Fan Noise Simulation. Source modelling (LES) and experiments (Christophe, Moreau,. Roger, Wang) for simplified fan geometry: Valeo Controlled Diffusion.
Computation of Airfoil Trailing Edge Noise Based on Steady Flow Data Sebastian Remmler von Karman Institute for Fluid Dynamics, Rhode-St-Gen` ese (Belgium) Dresden University of Technology (Germany)
EPFDC Presentation, 15th July 2009
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Introduction
Theory
Validation
Application
Outline Computation of Airfoil Trailing Edge Noise Based on Steady Flow Data 1
Introduction: Aeroacoustics of fans and airfoils
2
Theoretical Background
3
Validation
4
Application to CD Airfoil
5
Conclusion and Perspective
Conclusion
Outline
Introduction
Theory
Validation
Outline
1
Introduction: Aeroacoustics of fans and airfoils
2
Theoretical Background
3
Validation
4
Application to CD Airfoil
5
Conclusion and Perspective
Application
Conclusion
Outline
Introduction
Theory
Validation
Application
Fan noise generation mechanisms Tonal noise:
Broadband noise:
thickness noise (for M 0)
turbulence-interaction noise (leading edge)
load noise
boundary layer noise (trailing edge)
rotor-stator interaction noise
Conclusion
Outline
Introduction
Theory
Validation
Application
Fan noise generation mechanisms Tonal noise:
Broadband noise:
thickness noise (for M 0)
turbulence-interaction noise (leading edge)
load noise
boundary layer noise (trailing edge)
rotor-stator interaction noise
Conclusion
Outline
Introduction
Theory
Validation
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Conclusion
Fan Noise Simulation Source modelling (LES) and experiments (Christophe, Moreau, Roger, Wang) for simplified fan geometry: Valeo Controlled Diffusion airfoil in uniform flow Analogies: extended Amiet, FW-Hall, FW-Hawkings
Objective: complex geometries → simpler methods necessary
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Application
Acoustic Source Modelling
DNS
LES
U-RANS
RANS stochastic or statistical model
pressure fluctuations = acoustic sources
broadband and tonal noise
tonal noise
broadband modelled
Conclusion
Outline
Introduction
Theory
Validation
Outline
1
Introduction: Aeroacoustics of fans and airfoils
2
Theoretical Background
3
Validation
4
Application to CD Airfoil
5
Conclusion and Perspective
Application
Conclusion
Outline
Introduction
Theory
Validation
Application
Statistical Model: General Idea Assumption: streamwise and spanwise homogeneity ∂ hu1 i (x2 )/∂x2 , u02 (x2 ), Λ(x2 ) (from RANS computation) R22 (˜ r1 , r˜3 ), Uc , α (modelled) ⇓ Statistical model explicit expression computed by Monte Carlo integration ⇓ Φpp (ω): surface pressure spectrum equivalent acoustic source
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Statistical Model of Panton & Linebarger [5] Wavenumber spectrum of pressure: ∞ 2 RRR k1 −k(x2 +ˆ x2 ) x2 ) S ∂hu1 i(x2 ) ∂hu1 i(ˆ φpp (k1 ) = 8ρ2 e dx2 dˆ x2 dk3 22 ∂x2 ∂x ˆ2 k2 0
Fourier transform of vertical velocity covariance: RR∞ u0 (x )u0 (ˆ x ) R22 cos(αk˜1 r˜1 ) cos(k˜3 r˜3 )d˜ r1 d˜ r3 S22 = 2 2π2 2 2 Λ2 0
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Conclusion
Statistical Model of Panton & Linebarger [5] Wavenumber spectrum of pressure: ∞ 2 RRR k1 −k(x2 +ˆ x2 ) x2 ) S ∂hu1 i(x2 ) ∂hu1 i(ˆ φpp (k1 ) = 8ρ2 e dx2 dˆ x2 dk3 22 ∂x2 ∂x ˆ2 k2 0
Fourier transform of vertical velocity covariance: RR∞ u0 (x )u0 (ˆ x ) R22 cos(αk˜1 r˜1 ) cos(k˜3 r˜3 )d˜ r1 d˜ r3 S22 = 2 2π2 2 2 Λ2 0
Modelled terms Velocity fluctuation correlation coefficient: √ 2 2 ˜ 2 R22 = 1 − √ 2 r˜ ˜ 2 e− r˜ +(˜x2 −xˆ2 ) 2
r˜ +(˜ x2 −x ˆ2 )
Ratio of streamwise to spanwise turbulence scales: 1 . α . 3 Convecion velocity (Φpp (ω) = φpp (ω/Uc )/Uc ): 0.5 . Uc /U∞ . 1
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Application
