PAMM · Proc. Appl. Math. Mech. 8, 10669 – 10670 (2008) / DOI 10.1002/pamm.200810669

Aerodynamic and aeroacoustic phenomena due to non-uniform flow ofwind turbines

Alexandru Dumitrache, Horia Dumitrescu, Vladimir Cardos1,∗1 Institute of Mathematical Statistics and Applied Mathematics, P.O. Box 1-24, 010145 Bucharest, Romania

Stall control and pitch control are the most commonly used methods of regulating power. However, through the opportunitiespresented by the flexible (or teetered) hub of a two-bladed teetered rotor one can also utilize yaw control to regulate power.This is achieved by adjusting the capture area of the rotor disk relative to the prevailing wind direction. This paper presentsthe aerodynamic and aeroacoustic results obtained from theoretical models for a rotor when is yawed to the undisturbed flow.

c© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Modern in utility-sized wind turbines rely mainly on two mechanisms for power control at high wind speeds, namely pitchand stall control [1]. Small wind turbines, with rated power values in the 0.5 − 20 kW range, however, mainly utilize furling(or yawing) as their mechanism for power regulations. This is achieved by adjusting the capture area of the rotor disk relativeto the dominant wind direction. In the present work a modified strip theory approach has been used to determine the effectsof non-axial flow on the power performance and sound pressure level.

2 Aerodynamic Analysis

Blade element momentum theory is the standard computational technique for the prediction of power curves of wind tur-bines; it is based on the 2-D aerodynamic characteristics of airfoil blade elements and some corrections accounting for 3-Dwing aerodynamics. Before being able to calculate the force and moments on a blade, it is necessary to derive the velocitycomponents of the air flow relative to any point on the blade and also the induced velocity components.

2.1 Velocity components at the blade. The effects of the axially, radial and rotationally induced velocities can now beincluded giving the components of relative wind in the blade based coordinate system as normal to rotation plane and span,chordwise and spanwise:

Wi = Vwcosγcosβ − vacosβ − (Vwsinγsinψsinβ − vatan(χ/2)sinψsinβ), (1)

Wj = Ωrcosβ + vt − (Vwsinγcosψ − vatan(χ/2)cosψ, (2)

Wk = Vwsinγsinψcosβ − vacosβ − vatan(χ/2)sinψcosβ + Vwcosγsinβ − vasinβ, (3)

where γ is the yaw angle, χ is the wake skew angle and β is the conning angle. Further on we define the non-dimensionalvelocities: a = va/(Vwcosγ), a′ = vt/(Ωrcosβ), λ = ΩRcosβ/Vw, X = Ωrcosβ/Vw, Ci = Wi/(Vwcosγ), Cj =Wj/(Vwcosγ) and Ck = Wk/(Vwcosγ), and substitute them into equations (1-3) to obtain

Ci = (1 − a)cosβ − (tanγ − atan(χ/2)))sinψsinβ, (4)

Cj = X/cosγ [1 + a′ − (sinγcosψ)/X + (acosγtan(χ/2)cosψ)/X] , (5)

Ck = tanγsinψcosβ − atan(χ/2)sinψsinβ + (1 − a)sinβ], (6)

Now equating the momentum theory with the blade element theory, one obtain

a

1 − a=σrcos

2β

8πf

∫ 2π

0

Cxsin2φ

dψ, (6)

a′

1 − a′=

σr8πf

∫ 2π

0

Cy

sinφcosφ[1 − tanφtanγcosψ

(1−a)cosβ]dψ. (7)

The non-dimensionalized resultant velocity relative to a blade element is given by

W 2/(V 2wcos

2γ) = [(1 − a)cosβ]2 + [X(1 + a′)]2 [cosγ + cosψ(atan(χ/2) − tanγ)]−2. (8)

∗ Corresponding author E-mail: alex dumitrache@yahoo.com, Phone: +00 40 21 3182433, Fax: +00 40 21 3182439

c© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

10670 Sessions of Short Communications 12: Waves and Acoustics

Fig. 1 Comparison between total predicted noise levelsand experimental data (γ = 0deg).

