Quantitative criteria to design optimal permeable trailing edges for noise abatement

An experimental study on broadband noise scattered by permeable trailing edges with different pore arrangements is performed. A NACA 0018 airfoil with chord c = 0.2 m is investigated at chord-based Reynolds numbers ranging from 1.4 × 10 5 to 3.8 × 10 5 and angles of attack of 0.2 and 5.4 degrees. Noise emission from five 3D-printed perforated trailing-edge inserts, with channels normal to the chord, is measured with a microphone antenna. For comparison, inserts manufactured with metallic foams, with comparable flow permeability K but more tortuous pore paths, are also analysed. All the inserts have a permeable extension s equal to 20% of the chord ( s / c = 0.2). It is shown that noise mitigation L p , computed as the difference between far-field noise scattering from solid and permeable edges, collapse when nondimensionalizing frequency as Strouhal number based on the chord. From the collapsed data, it is observed that the maximum noise attenuation reported for each insert L p ,max reaches an asymptotic value of 9.3 dB for increasing K . To parameterize such asymptotic behaviour, noise reduction levels are fitted to a newly proposed relation L p, max = γ1 tanh ( γ2 K ) (where γ 1 and γ 2 are fitting coefficients, that depend on the type of insert and angle of attack). Following this analysis, limit permeability values for perforated and metal foam inserts of K = 3.5 × 10 −9 and 1 × 10 −9 m 2 are found, respectively; above these thresholds, less than 1 dB additional noise mitigation is reported; below, a difference in L p ,max of up to 4 dB for a given K is measured depending on the pore organization. Consequently, the tortuosity of the permeable structure is identified as an additional parameter (to K ) controlling noise attenuation. It is also observed that the acoustic performance of lower-permeability edges is less sensitive to changes in the angle of attack. Tests for permeable lengths equal to s / c = 0.05 and 0.1 are performed: the change of L p ,max with increasing s / c is also properly described with a hyperbolic tangent, evidencing equally good performance in noise reduction for all measured extents. Finally, for the most permeable insert with periodic pore arrangement, an extremely loud tonal noise caused by vortex shedding (+30 dB higher than broadband levels) from the blunt solid-permeable junction at s/c = 0 . 8 is reported. Since applying a longer permeable surface, or increasing the permeability at the trailing edge decreases the aerodynamic performance of the blade, a permeable trailing edge with s / c = 0.05, K = 1 × 10 −9 m 2 and tortuosity of 1.15 is recommended to optimize broadband noise abatement and avoid shedding-related tones for the conditions explored in the current study. © 2020 The Authors. Published by Elsevier Ltd. ∗ Corresponding author. E-mail address: a.rubiocarpio@tudelft.nl (A. Rubio Carpio). https://doi.org/10.1016/j.jsv.2020.115596 0022-460X/© 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ ) 2 A. Rubio Carpio, F. Avallone and D. Ragni et al. / Journal of Sound and Vibration 485 (2020) 115596 This is an open access article under the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ )


a b s t r a c t
An experimental study on broadband noise scattered by permeable trailing edges with different pore arrangements is performed. A NACA 0018 airfoil with chord c = 0.2 m is investigated at chord-based Reynolds numbers ranging from 1.4 × 10 5 to 3.8 × 10 5 and angles of attack of 0.2 and 5.4 degrees. Noise emission from five 3D-printed perforated trailing-edge inserts, with channels normal to the chord, is measured with a microphone antenna. For comparison, inserts manufactured with metallic foams, with comparable flow permeability K but more tortuous pore paths, are also analysed. All the inserts have a permeable extension s equal to 20% of the chord ( s / c = 0.2). It is shown that noise mitigation L p , computed as the difference between far-field noise scattering from solid and permeable edges, collapse when nondimensionalizing frequency as Strouhal number based on the chord. From the collapsed data, it is observed that the maximum noise attenuation reported for each insert L p ,max reaches an asymptotic value of 9.3 dB for increasing K . To parameterize such asymptotic behaviour, noise reduction levels are fitted to a newly proposed relation L p, max = γ 1 tanh ( γ 2 K ) (where γ 1 and γ 2 are fitting coefficients, that depend on the type of insert and angle of attack). Following this analysis, limit permeability values for perforated and metal foam inserts of K = 3.5 × 10 −9 and 1 × 10 −9 m 2 are found, respectively; above these thresholds, less than 1 dB additional noise mitigation is reported; below, a difference in L p ,max of up to 4 dB for a given K is measured depending on the pore organization. Consequently, the tortuosity of the permeable structure is identified as an additional parameter (to K ) controlling noise attenuation. It is also observed that the acoustic performance of lower-permeability edges is less sensitive to changes in the angle of attack. Tests for permeable lengths equal to s / c = 0.05 and 0.1 are performed: the change of L p ,max with increasing s / c is also properly described with a hyperbolic tangent, evidencing equally good performance in noise reduction for all measured extents. Finally, for the most permeable insert with periodic pore arrangement, an extremely loud tonal noise caused by vortex shedding (+30 dB higher than broadband levels) from the blunt solid-permeable junction at s/c = 0 . 8 is reported. Since applying a longer permeable surface, or increasing the permeability at the trailing edge decreases the aerodynamic performance of the blade, a permeable trailing edge with s / c = 0.05, K = 1 × 10 −9 m 2 and tortuosity of 1.15 is recommended to optimize broadband noise abatement and avoid shedding-related tones for the conditions explored in the current study.
This is an open access article under the CC BY license.

