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JIM Jk 


AIAA 2002-2477 

Trailing Edge Noise Prediction Based 
on a New Acoustic Formulation 

J. Casper and F. Farassat 
NASA Langley Research Center 
Hampton, VA 


8th AIAA/CEAS Aeroacoustics 
Conference and Exhibit 

June 17-19, 2002 / Breckenridge, CO 


For permission to copy or republish, contact the copyright owner named on the first page. For AlAA-held copyright, 
write to AIAA Permissions Department, 1801 Alexander Bell Drive, Suite 500, Reston, VA 20191-4344. 



AIAA 2002-2477 


TRAILING EDGE NOISE PREDICTION BASED 
ON A NEW ACOUSTIC FORMULATION 1 


J. Casper* and F. Farassat 1 
NASA Langley Research Center 
Hampton, VA 23681 


Abstract 

A new analytic result in acoustics called “Formula- 
tion IB,” proposed by Farassat, is used to compute 
broadband trailing edge noise from an unsteady surface 
pressure distribution on a thin airfoil in the time do- 
main. This formulation is a new solution of the Ffowcs 
Williams-Hawkings equation with the loading source 
term, and has been shown in previous research to pro- 
vide time domain predictions of broadband noise that are 
in excellent agreement with experiment. Furthermore, 
this formulation lends itself readily to rotating reference 
frames and statistical analysis of broadband trailing edge 
noise. Formulation IB is used to calculate the far field 
noise radiated from the trailing edge of a NACA 0012 
airfoil in low Mach number flows, using both analyti- 
cal and experimental data on the airfoil surface. The 
results are compared to analytical results and experi- 
mental measurements that are available in the literature. 
Good agreement between predictions and measurements 
is obtained. 

Nomenclature 

b = airfoil semi-span (m) 

C = airfoil chord (m) 

Co = ambient sound speed (m/sec) 

/ = frequency (Hz) 

/ = geometry function for airfoil surface (Fig. 1) 
E* — combination of Fresnel integrals (Eq. (2c)) 
g = surface pressure transfer function 
k c = u>/U c , convective wave number (m _1 ) 

£2 = spanwise correlation length (m) 

M = V/co, Mach number vector 
M r = M ■ rjr Mach number in radiation direction 
M v = M ■ v Mach number in direction of v 
AP = unsteady airfoil surface pressure jump (Pa) 
p = unsteady airfoil surface pressure (Pa) 
p' = sound pressure radiated to observer (Pa) 
dp/ds = surface pressure gradient in the direction of V 
qo = poll 2 /2, dynamic head (kg-m/s 2 ) 

Copyright © 2002 by the American Institute of Aero- 
nautics and Astronautics, Inc. No copyright is asserted in 
the United States under Title 17, U.S. Code. The U.S. Gov- 
ernment has a royalty-free license to exercise all rights under 
the copyright claimed herein for Governmental Purposes. All 
other rights are reserved by the copyright owner. 

1 This manuscript has been revised since its original pre- 
sentation. Last revision July 19, 2002. 

* Research Scientist, Computational Modeling and Simula- 
tion Branch, AIAA Senior Member 

1 Senior Research Scientist, Aeroacoustics Branch, AIAA 
Assoc. Fellow 


r = x — y. sound radiation vector (m) 

S qq = surface pressure correlation function 
T =1//, acoustic period (sec) 
t = observer time (sec) 

U = uniform freestream speed (m/sec) 
u = unsteady streamwise velocity (m/sec) 

V = airfoil velocity vector 
x = [aq, X 2 , * 3 ] t , observer position 
S = [l/i 1 y~‘ 0]L surface source position 
0 = v/1 -M 2 

A = Co//, acoustic wave-length (m) 

/r = Mu/0 2 U 

ip = directivity angle (Fig. 4) 

v = unit inward facing normal on surface edge (Fig. 1) 
9 = angle between surface normal and f (Fig. 1) 
pa = ambient density (kg/m 3 ) 
t = t — r/co , source time (sec) 

= power spectral density of surface pressure 
q> = random phase variable (radians) 
uj = 2nf, circular frequency (radians/sec) 

Subscripts 

1,2,3 = Cartesian coordinate directions (Fig. 2) 

ret = evaluated at source time r 

1. Introduction 

Trailing edge (TE) noise has been the subject of ex- 
tensive research within the aeroacoustic community for 
decades, both experimentally and analytically. Areas 
of current research include the prediction of TE noise 
from rotating machinery and airframes. Research in the 
area of TE noise prediction has, in large part, been mo- 
tivated by the desire to incorporate the results of TE 
noise analysis into a design methodology. The present 
work is similarly motivated, and the resulting formula- 
tion should lend itself well to an engineering design tool 
suite when aeroacoustics plays a role in the design. 

The literature abounds with various theoretical ap- 
proaches to the prediction of TE noise. Howe 1 catego- 
rized the various theories of TE noise into three groups: 

(i) Theories based on the Lighthill 2 acoustic analogy, 
e.g., Ffowcs Williams and Hall 3 . 

(n) Theories based on the solution of special problems 
approximated by the linearized hydrodynamics equa- 
tions, e.g., Amiet 4 ’ 3 and Goldstein 6 . 

(iii) Ad hoc models, involving postulated source dis- 
tributions whose strengths and types are empirically de- 
termined. 

The present work falls into the first category. This 


1 

American Institute of Aeronautics and Astronautics 



new result, “Formulation IB,” is a solution of the loading 
source term of the Ffowcs Williams-Hawkings equation.' 
Such time domain methods provide for a total decou- 
pling of the acoustic signal from the aerodynamics. As 
such, these methods readily avail themselves to acoustic 
predictions that are based on input from experimental 
measurements or computational fluid dynamics (CFD) 
solutions. For example, Singer et alf used a solution of 
the Ffowcs Williams-Hawkings equation to predict TE 
noise from sources that were modeled with CFD simula- 
tions. The acoustic formulation in their work 8 is known 
as “Formulation 1A.” 9 What distinguishes Formulation 
IB from this prior formulation is its relative simplicity, 
which makes it highly suitable for rotational reference 
frames and statistical analysis of broadband TE noise. 

In Ref. 10, Formulation IB was applied to the pre- 
diction of far field noise due to incident turbulence on a 
NACA 0012 airfoil at tunnel speeds ranging from 40 m/s 
to 165 m/s, and compared to the experimental results 
of Paterson and Amiet. 11 The time dependent surface 
pressure required as input to Formulation IB was gener- 
ated by stochastic modeling of the incident turbulence 
and approximation of the airfoil response with a re- 
sult from thin airfoil theory. Formulation IB was then 
used to predict the acoustic pressure as a function in 
time at a prescribed microphone location. The time do- 
main results were then Fourier analyzed to determine 
the spectral density of the far field noise. The far field 
spectra were found to be in excellent agreement with 
the frequency domain predictions and experimental mea- 
surements of Paterson and Amiet 11 . 

