Trailing edge noise prediction based on wall pressure spectrum models for NACA0012 airfoil

Journal of Wind Engineering & Industrial Aerodynamics 175 (2018) 305–316

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Trailing edge noise prediction based on wall pressure spectrum models for NACA0012 airfoil Yakut Cansev Küçükosman *, Julien Christophe, Christophe Schram Environmental and Applied Fluid Dynamics Department, von Karman Institute for Fluid Dynamics, Sint-Genesius-Rode, B-1640, Belgium

A R T I C L E I N F O Keywords: Aeroacoustics Trailing-edge noise Wall-pressure spectra models Amiet's theory

A B S T R A C T

This paper compares several approaches for the prediction of the noise emitted by a NACA0012 airfoil at 0∘ and 4∘ of angle of attack and a Reynolds number Re ¼ 1:5  106 . Amiet's semi-analytical model for trailing-edge noise is combined with two-dimensional Reynolds-Averaged Navier-Stokes (RANS) computations. The wall-pressure spectrum, which constitutes the cornerstone of Amiet's model, is obtained by processing the boundary layer data extracted from the simulations. The specific contribution of the paper is a comparison of two families of prediction methods: semi-empirical wall-pressure models that are fitted to experimental databases, and statistical approaches based on the integration of the Poisson equation across the boundary layer profile. The semi-empirical models that were calibrated on airfoil databases provide better predictions with experiments, while the models based on flat-plate boundary layer data fail to reproduce the measured spectra. Considering the statistical approach, it was shown to predict the general spectral features, but with an overall under-prediction of about 3 dB. It can be concluded from this study that the statistical approach proves indeed more robust than semiempirical models when the latter were not precisely calibrated for the flow under consideration. Further improvements of the statistical approach are suggested for future work.

1. Introduction The increasing need for sustainable and clean energy resources is a strong incentive in the field of wind power. However, a main issue with the implementation of wind turbines is the acoustic disturbance they cause in their immediate environment (Bockstael et al., 2011; Waye and € Ohrstr€ om, 2002; Schmidt and Klokker, 2014). Therefore, low-cost and precise noise prediction tools are needed in the process of wind turbine design and wind farm planning. The predominant wind turbine noise production mechanism is associated with the turbulence that develops along the blade surface and scatters at the trailing edge as acoustic waves (Barone, 2011; Oerlemans et al., 2009). Trailing-edge noise prediction approaches can be distinguished along three categories; semi-empirical, direct and hybrid methods. The application of the semi-empirical models (Brooks et al., 1989) are limited since the models are calibrated against experimental data which can lead to poor prediction for other airfoil profiles and flow conditions (Moriarty and Migliore, 2003). The direct methods (Sanderg and Sandham, 2008; Gloerfelt and Le Garrec, 2009) provide accurate and reliable predictions and are applicable for industrial applications.

However, when these high-fidelity methods are utilized as a design and optimization tool, they demand high computational cost (Herr et al., 2015). Hybrid methods offer an interesting compromise in terms of accuracy vs. CPU cost, by decoupling the flow and acoustic calculations (Redonnet, 2014). Hybrid methods usually consist of the following two steps: first, the unsteady flow field is computed in the region of the source term; secondly, an acoustic propagation method is used to compute the acoustic source radiation towards the far-field. In order to further reduce the computational cost, Reynolds-Averaged Navier-Stokes (RANS) simulations can be preferred over scale-resolved simulations to provide a source model. In that case, complementary stochastic methods are necessary to synthesize the missing unsteady information about the flow. The Stochastic Noise Generation and Radiation (SNGR) (Ewert, 2008; Herr et al., 2015) and Random Particle-Mesh (RPM) (Ewert, 2007) were developed to this end. Finally, purely statistical methods (not involving any stochastic reconstruction) offer the cheapest solution amongst the hybrid methods. The RANS-based Statistical Noise Model (RSNM) (Doolan et al., 2010) follows this path; the acoustic far field is computed using a semi-infinite half plane Green's function combined with a model for the turbulent velocity cross-spectrum in the vicinity of the

* Corresponding author. E-mail addresses: [email protected] (Y.C. Küçükosman), [email protected] (J. Christophe), [email protected] (C. Schram). https://doi.org/10.1016/j.jweia.2018.01.030 Received 18 July 2017; Received in revised form 8 November 2017; Accepted 16 January 2018 0167-6105/© 2018 Published by Elsevier Ltd.

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Journal of Wind Engineering & Industrial Aerodynamics 175 (2018) 305–316

librium parameter, βc . Therefore, the purpose of this study is threefold:

trailing-edge. Alternatively, the wall-pressure based models compute the acoustic far field using a diffraction analogy technique (Chandiramani, 1974) or Amiet's theory (Amiet, 1975). Amiet's theory requires the wall-pressure spectra information which can be obtained directly from Scale-Resolving Simulation (SRS). However, SRS computations require significant computational cost that is unappealing for industrial design and optimization tools. Kraichnan (1956) was the first to express the wall-pressure fluctuations for a flat plate based on the solution of the Poisson equation. The method expresses the pressure fluctuations in terms of the two-point correlation of the wall normal velocity fluctuations and the mean velocity profile. Following this approach the TNO model was developed by Parchen (1998), which is based on the turbulent boundary layer and the wall-pressure wavenumber frequency spectrum where Blake's equation (Blake, 1986) is used for the prediction of the wall-pressure wavenumber frequency spectrum. This model was observed to yield an under-prediction of the noise level compared to some experimental results (Kamruzzaman et al., 2011; Bertagnolio, 2008), even though it shows a correct behavior with respect to incoming velocity and angle of attack. Lilley and Hodgson (1960) developed an extended version of the Kraichnan (1956) method by considering the pressure gradient in the streamwise direction with empirically obtained inputs. Later, Panton and Linebarger (1974) expressed these inputs by empirically determined analytical expressions, yet this was insufficient to apply for more complex non-equilibrium flows. Lee et al. (2004) showed that the Kraichnan model is still applicable for more complex flows by obtaining the input parameters through RANS simulations of the reattachment after a backward-facing step. Lately, Remmler et al. (2010) applied this technique to zero and adverse pressure gradient flows. Besides simplified theoretical approaches, the development of the semi-empirical relationships has served to describe the pressure fluctuations beneath the boundary layer based on a theoretical basis. These models are derived by fitting the experimental wall-pressure spectra rescaled with the boundary layer variables. The model proposed by Schlinker and Amiet (1981) used the external variables to fit the experimental data obtained from Willmarth and Roos (1965). Later, Later, Howe (1998) reformulated the wall-pressure model proposed by Chase (1980) by re-scaling with the mixed boundary layer variables. The model exhibited better performance by capturing the ω1 decay at high frequencies. However, this model does not take into account the Reynolds number effects where the overlap region increases at the intermediate frequencies. Moreover, this model does not capture ω5 decay for the highest frequencies. Goody (2004) improved this model by adding a term in the denominator which satisfies the decay for high frequencies. He also added a non-dimensional variable that sets the overlap region depending on the Reynolds number. This model and earlier ones perform better for simple flows, however, they exhibit significant differences for Adverse Pressure Gradient (APG) and separated flows. Rozenberg et al. (2012) developed the Goody model for the APG flow by introducing two additional parameters which are Coles' wake, Π and Clauser's parameters, βc . Catlett et al. (2015) extended the Goody model for APG flows by introducing non-dimensional parameters involving the Reynolds number and the Clauser's parameter. Kamruzzaman et al. (2015) proposed another model based on the Goody model by using airfoil measurement data. Hu and Herr (2016) claimed that using the shape factor, H ¼ δ =θ is more suitable for characterizing APG flows. Moreover, they suggested that the proper scaling for the spectrum should be the dynamic pressure as a better fitting is observed with their experimental data. Later, Lee and Villaescusa (2017) extended the Rozenberg model by modifying some of the terms to provide a better universal approach. The models tend to focus on the accuracy of the wall-pressure spectrum predictions, however, there are few assessments of the far-field noise prediction. Furthermore, for the semi-empirical models, the determination of some of the required parameters is delicate, especially the boundary layer thickness in the presence of APG flows and the APG driven parameters such as Coles' wake parameter, Π and Clauser's equi-

