Broadband trailing edge noise from a sharp-edged strut
Danielle J. Moreau,a) Laura A. Brooks, and Con J. Doolan
School of Mechanical Engineering,
The University of Adelaide,
South Australia,
Australia 5005
Broadband noise from a sharp-edged strut 1
Abstract
This paper presents experimental data concerning the flow and noise
generated by a sharp edged flat plate at low-to-moderate Reynolds
number (Reynolds number based on chord of 2.0 × 105 to 5.0 × 105).
The data are used to evaluate a variety of semi-empirical trailing edge
noise prediction methods. All were found to under-predict noise at
lower frequencies. Examination of the velocity spectra in the near wake
reveals that there are energetic velocity fluctuations at low frequency
about the trailing edge. A semi-empirical model of the surface pressure
spectrum is derived for predicting the trailing edge noise at low-to-
moderate Reynolds number.
PACS numbers: 43.28.Ra
2
I. INTRODUCTION
Trailing edge noise is produced when boundary layer turbulence convects past a sharp
trailing edge.1,2 Turbulent trailing edge noise is important to a broad range of applications
but most studies conducted in the past have focused on high Reynolds number applications
such as commercial aircraft, compressors and turbo-machinery and wind turbines. Only a
few studies have investigated trailing edge noise at low-to-moderate Reynolds numbers3,4 and
as such, data sets at this Reynolds number range are relatively rare despite their importance
for understanding flow-induced noise generation from micro-wind-turbines, unmanned air
vehicles and underwater control surfaces. Furthermore, semi-empirical models5–8 provide
accurate predictions of trailing edge noise at high Reynolds numbers7–9 but there has been
little evaluation of these models at lower Reynolds numbers.
The approach to the prediction of trailing edge noise has typically been based on a
knowledge of the surface pressure fluctuations at the trailing edge. Chase10 was one of the
first to formulate theory describing the sound field radiated from an idealised trailing edge
in terms of the turbulence induced surface pressure fluctuations. This theory makes use of
an analytical Green’s function for a semi-infinite half plane. Chandiramani11 reformulated
the theoretical model developed by Chase10 by representing the sound pressure fluctuations
as a distribution of harmonic evanescent waves. This theory was later applied by Chase12
to the prediction of airfoil trailing edge noise. Howe13,14 extended the Chase-Chandiramani
diffraction theory to include the combined effects of finite chord, airfoil thickness and trailing
edge geometry on the trailing edge noise produced by a flat plate. Amiet6 developed a more
complete theoretical model for calculating the noise radiated from an airfoil trailing edge
using spectral characteristics of the wall pressure. This theory has been extended by Roger
and Moreau4,15 and Roger et al.16 to account for leading edge backscattering effects.
To validate trailing edge noise theory, experimental studies have largely focused on mea-
suring the airfoil surface pressure fluctuations and far-field noise spectra at high Reynolds
a)Electronic address: [email protected]
3
numbers (Reynolds numbers based on chord of approximately Rec > 1×106). Most of these
studies have concentrated on measurements on NACA0012 airfoils and flat plate models.
NACA airfoils are a family of airfoils whose shape is characterised using a series of digits.17
Symmetric airfoils are denoted NACA00xx, where ‘xx’ defines the thickness to chord ra-
tio. A flat plate is considered to be the limiting case of a symmetric airfoil. Brooks and
Hodgson3 measured the radiated trailing edge noise and wall pressure fluctuations for a
NACA0012 airfoil with varying trailing edge bluntness. Good agreement was found between
the measured airfoil noise spectra and that predicted using measured surface pressure fluc-
tuations and the theory developed by Howe.18 A number of experimental studies have been
conducted on the trailing edge noise generated by flow over asymmetric beveled trailing
edges at high Reynolds numbers.1,2,19,20 The data measured in these studies includes the
fluctuating surface pressure near the trailing edge, boundary layer profiles and the radiated
noise spectra at Rec > 1 × 106. Gershfeld et al.20 found some agreement between beveled
trailing edge noise spectra and that predicted using existing theoretical models of trailing
edge noise developed by Howe.18 Schinkler and Amiet21 conducted an experimental study
of helicopter rotor trailing edge noise, measuring boundary layer and far-field acoustic data
for a local blade segment over a range of Mach numbers, propagation angles and airfoil
angles of attack. Some agreement was found between measured and predicted noise levels
using a generalised description of the surface pressure fluctuations and the theoretical model
developed by Amiet.6 Roger and Moreau4 were one of the few to experimentally investigate
the trailing edge noise mechanism at low Reynolds numbers by applying an extension of
Amiet’s theory6 to the prediction of trailing edge noise from subsonic fans. Good agreement
was found between experimental data and predictions obtained using the measured surface
pressure fluctuations at Reynolds numbers of Rec < 3.5 × 105. Herr9 recently presented
an experimental database of far-field noise and unsteady surface pressure measurements
for a plate airfoil with variable chord at high Reynolds numbers. Accurate predictions of
the far-field noise were achieved using the same surface pressure approach as Brooks and
Hodgson.3
4
There are numerous semi-empirical models for predicting the far-field trailing edge
noise.5–8 Additionally, semi-empirical models of the fluctuating wall surface pressure
spectrum22–31 have been developed and can be used with existing theory6,18 to predict far-
field trailing edge noise. Brooks et al.5 developed a semi-empirical airfoil self noise prediction
model based on boundary layer thickness at the trailing edge. This well known empirical
model, named the BPM model, was derived from aerodynamic and acoustic data for two
and three-dimensional NACA0012 airfoil models at a wide range of Reynolds numbers.