The time domain approach that is described in Ref. 
10 is used in the current work to predict far field radia- 
tion from the trailing edge of an airfoil. In the following 
section, Formulation IB is briefly reviewed for the case 
of a flat surface in a general non-uniform motion. (For a 
formal derivation, see Ref. 10.) Some advantages of this 
new formulation relative to other solutions of the Ffowcs 
Williams-Hawkings equation are described. 

In Section 3, a model problem is considered in which 
an unsteady surface pressure that is comprised of a single 
frequency induces an acoustic source at the trailing edge 
of a flat plate in uniform motion. The unsteady surface 
pressure is an analytical result from thin airfoil theory 
that is taken from the work of Amiet. 4,5 * 12 Two simple 
test cases are presented for validation purposes. The 
directivity of the tone induced by this surface pressure 
is examined for qualitative correctness. The results of a 
velocity scaling exercise are shown to be consistent with 
the results of Ffowcs Williams and Hall. 3 

In Section 4, the surface pressure formulation intro- 
duced in Section 3 is used as the basis function of a linear 
superposition that provides an analytic source model 
for broadband TE noise. This stochastically modeled 
surface pressure is used as input to Formulation IB to 
predict broadband TE noise from a NACA 0012 airfoil in 
a low-turbulence uniform mean flow. The surface pres- 
sure correlations that are required in the aerodynamic 
model are taken from two sources: an empirical flat plate 
formulation 13,14 and experimental data. 15 The resulting 


calculations are compared to the acoustic predictions of 
Schlinker and Amiet 16 and the experimental measure- 
ments of Brooks and Hodgson. 17 


2. Acoustic Formulation 


Consider a flat, finite surface moving in the plane 
Xu = 0 along a velocity vector V . The velocity vector and 
the plate’s geometry are related to the coordinate axes 
as pictured in Fig. 1. Let f(xi,X2,t) denote a geometric 
function that is so defined that / = 0 on the surface edge i 
and / > 0 on the interior of the surface. Let i> = V / 
denote the unit inward geodesic normal that lies in the 
plane of the surface. Let x = [xi,X 2 ,xs] t denote the po- 
sition of an observer, and by y = [yi,y 2 , 0] T the position 
of a source point on the plate’s surface. The unsteady 
perturbation pressure p(y, r) on the surface gives rise to 
sound that radiates along r = x—y to the observer. This 
sound is described by p'(x, t), the perturbation pressure 
that arrives at the point (xi,X2,X3) at time t. 


The derivation of Formulation IB can be found in 
Ref. 10. However, for derivation purposes, both x and 
y frames of reference are considered fixed relative to the 
medium at rest. The resulting formulation contains a 
time derivative p that is evaluated relative to an ob- 
server that is fixed with respect to the medium at rest, 
e.g., as measured by a transducer just above the surface 
that remains stationary as the surface passes by it. This 
quantity p can be related to dp/dr, the time derivative 
of pressure in the reference frame of the moving surface, 
e.g., as measured by a transducer attached to the surface. 
This relation is 




where dp/ ds is the gradient of p in the direction of V, and 
V is the local magnitude of V. Here, s is in the direction 
of the velocity V of the surface in the reference frame 
fixed to the undisturbed medium. The final expression 
for the sound radiated to the observer is 


= j 

+ L 

-L 


( dp /dr - V dp/ds ) cos 8 


cor (1 
p cos 9 


/> o [r 2 {\-M r ) 
Mr p cos 9 
r ( 1 — M r ) J 


- M r ) 
dS 

ret 

dl. 


dS 


( 1 ) 


where co is the ambient sound speed, r is the magnitude 
of the radiation direction vector f from a point on the 
surface to the observer, M r is the Mach number in the 
direction of r, M„ is the Mach number in the direction 
of the inward-facing geodesic normal 0, and 9 is the an- 
gle subtended by the surface normal and the radiation 
vector r (See Fig. 1.). The subscript “ret” denotes eval- 
uation at retarded time r = t — r/c o- This is the source 
time at which a surface pressure fluctuation at the point 
(y i ,3/2,0) made its contribution to the signal detected by 
the observer at time t. Note that y , r, and 9 are pictured 
in observer time in Fig. 1. 

For a far field observer in a low Mach number flow, 
the first integral in Eq. (1) dominates the acoustic signal. 


2 

American Institute of Aeronautics and Astronautics 



This is because the second integral in Bq. (1) is propor- 
tional to 1/r 2 and the third integral is proportional to M. 
Perhaps most significant in regard to the form of Eq. (1) 
is that it is valid, as is, for rotating surfaces. Its predeces- 
sor, Formulation lAf is significantly more complicated 
in its rotational form, and cannot be approximated by 
only one surface integral in the far field for low Mach 
number flows, as can the present formulation. Such a 
significant simplification for far field calculations makes 
Formulation IB more suitable for statistical analysis of 
broadband noise for rotating surfaces. A statistical for- 
mulation based on Eq. (1) was derived in Ref. 10. 

3. Model Problem - Trailin g Edge Tone 

Any noise prediction made with Eq. (1) will be only 
as good as the input surface pressure p(y , r). The cur- 
rent thinking is that such time-dependent pressure data 
would result from experimental measurement or a com- 
putational fluid dynamics (CFD) calculation. However, 
in this section, a simplified analytic expression is used for 
p(y, t ) to serve as a model problem. A result from thin 
airfoil theory 4 will be used to describe the unsteady sur- 
face pressure that is produced by the passage of a single 
frequency disturbance past the trailing edge of a slender 
airfoil. This simple surface pressure formulation will be 
extended to a broadband source model in the following 
section. 

3.1 Surface Pressure from Thin Airfoil Theor y 

The airfoil for this model problem is a rectangular 
flat plate in the plane £3 = 0, undergoing a uniform 
rectilinear motion, as in Fig. 2. The velocity vector 
V = [— U, 0,0] r , where U is a constant subsonic speed. 
The plate’s surface and its boundary, f > 0, are defined 
by the rectangle { — C < x\ <0} x { — b < xz < 6}, 
with the trailing edge at £i = 0. An unsteady pressure 
distribution is assumed on this surface, and is analyt- 
ically prescribed from thin airfoil theory, as discussed 
below. 

