1.
Trailing-edge scattering of spanwise-coherent
structures
Alex Sano∗
, Petrˆonio A. S. Nogueira†
, Andr´e V. G. Cavalieri‡
Instituto Tecnol´ogico de Aeron´autica, S˜ao Jos´e dos Campos, SP, 12228-900, Brazil and
William R. Wolf§
Universidade Estadual de Campinas, Campinas, SP, 13083-970, Brazil
We study mechanisms of noise generation by an airfoil at zero angle of attack using
flow-acoustic correlations. We use a large-eddy simulation (LES) of the compressible flow
around a NACA0012 airfoil for M∞ = 0.115 and Rec = 408000. With this simulation we
analyse flow fluctuations around the airfoil, such as pressure and velocity, and relate them
to the far-field sound using standard correlation techniques. Similar to the results of Cava-
lieri et al.,3
who have obtained more significant flow-acoustic correlation for turbulent jets
when the axisymmetric mode was isolated, we have noticed much higher correlations in
the present problem when the two-dimensional mode, i.e. spanwise-averaged fluctuations,
was isolated from the turbulent flow and subsequently correlated to the acoustic pressure.
This result is justified theoretically by an analysis of the tailored Green’s function for a half
plane, where we find that a necessary condition for trailing edge scattering is |kz| < k, where
kz is the spanwise wavenumber of turbulent disturbances and k is the acoustic wavenum-
ber. Two-dimensional perturbations, associated to kz = 0, are always radiating. Isolation
of spanwise-coherent disturbances is thus a means of filtering non-radiating structures.
Another feature that tends to increase correlation coefficients is the use of the difference
between fluctuations in the upper and lower surfaces of the airfoil, which again is in line
with theory: when disturbances are in phase opposition between the two sides of the airfoil
acoustic scattering is maximal.
Nomeclature
δ Boundary-layer thickness
xi Cartesian coordinate of observer
yi Cartesian coordinate of source
L Characteristic length
c Chord
gij
Contravariant metric tensors
ui
Contravariant velocity components
gij Covariant metric tensors
ρ Density
δ∗
Displacement-thickness
R Distance between the source and observer
R Distance between the source image and the observer
ω Frequency
G Green’s function
qj Heat flux
∗Msc. Student, Divis˜ao de Engenharia Aeron´autica, Instituto Tecnol´ogico de Aeron´autica.
†MSc. Student, Divis˜ao de Engenharia Aeron´autica, Instituto Tecnol´ogico de Aeron´autica.
‡Assistant Professor, Divis˜ao de Engenharia Aeron´autica, Instituto Tecnol´ogico de Aeron´autica, AIAA Member
§Assistant Professor, Faculdade de Engenharia Mecˆanica, Universidade Estadual de Campinas, Campinas, SP, Brazil
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21st AIAA/CEAS Aeroacoustics Conference
22-26 June 2015, Dallas, TX
AIAA 2015-2384
Copyright © 2015 by Alex Sano. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
AIAA Aviation

2.
He Helmholtz number
Tij Lighthill tensor
M∞ Mach number
∆˜x Non-dimensional distance
n Surface normal vector
xi Observer position
vφ Phase velocity
Pr Prandtl number
p Pressure
˜p Fourier Transform of pressure
r Radius
γ Ratio of specific heats
Re Reynolds number
SG Lighthill’s tensor in the frequency domain
ao Sound speed
yi Source position
Pi Dipole strength
S Surface
T Temperature
E Total energy
t Time
˜t Non-dimensional time
u u-component of velocity in a Cartesian system
v v-component of velocity in a Cartesian system
w w-component of velocity in a Cartesian system
τij Viscous stress tensor
V Volume
λ Wavelength
k Wavenumber
I. Introduction
Noise emitted by airfoils and wings occurs due to the interaction of turbulence with the neighbouring
solid surfaces. This kind of problem is fundamental for aeroacoustics, and there are several applications
where this phenomenon is relevant, among which the design of low-noise aerodynamic profiles, rotor blades
and fans.
Brooks, Pope and Marcolini2
classify five different mechanisms of sound generation by an airfoil, namely:
turbulent boundary layer convected past the trailing edge noise; separation stall noise; laminar boundary
layer leading to vortex shedding noise; tip vortex formation noise; and trailing edge bluntness vortex shedding
noise. Most of the cases above correspond to the interaction between trailing edge and flow, which is a result
of the end of the profile inducing an abrupt change of boundary condition for pressure and/or velocity.
