Improving Accuracy of Airfoil Trailing Edge Noise Models with Turbulent Flow Anisotropy for Cambered Airfoil

Hussain Ali Abid 1 School of Engineering and Material Science, Queen Mary University of London Mile End Road, London, E1 4NS, UK

Annabel. P. Markesteijn 2 School of Engineering and Material Science, Queen Mary University of London

Mile End Road, London, E1 4NS, UK

Sergey A. Karabasov 3 School of Engineering and Material Science, Queen Mary University of London

Mile End Road, London, E1 4NS, UK

Zang B. 4 Department of Aerospace Engineering, University of Bristol Bristol, England, BS8 1TR, UK

Mahdi Azarpeyvand. 5

Department of Aerospace Engineering, University of Bristol Bristol, England, BS8 1TR, UK

ABSTRACT

The TNO type SP spectrum models are used to represent the surface pressure spectrum beneath a turbulent boundary layer, with a focus on adverse pressure gradient airfoil flows and associated trailing edge noise prediction. The input for the models is provided via the Wall Modelled Large Eddy Simulation (WM-LES). A cambered NACA65410 airfoil with an inflow Reynolds number of 4.22 × 10 5 and angles of attack of 0 and 4 degrees are studied. We show that the inconsistencies in the existing TNO model's surface pressure and far-field noise predictions are due to inaccurate modelling of turbulent anisotropy, which is corrected by including the anisotropy in the definition of the correlation length scales and velocity spectrum. The anisotropy is linked to the boundary layer integral parameters and one-point velocity spectrum, which improved the predictions

1 PhD. Student, School of Engineering and Materials Science, h.abid@qmul.ac.uk

2 Research Associate, School of Engineering and Materials Science / Director GPU-prime Ltd, Cambridge, UK

3 Professor of Computational Modelling, School of Engineering and Materials Science, AIAA Associate Fellow

4 Lecturer, Department of Aerospace Engineering

5 Professor in Aerodynamics and Aeroacoustics, Department of Aerospace Engineering

ENTER EMAIL HERE

compared to the experimental observations. The Amiet trailing edge noise model is used to predict far-field noise, and the impact of turbulence anisotropy on far-field noise prediction is assessed. The addition of turbulence anisotropy improves the noise prediction by 5 decibels.

1 INTRODUCTION

Flow-induced noise and vibrations are significant noise sources in turbomachinery applications such as in wind turbines, marine propellers, rotorcrafts, and aircraft airframes. The most dominant high-frequency broadband noise is the trailing edge (TE) noise which is one of the five types of airfoil self-noise. The TE noise is caused by the scattering of the hydrodynamic incidence surface pressure at the trailing edge into the acoustic waves propagated to the far-field observer. The hydrodynamic fluctuating surface pressure (SP) is caused by turbulence eddies in the boundary layer, which vary over spatial and temporal scales. Hence the resulting TE noise is broadband in nature. Reducing TE noise is difficult, and consistent and efficient methods for characterising these sources are required. Several suites of methods are available for noise quantification; however, recent focus has been on developing efficient, robust noise quantification methods that can be readily applied in the design optimisation loop to design aerodynamic bodies with reduced noise signature. In this case, noise prediction is a two-step process. In the first step, the source of the TE noise (incidence SP spectrum) is required to be analytically quantified, which is then propagated to the far-field observer through an acoustic analogy such as the semi-analytical Amiet TE noise model [1].

There are several low order empirical and semi-analytical SP spectrum models available in the literature, such as those by Chase [2], Goody [3], and Rozenberg et al. [4] models. These models are applicable to low Reynolds number flows with zero to mild pressure gradients. More recently, an advanced empirical SP spectrum model which accounts for the adverse pressure gradient (APG) boundary layer has been developed [5]; These models, however, do not account for the wide range of dynamic flow physics that occurs on the airfoil with strong surface curvature at high angles of attack (AoA). Hence, developing a physics-based SP spectrum model is paramount for accurate noise prediction modelling. The most widely employed physics-based model for TE noise modelling is known as the TNO-Blake [6], [7] model, which stems from the solution of the Poisson equation. These models are semi-analytical as they require boundary layer (BL) integral parameters such as time-averaged streamwise velocity profile, wall-normal Reynolds stresses and velocity and pressure correlation length scales. Additional parameters include the velocity spectrum of the turbulent velocity, which is required to be modelled. Abid et al. 2021 [8], [9] have shown the SP spectrum modelling is most dominantly influenced by the accurate modelling of the velocity spectrum. The velocity auto-spectrum has been modelled using the isotropic von-Kármán spectrum in the original formulations; however, the experimental observation [10]–[12] revealed inaccurate modelling of the velocity spectrum resulted in SP spectral discrepancies of 6-9dB.