Amiet 4 has proposed a formulation to model the re- 
sponse of an airfoil to the passage of a pressure distur- 
bance over its trailing edge. This formulation, formally 
derived in Ref. 12, is based on the theory of a thin airfoil 
of infinite span and models the moving disturbance as 
stationary in the variable Xi — U c t, where U c is the con- 
vection speed of the disturbance. The induced pressure 
jump on the airfoil surface can be written 

AP(ai,t) = 2Pog(xi,kc)e~ rkc( ' xl ~ v ‘ e> (2a) 

where k c = oj/U c is the streamwise convective wave num- 
ber, and Po is the amplitude of the disturbance. The 
factor of two in Eq. (2a) indicates that the pressure is 
assumed to be antisymmetric between the upper and 
lower surfaces, and this expression thereby accounts for 
the pressure on both sides of the airfoil, i.e., the pres- 
sure jump. Note that Eq. (2a) differs from the general 
form for the pressure jump in Ref. 10 because the explicit 
term e ~ lkcX1 [ n Eq. (2a) was incorporated into the trans- 
fer function g in Ref. 10. The formulation in Eq. (2a) is 
used here for consistency with the TE noise research of 
Schlinker and Amiet. 16 


The transfer function g(xi,k c ) is 

g(xi,k c ) = -1 + (1 + i)E~[— xi( k c +//(1 + A/))] (2b) 

where p = Mu>//3 2 U, = \/1 — A/ 2 , and the function 

E* is given by 

E*(0 = r —— du = C(0 - iS(0 (2c) 
Jo (2 tvu)2 

The quantities C(£) and 5(f) are the Fresnel cosine and 
sine integrals, and will be evaluated numerically by the 
formulas derived by Boersma. 18 The final representation 
for the unsteady surface pressure p{yi, r), assumed to be 
a real quantity, is 

P(y i, t) = »{— AP(i/i, r)} (2d) 

The pressure jump is negative in Eq. (2d) because the 
acoustic formulation in Eq. (1) is derived from a form 
of the Ffowcs Williams-Hawkings equation in which the 
unit surface normal <33 is assumed to point into the fluid, 
i.e., in the positive *3 direction on the upper surface, and 
in the negative *3 direction on the lower surface. There- 
fore, using the same positive surface normal on both sides 
of the airfoil, the sum of the pressure on both sides is 
P = Pupper — Power, and this expression is the negative of 
the conventional notion of a pressure jump. 

Note that the transfer function in Eq. (2b) represents 
the effect of the induced surface pressure only, and ne- 
glects the effect of the incident pressure. The neglect of 
the incident pressure field effect is not of concern here, as 
this model problem is presented for illustrative purposes 
only. After the initial derivation of this induced pres- 
sure formulation 4,12 Amiet later altered the formulation 
to include the effect of the incident pressure field.' 7 The 
effects of both induced and incident surface pressure will 
be employed in the broadband formulation in Section 4. 

3.2 Directivity Calculation 

Using Eqs. (2a)-(2d) as the input surface pressure in 
Eq. (1), the directivity of a single frequency source is 
now examined. The flat plate has a chord length C = 0.5 
meter, and a span 26 = 2.0 meters. The flow speed U is 
determined by a free stream Mach number M = 0.2, with 
co = 343 m/s. The disturbance amplitude Po is taken 
as one percent of the dynamic head qo = pou / 2, with 
po = 1.23 kg-m/s , and the convection speed is taken 
to be U c = 0.8 U. The initial surface pressure p(x i,0), 
— C < xi < 0, is shown in Fig. 3. This pressure profile 
represents the surface pressure over the entire span at 
observer time t = 0. Note, again, that the formulation 
in Eqs. (2a)-(2d) represents the induced surface pressure 
only. 

The radiated noise p'(x, t) is calculated at 360 equally 
spaced locations on a circular arc in the plane £2 = 0. 
The radius of this arc emanates from the mid-span lo- 
cation on the trailing edge, as shown in Fig. 4. The 
arc trajectory (r, «6) is determined by r = 2 meters and 
0 < V 7 < 27T. The surface discretization is a uniform grid 
of 100 x 400 surface elements. The directivity is deter- 
mined by the peak pressure amplitude calculated at each 


3 

American Institute of Aeronautics and Astronautics 



position, during one period T = 1// for a frequency of 
2.5 kHz, with 128 timesteps in a period. Fig. 5 shows 
the results, in polar form, where the notation ||p'|| is 
interpreted as 


The results are Fourier analyzed and compared with ex- 
perimental TE noise spectra. 1 ' 

4.1 Ex p eriment Descri p tion 


lb'll = 0 “ t ^ r b'OM)l 

The upstream directivity of the major lobes is consistent 
with the research of previous authors, e.g., Singer, et al 
[ 8 ], 


3.3 Velocit y Scalin g Law 

Attention is now turned to the way in which the in- 
tensity of the far field noise, as predicted by Eq. (1), will 
scale as a function of velocity, when the surface pressure 
is described by Eqs. (2a)-(2d). A scaling law' will be de- 
termined under the assumption that the acoustic source 
is noncompact, i.e. , A < C. Furthermore, the observer 
is assumed to be in the acoustic and geometric far field, 
i.e., r A and r ~S> C, respectivey. 

Because scaling laws are typically determined for low 
Mach number flow's, 3 ’ 19 the Mach number range of inter- 
est is 0.01 < M < 0.2. The surface pressure amplitude 
Po is one percent of the dynamic head. The plate’s phys- 
ical dimensions are the same as in the above directivity- 
problem. The observer is chosen at a distance of 10 me- 
ters, directly above the trailing edge, i.e., x = [0,0, 10] 7 
in meters. The calculations are performed on a 100 x 400 
uniform surface grid. 

The surface pressure in Eqs. (2a)-(2d), with a fre- 
quency of 2.5 kHz, is used as input to equation to Eq. (1) 
to predict the far field sound p'(x,t) to the observer. A 
separate calculation is run for each of 50 equally spaced 
Mach numbers between 0.01 and 0.2. Each calculation is 
performed for one period with 128 timesteps. The aver- 
age intensity I(x) of the acoustic signal at the observer 
x, assuming spherical spreading, is then calculated by 




[p'{x, £)] 2 dt 
poco 


The average acoustic intensities for this test case, as 
a function of Mach number, are represented as circles 
in Fig. 6. The slope of these results on a log-log plot 
can be visually determined by observing their proximity 
to the dotted line w'hose slope is exactly five. This U° 
proportionality is consistent with the result of Ffowcs 
Williams and Hall, 3 as expected from the idealized con- 
ditions placed upon the calculations. 


4. Broadband Predictions 

The analytic surface pressure in the previous section 
is extended to model a broadband trailing edge source 
on a slender airfoil at zero angle of attack. Following 
the approach of Schlinker and Amiet., 16 the surface pres- 
sure correlations required as input are evaluated by flat 
plate theory and by experimental measurements. This 
broadband surface pressure is used as input to Formula- 
tion IB to predict far field radiation in the time domain. 