Introduction
Turbulent-boundary-layer trailing-edge noise is a significant contributor to noise generated by wind turbines [1] , ventilation systems [2] or aircraft airframes [3] . This noise source is produced by turbulent eddies being diffracted at the aft edge of lifting devices [4] . The employment of permeable trailing edges can mitigate noise up to 25 dB [5] with respect to their equivalent solid configuration [6] . It is now well-established that permeable trailing edges mitigate the acoustic impedance jump at the edge, and promote noise scattering from different streamwise positions [7][8][9] . Such a process is controlled by the local permeability of the material [10] , that determines the degree of flow communication between both sides of the airfoil (thus the magnitude of the acoustic impedance jump). Consequently, the employment of more permeable structures usually yields larger noise reduction [11] .
Previous research on permeable trailing edges employed (open-cell) porous structures such as foams [12,13] , sintered granulates [10] , fiber felts [5] or metal meshes [14] . Materials with such a complex pore arrangement often entail additional fluid-dynamic and acoustic features. For instance, hydrodynamically rough surfaces produce high-frequency excess noise and increased turbulence levels close to the wall [15] ; these effects might partially mask their acoustic benefits [9,11] . Further aspects, such as the change of properties when subject to conventional machining processes, or repeatability and inhomogeneity issues also discourage their use for research. Recently developed manufacturing techniques such as additive manufacturing (or 3D-printing) [16] allow constructing permeable trailing edges with channels that connect suction and pressure sides of the airfoil [17] . Consequently, edges with tailored permeability can be manufactured by varying hole size, shape or spatial density.
In the current manuscript, five different permeable trailing edges for a NACA 0018 are acoustically characterized at angles of attack α of 0.2 and 5.4 degrees and chord-based Reynolds numbers Re c ranging between 1.8 × 10 5 and 4.5 × 10 5 . These permeable edges, with straight channels normal to the chord, allow for flow communication between suction and pressure side along the last 20% of the airfoil. The five perforated inserts have the same hole diameter d h but different permeability K , obtained by varying the spacing between holes l h . Noise scattering of such perforated trailing edges, with a periodic channel arrangement, is compared to that of open-cell metal foam inserts, with different (random) micro-structure but similar macroscopic permeability. The comparison yields general criteria that might serve as guidelines for the design of future permeable trailing edges. The current publication is organized as follows. The experimental set-up, the 3D-printed inserts and the acoustic phased array are described in Section 2 . Then, far-field noise results are discussed in five subsections: firstly, scaling laws for noise mitigation with flow speed or material properties are presented in Section 3.1 ; the tortuosity of a permeable edge is proposed to further control and optimize noise abatement in Section 3.2 ; the scaling of far-field acoustic pressure with flow speed is reported in Section 3.3 ; a brief analysis on the effect of varying the chordwise permeable length on noise mitigation is presented in Section 3.4 ; finally, the origin of a tone measured for the most permeable perforated insert is addressed in Section 3.5 . To conclude, in Section 4 a summary of the main findings is presented.

Wind tunnel and model
Experiments are carried out in the aeroacoustic vertical open-jet wind tunnel (A-Tunnel) at Delft University of Technology. The rectangular cross-section at the nozzle outlet is 0.4 × 0.7 m 2 (contraction ratio equal to 15:1). For the present investigation, experiments are performed at free-stream velocities of 12.5, 15, 20, 25, 30 and 34 m/s. Inflow conditions at the exit section are characterized by means of a Pitot tube and constant temperature anemometry ( Dantec Dynamics P11 probe; TSI IFA 300 bridge). The streamwise velocity is uniform within 0.6% at the measured cross-section, and the turbulence intensity is below 0.06% for the range of free-stream velocities specified above.
A modular NACA 0018 airfoil with chord c = 200 mm and span L = 400 mm ( Fig. 1 (a-c)) is placed at the potential core of the jet with the leading edge 500 mm downstream the nozzle outlet. The model is installed between two wooden side plates (with height equal to 1.2 m) flush-mounted to the nozzle outlet ( Fig. 1 (d)). The high span-to-chord ratio L / c = 2 suggests that three-dimensional flow effects are negligible along the midspan line; this aspect is further confirmed by the good agreement between experimental boundary layer data obtained with the same set-up [9] , XFOIL data [18] and highfidelity simulations on a two-dimensional model [19] .
The aluminium body of the airfoil is manufactured with numerical control machining (surface roughness: 0.05 mm). The last 20% of the airfoil can be replaced by permeable inserts made of different materials and permeability patterns. The inserts are simply attached to the airfoil by a dovetail construction, and their surface is flush to the solid airfoil surface. A piece of Scotch Crystal Tape (width: 19 mm; nominal thickness: 51 μm) is carefully applied at the junction between the solid body of the airfoil and the permeable/solid trailing-edge inserts in such a way that the tape covers 1 mm of the insert in the streamwise direction. This procedure ensures a smooth transition between both pieces. The trailing-edge inserts have a thickness of 15.7 mm at the root and 0.3 mm at the tip.
Boundary-layer transition to turbulence is forced at 20% of the chord on both sides of the airfoil. The tripping device, with a chordwise extent of 10 mm, has 0.84 mm-height carborundum elements randomly distributed along the entire span. The turbulent state of the boundary layer is confirmed by scanning the airfoil surface with a Brüel & Kjaer (B & K) 4134 microphone connected to a B & K 2619 preamplifier.
The streamwise-vertical-spanwise X − Y − Z coordinate system, employed in the remaining of the manuscript, is also depicted in Figs. 1 (a-c). Its origin is located at the intersection between the midspan plane and the trailing edge, X and Z axes are respectively aligned with chord and trailing edge, and the Y axis points to the microphone antenna.

