Inflow turbulence distortion for airfoil leading-edge noise prediction for large turbulence length scales for zero-mean loading Open Access

Noise exposure is harmful to the health and well-being of people, causing adverse psychological and physical effects.^1,2 Noise exposure also affects animal beings, e.g., marine mammals, birds, and fish species, that use sound to communicate, reproduce, search for prey, and navigate, among other sensory purposes.^3–5 Exposure to anthropogenic noise leads to alteration of behavior, foraging, reduction of communication ranges, and habitat avoidance in marine mammals and fish.³ Stringent regulations^6–9 are implemented to limit noise exposure suffered by people and wildlife. Flow-induced noise is an important contributor to anthropogenic noise in many applications, such as wind turbines, aircraft, and ships. Among the flow-induced noise sources for airfoils, leading-edge (LE) noise is the dominant mechanism for cases with significant turbulent inflow,^10,11 e.g., ship propellers¹² and fans.¹³ LE noise is generated by the interaction of unsteady turbulent inflow with the LE of an airfoil or hydrofoil.¹⁴ The understanding and modeling of airfoil LE noise are paramount to designing silent rotating blades.

Noise prediction models for LE noise are widely used in designing and optimizing propellers and fans. Amiet's LE noise prediction model¹⁵ is one of the most used models due to the good balance of accuracy of prediction, computational simplicity, and short computational turnaround time. This model considers an infinitely thin flat plate exposed to an inflow with turbulence which is modeled according to Taylor's frozen turbulence hypothesis.¹⁶ Amiet's model does not account for real airfoil geometric characteristics, i.e., finite thickness, camber, and angle of attack. The angle of attack effect on the LE radiated noise is negligible,^13,17,18 with the LE noise changing approximately 1 dB for angles of attack between 0º and 12º for a National Advisory Committee for Aeronautics (NACA) 0012 airfoil.¹⁷ Camber is also reported to have small effects on the radiated noise.^13,17 The airfoil thickness has the most significant impact on the LE far-field noise.^13,17,19 This impact is due to the inflow turbulence distortion.²⁰ The current work investigates how the turbulence distortion can be accounted for in Amiet's LE noise prediction model.

Paterson and Amiet¹⁸ measured the LE noise generated by a NACA 0012 subjected to inflow turbulence and compared the experimental data with analytical results from Amiet's LE model. They observed that the model overestimates the far-field noise for low Mach numbers and high frequencies. They attributed this mismatch to the finite airfoil thickness, as verified by Gershfeld.²¹ According to Paterson and Amiet,¹⁸ this occurs for frequencies where the gust wavelength is comparable to the airfoil thickness in the vicinity of the LE, i.e., in the high-frequency range. A similar discrepancy between the predicted and measured noise for high frequencies was observed by Moreau and Roger¹³ and Oerlemans and Migliore.²² According to Gill et al.,²⁰ the high-frequency mismatch between the predicted and the measured noise is due to the inflow turbulence distortion caused by the velocity gradients in the LE stagnation region. Ayton and Chaitanya¹⁹ observed that increasing the LE radius reduces the radiated noise generated by an airfoil of fixed thickness for high frequencies at low Mach numbers. Based on analytical results, they argued that the local LE geometry plays a key role in the LE noise mainly for thin airfoils at high frequencies, and for low Mach numbers this is mostly due to the distortion of the turbulence. Based on these research findings, it is clear that Amiet's model needs improvements to account for the airfoil finite thickness.

The efforts to account for the turbulence distortion in Amiet's model are focused on turbulence velocity spectrum modeling, which is an input to Amiet's model. The turbulence spectrum used generally is the von Kármán energy spectrum²³ for isotropic turbulence. This formulation assumes that the spectral level decays with the wavenumber to the power of −5/3 for high frequencies. To account for the turbulence distortion in the velocity spectrum formulation, Moreau and Roger¹³ proposed a correction to the incident velocity spectrum based on the asymptotic behavior of the inflow turbulence spectrum given by the rapid distortion theory (RDT) developed by Hunt.²⁴ Its asymptotic behavior shows that the one-dimensional turbulence spectrum tends to a −10/3 power law for high frequencies instead of the −5/3 from the von Kármán turbulence spectrum. Moreau and Roger¹³ showed that the far-field noise prediction for high frequencies approximated the measured noise well when the proposed velocity spectrum was used. The inflow turbulence characteristics used as input for the noise prediction model, i.e., root mean square of the velocity fluctuations $urms$ and longitudinal integral length scale $Λf$⁠, were evaluated experimentally at the LE location with the airfoil removed. This is a common approach in LE noise studies.^17,22,25 Using a similar methodology as Moreau and Roger,¹³ de Santana et al.²⁶ formulated a semi-empirical RDT-based velocity spectrum to model the turbulence distortion occurring in the vicinity of the airfoil LE. They measured the turbulence characteristics in the vicinity of a NACA 0012 airfoil LE. The RDT-based spectrum agreed better with the measured velocity spectrum in the vicinity of the airfoil LE than the isotropic von Kármán spectrum. They compared the measured LE noise to Amiet's noise prediction using the RDT-based velocity spectrum with $urms$ and $Λf$ values measured at $x/rLE=−0.12$ and at $x/rLE=−4$⁠, where x is the streamwise direction with x = 0 corresponding to the LE location and $rLE$ is the airfoil LE nose radius. They observed that using experimental values of $urms$ and $Λf$ too close (at $x/rLE=−0.12$⁠) or too distant (at $x/rLE=−4$⁠) from the leading-edge yields overpredicted or underpredicted sound levels, respectively. The position $x/rLE$ where $urms$ and $Λf$ should be obtained to result in accurate noise predictions is still an open question and it is discussed in this paper. Other experimental studies also focused on the turbulence distortion in the vicinity of an airfoil LE. However, these studies focused on understanding the turbulence-LE interaction for a porous LE^27,28 and a LE with serrations.²⁹ Analytical^30–33 and numerical^20,34,35 investigations were also carried out to comprehend further the effect of the inflow distortion on the LE radiated noise. In summary, the various studies show that the inflow turbulence distortion has a significant effect on the LE noise prediction. However, the understanding of how to account for the turbulence distortion effects in Amiet's noise prediction model to yield accurate noise estimates is still lacking in the literature.

