On the Use of Filtering Techniques for Hybdrid Methods in Computational Aero-Acoustics

mustardarchaeologistMécanique

22 févr. 2014 (il y a 3 années et 6 mois)

190 vue(s)

On the Use of Filtering Techniques for Hybdrid Methods in
Computational Aero-Acoustics
W.De Roeck,G.Rubio,W.Desmet
K.U.Leuven,Department of Mechanical Engineering,
Celestijnenlaan 300 B,B-3001,Leuven,Belgium
e-mail:wim.deroeck@mech.kuleuven.be
Abstract
Hybrid CAA-approaches are commonly used for aeroacoustic engineering applications.In this kind of com-
putational techniques,the numerical domain is split into a noise generating region,where an aerodynamic
eld generates the acoustic sources,and an acoustic propagation region.Nowadays a large variety of hy-
brid approaches exist differing from each other in the way the source region is modeled;in the way the
equations are used to compute the propagation of acoustic waves in a non-quiescent medium;and in the
way the coupling between source and acoustic propagation regions is made.The coupling between source
and propagation region is usually made using equivalent sources (acoustic analogies) or acoustic boundary
conditions (Kirchhoff's method).For certain applications both coupling approaches tend to give erroneous
results:acoustic analogies are inaccurate if the acoustic variables are of the same order of magnitude as the
ow variables,which is the case for ow-acoustic feedback phenomena such as cavity noise or when acous-
tic resonance occur which happens for duct aeroacoustics applications;acoustic boundary conditions are
sensitive to hydrodynamic pressure uctuations when a vortical owpasses through the Kirchhoff's surface.
These inaccuracies can be avoided by using appropriate ltering techniques where the solution in the source
domain is split into an acoustic and a hydrodynamic part.This paper illustrates the need for such ltering
techniques for CAA-applications and starts with the theoretical development of a new ltering technique
based on an aerodynamic-acoustic splitting.
1 Introduction
Aeroacoustics is a research area of research of growing interest and importance over the last decade.In the
transportation sector,the interest for this eld has emerged during the last few years,due to various reasons.
In aeronautics,for example,strict noise regulations around airports are forcing aircraft manufacturers to
reduce the noise emissions during landing and take-off operations.In automotive industry,customer surveys
identify wind noise as a regular complaint.
With the increase in computational power,the direct computation of aerodynamic noise has become feasible
for academic cases [1,2,3].Such a direct approach solves the compressible NavierStokes equations,which
describe both the ow eld and the aerodynamically generated acoustic eld.Due to the large disparity in
energy and length scales between the acoustic variables and the ow variables,which generate the acoustic
eld,and since acoustic waves propagate over large distances,the direct solution of the NavierStokes
equations (DNS) for computational aeroacoustics (CAA) problems is only possible for a limited number
of engineering applications [4].
In order to meet the required design times without excessive the costs,hybrid methods are proposed.In these
methods,the computational domain is split into different regions,such that the governing ow eld (source
region) or acoustic eld (acoustic region) can be solved with different equations,numerical techniques,and
computational grids.As such,prediction of the acoustic eld at large distances from the sound source is
enabled.There exists a large number of hybrid methodologies differing from each other in the type of
595
applied propagation equations or in the way the coupling between source region and propagation region is
etsablished.
The classical linear acoustic wave equation or the convective wave equation can be used as acoustic propa-
gation equations [5].Both of themmake assumptions about the mean ow eld:the acoustic wave equation
assumes no mean ow,while the convected wave equation can only be used when an irrotational mean ow
is present.In most engineering applications,these assumptions do not hold and more advanced propagation
models,based on a linearization of the Euler equations are used [6,7].These equations can be used for most
types of mean ow.
The coupling methods,that are commonly used for hybrid CAA-applications can be divided roughly into
two categories:one based on equivalent source formulations and the other based on acoustic boundary
conditions.The idea of using equivalent aeroacoustic sources was rst introduced by Lighthill [8].By
rewriting the NavierStokes equations in such a way that the left hand side equals the linear acoustic wave
equation,the well-known Lighthill stress tensor is obtained as aeroacoustic source termin the right hand side.
This idea is since then widely used for all kinds of other propagation equations.It is shown in this paper
that acoustic analogies fail to give accurate results for applications where the acoustic variables become
of the same order of magnitude as the aerodynamic uctuations,which is the case when the acoustic eld
is generated by a ow-acoustic feedback coupling (e.g.cavity noise) or when acoustic modes are present
(e.g.duct aeroacoustics).With a ltering of the source region results into an acoustic part and an purely
aerodynamic part,it should be possible to avoid these drawbacks.
The other coupling method,based on acoustic boundary conditions,assumes that the acoustic variables on
a surface surrounding the source region (Kirchhoff's surface) can be obtained from a proper ow domain
simulation.This acoustic information can then be used as boundary condition for various propagation equa-
tions [9,10],resulting in an acoustic continuation of the source region calculation.However,when a vortical
ow passes through the Kirchhoff's surface,aerodynamic uctuations,cause hydrodynamic pressure uctu-
ations to be present in the propagation region and results in'unphysical'or even unstable acoustic solutions
in the downstream region.It is clear that proper ltering techniques are needed to avoid these errors.The
theoretical development of such ltering procedures is elaborated in the present paper.
The rst section discusses the theory of various propagation equations and coupling techniques and shows
the need for ltering techniques for computational aeroacoustics.In the next section this is illustrated for
two applications:noise generation by a owover a rectangular cavity and by a square cylinder in cross-ow.
Next a rst (frequency-domain) ltering procedure proposed by Ovenden and Rienstra [11] is discussed.
The theoretical framework is developed for more general ltering techniques,based on an aerodynamic-
acoustic splitting approach,for subsonic CAA-applications.The main conclusions are summarized in the
nal section.
2 Theory of Hybrid CAA-techniques
In hybrid CAA-techniques,the computational domain can be split into two different regions (g.1):

Source region:This is the region where the ow disturbances generate noise.The uctuating ow
variables must be calculated with the NavierStokes equations.The computational grid size is driven
by the length scales of the ow eld and the method used to solve the source region.