 to investigate the sensitivity of the semi-empirical wall-pressure spectrum models on the boundary layer thickness determined by three different approaches and the APG driven parameters obtained by two different approaches;  the comparison of the wall-pressure spectra obtained by six different semi-empirical models: Goody, Rozenberg, Catlett, Kamruzzaman, Hu & Herr and Lee as well as one statistical model, Panton & Linebarger;  the comparison of the far-field noise prediction by Amiet's trailing edge model using the aforementioned wall-pressure spectrum models. Section 2 of this paper focuses on Amiet's theory. Section 3 gives a brief description of the wall pressure models that have been investigated. The numerical simulation details are given in Section 4 and are validated against experimental wake profiles in Section 5. The wall-pressure spectrum models in Section 6 and far-field noise predictions in Section 7 are compared with the experimental data and the conclusion is drawn in Section 8. 2. Amiet's analytical model for trailing edge noise Amiet's theory provides an analytical model to compute the broadband trailing-edge noise of an isolated airfoil. It is approximated as a flat plate with zero angle of attack embedded in a uniform flow. The main trailing-edge scattering is obtained by assuming that the chord is infinite in the upstream direction (Amiet, 1976). Later, a leading-edge back-scattering correction is performed by considering the finite length chord by Roger and Moreau (2005). The turbulent eddies convected by the mean flow at the trailing edge are assumed to be frozen. The far-field acoustic Power Spectral Density, PSD, ðSpp Þ for the trailing-edge for a large span airfoil and an observer located in the midspan plane at position, x ¼ ðR; θ; z ¼ 0Þ for a given angular frequency (ω) can be written as:  2
 sinθ 2 d  2 Spp ðx; ωÞ ¼ ðkcÞ L j ly ðωÞϕpp ðωÞ 2π R 2 

(1)

where c is the chord length, d is the span, ly is the spanwise correlation length, ϕpp is the wall-pressure spectrum and L ¼ L 1 þ L 2 is the aeroacoustic transfer function given by (Roger and Moreau, 2005) for super-critical, L 1 , and sub-critical, L 2 , gusts. It depends on geometrical parameters (chord and span) as well as on the pulsation ω. The spanwise correlation length is computed by the Corcos (1964) model as: ly ðωÞ ¼

b Uc

ω

(2)

where Uc is the convection velocity, ky is the spanwise wavenumber and a constant b ¼ 1:47. 3. Wall-pressure spectrum models 3.1. The semi-empirical model All the semi-empirical wall-pressure spectrum (WPS) models have the form of (Catlett et al., 2015; Hu and Herr, 2016; Lee and Villaescusa, 2017): ϕpp aðω Þb :  ¼ e c ϕ ½iðω Þ þ d þ ½fRg ω h

(3)

The shape of the spectra is modified through the parameters a  h given in Eq. (3). The overall amplitude of the spectra is altered by a. The

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Journal of Wind Engineering & Industrial Aerodynamics 175 (2018) 305–316

Zagarola-Smits's defect law provides a better collapse than the defect law and exhibits an auto-similarity of the velocity profile for the outer region, thus, Zagarola-Smits's parameter, Δ ¼ δ=δ , is chosen as a driving parameter for the APG. βc is used to quantify the local pressure gradient even though the tested boundary layers were non-equilibrium flows. Π represents the large-eddy structures in the outer region of the turbulent boundary layer. Coles (1956) modified the law of the wall with an additional wake parameter as:

slopes corresponding to different frequencies are adjusted by the parameters b, c, e and h. The parameter b determines the slope at low frequencies. The overlap region is modified by the parameters b, c and e. The high slope region is adapted by the parameters b and h. The onset of the transition between the overlap and high frequency region is adjusted by the parameters f and g combination with time-scale ratio R. The location of the low-frequency maxima is weakly dependent on the parameter d. Lastly, the parameter i is 1.0 for all except the Rozenberg model. In the case of Rozenberg, a constant of 4.76 is introduced when the boundary layer thickness is replaced by the displacement thickness by assuming Δ ¼ δ=δ ¼ 8. The scaling factor for the spectrum is ϕ and for the frequency it is ω . In this paper six different semi-empirical wall-pressure spectrum models are investigated: Goody, Rozenberg, Catlett, Kamruzzaman, Hu & Herr and Lee. The parameters and the scaling factors are summarized in Table 1 excepted for Lee's model that is an extension of the Rozenberg model. In the following section, the governing variables and the spectral behavior of each model are discussed.

π y 1   2Π uþ ¼ ln yþ þ C þ sin2 κ κ 2δ

(4)

where uþ ¼ u=uτ , y þ ¼ yuτ =ν, κ ¼ 0:41 is the von Karman constant, C ¼ 5:1 and the Coles wake parameter, Π, is obtained by solving the following equation implicitly: 2Π  lnðΠ þ 1Þ ¼


  κUe δ Ue  ln  κC  ln κ : uτ ν

(5)

However, they calculated Π through an empirical formula proposed by Durbin and Reif (2011):

3.1.1. Goody model The Goody model extends the overlap region by introducing the timescale ratio, RT , which accounts for Reynolds number effects for Zero Pressure Gradient (ZPG) boundary layers. The wall-pressure spectrum is scaled by mixed variables: δ is the boundary layer thickness, Ue is the velocity at the edge of the boundary layer, τw is the wall shear stress. The time-scale ratio is defined as the ratio of the outer time scale to the inner time scale, RT ¼ ðδ=Ue Þ=ðν=u2τ Þ where uτ is the friction velocity and ν is the kinematic viscosity. The frequency is scaled by δ=Ue . The spectrum has a slope of ω2 at low frequencies. It decays with a slope of ω0:7 at midfrequencies and ω5 at high frequencies. This model is accurate over a wide range of Reynolds numbers (Hwang et al., 2009). Furthermore, it is considered as a basis for the Adverse Pressure Gradient (APG) wall-pressure spectrum models.

Π ¼ 0:8ðβc þ 0:5Þ3=4 :

(6)

It is pointed out that Δ and Π are influenced by the boundary layer history whereas βc is a local parameter. As Δ decreases, the amplitude of the spectrum increases at mid and high frequencies and decreases for low frequencies. βc and Π are correlated, when they increase, the peak amplitude gets higher and the slope of the overlap region gets steeper. 3.1.3. Catlett model Catlett et al. (2015) developed a new empirical approach for APG boundary layers based on the Goody model by testing three different trailing edge configurations for a flat plate. The scaling factor for the wall-pressure spectrum and frequency are kept the same as Goody's model. Similar to the Rozenberg model, the local pressure gradient is defined in the form of Clauser's equilibrium parameter, βc . However, the length and pressure are scaled with outer boundary layer variables, βδ ¼

3.1.2. Rozenberg model Rozenberg et al. (2012) proposed a wall-pressure model based on the Goody model by considering the variations between ZPG and APG flows. Firstly, the scaling factor for both spectrum and frequency was replaced by the displacement thickness, δ instead of the boundary layer thickness, δ since the former was found to be more accurate. Secondly, the scaling for the pressure fluctuations was changed to the maximum shear stress along the normal distance, τmax . In addition, to characterize the effect of the APG, three parameters are defined: Zagarola-Smits's parameter (Zagarola and Smits, 1998), Δ ¼ δ=δ , Clauser's equilibrium parameter (Clauser, 1954), βc ¼ ðθ=τw Þðdp=dxÞ, where θ is the momentum thickness, and Coles' wake parameter (Coles, 1956), Π. It is found that

