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 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 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 NavierStokes 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 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 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.

SourceRegion

PropagationRegion

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 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 higher-order 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 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 NaviesStokes 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

/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 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 NavierStokes 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

(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 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 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 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 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 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 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 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 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 CFD-codes,offering LES solution schemes,

calculate the ow eld with lower-order 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 aero-acoustic 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 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 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 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 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 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 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 aerodynamic-acoustic 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 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 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 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 modied 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 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 left-running

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 low-Mach 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 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 oweld.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 satised by eq.(9).For the hydrodynamic pressure

uctuations it is sufcient 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

+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 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 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 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 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 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 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).

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610 PROCEEDINGS OF ISMA2006

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