Integral Methods in Computational

Aeroacoustics

-From the (CFD) Near-Field to the (Acoustic)

Far-Field �

Anastasios S. Lyrintzisy

School of Aeronautics and Astronautics

Purdue University

W. Lafayette, IN 47907-2023

Abstract

A review of recent advances in the use of integral methods 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 method, permeable (porous) surface Ffowcs-Williams

Hawkings (FW-H) equation allow the radiating sound to be 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 method (i.e. Lighthill, rigid body FW-H equation). A

numerical CFD method 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.

yProfessor, e-mail: lyrintzi@ecn.purdue.edu.

1

thus on the control surface. Di�usion and dispersion errors associated with wave propagation

in the far-�eld are avoided. The surface integrals and the �rst derivatives needed can be easily

evaluated from the near-�eld CFD data. Both methods can be extended in order to include

refraction e�ects outside the control surface. The methods have been applied to helicopter

noise, jet noise, propeller noise, ducted fan noise, etc. A simple set of portable Kirchho�/FW-

H subroutines can be developed to calculate the far-�eld noise from inputs supplied by any

aerodynamic near/mid-�eld CFD code.

1 Background

For an airplane or a helicopter, aerodynamic noise generated from uids is usually very impor-

tant. There are many kinds of aerodynamic noise including turbine jet noise, impulsive noise

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

boundary layer separated ow, etc. (e.g. Lighthill1). Accurate prediction of noise mechanisms

is essential in order to be able to control or modify them to comply with noise regulations, i.e.

Federal Aviation Regulations (FAR) part 36, and achieve noise reductions. Both theoretical and

experimental studies are being conducted to understand the basic noise mechanisms. Flight-

test or wind-tunnel test programs can be used, but in either case di�culties are encounted

such as high expense, safety risks, and atmospheric variability, as well as re ection problems

for wind tunnel tests. As the available computational power increases numerical techniques

are becoming more and more appealing. Although complete noise models have not yet been

developed, numerical simulations with a proper model are increasingly being employed for the

prediction of aerodynamic noise because they are low cost and e�cient. This research has led

to the emergence of a new �eld: Computational AeroAcoustics (CAA).

CAA is concerned with the prediction of the aerodynamic sound source and the transmission

of the generated sound starting from the time-dependent governing equations. The full, time-

dependent, compressible Navier-Stokes equations describe 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 di�culties in the prediction of both the sound generation and its

transmission.2�3 A review of aerospace application of CAA methods was given by Long et al.4

Aerodynamically generated sound is governed by a nonlinear process. One class of problems

is turbulence generated noise (e.g. jet noise). An accurate turbulence model is usually needed

in this case. A second class of problems involves impulsive noise due to moving surfaces (e.g.

helicopter rotor noise, propeller noise, fan noise etc.). In these cases an Euler/Navier Stokes

model or even a full potential model is adequate, because turbulence is not important.

Once the sound source is predicted, several approaches can be used to describe its prop-

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

Navier-Stokes equations far enough to encompass the location where the sound is to be calcu-

lated. However, if the objective 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 supersonic jet aeroacoustics was pointed out by Mankbadi et al.5

Furthermore, because the acoustic uctuations are usually quite small (about three orders of

magnitude less than the ow uctuations), the use of nonlinear equations (whether Navier-

Stokes or Euler) could result in errors, as pointed out by Stoker and Smith.6 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 been identi�ed.

Traditional Acoustic Analogy The �rst of these approaches is the acoustic analogy.7 In

the acoustic analogy, the governing Navier-Stokes equations are rearranged to be in wave-type

form. There is some question as to which terms should be identi�ed as part of the sound

source and retained in the right-hand side of the equation and which terms should be in the

left-hand side as part of the operator (e.g., Lilley8). The far-�eld sound pressure is then given in

terms of a volume integral over the domain containing the sound source. Several modi�cations

to Lighthill's original theory have been proposed to account for the sound- ow interaction

or other e�ects. The major di�culty with the acoustic analogy, however, is that the sound

3

source is not compact in supersonic ows. Errors could be encountered in calculating the

sound �eld, unless the computational domain could be extended in the downstream direction

beyond the location 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) equation9 was introduced to extend acoustic analogy in the

case of solid surfaces. However, when acoustic sources (i.e., quadrupoles) are present in the

ow�eld a volume integration is needed. This volume integration of the quadrupole source

term is di�cult to compute and is usually neglected in most acoustic analogy codes (e.g.

WOPWOP10). Recently, there have been some successful attempts in evaluating this term

(e.g. WOPWOP+11;12).

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

extend the CFD solutions to the far-�eld (e.g. Lim et al.13, Viswanathan and Sankar14, Shih et

al.15). The LEE equations employ a division of the ow �eld into a time-averaged ow and a

time-dependent disturbance which is assumed to be small. The hybrid (zonal) approach consists

of the near-�eld evaluation using an accurate CFD code (e.g. for jet noise the code is usually

based on Large Eddy Simulations: LES) and the extension of the solution to the mid-�eld

using LEE. Considerable CPU savings can be realized, since the LEE calculations are much

cheaper than the CFD calculations. This approach is very promising, because it accounts for a

variable sound velocity outside the near-�eld where usually an LES model is applied. However,

dissipation and dispersion errors still exist and an accurate description of propagating far-�eld

waves is compromised because of this. On the other hand, this method may be appropriate

for the an intermediate region in some problems, outside from the reactive near-�eld where the

speed of sound is still not constant, before moving to another integral method for the far-�eld.