δ dP q dx

where q ¼ 0:5ρU02 is the dynamic pressure and U0 is the local

free-stream velocity. They found that when the parameters, a and c  h, are plotted as a function of βδ;Δ Reδ;Δ or βδ H, they fit into a power-law function. Δ is the Rotta-Clauser parameter (Clauser, 1954; Rotta, 1953) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi defined as Δ ¼ δ ð2=Cf Þ, Cf ¼ τw =q is the skin friction coefficient, e are the Reynolds numbers and H ¼ δ =θ is the shape factor. Reδ;Δ ¼ ðδ;ΔÞU ν Contrary to all the semi-empirical models investigated in this paper, the boundary layer thickness is deduced by a percentage of the turbulent

Table 1 The parameters and scaling factors for the semi-empirical models.

a

a

Goody

Rozenberg

3.0

½2:82Δ2 ð6:13Δ0:75 




 4:2

Π Δ

Catlett  þ1

e

þ dÞ 

e

þ 3:0

b c

2.0 0.75

d

0.5

e

3.7

3:7 þ 1:5βc

 1:93ðβδ Re0:05 Þ δ

f

1.1

8.8

g

0.57

0.57

 2:57ðβδ Re0:05 Þ δ

7

pffiffiffiffiffiffi minð3; 19= RT Þ þ 7

38:1ðβδ H 0:5 Þ2:11  0:5424

h i R ϕ

1.0 RT τ2w δ=Ue ωδ=Ue

4.76 RT τ2max δ =Ue ωδ =Ue

0:35 Þ 0:797ðβΔ ReΔ 1.0 RT τ2w δ=Ue ωδ=Ue

ω a

2.0 0.75
0:75 ½0:375e  1 4:76 1:4 Δ

Kamruzzaman

0:131 7:98ðβΔ Re0:35 10:7 Δ Þ

2.0 20:9ðβδ Re0:05 Þ δ

2:76

0:35 Þ 0:328ðβΔ ReΔ

þ 0:912

0:310

þ 0:397

m 0:45½1:75ðΠ2 β2c Þ
0:3 H m ¼ 0:5 1:31

Hu þ 15

ð81:004d þ 2:154Þ⋅107

2.0 1.637

1:5ð1:169lnðHÞ þ 0:642Þ1:6

1.0

0.27

105:8⋅10

5

Reθ H0:35

0:628

þ 3:872

2.47

0:224

þ 2:19

1:152=7

1:13=ð1:169lnðHÞ þ 0:642Þ0:6 7.645

2/7

0.411

7

6

1.0 R0 T τ2w δ =Ue ωδ =Ue

1.0 Rτ q2 θ=uτ ωθ=U0

0:0724

The parameter a in the Lee and Villaescusa (2017) model is replaced by a . 307

þ 7:310

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kinetic energy, TKEðy ¼ δÞ ¼ 0:0002U02 (Schetz, 1993).

3.2. The statistical model

3.1.4. Kamruzzaman model Kamruzzaman et al. (2015) modified the Goody and Rozenberg models to present a new wall-pressure spectrum model for the airfoil trailing-edge noise prediction. Various airfoils with different angles of attack and Reynolds number were used to develop the model. They perform the same scaling factors for the normalization of the wall-pressure spectra and frequency as in the Rozenberg model except that τmax is replaced with τw . The spectral amplitude for all frequencies are modulated by the parameter a which is a combination of βc , Π and H while the rest of the parameters are kept constant by fitting to the experimental data. Furthermore, the definition of RT is modified to R0 T ¼

As an alternative to the previously described models, the following statistical model proposed by Panton and Linebarger (1974) and later by Remmler et al. (2010) is also used in the present paper. The incompressible Navier-Stokes equation is reformulated in the form of a Poisson equation for pressure to express the wall-pressure fluctuations beneath the turbulent boundary layer. With the assumption of statistically stationary and homogeneous flow in the streamwise and spanwise directions, the Poisson equation can be solved by using the Green's function technique as:

 2

δ uτ Ue ν , to account for the boundary layer loading effects. The βc

2 ∂hU1 i ∂hU1 i ∞ k1 ðωÞ ϕðk1 Þ ¼ 8ρ2 ∫ ∫ ∫ 0   2 ejkjðωÞðyþ~yÞ S22 ðy; ~y; ωÞ dyd~ydk3 k ðωÞ ∂y ∂~y

parameter is

obtained by a curve-fit proposed by Nash (1966): G ¼ 6:1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi βc þ 1:81  1:7;


where H ¼

1

G λ



where jkj ¼

qffiffiffiffiffiffiffiffiffiffi with λ ¼ 2 Cf :

S22 ðy; ~y; ωÞ ¼

Finally, the Coles wake parameter is computed from Eq. (6) which is the same way as in the Rozenberg model.

    u0 2 ðyÞu0 2 ð~yÞ 2 ∞ Λ ∬ 0 R22 cos αke1 ðωÞ re1 cos αke3 re3 dre1 dre3 2

π

(11)

where u0 2 is the root mean square of the wall-normal velocity fluctuations, Λ is the integral length scale and α is the scale anisotropy factor which is the ratio of streamwise to spanwise turbulent length scales which is defined as by Remmler et al. (2010):

3.1.5. Hu & Herr model Hu and Herr (2016) developed a new model based on Goody's model by measuring the unsteady pressure fluctuations on a flat plate. The adverse and favorable pressure gradients were created by a rotatable NACA0012 airfoil. Contrary to the other models, they argued that the scaling factors for the wall-pressure spectra, qu2τθ and the frequency, Uωθ0 are

8 k1 δ < 1 < 3; α ¼ 0; 1  k1 δ  5 : : 1; k1 δ < 5

more suitable for APG flow. As a consequence, a better agreement is observed when the time-scale ratio, RT , is replaced by Reτ . Moreover, it is acknowledged that the boundary layer profile is a main driving parameter for the wall-pressure fluctuations. Thus, the spectral slope at medium frequencies is characterized by the shape factor H, rather than the form of a Clauser's equilibrium parameter β, which fails at rapid pressure gradient alterations. Lastly, it is explained that the low frequency slope, ω2 does not hold in the case of non-frozen turbulence and is replaced by b ¼ 1:0. They also pointed out that the turbulence-turbulence term in the Poisson equation gains importance over the mean-shear source term and exhibits a plateau at low frequencies.

(12)

The velocity correlation function R22 is modeled according to (Panton and Linebarger, 1974): "

# pffiffiffiffiffiffiffiffiffi 2 2 ~r2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p R22 ¼ 1  e ~r þ~y 2 ~r 2 þ ~y2

(13)

2 2 with ~r 2 ¼ re1 þ re3 . Non-dimensionalisation of the integration coordinates and the wave numbers was performed by the integral length scale, rei ¼ ri =Λ, kei ¼ ki Λ and y~ ¼ ðy  b y Þ=Λ. The five dimensional inte-

gration in Eq. (10) is performed with a Monte Carlo method using importance sampling for enhancing convergence to reduce the computational cost of the numerical integration of the fivefold integration. The reader is referred to Ref. (Remmler et al., 2010) for further details on the Monte Carlo method. The vertical velocity fluctuations cannot be obtained from the k  ω turbulence model which assumes isotropic turbulence. However, the measurements performed by Klebanoff (1955) for a zero pressure gradient boundary layer demonstrates that the velocity fluctuation

3.1.6. Lee model Lee and Villaescusa (2017) suggested an extended version of the Rozenberg model to provide an accurate prediction for extensive applications. For convenience, the scaling factor used in the normalization of the spectra, τmax is replaced by τw while keeping the other scaling parameters the same. They found that the Rozenberg model performs an early transition from the overlap region to high frequency region. Furthermore, at high frequencies for zero and low pressure gradient flows, a rapid decay rate is observed. To overcome this, the parameter h is modified as follows:

2

components, u0 i , are not constant fractions of 2k. Instead, he proposed an 2

anisotropy factor for each component such that βi ¼ u0 i =ð2kÞ in which βi is not universal for all types of boundary-layer flows. As a first approximation, Remmler et al. (2010) proposed to use an anisotropy factor, which is obtained from a flat plate boundary layer simulation using a Reynolds stress transport model in which the velocity fluctuation components can be obtained directly. Panton and Linebarger (1974) have specified the length scale, Λ, 1.5 times the Prandtl mixing length, lm , Λ ¼ 1:5lm which can be calculated as lm ¼ LCm where L is the turbulence length scale. Cm is a turbulence model constant and was chosen as 1.9 for k  ω SST (Remmler et al., 2010). The turbulence length scale, L can be computed directly from the numerical simulations in which the turbulence model provides the turbulent kinetic energy and turbulent dissipation rate as:

(8)

and if h is equal to 12.35, the following expression is used:  pffiffiffiffiffiffi  h ¼ min 3; 19 RT þ 7 :

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k21 þ k23 , U1 is the streamwise velocity and y is the wall

normal direction. The energy spectrum of the vertical velocity fluctuations S22 is given as:

(7)

 pffiffiffiffiffiffi  h ¼ min 5:35; 0:139 þ 3:1043βc ; 19 RT þ 7 ;

(10)

(9)

The Rozenberg model exhibits higher amplitudes at low and midfrequencies for low pressure gradient flows. Thus, the parameter d in the denominator is modified as d ¼ maxð1:0; 1:5dÞ if βc < 0:5 to alter the trend. It should be noted that the parameter d in the calculation of the parameter a is kept as in the original model. Lastly, a correction is suggested for the parameter a to adjust the amplitudes for high β as a ¼ maxð1; ð0:25βc  0:52ÞaÞ.

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L ¼ Cμ

k3=2

ε

Journal of Wind Engineering & Industrial Aerodynamics 175 (2018) 305–316

:

(14)

Cμ is given as 0.09 and the turbulent dissipation, ε, is calculated as ε ¼ k⋅ω. The crucial parameters for the semi-empirical models are the boundary layer thickness, δ, the boundary layer edge velocity, Ue , the wall-shear stress, τw and the friction velocity, Uτ , which are the only variables in the Goody model. All the other non-dimensional parameters can be derived from the boundary layer variables. In addition, the Rozenberg, Kamruzzaman and Lee models used the Coles' wake parameter, Π and the Clauser's equilibrium parameter, βc . Similarly, Catlett used a form of the Clauser's parameter in their model. The appeal of these approaches is that only the velocity profile normal to the surface is needed. In addition to the velocity profile, the Panton & Linebarger model requires the turbulent kinetic energy and the specific dissipation profiles.

Fig. 1. The main domain configuration.

wall functions are used for k, ω and νt . Pressure is set to be zero at the inlet. For the top and bottom boundaries of the domain, symmetric boundary conditions are applied. The summary of the boundary conditions are given in Table 3. The simulations were performed using the incompressible OpenFOAM solver simpleFOAM with second order accurate schemes and applying under-relaxation factors of 0.3 for pressure and 0.7 for other variables.

4. Numerical setup 4.1. Airfoil configuration The simulations were performed for a NACA0012 airfoil with two different angle of attacks at the same Reynolds number. The operating conditions are chosen from the BANC benchmark (Herr et al., 2015). The measurements were performed in the Laminar Wind Tunnel (LWT) of the Institute of Aerodynamics and Gas Dynamics (IAG). The airfoil conditions are given in Table 2. The flow was forced to transition by using a tripping device located at 0:065% of the chord on both the suction side and the pressure side (Herrig et al., 2013).

4.3. Mesh properties The mesh was created with the ANSYS ICEM meshing tool. The mesh topology is based on the C-Grid type, which is here combined with an HGrid, to optimize the skewness of the cells. In total, the mesh consists of around 150; 000 hexahedral cells with a growth rate of 1.05. The average non-orthogonality of the cells is 10 with a maximum value of 50. It has been assured that the y þ value is less than 2 as shown in Fig. 2(b). Mesh sensitivity studies were conducted, though not reported in this paper for the sake of conciseness, and indicated that this mesh guarantees a satisfactory convergence of the results.

4.2. Computational configuration Two dimensional (2D) steady RANS computations are performed using the open source CFD solver OpenFOAM 4.0. The computational domain is shown in Fig. 1 has a 10c length in the crosswise direction, 5c upstream and 10c downstream in streamwise direction. The k  ω SST model, proposed by Menter (1994), has been proven to provide better predictions for flows with strong pressure gradients compared to the other models (Tulapurkara, 1997; Catalano and Amato, 2003; Kral, 1998). Furthermore, numerical simulations have been performed successfully for flows around isolated wings (Guilmineau et al., 1997; Sorensen et al., 2004) as well as for wind turbine aerodynamics (Pape and Lecanu, 2004; Guerri et al., 2006; Sørensen et al., 2002). Thus, the simulations were performed using the k  ω SST model assuming fully-turbulent flow. While this assumption may be questionable at the leading edge of the airfoil, it was shown by previous authors (Murman, 2011; Rumsey, 2007) that the k  ω SST model can predict a numerical laminarization of the boundary layer in the presence of a favorable pressure gradient at the leading edge, followed by a numerical transition just after, thereby mimicking the physical flow behavior. This is qualitatively observed from Fig. 2(a), showing the wall friction coefficient. A sudden decrease of Cf is indeed observed at the leading edge, followed by its growth just downstream of it. At the inlet, the flow properties are calculated based on a turbulence intensity of 0:1%. Zero flux boundary conditions are imposed at the outlet. On the airfoil surface, a no slip condition for velocity and adaptive

5. Wake characteristics The experimental wake profiles were extracted in the vicinity of the trailing edge of the suction side of the airfoil at x=c ’ 1:0038 (x=c ¼ 0 is at the leading edge). The Cp distribution along the airfoil was measured with pressure taps and the boundary layer profiles were extracted with a hot-wire (Herrig et al., 2013). The validation between the experimental data and the 2D RANS simulations are shown in Fig. 3 with non-dimensional parameters. The Cp distributions for both angles of attack show a good agreement with the experimental results as shown in Fig. 3(a), except for the position close to the tripping device location where a drop in the measurement is observed. The wake velocity profile for 0∘ angle of attack agrees well with the experimental data. However, a deviation of 2:2% is observed below (y=c < 0:024) the inflection point with respect to the experimental data (Fig. 3(b)). The turbulent kinetic energy profile is well predicted for the 0∘ angle of attack above y=c  0:01 but the peak point is under-estimated with respect to the experimental data. The trend as well as the peak point location and amplitude is poorly predicted for the 4o angle of attack (Fig. 3(c)). In contrast to the kinetic energy profiles, a better estimation was observed for the specific dissipation rate, except for the outer region, y=c  0:04 as shown in Fig. 3(d). Such discrepancies produced by one/two-equation turbulence models for APG flows (Menter, 1992) have already been observed by the authors, however no definitive explanations have been found so far. At most it can be conjectured that the model doesn't account properly for the decrease of the turbulence production that is normally the result of a smaller velocity gradient in the near-wall region, induced by a stronger APG. To overcome this issue, the Differential Reynolds Stress Model was proposed by W. M. Henkes (1998) for flows with an APG. Overall, it was observed that the profiles for 0∘ angle of attack are better predicted than

Table 2 Airfoil Configuration. Angle of Attack

chord

Re

U∞

ρ∞



ðdegÞ

m

–

m/s

kg=m3

α ¼ 0∘ α ¼ 4∘

0.4 0.4

1:5  106 1:5  106

56 54.8

1.181 1.190

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Fig. 2. Skin friction and mesh resolution; 0∘ (—), 4∘ (—) angle of attack.

determination of the boundary layer parameters required by the semiempirical models. Later, a comparison will be performed between the semi-empirical models namely Goody (2004) and Rozenberg et al. (2012) with the statistical model of Panton and Linebarger (1974).

