Implementation of Lighthill's Acoustic Analogy in a Finite/Infinite Elements Framework

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10th AIAA/CEAS Aeroacoustics Conference – 10-12 May 2004, Manchester, UK
Implementation of Lighthill’s Acoustic Analogy
in a Finite/Infinite Elements Framework
St´ephane Caro∗
, Paul Ploumhans and Xavier Gallez
Free Field Technologies SA –www.fft.be
16 place de l’Universit´e, B1348 Louvain La Neuve, Belgium
In this paper, we investigate the use of the variational formulation of Lighthill’s analogy
implemented in a Finite/Infinite Element framework. We show how this analogy can be
used to solve exterior and interior aeroacoustic problems in the frequency domain. For
exterior problems, infinite elements are used to enforce the Sommerfeld radiation boundary
condition. For interior problems, an acoustically non-reflecting exit boundary condition is
obtained by coupling the finite element domain with a semi-infinite duct, wherein pressure
fluctuations are represented by duct modes.
A derivation of the analogy is presented and is compared with the derivation of Curle’s
analogy. The implementation is described and its results on a test case are shown to be in
good agreement with others’ implementation. The equivalence with a Curle-like technique
is shown. Applications with synthetic sources are presented.
Nomenclature
Latin symbols
a0Speed of sound at rest
divT Divergence of the Lighthill’s tensor T,
divTi=∂Tij /∂xj
fFrequency
kWavenumber k=ω/a0
LChord length
MMach number
pPressure
rDistance from the source to the ob-
server, r=|x−y|
Re Reynolds number
tTime
t0Retarded time, t0=t−r/a0
TLighthill stress tensor
vFluid velocity vector
xPosition of the observer
yPosition of the source
Greek symbols
δIdentity matrix
γRatio of specific heats
λAcoustic wavelength, λ=a0/f
ωReduced frequency, ω= 2πf
ρFluid density
ρ0Fluid density in fluid at rest
ρaAcoustic density fluctuation
τViscous stress tensor
Notations
zVector z= z1
.
.
.
zm!
znNormal component of vector z,v·n
˜zFourier-transform of the variable z
∗Corresponding author: stephane.caro@fft.be
Copyright c
2004 by Free Field Technologies SA. Published by the American Institute of Aeronautics and Astronautics,
Inc. with permission.
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American Institute of Aeronautics and Astronautics Paper 2004-2891
10th AIAA/CEAS Aeroacoustics Conference AIAA 2004-2891
Copyright © 2004 by Free Field Technologies SA. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

I. Introduction
Aeroacoustics is governed by the compressible Navier-Stokes equations. A direct numerical simulation
(DNS) of these equations is impractical on most engineering problems, and this thus calls for an alternative
approach.
One alternative consists in using an acoustic analogy, as first proposed by Lighthill.1Acoustic analogies
rest on the assumption that noise generation and propagation are decoupled, that is, flow generated noise
does not impact the internal dynamics of the flow. In practice, using an acoustic analogy is a two steps
procedure. In the first step, an unsteady flow analysis is used to compute aerodynamic sources (i.e. Lighthill’s
tensor as defined in Section II.A). The second step consists in computing the propagation and radiation
of the aerodynamic sources. In the present work, the second step uses the Finite Element/Infinite Element
method.
In the first part of this paper, the derivation of the acoustic model is presented, starting from Lighthill’s
analogy. A first application for exterior acoustics is presented in the second part, using a point aerodynamic
source (i.e. a point quadrupole); the method is then compared with a Curle-like approach, and results are
also compared with those presented by Oberai2et al. The third and fourth parts present interior aeroacoustic
applications, in 2D and 3D, respectively.
The developments presented herein have been implemented in the commercial software package Actran.
II. Derivation of the acoustic model
The aeroacoustic module of Actran is based on Lighthill’s acoustic analogy implemented in its varia-
tional form following the approach first proposed by Oberai et al.3Moving walls, if any, are assumed to lie
outside the computational domain, Ω.
