the effect of boundary layer type on trailing edge noise...
TRANSCRIPT
The effect of boundary layer type on trailing edge noise
from sharp-edged flat plates at low-to-moderate
Reynolds number
Danielle J. Moreau∗, Laura A. Brooks, Con J. Doolan
School of Mechanical Engineering, The University of Adelaide, South Australia,
Australia, 5005
Abstract
This paper presents an experimental study of the trailing edge noise pro-
duced by sharp-edged flat plates with mixed laminar, transitional and tur-
bulent boundary layers at low-to-moderate Reynolds number. Mean and
unsteady velocity data in the wake and upstream of the trailing edge have
been measured using hot-wire anemometry at Reynolds numbers of Rec =
0.7×105−2.7×105, based on chord. These data are related to far-field noise
measurements to determine the flow mechanisms responsible for the trail-
ing edge noise in each flow regime. Attached turbulent boundary layers are
observed to yield the weakest noise generation mechanism while mixed tran-
sitional and laminar boundary layers at the trailing edge yield the strongest
mechanism producing both high levels of broadband noise and tonal noise
components. Vortex shedding in the wake is only observed when tonal noise
is produced and experimental results indicate that in this case, the tonal
∗Corresponding author. Tel: +618 8303 3252; fax: +618 8303 4367.Email addresses: [email protected] (Danielle J. Moreau),
[email protected] (Laura A. Brooks), [email protected](Con J. Doolan)
Preprint submitted to The Journal of Sound and Vibration February 6, 2012
noise is governed by vortex shedding processes at the trailing edge.
Keywords: Trailing edge noise, bevelled trailing edge, hot-wire
anemometry
1. Introduction
Trailing edge noise, which is produced by the scattering of boundary
layer vorticity at a sharp trailing edge [1, 2], is important to a broad range
of applications; however, most studies conducted in the past have focused
on high Reynolds number applications such as commercial aircraft, turbo-
machinery and wind turbines. Experimental datasets spanning different
flow regimes (i.e. laminar, transitional and turbulent) at low-to-moderate
Reynolds number are relatively rare despite their importance for under-
standing flow-induced noise generation from applications such as micro-wind-
turbines, unmanned air vehicles and underwater control surfaces.
At high Reynolds numbers (Rec > 1 × 106, based on chord), turbulent
boundary layers that are attached at the trailing edge generate broadband
noise. Depending on the degree of trailing edge bluntness, vortex shedding
may occur at the trailing edge, significantly altering the magnitude and spec-
trum of the radiated noise [1]. Experimental studies of turbulent trailing edge
noise at high Reynolds numbers (Rec > 1×106) have involved measuring the
surface pressure fluctuations, boundary layer properties and far-field noise
spectra for airfoil shapes [3, 4], asymmetric beveled trailing edges [5, 1, 6, 2]
and flat plate models [7]. These data have been used to validate predictions
of trailing edge noise calculated using the theory of Howe [8] and Amiet [9].
At low-to-moderate Reynolds number, the flow structure over the airfoil is
2
complex and highly varied for different airfoil geometries. Small alterations to
the airfoil angle of attack, shape or flow speed can lead to significant changes
in the statistics of the flow and in the radiated trailing edge noise. Previ-
ous experimental studies on trailing edge noise at low-to-moderate Reynolds
number (Rec < 5 × 105) have largely focused on understanding the mecha-
nism responsible for tonal noise production when the flow is laminar on one
or both sides of the airfoil at the trailing edge. Paterson et al. [10] conducted
one of the first experimental studies on airfoil self-noise at low-to-moderate
Reynolds number. By measuring the cross-correlation of boundary layer
and far-field acoustic data from NACA0012 and NACA0018 airfoils, they
concluded that airfoil tonal noise is governed by vortex shedding from the
trailing edge. Tam [11] analysed the measurements of Paterson et al. [10] and
proposed that tonal noise is instead produced by an aeroacoustic feedback
loop between the first point of boundary layer instability and a point in the
wake which acts as the noise source. The aeroacoustic feedback loop pro-
posed by Tam [11] has since been modified in experimental studies on airfoil
tonal noise at low-to-moderate Reynolds number by a number of researchers
[12, 13, 14, 15], who have suggested that a feedback loop between instabilities
in the laminar boundary layer and acoustic waves generated at the trailing
edge is responsible for tonal noise. Nash et al. [16] and Mc Alpine et al. [17]
measured far-field acoustic and boundary layer data for a NACA0012 airfoil
at low-to-moderate Reynolds number and proposed that airfoil tonal noise
is generated by the diffraction of boundary layer T-S waves at the trailing
edge that are strongly amplified by the inflectional mean velocity profile in
the separated shear layer at the trailing edge. In a recent study by the
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authors [18] on the tonal noise radiated by a sharp-edged flat plate at low-
to-moderate Reynolds number, far-field noise and unsteady velocity data in
the wake were measured simultaneously and indicated that the tonal noise
production process is governed by vortex shedding at the sharp trailing edge.
