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On the effect of acceleration on trailing edge noise
radiation from rotating blades
Samuel Sinayoko1 Mike Kingan2 Anurag Agarwal1
1University of Cambridge, UK
2University of Southampton, Institute of Sound and Vibration Research, UK
19th AIAA/CEAS Aeroacoustics Conference, Berlin, 29 May 2013
Outline
Background
Point source model
Distributed source model
Conclusions
2 of 28
Background: isolated airfoils
Amiet 1974, 1975, 1976
S pp = a |Ψ|2 lS Sqq
3 of 28
Background: rotating airfoils
Observer
Flow
γ
x
y
z
α
θΩ
Schlinker and Amiet (1981)
Spp(ω) =1
T
T
0Spp(ω, t)dt
4 of 28
Background: rotating airfoils
Observer
Flow
γ
x
y
z
α
θΩ
Schlinker and Amiet (1981)
Spp(ω) =1
T
T
0Spp(ω, t)dt
4 of 28
Amiet’s key idea
Lowson (1965)
p ∼ F+ aF M
F ∼ ω F M ∼ ΩM
5 of 28
Amiet’s key idea: ω Ω
Lowson (1965)
p ∼ F
→ Source in rectilinear motion
6 of 28
Previous work: theory
Sinayoko, Kingan and Agarwal (AIAA 2012, arXiv 2013)
Formulations for time averaged PSD
Without acceleration:
Spp =
A |Ψ|2 ly Sqq dγ
With acceleration:
Spp =
m
B |Ψ|2 ly Sqq
7 of 28
Previous work: results – Wind Turbine kC = 50
30
6090
120
150
0180
5959 4949
8 of 28
Previous work: results – Propeller Cruise kC = 50(2013)
30
6090
120
150
0180
5757 5151
9 of 28
Objectives
Range of validity?
Is error due to acoustic lift or acceleration?
10 of 28
Outline
Background
Point source model
Distributed source model
Conclusions
11 of 28
Theory
Rotating dipole
Solved exactly with or without acceleration
Source spectrum by Chou and George (1984)
Not using the acoustic lift
12 of 28
Directivity: propeller at cruise
ω/Ω = 100
30
6090
120
150
0180
9090 8080
13 of 28
Sound pressure level
10−5
10−3
10−1
101
|∆SPL| ∞
10−4 10−2 100 102 104
ω/Ω
14 of 28
Sound power level
10−7
10−5
10−3
10−1
101|∆
SPW
L|
10−4 10−2 100 102 104
ω/Ω
→ Works even at low frequencies...?
15 of 28
Sound power level
10−7
10−5
10−3
10−1
101|∆
SPW
L|
10−4 10−2 100 102 104
ω/Ω
→ Works even at low frequencies...?
15 of 28
Instantaneous PSD: theory
Inverse Fourier transform over of cross-correlation function
Rpp(ω,) = E[p(ω+/2)p(ω−/2)]
→ Spp(ω, t)
16 of 28
Instantaneous PSD: theory
Inverse Fourier transform over of cross-correlation function
Rpp(ω,) = E[p(ω+/2)p(ω−/2)]
→ Spp(ω, t)
16 of 28
Instantaneous PSD: wind turbine
ω/Ω = 10
0.0 0.2 0.4 0.6 0.8 1.0t/T
62.0
62.2
62.4
62.6
62.8
63.0
63.2
63.4
Inst
ant.
PSD
(dB
re2×
10−5)
-0.75
-0.55
-0.35
-0.15
0.05
0.25
0.45
0.65
0.85
Inst
ant.
PSD
min
usti
me
aver
aged
PSD
(dB
)
17 of 28
Instantaneous PSD: propeller at cruise
ω/Ω = 10
0.0 0.2 0.4 0.6 0.8 1.0t/T
84.60
84.65
84.70
84.75
84.80
84.85
84.90
84.95
85.00
Inst
ant.
PSD
(dB
re2×10
−5)
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
Inst
ant.
PSD
min
usti
me
aver
aged
PSD
(dB
)
18 of 28
Instantaneous PSD: propeller at cruise
ω/Ω = 1
0.0 0.2 0.4 0.6 0.8 1.0t/T
61.2
61.4
61.6
61.8
62.0
62.2
62.4
62.6
62.8
Inst
ant.
PSD
(dB
re2×
10−5)
-0.94
-0.74
-0.54
-0.34
-0.14
0.06
0.26
0.46
0.66
0.86
Inst
ant.
PSD
min
usti
me
aver
aged
PSD
(dB
)
18 of 28
Instantaneous PSD: propeller at cruise
ω/Ω = 0.1
0.0 0.2 0.4 0.6 0.8 1.0t/T
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
Inst
ant.
PSD
(dB
re2×
10−5)
-2.30
-1.80
-1.30
-0.80
-0.30
0.20
0.70
1.20
Inst
ant.
