Surface Integral Methods in Computational Aeroacoustics

Surface Integral Metho ds in Computational

Aeroacoustics

-From the CFD Near-Field to the

Acoustic Far-Field

Anastasios S. Lyrintzis

Scho ol of Aeronautics and Astronautics

Purdue University

W. Lafayette, IN 47907-2023

Abstract

A review of recent advances in the use of integral metho ds in Computational AeroAcoustics

CAA for the extension of near- eld CFD results to the acoustic far- eld is given. These

integral formulations i.e. Kirchho 's metho d, p ermeable p orous surface Ffowcs-Williams

Hawkings FW-H equation allow the radiating sound to b e evaluated based on quantities

on an arbitrary control surface if the wave equation is assumed outside. Thus only surface

integrals are needed for the calculation of the far- eld sound, instead of the volume integrals

required by the traditional acoustic analogy metho d i.e. Lighthill, rigid b o dy FW-H equa-

tion. A numerical CFD metho d is used for the evaluation of the ow- eld solution in the

Presented at the CEAS Workshop \From CFD to CAA" Athens Greece, Nov. 2002.

Professor, e-mail: [email protected]. 1

near eld and thus on the control surface. Di usion and disp ersion errors asso ciated with

wave propagation in the far- eld are avoided. The surface integrals and the rst derivatives

needed can b e easily evaluated from the near- eld CFD data. Both metho ds can b e ex-

tended in order to include refraction e ects outside the control surface. The metho ds have

b een applied to helicopter noise, jet noise, prop eller noise, ducted fan noise, etc. A simple

set of p ortable Kirchho /FW-H subroutines can b e develop ed to calculate the far- eld noise

from inputs supplied byany aero dynamic near/mid- eld CFD co de.

1 Background - Aeroacoustic Metho ds

For an airplane or a helicopter, aero dynamic noise generated from uids is usually very im-

p ortant. There are many kinds of aero dynamic noise including turbine jet noise, impulsive

noise due to unsteady ow around wings and rotors, broadband noise due to in ow turbu-

lence and b oundary layer separated ow, etc. e.g. Lighthill . Accurate prediction of noise

mechanisms is essential in order to b e able to control or mo dify them to comply with noise

regulations, i.e. Federal Aviation Regulations FAR part 36, and achieve noise reductions.

Both theoretical and exp erimental studies are b eing conducted to understand the basic

noise mechanisms. Flight-test or wind-tunnel test programs can b e used, but in either case

diculties are encounted such as high exp ense, safety risks, and atmospheric variability,

as well as re ection problems for wind-tunnel tests. As the available computational p ower

increases numerical techniques are b ecoming more and more app ealing. Although complete

noise mo dels have not yet b een develop ed, numerical simulations with a prop er mo del are

increasingly b eing employed for the prediction of aero dynamic noise b ecause they are low-

cost and ecient. This research has led to the emergence of a new eld: Computational

AeroAcoustics CAA.

CAA is concerned with the prediction of the aero dynamic sound source and the transmis-

sion of the generated sound starting from the time-dep endentgoverning equations. The full,

time-dep endent, compressible Navier-Stokes equations describ e these phenomena. Although

recent advances in Computational Fluid Dynamics CFD and in computer technology have 2

made rst-principle CAA plausible, direct extension of current CFD technology to CAA re-

quires addressing several technical diculties in the prediction of b oth the sound generation

and its transmission. A review of aerospace application of CAA metho ds was given by

Long et al.

Aero dynamically generated sound is governed by a nonlinear pro cess. One class of

problems is turbulence generated noise e.g. jet noise. An accurate turbulence mo del is

usually needed in this case. A second class of problems involves impulsive noise due to

moving surfaces e.g. helicopter rotor noise, prop eller noise, fan noise etc.. In these cases

an Euler/Navier Stokes mo del or even a full p otential mo del is adequate, b ecause turbulence

is not imp ortant.

Once the sound source is predicted, several approaches can b e used to describ e its

propagation. The obvious strategy is to extend the computational domain for the full,

nonlinear Navier-Stokes equations far enough to encompass the lo cation where the sound

is to b e calculated. However, if the ob jective is to calculate the far- eld sound, this direct

approach requires prohibitive computer storage and leads to unrealistic turnaround time.

The impracticality of straight CFD calculations for sup ersonic jet aeroacoustics was p ointed

out by Mankbadi et al. Furthermore, b ecause the acoustic uctuations are usually quite

small ab out three orders of magnitude less than the ow uctuations, the use of nonlinear

equations whether Navier-Stokes or Euler could result in errors, as p ointed out by Stoker

and Smith. One usually has no choice but to separate the computation into two domains,

one describing the nonlinear generation of sound, the other describing the propagation of

sound. There are several alternatives to describing the sound propagation once the source

has b een identi ed.

1.1 Field solution of Simpler Equations

Linearized Euler Equations LEE The rst alternative is to use simpler equations in

the acoustic far- eld. The Linearized Euler Equations LEE have b een used in order to 3

7 8

extend the CFD solutions to the far- eld e.g. Lim et al. , Viswanathan and Sankar , Shih

et al. . The LEE equations employ a division of the ow eld into a time-averaged ow and

a time-dep endent disturbance which is assumed to b e small. The hybrid zonal approach

consists of the near- eld evaluation using an accurate CFD co de e.g. for jet noise the co de

is usually based on Large Eddy Simulations: LES and the extension of the solution to the

mid- eld using LEE. Considerable CPU savings can b e realized, since the LEE calculations

are muchcheap er than the CFD calculations. This approachisvery promising, b ecause it

accounts for a variable sound velo city outside the near- eld where usually an LES mo del is

applied. This metho d may also b e appropriate for an intermediate region in some problems,

outside from the reactive near- eld where the sp eed of sound is still not constant, b efore

moving to another integral metho d for the far- eld.

Other Equations Hardin and Pop e hava prop osed a decoupling of the time- dep endent

incompressible ow and the compressible asp ects acoustics of the ow. This technique

was used successfully to predict the owover a two-dimensional cavity. A eld solution of

11 11

the wave equation can also b e used e.g. Freund . Freund claims that the eld solution

of the wave equation is cheap er than the surface integral solutions see section 1.2.2, when

the solution everywhere in the eld is sought. However, in most applications only a few

lo cations are needed to study directivity and compare with microphone measurements. Also,

for anynumerical solution of eld equations dissipation and disp ersion errors still exist and

an accurate description of propagating far- eld waves is compromised.