II.A. Lighthill’s analogy
The mass and momentum conservation equations governing the motion of a fluid under no external forces
can be written
∂ρ
∂t +∂
∂xi
(ρvi) = 0 (1)
∂
∂t (ρvi) + ∂
∂xj
(ρvivj) = −∂p
∂xi
+∂τij
∂xj
,(2)
where ρis the fluid density, vis the velocity, pis the pressure and τis the viscous stress tensor. By
combination of Eqs. (1) and (2), Lighthill’s analogy1is found as
∂2
∂t2(ρ−ρ0)−a2
0
∂2
∂xi∂xi
(ρ−ρ0) = ∂2Tij
∂xi∂xj
,(3)
where ρ0denotes the density at rest, a0the speed of sound at rest and Tis the Lighthill’s tensor defined as
Tij =ρvivj+(p−p0)−a2
0(ρ−ρ0)δij −τij .(4)
Eq. (3) and Eq. (4) are the starting point for the developments presented in Sections II.B and II.C. For a
Stokesian perfect gas like air, in an isentropic, high Reynolds number and low Mach number flow, Lighthill’s
tensor Tcan be approximated by
Tij 'ρ0vivj.(5)
Away from the source region, the density fluctuations, ρ−ρ0, correspond to acoustic density fluctuations.
This is highlighted by rewriting Eq. (3) as
∂2ρa
∂t2−a2
0
∂2ρa
∂xi∂xi
=∂2Tij
∂xi∂xj
,(6)
where ρa=ρ−ρ0. Additional details can be found e.g. in Lighthill1or Goldstein.4
In practice, to use Lighthill’s analogy in the two steps procedure described above, one considers that
there is no aeroacoustic coupling, that is the influence of the acoustics on the flow is weak, and Lighthill’s
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tensor (i.e. the right hand side of Eq. (3)) is considered to be independent from the density fluctuation,
ρ−ρ0. This makes Eq. (3) a true wave equation.
The present approach to treat aeroacoustic problems is intended to be used for isentropic flows where
the convection and refraction effects can be neglected with respect to the other effects, which is generally
true if the Mach number is low enough (below 0.2 - 0.3). Some studies on the flow convection effects at such
Mach numbers can be found e.g. in Jacob.5
II.B. Integral formulation of Lighthill’s analogy
The integral formulation of Lighthill’s analogy was first derived by Curle,6and is best known as Curle’s
analogy. The main steps of his developments are reproduced here to allow an easier comparison with the
developments of the variational formulation of Lighthill’s analogy that will be presented in Section II.C.
The wave equation (3) can be rewritten in its integral form (Stratton7),
ρ−ρ0=1
4πa2
0Z
Ω∂2Tij
∂xi∂xj
1
rt0
dy+1
4πZ
Γ1
r
∂
∂n (ρ−ρ0) + 1
r2
∂r
∂n (ρ−ρ0) + 1
a0r
∂r
∂n
∂
∂t (ρ−ρ0)t0
dy,(7)
where Γ denotes the boundary of the acoustic domain, r=|x−y|, and the integrals’ arguments are evaluated
at the retarded time t0=t−r/a0. Starting from this equation, Curle transforms the right-hand side by
integrating the volume integral by parts using Green’s theorem twice, and applies some transformations to
the surface integral to obtain
ρ−ρ0=1
4πa2
0
∂2
∂xi∂xjZ
ΩTij
rt0
dy+1
4πa2
0Z
Γni
1
r
∂
∂yjTij +a2
0(ρ−ρ0)δij t0
dy
+1
4πa2
0
∂
∂xiZ
Γnj
1
rTij +a2
0(ρ−ρ0)δij t0
dy.(8)
By substituting for Tij as given by Eq. (4), this becomes
ρ−ρ0=1
4πa2
0
∂2
∂xi∂xjZ
ΩTij
rt0
dy+1
4πa2
0Z
Γni
1
r
∂
∂yjρvivj+ (p−p0)δij −τij t0
dy
+1
4πa2
0
∂
∂xiZ
Γnj
1
r(ρvivj+ (p−p0)δij −τij )t0
dy.(9)
From the momentum conservation equation (2), we have that
ni
∂
∂yjρvivj+ (p−p0)δij −τij =−ni
∂
∂t ρvi.(10)
If each surface is fixed or vibrating in its own plane,
vn=vini= 0 .(11)
Equation (9) finally reduces to
ρ−ρ0=1
4πa2
0
∂2
∂xi∂xjZ
Ω
Tij y, t −r
a0
rdy+1
4πa2
0
∂
∂xiZ
Γ
Piy, t −r
a0
rdy,(12)
with
Pi=p δij −τij nj,(13)
where Pis the force per unit area exerted by the fluid on the solid boundaries. Equation (12) is known
as Curle’s analogy (it could also be termed the integral formulation of Lighthill’s analogy) and highlights
the influence of solid boundaries upon aerodynamic sound: ...one can look upon the sound field as the sum
of that generated by a volume distribution of quadrupoles and by a surface distribution of dipoles.6The
strength of the dipole per unit area is exactly equal to the force per unit area exerted by the fluid on the
solid boundaries.