Table 3 Boundary conditions. Inlet U

m=s

U∞

p 2 k=U∞

– –

rP ¼ 0 0.0000015 – 0.225

νt ωc=U∞

m=s2 –

Outlet

Airfoil

! rU ¼ 0 P¼0 rk ¼ 0 – rω ¼ 0

! U ¼0 rP ¼ 0 low-Re wall function low-Re wall function omega wall function

6.1. Determination of the boundary layer parameters In Section 3, it is noted that the commonly used approach to determine the boundary layer thickness, δ, is to find the location where the boundary layer edge velocity, Ue , is 0.99 % of the free-stream velocity Ue ¼ 0:99 U∞ . However, this approach is not applicable in the presence of the adverse pressure gradient due to the fact that the free stream velocity, U∞ , does not converge to a certain value. In the following, three different approaches for defining the boundary layer thickness are presented:

for 4∘ angle of attack. Moreover, the effect of the adverse pressure gradient on the suction side is more present when the angle of attack is increased. Therefore, the boundary layer gets thicker and the turbulent kinetic energy as well as the specific dissipation is increased as shown in Fig. 3(b) and (c).

1. Herr et al. (2015) proposed to determine the boundary layer thickness at the inflection point.

6. Wall-pressure spectrum This section focuses on the prediction of the wall-pressure spectra using the models described in Section 3. The first part is the

Fig. 3. Pressure coefficient along the airfoil and wake profiles at x=c ’ 1:0038 from the leading ) angle of attack edge; RANS 0∘ (——), 4∘ ( and the experimental data (Herr et al., 2015)0∘ (□) and 4∘ ( ) angle of attack.

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2. the boundary layer thickness is defined as the distance where the turbulent kinetic energy is equal to TKEðδÞ ¼ 0:0002U02 (Catlett et al., 2015). 3. the boundary layer thickness is defined as the position at which the total pressure is 99% of the maximum total pressure.

Table 4 Boundary layer thickness and boundary layer edge velocity obtained by three different methods.

The boundary layer profile normal to the surface is extracted at x=c ¼ 0:99 from the leading edge. Fig. 4 represents the position of the boundary layer thickness and the corresponding edge-velocity for NACA0012 at 0∘ angle of attack with three different approaches mentioned previously. The inflection point method in Fig. 4(a), where the second derivative of the streamwise velocity with respect to the wall-normal distance changes the sign, proposed by Herr et al. (2015) was challenging since the second derivative was noisy and over-estimated the value which is not shown in Fig. 4. Thus, the inflection point was determined from the minimum value of the first derivative. The second approach is to assign the boundary layer thickness from the kinetic energy where TKEðδÞ ¼ 0:0002U02 (Catlett et al., 2015) (Fig. 4(b)). This method is more straightforward since the right-hand side of the equation is a known value. The last method considered in this paper is to locate δ at which the total pressure is 99% of the maximum total pressure. Even though the approach seems similar to the commonly used method, Ue =U∞ ¼ 0:99, it guarantees that the total pressure becomes constant above the boundary layer thickness (Fig. 4(c)). The values obtained using each methods are given in Table 4. The next step is to investigate the sensitivity of the semi-empirical methods to the boundary layer thickness and the boundary layer edge velocity obtained from three different approaches. The momentum thickness, θ, and displacement thickness, δ , were computed for each method. The friction velocity, uτ , and Coles wake parameter, Π, were found by fitting the velocity boundary layer to the law of the wake. The effect of the different boundary layer thicknesses and edge velocities on the wall-pressure spectra predicted by the Goody, Rozenberg and Catlett models are shown in Fig. 5, and are compared with experimental data. The other models are found to be insensitive to these parameters and not presented in the paper. It is observed that the Goody

Method

δ=c

Ue =U∞

δ =c

θ=c

1 2 3

0.0252 0.0235 0.0229

0.9231 0.9202 0.9186

0.0058 0.0057 0.0057

0.0035 0.0034 0.0034

Fig. 5. Wall-pressure spectra prediction by different models at x=c ¼ 0:99 from ), Catlett ( ) models and the leading edge; Rozenberg (—), Goody ( experimental data from LWT ( ) (Herr et al., 2015); 1st method (—); 2nd method ) and 3rd method (—). (

and Catlett models exhibit a similar behavior. The models are insensitive to the boundary layer parameters at moderate and high frequency whereas 1 dB deviations occurs at low-frequencies. On the contrary, the Fig. 4. Determination of the boundary layer thickness and the corresponding boundary layer edge velocity with three different methods for 0∘ angle of attack at x=c ¼ 0:99 from the leading edge.

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6.2. Comparison of the wall-pressure spectrum models

Rozenberg model is sensitive to the boundary layer parameters. From the point of maximum of ϕ, up to the highest frequency a deviation of up to 2 dB occurs. The other models mentioned in Section 3 were disregarded since they do not exhibit any sensitivity to the boundary layer parameters. When the scaling parameters for the spectrum and the frequency are considered, Goody and Catlett models are the only ones who used the same scaling. This could be a possible explanation of their similar behavior even though their parametrization is completely different. For the Rozenberg model, the driving parameter for the trend is mainly Δ which appears only in this model. Overall, it is observed that the first method over-predicts the boundary layer parameters and underestimates the high frequency part for the Rozenberg model. The drawback of the Catlett et al. (2015) method is that when the simulations are performed by a two-equation turbulence model, it is not always possible to acquire the turbulent kinetic energy, kT accurately. Thus, in terms of ease and direct comparison to experimental data, it is suggested that the boundary layer thickness obtained by the total pressure method is more suitable. Consequently, this method has been applied to all the results presented herein after. Furthermore, the experimental data is dominated by the background noise after 15 kHz, thus the frequencies higher than this will be neglected for the following parts. The other discrepancy found in Section 3 regarding the semiempirical models is the determination of Π and βc . These parameters are only in the Rozenberg and Kamruzzaman models. In both models, Π is calculated from Eq. (6) whereas βc is calculated from Eq. (7) for the Kamruzzaman model. However, in the present paper, Π was obtained from solving Eq. (5) implicitly as well as finding the best fit with the law of the wake whereas the βc value was calculated directly from the definition of the Clauser's equilibrium parameter, which is the same method as the Rozenberg model. The corresponding values for two models are given in Table 5 and compared with the experimental data in Fig. 6. In the general form of the wall-pressure spectrum given in Eq. (3), the parameter a, which alters the amplitude of the spectrum, is driven by Π for the Rozenberg model while Π and βc for the Kamruzzaman model. The corresponding values are given in Table 1. Fig. 6(a) shows an increment of 1 dB over the whole range of the spectrum when the Π value is increased by 30%. For the Kamruzzaman model, even though βc value is reduced by 2%, which should result in a decrease on the amplitude of the spectra, an increment of less than 1 dB is observed since the increment of Π was more dominant. Furthermore, the comparison with the experimental data reveals that the amplitude for the low and mid frequencies are captured better with the newly calculated variables. However, the higher frequency part is over-estimated for both models. In conclusion, this analysis demonstrates that there is not a certain method that gives the most accurate prediction. Depending on the definition of the variable the spectrum of the amplitude changes around 1 dB. The wall-pressure spectra presented herein after are calculated by the methods considered in the current paper which provide better predictions for the higher frequencies. The summary of parameters required by the semi-empirical models for 0∘ angle of attack (only suction side due to the symmetry) and 4∘ angle of attack for both sides are given in Table 6. It can be interpreted that in the presence of the adverse pressure gradient, the integral boundary layer parameters increase as the velocity profile gets thicker, therefore, the friction velocity, uτ decreases.