Improved TNO models were developed to address this problem which used the 'turbulence anisotropy' stretching parameters methodology originally introduced by lynch et al. [13] to adjust for the anisotropic effect in the velocity spectrum. As a result, various researchers, such as Stalnov et al. [14] sought to construct an anisotropic stretching model. The developed anisotropic model was based on root-mean-square turbulent velocity in zero-pressure gradient boundary layers flow and was applied to the NACA0012 airfoil at zero angles of attack. The study showed an improved correlation between the SP spectrum model prediction and the measurement; however, the modelling was limited to zero pressure gradient boundary layer flows. Another model was developed by Bertagnolio et al. [15]. They linked the turbulence anisotropy to boundary layer parameters such as SP gradient, turbulent velocity, and boundary layer thickness. The aim was to

extend the anisotropy modelling to strong adverse pressure gradient flows. However, the resulting anisotropy parameters were inapplicable for airfoil turbulent boundary layer flows by [8]; who modified the turbulence anisotropy scaling to yield an appreciable comparison of the prediction with the measurement for NACA0012 and NACA0015 airfoil at the range of angle of attack. In addition, most of the existing research has been focused on establishing the turbulence anisotropy modelling for a symmetric airfoil, with little attention given to the modelling of the SP spectrum for the cambered airfoil. Hence in this paper, we extend the modelling capability of the TNO model to the NACA65410 cambered airfoil at 0 and 4-degree angles of attack. The boundary layer information for the TNO models will be obtained from the scale-resolving Wall Modelled Large Eddy Simulation (WM -LES). In conjunction, we compare noise prediction obtained using the Amiet TE noise model prediction with the measurement and investigate and the influence of the turbulence anisotropy modelling on the noise prediction using the Amiet TE noise model.

2 SURFACE PRESSURE SPECTRUM MODELS FOR TRAILING

EDGE NOISE

2.1 Surface pressure spectrum models

The trailing edge (TE) noise computation is a two-step process. In the first step, the source of the TE noise, i.e., SP spectrum, is computed using the TNO models in the close vicinity to the trailing edge. The SP spectrum is input into the Amiet TE noise model to obtain the far-field noise in the second step.

The acoustic source (unsteady surface pressure (SP) fluctuations) is modelled using semi- empirical TNO models and is compared with the experimental data and LES simulations. The TNO models' source variables are derived from the LES simulation. The TNO-Blake (TB) model, an isotropic variation of the TNO model, is given by Equation 1.

ϕ pp (k 1 , k 3 ; ω)

2 δ BL

2

2 ̅̅̅(x 2 )ϕ̃ 22 (k 1 , k 3 , Λ f (x 2 ))ϕ m e −2 | 𝐤 | x 2 dx 2 ( 1 )

2 k 1

2 ∫ C n Λ 22|2 (x 2 )( ∂U 1

(x 2 ))

= 4ρ o

u 2

2 + k 3

k 1

∂x 2

0

Equation 1

Here k 3 is the spanwise wavenumber, δ BL is boundary layer thickness, Λ f is isotropic classical integral length scale, U 1 is streamwise velocity profile, u 2

2 is mean square vertical turbulent velocity, ϕ ̃ 22 is normalised velocity spectra of vertical turbulent velocity and Λ 22|2 is vertical correlation length scale. ϕ m is a moving axis spectrum that defines the evolution of turbulence eddies in the boundary layer as they convect, with velocity U c , past the TE. ϕ m is modelled as a delta function ϕ m = δ(ω −U c k 1 ) in the current study.

In Equation 1, we included a multiplicative component C n to account for the correct definition of correlation length scale, as proposed by [15] in the new model referred to as the HS model. Readers are advised to refer to [8] for the derivation of the HS model and its application to NACA0012 [9] and NACA0015 airfoil.

The one-point SP spectrum Π w is obtained by integrating it in k 1 and k 3 wavenumber as follows:

+∞

( 2 )

Π w (ω) = ∬ϕ pp (k 1 , k 3 ; ω)dk 1 dk 3

−∞

Equation 2

Table 1 summarises the surface pressure spectrum models employed in the current study.