This was recognised early by Ffowcs Williams and Hall, who showed theoretically that a trailing edge
scatters the neighbouring turbulent fluctuations, leading to significant sound radiation. This phenomenon
has since been labelled trailing-edge noise. Unlike sound emission by compact surfaces, which tends to be
dominated by lift fluctuations as shown by Curle,6
trailing-edge noise approaches the non-compact limit,
modelled by Ffowcs Williams and Hall as a half plane with eddies on the vicinity of its edge. Follow-up
modelling work by Amiet1
considered that trailing-edge noise is due to induced dipoles in the vicinity of the
edge, due to advected turbulence, considered in statistical terms to remain stationary as it is advected.
The objective of the present work is to use flow simulation data to study in more detail the relevant
sound-production mechanisms in the flow around an airfoil, in particular with a view to extracting the
relevant turbulent scales to the trailing-edge noise problem. To do this we study the large-eddy simulation
(LES) of Wolf, Azevedo and Lele,13
of a NACA0012 airfoil at zero angle of attack, Mach number M∞ = 0.115
and Reynolds number based on the airfoil chord equal to Rec = 408000. We use the full simulation data
to calculate flow-acoustic correlations in order to examine the relevant fluid motions related to acoustic
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3.
radiation.
Some previous work have also used numerical simulations to study trailing-edge noise. As shown by
Oberai, Roknaldin and Rughes9
using an LES, the source of acoustic noise in major part is concentrated on
trailing edge due the interaction of the turbulent flow and the end of the profile. Sandberg and Sadham10
have conducted a direct numerical simulation (DNS) to investigate the interaction of the turbulent flow past
a trailing edge and shown that two-dimensional theory for low frequencies predicts the far field acoustic
pressure due the significant spanwise coherence of pressure disturbances, showing that in this case the noise
generation is predominately two-dimensional.
A theoretical analysis, using the tailored Green’s function for a trailing edge derived by Ffowcs Williams
and Hall, guides our investigation. We show in section II that only spanwise-elongated structures radiate
to the far-field, and subsequently we examine in section III how two-dimensional motions around the airfoil
are related to the far-field sound. In addition, we study what the influence on flow-acoustic correlations
when the quantities are manipulated doing a delta between the upper and lower surface, in other words, the
difference of the properties between upper and lower surface of the airfoil.
II. Theory
A. Basic equations
We review here the fundamental theory of noise generation by an airfoil. Lighthill8
rearranged the equations
of motion of a fluid in a way to obtain an inhomogeneous wave equation, with a source term called Lighthill
tensor, Tij:
∂2
ρ
∂t2
− a2
o
2
ρ =
∂2
Tij
∂xi∂xj
(1)
Tij = ρvivj + pij − a2
oρδij (2)
The solution of equation 1 neglecting solid boundaries makes use of the free-field Green’s function to the
problem, defined as
∂2
G
∂t2
− a2
o
2
G = δ(x − y)δ(t − τ) (3)
where the Green’s function G should satisfy the Sommerfeld boundary condition and also a causality condi-
tion. Manipulation of eqs. (1) and (3) leads to
ρ − ρo =
1
4πa2
o
∂2
∂xi∂xj
V
Tij(y, t − |x−y|
ao
)
|x − y|
dV (4)
The previous equation does not account for solid boundaries to the problem, which are relevant for
trailing-edge noise. Extension to Lighthill’s analogy accounting for surface effects are presented by Curle6
and Ffowcs Williams and Hall.12
Curle’s theory employs a more general resolution to equation 1, as follows
ρ − ρo =
1
4πa2
o
v
∂2
Tij
∂xi∂xj
dt
|x − y|
+
1
4π
S
1
r
∂ρ
∂n
+
1
r2
r
n
ρ +
1
aor
∂r
∂n
∂ρ
∂t
dS(y) (5)
With further manipulation, Curle rewrote the solution as
ρ − ρo =
1
4πa2
o
∂2
∂xi∂xj
V
Tij(y, t − r
ao
)
r
dy −
1
4πa2
o
∂
∂xi
S
Pi(y, t − r
ao
)
r
dS(y) (6)
where
Pi = −njpij (7)
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4.
The term Pi corresponds to a surface distribution of dipoles and nj is the direction outward over the airfoil
surface.
Without knowledge of the normal tension Pi on the surface (approximately equal to the pressure for high-
Re flows), eq. (6) is actually an integral equation, which should be solved numerically using a boundary
element method. However, a simplification is possible if instead of the free-field Green’s function G in eq.
