grid used for the high frequencycase provides the contours plotted in Fig. (8) of the minimum and maximum numbers of grid nodes peracoustic wavelength. These have been computed, for each computational stencil, by accounting for the localmean-flow velocity and temperature, and thus correspond to upstream and downstream propagating waves.

Figure 8. Minimum (left) and maximum (right) numbers of grid nodes per local acoustic wavelength (kr2 = 110).

The far-field SPL directivity is computed at a distance of 100 m from the aft cone vertex, along an arc of

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180 deg. Two different integral formulations are used: when a potential field is available (Pierce’s equation), aporous FW-H formulation, derived from a 3D generalization of Lockard’s approach,14 is used, whereas, whena pressure perturbation field is available (Lilley’s equation), a Kirchhoff formulation, obtained by applyingFourier and Galilean transformations to the time-domain solution by Farassat & Myers,15 is used. For thepresent case study, the integration surface is a coaxial cylinder of radius 2.7 m spanning the whole axialextension of the computational domain, including the PML buffers.

All the simulations have been carried out by considering an inlet power level corresponding to 120 dB(reference power 1 · 10−12).

IV.B. Comparison between Pierce’s and Lilley’s wave equations

This subsection addresses the main topic of the present paper: how much the predicted noise levels areaffected by the employed wave model. The second- and third-order wave equations (4) and (2) are solvedwith the same computational grid. Since no one of these equations can be rigorously applied to the presentcase study, the difference between the corresponding numerical solutions provides a conservative estimationof the modeling error. Indeed, as pointed out in a previous work,6 the discretization of third-order derivativeswith the GFD method, does not increase significantly the discretization error.

The technique used to quantify the modeling error consists in testing the Pierce’s pressure field pP againstthe Lilley’s linear system

[AL]{p′/p} = {b}, where p′/p is the unknown column. The resulting residue is

divided by the local magnitude of pP , i.e.:

{R} =

∣∣[AL]{p′/p} − {b}

∣∣|{pP }|

. (6)

Figs. (9) and (10) show results for the lower frequency case. The Pierce’s solution in Fig. (9) exhibitsa more prominent radiation lobe (contour band 120 − 140 dB). This is confirmed by the SPL directivity inFig. (10), where the main radiation peak for Pierce’s solution is about 4 dB higher than the one for Lilley’ssolution. This significant difference can be due either to the effects of axial mean-flow gradients in Eq. (2),or to the relatively low ratio between the mean-flow variational scale and the acoustic wavelength. Indeed,the relative residue plotted in Fig. (10) exhibits higher values in regions where higher values of the axialmeanflow derivatives are expected, as well as in the bypass shear layer.

Figure 9. Comparison between Lilley’s (left) and Pierce’s (right) SPL contours for the case kr2 =55. Inlet mode m=52and n=1.

Figs. (11) and (12) show results for the higher frequency case. Due to the multiple wave diffractions andreflections and to the wave refraction across the shear layer, is not easy to appreciate macroscopic differencesbetween the Pierce’s and Lilley’s SPL contours plotted in Fig. (11). The relative residue distribution shownin Fig. (12) reveals that the residue is higher in the boundary layer and in the shear layer. This argumentcould be in favour of the thesis according to which the modeling error due to the presence of concentratedmean-flow vorticity exceeds the modeling error due to the mean-flow axial gradients. Therefore, since themodeling error of Pierce’s equation decreases as the frequency increases, smaller noise differences betweenthe two solutions are expected for this case. Unfortunately, this is not fully confirmed by the SPL directivitypatterns plotted in Fig. (12). In fact, significantly smaller differences can be observed only for the secondradial mode. For the other radial modes, the differences of the highest peak values are of the same order asthose observed for the lower frequency case in Fig. (10). We can conclude this investigation by pointing out

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0 30 60 90 120 150 180θ (deg)

20

30

40

50

60

70

80

90

SPL

(dB

)

Figure 10. Comparison between Lilley’s and Pierce’s (right) solutions for the case kr2 =55. Relative residue contourson the left. SPL directivity on the right: Lilley (solid line), Pierce (dashed line). Inlet mode m=52 and n=1.

Figure 11. Comparison between Lilley’s (left) and Pierce’s (right) SPL contours for the case kr2 = 110. Inlet modem=52 and n=1 (top), n=2 (middle), n=3 (bottom).

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0 30 60 90 120 150 180θ (deg)

20

30

40

50

60

70

80

SPL

(dB

)

0 30 60 90 120 150 180θ (deg)

20

30

40

50

60

70

80

SPL

(dB

)

0 30 60 90 120 150 180θ (deg)

20

30

40

50

60

70

80

SPL

(dB

)

Figure 12. Comparison between Lilley’s and Pierce’s (right) solutions for the case kr2 =110. Relative residue contourson the left. SPL directivity on the right: Lilley (solid line), Pierce (dashed line). Inlet mode m = 52 and n = 1 (top),n=2 (middle) n=3 (bottom).

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that, although the noise level discrepancies between the two models does not exceed 4 dB for the addressedcases, a deeper analysis is necessary in order to understand which acoustic model is better suited for arealistic bypass CAA prediction.

IV.C. Concept for active noise control

A preliminary study has been carried out in order to individuate one of the best locations for wall mountedactuators in proximity of the bypass lip. The idea is in fact to develop an active control system integratedwith the bypass chevrons.

The most natural way to actively control a spinning mode of circumferential order transmitted through aduct consists in using a ring of actuators with vibration velocity V j

n = Vn exp(i m∆θj), where j is the actuatorindex. Assuming a continuous circumferential distribution of actuators, axi-symmetric computations can beperformed in order to determine the proper value of Vn for each transmitted mode (magnitude and phase)and for each frequency of interest. These values are finally synthesized into a control law to be successivelyverified and possibly refined by performing 3D computations based on the real actuators shape and locationand the real nozzle shape.