The present paper aims to experimentally study the turbulence in the vicinity of an airfoil LE in order to investigate how the turbulence distortion can be considered in Amiet's model. Experiments were performed at zero angle of attack for a NACA 0008 and a NACA 0012 airfoil subjected to large turbulence length scales (⁠$10<Λf/rLE<43$⁠). The paper is organized as follows. In Sec. II, Amiet's LE noise model is reviewed together with the von Kármán and the RDT-based velocity spectra. In Sec. III, the experimental method to evaluate the turbulence distortion and to measure the LE far-field noise is discussed. In Sec. IV, the turbulence characteristics in the LE vicinity are analyzed. In Sec. V, predictions of the LE far-field noise are compared to measurements. In this section, it is also discussed how to account for the turbulence distortion in Amiet's model. Finally, the main conclusions of this study are stated in Sec. VI.

Amiet¹⁵ proposed a semi-analytical model to predict the far-field noise generated by the interaction of a turbulent uniform flow with a flat plate of infinitely large span and infinitely small thickness. The model assumes a stationary observer. The two-sided power spectral density (PSD) of the far-field noise at an observer position $xo→=(xo,yo=0,zo)$ for a flat plate of chord c and span d is

where $ω=2πf$⁠, f is the frequency, ρ is the fluid density, c_o is the speed of sound, $σ2=xo2+(1−M2)(yo2+zo2)$⁠, M is the Mach number, $U∞$ is the free-stream velocity, $Kx=ω/U∞, Φww$ is the two-dimensional turbulence spectrum. The reference coordinate system considers the streamwise-direction axis x, the spanwise direction axis y, and z is the direction normal to x and y. The aeroacoustic transfer function $L$ is determined from de Santana (Ref. 36, pp. 155, 166, 168). Equation (1) is converted to a one-sided spectrum G_pp written in terms of f as $Gpp(xo→,f)=(2×2π)Spp(xo→,ω)$⁠.

The main input for Amiet's prediction model is the turbulence spectrum $Φww$⁠. Two formulations for $Φww$ are used in this research: a modified-von Kármán turbulence spectrum and an RDT-based turbulence spectrum. These two formulations are discussed in Secs. II A and II B.

The inflow turbulence spectrum in Amiet's model is usually assumed to be represented by the two-dimensional von Kármán turbulence spectrum.¹⁵ A modified-von Kármán spectrum is used in the current study, given as³⁷ 

where k is the wavenumber, k_x and k_y are the wavenumbers in the x and y directions, respectively, $kx̂=kx/ke, kŷ=ky/ke$⁠, and k_e is the wavenumber scale of the largest eddies,¹⁴ 

where $Γ( )$ is the gamma function. $fη$ is a function to model the dissipation range for high frequencies proposed by dos Santos et al.³⁷ and η is the Kolmogorov length scale. See dos Santos et al.³⁷ for more details about this model. η is determined from the dissipation frequency f_d, which is the frequency at which the dissipation range starts. Dos Santos³⁷ showed that f_d depends on the flow speed U and the root mean square of the streamwise velocity fluctuations $urms$ as³⁷ 

The one-dimensional inflow turbulence spectrum for the streamwise velocity $Φuu(kx)$ is also important because the experimental spectrum can be compared with this formulation. This turbulence spectrum is given as³⁷ 

It is clear from this equation that the von Kármán spectrum follows a −5/3 power law in k_x at high frequencies.

