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11th AIAA/CEAS Aeroacoustics Conference, 23-25 May 2005, Monterey CA, USA
Aeroacoustic Simulation of the Noise radiated by an
Helmholtz Resonator placed in a Duct
St´ephane Caro∗
, Paul Ploumhans, Xavier Gallez, Free Field Technologies SA –fft.be
16 place de l’Universit´e, B1348 Louvain-la-Neuve, Belgium
Friedrich Brotz, Michael Schrumpf, BEHR GmbH & Co. –behr.de
Siemensstrasse 164, D70469 Stuttgart, Germany
Alex Read, Fred Mendon¸ca, CD-adapco –cd-adapco.com
200 Shepherds Bush Road, London W6 7NY, UK
The quintessential aeroacoustic test case of an Helmholtz resonator was studied, de-
ploying the recently released coupling between the Engineering design tools STAR-CD (a
CFD code) and Actran/LA (a CA code). The objective, to accurately model both the
frequency and magnitude of the aeroacoustic resonance phenomenon in the geometry as
compared with both experimental and analytical data, was achieved. This paper presents
the methodology used, demonstrates the importance of accurate representation of com-
pressible effects in CFD, and presents methods to optimise the data transfer between the
codes by focusing on the dominant source regions.
I. Presentation of the Helmholtz resonator
A. Description of the test case
(a) General view (b) Dimension (c) Shape
Figure 1. Helmholtz resonator placed in a tube in a presence of a flow.
An area of considerable interest in the design of HVAC units, in this case from the automotive sector, is
the noise they generate at high flow rates. In order to demonstrate the ability of the commercial Engineering
design software packages STAR-CD and Actran/LA to, in combination, accurately model the requisite
∗Corresponding author: stephane.caro@fft.be
Copyright c
2005 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc. with
permission.
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American Institute of Aeronautics and Astronautics Paper 2005-3067
11th AIAA/CEAS Aeroacoustics Conference (26th AIAA Aeroacoustics Conference)
23 - 25 May 2005, Monterey, California AIAA 2005-3067
Copyright © 2005 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
physical phenomena of aeroacoustics, a simple test case was chosen: the Helmholtz resonator. This test case
has a number of advantages. First, it exhibits strong acoustic resonance effects induced by fluid dynamics:
in that respect it is the quintessential aeroacoustics test case. Second, the frequency of this resonance can
be estimated by a simple, flow velocity independent (Eq. (1)). Third, the uncomplicated geometry makes
both numerical, and experimental evaluations relatively simple to perform, allowing all efforts to be focused
on understanding the complex fluid/acoustic interactions and how best to model them. Finally, there is no
commercial sensitivity associated with the results or geometry, enabling their communication.
It is possible to demonstrate that the resonance, on such a system, should occur at a frequency f0given
by:
f0=a
2πsπr2
V·(L+ ∆L)(1)
where ais the speed of sound, ris the radius of the tube of the resonator neck, Vis the volume of the
resonator, Lis the length of the tube and ∆Lis the orifice correction (see for example Mechel1). This gives,
in the present case, f0= 358 Hz.
B. Acoustic measurements
The acoustic measurements were carried out in the reverberation chamber of the Acoustic Laboratory at
Behr, Stuttgart. The air flows through a settling chamber directly into the duct containing the resonator.
The acoustic signal is measured using a microphone, positioned one meter away from the duct outlet and
0.5 m above the upper part of the duct (Figure 1).
The reverberation chamber is made of hard, oblique-angled walls. The sound field in the room is diffuse,
so that the measurement is independent from the microphone position. In addition, brief noise events are
blurred. The Figure 2shows the experimental setup.
Figure 2. Experimental setup: the resonator is placed in a reverberant room for measurements
Experiments conducted confirm that the resonance frequency is independent of the inlet velocity and the
position of the microphone. The measured frequency, 358 Hz, is consistent with the analytic formula above.
In the remaining of this paper, the measured acoustic power will be compared with the simulated acoustic
power. Indeed, as the acoustic field is homogeneous in the whole reverberant room, the acoustic power seems
the best quantity to compare. Actran/LA gives direct access to the radiated power. In order to compute
the radiated power, from the acoustic pressure level measured in the room, one uses the Eyring’s formula:2
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American Institute of Aeronautics and Astronautics Paper 2005-3067
T=−0.07 V
Slog (1 −α)(2)
where Tis the (measured) reverberation time, Vthe volume of the room, Sthe sum of the wall surfaces,
and αthe absorption coefficient (assuming that it does not depend on the wall considered). When T= 4.8
s, V= 113 m3and S= 75 m2,α= 0.0224. Sabine’s formula3would give αs= 0.0221.
