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Numerical prediction of wind turbine noise

A. Tadamasa*, M. Zangeneh Department of Mechanical Engineering, University College London, Torrington Place, London WC1E 7JE, United Kingdom article i nf o

Keywords: Wind turbine noise FWeH equation NREL Phase VI blade abstract

1. Introduction

The main aim of the paper is to develop and validate a numerical methodologyforpredictingthenoiseradiatedfromthewindturbine blades to the far-ﬁeld. The hybrid methodology is used where Reynolds-averaged NaviereStokes (RANS) based CFD solver is used to calculate the aerodynamic noise sources and its propagation to the far-ﬁeld is calculated using Ffowcs WilliamseHawkings (FWeH) equation[7].TherearetwotypesofFWeHapproachesforpredicting far-ﬁeld noise. In the original FWeH approach the sources of ﬂow perturbations on the solid surfaces of the blades are considered. The usual formulation of this approach is to use a surface integral on the blade to estimate the so-called loading noise and thickness noise.

However, this approach fails to predict the quadruple noise as in this

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2. Background theory 2.1. FWeH equations

The original FWeH equation was developed in 1969 from

Lighthill’s acoustic analogy [8] by including the effect of the moving solid body [7]. This equation is a rearrangement of continuity equation and NaviereStokes (NeS) equations into an inhomogeneous wave equation with sources of sound. It is ﬁrst derived by representing the blade surface as a moving control surface, which introduces discontinuity in the unbounded ﬂuid domain. The shape and the motion of the control surface is deﬁned byfð x!;tÞ¼ 0, with f < 0 for its interior and f > 0 for its exterior. It has been assumed that the ﬂow inside this control surface have the same ﬂuid state as the undisturbed medium and outside as the real state including the inﬂuence from the body [4]. The FWeH equation is derived by obtaining the equation that can be applied in the entire unbounded domain, both inside and outside the control surface. This is done by using the generalised functions to describe the ﬂow ﬁeld. The generalised variables (shown with tilde on the top of the variable) for density, momentum and compressive stress tensor respectively can be written as [9]

~r ¼ r0HðfÞþ r0 ~r~ui ¼ ruiHðfÞ

~pij ¼ p0ijHðfÞþ p0dij

where H(f) is Heaviside function, HðfÞ¼

Subscript, 0,d eﬁne the value in undisturbed medium and the primed value represents difference between the value in real state and in undisturbed medium ðe:g: r0 ¼ r r0Þ. The generalised variables (1) are substituted to ordinary conti- nuityequationtoobtain the generalised version of theequation [10], vt þ v~r~ui vxi vt þ rui vxi dðfÞ; (2) where v=vt indicates generalised differentiation. d(f) is Dirac’s delta function, which is the derivative of Heaviside function, vHðfÞ

Similarly, the generalised conservation of momentum equation is obtained by substituting the generalised variables into the ordinary equation, vt þ vxj

¼ rui ruiuj þ p0 ij vxj vf=vxi and vf=vt in the Eqs. (2) and (3) are replaced by n i and vn respectively, wheren i is the unit vector in the direction normal

(outward) to the control surface and vn is the velocity of control surface in the normal direction. Then, these equations are combined and rearranged, by assuming no ﬂuid ﬂow through the control surface, to give differential FWeH equation [1], vxi ½lidðfÞ þ v2 vxivxj where c0 is the speed of sound in undisturbed medium, li ¼p0ijnj is thelocalforcevectorcomponentsexertedbythesurfaceonﬂuid,p0 is theacousticpressurewherep0 ¼c20ðr r0ÞandTij ¼ruiujþpij c20rdij is the Lighthill stress tensor.

A ﬁrst term on the right hand side of Eq. (4) represents the source that is proportional to the local rate of mass injection into the exterior of control surface and thickness noise can be represented by this type of source. A second term represents the source proportional to the local force intensity and the loading noise can be represented by this type of source. A third term represents quadrupole type source. As ﬁrst 2 terms has d(f) function, which is zero for all f except at the control surface, f ¼ 0, they can be determined as surface sources. Similarly the third term has H(f) function, which is equal to unity only in the region exterior to the surface, it can be determined as a volume source. Due to the complexity of volume integral calculation, the third term is often assumed negligible. However, Di Francescantonio [12] developed a permeable type FWeH equation, which can include the quadrupole source without this complex volume integral calculation (explained later in Section 2.3).

The main advantage of FWeH equation is being able to separate each source terms and can determine which type of noise is dominant. The different characteristics of these sources can also determine the frequency and directional characteristics of the generated sound ﬁeld. This can help to arrive at useful guidelines for the design of reduced noise wind turbine blades.