Propagation region:It can be assumed that the oweld does not generate any sound in this part of the
computational domain.Only the propagation of the acoustic waves,generated in the source region is
inuenced.If convection and refraction of sound waves do not occur or have a negligible inuence (the
acoustic far-eld),one can use the conventional acoustic wave equation to compute the propagation of
the acoustic waves.If these effects become important (in the acoustic near-eld or for duct acoustics),
the convective wave equation can be used in a limited number of applications,where the mean ow
596 PROCEEDINGS OF ISMA2006
eld is irrotational.In general,more accurate solutions can be obtained by using equations based on
a decomposition of the Euler equations in a mean part and a uctuating part,like e.g.the linearized
Euler equations (LEE) [6].The computational grid size is driven by the smallest acoustical wavelength
of interest,which is typically larger than the grid size needed for owcalculations in the source region.
Source￿Region
Propagation￿Region
Figure 1:Sketch of the different computational domains for free-eld (left) and ducted (right) aeroacoustic
applications.
2.1 Propagation Equations
Linearized Euler equations (LEE) are commonly used to describe the near-eld propagation of acoustic
waves in the presence of a non-uniform mean ow [6].Refraction and convection effects of the mean
ow on the acoustic eld are taken into account by the LEE.Acoustic dissipation due to the viscosity of
the mean ow eld is not taken into account.The LEE are obtained by decomposing the ow variables
(density,velocity and pressure) of the NavierStokes equations into their mean values (ρ
0
,u
i0
,p
0
) and their
uctuating (acoustic) parts (ρ
￿
,u
￿
i
,p
￿
) and by neglecting viscosity and higher-order terms,
∂ρ
￿
∂t
+

∂x
i

￿
u
i0

0
u
￿
i
) = Φ
cont
(1)
∂ρ
0
u
￿
i
∂t
+

∂x
j

0
u
j0
u
￿
i
+p
￿
δ
ij
) +(ρ
￿
u
j0

0
u
￿
j
)
∂u
i0
∂x
j
= Φ
mom,i
(2)
∂p
￿
∂t
+

∂x
i
(u
i0
p
￿
+γp
0
u
￿
i
) +(γ −1)(p
￿
∂u
i0
∂x
i
−u
￿
i
∂p
0
∂x
i
) = Φ
ener
(3)
where Φ
cont

mom,i
and Φ
ener
are the source terms for,respectively,the continuity,momentum and energy
equation,which can contain non-linear,viscosity and temperature effects and can be calculated based on
time-dependent source domain results.The mean ow variables can be easily obtained by calculating the
Reynolds-Averaged NaviesStokes equations (RANS).
Although the LEE describe the propagation of acoustic waves in a non-quiescent medium,they also support
the propagation of vorticity and entropy waves,which can lead to unphysical or even unstable acoustic
solutions if the source terms Φ excite the entropy or vorticity modes of the LEE [7].This can be avoided
by assuming that the acoustic eld is irrotational (
￿
ω
￿
= 0) (where
￿
ω
￿
is the vorticity vector) and isentropic
(dp = c
2
0
dρ with c
0
=
￿
γp
0

0
the speed of sound).Under these assumptions,the LEE can be rewritten as:
∂p
￿
∂t
+c
2
0

∂x
i
(
p
￿
u
i0
c
2
o

0
u
￿
i
) = c
2
0
Φ
cont
(4)
AEROACOUSTICS AND FLOW NOISE 597
∂u
￿
i
∂t
+

∂x
i
(u
j0
u
￿
i
+
p
￿
ρ
0
) =
1
ρ
0
Φ
mom,i
(5)
This set of equations is also known as the Acoustic Perturbation Equations (APE) [7].It can be proven that
these equations render perfectly stable solutions and they are,from a computational point of view,more
efcient than the LEE,since they solve one equation less due to the fact that the isentropic relation between
pressure and density is inherently satised.If only acoustic modes are excited by the source vector Φ,and if
the acoustic eld can be assumed to be irrotational,LEE and APE are identical.
2.2 Coupling Strategies
A distinction between two coupling strategies between the two regions can be made for the most commonly
used CAA-techniques:

Equivalent sources:Equivalent acoustic sources,which appear in the right hand side of the various
propagation equations,can be calculated based on the computation of the source region.Lighthill [8]
rst introduced this kind of acoustic analogy in the early 1950's and nowadays,there exist a large
variety of aeroacoustic source formulations.When equivalent sources are used,the source region is
part of the propagation region.

Acoustic boundary conditions:The acoustic pressure,velocity and density uctuations on a surface
surrounding the dominant aeroacoustic sources can be introduced as a boundary condition for the
propagation equations.Kirchhoff's method [10] or the method introduced by Ffowcs-Williams and
Hawkings [9] are most commonly used as boundary condition.In this way,the propagation region
does not overlap with the source region.
2.2.1 Equivalent Sources
Lighthill [8] introduced the use of acoustic analogies.By rewriting the NavierStokes equations in such
a way that the left-hand side equals the linear acoustic wave equation without a mean ow,whereas all
other terms are treated as right-hand side source terms,Lighthill obtained,for subsonic,isentropic ows,the
Lighthill stress tensor as equivalent source term:
Φ = −