Since the sound field is assumed to be diffuse, the intensity is given by a relation of the type p2/4ρ0a0. The
radiated energy equals the absorbed energy, thus the source (i.e. radiated) energy is given by p2
4ρ0a0·αS.
One can write:
W=p2·αS
4ρ0a0
Lw=Lp+ 10 log10 αS ·p2
ref
4ρa0Wref !(3)
where Wis the radiated power which corresponds to pthe acoustic pressure measured in the diffuse sound
field medium, and Lwis the power level associated with the measured acoustic level Lp. For practical
reasons, the radiated power L0
wis computed as if in the free field. The comparison with the measured Lpis
then done by computing the equivalent Lwusing Eq. (3).
II. CFD study
Two simulations have been conducted to compute the unsteady flow and the consequently the acoustic
source terms for Actran/LA. Both sets of results are presented here. The results from the first calculation
were used to understand the minimum required extent of the acoustic source region output, the results for
which were then applied for the second case. The first study also demonstrates the importance of certain
parameters - the solution of temperature - which in a typical fluids calculation, may be neglected.
A. First computation
The CFD mesh is predominantly hexahedral mesh, consisting of 555 000 cells (Figure 3).
The mesh has been generated using ICEM-CFD Tetra (green part), the tetras have then been converted
into hexa elements (still in ICEM-CFD Tetra). Local refinements were done afterwards in pro-STAR with
Cmrefine. This meshing methodology is favored over tetrahedral meshing due to the numerical dissipation
associated with tetrahedral meshes. This is of particular importance both generally for LES calculations and
specifically for aeroacoustics. Numerical dissipation kills the turbulent fluctuations that LES resolves and
over-predicts the rate of turbulent energy decay, resulting in under-prediction of high frequency broadband
noise. An area of further investigation for the authors is the use of “arbitrary polyhedra”, which at the
time of writing have recently become available, see Peric.4These combine the ease-of-use associated with
tetrahedral mesh generators, whilst largely obviating their detrimental numerical features. The mesh is
refined in the resonator neck region in order to accurately resolve the unsteady flow features - and acoustics
sources - in this region. The DES/k−εturbulence model with hybrid wall functions were used due to their
proven suitability to industrial aeroacoustic calculations. The temporal discretisation was Crank-Nicholson;
the simulation ran for 10 000 time steps with a time step size of δt = 2.10−5s. Part of the motivation for
using STAR-CD was the speed of its transient solver, which has repeatedly been shown to be up to an order
of magnitude faster than alternative codes. In the first case, the flow is solved in a partially-compressible
mode with the fluid assumed to obey the ideal gas law with respect to pressure, but not temperature. A fixed
pressure boundary condition was used at the outlet, a fixed velocity condition at the inlet. Both boundaries
are considered to be adequately far from the source region. The initial field for the transient calculation was
the flow field calculated in a prior steady-state run.
Snapshots of velocity and pressure, as well as pressure fluctuations with time for a selected location, are
shown on Figure 4. It can be seen that the main resonance from this simulation occurs at 305 Hz, which
does not correspond to the expected value (358 Hz). Although the amplitude and phase of the fluctuation
varies between different monitor locations, the frequency does not. Further tests, undertaken at Behr show
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American Institute of Aeronautics and Astronautics Paper 2005-3067
Figure 3. Hexadominant CFD mesh generated using ICEM-CFD Tetra: 555 000 cells with a mesh refinement
in the neck region
Figure 4. Velocity distribution at a given time step, and pressure fluctuations for a selected location, result
related to CFD#1
that the inlet velocity does not effect this resonance frequency. This illustrates that the frequency obtained
is the simulation’s manifestation of the Helmholtz resonance frequency, thus it is independant of the velocity.
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B. Second computation
Even in cases where the variation in temperature is very small, when the temperature equation is not solved
for, the speed of sound is not correctly computed (details are given in appendix, see Section IV), resulting
in resonance occuring at the wrong frequency.
This simulation was the re-run with the same parameters, but including the solution of temperature.