To account for the turbulence distortion, de Santana et al.²⁶ proposed a modification to the turbulence energy spectrum based on the asymptotic results of the RDT developed by Hunt²⁴ for turbulent flow around two-dimensional bluff bodies. In the proximity of an upstream cylinder wall, the decay of the one-dimensional turbulence energy spectrum $Φww(kx)$ tends to a −10/3 power law at high frequencies. De Santana et al.²⁶ proposed an energy spectrum formulation that results in an expression for $Φww(kx)$ that follows this −10/3 power law at high frequencies. In addition, they assumed that the turbulence distortion only redistributes the energy among the turbulent scales, without any net increase or decrease in the integrated spectrum. The resulting turbulence spectrum is given as²⁶ 

The one-dimensional inflow turbulence spectrum for the streamwise velocity is

This equation was obtained from the integration of E(k) proposed by de Santana.²⁶ From Eq. (7), it is clear that the wavenumber for $Φuu(kx)$ follows a −10/3 power law at high frequencies as $Φww(kx)$ showed by Hunt.²⁴ 

In this study, Eqs. (2) and (6) were used as input for Amiet's noise prediction model [Eq. (1)]. Equations (5) and (7) were compared with the velocity spectrum obtained from hot-wire measurements.

The experiments were performed in the Aeroacoustic Wind Tunnel of the University of Twente, which is an open-jet, closed-circuit facility with a contraction with a ratio of 10:1. After the contraction, the flow enters a closed test section and subsequently an open test section. The test section dimensions are 0.7 m × 0.9 m. More information about the wind tunnel can be found in de Santana et al.³⁸ The experiments were performed in the open test section. The wind tunnel maximum operating velocity is 60 m/s in the open-jet configuration with turbulence intensity below 0.08%.³⁸ An anechoic chamber of 6 m × 6 m × 4 m encloses the test region. The empty chamber is anechoic from 160 Hz. The flow temperature was controlled at approximately 20 °C. The origin of the coordinate reference system is at the mid-span airfoil leading edge.

A NACA 0008 airfoil with c = 300 mm chord and d = 700 mm span was used. This airfoil has a LE radius of $rLE=2.1$ mm and a maximum thickness of 24 mm at 30% of the chord. This airfoil was instrumented with pressure ports along the chord and span. The pressure ports enabled the measurement of the static pressure and the wall-pressure fluctuations (WPFs) using the remote microphone probe (RMP) technique.^39,40 The WPF measurements are described in Sec. III E. The angle of attack was 0° for all experiments. Measurements with a NACA 0012 airfoil (c = 200 mm and $rLE=3.2$ mm) were also performed to confirm that the results obtained for a NACA 0008 airfoil regarding the noise prediction are also valid for a different airfoil geometry. These results are discussed in Sec. V C. The turbulence characteristics close to the LE for the NACA 0012 are not discussed for conciseness since similar results are obtained for the NACA 0008.

The inflow turbulence was generated by a square mono-planar grid and a rod; see Fig. 1. The grid with a 140 mm mesh and an open area ratio of 60% was installed approximately 2 m upstream of the airfoil LE location in the wind tunnel closed test section. Two-component hot-wire measurements of turbulence quantities, spectra, and space-time correlations were performed for the grid without the airfoil in the flow. These measurements are discussed in Refs. 37 and 41. They showed that the grid generated turbulent flow is isotropic with a spectrum accurately described by the von Kármán turbulence spectrum in the energy-containing range and the inertial subrange, which is also discussed in Sec. IV. A rod with a diameter of 40 mm diameter was installed vertically approximately 680 mm upstream of the airfoil LE. The width of the wake generated by the rod at the LE position (at x = 0) was determined using hot-wire measurements and is approximately 20 times larger than the airfoil maximum thickness. This ensures that the airfoil was completely submerged in the turbulent inflow. The spectral characteristics of the rod induced turbulence are discussed in Sec. IV. Table I lists the turbulence parameters for the grid and the rod at the LE location when the airfoil is not present, where $Re=cU∞/ν$ and ν is the fluid kinematic viscosity. The turbulence parameters were obtained from hot-wire measurements (Sec. III D). The maximum free-stream velocity with the grid installed is 30 m/s (⁠$Re=6×105$⁠) due to the pressure drop induced by the grid. The length scales investigated are large compared to the airfoil LE radius: $10<Λf/rLE<43$⁠.

The turbulent inflow was evaluated using hot-wire anemometry; see Fig. 1. A single-wire probe (Dantec Dynamics model 55P15) of 5 µm diameter and 1.5 mm length was used to measure the streamwise velocity. The hot-wire probe was mounted in a Dantec Dynamics 55H22 probe installed in a symmetric airfoil, which was fixed in a 3D traverse system that performed the probe translation with a resolution of 6.5 µm. The hot-wire data were acquired by the Dantec StreamLine Pro CTA system and the Dantec StreamWare software in combination with the National Instruments 9215 A/D converter. For each measurement, the data were recorded for 30 s with a sampling frequency of 65 536 Hz and an anti-aliasing filter with a cut-off frequency of 30 kHz. A high-pass filter with a cut-off frequency of 5 Hz was used during the post-processing to eliminate the effect of the flow buffeting instability naturally present in an open test section wind tunnel,⁴² which has a direct influence on the length scale calculation. The hot-wire measurements were performed for the flow conditions shown in Table I.