Roger and Moreau [19] have measured some of the only available experi-
mental trailing edge noise data in different flow regimes at Reynolds numbers
of Rec < 3.5×105. In their study, the far-field noise and wall pressure fluctua-
tions close to the trailing edge of a Valeo airfoil were measured simultaneously
for turbulent, separated and laminar boundary layers. These various bound-
ary layer types were obtained by changing the airfoil angle of attack. These
data were then used to validate theoretical predictions of trailing edge noise
calculated using an extension of the theory developed by Amiet [9]. Wang
et al. [20] used LES (Large-Eddy Simulation) to compute the wall pressure
fluctuations and noise produced by flow over the cambered airfoil studied by
Roger and Moreau [19] at a Reynolds number of Rec = 1.5×105. Predictions
of surface pressure fluctuations in the trailing edge region and the far-field
noise calculated using the Ffowcs-Williams and Hall solution to the Lighthill
equation showed agreement with the experimental measurements of Roger
and Moreau [19].
Limited published data exist comparing the noise generated by an airfoil
or a flat plate with different flow regimes on the upper and lower surface
simultaneously at low-to-moderate Reynolds number. As part of their recent
study on airfoil tonal noise, Golubev et al. [21] investigated the effect of single
sided tripping on the noise radiated by a NACA 0012 airfoil at a range of
angles of attack and Reynolds numbers of Rec = 0.5 × 105 − 3 × 105. They
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found that the level of contribution of pressure and suction side boundary
layer instabilities to the radiated noise depends on flow velocity. This paper
presents an experimental study of the trailing edge noise radiated by three flat
plate models with mixed boundary layers (mixed turbulent, transitional and
laminar boundary layers on the upper and lower plate surfaces) at Reynolds
numbers of Rec = 0.7 × 105 − 2.7 × 105. Trailing edge flow and noise data
for each of the three flat plate models are presented and compared to give
insight into the flow mechanisms responsible for the production of trailing
edge noise in different flow regimes. The paper is structured as follows:
Section 2 provides details of the experimental facilities and method; Section
3 presents the experimental results including mean and rms velocity profiles
at the trailing edge, velocity spectra in the wake and far-field acoustic data;
and the paper is concluded in Section 4.
2. 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 250 Hz.
The anechoic wind tunnel contains a contraction outlet that is rectangular
in cross section and has dimensions of 75 mm (height) × 275 mm (width).
The maximum flow velocity of the free jet is ∼ 40 m/s and the free-stream
turbulence intensity was measured to be 0.3% at the outlet plane [22].
The three flat plate models used in this experimental campaign have a
chord of c = 200 mm, a span of s = 450 mm and a thickness of h = 5
5
Figure 1: Schematic diagram of the flat plates (chord = 200 mm and span = 450 mm).(a) Plate One, (b) Plate Two and (c) Plate Three.
mm. The leading and trailing edge geometries for each of the three flat
plate models are shown graphically in Fig. 1 and summarised in Table 1. It
will be shown in Section 3.1 that the leading and trailing edge geometries
of these three plates, combined with the on-coming flow conditions, result
in two different cases of mixed boundary layers at low-to-moderate Reynolds
number: turbulent and turbulent on the top and bottom surfaces of Plate One
and transitional and laminar on the top and bottom surfaces of Plates Two
and Three. The flat plate models were each secured between two sideplates
attached to the contraction flange in the wind tunnel test section at zero
angle of attack, as shown in Fig. 2. End-effects were eliminated by extending
the span of the flat plate models well beyond the width of the contraction
outlet.
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Table 1: Leading and trailing edge geometries of the three flat plate models.
Plate Leading edge (LE) Trailing edge (TE)
One Circular with radius of 2.5 mm Symmetrical wedge shapedwith an apex angle of 12◦
Two Elliptical with semi-major axisof 8 mm and semi-minor axisof 2.5 mm
Asymmetrical wedge shapedwith an apex angle of 6◦
Three Elliptical with semi-major axisof 8 mm and semi-minor axisof 2.5 mm
Asymmetrical wedge shapedwith an apex angle of 12◦
Housing
TELE
Flat plate
Side plate
Contraction outlet
Flow
200 mm
(a)
Flat plate5 mm275 mm
450 mm
75 mm
Contraction outlet
Contraction flange
Contraction flange
(b)
Side plate
Figure 2: Schematic diagram of the flat plate model secured in the housing and attachedto the contraction outlet. (a) Side view and (b) front view.
7
Acoustic measurements were taken using two B&K 1/2” microphones
(Model No. 4190): one 583 mm directly above and one 583 mm directly
below the trailing edge. To provide isolation from wind noise, wind socks
were placed on all the microphones prior to data collection. The method for
extracting and analysing trailing edge noise developed by Moreau et al. [23]
was implemented. This method removes extraneous noise sources from the
far-field noise measurements using the two phase-matched microphones lo-
cated 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.
Hot-wire anemometry was used to obtain boundary layer profiles and
unsteady velocity data about the plate trailing edge and in the wake. A
TSI 1210-T1.5 single-wire probe with a wire length of 1.27 mm and a wire
diameter of 3.81 µm was connected to a TSI IFA300 constant temperature
anemometer system and positioned using a Dantec automatic traverse with
6.25 µm positional accuracy. The traverse allowed continuous movement in
the streamwise (x) and vertical (y) directions. Note that +y and −y are
denoted ‘above’ and ‘below’ the plate trailing edge, respectively. Both the
far-field noise measurements and the velocity data were collected using a
National Instruments board at a sampling frequency of 215 Hz for a sample
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time of 8 s.