2

0
u
i
u
j
)
∂x
i
∂x
j
(6)
The approach of replacing the whole noise generating oweld by an equivalent source termis appealing due
to its simplicity and can be used to identify possible aero-acoustic source phenomena.Furthermore,these
methods require less accurate source calculations,since they are based on aerodynamic uctuations [12],
and results obtained from incompressible computations or RANS-calculations,when the turbulent eld is
stochastically reconstructed,can be used [6].
If a similar approach is used for rewriting the LEE,there would be only a contribution of the non-linear terms
of the decomposed NavierStokes equations in the source term formulation [6].These terms are in most
applications even neglected in the Lighthill stress tensor and thus cannot be seen as an accurate representation
of the source generating mechanism.Ideally,the decomposition of the NavierStokes equation should be
carried out with two uctuating variables instead of one:an acoustic uctuating part (ρ
￿
ac
,u
￿
i,ac
,p
￿
ac
) and an
aerodynamic,or turbulent,uctuating part (ρ
￿
turb
,u
￿
i,turb
,p
￿
turb
).
The uctuating aerodynamic variables can be considered to be obtained from the ow domain calculation
and are thus no unknowns for the LEE,which need to be solved for the acoustic uctuating part.For such de-
composition all terms containing the turbulent uctuating variables should be treated as source terms,while
598 PROCEEDINGS OF ISMA2006
the terms containing the acoustic uctuating part should remain in the left-hand side.Nevertheless,in a large
number of aero-acoustic applications,the turbulent variables in the source region are orders of magnitude
larger than the acoustic variables,which reduces the necessity of such a decomposition.This results,for low
Mach number and isentropic applications,in a source termcontribution only in the momentumequations.
Φ
mom,i
= −
∂(ρ
0
u
i
u
j
)
￿
∂x
i
(7)
This source termis similar to the one proposed by Lighthill.It should be noted that for incompressible source
region calculations or sound generation by purely aerodynamic phenomena,where no ow-acoustic feedback
is present,the uctuating part contains no only a minor acoustic uctuating part and can thus be seen as an
accurate source termdenition.If,on the other hand,the acoustic variables are not negligible with respect to
the aerodynamic uctuating part,the source termcontains a spurious contribution (ρ
0
u
i0
u
￿
j,ac