3D-Printed trailing edges
Channelled trailing-edge inserts are manufactured with an EnvisionTEC 's Perfactory 4 Standard (resolution: 25 μm), a Digital Light Projector (DLP) printer. The inserts (surface roughness: 0.03 mm) are printed from root to tip using EnvisionTEC 's HTM 140 V2, a high-temperature photopolymer [20] . The total chordwise length of the inserts is 60 mm, of which only s / c = 0.2 = 40 mm are exposed to the flow ( Fig. 1 (b)). The maximum span of each insert is limited by the build envelope of the printer (160 × 100 × 180 mm 3 ) to 100 mm; hence, 4 inserts are assembled to build the entire trailing edge.
The inserts contain straight cylindrical channels normal to the chord connecting suction and pressure side. Holes are distributed according to a staggered square lattice, as shown in Fig. 2 . In the present investigation, d h is equal to 0.8 mm to ensure that the channels are open after the printing process and to avoid low-frequency acoustic tones [11] . The flow permeability K of the inserts is thus varied by changing the hole spacing l h . Five different perforated inserts, with homogeneous hole spacing l h of 1.5, 2, 2.5, 3 and 5 mm, are studied in the present manuscript. Hole spacings are chosen in order to obtain perforated edges with similar permeability to those manufactured with metallic foams (the permeability characterization is discussed more in detail later in Section 2.3 ). The smallest hole spacing ( l h = 1.5 mm) is determined by manufacturing constraints to guarantee 0.3 mm of solid material between orifices, which yields a satisfactory outcome of the 3D-printing process. Pictures of the inserts with l h = 1.5 and 5 mm are shown in Fig. 3 (a) and (b) respectively.  After the printing process it is observed that holes are not perfectly round due to limitations in the manufacturing technique [21] ; specifically, the most downstreamwise end of each hole is flattened, producing D-shaped channels. Optical microscopy ( Fig. 4 ) shows that these holes have a major diameter of approximately 800 μm and a shortened side with length of 700 μm. To account for this feature, an equivalent hole diameter of 754 μm, that yields a circle with area equal to the measured open area (of 4470 0 0 μm 2 ), is employed in Section 2.3 .

Flow permeability and form coefficient
Flow permeability K and form coefficient C of trailing-edge inserts are determined by measuring the static pressure drop p across perforated samples with similar hole distributions. The pressure drop across a sample of permeable material is determined by the Hazen-Dupuit-Darcy equation [22] where t is the thickness of the sample, ρ is the fluid density, μ is the dynamic viscosity and ν is the Darcian velocity (defined as the ratio of volumetric flow rate to sample cross-section area). To characterize the two hydraulic properties ( K and C ) of each hole pattern, 4 cylindrical samples with thickness t = 10 mm, representative of the average trailing-edge insert thickness, are manufactured. These samples have hole patterns with l h = 1.5 ( Fig. 5 (a)), 2.8, 4.5 and 6 mm (( Fig. 5 (b))), and an outer diameter of 55 mm. To yield the same D-shaped hole feature, samples are printed as the trailing-edge inserts, i.e., the printing direction is perpendicular to the channels' axes. K and C are then computed by least-squares fitting of Eq.
(1) to 20 pressure drop data, measured for Darcian velocities ranging between 0 and 1.1 m/s [23] .  Hydraulic properties for the hole distributions employed in the trailing edges are then estimated using Eq. (2) (a) and (b), is the porosity, defined as the ratio of fluid to solid volume. These empirical correlations are defined by Bae and Kim [24] for perforated plates with a similar hole arrangement and fully cylindrical holes. The original fit coefficients β 1 , β 2 and β 3 , as well as the ones retrieved with present data, are reported in Table 1 . To validate the results, correlations based on measured data are compared to the ones presented by Bae and Kim [24] in Fig. 6 (a) and (b). For completeness, K and C values computed from Eq. (1) (referred to as "Measurements"), and the ones interpolated at l h employed in trailing-edge inserts (referred to as "Interpolation") are shown. The relative error for the original correlations stated by the  [24] , together with the 3% relative error stated by the authors (grey shadowed area).

Table 2
Characteristics of permeable micro-structures measured on samples with t = 0.01 m. P stands for perforated and MF stands for metal foam. K and C values are interpolated from Eq. (2) (a) and (b) employing the corresponding l h value; measured values (computed from fit of experimental data to Eq. (1) ) were reported in a preliminary analysis [25] , hence the small discrepancies among them.
Similarly, the porosity σ is now computed with the corrected d h of 754 μm to account for D-shaped holes.
Type authors (equal to 3%) is also depicted as a grey shadowed area. Results show that both correlations are in good agreement, confirming the goodness of the permeability characterization. Finally, a summary of the characteristics of all perforated configurations created is presented in Table 2 , where the resistivity R = K/μ is also reported for completeness. Since the noise scattering analysis includes a comparison with inserts manufactured with metal foams, a summary of relevant parameters previously measured [9,23] for metal foams with d c = 450, 580 and 800 μm, is also included. For the metal foam with d c = 1200 μm, permeability data as reported by the manufacturer is included.