The experiment that is modeled in this section is re- 
ported by Brooks and Hodgson. 1 ' A NACA 0012 airfoil is 
placed between two plates at zero angle of attack in the 
test section of an open jet wind tunnel. A schematic of 
this experimental setup is shown in Fig. 7. Noise prop- 
agates from the test section into an anechoic chamber 
that is instrumented with microphones. 

The airfoil has a chord length of 0.6096 m and a 
span of 0.46 m. The tunnel speeds of interest here 
are 38.6 m/s and 69.5 m/s. The chord-based Reynolds 
numbers are 1.57 million and 2.82 million, respectively. 
Boundary layer tripping was applied at 15 percent chord 
downstream of the leading edge to ensure a spanwise uni- 
form transition location and a fully developed turbulent 
boundary layer at the trailing edge. 

For radiated noise measurements, eight microphones 
are located in the plane perpendicular to the airfoil 
midspan. The presence of extraneous noise sources pre- 
cludes direct mearurement of TE noise by a single mi- 
crophone. Therefore, to evaluate the TE noise, a cross- 
spectral analysis of pairs of microphones w'as employed 
in a manner consistent with the coherent output power 
method. 20,21 The microphone pictured in Fig. 7 repre- 
sents the location for which the current predictions are 
made, at a distance of 1.22 m directly above the airfoil 
trailing edge. Note that a shear layer forms dowmstream 
of the nozzle lip, between the airfoil and the microphone. 
Although both the directivity and the amplitude of the 
TE noise are affected by refraction through this shear 
layer, the corrections for the microphone at this location 
are small enough to ignore (see Ref. 17). 

4.2 Broadband Anal y sis 

For prediction purposes, the airfoil is modeled as a 
flat plate in order to evaluate the unsteady surface pres- 
sure w'it.h a broadband extension of the analytic formu- 
lation in Section 3. The airfoil geometry is oriented 
with respect to the coordinate axes as in Fig. 2, with 
{ — C < *i < 0 } x { — b < X 2 < b}, where C = 0.6096 
m and 2b = 0.46 m. 

The surface pressure arises from boundary layer tur- 
bulence that is assumed to convect in a frozen pattern 
along the airfoil surface towards the trailing edge. Unlike 
the single frequency source in Section 3, the broadband 
nature of the surface pressure in the present case requires 
consideration of both chordw-ise and spanw'ise wave num- 
bers, k i and fe, respectively. Each Fourier component of 
this broadband surface pressure jump is associated with 
a wave number pair (k\, k-i) and can be written 

AF(/ci,fe; n,X2,l) = (3) 

2P(k u k2)g(x 1 M,k2)e- l[kl{xi - Uct)+k * X2] 

where P(ki, fe) is the amplitude of the pressure jump as- 
sociated with the wave number pair (ki,ko)- Because of 


4 

American Institute of Aeronautics and Astronautics 



the assumed convective nature of the turbulence, the no- 
tation for the chordwise wave number fo will be replaced 
by fo to emphasize its dependence on the convection 
speed U c and to avoid confusion with the conventional 
notion of fo = ui/U. In an exact sense, there are in- 
finitely many combinations of frequency and convection 
speed whose ratio uj/U c yields a given value of fo . How- 
ever, it is assumed here that the acoustically relevant 
structures in the turbulent boundary layer are frozen 
with respect to a single convection speed that is taken 
as U c = 0.8 U. 

The complete broadband spectrum for the surface 
pressure jump AP(xi, x 2 ,t) is obtained by summing all 
Fourier components in Eq. (3): 

AP(xi,X 2 ,t)= (4) 

poo poo 

2 / / P(kc,k 2 )9(x 1 ,k c ,k2)e- ilk ^ 1 - Uct)+k2X2] dkcd^ 

J — ooJ—oo 


A straightforward approach for predicting the desired 
broadband far field measurements is to use the real part 
of Eq. (4) as input to Formulation IB. This approach 
requires knowledge of a two-component surface pres- 
sure spectrum and a dual wave number transfer function 
g(xi, k € , fo). However, because one of the objectives of 
the current work is to reproduce the results of Schlinker 
and Amiet 16 from a time domain perspective, an ap- 
proach similar to that taken in Ref. 16 will be used to 
model the surface pressure. 

The analysis for the general formulation in Ref. 16 
comes from previous work 22 in which Amiet argues that, 
within certain limitations, integration over all spanwise 
wave numbers is not required. His conclusion, derived 
mathematically in the frequency domain, is that only 
one spanwise wave number contributes to the sound de- 
tected by an observer in a given location. In particular, 
Amiet focuses on an observer in a spanwise symmetric 
location, for which only the zero spanwise wave num- 
ber needs to be considered. This result is argued to be 
exact in the limit of infinite span and a good approx- 
imation for an airfoil of finite span that responds to a 
high frequency disturbance. Although Amiet.’s analysis 
was initially presented to derive an acoustic formulation 
for incidenct turbulence noise, the result pertaining to 
spanwise wave numbers is sufficiently general to apply 
to the present trailing edge problem. 

The derivation of Amiet’s analytical result can be gen- 
erally described as follows. First, Eq. (4) is transformed 
into Fourier space. Then, a two-point cross-correlation 
function is formed and related to the far field power spec- 
trum through Kirchhoff ’s formula 23 and Curie’s result.. 19 
In order to follow a similar line of reasoning in the time 
domain, Eq. (4) itself must be related to the far field 
acoustic pressure through Formulation IB. In the case 
of a distant observer directly overhead of a finite-span 
airfoil, the terms r, M r , and 9 in Eq. (1) are weak 
functions of y\ and y 2 on the airfoil surface, and there- 
fore will be considered constants. Furthermore, for the 
observer position considered here, the differences in re- 
tarded time, as a function of airfoil surface location, can 


be neglected. These assumptions are consistent with the 
acoustic model employed by Amiet. 22 For the present 
problem, including the above assumptions, Eq. (1) is ap- 
proximated by 

4 np(x,t)~ (5) 


cos# 

Cor (l -M r ) 


a 


b L 


§=p(y,f) + u-^-p(y,f) 


dy 


where the over-bars on 9, r, and M r denote mean values 
over the airfoil surface, and therefore the retarded time 
t = t — f/co is constant for fixed t. Recall that only the 
first integral in Eq. (1) is significant under the present 
assumptions of a far field observer in a low Mach number 
flow. 