Kirchho� Method Another alternative is the Kirchho� method which assumes that the

sound transmission is governed by the simple wave equation. Kirchho�'s method 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 be obtained in terms of a surface integral of the surface

pressure and its normal and time derivatives. This approach has the potential to overcome

some of the di�culties associated with the traditional acoustic analogy approach. The method

is simple and accurate and accounts for the nonlinear quadrupole noise in the far-�eld. Full

di�raction and focusing e�ects are included while eliminating the propagation of the reactive

near-�eld.

This idea of matching between a nonlinear aerodynamic near-�eld and a linear acoustic

far-�eld was �rst proposed by Hawkings16. The use of Kirchho�'s method has increased sub-

stantially the last 10 years, because of the development of reliable CFD methods that can be

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 methodology for each. We have

been referring to this technique as the \Kirchho� method." It has been used to study various

aeroacoustic problems, such as propeller noise, high-speed compressibility noise, blade-vortex

interactions, jet noise, ducted fan noise, etc. An earlier review on the use of Kirchho�'s method

was given by Lyrintzis.17

Porous FW-H equation A �nal alternative is the use of permeable (porous) surface FW-H

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

a solid body and is impenetrable. However, if the surface is assumed to be porous, a general

equation can be derived (as shown in the original reference 9 and in reference 18). The porous

surface can be used as a control surface in a similar fashion as the Kirchho� method explained

above. Thus the pressure signal in the far-�eld can be found based on quantities on the control

surface provided by a CFD code.

Farassat in a recent review article19 reviewed all the available FW-H and Kirchho� equa-

tions for application to noise evaluation from rotating blades. The current article focuses only

5

on control surface methods (i.e. Kirchho�, porous FW-H) and discusses issues with their ap-

plication in various types of aerocoustic problems including rotor noise, jet noise, ducted fan

noise, airfoil noise etc.). At �rst the main formulations will be reviewed, advantages and dis-

advantages of each method will be discussed. Then we will present several algorithmic issues

and various application examples.

2 Kirchho�'s Method Formulations

Kirchho�'s method is an innovative approach to noise problems which takes advantage of

the mathematical similarities between the aeroacoustic and electrodynamic equations. The

considerable body of theoretical knowledge regarding electrodynamic �eld solutions can be

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

published in 188220. 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. Morgans21 derived a

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

used for the derivation of an extended Kirchho� formulation. A �eld function is de�ned to be

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 Hawkings9 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 respect to the observation coordinates

and time and that is di�cult to use in numerical computations. Farassat and Myers22 derived

a Kirchho� formulation for a moving, deformable, piecewise smooth surface. The same partial

derivatives were taken with respect to the source coordinates 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.

It should be noted that Morino and his co-workers23�27 have developed several formulations

for boundary element methods using the Green's function approach, which are equivalent to

Kirchho� formulations. Morino's formulations were derived with aerodynamic applications

in mind, so the observer is in the moving coordinate system. However, they can be used

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

constant speed (e.g., wind tunnel experiments), as mentioned in reference 17. Their latest

formulation 27 appears to provide an integrated boundary element framework for Aerodynamics

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,

22� =

1

a2�

@2�

@t2�

@2�

@xi@xi= 0 (1)

is valid for some acoustic variable �, and sound speed a�, in the entire region outside of a closed

and bounded smooth surface, S.

The signal, in the stationary coordinate system, is evaluated with a surface integral over

the control surface, S, of the dependent variable, its normal derivative, and its time derivative

(�gure 1). S is allowed to move in an arbitrary fashion. The dependent variable � is normally

taken to be the disturbance pressure, but can be any quantity which satis�es the linear wave

equation.

4��(~x; t) =ZS

"E1

r (1 �Mr)

#ret

dS +ZS

"�E2

r2 (1�Mr)

#ret

dS (2)

where

E1 = (M2n � 1) @�

@n+Mn

~Mt � r2��Mn

a�_�+ 1

a�(1�Mr)2

h_Mr (cos � �Mn) �

i+ 1

a�(1�Mr)

��_nr � _Mn � _nM

��+ (cos � �Mn) _�+ (cos � �Mn)�

�(3)

E2 =(1�M2)(1�Mr)2

(cos � �Mn) (4)

7

Here (~x; t) are the observer coordinates and time, and (~y; � ) are the source (surface) coordinates

and time. Mi is the Mach number vector of the surface, r is the distance from source to observer,

� is the source emission angle, and n̂ is the control surface normal vector (cos � = br � bn). ~Mt

is the Mach number vector tangent to the surface, and r2 is the surface gradient operator. A

dot indicates a source time derivative, with the position on the surface kept �xed. Also,

_Mr = _Mibri _nr = _nibri _Mn = _Mibni _nM = _niMi (5)