Na me

Model Name

Multipl ier C n

Vertical correlation length scale

Anisotro

Integr al length

Scaling with Kinematic

py Stretching coefficients

scale in velocity

Λ 22|2

Shape Factor

β 1,2,3

spectra

ϕ ̃ 22

TB Original TNO-Blake

1 Λ 22|2 (x 2 ) β 1,2,3 = 1 Λ f (x 2 ) NO

(Parchen)

HS Abid et al. [8]

2 Λ 22|2 (ω, x 3 ) Equation

Λ f (ω, x 2 ) YES

10

Table 1 Summary of the wall pressure spectrum models

The SP spectrum from LES and experiment can be computed by obtaining a single point pressure–time signal at a point closest to the airfoil surface. The pressure-time signal is post- processed to obtain pressure auto-spectra based on the Welch method of periodogram. The pressure signal obtained from the LES is subdivided into the sample divided by 16 subblocks. The welch method is based on FFT and is combined with the Hanning window with a 50 per cent overlap.

2.1.1 Surface Pressure Spectrum Source Modelling.

The mean flow quantities in the integral equation of the SP spectrum in Equation 1 need to be quantified to evaluate the SP spectrum. In this paper, we employ the LES simulation to obtain these parameters for NACA65410. The LES simulation provides the time-averaged streamwise velocity profile U 1 which can be utilised to obtain displacement thickness δ ∗ , momentum thickness θ , kinematic shape factor H = δ ∗ /θ and mean shear ∂U 1 / ∂x 2 using the following Equation 3 and Equation 4 .

δ BL

δ ∗ = ∫(1 − U 1

( 3 )

) dx 2

U e

0

Equation 3

δ BL

δ ∗ = ∫ U 1

(1 − U 1

( 4 )

) dx 2

U e

U e

0

Equation 4

Here, δ BL is boundary layer (BL) thickness defined as the point in the space where the mean velocity reaches 95% of the free stream velocity. Furthermore, the vertical turbulent Reynolds stress u 2

2 is obtained directly from the LES simulation. Λ 22 |2 is a vertical correlation length scale computed using the integral of the two-point correlation of the turbulent velocity given by Equation 5 .

For the isotropic flow, the Λ 22|2 can be related to Λ f using Equation 6 with v = 1/3 for the von- Kármán spectrum.

δ

u i (x 2 ).u i (x 2 + Δx j ) ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅

( 5 )

Λ 22|2 (x 2 ) = ∫

d(Δx 2 )

2 (x 2 ). u i

2 (x 2 + Δx j )

√u i

0

Equation 5

Γ(v)Λ 2|22 (𝑥 2 )

√ πΓ(v+ 0.5) ( 6 )

Λ f (𝑥 2 ) =

Equation 6

For the HS model, we need the frequency-dependent length scales such as Λ 22|2 (x 2 ,ω) and Λ f

′ (ω,x 2 ) obtained using Equation 7 and Equation 8

55Γ ( 1

3 )

3 + 11 ( β 1 Λ f k c ) 2

1

√1 + (β 1 Λ f k c ) 2 ] ( 7 )

Λ 22|2 (x 2 , ω) = [(

)Λ f (x 2 )β 2

108√πΓ( 17

3 + 8(β 1 Λ f k c ) 2

6 )

Equation 7

Γ(v)Λ 2|22 (x 2 ,ω)

′ (ω,x 2 ) = H

√ πΓ(v + 0.5) ( 8 )

Λ f

Equation 8

Here k c is convective wavenumber defined as ω/U c . The SP spectrum also requires a model for the normalised 2D velocity spectrum of the wall-normal turbulent velocity ϕ ̃ 22 . The velocity spectrum is modelled using the von-Kármán spectrum given by Equation 9

( β 1 Λ f k 1 ) 2 + ( β 3 Λ f k 3 ) 2

( v )( v + 1 )

ϕ ̃ 22 (k 1 , k 3 , β 1 ,β 2 ) =

2 β 1 β 3

[1 + (β 1 Λ f k 1 ) 2 + (β 3 Λ f k 3 ) 2 ] v+2

𝜋 Λ f

( 9 )

Equation 9

Note that β 1,3 are anisotropic stretching parameters that directly influence the spectrum's shape and amplitude. Note that for the isotropic case β 1,2,3 = 1 . Additionally, for the HS model, the Λ f is replaced with a scale Λ f

′ given by Equation 8.

