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Encyclopedia of Aerospace Engineering
,Online ©2010 John Wiley &Sons,Ltd.
This article is ©2010 John Wiley &Sons,Ltd.
This article was published in the
Encyclopedia of Aerospace Engineering
in 2010 by John Wiley &Sons,Ltd.
Computational Aeroacoustics
Eric Manoha
1
,St
´
ephane Redonnet
1
and St
´
ephane Caro
2
1
CFD and Aeroacoustics Department,ONERA,Ch
atillon,France
2
Free Field Technologies SA,Axis Park LLN,Belgium
1 Introduction 1
2 Flow Noise Sources Mechanisms and Modelling 2
3 Overview of CAA Approaches 3
4 Finite Difference High Order Method 4
5 Discontinuous Galerkin Method 10
6 Summary and Future Need 14
References 15
1 INTRODUCTION
Due to increasing environmental constraints,acoustic criteria
are now considered by aircraft manufacturers in the early
stages of future aircraft projects.Inthis context,besides semi-
empirical tools based on analytical developments calibrated
using experimental data,innovative numerical methods take
a growing part in the industrial process.
Aircraft noise results fromthe contribution of a large num-
ber of aeroacoustic sources associated with (i) moving and
ßuctuating forces (e.g.,fans,propellers,rotors,turbines),
(ii) isolated turbulence (jets),or (iii) interaction of ßows
(most often turbulent) with any components or surfaces of
the airframe (landing gears,airfoil trailing edges,slats,ßaps,
cavities).
The acoustic waves generated by these aerodynamic
sources initially propagate in the near Þeld in a com-
plex medium,refracting through turbulence and mean ßow
gradients,reßecting on surfaces,and scattering on geometri-
cal singularities such as edges.Then,after travelling a certain
distance comparable to the aircraft overall dimension from
the sources,the acoustic waves radiate towards a far-Þeld
observer through a mediumwhich,up to reasonable distance
fromthe aircraft canbe consideredas a laminar uniformmean
ßow.
All the aeroacoustic mechanisms involved (source gen-
eration,acoustic propagation,refraction and scattering) are
relevant to the physics of ßuid dynamics,so they could be
theoretically simulated by solving the NavierÐStokes equa-
tions on a large domain extending from the source region to
the far Þeld.However,such an approach faces the difÞculty
of handling ßuctuations covering a very extended range of
length scales and amplitudes:turbulent eddies or structures
that generate noise have small length scales but high energy
amplitudes,whereas the resulting acoustic waves propagat-
ing away with the same frequency have comparatively very
long wavelengths but small energy amplitude.Consequently,
such direct computation requires large computing resources
and can only be achieved for academic cases.
For more realistic cases,it is often necessary to parti-
tion the problemand simulate separately (i) the noise source
generation,(ii) the acoustic propagation through a complex
non-uniform ßow,and (iii) the acoustic radiation through a
quiescent mediumtowards the observer (Figure 1).
For noise source generation,a wide panel of methods is
available,from the simplest semi-empirical models to the
most sophisticatedunsteadyßowsimulations basedonDirect
Numerical Simulation (DNS) or innovative developments
of DNS such as the Large Eddy Simulation (LES).The
applicability and potentialities of these methods obviously
directly depend on what kind of noise source is aimed at.
DOI: 10.1002/9780470686652.eae344
Encyclopedia of Aerospace Engineering
,Online ©2010 John Wiley &Sons,Ltd.
This article is ©2010 John Wiley &Sons,Ltd.
This article was published in the
Encyclopedia of Aerospace Engineering
in 2010 by John Wiley &Sons,Ltd.
2 Acoustics and Noise
Sources region
Turbulent flows around bodies
Fluctuating / moving forces
Low turbulence
Non-uniform
mean flow
Source simulation
Steady CFD (RANS)
+
stochastic models
Unsteady CFD (DNS, URANS, LES, DES, NLDE)
Acoustic
propagation
through non-
uniform flow
using Euler
equations
Integral methods
• FW-H
• Kirchhoff
• BEM
Farfield observer
Outer region
Uniform flow
Figure 1.
Hybrid approaches for the numerical simulation of aeroa-
coustic mechanisms generation and radiation.
In the present ÒAcoustics and NoiseÓ Section,the preceding
chapters are speciÞcally devoted to individual aircraft noise
sources (jet,turbomachinery,landing gear,high lift devices,
or propellers),and it is assumed that the numerical methods
available for the simulation of the aeroacoustic mechanisms
involved are extensively presented,compared,and evaluated
in detail in these subsections.In the present subsection,only
the main methods will be cited and brießy described.
On the other hand,the simulation of the acoustic radi-
ation through a uniform and non-turbulent mean ßow up
to an observer in the far Þeld generally relies on integral
methods suchas the Kirchhoff integral,the Ffowcs WilliamsÐ
Hawkings equation,or the Boundary Element Method
(BEM).The most interesting advantage of these methods is
that they all assume that the acoustic propagation is fully
modeled by a very simple GreenÕs function,the main conse-
quencebeingthat thepropagationmediumdoes not needtobe
meshed.Another consequence is that the CPU and memory
requirements are reasonable,which explains why these meth-
ods have been widely used since the 1960s.Practical applica-
tionof theseintegral methods tospeciÞcaircraft noisesources
is extensively detailed in previous subsections of this chapter.
More recently,roughly in the last 15 years,under the
generic name of ÒComputational AeroAcousticsÓ (CAA),a
considerable amount of research effort has been dedicated to
the development of reliable numerical methods for the sim-
ulation of acoustic propagation in an Òintermediate domainÓ
where the turbulence rate is low,so the domain is actually
free of any noise source,but the mean ßow gradients are
signiÞcant,which means that the acoustic propagation may
be subject to strong refraction effects (Colonius,1997).This
research has unveiled speciÞc accuracy and non-dispersion
requirements,which have rapidly generalized the use of
(often linearized) Euler equations discretized with Þnite
difference high-order spatial schemes on multi-block struc-
tured grids.More recently,additional work was devoted to
develop other methods,with the same requirements,but
working on unstructured grids,which are known to be much
more convenient to model complex geometries.The best-
known method in this category is the Discontinuous Galerkin
Method (DGM).
It should be noted that the generic name ÒCAAÓ was
Þrst introduced to denote this young and rapidly growing
discipline,devoted to the numerical simulation of acoustic
propagationincomplexßows.This speciÞc label is nowoften
used in a wider sense,including all the numerical methods
used for aeroacoustic application.This extension could be
seen as inappropriate,considering that most of the methods
used for the simulation of the noise source actually belong
more to the domain of Computational Fluid Dynamics (CFD)
than aeroacoustics or CAA.
The present Chapter is organized as follows.
Section 2 is devoted to the mechanisms that generate ßow
noise:it brießy recalls (i) the involved physical mechanisms
(classiÞed by source nature) and (ii) the methods (namely
CFD) that are typically used to model or simulate them.
Section 3 presents general aspects and requirements for
the simulation of acoustic propagation and radiation within
complex media,and introduces the two main families of
numerical approaches (namely CAA) used to simulate it,
which are developed in the next sections.
Section 4 is devoted to the block-structured CAA
approach,basedonÞnitedifferencehigh-order schemes oper-
ating on multi-block structured grids.The theory of the
method is detailed;then its numerical implementation is
presented.The performance is presented,based on several
examples,from an academic 2D case up to more complex
industrial 2D and 3D conÞgurations.
Section5describes the unstructuredCAAapproach,based
on the Discontinuous Galerkin Method.Again,the theory,
the implementation,and the performance of the method are
described;then several examples are given.
2 FLOWNOISE SOURCES MECHANISMS
AND MODELLING
2.1 Tonal noise sources
Most tonal sources in aircraft noise are generated by rotat-
ing machinery,either in propulsion systems (fan,propeller,
DOI: 10.1002/9780470686652.eae344
Encyclopedia of Aerospace Engineering
,Online ©2010 John Wiley &Sons,Ltd.