1.2 Integral Metho ds

1.2.1 Volume Integral Metho ds

Traditional Acoustic Analogy The rst integral approach for acoustic propagation is

the acoustic analogy. In the acoustic analogy, the governing Navier-Stokes equations are

rearranged to b e in wave-typ e form. There is some question as to which terms should b e 4

identi ed as part of the sound source and retained in the right-hand side of the equation and

which terms should b e in the left-hand side as part of the op erator e.g., Lilley . The far-

eld sound pressure is then given in terms of a volume integral over the domain containing

the sound source. Several mo di cations to Lighthill's original theory have b een prop osed to

account for the sound- owinteraction or other e ects. The ma jor diculty with the acoustic

analogy,however, is that the sound source is not compact in sup ersonic ows. Errors could

b e encountered in calculating the sound eld, unless the computational domain could b e

extended in the downstream direction b eyond the lo cation where the sound source has

completely decayed. Furthermore, an accurate account of the retarded time-e ect requires

keeping a long record of the time-history of the converged solution of the sound source,

which again represents a storage problem. The Ffowcs Williams and Hawkings FW-H

equation was intro duced to extend acoustic analogy in the case of solid surfaces. However,

when acoustic sources i.e., quadrup oles are present in the ow eld a volume integration is

needed. This volume integration of the quadrup ole source term is dicult to compute and

is usually neglected in most acoustic analogy co des e.g. WOPWOP . Recently, there

16;17

have b een some successful attempts in evaluating this term e.g. WOPWOP+ .

1.2.2 Surface Integral Metho ds

Kirchho Metho d Another alternative is the Kirchho metho d which assumes that the

sound transmission is governed by the simple wave equation. Kirchho 's metho d consists

of the calculation of the nonlinear near- and mid- eld, usually numerically, with the far-

eld solutions found from a linear Kirchho formulation evaluated on a control surface

surrounding the nonlinear- eld. The control surface is assumed to enclose all the nonlinear

ow e ects and noise sources. The sound pressure can b e obtained in terms of a surface

integral of the surface pressure and its normal and time derivatives. This approach has the

p otential to overcome some of the diculties asso ciated with the traditional acoustic analogy

approach. The metho d is simple and accurate and accounts for the nonlinear quadrup ole

noise in the far- eld. Full di raction and fo cusing e ects are included while eliminating the 5

propagation of the reactive near- eld.

This idea of matching b etween a nonlinear aero dynamic near- eld and a linear acous-

tic far- eld was rst prop osed byHawkings. The separation of the problem into linear

and nonlinear regions allows the use of the most appropriate numerical metho dology for

each. The terminology \Kirchho metho d" was intro duced by George and Lyrintzis. It

has b een used to study various aeroacoustic problems, such as prop eller noise, high-sp eed

compressibility noise, blade-vortex interactions, jet noise, ducted fan noise, etc. The use of

Kirchho 's metho d has increased substantially the last 10 years, b ecause of the development

of reliable CFD metho ds that can b e used for the evaluation of the near- eld. An earlier

review on the use of Kirchho 's metho d was given by Lyrintzis.

Porous FW-H equation A nal alternative is the use of p ermeable p orous surface FW-

H equation. The usual practice is to assume that the FW-H integration surface corresp onds

to a solid b o dy and is imp enetrable. However, if the surface is assumed to b e p orous, a

general equation can b e derived as shown in the original reference 14 and in reference 21.

The p orous surface can b e used as a control surface in a similar fashion as the Kirchho

metho d explained ab ove. Thus the pressure signal in the far- eld can b e found based on

quantities on the control surface provided by a CFD co de.

Farassat in a recent review article reviewed all the available FW-H and Kirchho

equations for application to noise evaluation from rotating blades. The current article

fo cuses only on control surface metho ds i.e. Kirchho , p orous FW-H and discusses issues

with their application in various typ es of aero coustic problems including rotor noise, jet

noise, ducted fan noise, airfoil noise etc.. At rst the main formulations will b e reviewed,

advantages and disadvantages of each metho d will b e discussed. Then we will present several

algorithmic issues and various application examples. 6

2 Surface Integral Formulations

2.1 Kirchho 's Metho d Formulations

Kirchho 's metho d is an innovative approach to noise problems which takes advantage of

the mathematical similaritiesbetween the aeroacoustic and electro dynamic equations. The

considerable b o dy of theoretical knowledge regarding electro dynamic eld solutions can b e

utilized to arrive at the solution of dicult noise problems. Kirchho 's formula was rst

published in 1882 . Itisanintegral representation i.e. surface integral around a control

surface of the solution to the wave equation. Kirchho 's formula, although primarily used

in the theory of di raction of light and in other electromagnetic problems, it has also many

applications in studies of acoustic wave propagation.

The classical Kirchho formulation is limited to a stationary surface. Morgans derived

a formula for a moving control surface using Green's functions. Generalized functions can

also b e used for the derivation of an extended Kirchho formulation. A eld function is

de ned to b e identical to the real ow quantity outside a control surface S and zero inside.

The discontinuities of the eld function across the control surface S are taken as acoustic

sources, represented by generalized functions. Ffowcs-Williams and Hawkings derived an

extended Kirchho formulation for sound generation from a vibrating surface in arbitrary

motion. However, in their formulation the partial derivatives were taken with resp ect to

the observation co ordinates and time and that is dicult to use in numerical computations.

Farassat and Myers derived a Kirchho formulation for a moving, deformable, piecewise

smo oth surface. The same partial derivatives were taken with resp ect to the source co ordi-

nates and time. Thus their formulation is easier to use in numerical computations and their

relatively simple derivation shows the p ower of generalized function analysis.

2630

It should b e noted that Morino and his co-workers have develop ed several formula-

tions for b oundary element metho ds using the Green's function approach. These are based

on the solution of the wave equation and hence, the integral expressions are the same as in

Kirchho 's metho d. However, the formulation in in terms of the velo city p otential. This 7

has advantages e.g., the b oundary condition is simple as well as disadvantages e.g. the

pressure of the wake. Morino's formulations were derived with aero dynamic applications

in mind, so the observer is in the moving co ordinate system. However, they can b e used

for aeroacoustics, for example when b oth the control surface and the observer move with a

constant sp eed e.g., wind tunnel exp eriments, as mentioned in reference 20. Their latest

formulation app ears to provide an integrated b oundary element framework for Aero dy-

namics and Aeroacoustics. A detailed discussion ab out the di erences in the aero dynamics

and the aeroacoustics of their various formulations can b e found in reference 30.

2.1.1 Farassat's Formulation

Farassat's Kirchho formulation gives the far- eld signal, due to sources contained within

the Kirchho surface. Assume the linear, homogeneous wave equation,

2 2

@ @ 1

= =0 1

2 2

a @t @x @x

i i

is valid for some acoustic variable , and sound sp eed a , in the entire region outside of a

closed and b ounded smo oth surface, S .

The signal, in the stationary co ordinate system, is evaluated with a surface integral

over the control surface, S , of the dep endentvariable, its normal derivative, and its time

derivative gure 1. S is allowed to move in an arbitrary rigid-b o dy fashion. The dep endent

variable is normally taken to b e the disturbance pressure, but can b e any quantity which

satis es the linear wave equation.

