Integral Methods in Computational Aeroacoustics

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 equation. A

numerical CFD metho d is used for the evaluation of the ow- eld solution in the near eld and

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

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

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 extended 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

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

tant. 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 turbulence 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,aswell 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 transmission

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 made

rst-principles CAA plausible, direct extension of current CFD technology to CAA requires 2

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

23 4

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 prop-

agation. 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 calcu-

lated. 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 impracticalityof

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 gen-

eration 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.

Traditional Acoustic Analogy The rst of these approaches 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 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 3

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

beyond 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 have b een some successful attempts in evaluating this term

11;12

e.g. WOPWOP+ .

Linearized Euler Equations LEE The second alternative is to use LEE in order to

13 14

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 much

cheap 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. However,

dissipation and disp ersion errors still exist and an accurate description of propagating far- eld

waves is compromised b ecause of this. On the other hand, this metho d may b e appropriate

for the 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.

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 4

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 propagation of the reactive

near- eld.

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

far- eld was rst prop osed byHawkings . The use of Kirchho 's metho d has increased sub-

stantially 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 separation of the problem into linear and nonlin-

ear regions allows the use of the most appropriate numerical metho dology for each. Wehave

b een referring to this technique as the \Kirchho metho d." 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. 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 9 and in reference 18. 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 equa-

tions for application to noise evaluation from rotating blades. The current article fo cuses only 5

on control surface metho ds i.e. Kirchho , p orous FW-H and discusses issues with their ap-

plication 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 dis-

advantages of each metho d will b e discussed. Then we will present several algorithmic issues

and various application examples.

2 Kirchho 's Metho d Formulations

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

the mathematical similarities b etween 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 . It is an integral 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 ordinates and time. Thus their formulation

is easier to use in numerical computations and their relatively simple derivation shows the 6

power of generalized function analysis.

2327

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

for b oundary element metho ds using the Green's function approach, which are equivalentto

Kirchho formulations. 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 17. Their latest

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

and Aeroacoustics.

2.1 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 @ @

2 = =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 fashion. The dep endentvariable 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

S S

r r

ret ret

where

i h

M 1

~ _ _

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

t 2 r n 1 n

@n a a 1M

n_ M n_ + cos M + cos M 3 +

r n M n n

a 1M

E = cos M 4

2 n

r 7

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

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

b b

is the source emission angle, andn ^ is the control surface normal vector cos = r n. M

is the Machnumber vector tangent to the surface, and r is the surface gradient op erator. A

dot indicates a source time derivative, with the p osition on the surface kept xed. Also,

_ _ _ _

b b b

M = M r n_ =_n r M = M n 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 presented in

1 2 2

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

=0 6 g = t +

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 Machnumber

in the radiation direction reaches the sonic p oint. This is a ma jor limitation of the retarded

29;30

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, the 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

Kirchho formula is then

Z Z

1 1 [] @

ret

dS 7 4~x;t= dS + cos

r a @n r

S S

ret 8

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! @ 1 cos

i!r=a

b b

4 ~x; ! = e cos + dS 8

r a @n r

where is the Fourier transform of , and ! is the cyclic frequency. An equivalent to equa-

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

31 32

Mankbadi and Pilon:

33 34

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.

Finally, for completeness we should mention that for the case where the Kirchho control

surface S coincides with the b o dy surface, there are some nonuniqueness 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

37 38

Kirchho surface byWu and Pierce and for moving 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, so the issues mentioned in this paragraph are

not relevant.

2.2 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 control

4044

surface. The mo di cations to the traditional Kirchho metho d consist of an additional 9

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

variable

" " "

Z Z Z

@ T E 1 E

ij 2 1

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

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

S S

r r r i j

ret ret ret

where

0 2

10 T = u u + p p a

ij ij i j ij o

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

i ij

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 gen-

erally 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.

2.3 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

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

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

Francescantonio we de ne new variables U and L as

i i

U = 1 11 +

i i

o o 10

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 velo city, and P is the compressive stress tensor with the

o ij

constant p subtracted. Nowby taking the time derivative of the continuity equation and

o ij

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

S S

r r

ret ret

and p ~x; t can b e determined byany metho d currently available e.g., references 10, 11.

In equations 14 and 15 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 surfaces:

0 0 0

p ~x;tisknown as thickness noise, p ~x; t is called loading noise and p ~x; t is called

T L Q

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

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

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

S S

r r

ret ret

i i h h

R R

L L 0

L 1

dS + ds ~x;t= 4p

2 2 2

S S L

c r 1M r 1M

r r

ret ret

h i

R 2

L r M +cM M

1 r r r

+ dS 17

2 3

c r 1M

ret

This is now an extension of Farassat's formulation 1A. ehere the dot over a variable implies

source-time di erentiation of that variable, L = L M , and a subscript r or n indicates It

M i i 11

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 havetobeevaluated 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. Formulation I was used by Strawn et

al. for rotorcraft noise predictions using a nonrotating control surface with very go o d results.