The anisotropic parameters employed in the velocity spectrum of the HS model are defined by Equation 10 and Equation 11. Note that the methodology only provides spanwise β 2 and wall- normal stretching coefficients β 3 . The streamwise stretching coefficient will be obtained by comparing the one-point spectrum obtained via LES and the analytical model, as discussed in Section 3.2.

β 2 = 1

1 6

3 γ

( 10 )

β 3 = (β 2 ) 1/2

Equation 10

1 3

2

[ ( ∂P

]

∂x 1 )

γ = δ Bl

( 11 )

u τ

ρ 0 μ

Equation 11

2.2 Amiet Trailing-Edge Noise Model

As the second step in the noise modelling for the TNO model derivation, the far-field noise spectrum can be deduced from the SP spectrum using the Amiet TE noise model. For the observer located in the midspan at a distance of x 2 in the wall-normal direction, perpendicular to the trailing edge x 1 = 1 , the far-field noise is given by Equation 12

S pp (x 1 , x 2 , x 3 = 0, ω)

2 L 3

2

= ( ωcx 2

2 |ℒ( ω

( 12 )

Λ p|3 (ω)Π w (ω)

4πc 0 σ 2 )

, k 3 = 0,x 1 , x 3 , U ∞ , U c )|

U c

Equation 12

ω

In Equation 12, ℒ(

U c , k 3 = 0, x 1 , x 3 , U ∞ , U c ) is the acoustic transfer function which convolutes

the acoustic source i.e. Π w (ω) to acoustic pressure in the far-field. We employ the Amiet transfer function for the TE noise computation, which is from Equation 13 to Equation 15

|ℒ( ω

, k 3 = 0, x 1 , x 3 , U ∞ , U c )|

U c

= − e 2iA

( 13 )

iA {(1 + i)e −2iA √2B Es ∗ [2(B−A)] −(1 + i)E ∗ [2B]}

Equation 13

Es ∗ [z] = E ∗ ( z )

; E ∗ (z) = 1 −i

2 erf(√iz)

( 14 )

√ z

Equation 14

A = ω

b + ωb

c o β 2 (M ∞ − x 1

σ ) ; B = ω

b + ωb

c o β 2 (M ∞ + 1)

U c

U c

( 15 )

Equation 15

Here c 0 is the speed of sound, U c is convection velocity of the turbulent eddies in the boundary layer, σ is flow corrected far-field observer location defined as σ 2 = x 1

2 + β 2 (x 2

2 + x 3

2 ) , β = 1 − M ∞

2 is the compressibility correction. x 1 and x 2 are observer coordinates in streamwise and vertical directions. Additionally, it is assumed that turbulence eddies convect the trailing edge undistorted, implying that radiation at radial frequency ω is due to convective wave number k 1 = k c = ω/U c . Here c is the chord length and L 3 is the airfoil wetted span length and Λ p|3 is a frequency-dependent

spanwise correlation length scale modelled using by Corcos [16] model given by Equation 16. Here U c is scaled with the freestream velocity and the scaling coefficient 𝛼 𝑐𝑣 = 𝑈 𝑐 /𝑈 ∞ is chosen as 0.65. b c is the Corcos constant with the value of 1.4.

Λ p|3 (ω) = b c U c

ω ( 16 )

Equation 16

3 DISCUSSION:

3.1 Evaluation of aerodynamic characteristics and boundary-layer profiles

Figure 1 and Figure 2 show the time-averaged pressure coefficient, streamwise wise velocity, and turbulent kinetic energy for NACA65410 at 0- and 4-degrees angle of attack, respectively. Furthermore, the Cp distribution is compared to the XFOIL prediction. The Cp distribution matches the experiment well at both angles of attack, demonstrating that the LES accurately captured the near-wall parameters at both angles. Furthermore, this trend is also depicted in the prediction of boundary layer profiles for mean streamwise velocity and turbulent kinetic energy. While the streamwise velocity is correctly captured, the turbulent kinetic energy profile for zero degrees angle of attack is slightly overestimated; this could be due to the overestimation of the streamwise turbulent velocity, which is generally observed for LES simulations of airfoil turbulent boundary layer flows.