The form of equation (2) and E1, E2 were given by Farassat and Myers22. E2 was presented in

the simpli�ed form shown here by Myers and Hausmann.28 The surface integrals are over the

control surface S, subscript ret indicates evaluation of the integrands at the emission (retarded)

time, which is the root of

g = � � t+j~x� ~yj

a�= 0 (6)

If the frame velocity is subsonic at the surface, then equation (6) has a unique solution. However,

equation (2) is still valid for supersonically moving surfaces. As we can see from equations 2

through 5, the (1 �Mr) term can produce a singularity in the case where the Mach number

in the radiation direction reaches the sonic point. This is a major limitation of the retarded

time formulation. Farassat and co-workers29;30 have recently presented a formulation that is

appropriate for supersonically moving surfaces (i.e. formulation 4) and veri�ed by application

to benchmark problems. Since, the supersonic formulation has not yet been applied to practical

problems it will not be presented here in the interest of brevity.

The above formulation is valid when the observer is stationary and the surface is moving at

an arbitrary speed. However, for the case of an advancing blade the observer is usually moving

with the free ow speed (e.g. rotor in a wind tunnel with a free stream not equal to zero). The

formulation can be adjusted for this case by allowing x(t) to move with the free stream instead

of being stationary in equation (6) for the retarded time.

It is possible to write equation (2) in a simple form valid for stationary surfaces. The

Kirchho� formula is then

4��(~x; t) =ZS

1

r

"1

a�_� cos � �

@�

@n

#ret

dS +ZS

[�]retr2

dS (7)

8

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

(7) can be expressed in the frequency domain (i.e. starting from Helmholtz equation) as

4� b�(~x; !) = ZSei!r=a�

"1

r

��i!

a�cos � b�� @ b�

@n

�+b� cos �r2

#dS (8)

where b� 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

Mankbadi31 and Pilon:32

Two-dimensional formulations can also be developed (Pilon33, Scott et al.34). Atassi et

al.34 developed a two-dimensional frequency domain formulation that uses a modi�ed Green's

function in order to avoid the evaluation of normal derivatives. Mankbadi et al.35 developed a

modi�ed Green's function for a cylinder control surface that was applied in jet noise predictions.

Hariharan et al.36 developed 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 body surface, there are some nonuniqueness di�culties 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 by Wu and Pierce37 and for moving Kirchho� surfaces by Wu38. Finally,

Dowling and Ffowcs Williams39 included the e�ects of in�nite plane walls in the stationary

Kirchho� formulation. However, in this paper 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� Method

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 be

possible for some cases. Therefore, additional nonlinearities can be added outside the control

surface.40�44 The modi�cations to the traditional Kirchho� method consist of an additional

9

volume integral. Thus equation(2) now becomes:43 (pressure is used here as the dependent

variable)

4�p0(~x; t) =ZS

"E1

r (1 �Mr)

#ret

dS +ZS

"�E2

r2 (1�Mr)

#ret

dS +ZV

"1

r(1 �Mr)

@2Tij@yi@yj

#ret

dV (9)

where

Tij = �uiuj � �ij +�(p� po)� a2

��0��ij (10)

where ui is the uid velocity, � is the density, �0 the density perturbation, and �ij is the 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 Tij becomes zero. Through the use of

Fourier transforms, equation (9) can also be expressed in the frequency domain.

Isom et al. 45 developed a nonlinear Kirchho� formulation (Isom's formulation) for some

special 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 above. However, they showed that for

the above special cases the nonlinear e�ects can be reduced to a surface integral.

2.3 The Porous Ffowcs Williams { Hawkings equation

A modi�ed integral formulation for the porous surface FW-H equation18 is needed because

the usual practice is to assume that the FW-H integration surface corresponds to the body

and is impenetrable. A convenient way to formulate this is as an extension of Farassat's

formulation 146 which was originally developed for the rigid surface FW-H equation. Following

Francescantonio42 we de�ne new variables Ui and Li as

Ui =

1�

�

�o

!�i +

�ui�o

(11)

10

and

Li = Pij n̂j + �ui(un � �n) (12)

where subscript o implies ambient conditions, superscript 0 implies disturbances (e.g. � =

�0 + �o), � is the density, u is the velocity, and Pij is the compressive stress tensor with the

constant po�ij subtracted. Now by taking 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 be written as (Formulation I)

p0(~x; t) = p0T (~x; t) + p0L(~x; t) + p0Q(~x; t) (13)

where

4�p0T (~x; t) =@

@t

ZS

"�oUn

rj1 �Mrj

#ret

dS (14)

4�p0L(~x; t) =1

a�

@

@t

ZS

"Lr

rj1 �Mrj

#ret

dS +ZS

"Lr

r2j1�Mrj

#ret

dS (15)

and p0Q(~x; t) can be determined by any method currently available (e.g., references 10, 11).

In equations (14) and (15) a dot product of the vector with the unit vector in the radiation

direction r̂ or the unit vector in the surface normal direction n̂i, respectively.

It should be noted that the three pressure terms have a physical meaning for rigid surfaces:

p0T (~x; t) is known as thickness noise, p0L(~x; t) is called loading noise and p0Q(~x; t) is called

quadrupole noise. For a porous surface the terms lose their physical meaning, but the last term

p0Q(~x; t) still denotes the quadrupoles outside the control (porous) surface S.