(3) one manipulates the corresponding equations using a tailored Green’s function Gt that satisfies the same
boundary conditions as the pressure. For the trailing-edge noise problem, the Green’s function in a domain
bounded by a rigid, semi-infinite flat plate was derived by Ffowcs Williams and Hall12
in the frequency
domain as
Gt(x, y, ω) =
e
1
4 iπ
√
π
e−ikR
R
uR
−∞
e−iu2
du +
e−ikR
R
uR
−∞
e−iu2
du , (8)
where uR and uR are given by
uR = 2
krr0
D + R
1
2
cos
θ − θ0
2
, (9)
uR = 2
krr0
D + R
1
2
cos
θ + θ0
2
, (10)
R is the separation between source (r0, θ0, z0) and observer (r, θ, z) and R is the distance between the
source’s image (r0, −θ0, z0) and the observer in cylindrical coordinates (z and z0 are equivalent to x3 and
y3, respectively). The far-field assumption for the observers was taken into account and the relationship
between the two coordinate systems can be seen in Figure 1.
Trailing
Edge
∞
∞
∞
z0
θ0
r0
θ
r
Plate
z
x1
y1
x2
y2
x3
y3
Figure 1. Relationship between different coordinate systems of the problem
The distances R, R and D can be obtained in cylindrical coordinates as
R = r2
+ r2
0 − 2rr0 cos (θ − θ0) + (z − z0)2
1
2
, (11)
R = r2
+ r2
0 − 2rr0 cos (θ + θ0) + (z − z0)2
1
2
, (12)
D = (r + r0)2
+ (z − z0)2
1
2
. (13)
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5.
Using the hypothesis of source near the plate (kr 1, i.e. the distance between trailing edge and source
is much smaller than the acoustic wavelength), the leading-order Taylor-series approximation of the Green’s
function (labelled here as Gs) is shown by Ffowcs-Williams and Hall to be
Gs(x, y, ω) =
2e
1
4 iπ
√
π
2krr0
r2 + (z − z0)2
1
2
cos
θ
2
cos
θ0
2
e−ikR
R
(14)
Using either form of the tailored Green’s function, Gt or Gs, the far-field sound at frequency ω can be
obtained as
ˆp(x, ω) =
1
4π y1 y2 y3
SG(y, ω)G(x, y, ω)dy3dy2dy1 (15)
with Lighthill’s tensor in the frequency domain,
SG(y, ω) =
ˆ∂2Tij
∂yi∂yj
(y, ω). (16)
The source can be defined throughout the domain, being defined as zero outside the turbulent region, so
that the integration limits eq. (15) go from −∞ to ∞
B. Radiating and non-radiating spanwise wavenumbers in the trailing-edge noise problem
Further insight on the dominant turbulent structures leading to trailing-edge noise can be obtained by
evaluating the contributions of different spanwise wavenumbers to the far-field sound. The source term can
be defined as a function of its Fourier transform in the spatial spanwise direction y3:
SG(y1, y2, y3, ω) =
1
2π
∞
kz=−∞
ˆSG(y1, y2, kz, ω)e−ikzy3
dkz (17)
Replacing in equation (15) we obtain
ˆp(x, ω) =
1
8π2
∞
y1=−∞
∞
y2=−∞
∞
kz=−∞
ˆSG(y1, y2, kz, ω)
∞
y3=−∞
G(x, y, ω)e−ikzy3
dy3 dkzdy2dy1 (18)
where the expression inside the brackets can be defined as:
∞
y3=−∞
G(x, y, ω)e−ikzy3
dy3 = ˆG(x, y1, y2, −kz, ω) (19)
Replacing this result in equation (18) we can obtain the final expression for the pressure field in terms
of the spanwise Fourier transform of both Green’s function and source, as
ˆp(x, ω) =
1
8π2
∞
y1=−∞
∞
y2=−∞
∞
kz=−∞
ˆSG(y1, y2, kz, ω) ˆG(x, y1, y2, −kz, ω)dkzdy2dy1 (20)
Equation (20) shows that each spanwise wavenumber kz in the source, ˆS(y1, y2, kz, ω), will have its
contribution to the far-field sound weighted by a corresponding wavenumber −kz in the Green’s function,
ˆG(x, y1, y2, −kz, ω). We will here evaluate analytically the Fourier transform in eq. (19) so as to determine
the said weighting factors, which will show us the dominant source wavenumbers for far-field radiation.
To apply the spatial Fourier transform along the z axis for the source, we need to evaluate the integral
ˆGs(x, y1, y1, kz, ω) =
∞
−∞
Gs(z0)eikzz0
dz0 (21)
considering that only z0 varies (the observers are fixed, as well as the other two components of the source).
Replacing equation 14 into 21, taking out of the integral the terms with no z0 dependency and reorganizing,
we obtain:
ˆGs(x, y1, y1, kz, ω) =
2e
1
4 iπ+ikzz
√
π
(2krr0)
1
2
cos
θ
2
cos
θ0
2
∞
−∞
1
R (r2 + (z − z0)2)
1
4
e−ikz(z−z0)−ikR
dz0 (22)
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6.