0
u
j0
u
￿
i,ac
)
fromthe acoustic eld.
For applications where the acoustic eld cannot be neglected another source term formulation is thus re-
quired.A possible alternative is proposed by Powell [13].He proposed to consider only the rotational part
of the owvariables as a source of sound.Since for most applications the acoustic variables can be assumed
to be irrotational,the rotational uctuating part can be considered as purely turbulent.The source termin the
momentumequations,for a low Mach number isentropic ow,can then be written as:
Φ
mom,i
= −ρ
0
(￿ω ×￿u)
￿
i
= −ρ
0
L
￿
i
(8)
The major vortex source term is thus the uctuating Lamb vector −
￿
L
￿
.The same source term is obtained
when the NavierStokes equations are rewritten in such a way that the left-hand side equals the APE [7].For
applications where the acoustic variables inside the source region are of the same order of magnitude as the
turbulent uctuations,there might still be some inuence (
−→
ω
0
×
−−→
u
￿
i,ac
) of the acoustic uctuations inside the
source termif the mean ow is not irrotational.
Hardin and Pope [14] proposed to use a viscous-acoustic splitting technique for hybrid CAA-applications,
to avoid these drawbacks.Their method,also referred to as expansion about incompressible ow(EIF),uses
equivalent source terms obtained with an incompressible source calculation.This equivalent source terms
contains no acoustic uctuations,since these uctuations are only generated by the compressibility of the
ow eld.However,it should be noticed that for applications where the traditional acoustic analogies fail,
the acoustic uctuations may signicantly change the ow eld inside the source region,which makes it
impossible for an incompressible source region calculation to render accurate ow domain results.
2.2.2 Acoustic Boundary Bonditions
Another way of coupling the results from the source region with the acoustic propagation equations,is
through the use of the uctuating density,pressure and velocity eld as acoustic boundary conditions for
the propagation equations.No source terms are required for this type of coupling.This imposes strong
restrictions on the calculation of the source region.Commercial CFD-codes,offering LES solution schemes,
calculate the ow eld with lower-order fairly dissipative numerical schemes without avoiding spurious
reections at the boundaries.If not taken care of properly,these numerical schemes and boundary conditions
can introduce numerical noise inside the computational domain.These errors can become of the same order
of magnitude as the acoustic variables necessary for this type of coupling [4].All these elements make the
method of acoustic boundary conditions more sensitive to the accuracy of the source region modeling as
compared to the acoustic analogy approach.Furthermore,a compressible simulation is needed to capture
any acoustic uctuation,which is a serious computational disadvantage for low Mach number applications,
which are commonly solved in an incompressible way.
A problem that arises with the use of acoustic boundary conditions is that the surface on which the vari-
ables are calculated (the Kirchhoff surface) should be located far enough fromthe aero-acoustic sources and
AEROACOUSTICS AND FLOW NOISE 599
no turbulent ow should pass the boundary.If vortical outow occurs through the Kirchhoff's surface,the
velocity uctuations may contain vorticity components,and density and pressure uctuations may contain
hydrodynamic and entropy uctuations,which may excite the vorticity and entropy modes of the propaga-
tion equations.Near the Kirchhoff surface small instabilities may occur due to the fact that the uctuating
variables do not exactly satisfy the propagation equations.For this reason articial selective damping [15] is
needed.
Especially the APE can suffer from unstable solutions near the Kirchhoff surface when a turbulent ow
passes this boundary.A way of avoiding these problems is,like for the equivalent sources,to carry out a
ltering of the variables at the Kirchhoff surface in such a way that only purely acoustic uctuations are
used as boundary conditions.In this way,only acoustic variables are taken into account in the propagation
equations.
A major advantage of the use of acoustic boundary conditions is that,when the boundary variables only
contain acoustic uctuations,this method can be seen as an acoustic continuation of the LES in the regions
where no further noise sources are present [12].In this way,if the owdomain calculation is accurate,it can
be assumed that this way of coupling renders the most accurate results.Furthermore,the propagation region
does not contain the source region and thus a smaller propagation region needs to be considered as compared
to the acoustic analogy approach.
3 Validation Examples of Hybrid CAA-Approaches
The drawbacks of the various coupling techniques and the need for aeroacoustic ltering techniques is in
this section illustrated for two two-dimensional applications:cavity noise and the aerodynamically generated
noise by a square cylinder in cross ow.
3.1 Cavity Noise
The phenomenon of ow-induced noise radiation in cavities has been studied in numerous investigations in
the past [16] and has a broad range of aerospace and automotive applications.The noise spectrum of cavity
noise contains both broadband components,introduced by the turbulence in the shear layer,and tonal com-
ponents due to a periodical vortex shedding at the cavity leading edge (wake mode) or a feedback coupling
between the ow eld and the acoustic eld (shear-layer mode or Rossiter mode).Full details about the
results that are discussed in this section can be found in [17,18,19].
3.1.1 Wake Mode
A cavity,oscillating in wake mode,is characterized by a large-scale vortex shedding fromthe cavity leading
edge.The vortex reaches nearly the cavity size,dragging during its formation irrotational free-stream uid
into the cavity.The vortex is then shed from the leading edge,and violently ejected from the cavity.In this
case the boundary layer separates upstream during the vortex formation,and downstream,as it is convected
away.The ow eld is characterized by turbulent velocities which are orders of magnitude larger than the
acoustic uctuations.
The dominant frequency occurs at a Strouhal number of 0.064.This is in agreement with the DNS-results
of Rowley [20] and is the dominant frequency for cavities oscillating in wake mode,independently of Mach
number.The instantaneous pressure contours obtained with LEEand acoustic boundary conditions are shown
in the left of gure 2.The pressure contours show an acoustic propagation with a dominant radiation up-
stream of the cavity.Since the LEE support the propagation of vorticity waves,hydrodynamic pressure
uctuations,often referred to as pseudo-sound,are observed in the outow region near the walls.When
600 PROCEEDINGS OF ISMA2006
Source￿Region
(a)
(b)
Figure 2:Instantaneous pressure contours (a) and directivity pattern obtained with different coupling tech-
niques (b) for a cavity oscillating in wake mode (M = 0.5,L/D = 4,Re
D
= 1500) (L and D are,
respectively,length and depth of the cavity).
using the APE as propagation equations,these pressure uctuations are not supported,which leads to in-
stabilities in the nal solution since the hydrodynamic uctuations are inherently present in the boundary
condition values.As explained before,a ltering procedure could avoid these instabilities and would make
it possible to evaluate the acoustic eld in the outow region.
Since the source region contains mostly turbulent uctuations,the different equivalent source term formu-
lations eq.(7,8) are thus containing primarily contributions from the turbulent eld and only a minor,erro-
neous,contribution of the acoustic eld inside the source region.The directivity pattern at the resonance
frequency,obtained with the LEE,coupled with the different aero-acoustic source term formulations are in
good agreement with the results obtained with the acoustic boundary conditions (g.2 right).
3.1.2 Shear-Layer Mode
As the length L of the cavity,relative to the momentumthickness of the boundary layer at the leading edge,
or the Mach number is decreased,there is a substantial change in the pattern of the cavity oscillations [18].
Under these circumstances,a shear-layer mode,characterized by the roll-up of vorticity in the shear layer,
occurs inside the cavity.The vortices are convected with the mean ow until they hit the downstreamcavity
edge.At that moment,acoustic waves are generated that propagate upstream,exciting the shear layer at the
upstream cavity edge.The turbulent velocity uctuations are for this mode of the same order of magnitude
as the acoustic uctuations.
The left of gure 3 shows the instantaneous pressure contours obtained with LEE and acoustic boundary
conditions for a cavity oscillating in shear-layer mode.A shorter acoustic wavelength and thus a higher
resonance frequency is observed.The Strouhal number of the dominant resonance equals 0.19 which is
in agreement with the experimental results of Rossiter [21].Hydrodynamic pressure uctuations,near the
downstreamwall are still present but have a much lower amplitude,since there is a much smaller perturbation
of boundary layer downstreamof the cavity trailing edge.
The turbulent velocity uctuations are much smaller and acoustic uctuations are of the same order of mag-
nitude.This leads to a large contribution of acoustic variables inside the source term formulations.The
directivity pattern at the rst resonance frequency,shown in the right of gure 3,is inaccurately predicted.
For both the vorticity based source terms and Lighthills source terms,the dominant propagation direction is
shifted downstream,resulting in a different radiation pattern.
AEROACOUSTICS AND FLOW NOISE 601
Source Region
(a)
(b)
Figure 3:Instantaneous pressure contours (a) and directivity pattern obtained with different coupling tech-
niques (b) for a cavity oscillating in shear-layer mode (M = 0.6,L/D = 2,Re
D
= 1500).
3.2 Noise Generation by a Square Cylinder in Cross-Flow
The owaround a rectangular cylinder has been subject of intense experimental and numerical research in the
past.Although most of this research focuses on technical problems associated with energy conversion and
structural design (e.q.vortex owmeters,buildings,towers,...) this application is also relevant in the eld of
aero-acoustics.Practical examples can be found e.g.in automotive applications such as the noise generated
by a luggage carrier system or a side-wing mirror.When the Reynolds number of a ow around a square
cylinder exceeds a critical value,a time-periodic oscillation develops.This Benardvon Karman instability
is characterized by a periodic phenomenon,referred to as vortex shedding and an antisymmetric wake ow
pattern,usually referred to as the von Karman vortex street.In the laminar regime,which usually persists up
to Reynolds number of about 400,the vortex shedding is characterized by one pronounced frequency [22].
The noise radiation is primarily caused by the uctuating lift forces acting on the cylinder,resulting in a
typical dipole-type of radiation.All details about this calculation can be found in [12].
(a)
(b)
Figure 4:Instantaneous uctuating pressure eld ( p
￿
) obtained with DNS (a) and APE with acoustic bound-
ary conditions(b) for a square cylinder in cross ow (M = 0.5,Re
D
= 200)
The instantaneous uctuating pressure eld obtained with DNS is shown in gure 4 on the left.A dipole
602 PROCEEDINGS OF ISMA2006
radiation is observed at a non-dimensional frequency of 0.292,which is in good agreement with other re-
sults [22].In the wake of the cylinder a von Karman vortex street is observed with hydrodynamic pressure
uctuations (pseudo-sound) which have a much larger amplitude than the acoustic uctuations.When a
hybrid method,using acoustic boundary conditions as coupling technique,is used for the same problem
(g.4b) these hydrodynamic pressure uctuations create spurious acoustic waves in the downstream direc-
tion resulting in an erroneous prediction of the acoustic eld.
Looking at the results of these two applications,it is clear that a ltering technique is needed in order to
obtain accurate acoustic results with hybrid CAA-technologies.Acoustic analogies tend to fail as coupling
technique when the turbulent uctuations in the source region are of the same order of magnitude as the
acoustic uctuations,which is the case in applications dominated by a ow-acoustic feedback phenomenon
or when acoustic modes are presents.Acoustic boundary conditions on the other hand are only reliable when
no outow through the Kirchhoff's surface occurs.
4 Potential Filtering Techniques
In this section possible ltering techniques are theoretically described.The basic principle of these tech-
niques is shown in gure 5.When acoustic boundary conditions are used as coupling technique it is suf-
cient to obtain acoustic variables in a small region surrounding the most important aeroacoustic sources.If
an aerodynamic-acoustic splitting is needed to obtain ltered equivalent sources,the ltering region should
be equal to the source region.
Source￿Region
Propagation￿Region
Filtering￿Region
Figure 5:Sketch of the ltering region for free-eld (left) and ducted (right) aeroacoustic applications.
4.1 Mode Matching Strategies
A rst type of ltering techniques is based on mode matching techniques and is thus only applicable for
ducted ows or for applications that mathematically allowa representation of the solution by slowly varying
modes [11].The technique uses a small matching interface between the source region and the acoustic
region consisting of three or more axial planes.At the matching interface,the acoustic pressure uctuations
are obtained through a least squares t of the pressure uctuations,obtained in the source region,with the
acoustic modes of the duct.This method is often referred to as the multiple plane matching technique.
Rienstra [23] showed that,for cylindrical ducts with a slowly varying cross-section where a nearly uniform
ow is assumed,the pressure p(x,r) (with x the axial and r the radial position) can,independent of circum-
AEROACOUSTICS AND FLOW NOISE 603
ferential order,be written as a summation of left- and right-running modes:
p(x,r) =