Experiments were conducted at free-stream velocities of U∞ = 5 − 20
m/s, corresponding to Reynolds numbers of Rec = 0.7× 105 − 2.7× 105.
3. Experimental results
The experimental results are presented in Sections 3.1 to 3.4 as follows:
Section 3.1 presents mean and unsteady flow data in the very near trailing
edge wake of the three flat plate models; far-field noise measurements are
given in Section 3.2; velocity spectra measured in the wake of the three flat
plate models are presented in Section 3.3; and analysis of the tonal noise
mechanism of Plate Three is given in Section 3.4.
3.1. Mean and rms velocity profiles at the trailing edge
The mean velocity profiles (U/U∞) measured in the very near wake (∼
0.7 mm downstream of the trailing edge) of the three flat plate models at
U∞ = 20 and 15 m/s (corresponding to Reynolds numbers of Rec = 2.7×105
and 2.0× 105, respectively) are compared with the Blasius solution in Fig. 3.
This figure shows that for all three plates, the flow at the trailing edge is
approaching a more turbulent mean velocity profile at the higher flow speed
of U∞ = 20 m/s. Additionally, the width of the profiles and hence the
width of the wake of the three plates slightly increases as the flow speed is
reduced, as expected. The mean velocity profiles measured at speeds below
U∞ = 15 m/s (not shown for brevity) are qualitatively similar to the results
at U∞ = 15 m/s in Fig. 3 (b).
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The mean velocity profiles for Plate One (with symmetric trailing edge)
in Fig. 3 are largely symmetric about the trailing edge position at both flow
speeds. Any slight differences in the mean velocity profiles above and below
the trailing edge are likely due to small physical variations in the upper and
lower surfaces of the plate or slight upstream flow non-uniformities that can
cause transition to occur at different chordwise locations. The mean velocity
profiles at both flow speeds indicate that the flow is turbulent at the trailing
edge of Plate One.
The turbulent flow state at the trailing edge of Plate One is strongly
affected by the shape of the leading edge which has a large effect on the
initial development of the boundary layer and therefore on the flow over
the entire plate surface [24]. As shown in Fig. 1, Plate One has a circular
leading edge while the leading edges of Plates Two and Three are elliptical.
Flow separation is likely to occur just after the circular leading edge of Plate
One. A separated laminar shear layer is highly unstable and will undergo
transition to turbulence much faster than a laminar boundary layer attached
to the plate surface and will normally reattach to the surface in a turbulent
state [25]. Thus the circular leading edge acts as a type of boundary layer
trip on the surface of the airfoil.
At both 15 and 20 m/s, the mean velocity profiles below the trailing edge
of Plate Two in Fig. 3 are highly comparable to the theoretical Blasius pro-
file. On the top beveled surface, the mean velocity profiles deviate slightly
from the Blasius profile indicating transitional flow. This transition is associ-
10
ated with the adverse pressure gradient created by the beveled trailing edge
geometry. At both flow speeds, the flow at the trailing edge of Plate Two is
laminar on the lower flat surface and in the beginning stages of transition on
the top beveled surface.
The mean velocity profiles for Plate Three in Fig. 3 are highly asymmetric
about the trailing edge at both flow speeds. Below the trailing edge, the
profiles have a Blasius velocity profile indicating that the flow is laminar
through the boundary layer on the lower flat surface of the plate. The profiles
show that the flow is approaching a more turbulent mean velocity profile on
the top beveled surface than on the lower flat surface. The steep angled
geometry of the 12◦ bevel creates a sudden adverse pressure gradient and
an associated rapid change in flow velocity resulting in turbulent flow at the
trailing edge. Plate Three therefore has mixed boundary layers at both flow
speeds; a laminar boundary layer on the lower flat surface and a transitional
boundary layer on the top beveled surface. An incompressible numerical
simulation of Plate Three in uniform flow at a Reynolds number of Rec =
2×105 (identical to the experiment at U∞ = 15 m/s) has been performed by
Doolan et al. [26]. Numerical mean velocity data show that the flow separates
at the bevel on the top surface of the plate and then reattaches just upstream
of the trailing edge, forming a separation bubble. The near wake hotwire data
in Fig. 3 are therefore measured after the point of reattachment and not in
the separated flow region.
The mean velocity data can be used to obtain an overall description of
11
the mean flow for Plates Two and Three. At U∞ = 15 m/s, the Plate
Two near wake data indicate that flow separation has not occurred on the
top surface as the mean velocity profile closely resembles the Blasius profile.
The adverse pressure gradient has thickened the boundary layer (compared
with the lower surface boundary layer), but it retains its laminar profile
throughout. At U∞ = 20 m/s, the Plate Two near wake mean velocity
profile shows a laminar inner profile, with significant deviation in the outer
region of the boundary layer. This suggests that the laminar boundary layer
separates at the bevel leading edge and reattaches as a laminar boundary
layer on the bevel just upstream of the trailing edge. The deviation from
a Blasius profile in the outer region is associated with the growth of the
separated shear layer over the bevel. At both U∞ = 15 and 20 m/s, Plate
Three shows similar behaviour to Plate Two at U∞ = 20 m/s. There is a
laminar inner profile and thicker profile in the outer layers due to the growth
of the separated shear layer. For Plate Three, the extent of this region is
greater due to the stronger adverse pressure gradient and delayed shear layer
reattachment position.