Before the surface pressure p(y, t) is specified, Eq. (5) 
is further simplified. For convenience, the terms 9 and 
M r will be neglected, as they are small (M r ss 0 and 
cos 9 ~ 1) for a distant observer directly above the airfoil. 
With these additional simplifications, if — AP in Eq. (4) 
is substituted for p(y, t) in Eq. (5), the far field acoustic 
pressure can be approximated in the form 


2 f'C r-b poo poo 

4np’(x,t )& — z / / / / F(yi,k c , ki) 

c 'o r J 0 J_ b J_ oa J_ ao 


x e -'^-Ucf)+k 2 y 2 ] dk2dkcdy2dyi ( 6a ) 


where 


T{y\ , fo, fo) = 


(6b) 


-P(k c . fo) 


d 

ik c (U c - U)g(yi,k c ,k 2 )+U—g(yi,k c ,k 2 ) 


Sufficient conditions 24 on AP and its derivatives have 
been assumed for the commutation of integration and 
differentiation in Eqs. (6a) and 6(b). The y 2 integration 
in Eq. (6a) can now be explicitly evaluated, yielding 


4np(x, t) : 


co 


o „ . 

’ J o J — OG J — O 


2 sin (fob) 
fo 


T{yi, fo, fo) 


x e- ikAyi - u ‘ f ) dk 2 dkcdyi (7) 

Integrating with respect to fo, the term sin(fob) /fo acts 
like a Dirac delta function when integrating over an un- 
bounded domain, and the result is 

9 /'C poo 

4irp'(x, t.) ss — = / / 2nJ r (yi,kc,0)e~' kc(vi ~ UcT) dk c dyi 
co r J 0 J-oo 

( 8 ) 

Eq. (8) indicates that only the zero spanwise wave 
number contributes to the noise detected by the far field 
observer. Eqs. (7) and (8) are time domain analogies 
to Eqs. (15) and (17) in Ref. 22. Furthermore, Eq. (8) 
suggests that the acoustic source p(y, r) in Eq. (1) can be 
evaluated as the real part of a simplified pressure jump: 

/ OO 

P(k c , 0)g( Xl ,k c ,0)e~ ik ‘ (xi ~ Uct) dk c 


p(y,r ) = S{-AP(t/i, r)} 


( 9 ) 


5 

American Institute of Aeronautics and Astronautics 



Note that, when performing the actual calculation, 
the y-i integration will be explicitly performed when Eq. 
(9) is input to Eq. (6). Only the k 2 integration will be 
neglected. Furthermore, the evaluation of the surface 
pressure terms at retarded time r = t — r/co will be ex- 
ecuted in an exact fashion, as prescribed by Eq. (1). All 
three integrals in Eq. (1) will be evaluated for the predic- 
tions that follow, although the first integral is expected 
to dominate the signal. 

The evaluation of the surface pressure in Eq. (9) is 
accomplished by first recognizing the turbulent fluctu- 
ations as a stochastic process. This process can be 
approximated by a truncated series whose limit exhibits 
the required relationship between the autocorrelation 
and the power spectrum of that process (e.g., Ref. 25). 
This relationship is achieved by evaluating the pressure 
amplitudes P(k c ,k 2 ) as a function of 'hpp, the power 
spectral density (PSD) of the surface pressure. To this 
end, the infinite wave number domain, —00 < k c < 00 , 
in Eq. (9) is integrally discretized and truncated such 
that k c ,-N < k c , n < k c ,N . The largest convective wave 
number fe c ,jv represents an “upper cutoff” wave number, 
beyond which the surface pressure amplitude P(k c , 0) is 
considered negligible or is out of range of experimental 
measurement. The unsteady surface pressure jump in 
Eq. (9) is then approximated by 

N 

AP(xi,t) A u ,oe i ' l ’ n g(xi,kc,n,0)e~ ikc n(xi ~ Uct) 

n=-N 

(10a) 

kc,n = n A k c , n = 0, ±1, ±2, . . . , ±N 
A k c = k c ,N/N 

The discrete surface pressure amplitudes {.4„,o} are 
evaluated by 

A n , 0 = [$pp (&.,„, 0) Afc t ]i (10b) 

where <&pp(k c ,k2) is the two-component PSD of the sur- 
face pressure. Amiet 16 argues that the required single 
wave number spectrum <&p.p(fc t , 0) can be evaluated by 

$pp(fe, 0) = — 4 2 (w)S<, g (w,0) (10c) 

7T 

where t X2 (w) is the spanwise correlation length and 
(u,Ax 2 ) is the spanwise surface pressure correlation 
function. 

The phase angles {<j>n} are independent random vari- 
ables uniformly distributed on [0, ’2tv]. The transfer 
function in Eq. (2b) can be used for g(x\,k c , n , 0) with the 
following modification. As previously noted, Eqs. (2a) 
and (2b) represent the induced pressure jump. Amiet. 5 
has suggested that the incident pressure, i.e., that which 
results from turblent eddies that contact the trailing 
edge, can be accounted for by the addition of an ex- 
ponential convergence factor of the form e ekcXl , where 
£ is a positive parameter. For —C < xi < 0, this ad- 
ditional term will be significantly larger than zero only 
in the immediate vicinity of the trailing edge, provided 
that ek c C is large. Therefore, to include the effect of the 
incident pressure, the transfer function to be used in Eq. 


(10a) is the two-component function g(x i,k c ,ko) in Ref. 
16, with k -2 = 0. 

g (a i,fc c ,0) = e ekaX1 - 1 + (1 + i)E*[-x\{k c +g.{l+M))\ 

(lOd) 

where E* is the same complex combination of Fresnel 
integrals as in Eq. (2c). Amiet 5 was able to avoid the di- 
rect use of the parameter e because of the manner in 
which the transfer function in Eq. (lOd) was used in 
his analysis. Amiet used the transfer function to de- 
fine an unsteady lift response function that involved the 
chordwise integration of the transfer function with other 
terms. The result of this integration yields an expres- 
sion that, upon clever manipulation of limits, does not 
contain e but still provides an additional term to the lift 
response function that accounts for the incident pres- 
sure. In the present case, the transfer function in Eq. 
(lOd) must be explicitly used and therefore a value for 
t must be specified. This value e = 1.5 is chosen for 
reasons that are discussed in the following subsection. 

To illustrate the effect of including this incident pres- 
sure term in the surface pressure formulation, Fig. 8 
shows the same single frequency surface pressure case 
in Section 3, with and without the the incident pres- 
sure term. Clearly, the incident pressure term has a 
significant effect only near the trailing edge, as expected. 
However, for a given frequency, differing values of e will 
result in differing amounts of upstream chordlength to be 
so affected. Note that the addition of this term causes 
the pressure jump to vanish at the trailing edge for all 
time, i.e., the Kutta condition is satisfied. Note also 
t.he increase in spatial oscillation that is caused near the 
trailing edge when the effect of this incident pressure 
term is included. 