Z Z

E E

1 2

4~x; t= dS + dS 2

r 1 M r 1 M

r r

S S

ret ret

where

h i

1 M

_ _ ~

E = M 1 + M cos M + M M r

1 r n n t 2

@n a a 1M

h i

_ _

n_ M n_ + cos M + cos M 3 +

r n M n n

a 1M

cos M 4 E =

2 n 2

r 8

Here ~x; t are the observer co ordinates and time, and ~y; are the source surface co ordi-

nates and time. M is the Machnumber vector of the surface, r is the distance from source

to observer, is the source emission angle, andn ^ is the control surface unit normal vector

cos = rb nb. M is the Machnumber vector tangent to the surface, and r is the surface

t 2

gradient op erator. A dot indicates a source time derivative, with the p osition on the surface

kept xed. Also,

_ _ _ _

M = M rb n_ =_n rb M = M nb n_ =_n M 5

r i i r i i n i i M i i

The form of equation 2 and E , E were given byFarassat and Myers . E was

1 2 2

presented in the simpli ed form shown here byMyers and Hausmann. The surface integrals

are over the control surface S , subscript ret indicates evaluation of the integrands at the

emission retarded time, which is the ro ot of

j~x ~y j

g = t + =0 6

If the frame velo city is subsonic at the surface, then equation 6 has a unique solution.

However, equation 2 is still valid for sup ersonically moving surfaces. As we can see from

equations 2 through 5, the 1 M term can pro duce a singularity in the case where the

Machnumb er in the radiation direction reaches the sonic p oint. This is a ma jor limitation

29;30

of the retarded time formulation. Farassat and co-workers have recently presented a

formulation that is appropriate for sup ersonically moving surfaces i.e. formulation 4 and

veri ed by application to b enchmark problems. Since, this sup ersonic formulation has not

yet b een applied to practical problems it will not b e presented here in the interest of brevity.

The ab ove formulation is valid when the observer is stationary and the surface is moving

at an arbitrary sp eed. However, for the case of an advancing blade the observer is usually

moving with the free- ow sp eed e.g. rotor in a wind tunnel with a free stream not equal

to zero. The formulation can b e adjusted for this case by allowing xttomove with the

free stream instead of b eing stationary in equation 6 for the retarded time.

It is p ossible to write equation 2 in a simple form valid for stationary surfaces. The 9

Kirchho formula is then

Z Z

1 [] 1 @

ret

4~x;t= dS + cos dS 7

r a @n r

S S

ret

The retarded time for this case is t r=c. With the use of a Fourier transformation, equation

7 can b e expressed in the frequency domain i.e. starting from Helmholtz equation as

b b

i! @ cos 1

i!r=a

b b

cos + dS 8 4 ~x; ! = e

r a @n r

where is the Fourier transform of , and ! is the cyclic frequency. An equivalentto

equation 8, valid for surfaces and observers in rectilinear motion was presented by Lyrintzis

34 35

and Mankbadi and Pilon:

35 36

Two-dimensional formulations can also b e develop ed Pilon, Scott et al. . Atassi

et al. develop ed a two-dimensional frequency domain formulation that uses a mo di ed

Green's function in order to avoid the evaluation of normal derivatives. Mankbadi et al.

develop ed a mo di ed Green's function for a cylinder control surface that was applied in jet

noise predictions. Hariharan et al. develop ed a framework for Kirchho 's formulations

without the use of normal derivatives.

For completeness we should mention that for the case where the Kirchho control surface

S coincides with the b o dy surface, BIE-Boundary Integral Equations there are some non-

uniqueness diculties in the prediction of the radiated acoustic sound in the exterior region

whenever the frequency coincides with one of the Dirichlet eigenfrequencies. These problems

where analyzed for the stationary Kirchho surface byWu and Pierce and for moving

41 42

Kirchho surfaces byWu. Finally,Dowling and Ffowcs Williams included the e ects

of in nite plane walls in the stationary Kirchho formulation. However, in this pap er we

are reviewing the use of Kirchho 's equation for extenting near- eld results in the far- eld

BIR Boundary Integral Representation, so the issues mentioned in this paragraph are not

relevant. 10

2.1.2 The Extended Kirchho Metho d

Equation 2 works well for aeroacoustic predictions when the control surface is placed in

a region of the ow eld where the linear wave equation is valid. However, this might not

b e p ossible for some cases. Therefore, additional nonlinearities can b e added outside the

4348

control surface. The mo di cations to the traditional Kirchho metho d consist of an

additional volume integral. Thus equation2 now b ecomes: pressure is used here as the

dep endentvariable

Z Z Z

0 2

1 E E @ T

1 2 ij

dS + 4p ~x;t= dS + dV 9

1 M r 1 M r r 1 M @y @y

r r r i j

S S

ret ret

ret

where

2 0

T = u u + p p a 10

ij i j ij o ij

where u is the uid velo city, is the density, the density p erturbation, and is the

i ij

viscous stress tensor. It is easy to show that this equation reduces to the traditional Kirchho

integral if the control surface is placed in a fully linear region, as T b ecomes zero. Through

the use of Fourier transforms, equation 9 can also b e expressed in the frequency domain.

Isom et al. develop ed a nonlinear Kirchho formulation Isom's formulation for some

sp ecial cases i.e., stationary surface at the sonic cylinder of a rotor, high frequency approx-

imation and observer on the rotation plane. They have included in their formulation some

nonlinear e ects using the transonic small disturbance equation. The nonlinear e ects are

generally accounted for with a volume integral, as shown ab ove. However, they showed that

for the ab ove sp ecial cases the nonlinear e ects can b e reduced to a surface integral.

2.2 The Porous Ffowcs Willi ams { Hawkings equation

A mo di ed integral formulation for the p orous surface FW-H equation is needed b ecause

the usual practice is to assume that the FW-H integration surface corresp onds to the b o dy 11

and is imp enetrable. A convenientway to formulate this is as an extension of Farassat's for-

mulation 1 whichwas originally develop ed for the rigid surface FW-H equation. Following

Di Francescantonio we de ne new variables U and L as

i i

11 U = 1 +

i i

o o

and

L = P n^ + u u 12

i ij j i n n

where subscript o implies ambient conditions, sup erscript implies disturbances e.g. =

+ , is the density, u is the uid velo city, v is the velo city of the control surface,

and P is the compressive stress tensor with the constant p subtracted. Nowby taking

ij o ij

the time derivative of the continuity equation and subtracting the divergence of momentum

equation, followed with some rearranging, the integral form of FW-H equation can b e written

as Formulation I

0 0 0 0

p ~x; t=p ~x; t+p ~x; t+p ~x; t 13

T L Q

where

@ U

o n

4p ~x; t= dS 14

@t r j1 M j

ret

Z Z

@ L L 1

r r

dS + dS 15 4p ~x;t=

a @t r j1 M j r j1 M j

r r

S S

ret ret

and p ~x; t can b e determined byany metho d currently available e.g., references 16, 17.

In equations 14 and 15 a subscript r or n indicates a dot pro duct of the vector with the

unit vector in the radiation directionr ^ or the unit vector in the surface normal directionn ^ ,

resp ectively.