An alternative way42 is to move the time derivative inside the integral: (Formulation II)

4�p0T (~x; t) =ZS

"�o( _Un + U _n)

r(1 �Mr)2

#ret

dS +ZS

"�oun(r _Mr + c(Mr �M2))

r2(1�Mr)3

#ret

dS (16)

4�p0L(~x; t) =1c

RS

h_Lr

r(1�Mr)2

iret

dS +RS

hLr�LM

r2(1�Mr)2

iret

ds

+1c

RS

hLr(r _Mr+c(Mr�M2))

r2(1�Mr)3

iret

dS (17)

This is now an extension of Farassat's formulation 1A.47 ehere the dot over a variable implies

source-time di�erentiation of that variable, LM = LiMi, and a subscript r or n indicates It

11

appears that Formulation I (equations 14, 15) has less memory requirements, because it does

not require storage of the time derivatives, and requires less operations per integral evaluation.

However, in general, integrals have to be evaluated twice in order to �nd the time derivative. In

the special case of a stationary control surface, or a �xed microphone location, i.e. " yover," the

integral can be reused at the next time step. Since memory appears to be more important for

these type of calculations, Formulation I is a good choice. Formulation I was used by Strawn et

al.48 for rotorcraft noise predictions using a nonrotating control surface with very good results.

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

for a Formulation II (equations 16, 17) might be more robust for a moving control surface.

Formulation II was used for rotorcraft noise prediction by Brentner and Farassat44 with a

rotating control surface with very good results. However, a more detailed comparison of the

two formulations would be very helpful.

For a stationary surface Formulation I reduces to:

4�p0T (~x; t) =@

@t

ZS

��oUn

r

�ret

dS (18)

4�p0L(~x; t) =1

a�

@

@t

ZS

�Lr

r

�ret

dS +ZS

�Lr

r2

�ret

dS (19)

and Formulation II becomes:

4�p0T (~x; t) =ZS

"�o _Un

r

#ret

dS (20)

4�p0L(~x; t) =1

a�

ZS

"_Lr

r

#ret

dS +ZS

�Lr

r2

�ret

dS (21)

With the use of a Fourier transformation both formulations (for a stationary surface) can

be written in the frequency domain as49

4� bp0T (~x; !) = �i!ZS

ei!r=a��o bUn

rdS (22)

4� bp0L(~x; !) = �i!

a�

ZS

ei!r=a�bLr

rdS +

ZS

bLr

r2dS (23)

12

where bp0, bUn, and bLr are the Fourier transforms of p0, Un, and Lr, respectively and ! is the

cyclic frequency. It should be noted that both time formulations reduce to the same frequency

formulation for a stationary control surface.

Time and frequency formulations for a uniform rectilinear motion can be found in reference

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

oped in the past (see, for example, references 51, 52) and can be readily extended to a porous

surface. Finally, a supersonic formulation can also be found in reference 30.

2.4 Comparison of Kirchho� FW-H Methods

Both the above formulations provide a Kirchho�-like formulation if the quadrupoles outside the

control surface (p0Q(~x; t) term) are ignored. The equivalence of the porous FW-H equation and

Kirchho� formulation was proven Pilon & Lyrintzis43 and Brentner & Farassat.44 They showed

that, for a surface placed in a linear region, the porous surface FW{H formulation is equivalent

to the linear Kirchho� formulation, plus a volume integral of quadrupoles (�uiuj). (Pilon and

Lyrintzis43 also claim that the control surface need not be placed in an entirely linear region.

The nonlinearities can be accounted for with the use of � = a2��0 as the dependent variable,

and the volume integral of quadrupoles, Tij.)

The major di�erence between Kirchho�'s and FW-H formulation is that Kirchho�'s method

needs p0; @p@n; @p@t

as input whereas the porous FW-H needs p0; �; �ui. Also, the porous FW-H

method allows for nonlinearities on the control surface, whereas the Kirchho� method assumes

a solution of the linear wave equation on the surface. Thus if the solution does not satisfy

the linear wave equation on the control surface the results from the Kirchho� method change

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

Kirchho� method. This was shown in reference 44 for a rotorcraft noise problem (see section

5.2). Another way to look at this di�erence is to state that the Kirchho� method puts more

stringent requirements to the CFD method to reach to the linear acoustic �eld before dissipation

and dispersion errors due to coarsening in the far-�eld take over.

13

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

control surface presents a challenge. A major motivation for the use of Kirchho�/porous FW-H

methods is the lack of volume integrations, which reduces necessary calculations by an order of

magnitude. However, the recently developed code WOPWOP+11;12 provides an e�cient means

of accounting for the quadrupoles in FW-H calculations that can be used for both methods,

because the quadrupole term is the same.

2.5 Mean Flow Refraction Corrections for Jet Noise

The Kirchho� and the FW-H formulas presented above can e�ciently and accurately predict

aerodynamically generated noise, as long as the control surface surrounds the entire source

region. In jet noise predictions, however, it is usually impossible, with current numerical

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

imposed by the computer architecture, as well as dispersion 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 be 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 points 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 model to determine sound produced by point acoustic sources within circular jets (e.g.,

Amiet53). This approach was adopted by Lyrintzis and co-workers 49;54 and in order to account

for refraction e�ects in the Kirchho� and the porous FW-H method. A real jet has non{zero

radial velocity, but the refracting e�ect of this component is minimal, and can safely be ignored.