Tortuosity
A preliminary comparison between metal foams and straight periodically channelled inserts [25] showed that trailing edges with diverse flow permeability and form coefficient might abate noise similarly; hence, although it is widely recognized that these two properties have a significant impact [5,11] , they might not be sufficient to fully characterize the noise mitigation capabilities of structures with different pore arrangements. In the present manuscript, tortuosity τ is proposed to account for the very different micro-structure of metal foam and channeled edges. Tortuosity τ ≡ l p / t is usually defined 1 as the ratio between the average pore length l p to the thickness of the porous medium t [28] . From this definition, it stems that straight channels have τ Y = 1. For the metallic foams, the tortuosity of the pore phase is statistically computed with random walk models [29] . To this aim, pytrax [30] , an open-source Python suite is employed on CT-scanned samples (size: 1 Note that a distinction between tortuosity τ , as defined in the present manuscript, and tortuosity factor κ = τ 2 should be made [26] . For more information about the physical interpretation of the latter quantity, the reader is referred to the excellent review of Tjaden et al. [27] . 10 × 10 × 5 mm 3 ; resolution: 12 μm) of metal foams with d c = 450, 580 and 800 μm. Random walk processes mimic the diffusion process of non-sorbing particles, i.e. walkers, randomly distributed within porous media [31] . The algorithm begins a time sequence in which each walker randomly selects a neighbouring voxel; if the chosen voxel is also void, the walker then migrates to that position; if not, the walker does not move at that time-step [27] . The mean squared displacement of the ensemble of random walkers r(t ) 2 is then computed as where n w is the total number of walkers, and r X,i , r Y,i and r Z,i refer to the displacement of the walker i along X, Y and Z respectively. The mean squared displacement within porous media r(t ) 2 p is then lower than the one obtained in a fully void space r(t ) 2 v . The degree of reduction, i.e. τ , is then computed as [32] where ˙ r refers to the time derivative of r . An accurate computation of τ is dependant on both the number of walkers and time-steps n t . To obtain reliable τ values, 50,0 0 0 walkers and 50,0 0 0 time-steps are employed after performing a parametric study. From Eq. (3) , it can be inferred that considering only axial mean square displacements, the tortuosity in a specific direction ( τ X , τ Y , τ Z ) can be also computed. Results, presented in Table 3 , are obtained from an ensemble of 100 simulations; reported values correspond to the ensemble average, and the uncertainty is calculated employing a 95% confidence level. From the results, it stems that metal foams are not perfectly homogeneous. To limit the uncertainty on the aeroacoustic analysis derived from using materials with somewhat heterogeneous properties, the same batch of foam material was used for the tortuosity computation and the insert construction. Similarly, the permeability characterization is carried out in such a way that K ≡ K Y and C ≡ C Y .
The tortuosity values of Table 3 are in line with those previously reported in the literature [33][34][35] for open-cell metal foams with similar characteristics. Finally, due to the lack of availability of a CT-scan sample for the metal foam with d c = 1200 μm, its tortuosity is obtained by extrapolating linearly metal foam data based on the cell diameter.

Open-jet correction for angle of attack
Experiments are performed at geometric angles of attack of 0 and 8 degrees. However, the open-jet configuration of the experimental set-up decreases the effective angle of attack α. For this reason, α is computed by analysing the measured pressure distribution along the chord. Static pressure measurements are obtained through 15 differential pressure Honeywell TruStability transducers (range: ± 2.5 kPa; accuracy: 12.5 Pa), connected to 30 pressure taps (with diameter of 0.4 mm). Taps are distributed within X / c = -0.99 to -0.34 along a plane inclined 15 degrees and with an offset of 20 mm with respect to the midspan plane ( Fig. 1 (a-c)). This feature prevents flow interference between consecutive holes.
Pressure data are sampled for 15 s at a sampling rate of 50 Hz. α is determined by comparing the static pressure distribution for the reference case against XFOIL data [18] , which yields results for an airfoil in free-air conditions. The pressure , is plotted as a function of the chordwise position in Fig. 7 for a wind tunnel angle of attack of 8 degrees; for the present experiments, the effective angles of attack α are 0.2 and 5.4 degrees respectively. Additionally, the suction peak close to the leading edge confirms that blockage effects are negligible [36] , as expected from the low ratio of airfoil thickness to jet width (equal to 0.05). In Fig. 7 , the C p distributions obtained for perforated inserts with l h ranging from 1.5 to 5 mm are also plotted. No significant differences between data measured for solid or any of the permeable inserts are observed; the previously reported [14,[37][38][39] loss of lift must be therefore restricted to the contribution produced by the solid extent that is replaced by permeable material. For the current experimental set-up, that contribution amounts to only 4% of the overall lift.   Data are sampled for 20 s at a sampling rate of 50 kHz. Acoustic data are divided in time blocks of 50 0 0 samples ( t = 0.1 s), thus resulting in a frequency resolution of 10 Hz, and windowed using a Hanning weighting function with 50% overlap. The Cross Spectral Matrix of the measured acoustic pressure is obtained by averaging spectra computed for each sample block [41] . The refraction of sound waves within the shear layer of the jet is corrected using the method proposed by Sijtsma [42] . Conventional frequency domain beamforming [43,44] is performed on a square grid ranging between −2 < X/c < 2 and -2 < Z / c < 2 and distance between grid points of 10 mm. In order to minimize the effect of extraneous noise sources, integration of the source map within −0 . 3 < Z/c < 0.3 and −0 . 6 < X/c < 0.4 (red area in Fig. 8 ) is performed. A more detailed parametric study on the effects of varying the spanwise and chordwise extent of the integration region on the retrieved sound levels can be found in Merino-Martínez et al. [45] . Acoustic spectra include data from 400 Hz, which corresponds to the low-frequency bound for the anechoicity of the testing chamber. The frequency high-bound depends on the free-stream velocity, and is determined by the presence of the characteristic line source at the trailing edge within beamforming maps ( Fig. 9 (a-f)); specifically, data up to 1.4 and 3.8 kHz are reported for measurements at free-stream velocities of 12.5 and 34 m/s respectively. The aforedescribed acoustic data reduction is often employed in trailing-edge noise research, with satisfactory results [46] . Uncertainty is estimated in ± 1 dB based on a previously reported comparison with synthetic data [47] .