It should b e noted that the three pressure terms havea physical meaning for rigid

0 0 0

surfaces: p ~x; t is known as thickness noise, p ~x;t is called loading noise and p ~x;t

T L Q

is called quadrup ole noise. For a p orous surface the terms lose their physical meaning, but

the last term p ~x; t still denotes the quadrup oles outside the control p orous surface S .

An alternativeway is to move the time derivative inside the integral: Formulation I I

" "

Z Z

_ _

U + U u r M + cM M

o n n_ o n r r

4p ~x;t= dS + dS 16

2 2 3

r 1 M r 1 M

r r

S S

ret ret 12

h i h i

R R

L L

L 1

r M

4p ~x; t= dS + ds

2 2 2

c S r 1M S r 1M

r r

ret ret

h i

L r M +cM M

r r r

+ dS 17

2 3

c r 1M

ret

This is now an extension of Farassat's formulation 1A also originally develop ed for the

rigid surface FW-H equation. where the dot over a variable implies source-time di eren-

tiation of that variable, L = L M , and a subscript r or n indicates a dot pro duct of the

M i i

vector with the unit vector in the radiation directionr ^ or the unit vector in the surface

normal directionn ^ , resp ectively.

Comparing the two FW-H formulations, it app ears that Formulation I equations 14, 15

has less memory requirements, b ecause it do es not require storage of the time derivatives,

and requires less op erations p er integral evaluation. However, in general, integrals haveto

be evaluated twice in order to nd the time derivative. In the sp ecial case of a stationary

control surface, or a xed microphone lo cation, i.e. \ yover," the integral can b e reused

at the next time step. Since memory app ears to b e more imp ortant for these typ e of

calculations, Formulation I is a go o d choice for stationary surfaces. Formulation I was used

by Strawn et al. for rotorcraft noise predictions using a non-rotating control surface with

very go o d results. On the other hand, taking the time derivative inside could prevent some

instabilities. Thus Formulation I I equations 16, 17 might b e more robust for a moving

control surface. Formulation I I was used for rotorcraft noise prediction by Brentner and

Farassat with a rotating control surface with very go o d results. However, a more detailed

comparison of the two formulations would b e very helpful.

For a stationary surface Formulation I reduces to:

U @

o n

dS 18 4p ~x;t=

@t r

ret

Z Z

1 @ L L

r r

dS 19 4p ~x; t= dS +

a @t r r

S S

ret ret

and Formulation I I b ecomes:

o n

dS 20 4p ~x; t=

ret 13

Z Z

1 L L

r r

4p ~x; t= dS + dS 21

a r r

S S

ret

With the use of a Fourier transformation b oth formulations for a stationary surface

can b e written in the frequency domain as

o n

0 i!r=a

dS 22 4 pb ~x; ! =i! e

Z Z

b b

L i! L

r r

i!r=a 0

~x; ! = e 4 pb dS + dS 23

a r r

S S

0 0

b b

where pb , U , and L are the Fourier transforms of p , U , and L , resp ectively and ! is

n r n r

the cyclic frequency. It should b e noted that b oth time formulations reduce to the same

frequency formulation for a stationary control surface.

Time and frequency formulations for a uniform rectilinear motion can b e found in ref-

erence 54. Two-dimensional formulations for a solid surface FW-H equation have already

b een develop ed in the past see, for example, references 55, 56 and can b e readily extended

to a p orous surface. Finally, a sup ersonic formulation can also b e found in reference 33.

2.3 Comparison of Kirchho FW-H Metho ds

Both the ab ove formulations provide a Kirchho -like formulation if the quadrup oles outside

~x; t term are ignored. The equivalence of the p orous FW-H equa- the control surface p

43 44

tion and Kirchho formulation was proven Pilon & Lyrintzis and Brentner & Farassat.

They showed that, for a surface placed in a linear region, the p orous surface FW{H formula-

tion is equivalent to the linear Kirchho formulation, plus a volume integral of quadrup oles

u u . Pilon and Lyrintzis also claim that the control surface need not b e placed in an

i j

2 0

entirely linear region. The nonlinearities can b e accounted for with the use of = a as

the dep endentvariable, and the volume integral of quadrup oles, T .

One di erence b etween Kirchho 's and FW-H formulation is that Kirchho 's metho d

@p @p

0 0

needs p ; ; 3 variables as input, whereas the p orous FW-H needs p ;;u 5 variables,

@n @t 14

or U and L 4 variables, or their time derivatives for formulation I I. Thus the p orous FW-

n i

H metho d requires more memory, which can b e signi cant for large LES runs. The CPU

time is ab out the same. However, the ma jor di erence is that the p orous FW-H metho d

allows for nonlinearities on the control surface, whereas the Kirchho metho d assumes a

solution of the linear wave equation on the surface. Thus if the solution do es not satisfy the

linear wave equation on the control surface the results from the Kirchho metho d change

dramatically. This leads to a higher sensitivity for the choice of the control surface for the

Kirchho metho d in practical cases when the wave equation is not satis ed on the control

surface due to numerical errors or non-uniform velo cities outside the control surface. This

was shown in reference 47 for a rotorcraft noise problem see section 5.2. Another wayto

state this di erence is to state that the Kirchho metho d puts more stringent requirements

to the CFD metho d to reach to the linear acoustic eld b efore dissipation and disp ersion

errors due to coarsening in the far- eld takeover.

The volume integral of quadrup ole sources that arises in the non-linear region outside of

the control surface presents a challenge. A ma jor motivation for the use of Kirchho /p orous

FW-H metho ds is the lackofvolume integrations, which reduces necessary calculations by

16;17

an order of magnitude. However, the metho ds used in WOPWOP+ provide an ecient

means of accounting for the quadrup oles in FW-H calculations that may b e used in b oth

metho ds, b ecause the quadrup ole terms are similar.

2.4 Mean Flow Refraction Corrections for Jet Noise

The Kirchho and the FW-H formulas presented ab ove can eciently and accurately predict

aero dynamically generated noise, as long as the control surface surrounds the entire source

region. In jet noise predictions, however, it is usually imp ossible, with currentnumerical

metho ds, to determine the entire source region. This is due to time and memory limitations

imp osed by the computer architecture, as well as disp ersion and dissipation constraints.

Thus, a signi cant nonlinear source region, as well as a steady mean ow, will exist outside

of the control surface. Even if the unsteady sound sources outside of the control surface 15

can b e ignored, there is still a substantial steady mean ow in the region near the jet axis,

downstream of the control surface. Thus, some means of approximating the e ects of this

steady shear ow are required if an acoustic prediction is desired for observer p oints lying

near the jet axis.