Also, the lack of azimuthal variation in the parallel shear ow approximation has a very small

e�ect. The value of the axial velocity to be used in the shear ow approximation can be taken

directly from the CFD numerical simulation, at the downstream end of the control surface, as

an average of the time dependent axial velocity at each radial grid point.

14

The refraction problem now consists of a collection of point acoustic sources (the integrands

of equations (8) and (22) acting at radial location R, and the parallel shear ow with U

determined at each R). If the acoustic wavelength, � = 2�a�=!, is assumed to be small

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

If the steady velocity at the downstream end of the Kirchho� surface is denoted Us, the

sound emission angle with respect to the jet axis #s, and the propagation angle in the stagnant,

ambient air is denoted #�, then the axial acoustic phase speeds are preserved by the strati�ed

ow

a�cos#�

= Us +a�

cos #s(24)

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

This equation can be rearranged to show that there is a critical angle, #c de�ned by

#c = cos�1�

1

1 +Ms

�(25)

If the the observer angle #� is greater than #c 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 Ms is the Mach number 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.

Imposing the local \zone of silence" condition described above can allow a surface source at a

relatively large radial location to radiate sound into and through the shear ow. This is because

the local \zone of silence" decreases in size with the radial location of the source, due to the

decrease in source Mach number. The simple correction is to set the source strength to zero

if the observation point is located closer to the jet axis than the source point 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 located far enough downstream

of the jet potential 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� method (equation 8). In reference 49 an amplitude correction as

recommended by Amiet53 (but not included in reference 54) was added and the methodology

was applied to both Kirchho� and FW-H methods (equations 8 and 22).

2.6 Open Control Surface

Freund et al.55 developed a way to improve the accuracy of Kirchho� evaluations of sound �elds

for an open Kirchho� control surface. Asymptotic analysis was used to provide correction terms

which partially account for the missing portion of the integral surface. It was shown that the

major contribution comes from a point on the surface that intersects the line between 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 happens 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 be found in the original reference.

3 Algorithmic Issues

Some algorithmic issues are discussed below. Additional information for numerical algorithms

for acoustic integrals, in general, is given by Brentner.56

3.1 Choice of control surface

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

Since Kirchho�'s method assumes that the linear wave equation is valid outside the closed

control surface S, S must be chosen large enough to include the region of all nonlinear be-

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

surrounding the moving blade because of the increase of mesh spacing in CFD codes. Thus a

16

judicious choice of S is required for the e�ectiveness of the Kirchho� method. For example, in

the case of airfoil/rotor noise the control surface is typically located a couple of chordlengths

away from the airfoil/rotor surface.

For a porous FW-H formulation no normal derivatives are required and (because nonlin-

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

control surface,44 as will be shown in section 5.2. Thus the CFD requirements for the FW-H

are less strigent, making the method more attractive. Singer et al.57 used a FW-H method for

the analysis of slat trailing-edge ow. The interesting thing about this application is that part

of the control surface is solid and another part is porous.

3.2 Quadrature

For su�cient accuracy in the far-�eld calculations, high order quadrature should be used to solve

the surface integrals in equation (2). The predicted surface quantities (p0, @p=@n, @p=@t) should

also be very accurate. This can be achieved through the use of a very �ne mesh in the CFD

calculations. However, memory and time constraints often make this impractical. Meadows

and Atkins58 have shown that it is possible to obtain highly accurate Kirchho� predictions from

relatively coarse{grid CFD solutions. Through an interpolation process, more spatial points are

added to the Kirchho� quadrature calculations without additional e�ort in the CFD process.

This has the e�ect of re�ning the CFD mesh with almost no additional cost. They refer to this

process as \enrichment". High order quadrature, temporal interpolation, and enrichment are

important for accurate far-�eld noise predictions with the Kirchho� method, especially 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) method can be used to solve this nonlinear equation.

This method has been the basis of several Kirchho� codes (e.g. Lyrintzis & Mankbadi31, Strawn

17

et al.59, Lyrintzis et al.60 Polacsec & Prier61). The algorithm can be easily parallelized (e.g.

Wissink et al.62, Strawn et al.63) by partitioning the control surface and distributing to di�erent

processors. Since the only communication is the �nal global summation the parallel e�ciency of

the code is very high. Lockard50 discussed parallelization of FW-H codes. Long and Brentner64

proposed a master-slave approach for load balancing.

However, it is di�cult to write a versatile code for various mesh topologies used by current

CFD codes, including unstructured grids, based on this approach. In addition, when these codes

are extended to supersonically moving surfaces, the retarded time equation will have multiple

roots that will be di�cult to evaluate. Also, the codes sometimes require signi�cant memory.

Finally58, the variation of the source strength over a surface element in the retarded time can be

very high at certain observer locations (r̂ � n̂! 0) and near sonic velocities (Mr ! 1) requiring

a large number of points per wavelength.