Normalised rms velocity fluctuations (u′/U∞) measured in the very near
wake of the three flat plate models are shown in Fig. 4 for U∞ = 20 and
15 m/s. As the flow speed is reduced, an increase in the turbulent kinetic
energy about the trailing edge and a wider wake are observed, which is in
agreement with the mean velocity profiles of Fig. 3.
At both flow speeds, the rms velocity fluctuations in Fig. 4 for Plate One
12
display a local minimum at the trailing edge position and show that most of
the turbulent energy is contained in regions just above and below the trailing
edge. The rms velocity fluctuations of this plate are symmetric about the
trailing edge indicating that a symmetric wake is formed.
The rms velocity fluctuations in Fig. 4 for Plates Two and Three also
display a local minimum at the trailing edge position at both flow speeds.
Most of the turbulent energy is contained in regions just above and below
the trailing edge with turbulent energy levels being much higher in the more
turbulent flow on the top beveled surface than in the laminar flow on the
lower flat surface. Additionally, the fluctuating energy levels on the top
beveled surfaces of these two plates are higher than those observed in the
turbulent boundary layers of Plate One. The mean velocity and rms velocity
profiles indicate that the wake is highly asymmetric and much thicker above
the trailing edge than below it. This finding is in agreement with previous
studies [27, 28, 29, 30].
3.2. Far-field noise measurements
Spectral maps of the far-field acoustic data for the three plates at flow
speeds between U∞ = 5 and 20 m/s are shown in Fig. 5. This figure shows
that the lowest noise levels are radiated by Plate One with turbulent bound-
ary layers on both surfaces at the trailing edge. The noise produced by both
Plate One with turbulent trailing edge flow and Plate Two with mixed lam-
inar and transitional boundary layers at the trailing edge is entirely broad-
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(a) (b)
Figure 3: Normalised mean velocity profiles measured in the very near wake of the threeplates in the vertical (y) direction at x/c = 0.0035 for U∞ of (a) 20 m/s (Rec = 2.7× 105)and (b) 15 m/s (Rec = 2.0× 105).
(a) (b)
Figure 4: Normalised rms velocity fluctuations measured in the very near wake of the threeplates in the vertical (y) direction at x/c = 0.0035 for U∞ of (a) 20 m/s (Rec = 2.7× 105)and (b) 15 m/s (Rec = 2.0× 105).
band in nature and displays an oscillation in magnitude that varies with
frequency (see Figs. 5 (a) and (b)). At frequencies below 1 kHz, the oscilla-
tion in magnitude observed in the sound spectra is likely due to the acoustic
performance of the anechoic wind tunnel chamber. Standing waves produced
by weak sound reflections off hard walled surfaces in the anechoic chamber,
14
such as the steel collector, are responsible for this low frequency magnitude
oscillation. At higher frequencies (above 1 kHz) the oscillation in magnitude
is attributed to the scattering of acoustic waves from the leading edge. In
Moreau et al. [31], the far-field noise spectra for Plate One is compared with
the trailing edge noise predicted using a surface pressure approach [3] and
Howe’s [33] correction for multiple scattering from the leading and trailing
edges. The oscillation in magnitude introduced to the noise prediction by
Howe’s leading edge scattering correction corresponds well to that observed
in the experimental data at frequencies above 1 kHz.
Figure 5 (c) shows that below U∞ = 20 m/s, Plate Three with mixed
transitional and laminar boundary layers at the trailing edge radiates the
highest noise levels of all three plates. Additionally, high magnitude tones
are observed in the far-field noise spectra at speeds between U∞ = 9 and
17 m/s. These tones are characteristic of the self-induced discrete frequency
noise generated by airfoils at low Reynolds numbers [11]. Figure 6 shows
the frequencies of the dominant tones, f2 - f5, radiated by Plate Three as
a function of free-stream velocity. This figure shows that the frequencies of
the tones increase with an increase in flow velocity and display a ladder type
frequency behaviour consistent with the results of previous researchers who
have investigated airfoil tonal noise at low Reynolds number [10, 15, 16]. The
frequencies of the flat plate tones on each ‘rung’ of the ladder structure scale
with free-stream velocity according to U1.25∞
. Paterson et al. [10] observed
that the frequencies of the tones radiated by NACA 0012 and NACA 0018
15
airfoils increased according to U0.8∞
for small increases in flow speed and that
the power law of U1.5∞
described the average frequency behaviour of the tones.
These scaling laws derived by Paterson et al. [10] for an airfoil do not describe
the ladder structure of the flat plate tonal noise frequencies. The discrepancy
in the frequency scaling laws is attributed to significant differences in the
geometry of the NACA 0012 and NACA 0018 airfoils used by Paterson et
al. [10] and the flat plate studied here which possibly results in a different
noise generation mechanism for Plate Three.
For closer inspection, the far-field acoustic spectra for the three flat plate
models at free-stream velocities between U∞ = 15 and 20 m/s are compared
with background noise spectra in Fig. 7. For the data in this figure, the
background noise has been measured with the top trailing edge microphone.
