On the Use of Filtering Techniques for Hybdrid Methods in
Computational AeroAcoustics
W.De Roeck,G.Rubio,W.Desmet
K.U.Leuven,Department of Mechanical Engineering,
Celestijnenlaan 300 B,B3001,Leuven,Belgium
email:wim.deroeck@mech.kuleuven.be
Abstract
Hybrid CAAapproaches 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 nonquiescent 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 owacoustic 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 CAAapplications and starts with the theoretical development of a new ltering technique
based on an aerodynamicacoustic 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 takeoff 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 NavierStokes 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 NavierStokes
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 CAAapplications 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 NavierStokes equations in such a way that the left hand side equals the linear acoustic wave
equation,the wellknown 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 owacoustic 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 crossow.
Next a rst (frequencydomain) 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 CAAapplications.The main conclusions are summarized in the
nal section.
2 Theory of Hybrid CAAtechniques
In hybrid CAAtechniques,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 NavierStokes 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 oweld 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
inuenced.If convection and refraction of sound waves do not occur or have a negligible inuence (the
acoustic fareld),one can use the conventional acoustic wave equation to compute the propagation of
the acoustic waves.If these effects become important (in the acoustic neareld 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.
SourceRegion
PropagationRegion
Figure 1:Sketch of the different computational domains for freeeld (left) and ducted (right) aeroacoustic
applications.
2.1 Propagation Equations
Linearized Euler equations (LEE) are commonly used to describe the neareld propagation of acoustic
waves in the presence of a nonuniform 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 NavierStokes 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 higherorder terms,
∂ρ
∂t
+
∂
∂x
i
(ρ
u
i0
+ρ
0
u
i
) = Φ
cont
(1)
∂ρ
0
u
i
∂t
+
∂
∂x
j
(ρ
0
u
j0
u
i
+p
δ
ij
) +(ρ
u
j0
+ρ
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
,Φ
mom,i
and Φ
ener
are the source terms for,respectively,the continuity,momentum and energy
equation,which can contain nonlinear,viscosity and temperature effects and can be calculated based on
timedependent source domain results.The mean ow variables can be easily obtained by calculating the
ReynoldsAveraged NaviesStokes equations (RANS).
Although the LEE describe the propagation of acoustic waves in a nonquiescent 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
/ρ
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
+ρ
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
efcient than the LEE,since they solve one equation less due to the fact that the isentropic relation between
pressure and density is inherently satised.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 CAAtechniques:
•
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 FfowcsWilliams 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 NavierStokes equations in such
a way that the lefthand side equals the linear acoustic wave equation without a mean ow,whereas all
other terms are treated as righthand side source terms,Lighthill obtained,for subsonic,isentropic ows,the
Lighthill stress tensor as equivalent source term:
Φ = −
∂
2
(ρ
0
u
i
u
j
)
∂x
i
∂x
j
(6)
The approach of replacing the whole noise generating oweld by an equivalent source termis appealing due
to its simplicity and can be used to identify possible aeroacoustic 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 RANScalculations,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 nonlinear terms
of the decomposed NavierStokes 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 NavierStokes 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 lefthand side.Nevertheless,in a large
number of aeroacoustic 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 owacoustic feedback
is present,the uctuating part contains no only a minor acoustic uctuating part and can thus be seen as an
accurate source termdenition.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
+ρ
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 NavierStokes equations are rewritten in such a way that the lefthand 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 inuence (
−→
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 viscousacoustic splitting technique for hybrid CAAapplications,
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 signicantly 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 CFDcodes,offering LES solution schemes,
calculate the ow eld with lowerorder fairly dissipative numerical schemes without avoiding spurious
reections 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 aeroacoustic sources and
AEROACOUSTICS AND FLOW NOISE 599
no turbulent ow should pass the boundary.If vortical outow 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 articial 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 CAAApproaches
The drawbacks of the various coupling techniques and the need for aeroacoustic ltering techniques is in
this section illustrated for two twodimensional applications:cavity noise and the aerodynamically generated
noise by a square cylinder in cross ow.
3.1 Cavity Noise
The phenomenon of owinduced 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 (shearlayer 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 largescale vortex shedding fromthe cavity leading
edge.The vortex reaches nearly the cavity size,dragging during its formation irrotational freestream 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 DNSresults
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 pseudosound,are observed in the outow region near the walls.When
600 PROCEEDINGS OF ISMA2006
SourceRegion
(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 outow 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 aeroacoustic source term formulations are in
good agreement with the results obtained with the acoustic boundary conditions (g.2 right).
3.1.2 ShearLayer 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 shearlayer mode,characterized by the rollup 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 shearlayer 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 shearlayer mode (M = 0.6,L/D = 2,Re
D
= 1500).
3.2 Noise Generation by a Square Cylinder in CrossFlow
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
aeroacoustics.Practical examples can be found e.g.in automotive applications such as the noise generated
by a luggage carrier system or a sidewing mirror.When the Reynolds number of a ow around a square
cylinder exceeds a critical value,a timeperiodic oscillation develops.This Benardvon 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 dipoletype 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 nondimensional 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 (pseudosound) 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 CAAtechnologies.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 owacoustic feedback phenomenon
or when acoustic modes are presents.Acoustic boundary conditions on the other hand are only reliable when
no outow 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 aerodynamicacoustic splitting is needed to obtain ltered equivalent sources,the ltering region should
be equal to the source region.
SourceRegion
PropagationRegion
FilteringRegion
Figure 5:Sketch of the ltering region for freeeld (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 crosssection 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 rightrunning 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 rightrunning (µ > 0)
and leftrunning (µ < 0) slowly varying pressure modes,A
µ
are the modal amplitudes,k
µ
the modied
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 simplied,for frequencies below the transversal cutoff 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 modied right
and leftrunning 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 innite 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 simplied 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 sufcient
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 leftrunning
acoustic waves can be made,which makes it possible to exclude the reected 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 lowMach number
604 PROCEEDINGS OF ISMA2006
applications,it can be expected that acoustic pressure uctuations are of low amplitude and hence difcult
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 threedimensional 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 rightrunning mode and one acoustic leftrunning 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 oweld.If viscous dissipation
does not occur and under assumption of a uniform onedimensional 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 leftrunning mode.
The conservation of the acoustic wave amplitude is satised by eq.(9).For the hydrodynamic pressure
uctuations it is sufcient to add,for a onedimensional 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
CAAapplications,hence other ltering strategies are required.
4.2 An AerodynamicAcoustic 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 wellknown [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
+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 twodimensional and freeeld 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 rstorder differential equations.
For both problems,a proper set of boundary conditions has to be dened.
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
+ρ
0
= 0 (39)
The isentropic relation between pressure and density allow to determine the acoustic pressure uctuations,
dp
ac
= c
2
0
dρ
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 rdirection.
Further research is focused on the implementation of these timedependent set of boundary conditions and
the systemof coupled equations and on the validation of this aerodynamicacoustic 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 owacoustic 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 outow 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 pseudosound.
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
setup can be such that a owacoustic feedback occurs (shearlayer 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
crossow is difcult to predict with acoustic boundary conditions since a turbulent outow 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.Arst 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 timeconsuming.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
problemtimedependent.In theory this aerodynamicacoustic 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).
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AEROACOUSTICS AND FLOW NOISE 609
610 PROCEEDINGS OF ISMA2006
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