Here, the Helmholtz resonance frequency occurs at approximately 340 Hz, a dramatic improvement on the
first calculation. It should be noted that the remaining difference between this result and that given by the
analytical formula and experiment is purely a function of the accuracy with which the pressure waves are
being propagated within the CFD calculation. A further simulation (beyond the scope of this paper) was
undertaken on the same mesh, but with a finer time-step size. This yielded a Helmholtz resonance frequency
of 352 Hz - an impressive result given the comparitively coarse mesh used. This demonstrates that STAR-CD
can accurately capture these resonance phenomena.
III. Acoustic study
A. Theory and coupling with STAR-CD
1. Theory
The theory behind the formulation used in Actran/LA has already been presented in papers by Oberai5, 6
et al. and Caro7et al., and is briefly reviewed here. Starting from the mass and momentum conservation
equations, it is possible to derive Lighthill’s equation without any assumptions, as in the original paper.8
Then, some classical assumptions, valid only in low Mach number, high Reynolds number flows where
isentropic assumptions are reasonable from an acoustic point of view, lead to dramatic simplifications. The
final equation is a true wave equation whose right-hand side term is the simplified Lighthill’s tensor:
∂2ρa
∂t2−a2
0
∂2ρa
∂xi∂xi
=∂2Tij
∂xi∂xj
where ρais the acoustic density, a0the speed of sound at rest, and Tis Lighthill’s tensor, simplified as
follows:
Tij 'ρ0vivj
where vis the velocity vector and ρ0the mean density. In the frequency domain, the variational formulation
of Lighthill’s analogy is:
Z
Ω −k2˜ρaδρa+∂˜ρa
∂xi
∂δρa
∂xi
+1
a2
0
∂˜
Tij
∂xj
∂δρa
∂xi!dΩ = 0 ∀δρa
where Ω is the computational volume, the tilde denotes a quantity computed in the frequency domain, and δρa
is the test function used in the finite elements. Details can be found in Caro7et al. and in the Actran/LA
manual.9The only missing quantity is the source term, represented by the divergence of Lighthill’s tensor
in the frequency domain (in red in the equation above)). This quantity is computed in the time domain
by STAR-CD and a Fourier transform will then be used before the field is passed to Actran/LA. The
Fourier transform is performed using a dedicated package which comes with the standard distribution of
Actran/LA. The following sections contain some details on these steps.
2. Files exchanged between STAR-CD and Actran/LA
The process of the data exchange is done as follows:
1. The user produces a file with the node coordinates of the subset of the acoustic mesh where he wants
the sources to be accounted for;
2. STAR-CD reads this file; then, during the unsteady CFD simulation, at each time step, STAR-CD
writes
^
divT at all these nodes’ locations; this is a 0-order interpolation;
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3. The user performs the Fourier transform of the result, using filters if needed;
4. The user launches the Actran/LA simulation, which gives direct access to all acoustic fields in the
finite and infinite elements, including some energy indications.
The files produced for Actran/LA by STAR-CD are written in the Hierarchical Data Format (HDF),
an open-source format used to store named matrices and developed by NCSA (see http://hdf.ncsa.uiuc.
edu/index.html). It is an OS-independent, binary format. The Scientific DataSets (SDS) of HDF version
4 are used. The files are compatible with Matlab, Mathematica, and Tecplot among others.
Each HDF file contains several datasets (i.e. arrays). Each dataset has attributes associated to it; there
is one dataset for each time step, and the associated time is stored as an attribute of the dataset. The
complete description of the exchange protocol, which is common for all the CFD codes, has been defined by
FFT.10
3. Fourier transform
The Fourier transform process has been optimized to cope with the specific difficulties of the CAA process:
both the number of CFD time steps and nodes can be very large (several thousands). Therefore, it is
not realistic to perform the Fourier transform on all points at once. The procedure, based on a modified
Slatec routine, pre-computes the sine and cosine terms of the Fourier transform, and then, only vector-vector
multiplications are required.
It is also possible to filter the time-domain values; this requires more time steps, as it will be needed
to filter only overlapping subsets of the total time series, make the Fourier transform on the subseries and
average the results afterwards. Details can be found in the Actran/LA User’s manual.9
B. Resonances of the system
A preliminary study is conducted with normalized sources put at an arbitrary location near the neck of the
resonator (the exact location has very little influence on the result). The acoustic mesh, shown in Figure 5, is
designed to allow acoustic studies up to 3 kHz; it is made of linear tetrahedral finite elements and triangular
infinite elements, the total number of degrees of freedom (dof ) is approximately 33 000.