A suitable approximation to the downstream shear ow is necessary, in order to de-

termine the refraction e ects. In the past, several researchers have used an axisymmetric

parallel shear ow mo del to determine sound pro duced by p oint acoustic sources within cir-

57 53;58

cular jets e.g., Amiet . This approachwas adopted by Lyrintzis and co-workers and

in order to account for refraction e ects in the Kirchho and the p orous FW-H metho d. A

real jet has non{zero radial velo city, but the refracting e ect of this comp onent is minimal,

and can safely b e ignored. Also, the lack of azimuthal variation in the parallel shear ow

approximation has a very small e ect. The value of the axial velo city to b e used in the

shear ow approximation can b e taken directly from the CFD numerical simulation, at the

downstream end of the control surface, as an average of the time dep endent axial velo city

at each radial grid p oint.

The refraction problem now consists of a collection of p oint acoustic sources the inte-

grands of equations 8 and 22 acting at radial lo cation R, and the parallel shear ow

with U determined at each R. If the acoustic wavelength, =2a =! , is assumed to b e

small compared to the shear layer thickness , then geometric acoustics principles hold.

If the steady velo city at the downstream end of the Kirchho surface is denoted U ,

the sound emission angle with resp ect to the jet axis , and the propagation angle in the

stagnant, ambient air is denoted , then the axial acoustic phase sp eeds are preserved by

the strati ed ow

a a

= U + 24

cos cos

It is assumed that the sp eed of sound at the source is equivalent to that in the ambient air.

This equation can b e rearranged to show that there is a critical angle, de ned by

25 = cos

1+M

s 16

If the observer angle is greater than then no sound emitted at the source on the

Kirchho surface can reach the observer. This criterion is easily added to a stationary

surface Kirchho program. Note that M is the Machnumb er of the mean shear ow, and

not the Kirchho surface, which is assumed stationary.

An additional correction is necessary to accurately account for the mean ow refraction.

Imp osing the lo cal \zone of silence" condition describ ed ab ove can allow a surface source at

a relatively large radial lo cation to radiate sound into and through the shear ow. This is

b ecause the lo cal \zone of silence" decreases in size with the radial lo cation of the source,

due to the decrease in source Machnumb er. The simple correction is to set the source

strength to zero if the observation p oint is lo cated closer to the jet axis than the source

p oint on the Kirchho surface.

Finally, the geometric acoustics approximation is only valid for = > 1. It is assumed

here that the downstream end of the cylindrical Kirchho surface is lo cated far enough

downstream of the jet p otential core that the shear layer thickness is large compared with

the acoustic wavelength.

In reference 58 the mean ow refraction corrections were applied to the frequency domain

version of the Kirchho metho d equation 8. In reference 53 an amplitude correction as

recommendedby Amiet but not included in reference 58 was added and the metho dology

was applied to b oth Kirchho and FW-H metho ds equations 8 and 22-23.

2.5 Op en Control Surface

Freund et al. develop ed a way to improve the accuracy of Kirchho evaluations of sound

elds for an op en Kirchho control surface. Asymptotic analysis was used to provide cor-

rection terms which partially account for the missing p ortion of the integral surface. It

was shown that the ma jor contribution comes from a p oint on the surface that intersects

the line b etween observer and source. A correction term was estimated to account for the

missing parts of the Kirchho surface. The study is restricted to the case where the mean 17

ow is parallel to the available surface, as happ ens for example, for jet noise problems when

the downstream surface vertical to the jet axis is missing. The corrections are limited to

observers away from the jet axis. More details can b e found in the original reference. It

app ears that this study can b e extended to the p orous FW-H equation, as well.

3 Algorithmic Issues

Some algorithmic issues are discussed b elow. Additional information for numerical algo-

rithms for acoustic integrals, in general, is given by Brentner.

3.1 Choice of control surface

The Kirchho scheme requires stored data for pressure and pressure derivatives on a surface.

Since Kirchho 's metho d assumes that the linear wave equation is valid outside the closed

control surface S , S must b e chosen large enough to include the region of all nonlinear

b ehavior. However, the accuracy of the numerical solution is limited to the region immedi-

ately surrounding the moving blade b ecause of the increase of mesh spacing in CFD co des.

Thus a judicious choice of S is required for the e ectiveness of the Kirchho metho d. For

example, in the case of airfoil/rotor noise the control surface is typically lo cated a couple of

chordlengths away from the airfoil/rotor surface.

For a p orous FW-H formulation no normal derivatives are required and b ecause nonlin-

earities are allowed on the control surface the results are less sensitive to the choice of the

control surface, as will b e shown in section 5.2. Thus the CFD requirements for the FW-H

are less strigent, making the metho d more attractive. Singer et al. used a FW-H metho d

for the analysis of slat trailing-edge ow. The interesting thing ab out this application is

that part of the control surface is solid and another part is p orous. 18

3.2 Quadrature

For sucient accuracy in the far- eld calculations, high order quadrature should b e used

to solve the surface integrals in equation 2. The predicted surface quantities p , @ p=@ n,

@ p=@ t; ; u should also b e very accurate. This can b e achieved through the use of a very

ne mesh in the CFD calculations. However, memory and time constraints often make

this impractical. Meadows and Atkins have shown that it is p ossible to obtain highly

accurate Kirchho predictions from relatively coarse{grid CFD solutions. Through an

interp olation pro cess, more spatial p oints are added to the Kirchho quadrature calculations

without additional e ort in the CFD pro cess. This has the e ect of re ning the CFD mesh

with almost no additional cost. They refer to this pro cess as \enrichment". High order

quadrature, temp oral interp olation, and enrichment are imp ortant for accurate far- eld

noise predictions for b oth the Kirchho and the FW-H equation metho ds, esp ecially if the

CFD grid resolution is somewhat coarse.

3.3 Retarded or Forward Time

The retarded time equation 5 has a unique solution when the surface moves subsonically.

A Newton-Raphson or divide and conquer metho d can b e used to solve this nonlinear

equation. This metho d has b een the basis of several Kirchho co des e.g. Lyrintzis &

34 63 64 65

Mankbadi , Strawn et al. , Lyrintzis et al. Polacsec & Prier . The algorithm can b e

66 67

easily parallelized e.g. Wissink et al. , Strawn et al. by partitioning the control surface

and distributing to di erent pro cessors. Since the only communication is the nal global

summation the parallel eciency of the co de is very high. Lo ckard discussed parallelization

of FW-H co des. Long and Brentner prop osed a master-slave approach for load balancing.

However, it is dicult to write a versatile co de for various mesh top ologies used by

current CFD co des, including unstructured grids, based on this approach. In addition, when

these co des are extended to sup ersonically moving surfaces, the retarded time equation will

havemultiple ro ots that will b e dicult to evaluate. Also, the co des sometimes require 19

signi cant memory. Finally , the variation of the source strength over a surface elementin

the retarded time can b e very high at certain observer lo cations r ^ n^ ! 0 and near sonic

velo cities M ! 1 requiring a large numb er of p oints p er wavelength.