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II.C. Variational formulation of Lighthill’s analogy
The variational formulation of Lighthill’s analogy was first derived by Oberai3,2et al. It is here re-developed
to allow an easy comparison with the development of Curle’s analogy presented in Section II.B. The strong
variational statement associated to Eq. (3) can be written:
Z
Ω∂2
∂t2(ρ−ρ0)−a2
0
∂2
∂xi∂xi
(ρ−ρ0)−∂2Tij
∂xi∂xjδρ dx= 0 ∀δρ (14)
where δρ is a test function. The spatial derivatives are integrated by parts using Green’s theorem, to obtain
the weak variational form:
Z
Ω∂2
∂t2(ρ−ρ0)δρ +a2
0
∂
∂xi
(ρ−ρ0)∂δρ
∂xi
+∂Tij
∂xj
∂δρ
∂xidx=Z
Γa2
0
∂
∂xi
(ρ−ρ0)ni+∂Tij
∂xj
niδρ dΓ(x).
(15)
By substituting the right hand side of Eq. (4) for Tij in the surface integral, Eq. (15) becomes
Z
Ω∂2
∂t2(ρ−ρ0)δρ +a2
0
∂
∂xi
(ρ−ρ0)∂δρ
∂xi
+∂Tij
∂xj
∂δρ
∂xidx=Z
Γ
∂
∂xj
(ρvivj+ (p−p0)δij −τij )niδρ dΓ(x).
(16)
The righthand side of Eq. (16) is the natural boundary condition associated with the weak variational
problem. From the momentum conservation equation (2), we have that
ni
∂
∂xj
(ρvivj+ (p−p0)δij −τij ) = −ni
∂
∂t (ρvi).(17)
If each surface is fixed or vibrating in its own plane, this term vanishes (see Eq. (11)), which corresponds to
the natural boundary condition associated with the weak variational problem. Equation (16) finally reduces
to: Z
Ω∂2
∂t2(ρ−ρ0)δρ +a2
0
∂
∂xi
(ρ−ρ0)∂δρ
∂xi
+∂Tij
∂xj
∂δρ
∂xidx= 0.(18)
This is the variational formulation of Lighthill’s analogy.
II.D. Comparison of the integral and variational formulations of Lighthill’s analogy
From Section II.B and II.C, it is seen that the developments of the integral (i.e. Curle’s analogy) and
variational formulation of Lighthill’s analogy proceed along similar lines and involve similar steps. Both
formulations are exact and contain no approximation.
The starting point is however different in both cases. Curle’s analogy is derived from an integral equation
while Eq. (18) is derived by applying a weighted residual procedure to Lighthill equation. These two different
starting points also correspond to two different classes of numerical methods: integral (or boundary element)
method and finite element method, respectively. Consequently, Curle’s analogy is the natural starting point
for an implementation in a boundary element method, whereas the variational formulation of Lighthill’s
analogy is the natural starting point for finite element methods.
Curle’s analogy clearly highlights the distinct roles of the bulk of the fluid (volume integral of quadrupole
distribution) and of solid boundaries (surface integral of dipole distribution) on the aerodynamic sound,
while the variational formulation of Lighthill’s analogy does not separate the contributions.
The treatment of the solid surfaces is however different in the two methods. On one side, Curle’s analogy,
Eq. (12), is purely explicit and uses the free field Green’s function. Reflection of the aerodynamic sound
by solid surfaces must be accounted for by the surface source term, P. On the other side, the variational
formulation of Lighthill’s analogy is implicit and its practical use requires a discretization followed by a system
resolution (finite element method). The interactions between the solid surfaces and the aerodynamic sound
are accounted for by the acoustic solver. If these effects are significant (e.g. for non-compact solid boundaries,
see Section III, and interior aeroacoustic problems, see Section IV), this can be a crucial advantage of the
variational formulation of Lighthill’s analogy over Curle’s analogy. The reason is that usually source terms
are obtained from a CFD simulation that does not represent acoustic pressure fluctuations accurately enough.