Hot-wire static calibration was performed in situ with a Prandtl tube as reference. The calibration consisted of 21 logarithmically distributed points ranging from 2.5 to 60 m/s. The hot-wire dynamic calibration performed by a square wave test demonstrated that the system had a maximum frequency response of approximately 75 kHz. The velocity measurements had a maximum system uncertainty of 5% with a 95% confidence interval. This uncertainty was computed following the guidelines provided by Dantec Dynamics.⁴³ 

The streamwise velocity was measured for locations upstream and in the vicinity of the airfoil LE (⁠$x/rLE∈[−100,−1.8]$⁠) at mid-span, along the stagnation line. The measurements were performed for two configurations: with and without the airfoil in the test section. This was done to be able to determine the turbulence distortion induced by the airfoil in positions close to the airfoil LE. Therefore, the measurement locations (x, y, z) were exactly the same for both configurations.

The WPFs were measured using the RMP technique. This technique consists of installing the microphone outside the airfoil. The microphone is connected to the airfoil wall by a metal tube and a pinhole on the surface, i.e., the pressure ports. A long tube attached after the microphone is used as anechoic termination to damp the pressure waves inside the tubes. For more information about this technique, see Ref. 39. The technique was used because it yields a higher spatial resolution and due to the small thickness of the airfoil which leaves little space for installing microphones inside it. The RMP calibration was performed using an in-house calibrator equipped with a GRAS 40PH microphone as a reference and a speaker. A two-step calibration procedure was used. In the first step, the unsteady pressure generated by the speaker is measured by the reference microphone in the calibrator and the RMP, allowing the calculation of the transfer function between these two microphones. In the second step, the unsteady pressure generated by the speaker is measured by the reference microphone and a flush-mounted microphone, resulting in the transfer function between these two microphones. The final transfer function relating the RMP measurement to the unsteady pressure at the surface is obtained from the two intermediate transfer functions. The excitation generated by the speaker was white noise.

The RMPs used in this research were distributed in the spanwise direction at chordwise positions $x/rLE∈{0.01,0.50,1.00,2.00}$⁠. Each chordwise position has RMPs along the span at $|y/d|∈{0.014,0.018,0.022,0.049,0.100}$⁠. The WPFs were measured by FG-23329-P07 Knowles microphones for the flow conditions shown in Table I. The WPFs were acquired for 30 s at a sampling frequency of 65 536 Hz by a PXIe-4499 Sound and Vibration module installed in a NI PXIe-1073 chassis.

A phased-microphone array was used to localize and quantify the aeroacoustic noise sources for the rod generated inflow turbulence; see Fig. 1. The far-field noise was not measured for the cases with the grid because the noise generated by the grid was significantly louder than the noise radiated by the airfoil. The far-field noise was measured for the Reynolds numbers showed in Table I for the rod case. A circular array with 1 m diameter consisting of 62 GRAS 40PH microphones distributed in a Vogel spiral arrangement was used. This array geometry yields a flat mainlobe-to-sidelobe ratio over a wide frequency range. More information about the microphone array can be found in Sanders et al.⁴⁴ The sensitivity of each microphone was calibrated using the pistonphone GRAS 42AG Multifunction Sound Calibrator with a sound pressure level of 94 dB and frequency of 1 kHz. The microphone array plane was parallel to the plane composed of the airfoil chord-span lines. The distance between the microphone and the airfoil was 1.5 m with the array's center aligned with the airfoil center. A PXIe-4499 Sound and Vibration module installed in a NI PXIe-1073 chassis was used to acquire the far-field noise for 30 s at a sampling frequency of 65 536 Hz. The cut-off frequency of the anti-aliasing filter was 32 178 Hz.

The noise localization and quantification were done using an in-house beamforming code which has been benchmarked against the Arraybenchmark database.⁴⁵ The cross-spectral matrix (CSM) was estimated using Welch's method, with parameters described in Sec. III G. Diagonal removal was applied to the CSM. All microphone signals had equal weight in the beamforming process. The frequency response of each microphone was accounted for in the CSM calculation.

Conventional beamforming in the frequency domain was performed on a search grid in the range $x/c∈[−2,3]$ and $y/d∈[−0.71,0.71]$ in the plane composed by the airfoil chord-span lines. The grid resolution was 30 mm. The LE noise radiated by the airfoil was isolated using the source power integration (SPI) technique.⁴⁶ The region of integration (ROI) was $x/c∈[−0.5,0.5]$ and $y/d∈[−0.3,0.3]$⁠. The PSD from the SPI is given at a reference location that corresponds to the array center. The uncertainty of the sound power estimated by conventional beamforming in the frequency domain method is approximately 1 dB.⁴⁵ 

The PSD of the streamwise velocity $Φuu$ and the CSM were calculated based on Welch's method. The data were averaged using blocks of 8192 samples (125 ms) and windowed by a Hanning windowing function with 50% overlap, resulting in a frequency resolution of 8 Hz. The coherence between different RMPs was also computed using the same parameters. The uncertainty of the PSD is approximately 9% with a 95% confidence interval, calculated as presented in Sec. 11.6 of Glegg and Devenport.¹⁴ The spectral levels for the pressure measurements are shown in decibel, calculated as in Sec. 8.4 of Glegg and Devenport.¹⁴ The reference pressure and frequency used were $pref=20$µPa and $Δfref=1$ Hz.