With the integral in this form, we can apply the following transformation on z0:
z − z0 = q tan α, (23)
dz0 = −
q
cos2 α
dα, (24)
with q defined as
q = r2 + r2
0 − 2rr0 cos (θ − θ0) = (x1 − y1)2 + (x2 − y2)2. (25)
Since −∞ < z0 < ∞, α is a real number between −π/2 and π/2. If we apply the transform 23 on
equation 22 and make the necessary changes, we obtain:
ˆGs(x, y1, y1, kz, ω) = C
π/2
−π/2
F(α)eiqψ(α)
dα (26)
where
C =
2e
1
4 iπ+ikzz
√
π
(2krr0)
1
2
cos
θ
2
cos
θ0
2
(27)
F(α) =
1
r2 + q2 tan2
α
1
4
cos α
(28)
ψ(α) = −
k − kz sin α
cos α
(29)
It is easy to see that both F and ψ are continuous in the integration domain, as well as their first and
second derivatives. The far-field hypothesis corresponds to q → ∞; this allows an asymptotic solution to
the integral using the stationary phase method.5
The points αs leading to constant phase at eiqψ(α)
, which
have the leading-order contribution to the integral, must satisfy ψ (αs) = 0. The only root is given by
αs = arcsin
−kz
k
, (30)
and since α is a real number satisfying −π/2 < α < π/2, in order that the stationary phase point lie in the
integration path we must have
|kz| < k (31)
which means that the only wave numbers that radiate sound correspond to supersonic phase velocities along
the z axis, i.e. in the spanwise direction. This result is well known for the free-field Green’s function,4,11,7
and is here extended to the tailored Green’s function related to the edge scattering problem.
Using the far-field approximation (q → ∞) on integral 26, the stationary phase method gives an asymp-
totic approximation:
ˆGs(x, y1, y1, kz, ω) ≈ C
2π
q|ψ (α0)|
1
2
F(α0)eiqψ(α0)− 1
4 iπ
(32)
where
F(α0) =
1
r2 + q2 k2
z
k2−k2
z
1
4
√
k2−k2
z
k
(33)
ψ(α0) = − k2 − k2
z (34)
ψ (α0) = −
k2
k2 − k2
z
(35)
and C is defined by equation 27.
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7.
From the results showed above we can deduce some features of the sound generation in this problem.
Firstly, from equation 31 we find that the only radiating turbulent disturbances are the ones with |kz| < k
(or the perturbations with supersonic phase velocity, since we can define kz = ω
vφ
, where vφ is the phase
velocity). In second place, the wavenumber kz for the source term is associated with the incidence angle of the
perturbations on the trailing edge of the plate: for waves normal to the trailing edge we have kz = 0 (there
is no component of the perturbation in z direction), with increasing wavenumber for increasing incidence
angles. If this incidence angle is related to a wavenumber outside the supersonic region, only evanescent
waves will be generated, with zero contribution to the far-field sound. It is important to notice that two-
dimensional disturbances, associated to kz = 0, are always radiating, whereas perturbations associated with
other wavenumber may not generate sound at a given frequency.
In the following section we will use this theoretical prediction to perform flow-acoustic correlations in
a different manner. Besides taking the usual two-point correlation, we will isolate the two-dimensional
component of the flow fluctuations, corresponding to kz = 0. This is always radiating, and by the evaluation
of the correlation between kz = 0 components in the turbulence and in the acoustic field we expect to
highlight the dominant mechanism of trailing edge noise.
III. Flow-acoustic correlations
A. Database
In this section we present the results of flow-acoustic correlations in order to discern the regions in the flow
most relevant for sound radiation. The large-eddy simulation (LES) of Wolf et al. of a NACA0012 airfoil at
zero angle of attack, M∞ = 0.115 and Rec = 4.08 · 105
was used for that matter. The simulation attempts
to represent an infinite-span airfoil by considering periodic boundary conditions at the spanwise extremities.
This spanwise periodicity leads to discrete values of the spanwise wavenumber kz, given as
kzc =
2πn
Lz
c, (36)
where n is an integer number. The simulation domain has Lz = 0.1c, which is large compared to the boundary
layer thickness, and was seen by Wolf et al. to be sufficient to obtain a decaying two-point coherence along
the span. With this computational domain, besides the two-dimensional mode kz = 0 the first non-zero
wavenumber corresponds to kzc = 20π, which is large in terms of the airfoil chord, but small in terms of the
boundary layer thickness. Table 1 shows the wavenumber in terms of boundary layer thickness.
x/c 0.8540 0.8980 0.9170 0.9520 0.9700 0.9830
δ∗
0.0039 0.0045 0.0048 0.0055 0.0059 0.0065
kzδ∗
0.2471 0.2806 0.2995 0.3477 0.3728 0.4063
Table 1. Calculation of the wavenumber in terms of the boundary layer thickness.