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II.E. Harmonic perturbations
The implementation uses a frequency-domain formulation. Harmonic perturbations are considered, such
that any perturbed quantity pcan be written:
p(x, t) = Re˜p(x)eiωt.(19)
Lighthill’s equation Eq. (6) can be rewritten
∂2˜ρa
∂xi∂xi
+k2˜ρa=−1
a2
0
∂2˜
Tij
∂xi∂xj
,(20)
where k=ω/a0is the acoustic wavenumber. Equation (20) shows that using Lighthill’s analogy in a
frequency domain computation requires the Fourier transform of Tin the source term. In the frequency
domain, the product vivjbecomes a convolution: every frequency component of Tis thus a function of all
frequency components of v.
The weak variational statement associated to Eq. (20) is
Z
Ω k2˜ρaδρ −∂˜ρa
∂xi
∂δρ
∂xi−1
a2
0
∂˜
Tij
∂xj
∂δρ
∂xi!dx= 0 .(21)
In practice, if a CFD simulation is used to compute T, the derivative ∂˜
Tij /∂xjis best computed by the
CFD code on the CFD mesh. The reason is that CFD simulations generally use the finite volume method,
not the finite element method, and the way to compute derivatives differ in the two methods. Moreover, the
CFD mesh is usually finer than the acoustic mesh, which leads to a better approximation of derivatives. The
complex vector
^
divT is defined for each node of the acoustic mesh in the frequency domain, and is used to
describe the sources in the frequency domain:
]
divT i=∂˜
Tij
∂xj
.(22)
II.F. Treatment of boundary conditions
II.F.1. Natural boundary condition
When using the variational formulation of Lighthill’s analogy, the natural boundary condition associated to
Eq. (16) is applied on the solid boundaries that are in contact with the aeroacoustic sources region.
II.F.2. Admittance boundary condition
If acoustic liners are present, they are accounted for with a normal admittance boundary condition (also
called soft-wall boundary condition) applied on part of the solid boundary, which relates the normal velocity,
˜vn, to the acoustic pressure:
˜vn=A˜p . (23)
II.F.3. Radiation boundary condition
For applications to external aeroacoustic problems, 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 is an extension of a variable order Legendre
polynomial formulation whose numerical performance has been extensively studied (Astley and Coyette8,9).
More details on the numerical implementation can be found in the Actran User’s manual.10
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II.F.4. Coupling with a modal representation of a duct
Infinite elements are used as non-reflecting boundary condition for exterior problems. For interior problems,
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, according to
˜p(x0, y0, z0) =
∞
X
s=0
˜ps(x0, y0)A+
se−ik+
zsz0+A−
se−ik−
zsz0,(24)
where x0are the coordinates in a system of reference whose vector ezis aligned with the duct axis and points
outwards the finite element region; ˜psare the eigenfunctions of the Laplace operator reduced to the duct
section; A+
sand A−
sare the transmitted and reflected mode amplitudes, respectively; and k+
zs and k−
zs are the
transmitted and reflected axial wavenumbers, respectively. The eigenfunctions are normalized according to
Z
S
˜ps(x0, y0) ˜pr(x0, y0)dS =δsr .(25)
The exact expression of duct modes are detailed hereafter for a two-dimensional rectangular duct, and for
a circular duct, as these two types of ducts will be used in applications hereafter (see Section IV and V,
respectively).
When a finite element region is coupled to a semi-infinite duct to obtain a non-reflecting boundary
condition, reflected mode amplitudes are set to zero and transmitted mode amplitudes are found as part of
the problem’s solution.