Acoustic measurements
Beamforming results are included in Sections 3.1, 3.2 and 3.4 . Yet, given the low-frequency nature (below 400 Hz) of the fundamental tones described in Section 3.5 , the resolution of the array is too low to distinguish trailing-edge noise from other noise sources. For this reason, far-field acoustic spectra are obtained by averaging spectra acquired by the 64 microphones [48] . To accurately resolve tones, the number of samples in each sample block is increased to 65536, thus yielding a frequency resolution of 0.76 Hz. Acoustic data corrupted by tones also present in background noise measurements are removed from spectra and replaced by linear interpolation based on adjacent points. Noise levels are not corrected for sound wave reflections; therefore their magnitude is not analysed.

Scaling laws for noise mitigation
In Fig. 10 Fig. 10 (b), where positive values refer to noise abatement with respect to the fully solid airfoil. Similarly to other flow permeable structures with a more complex pore arrangement, such as foams [11] , sintered granulates [10] or felts [5,14] , perforated inserts yield low-frequency noise attenuation, with higher noise abatement being measured for inserts with higher flow permeability. Specifically, the largest noise attenuation (equal to 11 dB) is reported for the most permeable insert ( l h = 1.5 mm) at St c = 6.3. Also in line with other studies [11] , acoustic spectra for permeable trailing edges show cross-over Strouhal numbers St * c , which set the high-frequency bound for noise mitigation. St * c depends on the type of insert 2 Specifically, a higher St * c is computed for perforated inserts with higher permeability, i.e. increasing the permeability of the trailing edge not only yields higher abatement but also extends the frequency range at which noise mitigation is observed. Contrarily, a decrease in St * c with increasing pore size, hence permeability, has been previously reported for porous edges [9,11] (note that for foams these two properties are usually linked). This finding suggests that different mechanisms are responsible for excess noise in porous and perforated edges. This aspect will be further investigated later in this section.
In Fig. 11 (a) and (b), far-field acoustic spectra for a free-stream velocity U ∞ = 20 m/s and α = 5.4 degrees are shown.
Similarly to data for lower lift conditions ( Fig. 10 (a) and (b)), perforated trailing edges with higher permeability yield higher noise mitigation below St * c . Main differences with respect to the previous configuration concern a decrease of the highest ) b ( ) a ( noise abatement (for instance, the most permeable insert now yields L p ,(1/3) = 8 dB at St c = 12.5) and an overall increase of St * c . In Fig. 12 (a), narrow-band noise attenuation L p for U ∞ of 12.5, 15, 20, 25, 30 and 34 m/s are plotted as a function of f . Only data corresponding to inserts with l h = 2, 3 and 5 mm are shown for the sake of clarity. For the same reason, only one every ten points are depicted ( f = 100 Hz). As seen in Fig. 12 (b), L p data measured for different free-stream velocities collapse when plotted as a function of St c ; for L p , the relationship between frequency and U ∞ is thus linear 3 . After collapsing, data for inserts with different flow permeabilities yield curves that define the noise attenuation performance of each design independently of U ∞ . In line with results presented above, characteristic curves for inserts with higher permeability allow for higher maximum noise attenuation and St * c . This finding allows to predict the frequency for maximum noise attenuation, or the frequency range for noise reduction for different velocities. Although not shown here for the sake of brevity, the same methodology yields a satisfactory collapse of data measured at α = 5.4 degrees. For completeness, these characteristic curves have been compared to noise mitigation data predicted by the lower error symbolic regression models described in Sarradj and Geyer [49] . However, the overall agreement is not satisfactory; there is no collapse of predicted noise mitigation data with St c , and higher noise mitigation maxima are predicted for materials with decreasing permeability, contrarily to experimental data presented in this manuscript.   Fig. 13 , are analysed in the following. To compute these properties, cubic B-splines [50] with 8 knots and C2 continuity [51] are fitted to experimental data. In the test cases where St c or St * c are not present within the measured St c range, the spline is extended with constant slope, as shown in Fig. 13 (thin dotted line).
Firstly, the change in maximum noise attenuation L p ,max with permeability K , shown in Fig. 14 (a), is analysed. L p ,max data for metal foam edges, which also collapse with St c are also presented for comparison. To facilitate the interpretation of results, L p ,max data are fitted 4 with a hyperbolic tangent function, as defined in Eq. (5) L p, max = γ 1 tanh ( γ 2 K ) (5) where γ 1 defines the asymptotic L p ,max value and γ 2 the slope of the function within the low permeability range. The reasons to choose this function are twofold: firstly, it adequately mimics the asymptotic character of experimental data for increasing permeability of the edges; secondly, it passes through the point K = 0 m 2 , L p ,max = 0 dB, properly describing baseline data. These curves yield valuable information on the type of microstructure that maximizes noise mitigation. It is observed that, for perforated inserts at α = 0.2 degrees, increasing the permeability to 3.5 × 10 −9 m 2 causes up to