In order to overcome the limitations stated ab ove, another approach can b e develop ed

which accumulates signals with time moving forward from each surface elementtoan

observer, thus it avoids the retarded time calculation. Computer memory requirements are

reduced dramatically and the algorithm is inherently parallel. In this approach, the nal

overall observer acoustic signal is found from the summation of the acoustic signal radiated

from each source element of control surface during the same source time. The observer

time is a straight forward calculation using equation 6. For each surface element time is

moved forward from the source emission to the observer time. Since a di erent surface

element will result in a di erent observer time, interp olation techniques are required when

the integration is p erformed to obtain the overall acoustic signal at the observer p osition

Both linear interp olation and spline subroutines can b een used For high frequencies a digital

lter may b e used to increase accuracy e.g. Glegg . This metho d has b een used by several

7072 73 74

investigators, e.g. Ozyoruk and Long, Lyrintzis and Xue, and Rahier and Prier,

75 76 77

Algermissen and Wagner, , Delriex et al., and Kim et al. Finally, a marching-cub es

algorithm can b e used to provide an ecient algorithm that is easy to parallelize for the

evaluation of the propagation from an emission surface.

3.4 Rotating or Nonrotating Control Surface

For rotor applications b oth a rotating and a nonrotating formulation can b e used. A non-

rotating formulation uses a nonrotating control surface that encloses the entire rotor e.g.

79 80 81

Forsyth and Korkan , Strawn and Biswas , Baeder et al. . A rotating Kirchho for-

mulation allows the control surface to rotate with the blade aligning with the CFD lines,

82 64 65

e.g. Xue and Lyrintzis , Lyrintzis et al., ,Polacsec and Prier . No transformation of

data is needed since the CFD input is also rotating. A comparison of the rotating and the

nonrotating Kirchho metho ds showed that b oth metho ds are very accurate and ecient 20

Strawn et al. . For the p orous FWH metho d there are fewer applications. A rotating

metho d was used in references 45, 47, 76 and 83 and a nonrotating metho d in reference 52.

It should b e noted that the nonrotating formulation requires reliable data out to a

nonrotating cylinder i.e. the control surface surface that is usually farther out than a

rotating surface. Therefore, more accuracy of the CFD results is needed. Thus the nonro-

tating metho d has b een used in conjunction with Euler/Navier Stokes co des e.g., TURNS

84;85 86

co de ,OVERFLOWcode whereas the rotating Kirchho metho d has b een used with

87;88

full p otential co des e.g. FPR co de , as well. However, a drawback of the rotating

metho d is that the rotating sp eed of the tip of the rotating surface needs to remain sub-

sonic, to use the subsonic formulas shown in section 2, b ecause of the singularities app earing

at some terms. For high tip sp eeds e.g. M =0.92 the sup ersonic formulation of Farassat

tip

32;33

et el. can b e employed. For forward time algorithms the singularities disapp ear and a

simple metho d for sup ersonic rotation sp eeds has b een develop ed by Delrieux et al.

4 Validation Results

Both Kirchho and FWH formulations have b een validated using mo del problems. The

rst thing to do is, of course, check that the signal b ecomes zero inside the control surface.

The numb er of p oints p er p erio d and the numb er of p oints p er wave length should also b e

34;53

studied.

34;53

A stationary or translating p oint source have b een used by Lyrintzis et al., Myers

31 54 64

& Hausmann, and Lo ckard and a rotating p oint source by Lyrintzis et al. and Berezin

89 35;43;44;46

et al. Exp onential source distributions have b een used by Pilon and Lyrintzis. Hu

et al. used a line monop ole source and a Gaussian pressure and vorticity pulse category

3 b enchmark problem toverify their two-dimensional FW-H formulation. Farassat and

Farris used dip ole distributions on a at surface and a sphere to validate the sup ersonic

formulation i.e. formulation 4. Singer et al. used a line vortex around an edge. Meadows

and Atkins used an oscillating sphere and studied the e ects of quadrature see section 3. 21

Ozyoruk and Long have used the scattering problem of sound by a sphere gure 2. The

spherical sound waves are generated by a partially distributed Gaussian mass source. The

results from an exact solution and a direct Euler solver are also shown. Note that near 180

the Kirchho results are b etter than the direct calculation, b ecause of numerical dissipation

as the waves travel longer distances to arrive at the observer lo cations.

5 Aeroacoustic Applications

Kirchho 's formula has b een extensively used in light di raction and other electromagnetic

problems, aero dynamic problems, i.e. b oundary-elements e.g. Morino et al. , as well as in

problems of wave propagation in acoustics e.g. Pierce . Kirchho 's integral formulation

has b een used extensively for the prediction of acoustic radiation in terms of quantities

on b oundary surfaces the Kirchho control surface coincides with the b o dy. Kirchho 's

metho d has also b een used for the computation of acoustic scattering from rigid b o dies

using a b oundary element technique with the Galerkin metho d. The solid surface FW-H

equation with its various forms has b een used in several problems including prop eller and

helicopter noise. Here we will concentrate on the use of \Kirchho ," and \p orous" FW-H

equation metho ds, i.e. using a nonlinear CFD solver for the evaluation of acoustic sources

in the near- eld and a Kirchho /p orous FW-H formulation for the acoustic propagation.

We will review some \real-life" aeroacoustic applications of b oth metho ds concentrating in

recent advances.

5.1 Prop eller Noise

Hawkings suggested a stationary-surface Kirchho formula to predict the noise from high-

sp eed prop ellers and helicopter rotors. Forsyth and Korkan calculated high-sp eed prop eller

18 94

noise using the Kirchho formulation of Hawkings. Jaeger and Korkan used a sp ecial

case of the Farassat and Myers formulation for a uniformly moving surface to extend the

calculation to advancing prop ellers. In the ab ove applications, the control surface S was 22

chosen to b e a cylinder enclosing the rotor.

5.2 Helicopter Impulsive Noise

Kirchho 's metho d has b een widely applied in the prediction of helicopter impulsive noise.

The Kirchho metho d for a uniformly moving surface was initially used in two-dimensional

transonic Blade-Vortex Interactions BVI to extend the numerically calculated nonlinear

9698

aero dynamic BVI results to the linear acoustic far- eld. Actually, the rst application of

18 19

Hawkings Kirchho Metho d was given by George and Lyrintzis, where the terminology

\Kirchho Metho d" was intro duced. The Kirchho metho d was used to test ideas for BVI

noise reduction Xue and Lyrintzis , The metho d was also extended to study noise due to

other unsteady transonic ow phenomena i.e. oscillating aps, thickening-thinning airfoil

100

by Lyrintzis et al. Later, the metho d was used for the two-dimensional BVI problem by

101;102

Lin and co-workers.

Kirchho 's metho d has also b een applied to three-dimensional High-Sp eed Impulsive

81 80

HSI noise. Baeder et al. and Strawn & Biswas used a nonrotating control Kirch-

ho surface that encloses the entire rotor. The Transonic Unsteady Rotor Navier Stokes

84;85

TURNS co de was used for the near- eld CFD calculations. An unstructured grid

103 86 104

was used by Strawn et al. and an overset grid co de OVERFLOW by Ahmad et al.