Figure 5. Acoustic model for the monopole: trace of the mesh and boundary conditions
1. Boundary conditions
The inlet is coupled to an analytic representation of a semi-infinite duct, which makes the boundary condition
perfectly non-reflecting: all acoustic waves coming from inside the Finite Element domain can leave the
domain with no reflection and no damping. The technique, specific to in-duct problems, was presented in
Caro7et al. and the theory is briefly reviewed here.
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A non-reflecting condition is obtained by coupling the finite element model with a modal representation
of a semi-infinite duct with constant section and solid boundaries. In the duct, the acoustic pressure is the
superposition of duct modes. All propagating modes (plus a few evanescent, i.e. non-propagating, ones)
must be taken into account. According to the general theory of duct modes, only a limited number of duct
modes must be considered at a given frequency. Tests have been performed to verify that using additional
evanescent modes does not improve the solution.
For the outlet boundary, a free field propagation toward the exterior of the Finite Element domain is
assumed, and again a non-reflecting boundary condition is mandatory. The physical domain is unbounded
and the pressure fluctuations must satisfy the Sommerfeld radiation condition at large distance from the
aeroacoustic sources. This is enforced through the use of infinite elements. They are based on the multipole
expansion of the solution of the wave equation. The order of the expansion directly governs the accuracy
of the boundary condition. The infinite element method implemented in Actran/LA is an extension of a
variable order Legendre polynomial formulation whose numerical performance has been extensively studied
in Astley and Coyette.11,12
More details on the numerical implementation of the Finite/Infinite Elements and their associated Bound-
ary Conditions can be found in the Actran/LA User’s manual.9
2. Monopole
Actran/LA allows the use of a (very) fast solver, based on a Krylov method. This solver can be used
for example with normalized monopole sources provided the sources do not depend on the frequency (white
noise). Such a tool is very useful for preliminary studies, because it allows for quickly searching the pure
acoustic resonances of the system without knowing anything from the flow patterns. The computational
time is approximately 84 minutes, on a standard PC (Pentium 4, 2.4 GHz with 750 MB RAM).
The resultant Mean Square Pressure, normalized, is shown on Figure 6; the resonance frequency is at 358
Hz, which is consistent with both the measurement and the analytic formula. The resonance is thus a pure
acoustic phenomena. The vortex structures created by the impingement of the flow on the resonator neck
corresponds to a broadband source; the above shows that the resonator can be seen as a transfer function
with which the broadband spectrum should be multiplied, and the resulting acoustic spectrum contains a
peak at 358 Hz. The general idea is shown on Figure 7: the acoustic result is the “product” of the sources
and a pure acoustic Transfer Function.
Figure 6. Mean Square Pressure obtained with a normalized monopole placed in the vicinity of the resonator
neck (no CFD)
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Figure 7. The resonator can be seen as a Transfer Function: source * TF = acoustic result
C. Parametric study on the source extent
1. First result
A preliminary acoustics study was performed using the results from the first CFD simulation in order to
evaluate the influence of the extent of the acoustic source region output from STAR-CD.
The acoustic mesh used for this study is presented on Figure 8. The same types of boundary conditions
are used as in the previous section (free modal basis at the inlet and infinite elements at the outlet). The
mesh is refined in the vicinity of the resonator, and the sources will be exported only in this region.
Figure 8. Acoustic mesh used with CFD #1: the mesh is refined in the vicinity of the resonator
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The data obtained in Section Atheoretically allows the computation of the sources in the frequency
range of 25 Hz to 25 kHz, with a frequency step of 25 Hz. However, the very high frequency and low
frequency range are not solved for here: the CFD mesh is insufficiently fine to accurately capture very high
frequency sources, and the signal length too short to obtain meaningful resolution of low frequencies. Thus,
the frequency spectrum of 25 to 1000 Hz, with a frequency step of 25 Hz are computed.
The problem contains 81 000 degrees of freedom (dof). Using Actran/LA’s embedded iterative solver,
Actran/LA needs 2 hours and 30 minutes on a standard PC to solve this problem for 60 frequencies
(Pentium 4, 2.4 GHz, needs 640 MB RAM).
