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

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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

)

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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

)

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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

)

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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