Kirchho 's metho d predicted the HSI hover noise very well using a small fraction of CPU

time of the straight CFD calculation.

Another Kirchho metho d used in helicopter noise is the rotating Kirchho metho d i.e.

the surface rotates with the blade. The metho d was used for three-dimensional transonic

BVI's for a hovering rotor by Xue and Lyrintzis. The near- eld was calculated using

87;88

the Full Potential Rotor FPR co de. The rotating Kirchho formulation allows the

Kirchho control surface to rotate with the blade; thus a smaller cylinder surface around

the blade can b e used. No transformation of data is needed b ecause the CFD input is also

rotating. Since more detailed information is utilized for the accurate prediction of the far- 23

eld noise this metho d is more ecient. Finally, the metho d was extended for an advancing

105 106;107 89

rotor and was applied to HSI noise and BVI noise. Berezin et al. showed that

sometimes sp ecial care is needed for cho osing the CFD grids, b ecause the highly stretched

grids used for aero dynamic applications may not provide accurate information on the control

Kirchho surface.

A comparison of the rotating and the nonrotating Kirchho metho ds showed that b oth

metho ds are very accurate and ecient. Figure 3 shows a comparison for an advancing HSI

noise case 1/7 scale AH-1 helicopter, hover tip Machnumber M =0:665, advance ra-

tio =0:258, which corresp onds to an advancing tip Machnumber of M =0:837.

84;85

TURNS is used for the CFD calculations. We see that b oth metho ds compare very

108

well with the exp eriments. Kirchho 's metho d has b ecome a standard to ol for rotorcraft

acoustic predictions. The metho d is currently implemented in the TRAC TiltRotor Aeroa-

coustic Co des system develop ed by NASA Langley RKIR co de, Lyrintzis et al., Berezin

89 63

et al. and is employed at NASA Ames AFDD Strawn et al. . In Europ e, additional

65;74;75;76;109;110

versions of rotating and nonrotating Kirchho co des have also b een develop ed.

Kirchho 's metho d results have also b een compared with the traditional acoustic analogy

solid surface FW-H equation. A comparison with the acoustic analogy co de WOPWOP

WOPWOP uses the solid surface FW-H equation without accounting for quadrup oles has

shown that Kirchho metho d is sup erior when quadrup ole sources are present Lyrintzis

111 81

et al. for advancing HSI cases. Baeder et al. also compared the results with a linear

i.e. monop ole plus dip ole sources on the rotating blade solid surface FW-H equation

metho d for hover HSI. The FW-H results were inaccurate for tip Machnumb ers higher

than 0.7, b ecause of the omission of quadrup ole sources. However, a further comparison of

16;17

the rotating Kirchho metho d to WOPWOP+ WOPWOP+ is a solid surface FW-H

equation metho d accounting also for quadrup oles with a volume integral has shown that

112

the two metho ds give ab out the same results Brentner et al. , but Kirchho metho d

uses only surface integrals and avoids the quadrup ole volume integration. It should b e

noted that the robustness of the Kirchho metho d improves with the use of a less stretched 24

89 81

grid Berezin et al. or an Euler co de, e.g. TURNS Baeder et al. .

49 113;114

Isom et al. , and Purcell used a mo di ed Kirchho metho d which also included

some nonlinear e ects for a stationary surface, to calculate hover HSI noise. Results not

shown here show go o d agreement with exp erimental data.

A p orous FW-H metho d based on Kirchho subroutines was also develop ed by Brentner

47 83 52 76

&Farassat FWH/RKIR co de, Morgans et al. , Strawn et al., and Delriex et al.

These co des do not include quadrup oles outside the control surface, b ecause it was found to

115

b e of minor imp ortance unless the Machnumb er is really high. Thus the p orous FW-H

equation is also based on surface integrals. The p orous FW-H formalism is more robust than

the traditional Kirchho metho d with regards to the choice of the control surface, as shown

in gures 4 and 5 for a hover HSI noise case 1/4 mo del UH-1H mo del helicopter, hovering

113 87;88

at M =0:88, exp eriments from Purcell . FPR was used for the CFD calculations.

5.3 Airfoils

37;116118

Atassi and his co-workers have used Kirchho 's metho d for the evaluation of acous-

tic radiation from airfoils in nonuniform subsonic ows. They employed rapid distortion the-

ory to calculate the near- eld CFD. A sample comparison for the far- eld directivity of the

acoustic pressure using the Kirchho metho d and the direct calculation metho d i.e. rapid

119121

distortion theory is given in gure 6 from reference 37 for a 3 thick Joukowski

airfoil in a transverse gust at k =!c=2V = 1 and M=0.1. The semi-analytical results

1 1

for a at plate encountering the same gust are also shown in gure 6 and are very close to

the results from Kirchho 's metho d. The gure indicates that the direct calculation metho d

is not accurate in the far- eld, as the direct simulation results are very di erent from the

semi-analytical and the Kirchho results. This is due to discretization errors. However, this

CFD co de is accurate in the near- eld and the Kirchho metho d should b e used instead in

the far- eld, as indicated in gure 6.

92;61

Singer et al. used a FW-H metho d for the evaluation of acoustic scattering from 25

a trailing edge and slat trailing edge. The interesting thing ab out the slat trailing edge

application is that part of the control surface is solid and another part is p orous.

5.4 Fan Noise

Kirchho 's metho d can also b e applied to ducted fan noise. Very go o d results were shown by

7072

Ozyoruk and Long for a control surface in rectilinear motion. A forward time parallel

122

algorithm was used. A p orous FWH metho d was used by Zhang with very go o d results.

5.5 Jet Noise

123

Kirchho 's metho d has also b een applied in the estimation of jet noise. Soh , Mitchel

124 125 126

et al. Zhao et al., and Billson et al. used the stationary Kirchho metho d equa-

34 127 128

tion 7 and Lyrintzis & Mankbadi Chyczewski & Long, Morris et al., Gamet and

129 130 131

Estivalezes, Choi et al., and Kandula and Caimi used the uniformly moving formula.

It should b e noted that most of the ab ove references use an LES co de for the CFD data.

However, a RANS co de can also b e used, as shown in reference 131, where OVERFLOW

was used. Lyrintzis & Mankbadi also compared time and frequency domain formulations.

Mankbadi et al. applied a mo di ed Green's function to avoid the evaluation of normal

132 133

derivatives. Balakumar and Yen used parab olized stability equations for the jet simu-

lation and a cylindrical i.e. two-dimensional Kirchho formulation for the noise evaluation

134

Shih et al. compared several Kirchho formulations with the acoustic analogy, extending

the LES calculations and using a zonal LES + LEE metho d. The results showed that the

Kirchho metho d is much more accurate than the acoustic analogy for the compact source

approximation used and muchcheap er than extending the LES or p erforming a zonal LES

+ LEE.