￿
µ=−∞
A
µ
Ψ
µ
(x,r) exp
￿
−i
￿
x
x
0
k
µ
(εσ)dσ
￿
(9)
where µ = 0 is excluded.The functions Ψ
µ
represent the basis functions for the right-running (µ > 0)
and left-running (µ < 0) slowly varying pressure modes,A
µ
are the modal amplitudes,k
µ
the modied
wavenumber,taking into account the mean ow and ε is a small parameter,which is the only cause of
variation of the mean ow.For a two dimensional straight duct with a uniform mean ow and hard wall
boundary conditions eq.(9) can be simplied,for frequencies below the transversal cut-off frequency,to:
p(x) = A
−1
exp(−ik
−1
x) +A
1
exp(−ik
+1
x) (10)
where,in absence of viscous dissipation,k
+1
= k/(1+M) and k
−1
= −k/(1−M) are the modied right-
and left-running wave number where k = ω/c
0
is the wavenumber and Mthe Mach number.
In case of a matching zone that consists of three axial planes x = x
0
,x
1
,x
2
(where x
0
< x
1
,x
2
),the Fourier
decomposition (for each frequency and circumferential mode) of the pressure data is obtained from the
source region calculation and are equal to ℘
0
(r),℘
1
(r),℘
2
(r) at the respective planes.If swirl or other types
of vorticity,causing hydrodynamic pressure uctuations,are not dominant in the mean ow,the following
equations can be written based on eq.(9),with N the trunction number of the innite summation:

0
(r) =
N
￿
µ=−N
A
µ
Ψ
µ
(x
0
,r) (11)

1
(r) =
N
￿
µ=−N
A
µ
Ψ
µ
(x
1
,r) exp
￿
−i
￿
x
1
x
0
k
µ
(εσ)dσ
￿
(12)

2
(r) =
N
￿
µ=−N
A
µ
Ψ
µ
(x
2
,r) exp
￿
−i
￿
x
2
x
0
k
µ
(εσ)dσ
￿
(13)
For the simplied case of the 2D straight duct this becomes for each Fourier component:

0
(r) = A
−1
+A
1
(14)

1
(r) = A
−1
exp[−ik
−1
(x
1
−x
0
)] +A
1
exp[−ik
+1
(x
1
−x
0
)] (15)