Two-dimensional rectangular duct In a two-dimensional duct, the representation in terms of duct
modes is given by
˜p(x0, z0) =
∞
X
s=0
A+
sNscos(kxsx0)e−ik+
zsz0,(26)
where ais the height of the duct. The transverse and axial wavenumbers, kxs and k+
zs, are given by
k+
zs =pk2−k2
xs (27)
and
kxs =sπ
a.(28)
The normalization factor Nsis given by
Ns=(1 for s= 0 (plane wave),
√2 for s6= 0.(29)
Circular duct The representation in terms of duct modes of the pressure inside a circular duct is given
by
˜p(r, θ, z0) =
∞
X
m=−∞
∞
X
n=1
A+
mnNmn J|m|(krmn r)eimθe−ik+
zmnz0,(30)
where Jmis the Bessel function of order m,krmn and k+
zmn are the radial and longitudinal wavenumbers,
respectively, and Nmn is a normalization factor. The axial wavenumber, k+
zmn, is given by
k+
zmn =pk2−k2
rmn ,(31)
where krmn is obtained by resolving the hard wall boundary condition in r=a,
J0
m(krmna) = 0 .(32)
For any particular value of m, the nth root of Eq. (32) provides the value related radial wavenumber krmn .
Notice that for m= 0, the first root of Eq. (32) is kr01 = 0, which corresponds to a plane wave. The
normalization factor Nmn is chosen to satisfy the normalization condition given by Eq. (25), so that
Nmn =


1 for krmn = 0 and m= 0 (plane wave)
1−m2
k2
rmna2J2
m(krmna)−1
2for krmn 6= 0 (33)
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III. First application: trailing-edge noise
III.A. Problem description
A first application for external acoustics is presented in this section. The profile studied is a 2D eppler387,
commonly used for glider wings. The chord length is L= 1 m. The convection effects are neglected, the
profile is thus assumed to be in a stationary medium. A single quadrupole source T12 is placed 8 mm
above the trailing-edge, at a place where the turbulence would normally create such sources. The source
is described using four dipoles, 1 mm apart from one another, situated around the quadrupole position as
depicted in Figure 1.
Figure 1. Problem definition: shape of the eppler387 profile and definition of the source
This problem is a simplified model for trailing edge noise modelling, in the sense that it uses a simplified
source while the acoustic propagation phenomena (reflection, diffraction) due to a a true source of sound are
also found in the simplified model. The use of an analytical source description is convenient here, because
the source region is known but no unsteady CFD results are available. The very same problem was treated
by Oberai et al. in a similar fashion.2
At sufficiently high frequencies, the sound is diffracted by the trailing edge of the profile, whereas at lower
frequencies the acoustic field is simply reflected by the profile, leading to a dipolar directivity. Simulations
have been performed for the same frequencies as Oberai:242.5, 85, 170, 340, 680 and 1360 Hz, corresponding
to λ/L values of 8 to 1/4.
The boundary conditions used in this section have been described in Section II.F: the natural boundary
condition is used along the profile, whereas infinite elements are used at the outer domain boundary.
III.B. Mesh refinement study
Figure 2. Topology and mesh related to mesh2
The meshes presented were generated using the commercial software package Patran and consist of
quadratic quadrangular elements. The size sof any element must be lower than λ/4, where λis the acoustic
wavelength at the considered frequency, f. At the highest frequency, f= 1360 Hz (i.e. for λ/L = 1/4), we
have that s= 0.0625 m.
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Furthermore, the mesh has to be fine enough to allow an accurate description of the sources. The source
used in the present case has an extent of 1 mm2. The mesh must be that fine at least in the source region.
In order to conduct a mesh convergence study, three finite elements/infinite elements meshes were gen-
erated. The three meshes share the same topology, as depicted in Figure 2, but differ in their element size,
as summarized in Table 1.mesh1 is nearly uniform, whereas mesh2 and mesh3 are refined only in the source
region. The difference between mesh2 and mesh3 is that all elements of the former mesh are smaller than
λ/4 whereas some elements of the latter mesh are not. A detailed view of the source region for mesh2 is
presented on Figure 3.
name size in source
region
size out of the
source region
nb. of
FE
nb. of
IE
order
of IE
mesh1 0.002 0.002 122 307 1027 20
mesh2 0.002 0.05 809 67 10
mesh3 0.003 0.07 468 48 10
Table 1. Description of the 3 meshes used: size of the elements in the source region, size of the elements
outside the source region, total number of Finite Elements (FE), number of Infinite Elements (IE) and order
of the IE
Figure 3. Detail of mesh2 near the trailing edge and location of the element used to describe the source with
4 dipoles
Results were post-processed with the commercial software package Matlab. Figure 4shows the mag-
nitude of the acoustic pressure (in Pa) on a circle of radius 30 m, centered about the trailing edge. In the
far field, the three meshes yield similar results. Results obtained with mesh2 are nearly identical to the
ones obtained with mesh1, which suggests that the mesh has to be fine only in the source region. Results
obtained with mesh3 show some differences, confirming that the mesh size criterium based on the wavelength
is fundamental.