This article is ©2010 John Wiley &Sons,Ltd.
This article was published in the
Encyclopedia of Aerospace Engineering
in 2010 by John Wiley &Sons,Ltd.
Computational Aeroacoustics
3
compressor,and turbine) or in lifting systems (helicopter
rotor).The most basic mechanism involved in these tonal
sources is the rotation of the steady forces (lift/drag) exerted
by the mean ßow on the blade sections.Such aeroacoustic
generation/radiation of tonal sources can thus be modeled
from the computation of these forces using steady CFD,
assuming that the incoming ßow is axisymmetric w.r.t.the
machinery axis,and then using an integral method such as
the Ffowcs WilliamsÐHawkings integration to compute the
noise radiated in free Þeld fromthe moving blades.
In reality,things are more complex,the Þrst reason being
that the ßow ingested by the rotating machinery is generally
not axisymmetric,which induces azimuthal ßuctuations on
the forces,which are not accessible with steady CFD.In this
case,one may resort to unsteady RANS computation with
Chimera rotating grids.
Helicopter noise generated by the main rotor is a special
case because the ßow ingested by the rotor fundamentally
depends on the ßight phase (cruise,hover,and descent).
In certain conditions,blades may intercept wake structures
generated by other blades,which can only be simulated
through speciÞc modeling of these structures,but still gener-
ate periodic pressure ßuctuations onthe blades,andthus tonal
noise.
2.2 Broadband noise sources
Turbulence is by essence a random and broadband mecha-
nism,so whenever turbulence is involved,broadband noise
may be generated.Turbulence may result from the natural
evolution of the ßow,or be generated and possibly ampliÞed
by aircraft components the ßow interacts with.
Turbulence is generated in sheared ßows,either in jets or
in wall boundary layers and wakes.Free turbulence struc-
tures generate noise,but this noise generation is even more
efÞcient when these structures interact with solid bodies,and
especially on singularities such as airfoil trailing edges.
Turbulence modeling has been an extensive topic of
research since the 1940s,and many prediction models have
been built for the noise generated by turbulence,possibly
interacting with solid boundaries.However,realistic turbu-
lencepredictions haveonlybeenpossiblewiththespectacular
progress of unsteady CFD,driven by the exponential CPU
and memory capabilities of computers.
The basis of unsteady CFD is Direct Numerical Simula-
tion (DNS),which solves the exact NavierÐStokes equations
on ßowgrids,without any hypothesis or modeling.With this
technique,the only lower limitation in the size of the resolved
turbulent structures is the grid Þneness.Consequently,this
method requires very high grid resolution,especially in the
highly sheared ßow regions such as wall boundary layers or
wakes,which leads to large CPUand memory requirements.
Innovative developments of DNS are rapidly progressing,
mostly based on modeling the turbulent structures smaller
than the grid cell (Subgrid Scale Model) such as in LES.
However,it appears that LES is very efÞcient in simulating
unsteady ßows with large detached or separated structures,
but that the method is still very expensive in simulating Þne
turbulence in attached ßows like boundary layers.This is
why new methods are rapidly growing,based on selectively
combining steady and unsteady ßow solutions in the same
computation.For example,in Detached Eddy Simulation
(DES),RANS is used in attached ßow regions and LES is
used in separated zones.
Another evolution of LES is the Non-linear Disturbance
Equations (NLDE) method,inwhichthe physical magnitudes
in the NavierÐStokes equations are split into (i) a mean non-
variable component and (ii) a perturbed component whose
evolutions are computed by LES.
An exhaustive presentation should also mention methods
based on reconstructing turbulence Þelds fromsteady RANS
computations,based on stochastic models relying on a wave
number spectral density,calibratedbythe computedturbulent
kinetic energy and its dissipation rate.
The applicability and potential of these methods obvi-
ously directly depend on what kind of noise source is being
considered.
3 OVERVIEWOF CAA APPROACHES
3.1 Hybrid approach
The speciÞc requirements of the numerical simulation of the
propagation of acoustic waves over a given aerodynamic
Þeld often lead to adopting a ÒhybridÓ CAA approach,the
underlying idea being to arbitrary split the physical quantities
(density,velocity,pressure) into (i) aerodynamics variables
(numerically solved with a traditional CFD) and (ii) the
acoustic (so-called ÒperturbedÓ) variables,whose time evo-
lution will be computed with a CAA solver.
In most practical cases,the aerodynamic Þeld will be con-
sidered as time independent.From an application point of
view,this simply implies that the possible retroactions of
acoustic waves onto the aerodynamic Þeld will be neglected.
In other words,the effects that the ßow can have on the
acoustic propagation (such as convection or refraction) will
be properly taken into account,but not the contrary.However,
this restriction is weakly limiting,since such feedback only
occurs in special situations (Òscreech toneÓ phenomenon,
etc.).
DOI: 10.1002/9780470686652.eae344
Encyclopedia of Aerospace Engineering
,Online ©2010 John Wiley &Sons,Ltd.
This article is ©2010 John Wiley &Sons,Ltd.
This article was published in the
Encyclopedia of Aerospace Engineering
in 2010 by John Wiley &Sons,Ltd.
4 Acoustics and Noise
3.2 Formulation and solution of the perturbed
problem
In order to handle a wide range of problems,it is required to
adopt a general (in terms of physics)
continuous
formulation,
which can be numerically solved with robust techniques.
Then the numerical solution of the problem requires
discretizing
this
continuous
formulation over a space-time
domain with
nite
extent.Actually,unless it has an analytical
solution,a physical problemcannot be exactlysolvedthrough
numerical methods.Indeed,although computer resources are
increasing constantly,they are not (and will never be) inÞnite.
Due to such limits in terms of information volume that can be
effectivelytreated,anymathematical model tobenumerically
solved in time and space has to be discretized (or sampled)
and limited to a Þnite domain;this obviously contrasts with
the initial physical problem,which is continuous and (semi)
inÞnite in space and time.
Regarding
discretization
,all previous studies demon-
strated that only Òhigh orderÓ approximations are able to
accurately simulate all the features of sound propagation phe-
nomena.As long as a Òhigh orderÓ criterion is respected,it
does not matter if the numerical method is based on a struc-
tured approach (such as the ÒFinite DifferenceÓ approach
presented in Section 4),or an unstructured approach (such
as the ÒFinite ElementÓ approach,called Discontinuous
Galerkin,presented in Section 5).Both techniques are cur-
rently employed,each one presenting its own advantages and
disadvantages.
For the space-time Þnite domain requirement,there are
several ways to approach the original (semi) inÞnite problem.
In terms of time duration,the causality principle naturally
introduces a bound to the calculation.In terms of spatial
extent,the use of speciÞc boundary conditions allows trun-
cation of the computational area.
When the problemis discretized and bounded in space and
time,the formulation can be solved numerically,the com-
puter memory capacity and execution speed being the only
limitations for the size of the simulations.
4 FINITE DIFFERENCE HIGHORDER
METHOD
This section presents the strategy that has been adopted at
ONERA,and implemented in a block-structured CAAsolver
able to account for inhomogeneous ßows and solid bodies
(Redonnet,Manoha and Sagaut,2001).Several examples of
calculations using this method and tool are provided as a
validation and an illustration of the potential offered by the
method.
4.1 Theory and formulation
4.1.1 Continuous formulation
Initially developed at ONERA,the
conservative formula-
tion of the perturbed non-linear Euler equations
(Redonnet,
Manoha and Sagaut,2001) is directly derived from the
NavierÐStokes equations (closed by the state equation of
perfect gas).Such a formulation allows simulation of the
development of a perturbed ÒacousticÓ Þeld (subscript ÔpÕ)
withinanyÒaerodynamicÓsteadybackgroundßowÞeld(sub-
script ÔoÕ),which veriÞes NavierÐStokes equations:
u
o(
t,x
)