To determine flow-acoustic correlations we have taken flow fluctuations throughout a body-fitted O-
grid block (Figure 11, referenced as grid 1). Fluctuations at all these positions were with a representative
position in the acoustic field, aligned with the trailing edge, at x/c = 1 and y/c = 2. Details on the numerical
simulation can be found in the Appendix.
The use of numerical data has the advantage of allowing the study of correlations using different quantities,
taken at various positions in the flow. Moreover, it is possible to extract the expected dominant low spanwise
wavenumbers satisfying kz < k, which would be an ambitious task to perform experimentally. However, the
numerical simulation has a shorter duration (˜t = 15.98, ˜t = tao/L) compared to usual time series taken from
experiments, which impact the calculation of correlations. The clearest effect of this is an overall noise level
in the correlations, whose order is 60%. We will nonetheless see that for some positions the correlation peaks
greatly exceed the background noise, and are taken as significant despite the relative short time series.
B. Two-dimensionality of the acoustic field
A first observation on the radiated sound can be made with regard to the power spectral density (PSD) at
the point taken in the acoustic field. The PSD was calculated at this position in two ways: the first is the
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8.
usual single-point PSD, and the second accounted solely for the kz = 0 component of the acoustic field; this
last approach amounts to taking a spanwise average prior to computing PSD. The results are shown in figure
2.
10
0
10
1
10
2
10
3
10
−20
10
−19
10
−18
10
−17
10
−16
10
−15
10
−14
10
−13
10
−12
10
−11
He
PSD[dB]
With spanwise averaged
Without spanwise averaged
He
n=0
He
n=1
He
n=2
He
n=3
Figure 2. Power spectral densities of pressure at x/c = 1 and y/c = 2
To help the analysis, the vertical lines in figure 2 show cut-on values of He for the successive spanwise
wavenumbers, according to the analysis of section II:
Hen = knc =
2πn
Lz
c (37)
Helmholtz numbers satisfying He > Hen (i.e. to the right of the corresponding vertical line) are such
that the nth spanwise wavenumber in the source is propagative. On the other hand, if He < Hen (left of the
corresponding line) the nth spanwise wavenumber leads solely to evanescent waves. The two-dimensional
mode kz = 0, or n = 0, is propagative for all He.
Note that there is virtually no difference between spectra with and without spanwise averaging up to He
corresponding to the cut-on condition for n = 1, which is consistent with the theoretical analysis. Above
that value some differences appear, but the energy-containing part of the acoustic spectrum is clearly two-
dimensional. It is interesting to notice that above the cut-on He for n = 2 a new peak also emerges in the
PSD.
These results are in line with the theory presented in section II, and suggest that the predominantly
two-dimensional acoustic field is generated by corresponding two-dimensional flow fluctuations. In a similar
manner to the approach of Cavalieri et al.,3
who have obtained more significant flow-acoustic correlation for
jets when the axisymmetric mode was isolated, we compare in what follows the usual two-point statistics
with correlations where the two-dimensional component was extracted by taking spanwise averages in the
turbulent and acoustic fields.
C. Correlations between pressure in the near and acoustic fields
We start the evaluation of correlation coefficients by looking solely at pressure fluctuations, with correlations
between near-field and acoustic pressure. The peak correlation, with acoustic pressure taken in the reference
point, is shown in Figure 3, where both quantities were spanwise averaged. To aid the analysis the boundary-
layer thickness and displacement thickness were plotted in the trailing edge region. Determination of the
boundary layer thickness close to the trailing edge becomes difficult due to the significant gradients of the
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9.
outer velocity, and hence we show only values of δ up to x/c ≈ 0.94. There is a significant correlation
peak in outer portions of the figure, but this occurs outside the boundary layer, and is hence attributed to
correlations between acoustic waves.
Figure 3. Maximum correlation between near-field and acoustic pressure. Both quantities are spanwise
averaged.
A zoom of figure 3 in the trailing edge region is shown in figure 4 (a). We observe high correlation
peaks inside the boundary layer, extending all the way from the wall to the boundary layer thickness δ.
The correlation coefficients are roughly 40%. On the other hand, correlations without taking the spanwise
average of the pressure, shown by figure (b), do not display visible peaks above the noise level.