PSD
min
usti
me
aver
aged
PSD
(dB
)
18 of 28
Instantaneous PSD: propeller at cruise
ω/Ω = 10
0.00 0.25 0.50 0.75t/TΩ
−4
−3
−2
−1
0
1
2
3
4
Inst
ant.
acce
lera
tion
/ave
rage
dPS
D×10−2
19 of 28
Instantaneous PSD: propeller at cruise
ω/Ω = 1
0.00 0.25 0.50 0.75t/TΩ
−0.15
−0.10
−0.05
0.00
0.05
0.10
0.15
Inst
ant.
acce
lera
tion
/ave
rage
dPS
D
19 of 28
Instantaneous PSD: propeller at cruise
ω/Ω = 0.1
0.00 0.25 0.50 0.75t/TΩ
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
0.4
Inst
ant.
acce
lera
tion
/ave
rage
dPS
D
19 of 28
Outline
Background
Point source model
Distributed source model
Conclusions
20 of 28
Directivity: propeller at cruise
ω/Ω = 100
30
6090
120
150
0180
6060 5454
21 of 28
Sound pressure level
10−1
100
101
|∆SPL| ∞
101 102 103 104
ω/Ω
22 of 28
Sound power level
10−3
10−2
10−1
100
|∆SPW
L|
101 102 103 104
ω/Ω
23 of 28
Instantaneous PSD: propeller at cruise
ω/Ω = 10
0.0 0.2 0.4 0.6 0.8 1.0t/T
57.0
57.5
58.0
58.5
59.0
59.5
60.0
Inst
ant.
PSD
(dB
re2×
10−5)
-1.31
-0.81
-0.31
0.19
0.69
1.19
1.69
Inst
ant.
PSD
min
usti
me
aver
aged
PSD
(dB
)
24 of 28
Instantaneous PSD: propeller at cruise
ω/Ω = 1
0.0 0.2 0.4 0.6 0.8 1.0t/T
55.0
55.2
55.4
55.6
55.8
56.0
56.2
56.4
56.6
56.8
Inst
ant.
PSD
(dB
re2×
10−5)
-0.85
-0.65
-0.45
-0.25
-0.05
0.15
0.35
0.55
0.75
0.95
Inst
ant.
PSD
min
usti
me
aver
aged
PSD
(dB
)
24 of 28
Instantaneous PSD: propeller at cruise
ω/Ω = 0.1
0.0 0.2 0.4 0.6 0.8 1.0t/T
48.0
48.5
49.0
49.5
50.0
50.5
Inst
ant.
PSD
(dB
re2×
10−5)
-1.55
-1.05
-0.55
-0.05
0.45
0.95
Inst
ant.
PSD
min
usti
me
aver
aged
PSD
(dB
)
24 of 28
Outline
Background
Point source model
Distributed source model
Conclusions
25 of 28
Conclusions
1. Schlinker and Amiet’s theory is applicable even
at high subsonic speeds
2. It is most accurate when ω Ω
3. It remains accurate at much lower frequencies
ω ≈ Ω
4. The limiting factor is likely to be the acoustic lift,
not the effect of acceleration.
26 of 28
Conclusions
1. Schlinker and Amiet’s theory is applicable even
at high subsonic speeds
2. It is most accurate when ω Ω
3. It remains accurate at much lower frequencies
ω ≈ Ω
4. The limiting factor is likely to be the acoustic lift,
not the effect of acceleration.
26 of 28
Conclusions
1. Schlinker and Amiet’s theory is applicable even
at high subsonic speeds
2. It is most accurate when ω Ω
3. It remains accurate at much lower frequencies
ω ≈ Ω
4. The limiting factor is likely to be the acoustic lift,
not the effect of acceleration.
26 of 28
Conclusions
1. Schlinker and Amiet’s theory is applicable even
at high subsonic speeds
2. It is most accurate when ω Ω
3. It remains accurate at much lower frequencies
ω ≈ Ω
4. The limiting factor is likely to be the acoustic lift,
not the effect of acceleration.
26 of 28
Thank you!
Instantaneous PSD: propeller at cruise
ω/Ω = 10
0.00 0.25 0.50 0.75t/TΩ
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
Inst
ant.
acce
lera
tion
/ave
rage
dPS
D
28 of 28
Instantaneous PSD: propeller at cruise
ω/Ω = 1
0.00 0.25 0.50 0.75t/TΩ
−0.2
−0.1
0.0
0.1
0.2
0.3
0.4
Inst
ant.
acce
lera
tion
/ave
rage
dPS
D
28 of 28
Instantaneous PSD: propeller at cruise
ω/Ω = 0.1
0.00 0.25 0.50 0.75t/TΩ
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
Inst
ant.
acce
lera
tion
/ave
rage
dPS
D
28 of 28
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