t
u
p
(
u
o
)
+∇·
F
p
(
u
o
,
u
p
)
=
q
s

u
p(
t,x
)
(1)
Here,vectorial quantities
u
p
,
F
p
,respectively,denote the per-
turbed Þeld and ßux,whose detailed expression can be found
in Redonnet,Manoha and Sagaut (2001).The time derivative
and the spatial divergence are denoted by (

t
) and (
∇·
).
u
o
and
q
s
,respectively,denote the background mean ßowquan-
tities and the source term,all being known,and prescribed as
an input for the problem.
This formulation is of conservative nature and free of any
particular assumption (isentropy,incompressibility,or lin-
earity),so it guarantees physical generality and numerical
robustness.
4.1.2 Discrete formulation
As said above,it is necessary to turn the previous exact Òcon-
tinuousÓforminto an approximate ÒdiscreteÓform.The time
discretization leads to an approximate time evolution (
÷
τ
) of
the perturbed Þeld given by:
÷
τ
(

te)
[
u
p
]
=
[
u
p
]
+
t

M

m
=
0
b
m
{
w
}
m

(2)
where the term(
w
) results fromthe spatial discretization:
w
=
q
s

÷
∇·
F
p
(
u
o
,
u
p
) (3)
for which the approached divergence (
÷
∇·
) is deÞned by:
÷
∇·
[ ]
=