Amiet’s theory Analytical formula for the sound generated by the interaction of boundary layer turbulence with the trailing edge of an airfoil, developed by Amiet [1] and extended by Roger & Moreau [6] Sound radiated sound in the midspan plane (x = [R, θ, 0]): L sin θ 2 (k c)2 |I|2 Φpp (ω) ly (ω) Spp (x, ω) = 2π R 2 Φpp (ω) power spectral density of surface pressure fluctuations near trailing edge ly (ω) spanwise correlation length near trailing edge I transfer function for airfoil geometry c chord length L span
Conclusion
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Introduction
Theory
Validation
Outline
1
Introduction: Aeroacoustics of fans and airfoils
2
Theoretical Background
3
Validation
4
Application to CD Airfoil
5
Conclusion and Perspective
Application
Conclusion
Outline
Introduction
Theory
Validation
Application
Conclusion
Test Case I: BL with Adverse Pressure Gradient Boundary layer profiles available from Bradshaw [2], no RANS computation required. Experimental data directly used as input for statistical model: 1
0.12
〈u1〉 / U∞
0.08 0.6 0.06 0.4 0.04 〈u1〉 / U∞ u'2 / U∞ lm / δ
0.2 0 0
0.2
0.4
0.02 0 0.6
x2 / δ
0.8
1
u'2 / U∞ , lm / δ
0.1
0.8
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Test Case I: BL with Adverse Pressure Gradient Scale anisotropy factor α: -40 -45
10 log[Φ(ω) / U3∞ ρ2 δ*]
-50 -55 -60 -65 -70 experiment (Bradshaw) spectral model (α = 1) spectral model (α = 2) spectral model (α = 3) spectral model (α = α(k1δ))
-75 -80 -85 0.1
1 ω δ* / U∞
10
α = 3 for low wavenumbers, α = 1 for high wavenumbers, linear blending inbetween
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Test Case I: BL with Adverse Pressure Gradient Convection velocity Uc :
-40
-40
-45
-45
-50
-50 10 log[Φ(ω) / U3∞ ρ2 δ*]
10 log[Φ(ω) / U3∞ ρ2 δ*]
Scale anisotropy factor α:
-55 -60 -65 -70 experiment (Bradshaw) spectral model (α = 1) spectral model (α = 2) spectral model (α = 3) spectral model (α = α(k1δ))
-75 -80 -85 0.1
1 ω δ* / U∞
-55 -60 -65 -70 -75 experiment (Bradshaw) spectral model (a) Uc / U∞ = 0.9 spectral model (b) Uc / U∞ = f(k1δ) spectral model (c) Uc / Uτ = f(k1δ)
-80 -85 10
α = 3 for low wavenumbers, α = 1 for high wavenumbers, linear blending inbetween
0.1
1 ω δ* / U∞
Uc /U∞ = 0.9 for low wn, Uc /U∞ = 0.55 for high wn, linear blending inbetween
10
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Conclusion
Test Case II: BL with Zero Pressure Gradient mean velocity
velocity fluctuations
25
mixing length
5 experiment (Schewe) DNS (Spalart) RANS (K-ω) RANS (v2f) RANS (RSTM)
20
0.14
4.5 0.12 4 0.1
3.5
10
3
0.08 lm / δ
u+
100 u'2 / U∞
15
2.5
0.06
2 1.5
5
0.04
1
experiment (Schewe) RANS (K-ω) RANS (v2f) RANS (RSTM)
0.5 0
0 1
10
100 +
y
1000
lm (Prandtl) lm = 1.9 L (K-ω) lm = 0.65 L (v2f) lm = 1.9 L (RSTM)
0.02
0 0
0.2
0.4 0.6 x2 / δ
0.8
1
0
0.2
0.4 0.6 x2 / δ
0.8
⇒ rather similar RANS results for all used turbulence models (K-ω SST, v 2 f , RSTM)
1
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Conclusion
Test Case II: BL with Zero Pressure Gradient -45
-55
2 3
10 lg[Φ(ω) / ρ U∞δ*]
-50
-60
-65 experiment (Schewe) statistical model; RANS (K-ω) statistical model; RANS (v2f) statistical model; RANS (RSTM)
-70
-75 0.1
1 ω δ* / U∞
10
⇒ spectral results better with v 2 f and RSTM, compared to K-ω
Outline
Introduction
Theory
Validation
Outline
1
Introduction: Aeroacoustics of fans and airfoils
2
Theoretical Background
3
Validation
4
Application to CD Airfoil
5
Conclusion and Perspective
Application
Conclusion
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Introduction
Theory
Validation
Application
Conclusion
CD Airfoil in Jet Wind Tunnel
Experimental data: hot-wire and wall pressure spectra (ECL and MSU) [4] LES computations: Christophe [3] and Wang [7] BL data extracted on suction side near trailing edge Trailing edge noise: measured and computed by Amiet’s theory
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CD Airfoil: RANS Results mean velocity
velocity fluctuations
18
mixing length
3 experiment RANS (v2f) RANS (K-ω) RANS (RSTM)
16 2.5
RANS (v2f) RANS (K-ω) Prandtl RANS (RSTM)
1
14 0.8