For this preliminary study, only one duct acoustic mode (m = 52, n = 1) and one frequency (f =2162.59 Hz) have been considered. The first important conclusion is that the actuators must be locatedinside the duct and sufficiently far from the edge, in order to allow the anti-sound field to have a similar edgediffraction pattern as the transmitted acoustic mode. Fig. (13) shows the pressure perturbation field to becontrolled (left), and the anti-sound field (right) obtained by solving along the actuated boundary segmentthe equation ∂p′/∂n = 1. The optimal value:

Vn ' 1200 ei 0.94π

ρacai k(7)

has been determined by looking at the far-field acoustic pressure directivity plotted in Fig. (14). The resultingpressure perturbation field shown in the same figure exhibits vanishing radiated acoustic levels.

Figure 13. Acoustic pressure contour plots (Pa) for the case kr2 = 55. Inlet mode m = 52, n = 1 (left), unitary controlamplitude (right).

IV.D. Computational cost assessment for 3D computations

3D CAA computations are required in order to evaluate the effectiveness of a passive/active noise mitigationdevise in the presence of circumferential accidents due to the device itself, as in the presence of liner splices16

or nozzle chevrons, or due to nacelle structural elements like the nacelle pylon.In order to achieve realistic high frequencies (kr > 50) 3D results within lower CPU-time, suited for

parametric and optimization analyses to be executed on CIRA scalar and vectorial super-computers, andwith lower memory occupations, a full redesign of the code architecture is required. It is therefore useful toperform an assessment of the current status before starting the implementation activities.

The selected case for this assessment is the lower frequency bypass noise propagation case addressedin this section. The computational grid has been generated by circumferential extrusion and consists ofabout 9 millions of nodes. The peak memory occupation for this case is about 17GB, with most of thepost-processing features active, i.e., far field extrapolation and computation of the perturbation velocity and

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-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

50 60 70 80 90 100 110 120 130

p’

(Pa

)

Observation angle (deg)Figure 14. Acoustic pressure (Pa) for the case kr2 = 55. On the left: inlet mode m = 52, n = 1 directivity (solid line)versus the tuned anti-control directivity (dashed line). On the right: contour plots for the controlled case.

acoustic intensity vectors. Concerning the computational time, a comparison between the correspondingaxi-symmetric case (2.5D) and the 3D case is made in table (2).

Dimension Nodes CPUs Iterations Relative residue CPU time (hours)2.5D 34 412 1 15000 2.3 · 10−6 0.23D 8 809 472 8 2000 5 · 10−5 12

Table 2. Mean-flow properties for the idealized bypass configuration (quantities in SI units).

Computations have been carried out on a NEC-TX7 computer system that is based on Itanium 2 scalarprocessors. The axi-symmetric simulation has been converged over 15000 iterations of the linear systemsolution, reaching a residue of 2.3 · 10−6. The 3D computation has been stopped after 2000 iterations(about 12 hours) regardless the convergence level. The corresponding SPL directivity patterns are plottedin Fig. (15). It can be observed that, even though the peak values are fairly well predicted, the lowerconvergence level of the 3D solution dramatically affects the dynamic ratio of the solution.

This assessment has confirmed that 3D results at Helmholtz numbers of the order of 50 can be achievedwithin reasonable times. However, in order to carry out parametric computations and optimizations, signif-icant improvements are still necessary, in particular in the memory occupation.

0 30 60 90 120 150 180θ (deg)

20

30

40

50

60

70

80

SPL

(dB

)

Figure 15. SPL directivity for the kr2 = 55. Comparison between axi-symmetric (solid line) and 3D (dashed line)results. Inlet mode m=52, n=1.

V. Concluding remarks

Progresses have been made in CIRA capabilities in the field of nacelle noise prediction. The presenteffort, which is in the continuity line of previous ones, has confirmed that the flexibility and accuracy of thewave-based unstructured finite difference scheme permits to easily discretize complex wave operators andboundary conditions. Such a peculiarity has been exploited to investigate the influence of the wave model on

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the prediction of aft noise radiation from a realistic turbofan exhaust. Two wave models have been compared:a second-order acoustic potential model valid for rotational mean flows in the high frequency limit, and athird-order pressure perturbation model valid for unidirectionally sheared mean flows. The residual analysishas shown that the discrepancies between the two models tends to be confined in the boundary layer andin the shear layer as the acoustic frequency is increased. However, this encouraging result is not confirmedby the far-field noise peak levels that seem to be affected by the same incertitude level at two values of theacoustic frequency. Further analyses are therefore necessary in order to definitively state which wave modelis best suited for turbofan aft noise predictions.

Additional outcomes of the gained confidence level in CIRA CAA methodologies are: the generation ofbenchmark solutions for an idealized bypass duct with a buried inner nozzle, and a preliminary feasibilitystudy of an active control system. Future efforts will be focused on the reduction of CPU-time and memoryoccupation through a full redesign of the code.

Acknowledgements

The authors are grateful to Dr Roberto Mella of CIRA Informatic Systems Laboratory for his kindsupport to the present computational activity.

References

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3Zhang, X., Chen, X. X., Morfey, C. L., and Nelson, P. A., “Computation of Spinning Modal Radiation from an UnflangedDuct,” AIAA Journal , Vol. 42, No. 9, 2004, pp. 1795–1801.

4Leneveu, R., Schiltz, B., Manera, J., and Caro, S., “Parallel DGM Scheme for LEE Applied to Exhaust and BypassProblems,” AIAA Paper 2007-3510 , May 2007.

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