This result agrees with the theoretical considerations of section II, showing that higher correlations are
obtained when kz = 0 is isolated in the turbulence field. For the energy-containing part of the acoustic
spectrum only the two-dimensional mode is radiating. Furthermore we observe that the source of sound is
concentrated in the neighbourhood of the trailing edge (centred around x/c = 0.96), which is also predicted
by the theory12
.
Correlation coefficients were seen to be oscillatory, and besides presenting a positive peak they also present
negative peaks. In order to quantify these negative peaks, Figure 5 shows the minimum of flow-acoustic
correlation, again with a prior spanwise average of fluctuations. To facilitate the comparison with figures
of maximal correlation, the minimal were plotted using a reversed color scale. The results in Figure 5 also
show significant peaks (this time negative) of the correlation inside the boundary layer, in the vicinity of
the trailing edge. This negative peak occurs around x/c = 0.93, which is slightly upstream of the region of
positive peak. Furthermore, we also observe a region of significant correlation further downstream, in the
near-field of the wake.
We have seen that taking spanwise averages of flow fluctuations greatly enhances the correlation coeffi-
cients. If we now consider that trailing edge scattering occurs due to advected disturbances in both upper
and lower airfoil surfaces, it makes sense to extract the difference between upper- and lower-surface fluctu-
ations prior to the calculation of correlations. The underlying idea is that identical disturbances at both
sides of the trailing edge would lead to destructive interference and hence low acoustic scattering. This
idea is pursued in Figure 6, where we show the correlations of the pressure difference between upper and
lower surfaces of the airfoil to the acoustic pressure; spanwise-averaged quantities were taken. Similar to the
results previously presented, we observe high correlations inside the boundary-layer and close to the trailing
edge. This time, correlation coefficients are even higher, getting close to approximately 80%.
We have seen that correlation coefficients are greatly increased if spanwise averages and differences
between upper and lower surfaces are taken in the near-field, prior to the determination of correlations. This
suggests that the relevant sound-producing fluctuations are two-dimensional and in phase opposition in the
trailing edge region. This will be further pursued in the next section, where velocity-pressure correlations
are shown.
D. Velocity-pressure correlations
The results of the preceding section were of pressure-pressure correlations. Since velocity disturbances are
more often studied in analyses of the turbulent field, and are closely related to the source terms in acoustic
analogies, we present here similar correlations, this time taken between components of the velocity vector
with acoustic pressure.
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10.
(a) With spanwise average
(b) Without spanwise average
Figure 4. Maximum correlation between grid 1 pressure and far-field pressure.
Figure 5. Minimum correlation between grid 1 pressure and far-field pressure, both quantities spanwise
averaged. We plot (−1) times the minimal correlation coefficient to allow direct comparison with the color
scales of Figure 4 and other similar plots.
Figure 7 shows the minimum velocity-pressure correlation, where the u component was taken. Fluctua-
tions were spanwise-averaged as in the analysis of the previous section. No significant positive correlation
peaks were observed, hence we focus on the minimum correlation (which displays the negative peaks); again,
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11.
Figure 6. Maximum correlation between grid 1 pressure difference between the upper and lower surface of
the airfoil and far-field pressure; spanwise-averaged quantities.
an inversion of the color scale was done to allow direct comparison with the other plots. We observe a thin
region in the neighborhood of the trailing edge with high correlation with the radiated sound. This region
is inside the boundary layer, around the displacement thickness.
Figure 7. Minimum correlation between u-component and acoustic field pressure. Fluctuations were spanwise-
averaged. We plot (−1) times the minimal correlation coefficient to allow direct comparison with the color
scales of Figure 4 and other similar plots.
When the spanwise average is not taken, the correlation coefficients become negligible, as shown in Figure
8. This again shows that the two-dimensional part of the turbulent field is most relevant for sound radiation.
If the difference of the u-component between upper and lower surfaces is taken before correlation, we
obtain the results of Figure 9. As was observed with the pressure, the correlation coefficients become more
significant in the vicinity of the trailing edge.
The two-dimensional part of the acoustic field is the most correlated with the acoustic pressure, in spite
of its relative low kinetic energy. This is shown in Figure 10, which compares the root-mean-square (RMS)
of the u-component near the airfoil. We observe that downstream of the laminar-turbulent transition, in
a region with approximately 0.2 ≤ x/c ≤ 1.2, we have RMS(u2D) << RMS(u3D). The two-dimensional
part of the velocity field is thus a small fraction of the overall turbulence; however, these low-amplitude
two-dimensional structures are closely related with trailing-edge noise.
Correlations using v- and w-components are not presented here due to lack of expressive flow-acoustic
correlations.
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12.
Figure 8. Minimum correlation between u-component and acoustic field pressure. Fluctuations were not
spanwise-averaged. We plot (−1) times the minimal correlation coefficient to allow direct comparison with the
color scales of Figure 4 and other similar plots.