i
(
÷

i
[ ])
.
e
i
and
÷

i
[ ]
=
1

x
i
N

j
=−
N
a
j
τ
(
jx
i
e
i
)
[ ]
(4)
where
e
i
,

x
i
denote the unitary vector and the spacing in
the
i
th direction.
DOI: 10.1002/9780470686652.eae344
Encyclopedia of Aerospace Engineering
,Online ©2010 John Wiley &Sons,Ltd.
This article is ©2010 John Wiley &Sons,Ltd.
This article was published in the
Encyclopedia of Aerospace Engineering
in 2010 by John Wiley &Sons,Ltd.
Computational Aeroacoustics
5
The previous expressions involve linear combination of
translation operators (
τ
) that are weighted by coefÞcients
a
j
and
b
m
These coefÞcients are carefully chosen so that the
overall approximation is of Òhigh-orderÓ (Lele,1997;Tam,
1995;Lockard,Brentner and Atkins,1995) or presents good
properties of low numerical dissipation and dispersion.
In particular,a fair compromise between the scheme per-
formances andthe CPUrequirements is torelyon7-point (6th
order) stencil spatial derivatives and a four-step (3rd order)
time marching scheme.
4.2 Filtering
It is mandatory with Finite Difference High-Order schemes
to artiÞcially avoid all the spurious waves that are inherent
to high-order schemes,but without altering the quality of the
calculated solution.This can either be performed by adding
tothe discrete forma selective dampingterm(TamandWebb,
1993),or (as done here) by regularizing the computed solu-
tion via a high-order low-pass Þlter (Lele,1996).Applied to
the perturbed Þeld along each
i
th spatial direction,such a Þl-
ter is deÞned by a linear combination of translation operators
(
τ
) with weighting coefÞcients
f
j
:
ø
u
p
=

i
F
i
[
u
p
] with
F
i
[ ]
=
L

j
=−
L
f
j
τ
(
j
x
i
e
i
)
[ ] (5)
A fair compromise between the Þlter performances and the
CPUrequirements is to rely on
L
=
5 or 11-point (10th order)
stencil Þlters.
4.3 Boundary conditions
4.3.1 Solid boundary conditions
To efÞciently simulate the presence of a solid wall,a classi-
cal Òimage sourceÓ procedure is used.It simply consists in
reproducing (with a symmetry condition) in the interior of
the body the perturbation Þeld at the exterior.To deal only
with centered stencils (known to be less unstable than biased
ones) several rows of ghost points are thus created inside the
solid obstacle.
4.3.2 Open domain boundary conditions
As said above,open domain boundary conditions result from
the necessity of correctly simulating an inÞnite free Þeld
beyond a calculation grid of Þnite dimensions.Such a con-
dition has to be able to let the computed waves leave the
calculation domain without generating any reßection at the
boundaries.Several techniques such as characteristics based
conditions (Thomson,1990;Giles,1990),radiation condi-
tions (Tam and Webb,1993),or ÒPerfectly Matched LayerÓ
(Hu,1996) have been developed and presented in the litera-
ture.But in the light of many comparative studies it appears
that,because of excessively restrictive requirements or even
ill-posed natures,none of these conditions can simultane-
ously guarantee generality,efÞciency,and security for any
calculation case.
In the present method,a simple and efÞcient technique is
used;originally proposed by Redonnet,Manoha and Sagaut
(2001) and precisely assessed and validated by Gu
«
enanff
and Terracol (2005),this technique is based on progressively
decreasing the spatial derivatives and Þlters accuracy order
(this being achieved through the reduction of the schemesÕ
stencil half-width).Coupled with a rapid grid stretching (over
six peripheral rows of ghost points),it allows the perturba-
tions to leave the calculation domain properly.
4.4 Curvilinear geometries
Realistic applications always involve curved geometries,so
it is obvious that the present Cartesian formulation should be
able to deal with curvilinear grids.This can be done with the
help of the metrics technique (Visbal and Gaitonde,2000),
whichonlyimplies rewritingthedivergenceof perturbedßux:
÷
∇·
F
p
=
|
J
|

ξ
·
F
ξ
p
(6)
where the curvilinear ßuxes and Jacobian matrix are,respec-
tively,given by
F
ξ
p
=
F
p
·
J
|
J
|
and
J
= ∇
ξ
(7)
Here,

ξ
and

,respectively,denote the divergence operator
according to the curvilinear (local) and the Cartesian (global)
basis.
4.5 Application and validation cases
4.5.1 Validation requirements
One major difÞculty with CAA is to Þnd cases or conÞg-
urations for which validation is possible,either with other
numerical methods or with experimental data.This is mainly
because analytical solutions or other numerical methods used
for acoustic simulations generally rely on assumptions that
restrict their domain of application to uniform mean ßows,
DOI: 10.1002/9780470686652.eae344
Encyclopedia of Aerospace Engineering
,Online ©2010 John Wiley &Sons,Ltd.
This article is ©2010 John Wiley &Sons,Ltd.
This article was published in the
Encyclopedia of Aerospace Engineering
in 2010 by John Wiley &Sons,Ltd.
6 Acoustics and Noise
Figure 2.
RANS steady mean ßow:(a) pressure contours;(b) TKE distribution.
whereas the speciÞc feature of CAA is to propagate sound
over non-uniformmean ßows.Moreover,it is very challeng-
ing to collect experimental results with the accuracy required
for reliable comparison with CAA cases.
In the next paragraphs we present one 2DconÞguration in
the domain of airframe noise,and then two 3Dcases relevant
to acoustic engine installation effects.
For all three cases,one Þrst computation is achieved with-
out mean ßow,with the opportunity to compare the results
with the BEM,which solves the Helmholtz equation with
wall boundary conditions.Then a second acoustic compu-
tation is presented,performed over the actual non-uniform
mean ßow of the given conÞguration.This second compu-
tation has no validation basis,but it demonstrates that the
acoustic refraction effects are generally signiÞcant.
4.5.2 Slat noise radiation
The leading edge slat is known as a major airframe noise
source on large transport aircraft.The physical mecha-
nisms involved are complex as shown by several attempts
to compute the noise source generation via unsteady CFD
techniques (LES,DES,NLDE) (Terracol
et al.
,2003;Ben
Khelil,2004;Deck,2005).Interesting results can be derived
fromsimpler simulations of the noise radiated by a pulsating
source located inside the slat cove at selected locations where
the turbulent kinetic energy magnitude is signiÞcant.
A 2D high lift wing (Mincu
et al.
,2007) is considered
in the approach ßight conÞguration with a slat incidence of
23