In this section, the turbulence characteristics in the vicinity of the NACA 0008 airfoil LE are discussed. The results presented are based on the hot-wire and the WPF measurements. The streamwise velocity was measured at different upstream positions from the airfoil LE location for two cases named “no turbulence distortion” (NTD) and “turbulence distortion” (TD). For the NTD case, the airfoil was not present in the test section. Hence, the turbulence was not distorted by the mean flow caused by the airfoil. For the TD case, the airfoil was immersed in the flow. Thus, the turbulence was distorted by the velocity field in the LE stagnation region.

Figure 2 shows the mean streamwise velocity measured in the vicinity of the airfoil LE at the stagnation line. The streamwise velocity is nondimensionalized by the free-stream velocity $U∞$⁠. This velocity is taken as the velocity at $x/rLE=−30$⁠, where the influence of the airfoil on the mean flow is negligible. The velocity decays rapidly as the stagnation region near the LE is approached. The experimental data is compared with the potential flow analytical solution for a cylinder, given by

with the cylinder considered to have a radius equal to the airfoil LE radius because, according to Mish and Devenport,⁴⁷ in a region sufficiently close to the stagnation point, the inflow distortion produced by an airfoil is similar to that produced by a cylinder with a radius equal to the LE radius. The potential flow analytical solution does not agree well with the measured velocity in the vicinity of the LE. This was also observed by de Santana et al.²⁶ but with a smaller difference between the measurement and the analytical solution for $x/rLE>−0.6$⁠, where their results were obtained from particle image velocimetry measurements. Even though the measurements do not agree with the analytical formulation, the experimental curves show a good overlap for both inflow turbulence types and different Reynolds numbers.

The longitudinal integral length scale $Λf$ was computed as proposed by Hinze.⁴⁸ First, the turbulence time scale τ is determined as the time when the autocorrelation of the streamwise velocity reaches zero for the first time. In light of Taylor's frozen turbulence hypothesis,¹⁶ the longitudinal integral length scale is computed considering that the turbulence is convected with the mean flow velocity at the measurement location, which according to Pope⁴⁹ provides accurate results. Thus, the behavior of the longitudinal length scale in the vicinity of the airfoil LE depends on the behavior of the convection velocity, i.e., U, and the time scale. The free-stream values used to nondimensionalize Figs. 3 and 4, i.e., $τ∞, Λf,∞$⁠, and $urms,∞$⁠, were considered as the values at $x/rLE=−30$⁠.

Figure 3 shows the turbulence time scale and the integral length scale in the vicinity of the LE for inflow turbulence generated by the grid and the rod. Both the time and the length scales decay as the airfoil LE is approached. The time scale decay is due to the turbulence distortion. The integral length scale decay is due to the decrease in the convection velocity (see Fig. 2) and the turbulence time scale. This is an important observation because it shows that the length scale of the turbulence decreases as the stagnation point is approached, which is not only due to the velocity decrease but also due to the turbulence distortion. Furthermore, the integral length scale curves collapse well for both grid- and rod-generated inflow turbulence for the different Reynolds numbers tested.

Figure 4 shows the root mean square of the velocity fluctuations in the vicinity of the LE for inflow turbulence generated by the grid and the rod. This parameter also decays as the stagnation point is approached. For the grid generated turbulence, a higher dispersion of the data is observed. The $urms$ curves for the grid and the rod cases do not present a good agreement for $x/rLE>−3$⁠.

Figures 3 and 4 clearly show that the presence of the airfoil alters the turbulence characteristics in the LE vicinity. According to the RDT,⁵⁰ the behavior of $urms$ and $Λf$ in the stagnation region of a cylinder depends on the ratio between the cylinder radius r and the turbulence integral length scale. This dependency is also valid for the case of turbulence–airfoil interaction because, in a region sufficiently close to the stagnation point, the inflow distortion produced by an airfoil is similar to that produced by a cylinder with a radius equal to the LE radius.⁴⁷ According to the RDT,⁵⁰ for $r/Λf<0.4$⁠, the $urms$ values at the stagnation line decrease in the vicinity of the LE. For all the cases studied in this research, $rLE/Λf<0.4$⁠. Thus, the trend observed experimentally is supported by the RDT.