Figure 9. Minimum correlation between grid 1 u-component, with difference between the upper and lower
surfaces of the airfoil, and acoustic field pressure. See comments in the caption of Figure 7.
IV. Conclusion
We have used flow-acoustic correlations to identify regions and mechanisms most effective to generate
sound in a turbulent flow around a NACA 0012 airfoil with zero angle of attack and M = 0.115.. Theoretical
results for the trailing-edge noise problem, modelled using the tailored Green’s function of Ffowcs Williams
and Hall12
show that only wavenumbers satisfying kz < k, where kz is the spanwise wavenumber of turbulent
disturbances and k is the acoustic wavenumber, radiate sound. In particular, two-dimensional disturbances
(kz = 0) are always radiating.
With this in mind we show, using flow-acoustic correlations obtained using data from a large eddy sim-
ulation, that when spanwise averages of flow fluctuations are taken, flow-acoustic correlations significantly
increase. We have obtained significant correlations in the trailing edge region for spanwise-averaged quanti-
ties: when pressure-pressure correlations were taken, a region extending from the wall to the boundary layer
thickness was found to present high correlation peaks; and for correlations between streamwise velocity and
acoustic pressure, an elongated region around the displacement thickness was found to have significant cor-
relation peaks as well, indicating that in this region there are coherent structures inside the boundary-layer
associated with far-field sound. If the two-dimensional mode is not isolated, correlation coefficients are lower,
and even negligible in some cases. Flow-acoustic correlations also increase when a difference of fluctuations
between the upper and lower surfaces is taken, which agrees with theory: convection of disturbances in phase
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13.
(a) RMS of grid 1 u-component with spanwise-average
(b) RMS of grid 1 u-component with no spanwise-average
Figure 10. Comparison between the RMS of grid 1 u-component with spanwise-average and grid 1 u-component
with no spanwise-average
opposition across the trailing edge leads to maximal scattered sound.
Some features of the analysis are particular to the kind of numerical simulation used here, which is an
LES adopting periodic boundary conditions along the span. Similar simulations use the same approach
(e.g. Sandberg and Sandham10
), and, although validation of such simulations involve a demonstration of
coherence decay along the spanwise direction of the computational domain, an artefact persists: only discrete
values of spanwise wavenumbers are possible, and in some cases (such as in the simulation used in this work)
most of the sound radiation is two-dimensional, as kz = 0 is the only radiating wavenumber for most of the
acoustic spectrum. In experiments, we expect a much broader range of spanwise wavenumbers, and non-zero
kz would probably radiate sound. However, the |kz| < k condition implies that for most Helmholtz numbers
only spanwise-elongated structures, with characteristic dimension greater than the acoustic wavelength,
radiate sound to the far field. Further numerical and experimental work aiming at the characterization of
such structures is promising.
V. Acknowledgments
The authors acknowledge the financial support received from Conselho Nacional de Desenvolvimento
Cient´ıfico e Tecnol´ogico, CNPq, under grant No. 402233/2013-1 and 382381/2014-9. A. V. G. Cavalieri was
supported by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico, CNPq, through a research
scholarship. The authors also acknowledge the financial support received from Funda¸c˜ao de Amparo `a
Pesquisa do Estado de S˜ao Paulo, FAPESP, under Grant No. 2013/03413-4 and from Conselho Nacional de
Desenvolvimento Cient´ıfico e Tecnol´ogico, CNPq, under Grant No. 470695/2013-7. We thank Peter Jordan
for helpful discussions regarding the present work.
References
1R. K. Amiet. Noise due to turbulent flow past a trailing edge. Journal of Sound and Vibration, 47(3):387–393, 1976.
2T. F. Brooks, D. S. Pope, and M. A. Marcolini. Airfoil self-noise and prediction, volume 1218. National Aeronautics and
Space Administration, Office of Management, Scientific and Technical Information Division, 1989.
3A. V. G. Cavalieri, D. Rodr´ıguez, P. Jordan, T. Colonius, and Y. Gervais. Wavepackets in the velocity field of turbulent
jets. Journal of Fluid Mechanics, 730:559–592, 2013.
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4D. G. Crighton. Basic principles of aerodynamic noise generation. Progress in Aerospace Sciences, 16(1):31–96, 1975.
5D. G. Crighton, A. P. Dowling, J. E. Ffowcs-Williams, M. Heckl, and F. G. Leppington. Modern methods in analytical
acoustics: lecture notes. Springer, 1992.