III.C. Assessment of near field results
Near field results obtained with ACTRAN using mesh1 are compared to those from Oberai.2Figure 5shows
the amplitude of the acoustic pressure on a circle with radius 1.2 m centered about a point located at 90 %
of the chord. Both results are normalized with respect to their maximum value. The two sets of results seem
very consistent.
Figure 6and Figure 7show respectively the results in the far field and in the near field for all the
frequencies treated by Oberai et al.
The very slight differences in far field (not presented) can be attributed to the fact that the mesh used by
Oberai et al. was dedicated to an LES computation and was not designed specifically for acoustic purposes:
the elements used are linear triangles, and their shape in the source region is not well suited for accurate
acoustic computations. Moreover the Sommerfeld radiation condition was enforced using a DtN approach
in Oberai et al.while infinite elements were used in the present work.
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Figure 4. Module of the complex pressure field computed using Actran for λ/L = 1/4and for 3 different meshes
Figure 5. Module of the complex pressure field computed using Actran for λ/L = 1/4, and comparison with
Oberai’s result2
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(a) λ/L = 8 (b) λ/L = 4
(c) λ/L = 2 (d) λ/L = 1
(e) λ/L = 1/2 (f ) λ/L = 1/4
Figure 6. Far field directivity (amplitude of the acoustic pressure, in Pa) obtained with mesh2 for different
frequencies, at a distance of 30 m away from the trailing edge
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(a) λ/L = 8 (b) λ/L = 4
(c) λ/L = 2 (d) λ/L = 1
(e) λ/L = 1/2 (f ) λ/L = 1/4
Figure 7. Near field directivity (real part of the acoustic pressure, in Pa) obtained with mesh2 for different
frequencies, at a distance of 1.2 m away from the trailing edge
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III.D. Comparison with an explicit integral method
In their original work, Oberai et al. present a comparison of the directivity computed using the present
method, and the directivity computed using explicit integral methods like Curle’s6or Ffowcs-Williams and
Hall’s.11 In these comparisons, the authors make the assumption that the profile is compact and use a Green
function technique to solve the acoustic field in far field. The source terms used are the pressure fluctuations
along the profile as computed by their acoustic code.
Another computation of the far field noise can be made, using again the pressure fluctuations along the
profile as computed by Actran and an integral method based on the radiation of the surface term in Curle’s
analogy, but with no compacity assumption.
A comparison can be made between the diffracted part of the Actran result (full solution minus primary
quadrupolea) and this solution. Results are presented Figure 8. If the assumption of compacity is not made
when using an integral method, using accurate pressure data along the profile thus lead to the correct
diffracted acoustic field.
These results also show the growing importance of the scattered field as the wing profile becomes acous-
tically non-compact, that is, as the the chord length increases relative to the acoustic wavelength.
Figure 8. Module of the complex pressure field computed using Actran for λ/L = 1/4with mesh1: comparison
between the diffracted pressure (full solution minus primary quadrupole) and the result obtained using an
explicit integral method
aOnly the diffracted part of the primary quadrupole source is predicted with the integral method when the volume integral
is dropped
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IV. Second application: 2D pipe
IV.A. Problem description
A second 2D application is presented, using again a synthetic source, placed inside an infinite duct. The
source is placed near a diaphragm, which is typical of air conditioning problems: it represents the turbulence
sources that may exist near this diaphragm (Figure 9). The inlet and the outlet are coupled to an analytic
representation of a semi-infinite duct, wherein the pressure is represented by a series of duct modes (see
Section II.F.4).
Figure 9. Problem definition of the 2D pipe with free modes
A single cartesian mesh, consisting of quadratic quadrangles was generated with the software package
BoxProb. Both the acoustic wavelength and the extent of the source region were considered when setting
the element size, as explained in Section III. The mesh is presented on Figure 10. A two-dimensional model
such as the one presented below runs in a few minutes on a 2.8 GHz Pentium 4 processor.
