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: danielle.moreau@adelaide.edu.au

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