On the other hand taking the time derivative inside could prevent some instabilities. Thus

foraFormulation 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

~x;t= 4p dS 18

@t r

ret

Z Z

@ L L 1

r r

dS 19 dS + 4p ~x;t=

a @t r r

S S

ret ret

and Formulation I I b ecomes:

o n

4p ~x;t= dS 20

ret

Z Z

L 1 L

r r

dS + 4p ~x;t= 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

n o

i!r=a 0

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

Z Z

b b

L i! L

r r

0 i!r=a

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

a r r

S S

0 0

b b

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

n r n r

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 reference

50. Two-dimensional formulations for a solid surface FW-H equation have already b een devel-

op ed in the past see, for example, references 51, 52 and can b e readily extended to a p orous

surface. Finally, a sup ersonic formulation can also b e found in reference 30.

2.4 Comparison of Kirchho FW-H Metho ds

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

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

43 44

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 formulation is equivalent

to the linear Kirchho formulation, plus a volume integral of quadrup oles u u . Pilon and

i j

Lyrintzis also claim that the control surface need not b e placed in an entirely linear region.

0 2

as the dep endentvariable, The nonlinearities can b e accounted for with the use of = a

and the volume integral of quadrup oles, T .

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

@p @p

0 0

needs p ; ; as input whereas the p orous FW-H needs p ;;u . Also, the p orous FW-H

@n @t

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. This was shown in reference 44 for a rotorcraft noise problem see section

5.2. Another way to lo ok at 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. 13

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 an order of

11;12

magnitude. However, the recently develop ed co de WOPWOP+ provides an ecient means

of accounting for the quadrup oles in FW-H calculations that can b e used for b oth metho ds,

b ecause the quadrup ole term is the same.

2.5 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 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 determine

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 circular jets e.g.,

53 49;54

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. 14

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

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

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

If the 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. 15

In reference 54 the mean ow refraction corrections were applied to the frequency domain

version of the Kirchho metho d equation 8. In reference 49 an amplitude correction as

recommended by Amiet but not included in reference 54 was added and the metho dology

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

2.6 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 correction 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 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.

3 Algorithmic Issues

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

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 e-

havior. However, the accuracy of the numerical solution is limited to the region immediately

surrounding the moving blade b ecause of the increase of mesh spacing in CFD co des. Thus a 16

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.

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 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 with the Kirchho metho d, 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 & Mankbadi , Strawn 17

59 60 61

et al. , Lyrintzis et al. Polacsec & Prier . The algorithm can b e easily parallelized e.g.

62 63

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

50 64

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 signi cant memory.

Finally , the variation of the source strength over a surface element in 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 is develop ed which

accumulates signals time matched from each surface element to an 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

6567 68

acoustic signal at the observer p osition e.g. Ozyoruk and Long , Lyrintzis and Xue , and

69 70 71

Rahier and Prier , Algermissen and Wagner . 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. 18

3.4 Rotating or Nonrotating Control Surface

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

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

72 73 74

Korkan , Strawn and Biswas , Baeder et al. . A rotating Kirchho formulation allows the

control surface to rotate with the blade aligning with the CFD lines and rotate with the blade.

75 60 61

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 Strawn et al. .

For the p orous FWH metho d there are fewer applications. A rotating metho d was used in

references 42, 44 and 76 and a nonrotating metho d in reference 48.

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

rotating 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 nonrotating metho d

77;78

has b een used in conjunction with Euler/Navier Stokes co des e.g., TURNS co de ,OVER-

FLOWcode whereas the rotating Kirchho metho d has b een used with full p otential co des

80;81

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 subsonic, b ecause Farassat's formulation is currently limited

to subsonically moving surfaces. An extension to sup ersonically moving surfaces is needed.

This imp oses limits to the p osition of the tip of the rotating control surface in very high Mach

numb er cases e.g. M=0.92-0.95 for hover. However, the sup ersonic formulation formulation

29;30

of Farassat et el. can b e employed in the future for the rotating case for high-Machnumber

cases. 19

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 num-

31;49

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

31;49

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

28 50 60

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

82 32;40;41;43

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

84 30

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

85 58

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. 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. 20

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 in 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 acous-

tic propagation. We will review some \real-life" aeroacoustic applications of b oth metho ds

concentrating in recent advances.

5.1 Prop eller Noise

Hawkings used a stationary-surface Kirchho 's formula to predict the noise from high- sp eed

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

16 87

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 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 aero-

8992

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

16 77

Hawkings \Kirchho Metho d" was given by George and Lyrintzis. 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 by Lyrintzis et al. Later, the metho d was used for the two-

95;96

dimensional BVI problem by Lin and co-workers.