Figure 10. Mesh used for the 2D pipe with free modes (left), and zoom around the quadrupole (right)
At both sides of the duct, the acoustic pressure is expressed in terms of duct modes as described in
Section II.F.4. All propagating modes plus a few evanescent ones must be taken into account. According to
Eq. (26), only the values of sin the range [−smax, smax] must be considered, with
smax =E(2 ∗f∗h/a0)+1.(34)
In Eq. (34), Edenotes the truncated value and fis the considered frequency. Tests have been performed
to verify that using additional evanescent (i.e. non-propagating) modes does not improve the solution.
IV.B. Results
The results presented here have been postprocessed using Patran. Figure 11 and Figure 12 show the real
part of the acoustic pressure and its RMS value, respectively.
bBoxPro is a utility program embedded in Actran
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Figure 11. Real part of the acoustic pressure in Pa, f= 10 kHz
Figure 12. RMS of the acoustic pressure in Pa, f= 10 kHz
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V. Third application: 3D pipe
V.A. Problem description
The third application presented in this paper deals with sound propagation in a 3D geometry, typical of air
conditioning modules. The module consist of two circular inlet ducts and one square outlet duct. The first
inlet duct has a constant diameter of 7 cm, while the diameter of the second inlet duct increases from 4 cm
at the inlet plane to 7 cm through an abrupt contraction. The outlet is 10 cm wide.
A synthetic source (single quadrupole) is placed in the vicinity of the junction between the three ducts;
this source is a (very) rough modelling of the turbulent sources in this region. The source has a pure
frequency of 7 kHz.
As the inlet tubes are actually connected to an even more complex set of tubes where no sound generation
takes place, acoustic boundary conditions at the inlet planes are modelled by coupling each inlet duct to
a semi-infinite duct wherein the pressure is represented by outgoing duct modes (see Section II.F.4). In
contrast, as the outlet duct actually ends in the open air, far field sound propagation is modelled with
infinite elements, defined on a hemisphere connected to the outlet plane.
The acoustic domain is meshed with 420000 linear tetrahedra, according to the preference of the mesh
generator (Patran) for 3D problems. The basis of the infinite elements are linear triangles (5276). The
total number of nodes (degrees of freedom) amounts to 76265. The mesh is presented on Figure 13.
Figure 13. Mesh used for the 3D pipe problem
This problem is solved on a PC (2.8 GHz Pentium 4, 1 Go RAM) in 21 minutes. Memory requirements
are kept under control by using an out-of-core solver.10 A parallel version of solver exists and was presented
in Ploumhans12 et al.
V.B. Results
Results were post-processed using Patran. Figures 14 and 15 show the real part of the acoustic pressure
and its RMS value, respectively. It is visible on Figure 14 that only one low-order mode propagates inside
the inlet duct with the smallest cross-section (higher-order modes are actually cut-off by the contraction).
The spots seen on Figure 15 suggest that stationary waves may exist in this system.
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Figure 14. Real part of the acoustic pressure in Pa for the 3D pipe problem, f= 7 kHz
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Figure 15. Real part of the acoustic pressure in Pa for the 3D pipe problem, f= 7 kHz
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VI. Conclusions
The variational formulation of Lighthill’s analogy, first presented by Oberai2et al., was implemented in
the commercial software package Actran. The derivation of this analogy was re-presented in this paper
and was shown to involve the same steps as the derivation of Curle’s analogy.
The implementation was then used to treat exterior and interior aeroacoustic problems, in 2D and in 3D.
The method presented here, which makes use of synthetic sources, can be very useful during the design
stage, since the cost of an acoustic computation is usually much lower than the cost of an accurate unsteady
CFD. However, for quantitative predictions, unsteady CFD results are required to evaluate Lighthill’s tensor
and obtain realistic aeroacoustic sources. For this reason, interfaces with major CFD codes were developed.
Further improvements include the capability to perform coupled vibro-aero-acoustic simulations, and to
account for the convection effects in the acoustic propagation, making it possible to accurately treat problems
at higher Mach numbers.
Acknowledgments
The authors are greatly indebted to Dr Assad Oberai, Boston University, for many helpful discussions
on the subject of this paper.
References
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AIAA Journal, Vol. 40, 2002, pp. 2206–2216.
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due to Hydrodynamic Sources,” Comput. Methods Appl. Mech. Engrg., Vol. 190, 2000, pp. 345–361.
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Stuttgart, 2002.
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place de l’Universit´e, 1348 Louvain-la-Neuve, Belgium, 2004.
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