135;136

The uses of FW-H metho d in jet noise have b een sparse. Morris et al. and Uzun

137 90;138

et al. used the metho d with go o d results and Hu et al., used a two-dimensional

formulation of the p orous FW-H equation to evaluate noise radiation from a plane jet. 26

139

Rahier et al. compared the metho ds for numerical acoustic predictions for hot jets. Both

the Kirchho metho d based on pressure disturbance and the FW-H equation gave good

results, whereas the Kirchho metho d based on densitygave erroneous results.

Most of the ab ove approaches have used an op en control surface i.e. without the

downstream end in order to avoid placing the surface in a nonlinear region. Freund et al.

showed a means of correcting the results to account for an op en control surface, for cases

43;44;47

that the observer is close to the jet axis. Pilon and Lyrintzis develop ed a metho d to

account for quadrup ole sources outside the control surface. This approximation is based on

the assumption that all wave mo des approximately decay in an exp onential fashion. The

volume integral is reduced to a surface integral for a far- eld low frequency approximation

andaTaylor series expansion for axisymmetric jets. However, a simpler metho d suggested

140

in reference 53 is to just use an existing empirical co de e.g. MGB toevaluate the noise

using as in ow the CFD solution on the right side of the control surface. Thus MGB can

provide an estimate of the error of ignoring any sources outside the control surface of the

Kirchho /p orous FW-H metho d.

An approximate way to account for refraction e ects was develop ed by Lyrintzis and co-

53;58

workers , as explained ab ove in section 2.5. A typical result shown here gure 7 shows

the e ects of refraction corrections for a sup ersonic Machnumb er case excited, Mach2:1,

unheated T = 294K , round jet of Reynolds Number Re = 70000; the jet exit variables

were p erturb ed at a single axisymmetric mo de at a Strouhal number of St =0:20, the

amplitude of the p erturbation was 2 of the mean. Further development of refraction

corrections based on Lilley's equation, e.g. reference 141 is p ossible.

Finally, it should b e noted that for some complicated noise problems as, for example,

in jet noise several computational domains might b e needed: a complicated near- eld e.g.

using Large Eddy Simulations-LES, a simpli ed mid- eld with some nonlinear e ects, and

a linear Kirchho 's metho d for the far- eld. Kirchho 's formulation can b e the simplest

region of a general zonal metho dology. This idea has b een prop osed by Lyrintzis, but it

has not yet b een implemented. 27

5.6 Other Applications

Other applications of these surface integral metho ds have also b een attempted. For example,

in reference 142 a Kirchho metho d was applied to a cavity problem, with application to

vehicle noise predictions. Also the metho d is now part of the Fluent co de and has b een

77 143

applied to a wide range of problems and is currently implemented in the Star-CD co de

6 Concluding Remarks

Kirchho 's and p orous FW-H metho ds consist of the calculation of the nonlinear near-

and mid- eld numerically with the far- eld solutions found from a Kirchho /p orous FW-H

formulation evaluated on a control surface S surrounding the nonlinear- eld. The surface S

is assumed to include all the nonlinear ow e ects and noise sources. The separation of the

problem into linear and nonlinear regions allows the use of the most appropriate numerical

metho dology for each. The advantage of these metho ds is that the surface integrals and

the rst derivatives needed can b e evaluated more easily than the volume integrals and the

second derivatives needed for the evaluation of the quadrup ole terms when the traditional

acoustic analogy is used.

The p orous FW-H equation metho d is a newer idea and there fewer applications in the

literature. The metho d is equivalent to Kirchho 's metho d and has the same advantanges.

In comparison, it is more robust with the choice of control surface and do es not require

normal derivatives. Since the metho d also requires a surface integral, it is very easy to

mo dify existing Kirchho /solid surface FW-H co des. The metho d requires larger memory,

b ecause more quantities on the control surface are needed. However, we b elieve that the

robustness is more imp ortant and thus the p orous FW-H is the metho d we recommend.

The use of b oth metho ds has increased substantially over the last 10 years, b ecause of

the development of reliable CFD metho ds that can b e used for the evaluation of the near-

eld. The metho ds can b e used to study various acoustic problems, such as prop eller noise, 28

high-sp eed compressibility noise, blade-vortex interactions, jet noise, ducted fan noise, etc.

Some results indicative of the uses of b oth metho ds are shown here, but the reader is referred

to the original references for further details. The metho ds are b ecoming more p opular and

have b een coupled with pro duction co des, suchasOVERFLOW Fluent and Star-CD. We

b elieve that, a simple set of p ortable Kirchho /FW-H subroutines can b e develop ed to

calculate the far- eld noise from inputs supplied byany aero dynamic near/mid- eld co de.

Acknowledgements

The author was supp orted by the Indiana 21st Century Research and Technology Fund, and

the Aeroacoustics Research Consortium AARC, a government and industry consortium

mananged by the Ohio Aerospace Institute OAI.

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[1] Lighthill, M. J., "A General Intro duction to Aeroacoustics and Atmospheric Sound,"

ICASE Rep ort 92-52, NASA Langley Research Center, Hampton, VA 1992.

[2] Hardin, J., and Hussaini M. Y., Computational Aeroacoustics, Springer-Verlag, New

York, 1993.

[3] Tam, C. K. W., \Computational Aeroacoustics: Issues and Metho ds," AIAA Journal,

Vol. 33, No. 10, Oct. 1995, pp. 1788{1796 .

[4] Long, L. N., Morris, P. J., Ahuja, V., Hall, C., and Liu, J., \Several Aerospace Appli-

cations of Computational Aeroacoustics," Proceedings of the ASME Noise Control and

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Figure 1: Kirchho 's surface S and notation. 47

Figure 2: Sound scattering by a sphere. Comparison with exact solution from reference

70. 48

102

Figure 3: Comparison of acoustic pressures with exp erimental data at four di erent

microphone lo cations for an AH-1 blade with M =0:837. All microphones are in the

plane of the rotor from reference 63. 49

108

Figure 4: Comparison of Kirchho acoustic pressures with exp erimental data for an

observer in the plane of the rotor at 3; 4R from a UH-1H mo del rotor hovering at M =0:88

from reference 47. 50

108

Figure 5: Comparison of p orous FW-H acoustic pressures with exp erimental data for an

observer in the plane of the rotor at 3; 4R from a UH-1H mo del rotor hovering at M =0:88

from reference 47. 51

Figure 6: Comparison b etween far- eld directivity of acoustic pressure values using the

Kirchho metho d - - and the direct calculation metho d -- for a 3 thick Joukowski

airfoil in a transverse gust at k =1:0;M =0:1. The semi analytical results { for a at

plate encountering the same gust are also shown from reference 37. 52 50 No Corrections 40

10 j R / 0 R -10

-20

-30

-40 Refraction Corrections -50 0 10 20 30 40 50 60 70 80 90 100

x/Rj

2 0

Figure 7: Instantaneous contours of a =p . R>0: No refraction corrections. R<0:

Refraction corrections imp osed from reference 58. 53