Casper and Farassat7,8 developed a semi-empirical model named ‘Formulation 1B’ for the
time domain prediction of trailing edge noise from a known surface pressure distribution.
Noise levels estimated with Formulation 1B were found to be in good agreement with the
experimental data measured by Brooks and Hodgson.3 While accurate predictions of far-
field trailing edge noise have been obtained with semi-empirical models at high Reynolds
numbers7–9 (Rec > 1× 106), there is almost no validation of these models at lower Reynolds
numbers. In fact, few low Reynolds number data sets exist with all the necessary informa-
tion (e.g. boundary layer displacement thickness, surface pressure spectrum) required for
comparison with semi-empirical predictions.
The aim of this paper is to: (1) present new trailing edge flow and noise data for a flat
plate model at low-to-moderate Reynolds number; (2) evaluate a variety of semi-empirical
models of trailing edge noise for low Reynolds number conditions; and (3) present a new
semi-empirical surface pressure model for the prediction of trailing edge noise that is more
suitable for low-to-moderate Reynolds number. This paper is structured as follows: Section
II provides details of the anechoic wind tunnel facility and the experimental method; Section
III.A presents the acoustic results, their comparison with semi-empirical predictions and
development of a new surface pressure model for the prediction of trailing edge noise at
low-to-moderate Reynolds numbers; and Section III.B presents the aerodynamic results.
5
LE
TE
2.5 mm12◦
FIG. 1. Schematic diagrams of the flat plate model leading and trailing edge.
II. EXPERIMENTAL METHOD
Experiments were performed in the anechoic wind tunnel at the University of Adelaide.
The anechoic wind tunnel test chamber is cubic, approximately 8 m3 in size and has walls
that are acoustically treated with foam wedges. The test chamber provides a reflection free
environment (ideally) above 200 Hz. The anechoic wind tunnel contains a contraction outlet
that is rectangular in cross section and has dimensions of 75 mm x 275 mm. The maximum
flow velocity of the free jet is 40 m/s and the free-stream turbulence intensity is 0.3%.
The flat plate model used in these experiments has a chord of c = 200 mm, a span of
s = 450 mm and a thickness of h = 5 mm. The leading edge is circular with a radius of
2.5 mm while the trailing edge is a symmetric wedge shape with an apex angle of 12◦, as
shown in Fig. 1. The flat plate model was secured to a housing at zero angle of attack using
two side plates and this housing was in turn attached to the contraction flange, as shown in
Fig. 2. The span of the flat plate model extends beyond the width of the contraction outlet
to eliminate the noise produced by the interaction of the side plate boundary layers with the
model leading edge. As shown in Fig. 3, two extension plates made from 75× 75 mm steel
equal angle were attached to contraction flange and aligned with the top and bottom edges
of the contraction outlet. The extension plates essentially extend the contraction outlet past
the leading edge of the flat plate (see Fig. 2 (a)). The extension plates were added to the
contraction to minimise the interaction of the outlet shear layer with the plate trailing edge
region.