The wall-pressure spectra are predicted by six different semiempirical models and one statistical model for two different angle of attacks. Fig. 7 shows the comparison between the experimental data from IAG LWT (Herr et al., 2015) and the predictions. Only the suction side of the wall-pressure spectrum is shown for 0∘ angle of attack. For the 4∘ angle of attack case, it can be observed that below f < 5 kHz, the suction side contaminates the pressure spectrum and above this value, the pressure side contaminates the spectrum. The wall-pressure spectrum obtained from Goody's model exhibits a poor prediction for all frequency ranges for both cases. This behavior was expected since the Goody model was tuned for ZPG flows. The Catlett and Herr & Hu models are also unable to capture the behavior. They share a common point; both of them are based on a flat-plate where an APG flow was created with trailing-edge sections (Catlett et al., 2015) or a rotational airfoil (Hu and Herr, 2016) and obtained satisfactory results. Even though, they compared their model with Rozenberg's model, they haven't performed any predictions for airfoil flows. Therefore, it can be deduced that these models are unrepresentative for airfoils. Kamrazzuman's model, which is developed by considering airfoil flows, performs better compared to the previous models. For both cases, the model under-predicts below f < 4:5 kHz and over-predicts above this range. This spectral behavior can be regulated by reducing the parameter d in which the peak point of the spectrum shifts to the lower frequencies with an amplitude augmentation whereas the higher frequency range shifts downwards. When the formalization is considered, all the parameters are constant except the parameter a which is a function of Π and β and only regulates the spectrum amplitude. Thus, increasing the amplitude by changing those parameters will perform a better match for the peak region whereas worsen the higher frequencies. Thus, it can be concluded that this model provides a good prediction around 3 dB above the middle frequency range despite to its simplicity. The Rozenberg model performs the best for 0∘ angle of attack where the APG is mild. For the 4∘ angle of attack case, the model captures the behavior but under-predicts the amplitude by around 3 dB for the suction side where the APG is strong. Even though the model captures the amplitude and trend for low and middle range frequencies, it is unable to exhibit the decay for the higher frequencies. The last semi-empirical model performs as well as Rozenberg's model except for the 4∘ angle of attack on the suction side. The modification performed for parameter a, shifts the amplitude and performs a better prediction. The statistical model proposed by Panton & Linebarger, captures the trend for all the cases, however, it under-predicts the amplitude by around 3 dB. The possible reason for the under-estimation is due to the anisotropy factor applied for the wall-normal velocity fluctuations which are obtained from a flat-plate. However, for APG flows, it is expected to be higher. 7. Far-field noise prediction Amiet's far-field noise prediction model links the far-field sound to wall-surface pressure under the turbulent boundary layer through an aeroacoustics transfer function mentioned in Section 1. 1/3-octave band far-field trailing edge noise prediction with different wall-pressure spectrum models are given in Fig. 8 with experimental data conducted in DLR's Acoustic Wind-Tunnel Braunschweig (AWB), IAG Laminar Wind Tunnel (LWT) and UF Aeroacoustic Flow Facility (UFAFF) (Herr et al., 2015). For the scaling of the experimental data, Herr et al. (2015) indicates that a scatter band of 3 dB should be considered. The predictions were computed at an observer distance of r ¼ 1 m and a wetted span of b ¼ 1 m. The footprint of the wall-pressure spectrum can be recognized in Fig. 8. The Goody, Catlett and Herr & Hu models are unable to capture the trend of the experimental results for both cases. It can be observed that the predictions are insensitive to the APG flows. The deviation in the

Table 5 Values of Π and βc obtained from different equations. Models

Rozenberg Kamruzzaman

βc

Π

βc

Π

ðθ=τw Þðdp=dxÞ

Eq. (5)

Eq. (7)

Eq. (6)

5.2832

2.2931

– 5.1670

2.9834

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Fig. 6. The comparison of the wall-pressure spectra which is calculated by different definition of βc and Π; (–) is calculated by Π obtained from Eq. (5) and βc from ðθ=τw Þðdp=dxÞ; (—) is calculated by Π obtained from Eq. (6) for both models and βc is calculated from Eq. (7) for only Kamruzzaman model; experimental data from LWT( ) (Herr et al., 2015).

amplitude difference of less than 1 dB in the range of 3 < fc < 15 kHz between the two angle of attacks. One of the typical behaviors observed from both figures is when the far-field prediction is obtained from the semi-empirical models, they performed better at the higher frequencies, fc > 10 kHz, even though the wall-pressure spectrum predictions over-predicted in that range. Furthermore, the low and middle frequencies are under-predicted by mostly for all the wall-pressure spectrum models, however, the far-field noise prediction performed tolerably good results, especially for UAFF measurements. On contrary, the Panton & Linebarger model always under-predicted the wall-pressure spectrum but captured the middle frequency range for the far-field noise prediction. It should be noted again that the experimental data is scaled since the measurements were conducted in different facilities with slightly different operating conditions. The associated uncertainty on the scaled measurements is reported to be 3 dB (Herr et al., 2015) which is the only possible explanation why some models showing a good match with the measured wall pressure spectrum yield a poorer agreement with the far-field measurements, and reciprocally. Equation (1) does indeed indicate a direct relationship between the wall pressure spectrum and the far-field sound level.

Table 6 Boundary layer parameters. NACA0012

Ue =U∞ δ=c δ =c θ=c uτ =U∞ Δ RT Π βc H Δ

0∘

4∘

SS

SS

PS

0.9186 0.0229 0.0057 0.0034 0.0277 4.02 28.80 2.2931 5.2832 1.676 0.2056

0.9205 0.0293 0.0086 0.0047 0.0226 3.41 24.34 3.86 10.4288 1.830 0.3755

0.9207 0.0181 0.0039 0.0025 0.0318 4.68 29.89 1.4968 2.7998 1.56 0.1204

driving parameters is apparently not dominant. For 0∘ angle of attack in Fig. 8(a), it can be observed that the Kamruzzaman model performs better for the low and middle frequency ranges regarding to UAFF measurements and shows poor prediction for the higher frequencies. The Rozenberg and Lee models are exactly the same in this case and exhibit a good prediction with the experimental data. Even though, Panton & Linebarger's model under-estimates the wall-pressure spectrum by around 3 dB, the far-field noise measurements especially, LWT and AWB, shows a better match for the frequencies below f < 5 kHz. However, for the higher frequencies a difference around 2 dB is still present. For 4∘ angle of attack in Fig. 8(b), the experimentally obtained wallpressure spectrum shows that the frequencies below f < 3 kHz are dominated by the suction side and the above the pressure side is dominant. Even though the Kamruzzaman model under-predicts the wallpressure spectrum up to 4 dB below f < 3 kHz, a better match is observed for the far-field noise prediction. Furthermore, the wallpressure spectrum demonstrates a deviation for the higher frequencies which is not present in comparison with the far-field experimental data. Regarding the wall-pressure spectrum predictions, the Lee model performed better than the Rozenberg model. However, the far-field noise prediction appears to be over-estimated for low frequencies. The higher frequencies give the same results as Rozenberg's model since they are dominated by the pressure side in which the Lee model has the same formulation. Even though the Panton & Linebarger model under-predicts the wall-pressure spectra, a good match is observed below the middle frequencies, fc < 6 kHz. However, the difference between the experimental data for the higher frequencies is larger. It was mentioned in Fig. 3(a) that the kinetic energy for this case is 60% under-predicted compared to the experimental data and the difference between the two angle of attacks is around 20%. It can be concluded that with the underprediction of the kinetic energy as well as applying the same anisotopy factor, it is expected that the Panton & Linebarger model obtains a higher deviation compared to the experimental data while observing an