Figure 5 shows the streamwise velocity spectrum for the turbulence generated by the grid and the rod for the NTD and TD cases. The results for only one Reynolds number is shown for brevity, but similar results were obtained for the other Reynolds numbers. The frequency is nondimensionalized by the Strouhal number $St=fc/U∞$⁠. The experimental turbulence spectrum is compared with the modified-von Kármán and with the RDT-based spectrum expressions shown in Eqs. (5) and (7), respectively. The inputs (⁠$Λf, urms$⁠) for the turbulence models were obtained from the same data used to calculate the experimental spectral curve. For the comparison between the measured and modeled $Φuu$⁠, the relation $Φuu(f)=(2π/U)Φuu(kx)$ was used.^14,36

The measured spectral level for the TD case (⁠$Φuum|TD$⁠) reduces in comparison to the NTD case (⁠$Φuum|NTD$⁠) in the low- and high-frequency range. The experimental spectra for the TD and NTD cases overlap in the range $3<St<14$⁠. The low-frequency deviation is due to the turbulence distortion caused by the mean flow field induced by the airfoil. This decrease in level indicates that large turbulent structures in the streamwise direction lose energy as they are distorted. The high-frequency reduction is due to the turbulence dissipation range for the TD case, which starts at a lower frequency than for the NTD case. Due to the mean flow field induced by the airfoil, U and $urms$ decrease in the region close to the airfoil LE, resulting in a dissipation frequency f_d [see Eq. (4)] lower than for the NTD case. The von Kármán spectrum model in combination with the dissipation range modeling proposed by dos Santos et al.³⁷ can predict the velocity spectrum at $x/rLE=−2$ reasonably well, in particular for the grid-generated turbulence.

Figure 5 shows that $ΦuuRDT$ does not follow the measured turbulence energy content at $x/rLE=−2$⁠. In the RDT development, Hunt²⁴ showed that the slope of the one-dimensional velocity spectrum $Φww$ follows a −5/3 power law for high frequencies for positions close to the airfoil LE. However, the slope changes to −10/3 for a point extremely close to the stagnation location, $x/r→0$⁠. This variation can be seen in Fig. 12 in Ref. 24. The −10/3 slope is also observed for the one-dimensional velocity spectrum $Φuu$⁠; see Eq. (7). The behavior observed by Hunt²⁴ is also observed in the experimental results of de Santana et al.;²⁶ see Fig. 13 of Ref. 26. At $x/rLE=−1$⁠, the measured turbulence spectrum mostly follows the von Kármán spectrum. However, as the measurement point approximates the stagnation point, the measured spectrum agrees better with the RDT-based model. At $x/rLE=−0.12$⁠, the measured spectrum has a good overlap with the RDT-based spectrum. Therefore, the results of de Santana et al.²⁶ show that a better agreement between the RDT-based spectrum and the measured spectrum is obtained closer to the stagnation point, at $x/rLE=−0.12$⁠. The same behavior is expected for the current study. However, this cannot be confirmed because measurements closer to the airfoil LE were not possible due to the size constraints imposed by the hot wire. Thus, the measured velocity spectrum at $x/rLE=−2$ follows a −5/3 slope instead of a −10/3 slope for high frequencies because the measurement location is not yet close enough to the stagnation location.

The spanwise correlation length quantifies the length scales of the turbulent flow in the spanwise direction as a function of frequency. It is defined as^10,15,37

where the coherence (⁠$γ2$⁠) is computed between pairs of microphones at the same chordwise position spaced in the spanwise direction by η_y. Five microphones in the spanwise direction for each streamwise position were used to compute l_y, which results in 10 different η_y positions.

Figure 6 shows the computed spanwise correlation length for the grid-generated inflow turbulence at different chordwise positions located in the vicinity of the LE. This correlation length is nondimensionalized by the free-stream longitudinal length scale. The spanwise correlation length increases with the chordwise position. The largest variation occurs for $0.1≤x/rLE≤1$⁠. The correlation length remains practically the same for $1≤x/rLE≤2$⁠. Dos Santos et al.³⁷ evaluated the turbulence generated by the grid at the LE position without the airfoil in the flow. They showed that the largest spanwise correlation length, i.e., the lateral length scale Λ_g, corresponds to approximately 0.5 $Λf,∞$ for different flow speeds, characterizing a nearly isotropic flow. Based on the results shown in Fig. 6, the largest lateral length scale at $x/rLE=1$ is observed at $St≈0.6$⁠, corresponding to approximately 1.4 $Λf,∞$⁠. Thus, the lateral length scale on the airfoil surface increases considerably compared to the lateral length scale evaluated in the free stream. This increase is attributed to the turbulence distortion. Thus, based on the observations for the longitudinal and lateral length scales, the turbulent structures are distorted in the vicinity of the airfoil LE, resulting in a decrease in the longitudinal length scale and an increase in the lateral length scale. To better visualize this phenomenon, we can consider the turbulent inflow as a single vortex with a length Λ_f in the streamwise direction and a length Λ_g in the spanwise direction. For an isotropic turbulence in the free-stream, i.e., upstream of the airfoil LE, the vortex length is approximately $Λg=0.5Λf$ in the spanwise direction and $Λf$ in the streamwise direction. As the airfoil LE is approached, the vortex becomes smaller in the streamwise direction but it stretches in the spanwise direction.