To measure the far-field noise, three B&K 1/4” microphones (Model No. 4190) were
located in the anechoic wind tunnel: one above and one below the trailing edge and one
6
(a)
(b)
Flat plate
Flat plate
Contraction outlet
Contraction outlet
Contraction flange
Contraction flange
Side plate
Side plateExtension plate
Extension plateTE
TETE
LEFlow
275 mm
15 mm
450 mm
5 mm
75 mm
75 mm
200 mm
Housing
FIG. 2. Schematic diagram of the flat plate model secured in the housing and attached to
the contraction outlet. (a) Side view and (b) front view.
Contraction outlet Extension plate
Flat plateSide plate
FIG. 3. The flat plate model attached to the contraction outlet with the extension plates.
above the leading edge. The top and bottom trailing edge microphones were located at the
same radial distance from the trailing edge, perpendicular to the direction of the flow. The
positions of the three microphones relative to the plate leading and trailing edge are given in
7
Flat plate
Top TE micLE mic
Bottom TE mic
Flow
a
a
bf
d
e
LE TE
FIG. 4. Microphone positions relative to the flat plate model where a = 0.585 m, b = 0.600
m, d = 0.588 m, e = 0.618 m and f = 0.075 m.
Fig. 4. Each of the microphones were calibrated before commencing the acoustic tests. To
provide isolation from wind noise, wind socks were placed on all the microphones. Both the
microphone and velocity data (described later) were collected using a National Instruments
board at a sampling frequency of 215 Hz for a sample time of 8 s. All data are presented in
narrow band format with a frequency resolution of 1 Hz.
It is likely that the far-field noise measurements are contaminated with background noise
and thus the method for extracting and analysing trailing edge noise developed by Moreau et
al.32 has been used to process the far-field noise measurements. Extraneous noise sources are
removed from the far-field noise measurements using the two phase-matched microphones
located above and below the trailing edge. As the two microphones measure the trailing
edge noise to be equal in magnitude, highly correlated and 180◦ out of phase, subtracting the
out-of-phase signals isolates the trailing edge noise in the far-field noise measurements. An
offset value of 6 dB also needs to be removed from the corrected trailing edge noise spectra
when using this method. As sound produced at the leading edge has the same characteristics
as the trailing edge noise measured with the microphones above and below the trailing edge,
leading edge noise is not extracted using this technique. It will however, be shown in Section
III.A, that trailing edge noise is the dominant noise signal.
8
Hot-wire anemometry was used to obtain unsteady velocity data in the very near wake
of the trailing edge of the flat plate model. It was also used to measure the boundary layer
parameters (boundary layer height, displacement and momentum thickness). A TSI 1210-
T1.5 single wire probe with wire length of L = 1.27 mm and a wire diameter of d = 3.81
µm was used. The probe was connected to a TSI IFA300 constant temperature anemometer
system and positioned using a Dantec automatic traverse with 6.25 µm positional accuracy.
The probe was initially positioned at a location 0.6 mm downstream of the trailing edge
in line with the spanwise centre and then was traversed vertically perpendicular to the
chordline. Data were acquired over a vertical line spanning y = ±25 mm, where y = 0 is in
line with the trailing edge and a positive or negative y value indicates a position above or
below the trailing edge, respectively.
Far-field noise and trailing edge velocity data were recorded for the flat plate model
at the six free-stream velocities of U∞ = 38, 35, 30, 25, 20 and 15 m/s, corresponding to
Reynolds numbers: Rec ≈ 5.0× 105, 4.6× 105, 4.0× 105, 3.3× 105, 2.6× 105 and 2.0× 105,
respectively.
III. EXPERIMENTAL RESULTS
A. Far-field acoustic data
The far-field acoustic spectra for the flat plate model at free-stream velocities between
U∞ = 38 and 15 m/s are shown in Fig. 5 along with background noise spectra. The
background noise was measured with the top trailing edge microphone. At all free-stream
velocities, the corrected far-field spectra sit well above the background noise level, especially
at lower frequencies where the levels of trailing edge noise are high. Decreasing the flow
velocity from U∞ = 38 to 15 m/s has the expected effect of slightly decreasing the radiated
noise levels and this is particularly evident at lower frequencies.
Like trailing edge noise, sound produced at the leading edge and radiated to opposite
sides of the airfoil would be well correlated, equal in magnitude and 180◦ out of phase.
9
FIG. 5. Far-field acoustic spectra for the flat plate model for U∞ of (a) 38, (b) 35, (c) 30,
(d) 25, (e) 20 and (f) 15 m/s.