6N. Curle. The influence of solid boundaries upon aerodynamic sound. Proceedings of the Royal Society of London. Series
A. Mathematical and Physical Sciences, 231(1187):505–514, 1955.
7P. Jordan and T. Colonius. Wave packets and turbulent jet noise. Annual Review of Fluid Mechanics, 45(1), 2013.
8M. J. Lighthill. On sound generated aerodynamically. i. general theory. Proceedings of the Royal Society of London.
Series A. Mathematical and Physical Sciences, 211(1107):564–587, 1952.
9Assad A. Oberai, Farzam Roknaldin, and Thomas J. R. Hughes. Computation of trailing-edge noise due to turbulent
flow over an airfoil. AIAA journal, 40(11):2206–2216, 2002.
10Richard D. Sandberg and Neil D. Sandham. Direct numerical simulation of turbulent flow past a trailing edge and the
associated noise generation. Journal of Fluid Mechanics, 596:353–385, 2008.
11Christopher K. W. Tam. Supersonic jet noise. Annual Review of Fluid Mechanics, 27(1):17–43, 1995.
12J. E. Williams and L. H. Hall. Aerodynamic sound generation by turbulent flow in the vicinity of a scattering half plane.
Journal of Fluid Mechanics, 40(04):657–670, 1970.
13W. R. Wolf, J. L. F. Azevedo, and S. K. Lele. Convective effects and the role of quadrupole sources for aerofoil
aeroacoustics. Journal of Fluid Mechanics, 708:502–538, 2012.
A. Numerical Simulation
The Large-Eddy Simulations (LES) in this work are the result of Wolf et al.13
for which time-resolved data
is available for all points in the grid. In what follows we summarize some of the simulation characteristics.
To perform the numerical simulation two different grids were used. The first allows accurate resolution of the
airfoil boundary layer. This grid is body-fitted O-grid block composed by 1536 x 125 x 128 nodes (Figure 11
(b)). The second grid is more appropriate for acoustic waves emanating from the airfoil. This grid is a
Cartesian background grid block and it is composed by 896 x 511 x 64 (Figure 11 (a)).
Wolf et al. performed the solution of general curvilinear form of the compressible Navier-Stokes equations,
given as
∂ρ
∂t
+
∂(ρui
)
∂t
= 0, (38)
∂(ρui
)
∂t
+
(ρui
uj
+ gij
p−τij)
∂xj
= 0, (39)
∂E
∂t
+
∂[(E + p)ui
− τijgikuk
+ qj]
∂xj
= 0, (40)
To close the set of equations we define the total energy equation, E, express the heat flux using Fourier’s
law and the viscous stress tensor using the expression for a Newtonian fluid. We obtain thus
E =
p
γ − 1
+
1
2
ρgikui
uk
, (41)
qj = −
µ
RePr
gij ∂T
∂xi
(42)
τij =
µ
Re
(gjk ∂ui
∂xk
+ gik ∂uj
∂xk
−
2
3
gij ∂uk
∂xk
) (43)
Finally, the gas is taken as calorically perfect, hence
p =
γ − 1
γ
ρT (44)
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15.
(a) (b) (c)
Figure 11. Mesh details; (a)Full view of Cartesian grid with composed 896 x 511 x 64 nodes; (b)Body-fitted
O-grid block with composed 1536 x 125 x 128 nodes. (c)Detailed rounded trailing edge.
In Table 2 there is a summary of detailed properties used on current simulation. To force transition to
turbulence in the boundary layer, a boundary layer trip was simulated using artificial blowing and suction.
Figure 12 illustrates the region of boundary layer tripping.
Rec M∞ AoA[deg.] BL tripping ∆t NS NpS
408000 0.115 0 Top and bottom sides 0.0004 3 512
Table 2. Summary of flow configuration analyzed. Here, Rec is the Reynolds number based on the aerodynamic
profile chord, AoA is the angle of attack, δt is the dimensionless time step for time marching of the LES
equations, NS is the number of datasets for acoustic processing, and NpS is the number of samples in each
dataset.
Figure 12. Illustrative description of aerodynamic profile used on LES simulation and the region of boundary
layer tripping.
Figure 13 (a) shows a general view of the turbulent field around the airfoil by plotting the iso-surfaces of
λ2. Figure 13 (b) illustrates the boundary layer transition region and the noise generation due the airfoil.
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16.
(a) (b)
Figure 13. LES of flow past a NACA0012 airfoil at AoA = 0 deg, M∞ = 0.115 and Rec = 408000; (a) iso-surfaces
of λ2 colored by vorticity magnitude; (b) iso-surfaces of vorticity magnitude colored by streamwise momentum
with contours of dilatation in the background
(a) (b)
Figure 14. (a) Points used to correlate with center at x/c = 0.5; (b) Points used to correlate with center at
x/c = 1.0
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