Eqs. (10a)-(10d) represent the the complex- valued 
broadband surface pressure formulation to be used for 
the present TE noise predictions. The final representa- 
tion for the unsteady broadband pressure on the airfoil 
surface is then given by the real part of —A P in Eq. 
(10a). Using symmetry arguments and algebraic ma- 
nipulation, the indicial bounds for the surface pressure’s 
spectral representation are altered so that the domain in- 
cludes only positive wave numbers. As input to Eq. (1), 
the resulting real-valued surface pressure can be written 


p(lJU t) = -47T ^ -4»,o{ B n COs[fc c ,„(f/l — U C T ) + (j>n 


+ D n sin[fc c , n (j/i - U c t ) + <j>„] } (11a) 


where 


A, t ,o — 


— 4(tu„) S qq (uj n , 0) A k c 
7 r 


(11b) 


Bn = c «*‘.«» 1-1+C(£ b ) +$(£„) (11c) 

Dn = C(4)-«S(4) (lid) 

£,n = -yi[kc,n +Mn(l + M)\ (lie) 


and C(4) and S(£ i „) are the Fresnel cosine and sine inte- 
grals in Eq. (2c). Specific evaluations for the correlation 


6 

American Institute of Aeronautics and Astronautics 



lengths ti and surface pressure correlations S m will be 
discussed in the following section. 


4.3 Time Domain Predictions 

The lower and upper frequency bounds for both cal- 
culations are 25 Hz and 10 kHz, respectively. Therefore, 
/ = 25 Hz also serves as the fundamental frequency and 
the numerical bandwidth A/. Each calculation is per- 
formed for one period of the lowest frequency, T = 0.04 
s. The numerical solution is sampled at the Nyquist fre- 
quency, i.e. , Af = T/2N. The calculation is performed 
on a 500 x 100 surface grid with grid-point clustering 
near the trailing edge, as shown in Fig. 9. This trailing 
edge clustering allows for better resolution to account for 
the effect of the incident pressure term in Eq. (lOd), as 
previously shown in Fig. 8. As in the constant frequency 
case, because the surface pressure in Eqs. (lla)-(lle) 
is cast in only one spatial variable y%, and the observer 
location is symmetric relative to the airfoil span, the 
acoustic predictions are found to be relatively insensitive 
to the discretization in y 2 , and the primary concern for 
grid resolution is in the streamwise direction. With 500 
points in the streamwise direction and clustering near 
the trailing edge, a sufficient resolution of at least 10 
points per wavelength was obtained for the entire length 
of the chord. This conclusion was reached by inspection 
of surface pressure profiles for the highest frequency of 
10 kHz. 

The coordinate system for the calculation is such that 
the 212 -axis coincides with the center span line, so that 
the microphone position is in the plane X 2 = 0. The 
experimental microphone position for which comparisons 
are made is at a distance of 1.22 m from the model, 
and at an angle of 90 degrees relative to the chord and 
directly above the trailing edge. The measured observer 
position for the prediction is, then, x = [0, 0, 1.22 ] T in 
meters. 


The baseline prediction case under consideration is for 
a tunnel speed of U = 69.5 m/s. This flow condition, the 
above observer location, and airfoil geometry are incor- 
porated into an acoustic prediction using Eq. (1) with the 
surface pressure defined by Eqs. (lla)-(lle). Following 
the example of Schlinker and Amiet, 16 a first-cut pre- 
diction is performed using flat plate theory to evaluate 
the surface pressure correlations S m (ca,0) and spanwise 
corelation lengths £o(u) in Eq. (lib). The authors used 
empirical formulations for these quantities that they de- 
rived from previous analysis and boundary layer mea- 
surements of Corcos 26 and Willmarth and Roos. 2 ' The 
surface pressure correlations are approximated by 

x* 2 x 10" 5 

5w ( W, 0) ~ q ° U 1+CC + 0.217th 2 + 0.00562 th 4 

(12a) 

where go = poV 2 /2, 5 * is the trailing edge displacement 
thickness, and th = a >5*/U. The displacement thickness 
is also taken from a flat plate approximation for turbu- 
lent boundary layer thickness 5 on a flat plate, based on 
the chord Reynolds number Rec, i.e., 


5 0.37 c 5 * 

C C 


(12b) 


For the experiment of Brooks and Hodgson, 1 ' 
Schlinker and Amiet 16 used Eq. (12b) to compute the 
boundary layer thickness 5 and accounted for boundary 
layer tripping by taking the 15 percent chord station as 
the initial point of the calculation. Surface curvature 
was also accounted for in the downstream distance used 
in the calculation. The ratio 5/C used by Schlinker and 
Amiet for this experiment was reported as 0.0166 for U = 
69.5 m/s and 0.0187 for U = 38.6 m/s. The displacement 
thickness was then taken as 1/8 of the boundary layer 
thickness. The expression that Schlinker and Amiet) 6 
suggest for the spanwise correlation length is 

£a(u) » (12c) 

U) 


Fig. 10 shows the far field signal p'(x,t) that is pre- 
dicted by Formulation IB at the experimental micro- 
phone location, for a tunnel speed of 69.5 m/s. The 
surface pressure is modeled with Eqs. (lla)-(lle) and 
(12a)-(12c). The time signal p'(x,t) is Fourier ana- 
lyzed to determine a discrete set of spectral amplitudes 
{Pn}n=i- The far field sound pressure level (SPL) spec- 
trum is calculated by 


SPL(/„) = 20 log 


Pn_' 
Prrf . 


71 = 1,2,..., A (13) 


where the reference pressure is P re f = 20 yuPa. The SPLs 
are converted to a 1.0 Hz bandwidth by reducing the 
values in Eq. (13) by 10 log(A/). 


The resulting narrowband SPLs are compared with 
the prediction of Schlinker and Amiet 16 in Fig. 11. Also 
on this plot are the narrowband SPLs that were ex- 
perimentally measured by Brooks and Hodgson. 1 ' The 
predicted results of Schlinker and Amiet and the mea- 
surements of Brooks and Hodgson were obtained by 
digitizing the appropriate plots in Figure 34 of Ref. 16. 
Various values of the paramenter e in Eq. (lOd) were as- 
sessed in this comparison stage of the research. With 
an arbitrary parameter in the formulation, the value e 
= 1.5 was chosen for its agreement with the flat plate 
correlation results of Schlinker and Amiet. 16 This value 
of e is held fixed at 1.5 for all remaining calculations. 