The noise radiated by each plate in the three different flow regimes is ap-
proximately equal at U∞ = 20 m/s, as shown in Fig. 7 (a). At this flow speed,
all three plate spectra display higher levels of low frequency noise relative to
the rest of the spectra. A previous study by the authors [34] has shown that
the three plates also radiate broadband noise that is approximately equal at
higher flow speeds between U∞ = 20 and 38 m/s, where U∞ = 38 m/s was
the highest flow velocity used during testing.
When the flow speed is reduced to U∞ = 19 m/s (Fig. 7 (b)), Plate Three
with turbulent and laminar flow, on its top and bottom surfaces, respectively,
radiates broadband noise that is slightly higher in magnitude than that ra-
diated by the other two plates. Additionally, below 1 kHz, the spectra for
16
Plate Two with mixed transitional and laminar boundary layers sits just
above that of Plate One with turbulent flow at the trailing edge. Despite
slight differences in magnitude, the noise radiated by the three plates in all
three flow regimes has similar spectral content at this flow speed.
At U∞ = 18 m/s, broad peaks appear in the spectra for Plate Three
as shown in Fig. 7 (c). When the flow speed is reduced below U∞ = 18
m/s, these broad peaks develop into a number of high magnitude tones (see
Figs. 7 (d), (e) and (f)) and the frequency of these tones decreases as the
flow speed is reduced. Below U∞ = 18 m/s, the broadband noise radiated by
Plate Three is much higher in magnitude than that radiated by the other two
plate models at frequencies less than 1.5 kHz. Additionally, Plate Two with
mixed transitional and laminar boundary layers radiates broadband noise
below 1.5 kHz that is higher in magnitude than that radiated by Plate One
with turbulent flow at the trailing edge. The noise radiated by the three
plates above 1.5 kHz is approximately equal and close to background noise
levels.
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. Leading edge noise is therefore still present in the
far-field noise measurements processed with the trailing edge noise extraction
technique [23]. It is worth noting that Moreau et al. [34] have experimentally
analysed the cross-correlation of noise measured above the leading and trail-
ing edges of the three plates at U∞ = 20 and 15 m/s to show that trailing
17
edge noise significantly dominates the radiated sound field over the noise
produced at the leading edge.
Figure 5: Spectral maps of the far-field acoustic data at U∞ = 5 – 20 m/s for (a) PlateOne, (b) Plate Two and (c) Plate Three. (Color online and black-and-white in print)
3.3. Wake velocity spectra
According to Lighthill [35], fluctuating velocity is the source of all aero-
dynamic sound. The presence of solid boundaries makes these aerodynamic
sound sources more efficient and dipolar when the wavelength of sound is
greater than the chord [36]. When the wavelength of sound is significantly
18
Figure 6: Tonal frequencies of Plate Three scaled with free-stream velocity.
smaller than the chord, an edge diffraction process occurs, changing the na-
ture of the sound source so that it has a non-multipole character [37]. The re-
sults presented in this paper have frequency content in both of these regimes.
To gain insight into the trailing edge noise mechanism of the different flow
regimes, velocity data were measured in the wake of the three plates at the
selected free-stream velocity of U∞ = 15 m/s.
Figure 8 shows spectral maps of the fluctuating velocity (u′2/Hz) for the
three plates measured in the wake in the streamwise direction at y/c =
−0.0035 when U∞ = 15 m/s. Also shown for ease of comparison are the
single velocity spectra for the three plates at two selected positions in the
wake for U∞ = 15 m/s in Fig. 9. At the position further from the trailing edge
(at x/c = 0.25 in Fig. 9 (b)), the slope of the velocity spectra at frequencies
above 1 kHz agrees well with the -5/3 slope, showing the existence of the
inertial range. Closer to the trailing edge (at x/c = 0.01 in Fig. 9 (a)), the
inertial range is not present, showing that the turbulence is non-equilibrium
19
(a) (b)
(c) (d)
(e) (f)
Figure 7: Far-field acoustic spectra for the three plates compared to background noisespectra for U∞ of (a) 20, (b) 19, (c) 18, (d) 17, (e) 16 and (f) 15 m/s.
20
and that the classical cascade process has not yet established itself.
Figures 8 (a) and (b) show that the wake velocity field close to the trailing
edge of Plate One with turbulent trailing edge flow and Plate Two with
mixed laminar and transitional boundary layers is comprised of broadband
random velocity fluctuations. The velocity fluctuations close to the trailing
edge of Plate Two are higher in magnitude than those of Plate One. This
corresponds to the far-field noise spectra of Fig. 7 (f) which show that Plate
Two produces higher levels of broadband trailing edge noise than Plate One.
The high energy observed in the velocity spectra close to the trailing edge
of Plate Two is likely due to eddies or flow perturbations created in the
transitional boundary layer as it negotiates the adverse pressure gradient
on the top beveled surface of the plate. The broadband trailing edge noise
produced by the turbulent boundary layers of Plate One and the mixed
transitional and laminar boundary layers of Plate Two is governed by the
random velocity fluctuations in the vicinity of the trailing edge.
The spectral map for Plate Three with transitional and laminar flow on
its top and bottom surfaces respectively, shown in Fig. 8 (c), displays high
intensity velocity fluctuations at the same frequencies as the tones observed in
the corresponding far-field noise spectrum of Fig. 7 (f). These high intensity
velocity fluctuations in the wake are due to vortex shedding from the trailing
edge and are responsible for the production of tonal noise (see Section 3.4).