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

74 73

noise. Baeder et al. and Strawn & Biswas used a nonrotating control Kirchho surface that 21

77;78

encloses the entire rotor. The Transonic Unsteady Rotor Navier Stokes TURNS co de was

used for the near- eld CFD calculations. An unstructured grid was used by Strawn et al. and

79 98

an overset grid co de OVERFLOW by Ahmad et al. Kirchho 's metho d predicted the HSI

hover noise very well using a 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

ahovering rotor by Xue and Lyrintzis. The near- eld was calculated using the Full Potential

80;81

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- eld noise this metho d is more

ecient. Finally, the metho d was extended for an advancing rotor and was applied to HSI

99 100;101 82

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 ratio =0:258,

77;78

which corresp onds to an advancing tip Machnumber of M =0:837. TURNS is used for

102

the CFD calculations. We see that b oth metho ds compare very 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 Aeroacoustic Co des system develop ed by

60 82

NASA Langley RKIR co de, Lyrintzis et al. , Berezin et al. and is employed at NASA Ames

AFDD Strawn et al. . In Europ e, additional versions of rotating and nonrotating Kirchho

61;69;70;103;104

co des have also b een develop ed.

Kirchho 's metho d results have also b een compared with 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 22

105

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

11;12

sources. However, a further comparison of the rotating Kirchho metho d to WOPWOP+

WOPWOP+ is a solid surface FW-H equation metho d accounting also for quadrup oles with

avolume integral has shown that the two metho ds give ab out the same results Brentner et

106

al. , but Kirchho metho d uses only surface integrals and avoids the quadrup ole volume

integration. It should b e noted that robustness of Kirchho metho d improves with the use of

82 74

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

45 107;108

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

44 76 48

&Farassat FWH/RKIR co de, Morgans et al. and Strawn et al. . These co des do not

include quadrup oles outside the control surface, b ecause it was found to b e of minor imp ortance

109

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 at M =0:88, exp eriments from

107 80;81

Purcell . FPR was used for the CFD calculations.

5.3 Airfoils

34;110112

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

radiation from airfoils in nonuniform subsonic ows. They employed rapid distortion theory 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 distortion 23

113115

theory is given in gure 6 from references 34 and 101 for a 3 thick Joukowski airfoil in

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

1 1

encountering the same gust are also shown in gure 4 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.

85;57

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

6466

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

116

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

5.5 Jet Noise

There are some diculties with using the Kirchho and and the p orous FW-H metho ds for

some aeroacoustic problems. For an accurate prediction, the control surface must completely

enclose the aero dynamic source region. This may b e dicult or imp ossible to accomplish if

the source region is large. The validity of this metho d is also dep endent on the control surface

b eing placed in a region where the linear wave equation is valid. Additionally, the existence of

a shear ow outside the control surface will cause refraction of the propagating sound. Failure

to account for this refraction will also lead to errors when the observer lo cation is near the jet

axis.

117

Kirchho 's metho d has also b een applied in the estimation of jet noise. Soh and Mitchel 24

118 31

et al. used the stationary Kirchho metho d equation 7 and Lyrintzis & Mankbadi Chy-

119 120 121 122

czewski & Long , Morris et al. , Gamet and Estivalezes , Choi et al. and Kandula and

123

Caimi used the uniformly moving formula. It should b e noted that most of the ab ove refer-

ences use an LES co de for the CFD data. However, a RANS co de can also b e used, as shown

79 31

in reference 123, 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

124 125

avoid the evaluation of normal derivatives. Balakumar and Yen used parab olized stability

equations for the jet simulation and a cylindrical i.e. two-dimensional Kirchho formulation

126

for the noise evaluation 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

127;128 83

a zonal LES + LEE. Finally, Morris et al. used the p orous FW-H metho d and Hu et al. ,

used a two-dimensional formulation of the p orous FW-H equation to evaluate noise radiation

from a plane jet.

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 that the observer is

40;41;43

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 and a Taylor series expansion for

axisymmetric jets. However, a simpler metho d recommended in reference 49 is to just use an

129

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-

49;54

workers , as explained ab ove in section 2.4. Atypical result shown here gure 7 shows 25

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,

for example in reference 130 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.

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 is equivalent to Kirchho 's metho d and is very app ealing b e-

cause it is more robust with the choice of control surface and do es not require normal deriva-

tives. 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 use of b oth metho ds 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. 26

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

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Figure 1: Kirchho 's surface S and notation. 43

Figure 2: Sound Scattering by sphere. Comparison with exact solution from reference 65. 44

102

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

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

rotor from reference 59. 45

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 44. 46

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 44. 47

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

ho 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 36. 48 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 54. 49