and a ßap deßection of 17

.Figure 2 shows the steady
mean ßow(pressure contours) around this proÞle,computed
using a RANS solver (Spalart-Allmaras turbulence model).
The wing angle of attack is
α
Ð4

,the Mach number is 0.19,
and the Reynolds number (calculated using the proÞle chord)
is 2.5 millions.Figure 2a shows the pressure distribution in
the slat cove,andFigure 2bshows the turbulent kinetic energy
distribution.
For the acoustic computations,a hybrid Carte-
sian/curvilinear grid was used with a speciÞc overlapping
method developed by ONERA.Figure 3a shows the curvi-
linear multi-block structured grid,with details in the slat
cove region.This curvilinear grid (1) is immersed in a large
Cartesian grid (3),and the acoustic waves radiating in the
outward direction are transferred from the curvilinear grid
to the Cartesian grid through an overlap zone (2) using
high-order interpolation schemes.
Acoustic simulations are done with a pulsating source
located inside the slat cove near the slat cusp.This source
is a harmonic acoustic cylindrical (2D) monopole of given
frequency
ω
s
and spatial Gaussian envelope centered on the
Figure 3.
(a) Curvilinear multi-block structured grid;(b) Cartesian grid (1),overlap zone (2),and curvilinear grid (3).
DOI: 10.1002/9780470686652.eae344
Encyclopedia of Aerospace Engineering
,Online ©2010 John Wiley &Sons,Ltd.
This article is ©2010 John Wiley &Sons,Ltd.
This article was published in the
Encyclopedia of Aerospace Engineering
in 2010 by John Wiley &Sons,Ltd.
Computational Aeroacoustics
7
Figure 4.
Acoustic scattering of a pulsating source located near the slat cusp,without mean ßow:(a) Instantaneous perturbed pressure;
(b) directivity diagram:comparison with BEM.
excitation point
x
s
,generated with the following forcing
source term:





ρ
s
v
s
p
s





=





f
(
t
)
g
(
x
)
0
c
2
o
ρ
s





with
g
(
x
)
=
ε
s
e

α
s
|
x

x
s
|
2
and
f
(
t
)
=
sin(
ω
s
t
) (8)
in which
ε
s
=
0.01,
α
s
=
ln(2)/2,and the acoustic wavelength
2
￿
/
ω
s
is about 1/52 of the clean (slat/ßap retracted) wing
chord.
First,computation is done without the mean ßow.Figure 4
shows the distribution of instantaneous perturbed pressure
and a directivity diagram,with a comparison to a BEMcom-
putation.
Then another computation is achieved,taking into account
the non-uniform mean ßow around the high-lift wing.
Figure 5 presents the distribution of instantaneous perturbed
pressure and the inßuence of the mean ßowon the directivity
diagram.This result demonstrates that the acoustic refraction
effects are signiÞcant.
4.5.3 Aft fan noise shielding by airframe
This second simulation is relevant to acoustic installation
effects,a recent discipline dedicated to innovative engine
Figure 5.
Same as Figure 4,but with non-uniform mean ßow:(a) instantaneous perturbed pressure;(b) directivity diagram:comparison
with/without mean ßow.
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8 Acoustics and Noise
Figure 6.
(a) Artist and (b) schematic views of the Rear Fuselage Nacelle concept.Courtesy of Airbus-France.
installation on aircraft,with a view to beneÞt from engine
noise shielding by the airframe (fuselage,empennage).
Figure 6 shows a future concept for a low-noise aircraft
studied by Airbus,with the engines installed in Rear Fuselage
Nacelle (RFN) conÞguration,with the objective of shield-
ing the jet noise and the aft fan noise (radiated through the
exhaust) by the rear fuselage and empennage.As shown in
Figure 6,the purpose of the proposed simulation is to evalu-
ate the shielding of aft fan noise by a simpliÞed empennage
(Redonnet,Desquesnes and Manoha 2007).
The fan noise source is modelled inside the bypass duct
(with radius
R
) in the fan plane,under the formof an acous-
tic mode of an inÞnite annular duct with a uniform mean
ßow.Such a mode is a solution of the convected Helmholtz
equation with rigid wall conditions,any solution being char-
acterized by an azimuthal periodicity of order
m
and a given
radial distribution with order
n
.In the present case the mode
(
m
=
2,
n
=
2) is emitted with the reduced frequency
kR
=
ω
R
/
c
0
=
11.84,corresponding to one half of the Blade
Passing Frequency (BPF) for an engine at full thrust under
take-off conditions.
As in the slat simulation presented above,the grid gen-
eration strategy takes advantage of the Òoverlapped gridsÓ
technique,allowing the immersion of body-Þtted curvilinear
grids in a large Cartesian grid.Firstly,two body-Þtted 3D
meshes were built,for the nozzle and the airfoil.The noz-
zle axisymmetric grid has Þve domains and 206 258 nodes,
whereas the 2.5D (with a 4
R
wingspan extent) airfoil grid
contains 2 domains and 989 055 nodes.Finally,a third mono-
domain ÒBackground CartesianÓ mesh was constructed in a
box of approximately 5
R
×
4
R
×
4
R,
containing 9 049 830
nodes (Figure 7a).A Non Reßecting Boundary Condition
Figure 7.
(a) Body-Þtted grids immersed in Cartesian grid;(b) instantaneous aft fan noise radiation of the installed engine in static conditions
(mediumat rest).
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Computational Aeroacoustics
9
Figure 8.
Aft fan noise radiation of the installed engine in static
conditions (mediumat rest).Instantaneous pressure in two vertical
planes.Comparison of (a) Euler and (b) BEM.
is applied at the periphery of this domain,combined with a
radial grid stretching applied to the eight peripheral rows of
the domain.
Figure 7 shows the instantaneous pressure Þeld in the 3D
domain,after 2100 iterations or 15 source periods,which
means that the acoustic waves generated in the fan plane have
travelled over 15 acoustic wavelengths,about three times the
larger dimension of the computational domain (5
R
),which
ensures that the acoustic Þeld has reached a steady state.
For validation purposes,an unstructured three-
dimensional BEM grid of the rigid surfaces of the
nozzle and airfoil was built,containing 65 919 nodes and
131 830 (triangular) elements.The BEM computation was
achieved with the same fan noise mode (mode (
m
=
2,
n
=
2,
kR
=
11.84) by Airbus-France (ACTI3S code).Figure
8 compares Euler and BEM results in two vertical planes,
parallel and normal to the exhaust axis.Although contour
scales had to be adjusted,since sources are deÞned with
arbitrary amplitudes in both computations,the qualitative
agreement between both simulations is very good.
Finally,Figure 9a presents the RANS ßow computed
around the axisymmetrical engine and 2D airfoil at take-off
Mach number
M