Figure 1 Comparison of pressure coefficient 𝐶 𝑝 distribution between experimental measurement,

XFoil prediction and LES (a) 𝛼= 0 𝑜 (b) 𝛼= 4 𝑜

Figure 2 Comparison of Boundary layer profiles for streamwise velocity and turbulent kinetic

energy for NACA65410 at 𝑥 1 /𝑐 = 0.99 for (a) 𝛼= 0 𝑜 (b) 𝛼= 4 𝑜

3.2 Turbulent anisotropy parameters

The quantification of turbulence anisotropy is paramount for the accurate prediction of the SP spectrum. The anisotropy is incorporated in the modelling using the 'anisotropic stretching parameter' defined by Equation 10 and Equation 11. Figure 3 shows the spanwise and wall-normal anisotropy parameters and non-dimensional pressure gradient trend. It is illustrated that the anisotropy parameter increases as the flow approaches the TE. This is because the boundary layer becomes thicker while the friction velocity decreases as the TE is approached. While this method provides β 2 and β 3 . There is no such model available to quantify the streamwise anisotropic parameter β 1 .

Figure 3 Turbulence anisotropic modelling for spanwise and wall-normal stretching coefficients

for zero degrees angle of attack

We use the LES to get a pre-multiplied wavenumber spectrum and compare it to the isotropic von-Kármán spectrum's analytical model of the pre-multiple wavenumber spectrum to calibrate the streamwise anisotropic parameter. The analytical pre-multiple wavenumber spectrum is given by Equation 17

ϕ ̃ 22 (k 1 ) = Γ ( v + 0.5 )

β 1 Λ f [1 + (β 1 Λ f k 1 ) 2 ] v+0.5 [v + 1 −( v + 1 1 + (β 1 Λ f k 1 ) 2 )] ( 17 )

√ πΓ(v)

Equation 17

The ϕ ̃ 22 (k 1 ) is obtained from the LES simulation by taking the PSD of the single point of the turbulent velocity in the boundary layer using the Welch method of periodogram. ϕ 22 (k 1 ) is obtained using Taylor's frozen turbulence hypothesis as given by Equation 18

ϕ ̃ 22 (k 1 ) = U c ϕ 22 ( ω )

rms ) 2 ( 18 )

(u 2

Equation 18

ient y ; U. = 20.0m/s,a=0" + y- Lower surtece 3 Y-Upper Surace . ’ - en as ’ ace ee ’ fie pra) vsctropic tratching Content B.; U= = 0.0m, 020" ache ap 26 Seca eT Neate ee eae ee mec 2 © bus towerSurtece +) §ytowerSurtece S bi-Uppersurece + pr-Uppersurtace

Figure 4 shows the comparison of the analytical one-point spectrum for zero degrees angle of attack. Notably, the modelled pre-multiplied wavenumber spectrum has a peak at a smaller wavenumber than the calculated spectrum. Hence the peak is calibrated by reducing the β 1 , giving a value of 0.6. In previous studies [14], β 1 has been chosen to be 1 to minimise the empirical coefficients. However, the correct modelling of β 1 is critical for adverse pressure gradient (APG) SP predictions as β 1 shifts the spectral peak to a higher wavenumber (lower frequency). For APG flows, where the BL is thicker, the larger eddies are in the BL's outer layer, shifting the spectrum peak to the lower frequency. Reducing the β 1 as a comparable effect on the SP spectrum hence accurate modelling of β 1 is vital in accurately modelling the low-frequency SP spectrum and the far field noise.

Figure 4 Normalized pre-multiplied wavenumber spectrum for 𝛼= 0° [Left – comparison of modelled and computed one-point velocity spectrum] [Right – One point velocity spectrum shifted by 𝛽 1 =0.6 to match the peak wavenumber]

3.3 Evaluation of the surface pressure spectrum

Figure 5 shows the comparison of the prediction of the SP spectrum using the TNO based models with the LES predictions and the experimental measurement for NACA65410 at 0- and 4- degree angles of attack. The spectrum is obtained on the suction side of the airfoil for both angles. The peak frequency shifts towards lower frequency as the angle of attack increases, and the analytical models accurately predict this shift. The shifting of spectrum peak to lower frequency is observed because, with the increase of angle of attack, the pressure gradient on the suction side becomes more negative, resulting in a thicker boundary layer. Large eddies are in the outer layer in the thicker boundary layer, causing the peak to shift to a lower frequency. In addition, the LES simulation has also predicted the spectrum accurately for both angles; however, an earlier roll-off at a higher frequency is observed. This roll-off is related to the trip employed in the LES simulation, which causes additional mixing in the inner region of the boundary layer dominated by the small- scale eddies.