2
(r) = A
−1
exp[−ik
−1
(x
2
−x
0
)] +A
1
exp[−ik
+1
(x
2
−x
0
)] (16)
The amplitudes A
µ
can be determined by a least squares t of this overdetermined set of equations.For
numerical stability,it is preferable to rescale the basis functions in order to prevent exponentially large terms
at the zone ends fromunbalancing the least squares minimization [11].In principle two planes are sufcient
to determine the exact amplitudes.However,the overdetermination of the systemof equations is prefered to
avoid errors originating fromthe presence of small hydrodynamic pressure uctuations,which are not taken
into account in this method.
This technique is appealing due to its simplicity and is easy to implement.Some successful validations have
been preformed in the TurboNoiseCFD European project [11] for the acoustic propagation of aeroacous-
tic sources in turbofan engine bypass ducts.Furthermore,a distinction between the right- and left-running
acoustic waves can be made,which makes it possible to exclude the reected modes fromthe solution,mak-
ing this method less sensitive to the boundary conditions used for the source region computation.However
this approach has only a limited number of applications in which the acoustic pressure uctuations dominate
the hydrodynamic pressure uctuations.This ltering technique is useful for applications with a uniform
mean ow and where acoustic pressure uctuations are dominant.In a large number of low-Mach number
604 PROCEEDINGS OF ISMA2006
applications,it can be expected that acoustic pressure uctuations are of low amplitude and hence difcult
to obtain fromthe total pressure eld in the source region with this ltering technique.
In order to exclude hydrodynamic pressure uctuations,an extension can be made based on the character-
istic properties of the ow eld [24].A three-dimensional ow eld consists of ve characteristic modes,
each with their own characteristic velocity:two vorticity modes and one entropy mode that are convected
with the mean ow velocity
−→
v
0
;one acoustic right-running mode and one acoustic left-running mode with
characteristic velocities of c
0
+
−→
v
0
,respectively
−→
v
0
−c
0
.The hydrodynamic pressure eld is generated by
the vorticity modes and thus can be assumed to be convected with the mean oweld.If viscous dissipation
does not occur and under assumption of a uniform one-dimensional mean ow the hydrodynamic pressure
uctuations should be conserved along the characteristic line x+v
x,0
t = cte while the acoustic wave should
be conserved along the characteristic line x+(v
x,0
+c
0
)t = cte and x+(v
x,0
−c
0
)t = cte for respectively
the right- and left-running mode.
The conservation of the acoustic wave amplitude is satised by eq.(9).For the hydrodynamic pressure
uctuations it is sufcient to add,for a one-dimensional mean ow,the following equation to eq.(9):
p
turb
(x) = Bexp
￿
−i
￿
x
x
0
k
turb
(εσ)dσ
￿
(17)
where k
turb
= 2πf/v
(
x,0) is the turbulent wavenumber.For the 2D square duct the following equation can
then be obtained for each Fourier component of the total pressure eld:
p(x) = p
ac
(x) +p
turb
(x) = [A
−1
exp(−ik
−1
x) +A
1
exp(−ik
+1
x)] +[Bexp(−ik
turb
x)] (18)
To solve for the amplitudes of eq.(18) a minimum of three planes is needed.However it is useful to take at
least four planes and solve an overdetermined system of equations.Especially the hydrodynamic pressure
uctuations are,in most cases,not purely convected but also dissipated by viscous effects,which are not
taken into account in eq.(18).For this reason hydrodynamic pressure uctuations should still be of fairly low
amplitude.The ltering techniques based on mode matching strategies are not generally applicable for all
CAA-applications,hence other ltering strategies are required.
4.2 An Aerodynamic-Acoustic Splitting Technique
Another possible ltering technique is based on a decomposition of the velocity uctuations into an aerody-
namic or turbulent part (
−−→
v
￿
turb
) and an acoustic part (
−→
v
￿
ac
).The technique consists of taking the total (when
a ltering is needed to obtain accurate equivalent sources) or only a small part (if the coupling is carried out
using acoustic boundary conditions) of the source region.In this ltering region both velocity elds can then
be separated at every timestep of the computation and used to obtain ltered source terms [26] or acoustic
boundary conditions.
It is well-known [25] that each velocity eld (
−→
v
￿
) can be written as the sumof an irrotational (
−→
v
￿
ac
),solenoidal
(
−−→
v
￿
turb
) and both solenoidal and irrotational eld (
−→
u
￿
):
−→
v
￿
=
−→
v
￿
ac
+
−−→
v
￿
turb
+
−→
u
￿
(19)
where it can be assumed that the solenoidal and irrotational eld is negligibly small (
−→
u
￿
= 0).For most
isentropic,low Mach number,aeroacoustics problems,it can be assumed that the acoustic velocity uc-
tuations are inviscid and irrotational and that all compressible effects are purely acoustic.The continuity
equation (20) of the LEE can be considered for this kind of applications as purely'acoustical'.
∂ρ
￿
∂t
+
−→
v
0
∙ ￿ρ
￿

0
￿∙
−→
v
￿
= 0 (20)
AEROACOUSTICS AND FLOW NOISE 605
which results in the following conditions for the acoustic eld and turbulent eld at every timestep:
￿∙
−→
v
￿
ac
= ￿∙
−→
v
￿
= ￿
￿
(21)
￿×
−→
v
￿
ac
= 0 (22)
￿∙
−−→
v
￿
turb
= 0 (23)
￿×
−−→
v
￿
turb
= ￿×
−→
v
￿
=
−→
ω
￿
(24)
where ￿
￿
is the expansion ratio and
−→
ω
￿
the vorticity of the uctuating velocity eld in the source region,
which are both known fromthe ow domain calculation.
The distribution of the expansion ratio ￿
￿
is known from the source region calculation which leads to fol-
lowing potential formulation for the acoustic eld:
−→
v
￿
ac
= ￿φ (25)
￿
2
φ = ￿
￿
(26)
In a similar way the turbulent velocity eld can be written as:
−−→
v
￿
turb
= ￿× ￿χ (27)
￿×(￿× ￿χ) = ￿(￿∙ ￿χ) −￿
2
￿χ =
−→
ω
￿
(28)
or,since it can be assumed for two-dimensional and free-eld applications that ￿∙ ￿χ = 0:
￿
2
￿χ = −
−→
ω
￿
(29)
this leads to a systemof two coupled Laplace equations with the potential functions φ and χ as unknowns:
￿
2
φ = ￿
￿
(30)
￿
2
￿χ = −
−→
ω
￿
(31)
or in 2 dimensions in velocity formulation:
∂v
￿
x,ac
∂x
+
∂v
￿
y,ac
∂y
= ￿
￿
=
∂v
￿
x
∂x
+
∂v
￿
y
∂y
(32)
∂v
￿
y,ac
∂x