=
0.25.Figure 9b shows the aft fan noise
radiation (instantaneous pressure in the 3D domain) of the
installed engine for the same acoustic mode and frequency
as before.
4.5.4 Aft fan noise propagation from 3D exhaust
In this last calculation,aft fan noise is radiated through a
realistic 3D turbofan exhaust,including the pylon and the
internal bifurcation (Redonnet,Mincu and Manoha,2008;
Redonnet
et al.
,2009),accounting for representative ther-
modynamic conditions (mean ßow),and relevant fan noise
modal contents.
Figure 10 shows RANS computation of the mean ßow,
which corresponds to typical take-off conditions with Mach
number
M

=
0.26 at inÞnite and
M
inj
=
0.46 in the sec-
ondary exhaust just downstreamof the fan plane,rising up to
M
=
0.9 in the outer section of the secondary exhaust.There
are very large velocity and temperature gradients.These
strong mean ßow heterogeneities are due to (i) strong shear
layers at the outer of the primary and secondary exhausts and
(ii) the non axisymmetric nature of the geometry.
The following acoustic computations were done on a
multi-block structured grid containing about 12 millions cell
points,using a parallelized version of ONERAÕs CAAsolver,
running on either 64 or 128 cores of an
Itanium64 bit
parallel
computer.
Figure 9.
Aft fan noise radiation of the installed engine in take-off conditions:(a) RANS ßow around the axisymmetrical engine and 2D
airfoil;(b) instantaneous pressure in the 3D domain.
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10 Acoustics and Noise
Figure 10.
RANS computation of the mean ßow in take-off condition:(a) Mach number;(b) velocity Þelds.
Figure 11shows a transversal cut,at the outer sectionof the
secondary exhaust,of the acoustic Þeld radiated at the BPF
(reduced frequency normalized by the engine outer radius
R
:
kR
=
28.17) from the emission of an aft fan noise spinning
mode with azimuthal order
m
=
26 and radial order
n
=
1.
This Þrst computation is achieved without any mean ßow,for
the purpose of direct comparison of CAAcomputation (Fig-
ure 11a) with BEM results (Figure 11b).This comparison
shows a very good agreement.The acoustic Þeld exhibits a
complexpatternintheazimuthal andradial directions,mainly
resultingfromthedistortionof theacousticmodebytheexter-
nal (pylon) andinternal (bifurcation) enginecomponents.The
most noticeable effect is that the spinning motion of the mode
disappears downstreamof the nozzle.
The perturbed pressure Þeld presented in Figure 12 corre-
sponds tothe emission,at the BPF,of anaft fannoise spinning
mode with azimuthal/radial order (
m
,
n
)
=
(13,1) this time
accounting for the heterogeneous jet mean ßow shown in
Figure 10.In this case,the primary and secondary jet mean
ßow gradients induce strong refraction effects,resulting in a
deßection of the acoustic waves towards the radial direction.
Acoustic installation effects are still present,with a modal
de-structuring of acoustic waves within the duct (see Figure
12a top),leading to a strongly non axi-symmetric radiation
pattern (Figure 12b).
5 DISCONTINUOUS GALERKIN
METHOD
5.1 Introduction
The method presented in the previous subsection is based on
Þnite difference that allows the use of high-order schemes
Figure 11.
Root s pressure for aft fan noise radiation (
m
=
26,
n
=
1,BPF,
kR
=
28.17) for 3D engine in static conditions.Comparison of
(a) Euler;(b) BEM.
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Computational Aeroacoustics
11
Figure 12.
Acoustic Þeld from modal emission (
m
=
13,
n
=
1) at BPF in take-off conditions:(a) instantaneous pressure;(b) RMS
pressure.
and thus limits the dispersion and dissipation errors,so the
constraints on the possible boundary conditions are weak.
The stability issues are comparable to other methods.On
top of this,such methods can be computationally efÞcient
if enough care is taken during the choice of the numerical
methods,as explained above.
However,the Þnite difference method requires block-
structured grids,which can become tricky to generate for
complex geometries.One attractive workaround is the use
of hybrid techniques,as the one presented above,using both
Cartesian and body-Þtted grids,and an overlapping interface.
Such a technique also has a fewdrawbacks like the numerical
cost penalty if the accuracy of the scheme must be preserved.
Another alternative is to completely change the simulation
strategy and use simple unstructured meshes.
It is worth emphasizing that a similar development
occurred within the CFD community about one decade ago.
Block-structured meshes have long been used in CFD,but
unstructured grids are nowmore and more used,despite pos-
sible lower accuracy,because of the great simpliÞcation of
the grid generation process for complex geometries.
This possibility is offered,for example,by the use of the
DGM.The Linearized Euler Equations (LEE) can be imple-
mented on such a scheme,possibly in a conservative way.
All the required boundary conditions can be implemented.
The scalability observed so far is excellent on 10 to 100 pro-
cessors at least.And one very positive point of the method
is the
p
-adaptivity feature of the DGM schemes;in prac-
tice this means that the user does not need to reÞne the mesh
locallybecause the solver will adapt the order of eachelement
to correctly describe the physics.Consequently,the human
time is considerably reduced,without affecting the accuracy,
stability,or performance of the scheme.
The following paragraphs give an overview of the imple-
mentation of the Linearized Euler Equations Solver of Free
Field Technologies,called Actran/DGM.Afewexamples of
use are given,as well as some numbers on the performance
of this method.
5.2 Theory and implementation
5.2.1 Generalities
The equations to solve have been presented in the previ-
ous subsection.The theory of the Discontinuous Galerkin
elements,and some ways to have an efÞcient implemen-
tation,have been extensively described in the past,for
example in Atkins and Shu (1997),Atkins (1997),Cockburn
et al.
(2000),Remacle,Flaherty and Shephard (2003),and
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12 Acoustics and Noise
Chevaugeon
et al.
(2005).In comparison to the classical
Finite Element Method(FEM),the mainidea is tohave higher
orders,but the Þelds can be discontinuous at the interface
between two elements,and neighbouring elements do not
necessarily have the same order.
The strategy is explained in detail in Chevaugeon
et al.
(2005) and is summarized hereafter.
To obtain the DGformulation,one Þrst multiplies the LEE
with a test function,
ö
u
,and integrates over the domain