dB noise mitigation. Further increasing the permeability of the insert brings few additional noise attenuation benefits
(approximately 1 dB more, up to an asymptotic value of 9.3 dB at K = 6 × 10 −9 m 2 ). Furthermore, increasing α to 5.4 degrees penalizes noise mitigation: L p ,max values measured at a higher-lift condition are lower than those computed at 0.2 degrees independently of the permeability. Interestingly, lower-permeability variants are less sensitive to a change in α: for instance, for the least permeable perforated insert ( l h = 5 mm) increasing α yields a L p ,max loss equal to 0.5 dB, whilst for the most permeable insert ( l h = 1.5 mm) the decrease is equal to 2 dB. Given the lift decrease and drag increase when employing inserts with higher permeability [5,38] , as well as their higher sensitivity to changes in α, the employment of perforated inserts with moderate flow permeability (below a threshold of K = 3.5 × 10 −9 m 2 ) is recommended. For inserts manufactured with metal foams, similar features are found: maximum noise attenuation increases rapidly with permeability up to approximately 8.4 dB (for a K threshold of 1 × 10 −9 m 2 , less than one third of the value reported for perforated pores); further increasing K only yields a 1 dB gain in L p ,max , up to a maximum noise mitigation of 9.4 dB. Both metal foam and perforated trailing edges share a high bound for L p ,max of 9.3-9.4 dB; yet, for a given K (below 6 × 10 −9 m 2 ) an additional increase of up to 4 dB (for edges with K between 0.5 and 2 × 10 −9 m 2 ) can be achieved by employing a random pore arrangement. Similarly, to obtain similar L p ,max levels, perforated inserts must be more permeable than metal foam trailing edges. In view of these findings, the permeability of the trailing edge can be therefore considered as a good indicator for noise mitigation effectiveness, but it is insufficient to fully characterize the noise scattering of trailing edges with significant micro structural differences. Other parameters describing the different pore organization of permeable materials, such as tortuosity, need to be considered to fully characterize their noise scattering. This analysis will be presented in Section 3.2 . In Fig. 14 (b), the chord-based Strouhal number at which L p ,max occurs, St + c , is shown for perforated inserts at α = 0.2 and 5.4 degrees, and metal foam inserts at α = 0.2 degrees as a function of K . For perforated inserts at α = 0.2 degrees, an approximately linear decrease of St + c with K is measured: the least permeable insert produces maximum noise attenuation at St + c = 11, while the most permeable insert yields St + c = 7.5. Therefore, perforated trailing edges with increasing permeability not only yield higher L p ,max but also mitigate low-frequency noise more efficiently. Increasing the angle of attack to α = 5.4 degrees leads to a constant St + c value of 11 for all K ; similarly to L p ,max , inserts with higher flow permeability are more sensitive to a change in α. Regarding metal foams, low-permeability inserts ( K = 0.5 × 10 −9 m 2 ) yield maximum noise attenuation at St + c = 9.5, close to the value for perforated edges. However, increasing K to 1.5 × 10 −9 m 2 rapidly brings St + c to 5.7; for higher flow permeability St + c is approximately constant. Therefore, the choice of an adequate pore arrangement allows not only to obtain different maximum noise attenuation levels (below a certain K threshold), but also to target specific frequencies.
In Fig. 15 , the lower-bound cross-over Strouhal numbers St c , computed for perforated and metal foam inserts at α = 0.2 and 5.4 degrees, are plotted as a function of K . For perforated inserts at α = 0.2 degrees, St c increases from 0.04 to 2.44 within the measured permeability range. Increasing the angle of attack, or employing inserts with random pore distribution (i.e. metal foams) yield higher St c ; specifically, St c values between 0.96 and 2.68 are measured for the former case, and between 2.28 and 2.96 for the latter. It can be therefore concluded that to guarantee noise mitigation within the very low frequency range one must employ edges with low permeability and ordered pore arrangement.
In Fig. 16 (a), high-bound cross-over Strouhal number St * c data are presented. For this quantity, a linear-like increase of St * c with K is measured for perforated inserts at α = 0.2 degrees. Therefore, besides higher noise attenuation, inserts  with higher permeability also mitigate noise up to higher frequencies (in line with results shown in Fig. 10 (a) and (b)).
Additionally, increasing the angle of attack to α = 5.4 degrees further increments St * c . For metal foams, a different trend is reported: a lower St * c is measured for increasing permeability of the insert. The opposite trends for the variation of St * c with K indicate that different mechanisms are responsible for high-frequency noise in perforated and metal foam edges. For metal foam edges, high-frequency excess noise is caused by surface roughness [9,11] ; it has been previously shown that a larger pore size, that often translates in larger K , generates higher noise and becomes more relevant at lower frequencies [52] , in agreement with results in Fig. 16 (b). It is also important to remark that, for the present data-set, narrowband contributions are only found for the most permeable perforated insert; however, the low-frequency nature of these tones suggests that they are not driven by roughness. Their nature will be addressed in detail in Section 3.5 .
The resulting St c range for noise abatement St • = St * c -St c , measured for perforated and metal foam inserts at angles of attack of 0.2 and 5.4 degrees, is shown in Fig. 17 . Results are similar to those found for St * c , i.e., perforated edges with higher permeability yield acoustic benefits within a wider frequency range, and increasing α further extends it (approximately an increase of St • = 4 with α is measured independently of K ). For metal foam edges, a decrease in St • with K is observed instead; hence, contrarily to perforated edges, more permeable metal foam edges yield noise mitigation within a smaller frequency range. The similarity between Fig. 16 (a) and Fig. 17 suggests that St • is particularly sensitive to the appearance of noise generation mechanisms other than edge noise, that generate high-frequency excess noise.

Effect of tortuosity on noise mitigation
Following the discussion in Section 3.1 , tortuosity τ Y is proposed, in addition to K , as an additional indicator for the noise mitigation capabilities of permeable trailing edges. In Fig. 18 , L p ,max data are plotted as a function of flow permeability and tortuosity for perforated and metal foam trailing edges. It is observed that, for a given permeability, micro-structures with more tortuous pore paths, i.e. foams, gain up to 4 dB noise attenuation with respect to straight channelled inserts. To this end, highly tortuous pore arrangements, such as those derived from sintering [53] or casting around space-holders [54] manufacturing processes, might yield additional benefits for low-noise applications. For the current experimental set-up, a trailing edge with permeability of K = 1 × 10 −9 m 2 and tortuosity τ Y 1.15 is a good trade-off between aerodynamic and broadband acoustic performance.