10 8 6
lm [mm]
2 u'2 [m/s]
〈u1〉 [m/s]
12
1.5
0.6
0.4
1
4 2
experiment RANS (v2f) RANS (K-ω) RANS (RSTM)
0
0.2
0.5
0 0 1 2 3 4 5 6 7 8 9 10 x2 [mm]
0 0
1
2
3 4 5 x2 [mm]
6
7
8
0
1
2
3 4 x2 [mm]
5
⇒ similar results with K-ω and v 2 f , strong scattering of mixing length results, RSTM fails predicting the mean flow field
6
7
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CD Airfoil: Wall Pressure Spectra 75 70 65
Φ [dB re 2⋅10-5 Pa]
60 55 50 45 40 35 30 25 102
experiment statistical model; experimental flowfield statistical model; RANS (K-ω) statistical model; RANS (v2f) 103 f [Hz]
⇒ good spectral result with v 2 f , acceptable with K-ω
104
Conclusion
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CD Airfoil: Wall Pressure Spectra 75 70 65
Φ [dB re 2⋅10-5 Pa]
60 55 50 45 40 35 30 25 102
experiment LES (Christophe) LES (Wang) statistical model; experimental flowfield statistical model; RANS (K-ω) statistical model; RANS (v2f) 103 f [Hz]
104
⇒ good spectral result with v 2 f , acceptable with K-ω ⇒ differences due to turbulence model comparable to LES
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Application
CD Airfoil: Far-field Noise 40 30
SPL [dB re 2⋅10-5 Pa]
20 10 0 -10 -20 -30 -40 102
experiment statistical model; experimental flowfield statistical model; RANS (K-ω) statistical model; RANS (v2f) 103 f [Hz]
104
⇒ differences in wall pressure spectra are translated directly into similar differences in calculated far-field noise spectra
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CD Airfoil: Far-field Noise 40 30
SPL [dB re 2⋅10-5 Pa]
20 10 0 -10 -20 -30 -40 102
experiment LES (Christophe) LES (Wang) statistical model; experimental flowfield statistical model; RANS (K-ω) statistical model; RANS (v2f) 103 f [Hz]
104
⇒ differences in wall pressure spectra are translated directly into similar differences in calculated far-field noise spectra
Conclusion
Outline
Introduction
Theory
Validation
Outline
1
Introduction: Aeroacoustics of fans and airfoils
2
Theoretical Background
3
Validation
4
Application to CD Airfoil
5
Conclusion and Perspective
Application
Conclusion
Outline
Introduction
Theory
Validation
Application
Conclusion
Conclusion Spectral model was implemented and validated with literature test cases (flat plate boundary layer with and without adverse pressure gradient) Calculated spectrum strongly depends on RANS turbulence model, optimal choice of turbulence model is case-dependent
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Introduction
Theory
Validation
Application
Conclusion
Conclusion Spectral model was implemented and validated with literature test cases (flat plate boundary layer with and without adverse pressure gradient) Calculated spectrum strongly depends on RANS turbulence model, optimal choice of turbulence model is case-dependent Application to CD airfoil: Good agreement of wall pressure spectra with v 2 f turbulence model, K-ω also acceptable, RSTM did not work Noise computation with Amiet’s theory: agreement as good as wall pressure spectra
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Introduction
Theory
Validation
Application
Conclusion
Conclusion Spectral model was implemented and validated with literature test cases (flat plate boundary layer with and without adverse pressure gradient) Calculated spectrum strongly depends on RANS turbulence model, optimal choice of turbulence model is case-dependent Application to CD airfoil: Good agreement of wall pressure spectra with v 2 f turbulence model, K-ω also acceptable, RSTM did not work Noise computation with Amiet’s theory: agreement as good as wall pressure spectra
+ Quality of calculated spectra is not inferior to LES results, computational costs are lower by orders of magnitude - preliminary knowledge about flow field or validation of RANS required, restricted applicability due to assumptions
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Validation