∂v
￿
x,ac
∂y
= 0 (33)
∂v
￿
x,turb
∂x
+
∂v
￿
y,turb
∂y
= 0 (34)
∂v
￿
y,turb
∂x

∂v
￿
x,turb
∂y
=
−→
ω
￿
=
∂v
￿
y
∂x

∂v
￿
x
∂y
(35)
(36)
Decomposing the velocity eld into an acoustic uctuating part and a turbulent uctuating part thus requires
solving a coupled systemof two inhomogeneous Laplace equations or four rst-order differential equations.
For both problems,a proper set of boundary conditions has to be dened.
A rst boundary condition is imposed by the coupling of the two velocity elds:
v
￿
ac,x
+v
￿
turb,x
= v
￿
x
(37)
v
￿
ac,y
+v
￿
turb,y
= v
￿
y
(38)
606 PROCEEDINGS OF ISMA2006
A second set of boundary conditions is needed for the acoustic or turbulent uctuating velocity eld.If the
ltering region is bounded with rigid walls,both the acoustic and turbulent velocity uctuations should be
zero.The only type of boundary condition that remains are the boundaries of the ltering region,where no
coincidence with walls of the source region is occurring.
A possible boundary condition can be obtained by combining the continuity equation,with the irrotational
momentum equation of the LEE.Since the density uctuations can be assumed to be purely acoustical,the
uctuations can be obtained by:
∂ρ
￿
ac
∂t
+
−→
v
0
∙ ￿ρ
￿
ac

0
￿
￿
= 0 (39)
The isentropic relation between pressure and density allow to determine the acoustic pressure uctuations,
dp
￿
ac
= c
2
0

￿
ac
(40)
The irrotational momentum equation of the APE then provides a second set of boundary conditions for the
systemof equations
∂u
￿
i,ac
∂t
+