At this flow speed, vortices are shed into the wake at the far-field tonal noise
frequencies corresponding to the 2nd through 5th harmonics, f2 – f5, of the
21
fundamental frequency, f1 = 244 Hz. High intensity velocity fluctuations
are also observed in Fig. 8 (c) at the fundamental vortex shedding frequency,
f1, despite no tone being observed in the far-field noise spectrum at this
frequency. Further explanation of these experimental results is given in the
next section on the tonal noise mechanism of Plate Three.
Figure 7 (f), shows that the tonal noise produced by Plate Three is ac-
companied by an increase in broadband sound pressure level below 1.5 kHz.
Fig. 8 (c) shows that the random velocity fluctuations about the trailing edge
of Plate Three are more energetic than those of the other two plates. Thus,
vortex shedding in the wake of Plate Three is accompanied by high magni-
tude random velocity fluctuations about the trailing edge over the frequency
band at which vortex shedding occurs.
Paterson et al. [10] state that a necessary condition for the production
of airfoil tonal noise at low Reynolds number is that the boundary layer on
at least one surface of the airfoil (usually the pressure surface) is laminar
at the trailing edge. While the flow is laminar on the lower flat surface of
both Plates Two and Three, tonal noise is only produced by Plate Three
(see Fig. 7 (f)). Figure 8 (b) shows no evidence of vortex shedding from the
beveled trailing edge of Plate Two at the very shallow angle of 6◦. This
supports the theory that the tonal noise produced at a sharp trailing edge
in low-to-moderate Reynolds number flow is governed by vortex shedding in
the wake.
No vortex shedding is observed in the turbulent wake of Plate One in
22
Fig. 8 (a) and this plate does not radiate any tonal noise components (see
Fig. 7 (f)). As stated by Blake [38], vortex shedding is not expected to occur
when a turbulent boundary layer exists on one or both sides of the airfoil at
a sharp trailing edge and flow separation does not occur. As the symmetric
wedge-shaped trailing edge of Plate One has a small included angle of 12◦,
the turbulent boundary layers are expected to remain attached at the trailing
edge, preventing vortex shedding in the wake and thus tonal noise from being
produced.
Figure 10 shows spectral maps of the fluctuating velocity (u′2/Hz) mea-
sured in the vertical direction in the very near wake of the three flat plates
for U∞ = 15 m/s. For all three plates, the velocity spectra display a mini-
mum at the trailing edge position with most of the energy being contained
in a well defined region about the trailing edge. The spectra for Plate One
in Fig. 10 (a) indicate a symmetric wake results from the symmetric trailing
edge geometry. For Plates Two and Three in Figs. 10 (b) and (c), the energy
levels in the wake are observed to be much higher on the top surface of the
plate in the more turbulent flow than below it. This indicates a highly asym-
metric wake and is in agreement with the mean and rms velocity profiles
given in Figs. 3 and 4.
Large peaks of high energy are visible in the velocity spectra for Plate
Three (Fig. 10 (c)) at the same frequencies as the tonal components in the
far-field noise spectra. At this near-wake position, no peak is visible in the
velocity spectra at the fundamental vortex shedding frequency, f1. This
23
is consistent with the velocity spectra in Fig. 8 (c) which shows that high
intensity velocity fluctuations are not detected at f1 until x/c ≈ 0.05.
(a) (b)
(c)
Figure 8: Spectral maps of the power spectral density of the fluctuating velocity measureddownstream of the trailing edge in the streamwise (x) direction at y/c = −0.0035 atU∞ = 15 m/s (Rec = 2.0 × 105) for (a) Plate One, (b) Plate Two and (c) Plate Three.(Color online and black-and-white in print)
3.4. The tonal noise mechanism of Plate Three
As shown in Figs. 5 (c) and 6, the tonal noise radiated by Plate Three
displays the ladder-type frequency structure often associated with a feedback
loop mechanism involving convected disturbances in the boundary layer and
24
(a) (b)
Figure 9: Velocity spectra for the three plates measured in the wake for U∞ = 15 m/s(Rec = 2.0× 105) at y/c = −0.0035 and (a) x/c = 0.01 and (b) x/c = 0.25.
acoustic waves produced at the trailing edge [12, 13, 14, 15]. Arbey and
Bataille [15] have derived equations that describe the total phase change
around the aeroacoustic feedback loop and their analysis can be used to
determine whether an aeroacoustic feedback loop is responsible for the tonal
noise produced by Plate Three.
The aeroacoustic feedback loop length, L, between hydrodynamic distur-
bances in the boundary layer and acoustic waves produced at the trailing
edge is [15]
L =n+ 0.5
f(
1cv
+ 1(c0−U∞)
) , (1)
where n = 1, 2, 3, ..., c0 is the local speed of sound, cv is the convective
velocity of the hydrodynamic disturbances and f is the acoustic frequency.