8. Conclusions In this paper, the trailing-edge noise prediction is obtained using Amiets theory combined with RANS simulations. This theory requires the wall-pressure fluctuations of the turbulent boundary layer which is computed from either a semi-empirical or a statistical model. The semiempirical models are based on the re-scaling of the wall-pressure spectrum by the boundary layer variables. Several semi-empirical models are investigated. These models include Goody which assumes the zero pressure, Rozenberg, Catlett, Kamruzzaman, Herr & Hu and Lee which are developed for adverse pressure gradient flows. The statistical model proposed by Panton & Linebarger differs from the others in how the wallpressure spectra is obtained by the integration of the Poisson equation and thus requires the complete profiles. The simulations were performed on the NACA0012 airfoil with two different angle of attacks, 0∘ and 4∘ , at the same Reynolds number. The pressure distribution (Cp ) along the airfoil and the near wake profiles are compared with the experimental results from the IAG Laminar Wind Tunnel (LWT) for the validation. The Cp profile and the near wake profiles show a similar behavior as the experiments except for the turbulent kinetic energy for the 4∘ angle of attack under-predicted due to isotropic assumption. The sensitivity of the semi-empirical models on the wall-pressure spectrum with respect to the three different boundary layer thickness definitions is investigated. It is found that with a maximum increase of 10% of the boundary layer thickness leads to a decrease of around 1 dB for Goody, Catlett, Rozenberg and Lee models whereas the rest of the models are insensitive. Furthermore, the several methods to compute βc and Π are questioned. It is found that an increase of 30% in the Π value

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Fig. 8. Far-field noise prediction by different wall-pressure spectrum models at ), Rozenberg (–––), Catlett ( ), x=c ¼ 0:99 from the leading edge; Goody ( ), Herr & Hu ( ), Lee ( ), Panton & Linebarger ( ) Kamruzzaman ( and experimental data from LWT ( ), AWB ( ), UFAFF ( ) (Herr et al., 2015).

only increases the amplitude by 1 dB. After these analysis, a methodology is concluded to determine the required variables for the semi-empirical models. The wall-pressure spectra are computed with seven different models at x=c ¼ 0:99 from the leading edge. Even though five semi-empirical models out of six were developed for the APG flows, only three of them, Kamruzzaman, Rozenberg and Lee models, performed adequately. The drawback of the semi-empirical models is that they are tuned for specific experimental data, thus, they are not universal. Moreover, it is hard to obtain the required variables which are easily adjustable. Considering the success rate of the semi-empirical methods, it can be emphasized that the Panton & Linebarger model is equally predicting well by capturing the trend, however, under-estimating by the far-field noise by over 3 dB. Overall, the Lee model performed the best amongst the methods tested. Finally, the 1/3 octave band far-field noise prediction is performed by using Amiet's theory. Likewise, the predictions by the Goody, Catlett and Herr & Hu models for both angle of attacks exhibit the same behavior and performed poorly compared to the experimental data. The rest of the semi-empirical models exhibit a better prediction. Even though the Lee

Fig. 7. Wall-pressure spectra prediction by different models at x=c ¼ 0:99 from the leading edge; Goody ( ), Rozenberg (––), Catlett ( ), Kamruzzaman ), Herr & Hu ( ), Lee ( ), Panton & Linebarger ( ) and experi( mental data from LWT ( ), AWB ( ), UFAFF ( ) (Herr et al., 2015).

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Journal of Wind Engineering & Industrial Aerodynamics 175 (2018) 305–316 Guerri, O., Bouhadef, K., Harhad, A., 2006. Turbulent flow simulation of the NREL S809 airfoil. Wind Eng. 30 (4), 287–301. Guilmineau, E., Piquet, J., Queutey, P., 1997. Two-dimensional turbulent viscous flow simulation past airfoils at fixed incidence. Comput. Fluids 26 (2), 135–162. Henkes, R.A.W.M., 1998. Scaling of equilibrium boundary layers under adverse pressure gradient using turbulence models. AIAA J. 36 (3), 320–326. Herr, M., Ewert, R., Rautmann, C., Kamruzzaman, M., Bekiropoulos, D., Arina, R., Iob, A., Batten, P., Chakravarthy, S., Bertagnolio, F., jun 2015. Broadband trailing-edge noise predictions-overview of BANC-III results. In: 21st AIAA/CEAS Aeroacoustics Conference. American Institute of Aeronautics and Astronautics, Reston, Virginia, p. 2847. Herrig, A., Kamruzzaman, M., Würz, W., Wagner, S., 2013. Broadband airfoil trailingedge noise prediction from measured surface pressures and spanwise length scales. Int. J. Aeroacoustics 12 (1–2), 53–82. Howe, M.S., 1998. Acoustics of FluidStructure Interactions. Cambridge University Press, Cambridge. Hu, N., Herr, M., May 2016. Characteristics of wall pressure fluctuations for a flat plate turbulent boundary layer with pressure gradients. In: 22nd AIAA/CEAS Aeroacoustics Conference. Aeroacoustics Conferences. American Institute of Aeronautics and Astronautics, p. 2749, 0. Hwang, Y., Bonness, W.K., Hambric, S.A., 2009. Comparison of semi-empirical models for turbulent boundary layer wall pressure spectra. J. Sound Vib. 319 (1), 199–217. Kamruzzaman, M., Herrig, A., Lutz, T., Würz, W., Kramer, E., Wagner, S., 2011. Comprehensive evaluation and assessment of trailing edge noise prediction based on dedicated measurements. Noise Control Eng. J. 59 (1), 54. Kamruzzaman, M., Bekiropoulos, D., Lutz, T., Wrz, W., Krmer, E., 2015. A semi-empirical surface pressure spectrum model for airfoil trailing-edge noise prediction. Int. J. Aeroacoustics 14 (5–6), 833–882. Klebanoff, P.S., January 1955. Characteristics of Turbulence in Boundary Layer with Zero Pressure Gradient. Tech. rep. Kraichnan, R.H., 1956. Pressure fluctuations in turbulent flow over a flat plate. J. Acoust. Soc. Am. 28 (3), 378–390. Kral, L., 1998. Recent experience with different turbulence models applied to the calculation of flow over aircraft components. Prog. Aero. Sci. 34 (7), 481–541. Lee, S., Villaescusa, A., 2017. Comparison and assessment of recent empirical models for turbulent boundary layer wall pressure spectrum. In: 23rd AIAA/CEAS Aeroacoustics Conference. American Institute of Aeronautics and Astronautics, p. 3688. Lee, Y.-T., Blake, W.K., Farabee, T.M., Sep 2004. Modeling of wall pressure fluctuations based on time mean flow field. J. Fluid Eng. 127 (2), 233–240. Lilley, G.M., Hodgson, T., 1960. On Surface Pressure Fluctuations in Turbulent Boundary Layers. Tech. rep.. Advisory Group For Aeronautical Research and Development Paris(France). Menter, F.R., Aug 1992. Performance of popular turbulence model for attached and separated adverse pressure gradient flows. AIAA J. 30 (8), 2066–2072. Menter, F.R., Aug. 1994. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J. 32 (8), 1598–1605. Moriarty, P., Migliore, P.G., (U.S.), N. R. E. L, 2003. Semi-empirical Aeroacoustic Noise Prediction Code for Wind Turbines [electronic Resource]/P. Moriarty and P. Migliore. National Renewable Energy Laboratory Golden, Colo. Murman, S.M., 2011. Evaluating Modified Diffusion Coefficients for the SST Turbulence Model Using Benchmark Tests. AIAA Paper 3571. Nash, J.F., 1966. Turbulent-boundary-layer Behaviour and the Auxiliary Equation. HM Stationery Office. Oerlemans, S., Fisher, M., Maeder, T., K€ ogler, K., jun 2009. Reduction of wind turbine noise using optimized airfoils and trailing-edge serrations. AIAA J. 47 (6), 1470–1481. Panton, R.L., Linebarger, J.H., aug 1974. Wall pressure spectra calculations for equilibrium boundary layers. J. Fluid Mech. 65 (02), 261. Pape, A.L., Lecanu, J., 2004. 3D Navier-Stokes computations of a stall-regulated wind turbine. Wind Energy 7 (4), 309–324. Parchen, R., TNO-TH., T. P. D., 1998. Progress Report DRAW: a Prediction Scheme for Trailing Edge Noise Based on Detailed Boundary Layer Characteristics. TNO-report HAG-RPT. TNO Institute of Applied Physics. Redonnet, S., Jun. 2014. Aircraft noise prediction via aeroacoustic hybrid methods development and application of onera tools over the last decade : some examples. AerospaceLab 7, 1–16. Remmler, S., Christophe, J., Anthoine, J., Moreau, S., Sep 2010. Computation of wall pressure spectra from steady flow data for noise prediction. AIAA J. 48 (9), 1997–2007. Roger, M., Moreau, S., 2005. Back-scattering correction and further extensions of Amiet's trailing-edge noise model. Part 1: theory. J. Sound Vib. 286 (3), 477–506. Rotta, J., 1953. On the Theory of the Turbulent Boundary Layer. Rozenberg, Y., Robert, G., Moreau, S., Sep 2012. Wall-pressure spectral model including the adverse pressure gradient effects. AIAA J. 50 (10), 2168–2179. Rumsey, C.L., 2007. Apparent transition behavior of widely-used turbulence models. Int. J. Heat Fluid Flow 28 (6), 1460–1471. Sanderg, R.D., Sandham, N.D., jan 2008. Direct numerical simulation of turbulent flow past a trailing edge and the associated noise generation. J. Fluid Mech. 596. Schetz, J.A., 1993. Boundary Layer Analysis, 93. NASA STI/Recon Technical Report A. Schlinker, R.H., Amiet, R.K., 1981. Helicopter rotor trailing edge noise. In: Astrodynamics Specialist Conference. AIAA. Schmidt, J.H., Klokker, M., 12 2014. Health effects related to wind turbine noise exposure: a systematic review. PLoS One 9 (12), 1–28. Sørensen, N.N., Michelsen, J., Schreck, S., 2002. Navier-Stokes predictions of the NREL phase VI rotor in the NASA Ames 80 ft 120 ft wind tunnel. Wind Energy 5 (2–3), 151–169.