2.2. Solution of FWeH equation

The integral form of FWeH Eq. (4) is obtained by using Green’s function of wave equation in unbounded three-dimensional space; dðgÞ where is the distance between the noise source (yi) and the observer (xi), s is timewhen noise source is emitted, and t is timewhennoise reach observer.

Only the surface source terms are considered and the integral form of FWeH equation is obtained as [4], p0ð*x;tÞ¼ v vt r0vndðfÞdðgÞ

4pr d*yds

v vxi lidðf ÞdðgÞ

The changes of variables are required for integrating the delta functions. First, integration over s is performed. As the coordinates frame ﬁxed relative to the blade surface is used, d(f) is unaffected by s. Therefore, dsh dg

Secondly, the integrating over one space dimension is per- formed. It has been assumed f ¼ y3 and S describe surface on (y1, y2) plane. This gives relation d* yhdy1dy2df

After the integration of the delta functions, the retarded-time formulation of FWeH equation is obtained,

4pp0Tðx;tÞ¼ Z ret dS

ret dS (9)

ret dS ret dS

ret dS (10) where PT is thickness noise, PL is loading noise and Mr is the relative Mach number in radiation direction.

This is the most common solution to the FWeH equation and called Farassat’s formulation 1A. This formulation gives time history of the acoustic pressure. Due to the existence of Doppler factor, 1=ð1 MrÞ , this formulation is limited to the subsonic cases. The wind turbine works, in general, at low Mach number below 0.3

and this formulation will be applicable.

2.3. Permeable FWeH equation

Di Francescantonio [12] has developed a new formulation called permeable FWeH equation to distinguish it from original impenetrable FWeHE q. (4). In the original FWeH equation, the control surface,fð x!;tÞ¼ 0,wastakentocoincidewiththebladesurfacebut inthenewpermeableFWeHequation,thecontrolsurfaceistakenat a ﬁctitious surface at some distance from the blade surface enclosing thebladeandtheentirenoise-generatingregion.Astheﬂowcanpass through the control surface, it is called permeable FWeHe quation. This new equation has advantage of including the quadrupole noise as surface source, without complex volume integral calculation.

A permeable FWeH differential equation is derived similar manner to the original equation but the ﬂuid is allowed to pass across the control surface, v vxi p0ijn j þ ruiðun vnÞi dðfÞo þ v2 vxivxj where un is the ﬂuid velocity in direction normal to control surface and vn is the control surface velocity in direction normal to control surface.

The solution of permeable FWeH equation can be rearranged to give same format as the original FWeH equation for the comparison and computational simplicity. The third term on the Eq. (1) is a quadrupole source outside the control surface and this can be assumed small compared with the quadrupole source inside the control surface and can be neglected.

4pp’ðx;tÞ¼ Z ret dS ret dS ret dS ret dS ret dS (12) where

Fig. 1. Unstructured computational mesh of NREL Phase VI model.

r0 vi þ rui r0 and

Li ¼ Pijn j þ ruiðun vnÞ: (14)

3. Validation and results 3.1. Validations of CFD tool

The commercially available CFD solver, ANSYS CFX 1.0, has been used to calculate the aerodynamic ﬂow parameters, required as an input to the FWeH equations. The RANS (Reynolds-Averaged NaviereStokes Simulation) approach was used with Shear Stress Transport (SST) k-u based turbulence model. The validation of the ﬂow solver has been performed on NREL Phase VI HAWT wind turbine blade. This wind turbine blade model is widely used for validating the numerical codes for predicting the aerodynamics performances due to the availability of experimental data at various operating conditions [14,15]. The NREL Phase VI wind turbine is two-bladed which has 10.06 m diameter with power rating of 20kWandtheyarestall-regulated windturbinewith full-spanpitch control. The blade has S809 aerofoil cross-section, which is designed especially for use in the wind turbine blade. It is linearly tapered

Fig. 2. Pressure coefﬁcient distribution of NREL blade at wind speed of 7 ms at (a) 30% (b) 46.6% (c) 63.3% (d) 80% and (e) 95% of blade span.

from 0.737 m chord length at the blade root section to 0.356 m chord length at the blade tip section. It is nonlinearly twisted from 20.05 at the blade root to 2.0 at the blade tip (detailed blade geometry can be obtained in [13]). In this paper, the cases where operating in upwind and no yaw condition were chosen for the comparisonwith the experimental data. The blade is also pitched so that the tip chord is 3 towards the feather (towards wind direction) from the rotor plane. The tower height of NREL Phase VI wind turbine is 12.03 m. In the upwind conﬁguration, the inﬂuences of tower and nacelle on the rotor aerodynamics can be assumed to be negligible [14]. A model and the meshes were created using Pointwise Gridgen V15.0. Only one blade has been created and a periodic boundary condition has been applied to model the second blade. The no-slip wall boundary condition was applied on the blade surface. The computational domain was created from two regions; an inner region containing the blade which is rotating with 72 RPM

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