In order to overcome the limitations stated above, another approach is developed 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,

interpolation techniques are required when the integration is performed to obtain the overall

acoustic signal at the observer position (e.g. Ozyoruk and Long65�67, Lyrintzis and Xue68, and

Rahier and Prier69, Algermissen and Wagner70). Finally, a marching-cubes algorithm71 can

be used to provide an e�cient 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 both a rotating and a nonrotating formulation can be used. A nonrotating

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

Korkan72, Strawn and Biswas73, Baeder et al.74 ). A rotating Kirchho� formulation allows the

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

(e.g. Xue and Lyrintzis75, Lyrintzis et al.60, Polacsec and Prier61). No transformation of data is

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

Kirchho� methods showed that both methods are very accurate and e�cient (Strawn et al.59).

For the porous FWH method there are fewer applications. A rotating method was used in

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

It should be 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 method

has been used in conjunction with Euler/Navier Stokes codes (e.g., TURNS code77;78, OVER-

FLOW code79) whereas the rotating Kirchho� method has been used with full potential codes

(e.g. FPR code80;81), as well.

However, a drawback of the rotating method is that the rotating speed of the tip of the

rotating surface needs to remain subsonic, because Farassat's formulation is currently limited

to subsonically moving surfaces. An extension to supersonically moving surfaces is needed.

This imposes limits to the position of the tip of the rotating control surface in very high Mach

number cases (e.g. M=0.92-0.95 for hover). However, the supersonic formulation formulation

of Farassat et el.29;30 can be employed in the future for the rotating case for high-Mach number

cases.

19

4 Validation Results

Both Kirchho� and FWH formulations have been validated using model problems. The �rst

thing to do is, of course, check that the signal becomes zero inside the control surface. The num-

ber of points per period and the number of points per wave length should also be studied.31;49

A stationary or translating point source have been used by Lyrintzis et al.,31;49 Myers &

Hausmann,28 and Lockard50 and a rotating point source by Lyrintzis et al.60 and Berezin et

al.82. Exponential source distributions have been used by Pilon and Lyrintzis.32;40;41;43 Hu

et al.83 used a line monopole source and a Gaussian pressure and vorticity pulse (category 3

benchmark problem84) to verify their two-dimensional FW-H formulation. Farassat and Farris30

used dipole distributions on a at surface and a sphere to validate the supersonic formulation

(i.e. formulation 4). Singer et al.85 used a line vortex around an edge. Meadows and Atkins58

used an oscillating sphere and studied the e�ects of quadrature (see section 3). Ozyoruk and

Long65 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

better than the direct calculation, because of numerical dissipation as the waves travel longer

distances to arrive at the observer locations.

5 Aeroacoustic Applications

Kirchho�'s formula has been extensively used in light di�raction and other electromagnetic

problems, aerodynamic problems, i.e. boundary-elements (e.g. Morino et al. 25), as well as in

problems of wave propagation in acoustics (e.g. Pierce86). Kirchho�'s integral formulation has

been used extensively for the prediction of acoustic radiation in terms of quantities on boundary

surfaces (the Kirchho� control surface coincides with the body). Kirchho�'s method has also

been used for the computation of acoustic scattering from rigid bodies using a boundary element

technique with the Galerkin method.

20

The solid surface FW-H equation with its various forms19 has been used in several problems

including propeller and helicopter noise. Here we will concentrate in the use of \Kirchho�",

and \porous" FW-H equation methods, i.e. using a nonlinear CFD solver for the evaluation

of acoustic sources in the near-�eld and a Kirchho�/porous FW-H formulation for the acous-

tic propagation. We will review some \real-life" aeroacoustic applications of both methods

concentrating in recent advances.

5.1 Propeller Noise

Hawkings16 used a stationary-surface Kirchho�'s formula to predict the noise from high- speed

propellers and helicopter rotors. Forsyth and Korkan72 calculated high-speed propeller noise

using the Kirchho� formulation of Hawkings16. Jaeger and Korkan87 used a special case of

the Farassat and Myers22 formulation for a uniformly moving surface to extend the calculation

to advancing propellers. In the above applications, the control surface S was chosen to be a

cylinder enclosing the rotor.

5.2 Helicopter Impulsive Noise

Kirchho�'s method has been widely applied in the prediction of helicopter impulsive noise.88

The Kirchho� method for a uniformly moving surface was initially used in two-dimensional

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

dynamic BVI results to the linear acoustic far-�eld.89�92 Actually, the �rst application of

Hawkings16 \Kirchho� Method" was given by George and Lyrintzis.77 The Kirchho� method

was used to test ideas for BVI noise reduction (Xue and Lyrintzis.93 The method was also

extended to study noise due to other unsteady transonic ow phenomena (i.e. oscillating

aps, thickening-thinning airfoil) by Lyrintzis et al.94 Later, the method was used for the two-

dimensional BVI problem by Lin and co-workers.95;96

Kirchho�'s method has also been applied to three-dimensional High-Speed Impulsive (HSI)

noise. Baeder et al.74 and Strawn & Biswas73 used a nonrotating control Kirchho� surface that

21

encloses the entire rotor. The Transonic Unsteady Rotor Navier Stokes (TURNS) code77;78 was

used for the near-�eld CFD calculations. An unstructured grid was used by Strawn et al.97 and

an overset grid code (OVERFLOW)79 by Ahmad et al.98 Kirchho�'s method predicted the HSI

hover noise very well using a fraction of CPU time of the straight CFD calculation.