Fig. 2 Comparison between total predicted noise levelsand experimental data (γ = 60deg).

Once the induction factors are known as a function of the radial variable r the power coefficient for the complete bladecan be calculated from CP = 8λ2cosγ

∫ 1

0(r/R)3a′(1 − a)d(r/R). The elemental lift and drag forces per unit length are:

L = 1/2ρW 2cCL and D = 1/2ρW 2cCD, with W 2 given by Eq. (8).

3 Aeroacoustic analysis

The aerodynamic noise prediction method combines a model for predicting aerodynamic noise due to the effects of inflowturbulence upon on airfoil section, with prediction schemes for airfoil self-noise. The method is integrated into a programwhich averages the noise of the previous two bladed rotor our one revolution and gives as output a 1/3 octave, A-weightedspectrum at a user selectable location. Output spectra are predicted at a downwind location for various yaw angles andcomparisons with experiment are presented.

Inflow turbulence noise. The wind turbine acts to transform incoming turbulence to radiated noise, which is potentiallythe most important source of aerodynamic noise from wind turbine at high wind speeds. This model can be applied for bothhigh and low frequency, with smooth transition between the two regions: LP,INF = LHP,INF + 10log10Kc(1 +Kc)

where LHP,INF is the sound pressure level for high frequency region and Kc is the low frequency correction [2].Turbulent Boundary Layer - Trailing Edge noise (TBL-TE). As its name implies TBL-TE noise is caused by the flow

of a turbulent boundary layer over the impedance discontinuity existing at the trailing edge of the airfoil. The effect of theedge is to radically increase the efficiency of the acoustic radiation of the turbulence, particularly at lower speeds. The noiseis described as a function of local Mach number M , displacement thickness, δ∗, length of blade segment, ∆S, angle of attack,α and the distance of the source to observer position r. The total sound pressure level, in 1/3 octave band, is given by Brooks,Pope, Marcolini model as [3]: Lp,TBLTE = 10log10(10LP,α + 10LP,s + 10LP,p) with LP,α representing the effect of angleof attack, LP,s, the contribution of the suction side and LP,p the contribution of the pressure side of the airfoil.

Blunt Trailing Edge Noise. Trailing edge bluntness on an airfoil can result in the shedding of vortices into the wake,similar to that observed behind cylinders. If its thickness is too large, then very high tonal noise may occur. The total soundpressure level caused by a blunt trailing edge is also modelled by a scaling law proposed by Brooks et al. [3] LP,BTE =10log10(t∗M5.5∆sDh/r

2) + G3(t∗/δ∗avg, ψTE) + G4(t∗/δ∗avg, ψTE , St′/St′peak) with the Strouhal number based on the

trailing edge thickness t∗, St′ = ft∗/U.

4 Results and conclusions

In our paper, the noise prediction model developed in conjunction with the aerodynamic model captures the key features ofthe noise produced by a wind turbine rotor in yaw. Some comparisons between predicted noise from all sources (INT, TBL-TE, BTE) and experimental data show good agreement over all yaw angles up to 60 degrees (Figures 1 and 2). The modelaccurately predicts that the noise of the rotor is dominated by the tonal noise due to the trailing edge bluntness and this formof noise is controllable by simply sharpening of trailing edge.

References

[1] T. Burton, D. Sharpe, N. Jenkins, E. Bossanyi, Wind Energy Handbook, J. Wiley & Sons, Chichester, (1991).[2] R. K. Amiet, Acoustic Radiation from an airfoil in a turbulent stream, Journal of Sound and Vibration, 41, 407–420 (1975).[3] F. T. Brooks, D. S. Pope, M. A. Marcolini, Airfoil self-noise and prediction, NASA RP-1218, 6, 1–137 (1989).

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