Trailing edge noise for rotating blades

Analysis and comparison of two classical approaches

Samuel Sinayoko1 Mike Kingan2 Anurag Agarwal1

1University of Cambridge, UK2University of Southampton, Institute of Sound and Vibration Research, UK

18th AIAA/CEAS Aeroacoustics Conference, Colorado Springs, 6 June 2011

Motivation

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Outline

Isolated airfoil theory

Amiet’s theory for rotating airfoils

Kim-George’s theory for rotating airfoils

Results

Conclusions

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Review

Trailing edge for isolated airfoils

I Amiet 1974, 1975, 1976I Roger and Moreau 2005, 2009

Trailing edge for rotating airfoils

I Amiet 1976I Schlinker and Amiet 1981I Kim and George 1982I Blandeau and Joseph 2011

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Review

Trailing edge for isolated airfoils

I Amiet 1974, 1975, 1976I Roger and Moreau 2005, 2009

Trailing edge for rotating airfoils

I Amiet 1976I Schlinker and Amiet 1981I Kim and George 1982I Blandeau and Joseph 2011

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Outline

Isolated airfoil theory

Amiet’s theory for rotating airfoils

Kim-George’s theory for rotating airfoils

Results

Conclusions

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Isolated airfoil theory

I Amplitude

Spp = a |Ψ|2 ly Sqq

I Acoustic liftI Spanwise correlation lengthI Surface spectrum

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Isolated airfoil theory

I Amplitude

Spp = a |Ψ|2 ly Sqq

I Acoustic liftI Spanwise correlation lengthI Surface spectrum

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Isolated airfoil theory

I Amplitude

Spp = a |Ψ|2 ly Sqq

I Acoustic lift

I Spanwise correlation lengthI Surface spectrum

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Isolated airfoil theory

I Amplitude

Spp = a |Ψ|2 ly Sqq

I Acoustic liftI Spanwise correlation length

I Surface spectrum

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Isolated airfoil theory

I Amplitude

Spp = a |Ψ|2 ly Sqq

I Acoustic liftI Spanwise correlation lengthI Surface spectrum

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Outline

Isolated airfoil theory

Amiet’s theory for rotating airfoils

Kim-George’s theory for rotating airfoils

Results

Conclusions

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Rotating airfoil

Observer

Flow

γ

x

y

z

α

θΩ

Spp(ω) =

∫Spp(ω,γ)dt

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Rotating airfoil

Observer

Flow

γ

x

y

z

α

θΩ

Spp(ω) =

∫Spp(ω,γ)dt

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Change of reference frame

rotor plane

E

P

MSO

MFO

Observerz

θ

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Change of reference frame

rotor plane

E

PMSOMFO

Observerz

θ

MFS z′θ′

MOS

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Change of reference frame

P

Observer

MFS z′θ′

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Moving observer in wind tunnel

I p(x, t) : no effectI p(x,ω) : Doppler shift ω ′ → ω

PSD for moving observer

Spp(ω,γ) =ω ′

ωS ′pp(ω

′,γ)

Time for moving observer

dt =ω ′

ωdt ′

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Moving observer in wind tunnel

I p(x, t) : no effectI p(x,ω) : Doppler shift ω ′ → ω

PSD for moving observer

Spp(ω,γ) =ω ′

ωS ′pp(ω

′,γ)

Time for moving observer

dt =ω ′

ωdt ′

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Moving observer in wind tunnel

I p(x, t) : no effectI p(x,ω) : Doppler shift ω ′ → ω

PSD for moving observer

Spp(ω,γ) =ω ′

ωS ′pp(ω

′,γ)

Time for moving observer

dt =ω ′

ωdt ′

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Amiet’s theory for rotating blade

Spp(ω) =1

2π

∫ 2π0

(ω ′

ω

)2

S ′pp(ω

′,γ)dγ

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But... not everyone agrees

Spp(ω) =1

2π

∫ 2π0

(ω ′

ω

)eS ′pp(ω

′,γ)dγ

Source Exponent eAmiet (1976) 1Rozenberg et al (2010) 1Schlinker and Amiet (1981) 2Blandeau and Joseph (2011) -2

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Outline

Isolated airfoil theory

Amiet’s theory for rotating airfoils

Kim-George’s theory for rotating airfoils

Results

Conclusions

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Kim-George approach

p(x, t)

p(x,ω)

Spp(x,ω)

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Kim-George approach

I Amplitude

Spp =∑m

B |Ψ|2 ly Sqq

I Acoustic liftI Spanwise correlation lengthI Surface spectrum

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Kim-George approach

I Amplitude

Spp =∑m

B |Ψ|2 ly Sqq

I Acoustic liftI Spanwise correlation lengthI Surface spectrum

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Kim-George approach

I Amplitude

Spp =∑m

B |Ψ|2 ly Sqq

I Acoustic lift

I Spanwise correlation lengthI Surface spectrum

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Kim-George approach

I Amplitude

Spp =∑m

B |Ψ|2 ly Sqq

I Acoustic liftI Spanwise correlation length

I Surface spectrum

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Kim-George approach

I Amplitude

Spp =∑m

B |Ψ|2 ly Sqq

I Acoustic liftI Spanwise correlation lengthI Surface spectrum

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Summary

Amiet:

Spp =

∫A |Ψ|2 ly Sqq dγ

Kim-George:

Spp =∑m

B |Ψ|2 ly Sqq

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Outline

Isolated airfoil theory

Amiet’s theory for rotating airfoils

Kim-George’s theory for rotating airfoils

Results

Conclusions

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Blade elements

Blade Mach number

Pit

chan

gle

Wind Turbine

Open propellorat cruise

Open propellorat take-off

Cooling fan

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Wind turbine kc = 5

0

30

60

90

120

150

1807272 6262

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Open propellor takeoff kc = 5

0

30

60

90

120

150

1807878 6464

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Wind turbine kc=0.5

0

30

60

90

120

150

1807070 6060

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Wind turbine kc=5

0

30

60

90

120

150

1807272 6262

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Wind turbine kc=50

0

30

60

90

120

150

1805959 4949

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Cooling fan kc=0.5

0

30

60

90

120

150

1801010 55

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Cooling fan kc=5

0

30

60

90

120

150

1802727 1717

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Cooling fan kc=50

0

30

60

90

120

150

1802626 2121

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Open propellor take-off kc=0.5

0

30

60

90

120

150

1807171 6161

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Open propellor take-off kc=5

0

30

60

90

120

150

1807070 6060

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Open propellor take-off kc=50

0

30

60

90

120

150

1806767 5757

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Open propeller cruise kc=0.5

0

30

60

90

120

150

1807373 6868

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Open propeller cruise kc=5

0

30

60

90

120

150

1807373 6868

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Open propeller cruise kc=50

0

30

60

90

120

150

1807171 6666

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Outline

Isolated airfoil theory

Amiet’s theory for rotating airfoils

Kim-George’s theory for rotating airfoils

Results

Conclusions

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Conclusions

1. Right exponent in Amiet’s theory is 2

2. Amiet’s approach exact if Mch 6 0.85 andkc > 0.5

3. Applicable to a wide range of applications

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Conclusions

1. Right exponent in Amiet’s theory is 2

2. Amiet’s approach exact if Mch 6 0.85 andkc > 0.5

3. Applicable to a wide range of applications

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Conclusions

1. Right exponent in Amiet’s theory is 2

2. Amiet’s approach exact if Mch 6 0.85 andkc > 0.5

3. Applicable to a wide range of applications

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