Clearly, Fig. 11 shows that significant error exists be- 
tween the predictions and the measurements when flat 
plate formulations are used for the required surface pres- 
sure correlations. Fig. 12 sheds light on this error with a 
comparison of the flat plate formula in Eq. (12a) and the 
measured surface pressure correlations of Yu and Joshi. 
The notation S m denotes that the surface pressure cor- 
relations are normalized by go 5* [U. The measured data 
in Fig. 12 were obtained by digitizing the “average” plot 
in Figure 35(a) of Ref. 16. The normalized flat plate 
surface pressure correlations are significantly lower than 
the measured data, by as much as 7 dB. The reason that 
the flat plate approximation is so much in error is only in 
small part because of the the lack of pressure gradient. 
The most significant error made in the approximation 
in Eq. (12a) is the lack of a trailing edge; this empir- 
ical formulation is based on experimental measurement 
and analysis in which the flat plate is assumed to be infi- 
nite. Clearly, surface pressure correlations that are based 


7 

American Institute of Aeronautics and Astronautics 



on flat plate theory are inappropriate for predicting TE 
noise in this case. 

The experimental surface pressure correlations 28 in 
Fig. 12 will now be used in the surface pressure formu- 
lation to predict the TE noise associated with the two 
tunnel speeds of interest and compared with experimen- 
tal measurements. The modified formula for the surface 
pressure correlations is 

S qq (u,Q) « go jj S vq {u, 0) (14) 

where S M (w.O) denotes the normalized measured data 
in Fig. 12. The tabulated data obtained from digitiz- 
ing this information from Ref. 16 is stored in a file that 
is accessed and interpolated to obtain S qv (io, 0) for any 
frequency. Having altered the surface pressure correla- 
tion function by experimental data, the evaluation of the 
spanwise correlation length is now brought into question. 
However, it was concluded by Brooks and Hodgson 17 
that the function £a(w) for a flat plate and a thin airfoil 
are identical under suitable normalization. Therefore, 
the use of Eq. (12c) for £2 (w) will be retained for the 
remaining calculations. 

The predicted and measured far field SPLs for the 
two tunnel speeds are shown in Fig. 13. The experi- 
mental data in Fig. 13 were obtained by digitizing the 
measurements plotted in Figure 34 of Reference 16. The 
agreement with the measured data is significantly im- 
proved when the calculation includes surface pressure 
correlations that account for the trailing edge of an air- 
foil. In fact, it was concluded by Schlinker and Amiet 16 
that airfoil surface pressure correlations were absolutely 
necessary for realistic TE noise predictions. 

Concludin g Remarks 

The prediction of broadband trailing edge noise from 
rotating machinery and airframes is currently the sub- 
ject of intense research in aeroacoustics. The physics of 
broadband noise generation are well understood as the 
result of the pioneering research of Howe, 14 ’ 29 ' 30 Amiet 
and coworkers, 4 ’ 5 ’ 12 ' 10 ’ 22 and Brooks and coworkers. 17 ’ 31 
The previous work of these aeroacoust.icians, and many 
others, has clearly demonstrated that any successful 
broadband loading noise prediction requires an under- 
standing of two physical processes: the character of the 
time-dependent surface pressure that provides the acous- 
tic source, and the manner in which that source gives rise 
to an acoustic signal. 

Obtaining the fluctuating surface pressure distribu- 
tion analytically, numerically, or experimentally is itself 
a difficult problem. For this reason, past researchers have 
most often resorted to modeling the surface pressure, us- 
ing guidance from experiments to aid in the development 
of these models. Today, high resolution surface pressure 
fluctuations can be obtained from turbulence simula- 
tions in realistic situations where the airfoil geometry 
and kinematics are accurately modeled. Therefore, the 
improvement of the acoustic radiation model becomes an 
important research topic. In the past, acoustic radiation 
models were most often developed for airfoils in uniform 


rectilinear motion. In addition, other restrictive assump- 
tions, such as far field positioning of the observer, were 
often used to simplify the acoustic analysis. 

The present work further develops a simple and gen- 
eral acoustic result in the time domain, based on the 
solution of the loading noise term of the Ffowcs Williams- 
Hawkings equation. This new: solution, called Formula- 
tion IB, is, to date, the simplest analytical result for the 
prediction of loading noise and is suitable for statistical 
analysis of broadband noise for a surface in general mo- 
tion. The new formulation has been validated with time 
domain calculations that predict trailing edge noise on a 
NACA 0012 airfoil in a low r Mach number flow. The time 
domain predictions are found to be in excellent agree- 
ment with the frequency domain predictions of Schlinker 
and Amiet 16 as well as with the experimental measure- 
ments of Brooks and Hodgson. 17 These results are, to 
the authors’ knowledge, the first successful broadband 
trailing edge noise predictions in the time domain. 

The authors advocate the use of time domain meth- 
ods in the prediction of broadband noise. Because of 
the decoupling of the aerodynamics from the acoustics, 
the chief advantage of time domain methods is their po- 
tential for direct use of time-dependent surface pressure 
statistics from experiments or computer simulations. 

Acknowled g ements 

The authors would like to express their gratitude to 
Dr. Roy K. Amiet, whose input from his past experience 
was invaluable to this research. The authors are also 
grateful to Dr. Meelan Choudhari of NASA Langley Re- 
search Center for several enlightening discussions on the 
subject of stochastic modeling. 

References 

1. Howe, M. S., “A Review: of the Theory of Trailing Edge 
Noise,” Journal of Sound and Vibration, Vol. 61, 1978, 
pp. 437-465. 

2. Lighthill, M. J., “On Sound Generated Aerodynam- 
icallv. I. General Theory,” Proceedings of the Royal 
Society of London, A 211, 1952, pp. 564-587. 

3. Ffowcs Williams, J. E. and Hall, L. H., “Aerodynamic 
Sound Generation by Turbulent Flow in the Vicinity of 
a Scattering Half-Plane,” Journal of Fluid Mechanics, 
Vol. 40, 1970, pp. 657-670. 

4. Amiet, R. K., “Noise Due to a Turbulent Flow Past a 
Trailing Edge,” Journal of Sound and Vibration, Vol. 
47, No. 3, 1976, pp. 387-393. 

5. Amiet, R. K., “Effect of the Incident Surface Pressure 
Field on Noise Due to a Turbulent Flow Past a Trailing 
Edge,” Journal of Sound and Vibration, Vol. 57, No. 
2, 1978, pp. 305-306. 

6. Goldstein, M. E., “Scattering and Distortion of 
the Unsteady Motion on Transversely Sheared Mean 
Flows,” Journal of Fluid Mechanics, Vol. 91, No. 4, 
1979, pp. 601-632. 

7. Ffowcs Williams, J. E. and Hawkings, D. L., “Sound 
Generation by Turbulence and Surfaces in Arbitrary 


8 


American Institute of Aeronautics and Astronautics 



Motion,” Philosophical Transactions of the Royal So- 
ciety , A 264, 1969, pp. 321-342. 