10
Brooks and Marcolini33 experimentally analysed the cross-correlation of noise measured
above the leading and trailing edges of a flat plate and showed that noise produced at the
leading edge dominated the radiated sound field. It was suggested that the leading edge
noise was caused by the interaction of the turbulent boundary layer produced at the test
rig’s side plates, with the sharp leading edge. Fig. 6 shows the cross-correlation of the noise
measured with the top trailing edge microphone and the leading edge microphone in the
present study. The cross correlation functions at speeds between U∞ = 38 and 15 m/s differ
only in magnitude and so only the cross correlation functions at U∞ = 38 and 15 m/s are
shown. In Fig. 6, the cross-correlation functions have been normalised by the maximum
value of the cross-correlation function at U∞ = 38 m/s.
The time delays between sound radiated to the top trailing edge microphone and the
leading edge microphone from the trailing edge and leading edge respectively are as follows:
∆tTE =(a− b)c0
=(0.585− 0.600)
c0
= −4.4× 10−5s, (1)
and
∆tLE =(e− d)
c0
=(0.618− 0.588)
c0
= 8.7× 10−5s, (2)
where ∆tTE is the time delay between sound radiated to the top trailing edge microphone
and the leading edge microphone from the trailing edge, ∆tLE is the time delay between
sound radiated to the top trailing edge microphone and the leading microphone from the
leading edge and c0 is the speed of sound (343 m/s).
A peak is observed in the cross-correlation functions at ∆tTE in Fig. 6. The magnitude
of the cross-correlation function is significantly greater at ∆tTE than at ∆tLE, indicating
that trailing edge noise is the dominant noise mechanism. In these experiments, the span
of the plate extends beyond the width of the contraction outlet to reduce the interaction of
the flow with the side plates. The fact that trailing edge noise dominates the radiated sound
field in this case, supports the results of Brooks and Marcolini,33 that the leading edge noise
is produced by the turbulent boundary layer at the test rig’s side plates interacting with the
sharp leading edge.
11
A peak in the spectrum is observed at approximately 1.5 kHz, as shown in Fig. 5.
This peak is likely a facility induced effect caused by the acoustic interaction of sound waves
produced at the trailing edge with the extension plates. This conclusion was reached because
(1) the peak does not shift with flow speed, indicating that it is independent of the flow
conditions; (2) it is not present when the extension plates (detailed in Section II) are removed
(see other papers by Moreau et al.32,34) and (3) the near wake velocity spectra measured by
the hot wire shows no indication of high energy velocity fluctuations at this frequency (see
Section III.B). The results with the extension plates are retained in this paper because they
allow better resolution of the higher frequency noise components.
Comparison with trailing edge acoustic theory
Ffowcs Williams and Hall35 showed that for the idealized (non-compact) case of a semi-
infinite flat plate of zero thickness, the amplitude of the radiated trailing edge noise scales
proportional with M5. The far-field acoustic spectra for the flat plate model at free-stream
velocities between U∞ = 38 and 15 m/s are presented as one-third-octave band spectra and
normalised by M5 in Fig. 7. In this figure, the one-third-octave band spectra have been
normalised according to
Scaled Lp1/3 = Lp1/3 − 50log10(M), (3)
where Lp1/3 is the far-field spectra in one-third-octave bands. Data at centre frequencies
above 5 kHz have been removed from the spectra measured at U∞ = 15 m/s due to low
signal to noise ratio. Fig. 7 shows that the trailing edge noise scaling law of M5 gives a good
collapse of the far-field noise spectra for the three flat plate models, especially at frequencies
above 1 kHz. Ffowcs Williams and Hall35 theory was derived using the assumption that the
plate chord exceeds the acoustic wavelength of sound. The radiated sound is not expected
to scale according to the M5 scaling law at frequencies for which this assumption is invalid.
This corresponds to frequencies below 1.7 kHz for the plates used in these experiments.
12
FIG. 6. Cross-correlation between the top trailing edge microphone signal and the leading
edge microphone signal for U∞ of: (a) 38 and (b) 15 m/s (normalised to the overall maxima).
Microphone signals have been bandpassed between 800 and 104 Hz. The time delays between
sound radiated to the top trailing edge microphone and the leading edge microphone from
the trailing edge, ∆tTE, and the leading edge, ∆tLE, are shown with black dash-dot lines.
Values for ∆tTE and ∆tLE are calculated from geometry using Eqns. 1 and 2.