.The
divergence theorem is then applied to obtain the following
variational formulation:



t
u

ö
u
d
v
+


F
x
(
u

)

x
ö
u
d
v
+


F
y
(
u

)

y
ö
u
d
v
+


F
z
(
u

)

z
ö
u
d
v



f
ö
u
d
s
=


s
ö
u
d
v,

ö
u
(9)
where
f
(
u

)
=
Fu

·
n
is the normal trace of the ßuxes.
The physical domain

is discretized into a collection of
N
e
elements,
T
=
N
e

e
=
1
e
(10)
called a mesh.In each element
e
of
T
,each component of
u

is
discretized using polynomials.It is common,in the Þnite ele-
ment world,to distinguish reference coordinates,
ξ
,
η
,
ζ
,from
real coordinates
x
,
y
,
z
.Piecewise continuous approximations
are used on each element:
u

(
ξ,η,ζ
)
=
d

k
=
1
N
k
(
ξ,η,ζ
)
u

e
k
(11)
The coefÞcients
u

e
k
of the interpolation are associated with
the element.The interpolation is then discontinuous at
inter-element boundaries,contrary to classical Þnite element
methods.The interface term in the variational formulation
appears for each interface between elements,and for each of
them the normal trace of the ßux across this interface must
be computed.The ßux is not uniquely determined there,and
the choice of an appropriate numerical ßux is at the heart
of the properties of the method.For each interface in the
mesh,a unique normal to it is deÞned,
n
,and we note
u

L
and
u

R
the Þeld variables on the left and right sides of this
face,respectively,with
n
going from the left to the right.
The numerical ßux,
f
(
u

L
,
u

R
),is computed using a Riem-
man solver.The Riemann solver that we use is constructed by
computing characteristics of the left and right Þelds,and only
the upwind quantities are kept to compute the normal ßux.
Additional difÞculties occur when solving the linear
hyperbolic partial differential equations with non-constant
coefÞcients,to adapt to the non-uniform mean ßow.There
arethenseveral possibleimplementations that arenot detailed
here,as it is not the purpose of this work;the choice made for
Actran/DGMis to implement a quadrature-free Runge-Kutta
DGM.Again there are multiple possible choices,and great
carehas beentakenonthewaythis strategywas implemented.
A few other choices have been justiÞed in various papers
and will not be detailed here.For example,it has been found
that a conservative formulationcouldsometimes be anadvan-
tage,bothfor stabilityandfor performancereasons.Themean
ßow,computed in an external CFD code (typically a com-
mercial one),must also be mapped with a minimumof care.
Finally,as already stated above in the Finite Difference sec-
tion,great care must be taken with the boundary conditions:
damping zones of course,but also regular hard wall condi-
tions (curvilinear boundaries),impedance,modal basis,and
so on.In every case,a bad choice can have a dramatic inßu-
ence on the robustness of the tool,its performance,and of
course its accuracy.
Actran/DGM has been cross-veriÞed with other (non-
commercial) simulation codes and validated against
experiments,among others in the frame of the EU project
TURNEX (Tester
et al.
,2008).The accuracy of the tech-
nique is comparable to the one of more state-of-the-art codes
like FD schemes in the frequency domain.
5.2.2 Specicities of p-adaptivity
High performance is difÞcult to achieve on industrial-like
ßows and complex 3D geometries.Indeed,one must try to
conceal two antagonistic properties:
r
On the one hand,quadrature-free DGMcodes are usually
most efÞcient for high-order elements.To further keep the
time step as large as possible,it is advised to work with
meshes that consist of large,high-order elements.
r
On the other hand,ßow gradients and geometrical pecu-
liarities may require,locally,that some elements be very
small.
This negatively affects performance if all elements have the
same interpolation order.If the element order is kept high,
the time step does indeed decrease drastically.Alternatively,
if the element order is decreased,the number of elements
grows while the accuracy decreases (see dissipation and dis-
persionproperties of low-order elements inChevaugeon
et al.
(2005).
To overcome this difÞculty,a variable interpolation order,
adaptive algorithmis implemented in the code.As the inter-
polation order may vary from one element to the next,
acoustic variables,
q
,are projected,at the element interface
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Computational Aeroacoustics
13
q