The measured rod-airfoil far-field noise is analyzed and compared with Amiet's LE noise prediction. The effect of turbulence distortion on the far-field noise prediction is discussed, and new insights are given into turbulence distortion modeling for LE noise prediction models. The results are shown for $Re=4.8×105$ for brevity. Similar conclusions are also obtained from the results for the other Reynolds numbers tested.

Figure 7 shows the beamforming noise source maps for the noise generated by the airfoil when subjected to the inflow turbulence generated by the rod. From this figure, it is clear that the noise generated by the airfoil LE interacting with the inflow turbulence is dominant for Strouhal numbers up to 31.5 for $Re=4.8×105$⁠. Similar results were observed for the other Reynolds numbers tested. Trailing-edge noise is not observed at any frequency. Also, no noise contamination from the wall is observed in the source maps.

Figure 8 shows the measured and predicted LE far-field noise spectra. The measured noise is shown until $St≈30$ because the background noise was not dominant up to this frequency. The noise was predicted for three different cases using Eq. (1). See Table II for an overview of the three cases. For case (1), i.e., $GppAmiet(ΦwwvK(Λf|NTD,urms|NTD))$⁠, the far-field noise was calculated using the modified-von Kármán spectrum with as input the turbulence parameters measured at $x/rLE=0$ without the airfoil in the flow, i.e., NTD case. For case (2), i.e., $GppAmiet(ΦwwRDT(Λf|NTD,urms|NTD))$⁠, the far-field noise was computed using the RDT turbulence spectrum with as input the turbulence parameters measured at $x/rLE=0$ without the airfoil in the flow, i.e., NTD case. For case (3), i.e., $GppAmiet(ΦwwRDT(Λf|TD,urms|TD))$⁠, the far-field noise was computed using the RDT turbulence spectrum with as input the turbulence parameters measured at $x/rLE=−2$ with the airfoil installed in the test section, i.e., TD case. In short, the noise prediction for case (1) does not account for any turbulence distortion effects. For case (2), the turbulence distortion is taken into account only in the formulation of the turbulence spectrum. For case (3), the turbulence distortion is considered in the turbulence spectrum formulation and in the input values $urms$ and $Λf$⁠. The observer position is $(x0,y0,z0)=(0,0,1.5)$ m, based on the coordinate system for Amiet's noise prediction model.

For case (1), i.e., $GppAmiet(ΦwwvK(Λf|NTD,urms|NTD))$⁠, Fig. 8 shows that Amiet's noise prediction agrees fairly well with the measured far-field noise (⁠$Gppm$⁠) for low Strouhal numbers with a maximum difference of 4 dB but it overestimates the noise for $St ≳ 9$ with a maximum difference of 18 dB. The overestimation is attributed to the turbulence distortion caused by the finite airfoil thickness which effect is not taken into account in Amiet's noise prediction model.^{13,18,20,21,26}

For case (2), i.e., $GppAmiet(ΦwwRDT(Λf|NTD,urms|NTD))$⁠, Fig. 8 shows that the RDT-based noise prediction can capture the high-frequency behavior of the measured noise. This is expected because $ΦwwRDT$ captures the high-frequency decay of the turbulence energy when the turbulence is distorted. Even though Amiet's model using $ΦwwRDT$ can predict the trend of far-field noise for high frequencies well, a difference of 11 dB at St = 10 from the measured noise is still observed. Therefore, only using the spectrum formulation for accounting for the turbulence distortion does not lead to accurate noise predictions.

To accurately model the velocity spectrum, whether the turbulence is distorted or not, the turbulence parameters $urms$ and $Λf$ must be determined at the location of interest. Thus, the appropriate turbulence parameter inputs for $ΦwwRDT$ must be determined sufficiently close to the airfoil LE to account for the changes on $urms$ and $Λf$ caused by the turbulence distortion. Therefore, the noise prediction for case (3), i.e., $GppAmiet(ΦwwRDT(Λf|TD,urms|TD))$⁠, was computed using $ΦwwRDT$ with as input the values for $urms$ and $Λf$ measured at the airfoil LE vicinity, i.e., $x/rLE=−2$⁠. Figure 8 shows a really good match between the measured noise and the noise prediction for case (3) for $3≤St≤10$ with a maximum difference of 3 dB and a more accurate prediction for high frequencies with a maximum difference of 7 dB. This agreement demonstrates that the turbulence distortion must be considered not only in the turbulence spectrum but also in the turbulence input parameters, i.e., $urms$ and $Λf$⁠, for accurate modeling of the LE far-field noise.

From Fig. 8, it is clear that the most accurate noise prediction is obtained by using $ΦwwRDT$ with as input the values for $urms$ and $Λf$ measured at the airfoil LE vicinity at $x/rLE=−2$⁠. A sensitivity analysis was carried out to confirm that the better agreement between the measured and predicted noise occurs for the case that the turbulence parameters are taken at $x/rLE=−2$⁠.