Scaling of far-field acoustic pressure with free-stream velocity
The exponent m for the scaling of far-field mean-squared acoustic pressure with free-stream velocity is now analysed. This quantity is computed fitting the Overall Sound Pressure Level (OSPL) [55] , calculated as OSPL = 10 log 10 f c 10 L p(1 / 3) / 10 dB (6) to experimental data measured at different speeds. To isolate broadband noise mitigation, only frequency bands where noise abatement with respect to the baseline configuration is measured are considered. For the same reason, the most permeable perforated ( l h = 1.5 mm) and metal foam ( d c = 1200 μm) inserts are excluded from the analysis because, as will be seen in Section 3.5 , acoustic spectra for the former contain tones; for the latter, a low-frequency broadband hump is measured. The fit of experimental OSPL data to U ∞ is performed employing OSPL = m 10 log 10 with η being an additional fit coefficient and U ref = 1 m/s for convenience. For the baseline insert, the fit yields m = 5 . 1 , in line with theoretical [56,57] and experimental results [58] . The computed exponent m is plotted (together with the uncertainty, estimated as the 95% confidence bounds of the fit) as a function of K in Fig. 19 for perforated and metal foam inserts at α = 0.2 degrees. In spite of the many spectral differences described above, exponents m for micro-structures with similar flow permeability are comparable (within their uncertainty range) independently of the pore arrangement. This result suggests that although specific noise mitigation features (St c , St * c , L p ,max , etc.) for permeable inserts rely on certain micro-structural properties, they all act upon noise generation through the same mechanism, i.e. decreasing the acoustic impedance jump at the surface discontinuity and creating distributed noise scattering [9] . Data for both perforated and metal foam edges increase linearly from m = 5 . 1 (at K = 0 m 2 , solid case) to m = 5 . 9 (at K = 3 × 10 −9 m 2 ). Consequently, m values are fitted 5 to a line, defined as follows m = χ 1 K + χ 2 (8) The linear increase suggests intrinsic changes in the noise source nature (from a dipole over a non-compact to a compact surface), and might be explained by a more effective distribution of noise sources along the permeable extent for increasing K , that decreases the ratio of effective chord length to noise source extension, i.e., it promotes the compactness of the airfoil.

Effect of the relative permeable chord length on noise abatement
Tests are carried out to assess the effect of a change in the permeable chordwise extent s on noise mitigation. To this aim, besides data gathered for s / c = 0.2 (40 mm), additional experiments are carried out for perforated inserts with l h = 2 and 5 mm; specifically, these inserts are taped at both sides, leaving only s / c = 0.1 (20 mm) and s / c = 0.05 (10 mm) exposed to the flow. As seen in Figs. 20 (a) and (b), noise mitigation data for varying U ∞ also collapse when non-dimensionalizing the frequency as Strouhal number, yielding the curves already described in Section 3.1 . Similarly, relevant parameters such as St c , St * c , St • , L p ,max and St + c are computed fitting experimental data to a spline. For data corresponding to shorter permeable lengths, the lower-bound cross-over Strouhal number St c , is directly computed from measured data, i.e., no data extrapolation is required.
Results are shown in Fig. 21 (a) and (b), where St c , St * c , L p ,max and St + c are plotted as a function of the permeable length to chord ratio s / c for perforated inserts with l h = 5 and 2 mm respectively. For the perforated insert with l h = 5 mm, an overall decrease in all St-based parameters for increasing s / c is found ( Fig. 21 (a)); hence, a longer permeable insert with this pore arrangement does not broaden the frequency range for noise abatement, but is beneficial to tackle lower frequencies.  This result evidences that increasing s / c is not a convenient way to achieve noise mitigation throughout a wider frequency range (to this end, increasing K is more effective), but rather to enhance abatement upon lower frequencies.
Finally, maximum noise abatement L p ,max data are plotted in Fig. 22 (b). As in Section 3.1 , experimental data are fitted 6 with L p, max = ξ 1 tanh ξ 2 s c (9) 6 Fitting coefficients are reported in Appendix B . Results show that with s / c = 0.05, inserts with l h = 2 mm and 5 mm abate up to 1.6 and 6.7 dB respectively. Further increasing s / c yields negligible additional abatement (lower than 1 dB). Since a longer permeable extent would decrease lift and increase drag, this value represents a good trade-off between noise mitigation and aerodynamic performance. It is interesting to note that for the s / c = 0.05 configuration the permeable extent approximately coincides with the boundary layer thickness at the trailing edge for the baseline configuration (previous studies [9,12] report a value of 9.3 mm at U ∞ = 20 m/s); whether this observation can be generalized or is merely a coincidence for the current experimental set-up will be addressed in future research. It is also interesting to note that some of the features derived from decreasing the length of the permeable insert, such as the decrease in L p ,max , or the increase in St * c , are in line with those reported in Section 3.1 for increasing α. This might be related to the passage of steady flow through permeable media due to a pressure imbalance between pressure and suction side, illustrated, for instance, by Mößner and Radespiel [59] , that decreases the effective length of the permeable extent.