=

q
~
k
L
k
q

=

q
~
~
k
R
k
∂Ω
Figure 13.
Variable interpolation order adaptive algorithm.
level,on each element face onto its neighboursÕ functional
space,when needed,as depicted in Figure 13.
L
k
and
R
k
denote the face shape functions in the left-hand and right-
hand elements,respectively.
The code automatically computes the optimal element
order for every element in the mesh,based on the element
size,acoustic wavelength and local ßow conditions (the lat-
ter correctionincludes the variationof the speedof soundwith
ßow as well as Doppler effects),and the impact of the inter-
polation order on numerical dissipation,and dispersion (see
Chevaugeon
et al.
,2005).Finally,the time step is also com-
puted automatically to ensure that the CFLnumber (Courant,
Friedrichs and Lewy,1967) condition is satisÞed across the
entire mesh.
5.3 Examples:performance
The following example is a typical computation of the acous-
tic propagation of a fan acoustic mode generated inside an
Figure 15.
Non-uniformmean ßow (CFD code Fluent).
engine bypass duct at the fan blade passing frequency and
propagated through the engine exhaust.
Figure 14 shows the exhaust engine conÞguration (from
the EUproject TURNEX),with associated mesh and bound-
ary conditions.Thanks to the p-adaptivity,it is not necessary
to pay attention to the mesh reÞnements in the sheared zones,
as it would be with a more classical scheme.See details in
Manual (2009).
The non-uniformmean ßowwas computed by an external
CFD code (Fluent) in the TURNEX project (Figure 15).
Figure 16 shows the results (real part of the acoustic pres-
sure) of acoustic computations for two different modes at
different ßight conditions.
Figure 14.
Exhaust engine conÞguration Ð mesh and boundary conditions (fromEU project TURNEX).
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14 Acoustics and Noise
Figure 16.
Real part of the acoustic pressure for the mode
m
=
6,
n
=
1 at different ßight conditions and frequencies:(a) ßight cutback
conditions,
kR
=
10;(b) ßight sideline conditions,
kR
=
17.
Table 1.
EfÞciency of the parallel version on a real 3Dexhaust case.
Number of Computational RAMper Speed EfÞciency
CPU time (h) CPU (GB) up (%)
1 243 (10 d) 6 1 100
2 140 3.5 1.73 87
4 84 2.1 2.9 72
8 40 1.2 6.0 76
12 30 1.2 8.0 67
16 20 1.2 12.0 76
Concerning performances,typical computational times
for such 3D computations are given in Table 1.Thanks to
the good parallel scalability of this solver,it is mostly feasi-
ble to run the computation of 3D exhaust at the BPF within
less than 12 h.It should be noted that the conÞguration could
be made more complex (for example,including the pylon or
a scarf exhaust) without much more difÞculty.
5.4 Conclusions and perspectives for the
Discontinuous Galerkin Method
A real exhaust problem can be entirely prepared in a few
hours.The code is robust,easy to use,and fast enough for
industrial applications,and some expertise is available.The
success of the method is thus really encouraging.This DGM
scheme working in the time domain could well complement,
for some applications,the more classical Finite Element
Method tools currently in use for vibro-acoustics and for
CAA.The FEMstrategy is still much superior for most appli-
cations;however,switchingtoDGMinthetimedomaincould
open the door to treating much larger models.For example,in
the case of CAA,some good success has been observed with
Òclassical FEMÓtools working in the frequency domain,used
together with industrial CFD codes.So far,only automotive
applications have been treated:the Reynolds number is more
favorable,and the extent of the computational domains is
more limited.However,the great success of the DGMon the
one hand,and the progress of the LESeven for high Reynolds
numbers on the other hand,may motivate the development
of the right-hand side terms in Actran/DGM.
6 SUMMARY AND FUTURE NEED
This subsection is an overviewof the numerical methods that
are available for the simulation of aircraft noise.It deliber-
ately concentrates on the methods used to simulate the propa-
gation of acoustic waves on complex media characterized by
non-uniform mean ßows around solid bodies,a domain that
was originally denoted as ÒComputational AeroAcoustics,Ó
although this label nowoften includes also the simulation of
the noise sources (ßuctuating forces,turbulence).
It is shown that,due to a large range in the size and
amplitudes of the ßow structures involved in aeroacoustic
mechanisms,the simulation of a given problem generally
combines several methods applied in adjacent domains.It is
now admitted that all these numerical methods are generally
individually mature and efÞcient.The main challenge for the
future probably lies in the development of ÒbreakthroughÓ
computing technologies for smoothly and physically inter-
facing individual components to create efÞcient computation
chains (Leneveu
et al.
,2008).
The aeroacoustic community is now facing the challenge
of designing integrated computing tools allowing the com-
putation of the overall noise radiated by any new aircraft
concept,including (i) all possible noise sources,(ii) acous-
tic installation effects,and (iii) noise propagation in complex
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Computational Aeroacoustics
15
media from the aircraft to the ground.It is considered that
such tool will most probably rely on the concept of chaining
several solvers based on different methods and physics.
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DOI: 10.1002/9780470686652.eae344
Encyclopedia of Aerospace Engineering
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This article is ©2010 John Wiley &Sons,Ltd.
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DOI: 10.1002/9780470686652.eae344