Noise prediction with the BPM model
The one-third-octave band spectra for the flat plate model are compared to the noise
spectra predicted with the BPM model5 in Fig. 8. The BPM model is a semi-empirical
prediction method for estimating the noise generated by an airfoil encountering smooth
fluid flow. The BPM model incorporates the following self-noise mechanisms: (1) sepa-
13
FIG. 7. One-third-octave band spectra scaled with M5.
ration stall noise; (2) blunt trailing-edge vortex-shedding noise; (3) tip vortex formation
noise; (4) laminar-boundary-layer vortex-shedding noise; and (5) turbulent-boundary-layer
trailing-edge noise. The BPM model was derived from existing theory and aerodynamic and
acoustic data for two and three-dimensional NACA0012 airfoil models at a wide range of
Reynolds numbers. In Fig. 8, the noise spectra predicted with the BPM model have been
calculated using NAFNoise36 (NREL AirFoil Noise) at equivalent conditions to those used
in experiments here. NAFNoise calculates the boundary layer displacement thickness, δ∗,
required as input to the BPM model using XFOIL.37 While there are significant differences
in the spectral shape and level at frequencies below 2 kHz, the BPM model is in good agree-
ment with experimental data above 2 kHz. Poor predictions at low frequencies are likely
due to the fact that the BPM model was derived from experimental data that was at times
truncated at low and high frequencies. This truncation was done to eliminate the influence of
extraneous noise sources that were expected to significantly affect the noise levels in the low
and high frequency regions. Examining the results of Brooks et al.5 shows that experimental
data used in the derivation of the BPM model at α = 0 is well predicted at high Reynolds
numbers but underpredicted in the low frequency region at low Reynolds numbers. This is
the same result as observed here. This result shows that the BPM model is not particularly
14
FIG. 8. One-third-octave band spectra compared with one-third-octave band spectra pre-
dicted with the BPM model (dashed lines).
accurate at predicting trailing noise at low-to-moderate Reynolds numbers.
Noise prediction with semi-empirical surface pressure models
The far-field trailing edge noise can, instead, be predicted using a surface pressure ap-
proach. For the case of an orthogonal view angle, the far-field noise spectrum of a semi-
infinite plane, S∞(R,ω), can be calculated from the surface pressure frequency spectrum,
Φ(ω), upstream of the trailing edge according to3
S∞(R,ω) ∼= (1/2π2R2)(Ucs/c0)l3Φ(ω)/(1− Uc/c0), (4)
where l3 = Uc/ζzω, R is the observer distance, Uc is the convection velocity and ζz is the
spanwise coherence decay constant. The far-field noise spectrum in Eq. 4 was derived using
a semi-infinite rigid half-plane assumption. To account for the effects of finite chord length,
Howe38 derived a correction for multiple scattering from the airfoil leading and trailing edges
applicable to the far-field noise spectrum of a semi-infinite plane, S∞(R,ω). The far-field
noise spectrum for an airfoil with finite chord, S(R,ω), is given by38
S(R,ω) = S∞(R,ω)
∣∣∣∣∣ G(x,y, ω)
(G1(x,y, ω))half plane
∣∣∣∣∣2
, (5)
15
where G(x,y, ω) is the time harmonic Green’s function defined as
G(x,y, ω) = G1(x,y, ω) +GLE(x,y, ω) +GTE(x,y, ω). (6)
The components of G(x,y, ω) are
G1(x,y, ω) =−√κ0 × sin
12ψsin
(θ2
)ϕ∗(y)eiκ0|x|
π√
2πi |x|, (7)
GLE(x,y, ω) =
√κ0 × sin
12ψϕ∗(y)eiκ0(|x′|+csinψ)
iπ32 |x| (1 + e2iκ0csinψ/2πκ0csinψ)
×
F
2
√κ0csinψcos2
(θ2
)π
, and (8)
GTE(x,y, ω) =ϕ∗(y)eiκ0(|x|+2csinψ)
π2√
2ic |x| (1 + e2iκ0csinψ/2πκ0csinψ)×
F
2
√κ0csinψsin2
(θ2
)π
, (9)
where κ0 is the acoustic wavenumber, F denotes the Fresnel integral auxiliary function,
y is the source position and ϕ∗(y) is the velocity potential of ideal, incompressible flow
around the edge.38 Using Howe’s38 co-ordinate system, the observer position with respect
to the trailing and leading edges is x = (x1 = 0, x2 = a, x3 = 0) and x′ = (x1 + c =
c, x2 = a, x3 = 0) respectively. The angles between the observer and the trailing edge are
ψ = sin−1(r/|x|) = 90◦ and θ = cos−1(x1/r) = sin−1(x2/r) = 90◦, where r =√x2
1 + x22.