Appearance of shedding-related tones
In spite of the just reported benefits regarding broadband noise mitigation, for permeable inserts with periodic arrangement of the channels and high flow permeability, an undesired tone is measured independently of the angle of attack. Specifically, such a feature is found in far-field acoustic spectra for the most permeable perforated insert ( l h = 1.5 mm). In Fig. 23 , narrow band acoustic measurements for a free-stream velocity of 25 m/s and an angle of attack of 0.2 degrees are presented, together with background noise measurements and data for the l h = 5 mm and reference configurations. A strong (+30 dB over broadband noise levels) tonal component is measured at a frequency of 208 Hz, with harmonics at 416 and 624 Hz.
In Fig. 24 (a), narrow-band spectra for the l h = 1.5 mm insert at U ∞ = 15, 20, 25, 30 and 34 m/s are shown. Only at U ∞ = 15 m/s no tones are observed within the measured frequency range. When f is scaled employing the thickness of the airfoil at the solid-permeable junction h = 15.7 mm and the free-stream velocity, all tones collapse ( Fig. 24 (b)). The analysis yields a fundamental tone at St h ,1 = 0.13, with visible harmonics up to St h,i = 0.52. Such values are in line with those typically found for blunt trailing-edge noise [14,58] , in which a similar peak arises due to shedding from the blunt trailing edge. Therefore, the nature of this tone seems to be related to the appearance of periodic flow phenomena, i.e. vortex shedding from the apparent blunt end of the airfoil aluminium body, due to the high permeability and periodic pore arrangement of the insert. The introduction of a certain degree of randomness in the pore distribution might alter the organization and periodicity of vortices, thus preventing the appearance of strong tones. In agreement with this hypothesis, the metal foam with d c = 1200 μm and the perforated insert with l h = 1.5 mm have comparable flow permeability ( K 6 × 10 −9 m 2 ); yet, tones are solely reported for the latter insert.
This aspect is further addressed in an additional test in which the permeable insert is taped, leaving exposed to the flow a permeable chordwise extent equal to s / c = 0.1 = 20 mm. A similar analysis for this test case ( Fig. 25 (a) and (b)) (with a thickness h of 9.5 mm) reveals a strong peak at St h ,1 = 0.10, and subsequent harmonics. Note that for this configuration, additional weaker tones appear around the fundamental one. The origin of these might be in the taping procedure: if tape is not perfectly parallel to the trailing edge, or is applied differently at suction and pressure sides, one might expect slight variations in the solid-permeable junction thickness along the span; hence, some deviations from the fundamental tone frequency.
Finally, in Fig. 26 , both St h ,1 values are shown as a function of the thickness ratio h / δ * . Displacement thickness data are obtained from a previous boundary layer characterization [12] , performed at 20 m/s employing high-speed Particle Image Velocimetry; only data measured at this free-stream velocity are shown. Experimental data are compared to the empirical relation for St h ,1 proposed by Brooks et al. [58] , defined as where is the solid angle (in degrees) between the sloping surfaces upstream of the trailing edge. For permeable extents s / c = 0.1 and s / c = 0.2 the solid angles are respectively 21 and 19 degrees. For comparison, data reported in the original study for a flat plate ( = 0 deg.) and a NACA 0012 airfoil ( = 14 deg.) are also included. A fair agreement between measured data and predictions is found independently of the permeable extent; the shedding-related nature of the tone is therefore confirmed.

Conclusions
In this manuscript, acoustic measurements on a NACA 0018 airfoil (with chord c = 0.2 m) equipped with solid and permeable trailing-edge inserts are performed by means of a microphone antenna. To obtain criteria for maximizing noise abatement while limiting changes in aerodynamic performance, trailing edges with straight open channels that link suction and pressure sides are manufactured. For comparison, trailing-edge inserts manufactured with open-cell metal foams, with comparable flow permeability but random pore distribution, are also analysed.
Far-field noise spectra reveal the collapse of L p , defined as the difference between noise scattering from solid and permeable edges, when nondimensionalizing frequency with chord and free-stream velocity. Collapsed data produce sinelike L p curves that define the acoustic benefits of each trailing edge independently of flow speed. From this analysis, it stems that the change in maximum noise mitigation L p ,max with permeability K can be accurately described with a hyperbolic tangent ( L p, max = γ 1 tanh ( γ 2 K ) , where γ 1 and γ 2 are fitting coefficients), i.e. significant noise mitigation can be achieved with moderate K , but it saturates above certain K thresholds, that depend on the type of edge employed. Specifically, permeability thresholds of 3.5 × 10 −9 and 1 × 10 −9 m 2 are respectively obtained for perforated and metal foam inserts. The employment of edges with permeability equal to these thresholds is therefore recommended to minimize aerodynamic penalty (i.e., decrease in lift and increase in drag) due to the use of permeable materials in lifting devices, to maximize the frequency range for noise mitigation, and to prevent an excessive downgrade of the acoustic benefits when increasing the angle of attack. Furthermore, the tortuosity of the trailing edge is proposed to account for differences in noise mitigation reported for edges with similar permeability but different pore arrangement. This manuscript also stresses that increasing the chordwise extent of the permeable region s beyond s / c = 0.05 (which in the current study is in line with the boundary layer thickness at the trailing edge for the baseline configuration) does not extend the frequency range for noise mitigation, nor increases its magnitude significantly. Finally, it is also confirmed that inserts with high permeability and an ordered pore arrangement produce extremely loud tones, which are caused by vortex-shedding from the blunt edge of the airfoil solid body.
Taken together, these findings suggest that a permeable trailing edge insert with permeability equal to 1 × 10 −9 m 2 , with a tortuosity of 1.15 (which is in turn beneficial to avoid tones) and with s / c = 0.05 yields the best trade-off between broadband noise mitigation and aerodynamic loss. Future work will address optimal trailing-edge properties for more realistic conditions, i.e., cambered airfoils with longer chord, as well as the definition of the micro-structural and fluid features necessary to prevent shedding from permeable edges.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. In Section 3.1 , the collapse of noise mitigation data for different free-stream velocities when plotted as a function of St c is shown. In the analysis, the chord of the airfoil is employed as length scale for the sake of simplicity. However, an attempt to collapse data for different permeable inserts employing length scales based on hydraulic properties, such as √ K , 1/ C or F = KC was also performed.
Results are shown in Figs

Appendix B. Fitting Coefficients
For completeness, the coefficients employed for the fit of experimental data to Eqs. (5) , (8) and (9) are respectively presented in Tables B.1 , B.2 and B.3 , together with the coefficient of determination R 2 , that gives a measure of the goodness of the fit.