8. Singer, B. A., Brentner, K. S., Lockard, D. P., and 
Lilley, G. M., “Simulation of Acoustic Scattering from 
a Trailing Edge,” Journal of sound and Vibration, Vol. 
230, No. 3, 2000, pp. 541-560. 

9. Farassat, F. and Succi, G. P., “The Prediction of Heli- 
copter Rotor Discrete Frequency Noise,” Vertica, Vol. 
7, No. 4, 1983, pp. 309-320. 

10. Casper, J. and Farassat, F., “ Broadband Noise Pre- 
dictions Based on a New Aerocoustic Formulation,” 
AIAA Paper No. 2002-0802. 

11. Paterson, R. W., and Amiet, R. K., “Noise and Surface 
Pressure Response of an Airfoil to Incident Turbu- 
lence,” AIAA Journal of Aircraft, Vol. 14, No. 8, 1977, 
pp. 729-736. 

12. Amiet, R. K., “High Frequency Thin-Airfoil Theory 
for Subsonic Flow,” AIAA Journal, Vol. 14, No. 8, 
1976, pp. 1076-1082. 

13. Willmarth, W. W. and Roos, F. W., “Reslution and 
Structure of the Wall Pressure Field Beneath a Tur- 
bulent Boundary Layer,” Journal of Fluid Mechanics, 
Vol. 22, 1965, pp. 81-94. 

14. Corcos, G. M., “The Structure of the Turbulent Pres- 
sure Field in Boundary Layer Flows,” Journal of Fluid 
Mechanics, Vol. 18, 1964, pp. 353-378. 

15. Yu, J. C. and Joshi, M. C., “On Sound Radiation from 
the Trailing Edge of an Isolated Airfoil in a Uniform 
Flow,” AIAA Paper No. 79-0603, 1979. 

16. Schlinker, R. H., and Amiet, R. K., “Helicopter Rotor 
Trailing Edge Noise,” NASA Contractor Report No. 
3470, 1981. 

17. Brooks, T. F., and Hodgson, T. H., “Trailing Edge 
Noise Prediction from Measured Surface Pressures,” 
Journal of Sound and Vibration, Vol. 78, No. 1, 1981, 
pp. 69-117. 

18. Boersma, J., “Computation of Fresnel Integrals,” 
Mathematics of Computation, Vol. 14, 1960, p. 380. 

19. Curie, N., “The Influence of Solid Boundaries on Aero- 
dynamic Sound,” Proceedings of the Royal Society of 
London, A 231, 1954, pp. 505-514. 

20. Halvorsen, W. G. and Bendat, J. S., “Noise Source 
Identification Using Coherent Output Power Spectra,” 
Journal of Sound and Vibration, Vol. 9, 1975, pp. 15- 
24. 

21. Piersol, A. G., “Use of Coherence and Phase Data 
Between Two Receivers in Evaluation of Noise Envi- 
ronments,” Journal of Sound and Vibration, Vol. 56, 
pp. 215-228. 

22. Amiet, R. K., “Acoustic Radiation from an Airfoil in a 
Turbulent Stream,” Journal of Sound and Vibration, 
Vol. 41, 1975, pp. 407-420. 

23. Lamb, H., Hydrodynamics, Dover Publications, New 
York, 6th edition, p. 501. 

24. Courant, R. and John, F. Introduction to Calculus and 
Analysis, Vol. 2, Springer-Verlag, New York, 1989. 


25. Shinozuka, M. and Deodatis, G., “Simulation of 
Stochastic Processes by Spectral Representation,” Ap- 
plied Mechanics Review, Vol. 44, No. 4, 1991, pp. 

191-204. 

26. Corcos, G. M., “The Structure of the Turbulent Pres- 
sure in Boundary Layer Flows,” Journal of Fluid Me- 
chanics, Vol. 18, 1964, pp. 353-378. 

27. Willmarth, W. W. and Roos, F. W., “Resolution and 
Structure of the Wall Pressure Field Beneath a Tur- 
bulent Boundary Layer,” Journal of Fluid Mechanics, 
Vol. 22, 1965, pp. 81-94. 

28. Yu, J. C. and Joshi, M. C., “On Sound Radiation from 
the Trailing Edge of an Isolated Airfoil in a Uniform 
Flow,” AIAA Paper No. 79-0603, 1979. 

29. Howe, M. S., “Trailing Edge Noise at Low Mach Num- 
bers,” Journal of Sound and Vibration, Vol. 225, No. 
2, 1999, pp. 211-238. 

30. Howe, M. S., “Trailing Edge Noise at Low Mach Num- 
bers, Part 2: Attached and Separated Flows,” Journal 
of Sound and Vibration, Vol. 234, No. 5, 2000, pp. 761— 
775. 

31. Brooks, T. F., Pope, D. S., and Marcolini, A. M., “Air- 
foil Self-Noise and Prediction,” NASA RP-1218, July, 
1989. 


9 

American Institute of Aeronautics and Astronautics 



Bd ‘(o ,l K)d 



Fig. 4 Schematic for directivity calculation. Observer 
on circular path in plane X 2 = 0. 



Fig. 2 Schematic for the constant-frequency trailing 
edge noise problem in Section 3. 



90° 



Fig. 5 Directivity for a constant frequency source of 2.5 
kHz; observer distance 2 m; spacing between concentric 
circles on grid represents 0.25 Pa. 


Fig. 3 Initial surface pressure profile using Eqs. (2a)- 
2(d). 


10 

American Institute of Aeronautics and Astronautics 


Acoustic Intensity 



Fig. 6 Velocity scaling properties as determined by 
Formulation IB and the surface pressure in Eqs. (2a)- 
(2d). 



Fig. 8 Effect of incident pressure term on initial surface 
pressure profile using Eqs. (2a)-2(d), and modified with 
Eq. (lOd). 


microphone | 



Fig. 7 Schematic for trailing edge noise experiment of 
Brooks and Hodgson. 1 ' 


Fig. 9 Surface grid for prediction of experiment of 
Brooks and Hodgson; 1 7 every fourth point of in each di- 
rection is shown. 



11 

American Institute of Aeronautics and Astronautics 



Fig. 10 Predicted far-field signal, U = 69.5 m/s; mi- 
crophone at 90 u , 1.22 m above trailing edge. 



Fig. 11 Predicted and measured far field noise spectra, 
using surface pressure correlations from flat plate theory; 
U = 69.5 m/s; frequency domain prediction from Ref. 16; 
experimental data from Ref. 17. 



Fig. 12 Normalized surface pressure correlations; flat 
plate theory from Ref. 16; experimental measurements 
from Ref. 28. 



Fig. 13 Predicted and measured far-field noise spec- 
tra; predictions obtained with measured surface pressure 
correlations (Ref. 28); experimental SPLs from Ref. 17. 


12 

American Institute of Aeronautics and Astronautics