The semi-empirical models of the surface pressure frequency spectrum, Φ(ω), for zero-
pressure gradient turbulent boundary layers developed by Chase,24,27 Howe,29 Smol’yakov
and Tkachenko,28 Smol’yakov30 and Goody31 are used in Eq. 4 to predict the far-field trailing
edge noise. These models have been derived from surface pressure spectra measured at
various flow conditions and at a wide range of Reynolds numbers. The surface pressure
spectrum model derived by Chase24,27 is given by39
Φ(ω) =[a+γMα−3M (1 + µ2
Mα2M) + 3πCTα
−1T (1 + α−2
T )]×
ρ2u4∗ω−1, (10)
16
where α2M = α2
T = 1 + (βωδ/Uc), CM = 0.1533, CT = 0.00476, β = 0.75, µM = 0.176,
a+ = 2π(CM + CT ), γM = CM/(CM + CT ), δ is the boundary layer thickness and u∗ is the
skin friction velocity. In this model, the surface pressure frequency spectrum is constant at
low frequencies and decays according to ω−1 at high frequencies. An approximation of this
surface pressure model was derived by Howe29 and is given by
Φ(ω)Ueτ 2wδ∗ =
2(ωδ∗/Ue)2
[(ωδ∗/Ue)2 + 0.0144]1.5, (11)
where τw is the wall shear stress, δ∗ is the boundary layer displacement thickness and Ue
is the velocity at the boundary layer edge. This model, often referred to as the Chase-
Howe model, assumes that δ = 8δ∗ and Uc = 0.65Ue. Differences in the surface pressure
spectrum predicted with this model and the original version derived by Chase24,27 are in the
low frequency region where the spectrum is proportional to ω2.
Smol’yakov and Tkachenko28 derived the following surface pressure spectrum model
Φ(ω) =5.1(τ 2
wδ∗/U∞)
[1 + 0.44(ωδ∗/U∞)7/3], (12)
which predicts the spectrum to be constant at low frequencies and proportional to ω−7/3
at high frequencies. Smol’yakov30 also independently proposed a semi-empirical surface
pressure model expressed as
Φ(ω) =1.49× 10−5R2.74
θ ω̄2[1− 0.117R0.44θ ω̄0.5]
[u2∗/(τ
2wν)]
when ω̄ < ω̄0,
Φ(ω) =2.75ω̄−1.11
[1− 0.82e(−0.51(ω̄/ω̄0−1))
][u2∗/(τ
2wν)]
when ω̄0 < ω̄ < 0.2, or
Φ(ω) =
[1− 0.82e(−0.51(ω̄/ω̄0−1))
][u2∗/(τ
2wν)]
×
(38.9e−8.35ω̄ + 18.6e−3.58ω̄ + 0.31e−2.14ω̄)
when ω̄ > 0.2, (13)
17
where θ is the momentum thickness, ω̄ = ων/u2∗, ω̄0 = 49.35R−0.88
θ and Rθ = U∞θ/ν. At low
frequencies (ω̄ < ω̄0), this spectrum increases according to ω2. In the mid frequency region
(ω̄0 < ω̄ < 0.2), the spectrum peaks and decays proportional to ω−1.11 before decaying in
exponent form in the high frequency region (ω̄ > 0.2).
The most recent semi-empirical surface pressure model was derived by Goody31 and is
given by
Φ(ω)Ueτ 2wδ
=3(ωδ/Ue)
2
[(ωδ/Ue)0.75 + 0.5]3.7 + [(1.1R−0.57T )(ωδ/Ue)]7
, (14)
where the timescales ratio is RT = (u∗δ/ν)/√cf/2. This model incorporates Reynolds
number scaling through the timescales ratio, RT , and predicts the pressure spectrum to be
proportional to ω2 at low frequencies, ω−0.7 at mid frequencies and ω−5 at high frequencies.
Fig. 9 shows the far-field acoustic spectra for the flat plate model compared to estimates
of the far-field trailing edge noise predicted using Eqs. 4 and 5 and the semi-empirical models
of the surface pressure frequency spectrum given in Eqs. 10 - 14. As the skin friction velocity,
u∗, was not directly measured in experiments, it was calculated using the turbulent skin
friction coefficient, cf , according to u∗ = U∞√cf/2. For flow over a smooth flat plate, the
skin friction coefficient is approximated as cf = 0.0059/(Re1/5).40 The wall shear stress, τw,
was calculated from the skin friction velocity, u∗, as u∗ =√τw/ρ. The convection velocity
was approximated as Uc = 0.65Ue and the spanwise coherence decay constant ζz = 0.714
for a flat plate.3 The experimentally measured boundary layer parameters for Plate One are
given in Table II.