∂x
i
(u
j,0
u
￿
i,ac
+
p
￿
ac
ρ
0
) = 0 (41)
Another set of boundary conditions is based on the asymptotic behavior of the acoustic waves [27].If the
ltering region boundaries are located far enough from the source region following equation in spherical
coordinates (r,θ,ϕ) hold for the acoustic velocity uctuations:
1
V (θ,ϕ)
∂u
i,ac
(r,θ,ϕ)
￿
∂t
+
∂u
i,ac
(r,θ,ϕ)
￿
∂r
+
2
r
u
i,ac
(r,θ,ϕ)
￿
= 0 (42)
where the center of the coordinate systemis taken at the approximate position of the most important sources.
V (θ,ϕ) is the mean velocity of the acoustic waves (c + ￿u
0
) projected in the r-direction.
Further research is focused on the implementation of these time-dependent set of boundary conditions and
the systemof coupled equations and on the validation of this aerodynamic-acoustic splitting approach.
5 Conclusions
When using a hybrid computational approach to solve aeroacoustic problems,it is clear that in some appli-
cations proper ltering techniques are needed.Aeroacoustic analogies tend to fail when the source region,
where the aerodynamically generated sound eld needs to be solved,contains a turbulent uctuating eld
which is of the same order of magnitude as the acoustic eld.This is the case for applications in which
a ow-acoustic feedback phenomenon occurs (e.g.cavity noise) or when acoustic resonances are likely
to happen which as in ducted environments.The other coupling technique,which uses acoustic boundary
conditions,is unreliable if a vortical outow through the Kirchhoff's surface occurs.In this case hydro-
dynamic pressure uctuations,caused by the presence of vorticity,generate spurious acoustic waves in the
computational domain,also referred to as pseudo-sound.
In this paper the need for ltering techniques is illustrated by two examples:aerodynamically generated
sound by a ow over a rectangular cavity and over a square cylinder.For the rst application the numerical
set-up can be such that a ow-acoustic feedback occurs (shear-layer mode).In this case it is shown that
acoustic analogies do not give accurate results,while the same source termformulations give accurate results
when the cavity oscillates in wake mode.The downstream acoustic eld generated by a square cylinder in
cross-ow is difcult to predict with acoustic boundary conditions since a turbulent outow through the
Kirchhoff's surface takes places.
This paper proposes different ltering techniques,which are only theoretically developed;a validation of
these techniques is planned in future research.Arst class on ltering techniques is based on mode matching
AEROACOUSTICS AND FLOW NOISE 607
strategies.This technique can only be used in the frequency domain and when the hydrodynamic pressure
uctuations are lowin amplitude or do not occur,which is only the case for a limited number of applications.
The technique uses a least squares t of the pressure,obtained by the source region calculation,with a known
set of basis functions.The technique can be extended to incorporate the hydrodynamic pressure uctuations
by assuming that these uctuations are convected with the mean ow eld.Although appealing due to its
simplicity,this ltering technique is limited to a number of applications and does not solve the problems that
arise with acoustic analogies.
Another type of ltering technique is based on the decomposition of the velocity eld into an aerodynamic
uctuating part and a purely acoustic part.This technique is more general than the mode matching techniques
but is more time-consuming.The technique assumes an irrotational,inviscid acoustic eld and is only valid
for isentropic,low Mach number applications.It is based on a system of coupled Laplace equations which
need to be solved simultaneously.Proper boundary conditions still need to be developed and will make the
problemtime-dependent.In theory this aerodynamic-acoustic splitting technique solves the problems arising
for both coupling techniques although some extensive development and validation is necessary to verify the
validity and practical potential of this ltering approach.
Acknowledgements
The research work of Wim De Roeck is nanced by a scholarship of the Institute for the Promotion of
Innovation by Science and Technology in Flanders (IWT).
References
[1]
Colonius T.,Lele S.,Moin P., Sound generation in a mixing layer,Journal of Fluid Mechanics,
Vol.330,pp.375409 (1997).
[2]
Mitchell B.,Lele S.,Moin P., Direct computation of the sound generated by vortex pairing in an
axisymmetric jet,Journal of Fluid Mechanics,Vol.383,pp.113142 (1999).
[3]
Breuer M., Numerical and modeling inuences on large eddy simulations for the ow past a circular
cylinder,International Journal of Heat and Fluid Flow,Vol.19,pp.512521 (1998).
[4]
Tam C.K.W., Computational Aeroacoustics:Issues and Methods,AIAA-Journal,Vol.33,pp.1788
1796 (1995).
[5]
Goldstein M., Aeroacoustics,McGraw-Hill,New York,1976.
[6]
Bailly C.,Juv
´
e D., Numerical Solution of Acoustic Propagation Problems Using Linearized Euler
Equations,AIAA-journal,Vol.38,pp.2229 (2000).
[7]
Ewert R.,Schr
¨
oder W., Acoustic Perturbation Equations Based on Flow Decomposition via Source
Filtering,Journal of Computational Physics,Vol.188,pp.365398 (2003).
[8]
Lighthill M.J., On Sound Generated Aerodynamically;I.General Theory,Proc.Roy.Soc.(London),
Vol.211,pp.564587 (1952).
[9]
Ffowcs-Williams J.E.,Hawkings D.L., Sound Generation by Turbulence and Surfaces in Arbitrary
Motion,Phil.Trans.Roy.Soc.,Vol.A264,No.1151,pp 321342 (1969).
[10]
Lyrintzis A., The Use of Kirchhoff's Method in Computational Aeroacoustics,Journal Fluids Eng.,
Vol.116,pp.665676 (1994).
608 PROCEEDINGS OF ISMA2006
[11]
Ovenden N.C.,Rienstra S., Mode-Matching Stategies in Slowly Varying Engine Ducts,AIAA-Journal,
Vol.42,pp.18321840 (2004).
[12]
De Roeck W.,Rubio G.,Baelmans M.,Sas P.,Desmet W., The Inuence of Flow Domain Mod-
elling on the Accuracy of Direct and Hybrid Aeroacoustic Noise Calculations,Proc.12th AIAA/CEAS
Aeroacoustic conference,Cambridge,MA,USA,AIAA paper 2006-2419 (2006).
[13]
Powell A., Theory of Vortex Sound,Journal of the Acoustics Society of America,Vol.36,pp.177-195
(1964).
[14]
Hardin J.,Pope D.S., An Acoustic/Viscous Splitting Technique for Computational Aeroacoustics,
Theoretical and Computational Fluid Dynamics,Vol.6,pp.323-340 (1994).
[15]
Tam C.K.W.,Webb J.C.,Dong Z., A Study of the Short Wave Components in Computational Acous-
tics,Journal of Computational Acoustics,Vol.1,No.1,p.1-30 (1993).
[16]
Komerath N.M.,Ahuja K.K.,Chambers F.W., Prediction and Measurement Flows over Cavities a
survey,AIAA paper 82-022 (1987).
[17]
De Roeck W.,Rubio G.,Reymen Y.,Meyers J.,Baelmans M.,Desmet W., Towards Accurate Flowand
Acoustic Prediction Techniques for Cavity Flow Noise Applications,Proc.11th AIAA/CEAS Aeroa-
coustics Conference,Monterey,CA,USA,AIAA paper 2005-2978 (2005).
[18]
Rubio G.,De Roeck W.,Baelmans M.,Desmet W., Numerical Identication of Flow-Induced Oscil-
lation Modes in Rectangular Cavities using Large Eddy Simulation accepted for publication in Interna-
tional Journal for Numerical Methods in Fluids (2006).
[19]
De Roeck W.,Rubio G.,Baelmans M.,Desmet W., Towards Accurate Hybrid Prediction Techniques
for Cavity Flow Noise Applications,submitted for publication in AIAA-journal (2006).
[20]
Rowley C.W., Modeling,Simulation and Control of Cavity Flow Oscillations Ph.D.Dissertation,
California Institute of Technology,Pasadena,CA,USA (2002).
[21]
Rossiter J.E., Wind Tunnel Experiments on the Flow over Rectangular Cavities at Subsonic and Tran-
sonic Speeds,Royal Aircraft Establishment,technical report 64037 (1964).
[22]
Okajima A., Strouhal Numbers of Rectangular Cylinders,Journal of Fluid Mechanics,Vol.123,pp.
379398 (1982).
[23]
Rienstra S.W., Sound Transmission in Slowly Varying Circular an Annular Lined Ducts with Flow,
Journal of Fluid Mechanics,Vol.380,pp.279296 (1999).
[24]
Thompson K.W., Time Dependent Boundary Conditions for Hyperbolic Systems,Journal of Compu-
tational Physics,Vol.68,pp.124 (1987).
[25]
Batchelor G.K., An Introduction to Fluid Dynamics,Cambridge University Press (1967).
[26]
Ewert R.,Meinke M.,Schr
¨
oder W. Comparison of Source TermFormulations for a Hybrid CFD/CAA
Method,AIAA-paper 2001-2200 (2001).
[27]
TamC.K.W.,Dong Z., Radiation and OutowBoundary Conditions for Direct Computation of Acous-
tic and Flow Disturbances in a Non-Uniform Mean Flow,Journal of Computational Acoustics,Vol.4,
pp.175201 (1996).
AEROACOUSTICS AND FLOW NOISE 609
610 PROCEEDINGS OF ISMA2006