This aeroacoustic feedback model is applied to simultaneous measure-
ments of the far-field noise and flow on the top surface of the plate and in
the wake at the selected free-stream velocity of U∞ = 15 m/s. The far-field
25
(a) (b)
(c)
Figure 10: Spectral maps of the power spectral density of the fluctuating velocity measuredin the very near wake in the vertical (y) direction at x/c = 0.0035 at U∞ = 15 m/s(Rec = 2.0 × 105) for (a) Plate One, (b) Plate Two and (c) Plate Three. (Color onlineand black-and-white in print)
acoustic spectra for the flat plate at U∞ = 15 m/s is shown in Fig. 7 (f). As
stated in Section 3.3, the tones observed in the far-field noise spectra are the
2nd – 5th harmonics: f2 = 480 Hz; f3 = 729 Hz; f4 = 960 Hz and f5 = 1212
Hz, of the fundamental with frequency f1 ≈ 244 Hz.
Figure 11 shows a 2D map of the power spectral density of the fluctu-
ating velocity, u′2/Hz, measured in the horizontal streamwise direction from
a position 2 mm above the top flat surface of the plate (y/c = 0.035) when
26
U∞ = 15 m/s. In this figure, high intensity velocity fluctuations are visible at
the far-field tonal noise frequencies, f2 − f5, above the bevel upstream of the
trailing edge and in the wake. Figure 12 shows the phase difference between
the fluctuating velocity measured in the horizontal streamwise direction from
the position above the top flat surface of the plate and the far-field acoustic
noise at the peak tonal frequency f3 when U∞ = 15 m/s. The phase measure-
ments at tonal frequencies f2, f4 and f5 follow the same trend as those for f3
in Fig. 12 but are not shown for brevity. At locations where high intensity
velocity fluctuations are observed at f3 in the velocity spectra in Fig. 11,
the phase difference between the fluctuating velocity and far-field acoustic
signals at f3 in Fig. 12 varies linearly, confirming the development of strong
hydrodynamic fluctuations at this frequency. The convective velocity, cv, of
the hydrodynamic disturbances at tonal noise frequency f3 can be calculated
from the phase information in Fig. 12 according to
cv =1
m2πf, (2)
where m is the gradient of Fig. 12 given by m = ∆θd/∆x where θd is the
phase difference in radians and x is the probe position. Using Eq. (2) and
the gradient of Fig. 12 close to the trailing edge, flow disturbances at f3 have
a convective velocity of cv = 7.7 m/s. According to Eq. (1), the aeroacoustic
feedback loop length, L, for tonal frequency f3, where n = 3, is 36.4 mm.
Figure 13 (a) shows the power spectral density of the fluctuating velocity
27
measured in the horizontal streamwise direction from the position above
the top flat surface of the plate at f3 for U∞ = 15 m/s. In this figure,
high magnitude fluctuations at f3 are observed much closer to the trailing
edge than a distance of L from it. There is also no sudden increase in the
magnitude of the fluctuations at f3 at a distance L upstream of the trailing
edge as would be expected if acoustic waves produced at the trailing edge were
coupling with hydrodynamic fluctuations at this point. Figure 13 (b) shows
the coherence between the fluctuating velocity measured in the horizontal
streamwise direction above the top flat surface of the plate and the far-field
noise at f3 for U∞ = 15 m/s. This figure shows that very low coherence of
approximately 0.1 is measured between the acoustic and velocity signals at
f3 at a distance L upstream of the trailing edge. This is compared to high
coherence of approximately 0.7 between the acoustic and velocity signals
at f3 close to the trailing edge. The experimental measurements therefore
do not support a feedback loop between hydrodynamic fluctuations in the
boundary layer on the top surface of the plate and acoustic waves produced at
the trailing edge. It is worth noting that similar analysis has been performed
on experimental data measured below the lower flat surface of the plate (see
[39]) and no aeroacoustic feedback loop was found to exist in the boundary
layer on the plate lower surface.
The experimental results show that in this particular case and despite
some interesting similarities with other cases in the literature, the tonal noise
produced by Plate Three is generated without a feedback loop and that
28
vortex shedding is the mechanism responsible. Vortex shedding occurs in
the wake due to the sudden adverse pressure gradient produced by the steep
angled geometry of the beveled trailing edge. The high intensity velocity
fluctuations at tonal noise frequencies f2−f5 observed above the bevel and in
the wake in Fig. 11 are due to vortex shedding. Hydrodynamic fluctuations at
vortex shedding frequencies interact with the sharp trailing edge to produce
strong tonal noise. The numerical simulation of Plate Three in uniform flow
performed by Doolan et al. [26] also supports this hypothesis that vortex
shedding is the mechanism responsible for the tonal noise.
The wake velocity spectra measured in Fig. 8 (c) show high intensity
velocity fluctuations at the fundamental vortex shedding frequency, f1, de-
spite no tone being observed at this frequency in the far-field noise spectrum
(Fig. 7 (f)) or in the velocity spectra above the bevel upstream of the trailing
edge (Fig. 11). The fundamental peak in the velocity spectrum can be ex-
plained by the pairing of vortices shed from the upper and lower surfaces of
the plate. As the vortices form and pair they convect away from the trailing
edge, creating a moving velocity perturbation. The hot-wire probe is sensi-
tive to the streamwise component of this perturbation which occurs at half
the rate of vortex shedding. Results of the numerical simulation performed
by Doolan et al. [26] also confirm this model.