model showed a good agreement with the wall-pressure spectrum, the far-field noise prediction is over-estimated at low and middle frequencies. The Panton & Linebarger model performs better for the low and middle frequencies but under-estimates the higher frequencies. Considering the measurements obtained from different facilities, it can be concluded that the models predicted the far-field trailing-edge noise in the limit of 3 dB. The Panton & Linebarber model amongst the other models is found to be more robust and applicable for wide range of applications. A possible improvement to overcome the under-prediction for high frequencies could be to reconstruct the streamwise velocity fluctuations with an anisotropy factor developed for APG flows. Acknowledgement The authors acknowledge the support of the European Commission's Framework Program ”Horizon 202000 , through the Marie SkłodowskaCurie Innovative Training Networks (ITN) ”AEOLUS4FUTURE - Efficient harvesting of the wind energy” (H2020-MSCA-ITN-2014: Grant agreement no. 643167) to the present research project. The authors also acknowledge with thanks the WINERCOST TU1304 COST Action. Appendix A. Supplementary data Supplementary data related to this article can be found at https://doi. org/10.1016/j.jweia.2018.01.030. References Amiet, R., aug 1975. Acoustic radiation from an airfoil in a turbulent stream. J. Sound Vib. 41 (4), 407–420. Amiet, R., aug 1976. Noise due to turbulent flow past a trailing edge. J. Sound Vib. 47 (3), 387–393. Barone, M., 2011. Survey of techniques for reduction of wind turbine blade trailing edge noise. In: Prepared for Sandia National Laboratory. Bertagnolio, F., 2008. Trailing Edge Noise Model Applied to Wind Turbine Airfoils. Danmarks Tekniske Universitet, Ris Nationallaboratoriet for Bredygtig Energi ris-R1633(EN). Blake, W.K., 1986. Mechanics of flow-induced sound and vibration. In: General Concepts and Elementary Sources, vol. I. Academic Press. Bockstael, A., Dekoninck, L., De Coensel, B., Oldoni, D., Can, A., Botteldooren, D., 2011. Wind turbine noise: annoyance and alternative exposure indicators. In: Forum Acusticum 2011. European Accoustics Association (EAA), pp. 345–350. Brooks, T., Pope, D., Marcolini, M., 1989. Airfoil Self-noise and Prediction. NASA reference publication. National Aeronautics and Space Administration, Office of Management, Scientific and Technical Information Division. Catalano, P., Amato, M., 2003. An evaluation of rans turbulence modelling for aerodynamic applications. Aero. Sci. Technol. 7 (7), 493–509. Catlett, M.R., Anderson, J.M., Forest, J.B., Stewart, D.O., Dec 2015. Empirical modeling of pressure spectra in adverse pressure gradient turbulent boundary layers. AIAA J. 54 (2), 569–587. Chandiramani, K.L., 1974. Diffraction of evanescent waves, with applications to aerodynamically scattered sound and radiation from unbaffled plates. J. Acoust. Soc. Am. 55 (1), 19. Chase, D., may 1980. Modeling the wavevector-frequency spectrum of turbulent boundary layer wall pressure. J. Sound Vib. 70 (1), 29–67. Clauser, F.H., feb 1954. Turbulent boundary layers in adverse pressure gradients. J. Aeronaut. Sci. 21 (2), 91–108. Coles, D., 1956. The law of the wake in the turbulent boundary layer. J. Fluid Mech. 1 (2), 191–226. Corcos, G.M., mar 1964. The structure of the turbulent pressure field in boundary-layer flows. J. Fluid Mech. 18 (03), 353. Doolan, C.J., Gonzalez, C.A., Hansen, C.H., August 2010. Statistical estimation of turbulent trailing edge noise. In: Proceedings of 20th International Congress on Acoustics. American Institute of Aeronautics and Astronautics. Durbin, P., Reif, B., 2011. Statistical Theory and Modeling for Turbulent Flows. Wiley. Ewert, R., may 2007. RPM - the fast Random Particle-Mesh method to realize unsteady turbulent sound sources and velocity fields for CAA applications. In: 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference), vol. 4. American Institute of Aeronautics and Astronautics, Reston, Virigina, p. 1. Ewert, R., 2008. Broadband slat noise prediction based on CAA and stochastic sound sources from a fast random particle-mesh (RPM) method. Comput. Fluids 37 (4), 369–387. Gloerfelt, X., Le Garrec, T., 2009. Trailing edge noise from an isolated airfoil at a high Reynolds number. In: 15th AIAA/CEAS Aeroacoustics Conference (30th AIAA Aeroacoustics Conference), p. 3201. Goody, M., sep 2004. Empirical spectral model of surface pressure fluctuations. AIAA J. 42 (9), 1788–1794. 315

Y.C. Küçükosman et al.

Journal of Wind Engineering & Industrial Aerodynamics 175 (2018) 305–316 Willmarth, W.W., Roos, F.W., may 1965. Resolution and structure of the wall pressure field beneath a turbulent boundary layer. J. Fluid Mech. 22 (01), 81. Zagarola, M.V., Smits, A.J., 1998. Mean-flow scaling of turbulent pipe flow. J. Fluid Mech. 373, 3379.

Sorensen, N., Michelsen, J., et al., 2004. Drag prediction for blades at high angle of attack using CFD. Trans. Am. Soc. Mech. Eng. J. Sol. Energy Eng. 126 (4), 1011–1016. Tulapurkara, E., 1997. Turbulence models for the computation of flow past airplanes. Prog. Aero. Sci. 33 (1), 71–165. € Waye, K., Ohrstr€ om, E., 2002. Psycho-acoustic characters of relevance for annoyance of wind turbine noise. J. Sound Vib. 250 (1), 65–73.

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