A series of Actran/LA calculations were carried out using the same STAR-CD source data, the variable
being the extent of the source region used. Figure 9shows the different zones considered. In each successive
Actran/LA calculation the acoustic source region size was reduced, by removing zones in descending
numerical order. A comparison was then made of the computed acoustic power level from the different
source extents.
Figure 9. Acoustic mesh used with CFD #1 and zones where the sources are accounted for
Figure 10 shows these results. Clearly, the differences are negligible. This shows that only the sources
located in the direct vicinity of the resonator neck must be accounted for. It is interesting to note that
the sources inside the resonator itself do not contribute to the far field radiated sound. It then becomes
obvious that the other sources can be neglected. Indeed, the basis of this hybrid CFD-CA approach is the
assumption that the CFD code will not be able to accurately propagate acoustic waves much beyond the
acoustic source region itself: due to the relatively low magnitude of the acoustic wave and the fact that
industrial CFD codes, using unstructure meshes, are typically only second-order accurate. Therefore, it
would be counter-intuitive to then include regions comparatively far from the dominant acoustic sources in
the data transferred to the CA code. This exercise validates that assumption whilst, better still, informing
us that their influence is negligible, as opposed to detrimental.
D. Final result and comparisons (using CFD#2)
Based on the results above, a new acoustic mesh was generated; see Figure 11. Having understood the
requisite source region extent, the acoustic source mesh can be refined in the region close to the resonator
neck giving a mesh which corresponds to 83 000 dof.
In the second CFD simulation, more time steps were performed, giving a frequency resolution of 10 Hz.
The Actran/LA computational time, for 250 frequencies, was approximately 4 hours 45 minutes using a
single-CPU opteron and the iterative solver; it requires about 800 MB of RAM memory.
The result are compared again with the experimental measurements (Figure 12). Most importantly,
the amplitude and frequency of the Helmholtz resonance frequency are obtained, demonstrating the ability
of this methodology to accurately capture this important resonance phenomenon. The secondary peak
in the simulated result, at a slightly lower frequency than the ’true’ Helmholtz resonance frequency, is
the manifestation of the Helmholtz resonance as resolved by the CFD calculation. This illustrates the
requirement for careful consideration of both the CFD and CA data when looking at these results, since this,
easily inditifiable, second peak is purely a numerical feature that is a function of the accuracy with which
the CFD model propagates the acoustic waves in the resonator.
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Figure 10. Acoustic power level obtained with CFD #1 and different source subsets
Figure 11. Acoustic mesh used with CFD #2
The global shape and average levels are well represented. There is a small over-prediction of the broadband
levels, but their very low magnitude makes this acceptible.
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Figure 12. Acoustic result obtained with CFD #2 and comparison with the measurement
IV. Conclusions
An industrial-like interior aeroacoustic problem has been treated with STAR-CD and Actran/LA.
Good agreement with acoustic measurements has been obtained. Once again this shows that the methodol-
ogy applied is robust and that unsteady ’industrial’ CFD methodologies are sufficiently accurate to obtain
good results. The importance of compressibility - in what are traditionally considered incompressible fluid
problems - has been shown. Note, this only applies to cases with strong resonance effects.
Discrepancies between the simulated and experimental results have been both identified and quantified.
It has been shown that, although possible, it is not feasible to model the complete resonance problem in
CFD alone - once again justifying the added value of this combined CFD-CA approach.
Having achieved the key objective, to successfully model both the frequency and magnitude of the
Helmholtz resonance, little further work is required. However, this remains an interesting test case, on
which new computational methods can be validated: e.g. the use of polyhedral meshing methodologies.
This work represents a small part of the work undertaken - and continuing to be undertaken - by the authors
in the forum of industrial aeroacoustic simulation. Further areas of investigation include Actran/LA’s
ability to handle liners or geometry changes.
Appendix: Computation of the speed of sound
A compressible fluid behaves like an elastic solid: a displaced particle compresses and increases the
density of an adjacent particle, which moves and increases the density of the next particle, and so on: the
disturbance travels through the medium. Only small amplitude perturbations are called acoustic waves; they
travel at a given velocity to be identified here. There are several ways to achieve this goal; an alternative
explanation can be found e.g. in Kundu.13
The equation of state for a perfect gas is written:
p=ρrT , (4)
where pis pressure, ρis absolute density, Tis absolute temperature and ris the gas constant (r=R/M
with Rthe constant for a perfect gas and Mis the mass of one mole of gas).