Predictions of the far-field trailing edge noise using the surface pressure approach in
Fig. 9 show some agreement with experimental data at frequencies above 2 kHz. However,
some models consistently under predict the noise levels over all frequencies. Below 2 kHz, the
predicted noise levels are significantly less than the experimentally determined noise levels.
This is in agreement with the result achieved using the BPM model in Fig. 8, suggesting the
need for further development of semi-empirical prediction models of low Reynolds number
trailing edge noise.
18
Modified surface pressure model
To account for the low frequency mis-match between predicted spectra and experimental
data, a new surface pressure model is now proposed to approximate the low Reynolds number
experimental data presented in this paper. The proposed model is a modified version of
the surface pressure model developed by Smol’yakov and Tkachenko28. Without claiming
general validity, the experimental data can be approximated using the following surface
pressure model
Φ(ω) =76(τ 2
wδ∗/U∞)
[1 + 17(ωδ∗/U∞)1.1], (15)
which is a modification of Eq. 12. This model was formulated to give the least mean squared
error between predicted spectra and experimental data. The facility induced broadband
noise component centered at 1.5 kHz is specific to these experiments and so was not included
in the model. The surface pressure spectrum predicted with this model is constant and high
in amplitude at low frequencies and proportional to ω−1.1 at high frequencies. Acoustic
spectra calculated using Eqs. 4 and 5 and this modified surface pressure model are also
shown in Fig. 9 and give the best prediction of experimental data across the measured
frequency range at all flow speeds.
At low flow speeds (U∞ = 20 and 15 m/s), the new model provides a reasonably good
approximation at all frequencies (apart from the 1.5 kHz peak due to facility effects as
described previously). At higher flow speeds, the model under-predicts the radiated noise
below approximately 500 Hz. The reasons for this and other deviations from previously
published semi-empirical models can be explained by examining the flow velocity spectrum
in the region close to the trailing edge as detailed in Section III.B.
B. Velocity spectra
According to Lighthill,41 fluctuating velocity is the source of all aerodynamic sound. The
trailing edge makes these aerodynamic sound sources more efficient via an edge diffraction
19
FIG. 9. The far-field acoustic spectra for U∞ of (a) 38, (b) 35, (c) 30, (d) 25, (e) 20 and (f)
15 m/s compared with spectra predicted using Eqs. 4 and 5 and the semi-empirical mod-
els of surface pressure developed by Chase24,27 (Chase), Howe29 (Chase-Howe), Smol’yakov
and Tkachenko28 (Smol.-Tka.), Smol’yakov30 (Smol.) and Goody31(Goody) as well as the
modified version of the surface pressure model developed by Smol’yakov and Tkachenko28
(Smol.-Tka. M) to account for low frequency mis-match.20
process.35 Fig. 10 shows velocity spectra measured in the near wake of the trailing edge
of the flat plate model at free-stream velocities of U∞ = 38 and 15 m/s. These spectra
are measured at various y/c locations above the trailing edge. As the velocity spectra
for the flat plate model are symmetric about the trailing edge at all flow speeds, only the
spectra measured above the trailing edge are shown. The spectra measured at flow velocities
between U∞ = 35 and 20 m/s have not been included here as they follow the same trend as
the spectra measured at U∞ = 38 and 15 m/s. At U∞ = 38 m/s, Fig. 10 (a) shows that the
highest energy levels are recorded at the measurement position closest to the trailing edge
(y/c = 0.002). As the measurement position moves away from the trailing edge, the spectra
reduce in amplitude, especially at high frequencies. The spectra all show high energy at low
frequencies. Above 15 m/s (Fig. 10 (a)) and in the outer regions of the boundary layer (see
y/c = 0.03) a prominent broad hump is observed in the spectra below 500 Hz. This is also
observed in the acoustic far field measurements for the same test cases in Fig. 5. This low
frequency energy is most likely due to eddies or convected flow perturbations created at the
sharp change in slope upstream of the sharp trailing edge. This change in slope will create a
sudden adverse pressure gradient and an associated rapid change in flow velocity. However,
the same low frequency component is not observed in the velocity spectrum at 15 m/s.