Another Kirchho� method used in helicopter noise is the rotating Kirchho� method (i.e. the

surface rotates with the blade). The method was used for three-dimensional transonic BVI's for

a hovering rotor by Xue and Lyrintzis.75 The near-�eld was calculated using the Full Potential

Rotor (FPR) code.80;81 The rotating Kirchho� formulation allows the Kirchho� control surface

to rotate with the blade; thus a smaller cylinder surface around the blade can be used. No

transformation of data is needed because the CFD input is also rotating. Since more detailed

information is utilized for the accurate prediction of the far-�eld noise this method is more

e�cient. Finally, the method was extended for an advancing rotor and was applied to HSI

noise99 and BVI noise.100;101 Berezin et al.82 showed that sometimes special care is needed for

choosing the CFD grids, because the highly stretched grids used for aerodynamic applications

may not provide accurate information on the control Kirchho� surface.

A comparison59 of the rotating and the nonrotating Kirchho� methods showed that both

methods are very accurate and e�cient. Figure 3 shows a comparison for an advancing HSI noise

case (1/7 scale AH-1 helicopter, hover tip Mach numberMH = 0:665, advance ratio � = 0:258,

which corresponds to an advancing tip Mach number of Mat = 0:837). TURNS77;78 is used for

the CFD calculations. We see that both methods compare very well with the experiments.102

Kirchho�'s method has become a standard tool for rotorcraft acoustic predictions. The method

is currently implemented in the TRAC (TiltRotor Aeroacoustic Codes) system developed by

NASA Langley (RKIR code, Lyrintzis et al.60, Berezin et al.82) and is employed at NASA Ames

AFDD (Strawn et al.59). In Europe, additional versions of rotating and nonrotating Kirchho�

codes have also been developed.61;69;70;103;104

Kirchho�'s method results have also been compared with acoustic analogy (solid surface

FW-H equation). A comparison with the acoustic analogy code WOPWOP10 (WOPWOP uses

the solid surface FW-H equation without accounting for quadrupoles) has shown that Kirchho�

22

method is superior when quadrupole sources are present (Lyrintzis et al.105) for advancing HSI

cases. Baeder et al.74 also compared the results with a linear (i.e. monopole plus dipole sources

on the rotating blade) solid surface FW-H equation method for hover HSI. The FW-H results

were inaccurate for tip Mach numbers higher than 0.7, because of the omission of quadrupole

sources. However, a further comparison of the rotating Kirchho� method to WOPWOP+11;12

(WOPWOP+ is a solid surface FW-H equation method accounting also for quadrupoles with

a volume integral) has shown that the two methods give about the same results (Brentner et

al.106), but Kirchho� method uses only surface integrals and avoids the quadrupole volume

integration. It should be noted that robustness of Kirchho� method improves with the use of

a less stretched grid (Berezin et al.82) or an Euler code, e.g. TURNS (Baeder et al.74).

Isom et al.45, and Purcell107;108 used a modi�ed Kirchho� method which also included some

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

here) show good agreement with experimental data.

A porous FW-H method based on Kirchho� subroutines was also developed by Brentner

& Farassat44 (FWH/RKIR code), Morgans et al.76 and Strawn et al.48. These codes do not

include quadrupoles outside the control surface, because it was found to be of minor importance

unless the Mach number is really high.109 Thus the porous FW-H equation is also based on

surface integrals. The porous FW-H formalism is more robust than the traditional Kirchho�

method with regards to the choice of the control surface, as shown in �gures 4 and 5 for a hover

HSI noise case (1/4 model UH-1H model helicopter, hovering at MH = 0:88, experiments from

Purcell107). FPR80;81 was used for the CFD calculations.

5.3 Airfoils

Atassi and his co-workers34;110�112 have used Kirchho�'s method 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� method and the direct calculation method (i.e. rapid distortion

23

theory113�115 is given in �gure 6 (from references 34 and 101) for a 3% thick Joukowski airfoil in

a transverse gust at k1 = (!c=2V1) = 1 and M=0.1. The semi-analytical results for a at plate

encountering the same gust are also shown in �gure 4 and are very close to the results from

Kirchho�'s method. The �gure indicates that the direct calculation method 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 code is accurate in

the near-�eld and the Kirchho� method should be used instead in the far-�eld, as indicated in

�gure 6.

Singer et al.85;57 used a FW-Hmethod for the evaluation of acoustic scattering from a trailing

edge and slat trailing edge. The interesting thing about the slat trailing edge application is

that part of the control surface is solid and another part is porous.

5.4 Fan Noise

Kirchho�'s method can also be applied to ducted fan noise. Very good results were shown

by Ozyoruk and Long64�66 for a control surface in rectilinear motion. A forward time parallel

algorithm was used. A porous FWH method was used by Zhang116 with very good results.

5.5 Jet Noise

There are some di�culties with using the Kirchho� and and the porous FW-H methods for

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

enclose the aerodynamic source region. This may be di�cult or impossible to accomplish if

the source region is large. The validity of this method is also dependent on the control surface

being 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 location is near the jet

axis.