29
Figure 11: Spectral map of the power spectral density of the fluctuating velocity measuredin the horizontal (x) streamwise direction above the top flat surface of Plate Three (y/c =0.035) for U∞ = 15 m/s (Rec = 2.0× 105). (Color online and black-and-white in print)
Figure 12: Phase difference between the fluctuating velocity measured in the horizontalstreamwise direction above the top flat surface of Plate Three (y/c = 0.035) and thefar-field noise at f3 = 729 Hz for U∞ = 15 m/s (Rec = 2.0× 105).
4. Conclusion
This paper has presented results of an experimental investigation on the
noise generated by flat plate models with symmetric and asymmetric trailing
edge geometry and turbulent and mixed transitional and laminar boundary
30
(a) (b)
Figure 13: (a) Power spectral density of the fluctuating velocity measured in the horizontal(x) streamwise direction above the top flat surface of Plate Three (y/c = 0.035) and (b)the coherence between the fluctuating velocity and the far-field noise at f3 = 729 Hz forU∞ = 15 m/s (Rec = 2.0×105). The dashed vertical line indicates the theoretical feedbackloop length, L.
layers at low-to-moderate Reynolds number. The results include mean and
unsteady velocity data in the very near wake, velocity spectra measured
downstream of the trailing edge and far-field acoustic data. Below a Reynolds
number of Rec = 2.7×105, the trailing edge noise generated in three different
flow regimes differs significantly in magnitude and spectral content. Tonal
noise with a frequency ladder structure similar to that of airfoils at low
Reynolds numbers is produced by the flat plate with mixed turbulent and
laminar boundary layers at the trailing edge. Vortex shedding in the wake
occurs at the same frequencies as the tones in the far-field noise spectra and
analysis of the experimental results indicates that in this case, the tonal
noise is governed by vortex shedding processes at the trailing edge. An
aeroacoustic feedback loop was considered as the mechanism responsible for
the production of tonal noise, but the data were shown not be in agreeance
31
with such a mechanism. The noise radiated by the flat plate with mixed
laminar and transitional boundary layers is broadband in nature and higher
in magnitude than the broadband noise produced by the flat plate with
attached turbulent flow at the trailing edge. Velocity spectra measured in the
wake of these two plates indicate that the broadband noise is governed by the
small-scale random velocity fluctuations in the vicinity of the trailing edge.
The high levels of broadband trailing edge noise produced in transitional flow
are likely due to eddies or flow perturbations in the transitional boundary
layer.
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37
Figure Captions
Figure 1: Schematic diagram of the flat plates (chord = 200 mm and span = 450
mm). (a) Plate One, (b) Plate Two and (c) Plate Three.
Figure 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.
Figure 3: Normalised mean velocity profiles measured in the very near wake of
the three plates in the vertical (y) direction at x/c = 0.0035 for U∞ of
(a) 20 m/s (Rec = 2.7× 105) and (b) 15 m/s (Rec = 2.0× 105).
Figure 4: Normalised rms velocity fluctuations measured in the very near wake
of the three plates in the vertical (y) direction at x/c = 0.0035 for U∞
of (a) 20 m/s (Rec = 2.7× 105) and (b) 15 m/s (Rec = 2.0× 105).
Figure 5: Spectral maps of the far-field acoustic data at U∞ = 5 – 20 m/s for
(a) Plate One, (b) Plate Two and (c) Plate Three. (Color online and
black-and-white in print)
Figure 6: Tonal frequencies of Plate Three scaled with free-stream velocity.
Figure 7: Far-field acoustic spectra for the three plates compared to background
noise spectra for U∞ of (a) 20, (b) 19, (c) 18, (d) 17, (e) 16 and (f) 15
m/s.
Figure 8: Spectral maps of the power spectral density of the fluctuating veloc-
ity measured downstream of the trailing edge in the streamwise (x)
38
direction at y/c = −0.0035 at U∞ = 15 m/s (Rec = 2.0 × 105) for
(a) Plate One, (b) Plate Two and (c) Plate Three. (Color online and
black-and-white in print)
Figure 9: Velocity spectra for the three plates measured in the wake for U∞ = 15
m/s (Rec = 2.0 × 105) at y/c = −0.0035 and (a) x/c = 0.01 and (b)
x/c = 0.25.
Figure 10: Spectral maps of the power spectral density of the fluctuating velocity
measured in the very near wake in the vertical (y) direction at x/c =
0.0035 at U∞ = 15 m/s (Rec = 2.0× 105) for (a) Plate One, (b) Plate
Two and (c) Plate Three. (Color online and black-and-white in print)
Figure 11: Spectral map of the power spectral density of the fluctuating velocity
measured in the horizontal (x) streamwise direction above the top flat
surface of Plate Three for U∞ = 15 m/s (Rec = 2.0 × 105). (Color
online and black-and-white in print)
Figure 12: Phase difference between the fluctuating velocity measured in the hor-
izontal streamwise direction above the top flat surface of Plate Three
and the far-field noise at f3 = 729 Hz for U∞ = 15 m/s (Rec =
2.0× 105).
Figure 13: (a) Power spectral density of the fluctuating velocity measured in the
horizontal (x) streamwise direction above the top flat surface of Plate
Three and (b) the coherence between the fluctuating velocity and the
39
far-field noise at f3 = 729 Hz for U∞ = 15 m/s (Rec = 2.0× 105). The
dashed vertical line indicates the theoretical feedback loop length, L.
40