Similarly, the acoustic far-field spectrum does not display an excess in energy below 500 Hz.
Consequently, the improved semi-empirical model provides good agreement at 15 m/s. It is
possible that the low frequency component is present during the experiment, however the
shedding frequency was lower than 100 Hz, the lowest resolved frequency during the test.
IV. CONCLUSION
This paper has presented results of an experimental investigation on the noise generated
by a sharp-edged flat plate at low-to-moderate Reynolds numbers. The results include far-
field acoustic spectra and velocity spectra measured in the near wake of the trailing edge.
Examining the cross-correlation of noise measured above the leading and trailing edges
21
FIG. 10. Velocity spectra at various y/c locations for U∞ of (a) 38 and (b) 15 m/s.
demonstrated that while noise radiating from the leading edge contributed to the sound
field, trailing edge noise was the dominant noise mechanism. The trailing edge noise scaling
law of M5 was shown to give a good collapse of the far-field noise spectra at frequencies
above 1 kHz, demonstrating that the radiated sound level increases according to the M5
power law. Predictions of trailing edge noise using the BPM model and a semi-empirical
surface pressure approach were poor at low frequencies but in agreement with experimental
data above 2 kHz. To account for the low frequency mis-match, a modified version of the
surface pressure spectrum model developed by Smol’yakov and Tkachenko28 was proposed
to give more accurate prediction of the experimental data across the measured frequency
range at all flow velocities investigated.
22
Acknowledgments
This work has been supported by the Australian Research Council under grant
DP1094015 ‘The mechanics of quiet airfoils’.
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26
TABLE I. Boundary layer parameters for Plate One.
U∞, m/s δ ×10−3, m δ∗ ×10−3, m θ ×10−3, m
38 8.2 1.24 1.14
35 8.3 1.40 1.17
30 8.6 1.52 1.24
25 8.8 1.81 1.22
20 9.2 1.88 1.24
15 9.4 1.95 1.28
27
List of Figures
FIG. 1 Schematic diagrams of the flat plate model leading and trailing edge. . . . . 6
FIG. 2 Schematic diagram of the flat plate model secured in the housing and attached
to the contraction outlet. (a) Side view and (b) front view. . . . . . . . . . . 7
FIG. 3 The flat plate model attached to the contraction outlet with the extension
plates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
FIG. 4 Microphone positions relative to the flat plate model where a = 0.585 m,
b = 0.600 m, d = 0.588 m, e = 0.618 m and f = 0.075 m. . . . . . . . . . . . 8
FIG. 5 Far-field acoustic spectra for the flat plate model for U∞ of (a) 38, (b) 35,
(c) 30, (d) 25, (e) 20 and (f) 15 m/s. . . . . . . . . . . . . . . . . . . . . . . 10
FIG. 6 Cross-correlation between the top trailing edge microphone signal and the
leading edge microphone signal for U∞ of: (a) 38 and (b) 15 m/s (normalised
to the overall maxima). Microphone signals have been bandpassed between
800 and 104 Hz. The time delays between sound radiated to the top trailing
edge microphone and the leading edge microphone from the trailing edge,
∆tTE, and the leading edge, ∆tLE, are shown with black dash-dot lines.
Values for ∆tTE and ∆tLE are calculated from geometry using Eqns. 1 and 2. 13
FIG. 7 One-third-octave band spectra scaled with M5. . . . . . . . . . . . . . . . . 14
FIG. 8 One-third-octave band spectra compared with one-third-octave band spectra
predicted with the BPM model (dashed lines). . . . . . . . . . . . . . . . . . 15
FIG. 9 The far-field acoustic spectra for U∞ of (a) 38, (b) 35, (c) 30, (d) 25, (e)
20 and (f) 15 m/s compared with spectra predicted using Eqs. 4 and 5
and the semi-empirical models of surface pressure developed by Chase24,27
(Chase), Howe29 (Chase-Howe), Smol’yakov and Tkachenko28 (Smol.-Tka.),
Smol’yakov30 (Smol.) and Goody31(Goody) as well as the modified version
of the surface pressure model developed by Smol’yakov and Tkachenko28
(Smol.-Tka. M) to account for low frequency mis-match. . . . . . . . . . . . 20
28
FIG. 10 Velocity spectra at various y/c locations for U∞ of (a) 38 and (b) 15 m/s. . 22
29
30