Figure 9 shows Amiet's noise prediction using $ΦwwRDT$ with as input the turbulence parameters measured in the vicinity of the airfoil LE, i.e., TD case, at different $x/rLE$⁠. A clear tendency can be observed from the results in this figure: the predicted noise levels increase when the inputs for the prediction are determined at positions closer to the airfoil LE. This tendency is mainly due to the decrease in the integral length scale as the airfoil LE is approached because $Λf$ decreases more considerably than $urms$⁠; see Figs. 3 and 4. For inputs determined at $x/rLE<−2$⁠, the noise is underpredicted, whereas the noise is overpredicted for inputs determined at $x/rLE>−2$⁠. Therefore, these results show that $urms$ and $Λf$ should be extracted at $x/rLE=−2$ to obtain accurate noise predictions. These results also confirm that this finding is valid for a different airfoil geometry. The measured and predicted noise deviates slightly for St > 14 for both airfoil geometries.

Even though the experimental turbulence spectrum at $x/rLE=−2$ is better represented by the modified-von Kármán spectrum (see Fig. 5), the far-field noise is better predicted if the RDT-based turbulence spectrum is used in Amiet's model. The RDT-based turbulence spectrum mainly captures the high-frequency decay (⁠$k−10/3$⁠) of the far-field noise due to the turbulence distortion. However, the turbulence input parameters for this turbulence spectrum should be representative of the LE noise production mechanism, which might not necessarily be the turbulence parameters where the −10/3 power law decay is observed, i.e., $x/rLE$ tending to zero. Based on the results shown in this study, the representative turbulence parameters should be determined at $x/rLE=−2$ for an accurate LE noise prediction.

This paper discussed the turbulence distortion in the vicinity of an airfoil LE and the implications of this phenomenon on the measured and predicted LE noise. Turbulence in the near field, i.e., in the LE vicinity, was evaluated with hot-wire anemometry and WPF measurements. The measured far-field noise was compared with Amiet's LE noise prediction model. Experiments were performed at zero angle of attack for a NACA 0008 and a NACA 0012 airfoil subjected to large turbulence length scales.

The turbulence characteristics in the vicinity of the airfoil LE at the stagnation line were affected considerably by the mean flow induced by the airfoil. In particular, $urms$ and $Λf$ decreased as the airfoil LE was approached. This directly impacted the streamwise velocity spectrum in the low- and high-frequency range. In the vicinity of the airfoil LE, the longitudinal and the lateral length scales decrease and increase, respectively, due to the turbulence distortion caused by the airfoil mean flow field.

In this paper, it was demonstrated that Amiet's LE noise model can predict the far-field noise accurately when the turbulence distortion is considered. The trend of the far-field noise decay for high frequencies is captured by using the RDT-based turbulence spectrum in the model. However, a level difference between the measured and predicted noise of 11 dB at St = 10 is still observed. Thus, only using the RDT-based spectrum does not result in accurate noise predictions. A better agreement between the measured and predicted noise is obtained when the noise is predicted using the RDT-based turbulence spectrum with as input the turbulence parameters measured at the LE proximity, i.e., $x/rLE=−2$⁠. For this case, an excellent agreement is observed up to St = 10 and a more accurate prediction is observed for high frequencies. The position $x/rLE=−2$ is the location where the representative turbulence parameters should be determined to result in accurate LE noise predictions.

As the LE radius is considered to be the LE geometric parameter relevant for the turbulence distortion,⁴⁷ it is expected that a more accurate noise prediction is also obtained for other airfoil geometries when the turbulence parameters used as input for the RDT-based turbulence spectrum are extracted at the airfoil LE proximity. This paper is limited to two geometries. Therefore, thicker airfoil geometries should also be investigated to validate that the finding of this study, i.e., a more accurate noise prediction with RDT-based turbulence spectrum with as input the turbulence parameters extracted at $x/rLE=−2$⁠, holds for thicker geometries. Even though the results analyzed were for zero-mean loading, the noise prediction should be valid for non-zero loading because the angle of attack has negligible effects on the LE radiated noise.^13,17,18 Moreover, this study is limited to a large turbulence length scale compared to the airfoil LE radius (⁠$10<Λf/rLE<43$⁠). As the asymptotic RDT results depend on the ratio $Λf/rLE$⁠, accounting for the turbulence distortion effects on the noise prediction should also depend on this ratio. Thus, further investigations are required for length scales much smaller and comparable to the LE radius.

Part of this research received financial support from the European Commission through the H2020-MSCA-ITN-209 project zEPHYR (Grant Agreement No. 860101). The authors are grateful to Ing. W. Lette, ir. E. Leusink, S. Wanrooij, and others for their technical support. The authors would like to thank TNO and the Maritime Research Institute Netherlands (MARIN), particularly Dr. Roel Müller, Dr. Johan Bosschers, and Dr. Christ de Jong, for the insightful discussions and feedback.