Kirchho�'s method has also been applied in the estimation of jet noise. Soh117 and Mitchel

24

et al.118 used the stationary Kirchho� method (equation 7) and Lyrintzis & Mankbadi31 Chy-

czewski & Long119, Morris et al.120, Gamet and Estivalezes121, Choi et al.122 and Kandula and

Caimi123 used the uniformly moving formula. It should be noted that most of the above refer-

ences use an LES code for the CFD data. However, a RANS code can also be used, as shown

in reference 123, where OVERFLOW79 was used. Lyrintzis & Mankbadi31 also compared time

and frequency domain formulations. Mankbadi et al.35 applied a modi�ed Green's function to

avoid the evaluation of normal derivatives. Balakumar124 and Yen125 used parabolized stability

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

for the noise evaluation Shih et al.126 compared several Kirchho� formulations with the acoustic

analogy, extending the LES calculations and using a zonal LES + LEE method. The results

showed that the Kirchho� method is much more accurate than the acoustic analogy (for the

compact source approximation used) and much cheaper than extending the LES or performing

a zonal LES + LEE. Finally, Morris et al.127;128 used the porous FW-H method and Hu et al.83,

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

from a plane jet.

The above approaches have used an open control surface (i.e. without the downstream end)

in order to avoid placing the surface in a nonlinear region. Freund et al.55 showed a means

of correcting the results to account for an open control surface, for cases that the observer is

close to the jet axis. Pilon and Lyrintzis40;41;43 developed a method to account for quadrupole

sources outside the control surface. This approximation is based on the assumption that all

wave modes approximately decay in an exponential 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 method (recommended in reference 49) is to just use an

existing empirical code (e.g. MGB129) to evaluate 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�/porous FW-H method.

An approximate way to account for refraction e�ects was developed by Lyrintzis and co-

workers49;54, as explained above in section 2.4. A typical result shown here (�gure 7) shows

25

the e�ects of refraction corrections for a supersonic Mach number case (excited, Mach 2:1,

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

perturbed at a single axisymmetric mode at a Strouhal number of St = 0:20, the amplitude of

the perturbation was 2% of the mean). Further development of refraction corrections (based,

for example in reference 130) is possible.

Finally, it should be noted that for some complicated noise problems (as, for example, in jet

noise) several computational domains might be 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 method for the far-�eld. Kirchho�'s formulation can be the simplest region of a

general zonal methodology. This idea has been proposed by Lyrintzis,17 but it has not yet been

implemented.

6 Concluding Remarks

Kirchho�'s and porous FW-Hmethods consist of the calculation of the nonlinear near- and mid-

�eld numerically with the far-�eld solutions found from a Kirchho�/porous 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 methodology for each.

The advantage of these methods is that the surface integrals and the �rst derivatives needed

can be evaluated more easily than the volume integrals and the second derivatives needed for

the evaluation of the quadrupole terms when the traditional acoustic analogy is used.

The porous FW-H equation is equivalent to Kirchho�'s method and is very appealing be-

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

tives. Since the method also requires a surface integral, it is very easy to modify existing

Kirchho�/solid surface FW-H codes.

The use of both methods has increased substantially the last 10 years, because of the

development of reliable CFD methods that can be used for the evaluation of the near-�eld.

26

The methods can be used to study various acoustic problems, such as propeller noise, high-

speed compressibility noise, blade-vortex interactions, jet noise, ducted fan noise, etc. Some

results indicative of the uses of both methods are shown here, but the reader is referred to the

original references for further details. We believe that, a simple set of portable Kirchho�/FW-

H subroutines can be developed to calculate the far-�eld noise from inputs supplied by any

aerodynamic near/mid-�eld code.

Acknowledgements

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

the Aeroacoustics Consortium (AARC).

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AIAA paper No. 97-1599, Proceedings of the 3rd AIAA/CEAS Aeroacoustics Conference,

Atlanta, GA, May 1997 pp. 112{126.

42

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

43

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

44

Figure 3: Comparison of acoustic pressures with experimental data102 at four di�erent micro-

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

rotor (from reference 59).

45

Figure 4: Comparison of Kirchho� acoustic pressures with experimental data108 for an observer

in the plane of the rotor at 3; 4R from a UH-1H model rotor hovering at MH = 0:88 (from

reference 44).

46

Figure 5: Comparison of porous FW-H acoustic pressures with experimental data108 for an

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

(from reference 44).

47

Figure 6: Comparison between far-�eld directivity of acoustic pressure values using the Kirch-

ho� method (- -) and the direct calculation method (-�-) for a 3% thick Joukowski airfoil

in a transverse gust at k1 = 1:0;M = 0:1. The semi analytical results ({) for a at plate

encountering the same gust are also shown (from reference 36).

48

0 10 20 30 40 50 60 70 80 90 100x/Rj

-50

-40

-30

-20

-10

0

10

20

30

40

50

R/R

j

No Corrections

Refraction Corrections

Figure 7: Instantaneous contours of a2��0=p�. R > 0: No refraction corrections. R < 0:

Refraction corrections imposed (from reference 54).

49