Moreover, the predictive capabilities and influence of anisotropy inclusion can be observed by comparing spectrums obtained from HS and TB models. The HS model has an anisotropic stretching parameter that modifies the correlation length scale and the velocity spectrum, resulting in accurate prediction compared with the experiment. Moreover, the importance of including a frequency-dependent correlation length scale Λ f (ω, x 2 ) in the velocity spectrum can also be signified here. Since Λ f (ω, x 2 ) is a function of β 1 and β 3 , the inclusion of Λ f (ω, x 2 ) results in the

velocity spectrum becoming a function of all three anisotropy parameters, and combining these modifications results in appreciable prediction by the HS model with the experiment.

Figure 5 Surface Pressure Spectrum (PSD) computation using LES and WPS models on the

suction side of the airfoil. (a) 𝛼= 0 𝑜 (b) 𝛼= 4 𝑜

3.4 Evaluation of far-field noise and sensitivity of anisotropy parameter

In this section, the far-field noise is estimated using the Amiet TE noise model and compared to the experimental measurement for NACA65410 at 0 and 4 degrees angle of attack. Three noise spectra are illustrated for the Amiet model prediction. S pp

Amiet (Π LES ) is the prediction based on Equation 12 with Π LES obtained from LES. Similarly, S pp

Amiet (Π TB,HS ) implies that the source spectrum is derived from TB and HS model predictions. For both angles of attack, it can be seen that the Amiet model predicts the noise spectrum for the cambered airfoil with appreciable accuracy even though the model formulation assumes streamwise and spanwise homogeneity. Moreover, the significance of the anisotropy modelling for the SP spectrum is also illustrated shown my the difference in prediction by TB and HS informed Amiet model.

Figure 6 Farfield acoustic pressure computation for NACA65410 using Amiet TEN model (a)

𝛼= 0 𝑜 (b) 𝛼= 4 𝑜

The prediction by the HS model is quite close to the experimental measurement, while the TB model predictions underestimate by 10dB. The significance of including the anisotropy modelling can be further observed by analysing the sensitivity of anisotropic parameters to the far-field noise, as shown in Figure 7

Figure 7 Sensitivity analysis of HS model in the noise prediction (a) 𝛽 1 sensitivity (b) 𝛽 2

sensitivity

Figure 7 shows that anisotropic turbulence parameters significantly influence the shape and amplitude of the noise prediction. The streamwise stretching coefficient 𝛽 1 modifies the spectrum shape by increasing the spectral amplitude at low frequencies while having minimal influence on the high-frequency part of the spectrum. This can be further explained as the β 1 tends to increase the SP spectral levels at low frequencies, which translates into an increase in the far-field noise spectral level. Moreover, the wall-normal stretching coefficient solely influences the amplitude as increasing the β 2 from 0.4 to 1, shift the entire spectral level by 20dB. This can be explained as β 2 influences the amplitude of the Λ f (ω, x 2 ) in the velocity spectrum. This signifies the importance of including the anisotropy modelling to improve the trailing edge noise modelling accuracy.

4 CONCLUSION

The accuracy of the TNO-based trailing edge noise prediction model is highly dependent on accurate modelling of the boundary layer flow structure near the trailing edge. The cambered aerofoil’s boundary layer is subjected to a negative pressure gradient, and the eddy structure near the trailing edge is strongly anisotropic, necessitating accurate anisotropic modelling. Hence for the noise prediction of NACA65410, the turbulence anisotropy is related to boundary layer parameters such as non-dimensional pressure gradient. In addition, we relate the streamwise anisotropy parameter with a one-dimensional velocity spectrum and show that the streamwise anisotropy parameter is critical in accurate modelling of SP and farfield noise spectrum, especially at low frequency. Moreover, for anisotropic HS SP model, these anisotropy parameters alter the correlation length scales and velocity spectrum to produce the SP spectrum resulting in close agreement with the experimental data. The effect of anisotropic parameter sensitivity on farfield noise is investigated to demonstrate that successful noise prediction using the Amiet model necessitates appropriate turbulence anisotropy modelling.

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