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Perspectives Extension from one-point method to surface distributed pressure fluctuations ⇒ possible use of other analogies (e. g. Curle, Ffowcs Williams & Hawkings...) ⇒ more complex geometries like 3D fan blades ⇒ acoustical post-processing with standard tools, such as SYSNOISE
Outline
Introduction
Theory
Validation
Application
Conclusion
Perspectives Extension from one-point method to surface distributed pressure fluctuations ⇒ possible use of other analogies (e. g. Curle, Ffowcs Williams & Hawkings...) ⇒ more complex geometries like 3D fan blades ⇒ acoustical post-processing with standard tools, such as SYSNOISE
Trend verification ⇒ even if absolute noise levels are not always precisely predicted, at least trends must be reproduced by the statistical model ⇒ parametrical studies for reference velocity, angle of attack, incoming turbulence ⇒ useful for design of aeroacoustic applications
Outline
Introduction
Theory
Validation
Application
Conclusion
Perspectives Extension from one-point method to surface distributed pressure fluctuations ⇒ possible use of other analogies (e. g. Curle, Ffowcs Williams & Hawkings...) ⇒ more complex geometries like 3D fan blades ⇒ acoustical post-processing with standard tools, such as SYSNOISE
Trend verification ⇒ even if absolute noise levels are not always precisely predicted, at least trends must be reproduced by the statistical model ⇒ parametrical studies for reference velocity, angle of attack, incoming turbulence ⇒ useful for design of aeroacoustic applications
Include wall functions to make the statistical model work also with coarse RANS simulations
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Thank you for your attention. Questions?
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References R. K. Amiet. Acoustic radiation from an airfoil in a turbulent stream. J. Sound Vib., 4(4):407–420, 1975. P. Bradshaw. The turbulence structure of equilibrium boundary layers. J. Fluid Mech., 29(4):625–645, 1967. J. Christophe, J. Anthoine, and S. Moreau. Trailing edge noise of a controlled-diffusion airfoil at moderate and high angle of attack. In 15th AIAA/CEAS Aeroacoustics Conference, 2009. S. Moreau, D. Neal, Y. Khalighi, M. Wang, and G. Iaccarino. Validation of unstructured-mesh LES of the trailing-edge flow and noise of a controlled-diffusion airfoil. In Proceedings of the Summer Program 2006, pages 1–14. Center for Turbulence Research, 2006. R. L. Panton and J. H. Linebarger. Wall pressure spectra calculations for equilibrium boundary layers. J. Fluid Mech., 65(2):261–287, 1974. M. Roger and S. Moreau. Back-scattering correction and further extensions of Amiet’s trailing-edge noise model. Part 1: Theory. J. Sound Vib., 286:477–506, 2005. M. Wang, S. Moreau, G. Iaccarino, and M. Roger. LES prediction of pressure fluctuations on a low speed airfoil. In Annual Research Briefs. Center for Turbulence Research, 2004.
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Appendix: Turbulence data from RANS solution Turbulence length scale Λ general:
Λ = 1.5 lm = 1.5 Cm L
K-ε, K-ω, RSTM:
L = Cµ K ε
v2f :
L = CL max
3/2
=
Cµ K 1/2 β ω
K 3/2 ε ;
Cη
ν3 ε
1/4
Wall-normal velocity fluctuations u2 √ K-ω: u2 = 2 α2 K α2 - empirical anisotropy factor √ v 2 - wall-normal Reynolds-stress v2f : u2 = v 2 RSTM:
u2 ← hui uj i
hui uj i - Reynolds-stress tensor
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Appendix: Monte Carlo Method Monte Carlo integral for one dimension (N random input points xi ): b
Z
f (x)dx = (b − a) hf i =
I=
i=1
1
1
0.8
0.8
0.6
0.6
f(x, y)
f(x)
a
N →∞ (b − a) X f (xi ) N
0.4 f(x) theoretical f(x) average f(xi)
0.2 0 0
0.2
0.4
0.6 x
0.4 f(x, ymax) theoretical f(x, ymin) theoretical f(x, y) average f(xi, yi)
0.2 0 0.8
1
0
0.2
0.4
0.6 x
convergence: ∆I ∝ √σN ⇒ well suited for multidimensional integration (coordinate transform necessary to reduce σ).
0.8
1