If the process of the acoustic propagation is considered as isentropic, then:
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American Institute of Aeronautics and Astronautics Paper 2005-3067
p
ργ= constant,(5)
where γ=Cp/Cvis the ratio between the specific heats at constant pressure and volume, respectively:
Cp=∂h
∂T p(6)
Cv=∂e
∂T v(7)
γ=Cp
Cv
,(8)
where his the enthalpy and ethe energy of the fluid. For a diatomic gas like air, γ= 1.4. One can finally
define aand write the following equations:
a2=p
ρ=γp0
ρ0
=γrT. (9)
It is then possible to demonstrate that acorresponds to the speed of sound. In one dimension, starting
from the mass and momentum equations:
∂ρ
∂t +ρ0
∂v
∂x = 0 (10)
ρ0
∂v
∂t +∂p
∂x = 0 (11)
The last three equations can be combined together and after some manipulations one obtains the following
system of equations:
u=∂Φ
∂x (12)
p=a2ρ(13)
∂2Ψ
∂x2−1
a2
∂2Ψ
∂t2= 0 ∀Ψ⊂ {p, ρ, u, Φ}.(14)
This gives a physical meaning to a, which is the celerity of the (acoustic) pressure wave and is called
the speed of sound. The same kind of demonstration is possible in 3D and is available in numerous acoustic
synopsis. The main result of this section is the value of the speed of sound given by Eq. (9):
a=pγr T . (15)
In the case of an isothermal process, the equation of state can be simplified:
p=ρrT ⇔p
ρ= constant,(16)
and finally:
aT=√rT =a/√γ. (17)
This is actually what Isaac Newton found when he first tried to compute the speed of sound: he assumed
that the process was isothermal instead of isentropic. This demonstrates that when using a CFD computation
to propagate an acoustic information, fully-compressible mode is required (solution of both temperature and
pressure). In particular, if there is an aeroacoustic coupling, whereby the acoustics influence the flow, then
it can be critical to do so. In cases no coupling occurs, however, this is probably not mandatory.
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References
1Mechel, F., Formulas of Acoustics, Springer Verlag, 2002, ISBN: 3540425489.
2Eyring, C., “Reverberation Time in ’Dead’ Rooms.” J. Acous. Soc. Am., Vol. 1, No. 1, 1930, pp. 217–241.
3Sabine, W., “Collected Papers on Acoustics,” 1993, (originally 1921).
4Peric, M., “Flow simulation using control volumes of arbitrary polyhedral shapes,” ERCOFTAC bulletin - No. 62 , 2004.
5Oberai, A., Ronaldkin, F., and Hughes, T., “Computational Procedures for Determining Structural-Acoustic Response
due to Hydrodynamic Sources,” Comput. Methods Appl. Mech. Engrg., Vol. 190, 2000, pp. 345–361.
6Oberai, A., Ronaldkin, F., and Hughes, T., “Computation of Trailing-Edge Noise due to Turbulent Flow over an Airfoil,”
AIAA Journal, Vol. 40, 2002, pp. 2206–2216.
7Caro, S., Ploumhans, P., and Gallez, X., “Implementation of Lighthill’s Acoustic Analogy in a Finite/Infinite Elements
Framework,” AIAA Paper 2004-2891, 10th AIAA/CEAS Aeroacoustics Conference and Exhibit, 10-12 May 2004, Manchester,
UK.
8Lighthill, M., “On Sound Generated Aerodynamically,” Proc. Roy. Soc. (London), Vol. A 211, 1952.
9Free-Field-Technologies-S.A., Actran 2004 Aeroacoustic Solutions: Actran/TM and Actran/LA - User’s Manual, 16,
place de l’Universit´e, 1348 Louvain-la-Neuve, Belgium, 2004.
10SA, F. F. T., “Coupling between Actran and Star-CD: Coupling Procedure,” 2004, Confidential Document.
11Astley, R. and Coyette, J., “Conditioning of infinite element schemes for wave problems,” Commun. Numer. Meth.
Engng., Vol. 17, 2001, pp. 31–41.
12Astley, R. and Coyette, J., “The performance of spheroidal infinite elements,” International Journal for Numerical
Methods in Engineering , Vol. 52, 2001, pp. 1379–1396.
13Kundu, P., Fluid Mechanics, Academic Press, Inc., 1990.
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