Downstream porosity for the reduction of turbulence-aerofoil interaction

This paper is a predominantly experimental study into the use of porosity located downstream of an aerofoil leading edge for the reduction of turbulence interaction noise. Locating the porosity downstream of the leading edge has been shown to be beneficial in reducing the aerodynamic performance penalty compared with locating it directly at the leading edge [1], where most of the lift is generated. Noise measurements on a flat plate with downstream porosity are compared against the case of two flat plates in a tandem configuration. In both cases, the noise reduction spectra exhibit peaks of strong noise reduction at non-dimensional frequencies of fl d /U c = n , where l d is the distance between the leading edge and the downstream edge, U c is the convection velocity and n is an integer. To explain this behaviour requires a mechanism to be present in which a phase shift of 180 ◦ is introduced in the interaction process. In the paper we argue that the origin of this phase shift is due to secondary vorticity generated at the leading edge. Another key finding of this paper is that overall noise reductions are due to an effective shortening of the chord in which most of the radiation is produced by the section of the flat plate upstream of the porous section, leading to generally weaker radiation. Neither of these mechanisms have been reported previously in the literature. The paper concludes with noise measurements on a thin aerofoil with downstream porosity included, in which overall noise reductions of up to 2.8dB are achieved.

Geyer et al. [10] manufactured fully porous SD7003 aerofoils with commercially available porous materials, which they characterised by their airflow re- 10 sistivity. The porous aerofoil was located within a turbulent flow, where it was shown that noise reductions generally increased with increasing flow resistivity (higher permeability). However, it was found that the use of porosity over the entire chord incurred significant penalties in aerodynamic performance (lift and drag). In a later paper Geyer et al. [11] investigated the use of porous leading 15 edge (LE) inserts with inclined circular perforations limited to 5% of the chord.
This design produced a smaller aerodynamic penalty at low angles of attack while achieving noise reductions of up to 8 dB at some frequencies. An increase of noise (4-5 dB) at high frequencies was also observed, which was attributed to the surface roughness due to the pores. More recently, Ocker et al. [12] in- An alternative design for porous aerofoils was proposed by Roger et al. [4] consisting of a porous NACA-0012 aerofoil comprising a composite layered struc-25 ture of wire mesh and metal foam filled with steel wool. Measured noise reductions of 5 dB were achieved without attempting an optimization of the design parameters. Experimental studies with similar aerofoil designs consisting of a porous exoskeleton filled with foam or metal wool and a solid centre plate have been reported by Zamponi et al. [17] and Bampanis et al. [13], where noise 30 reductions of 4 and 6 dB were measured respectively. The latter design was reported to increase the drag by 15%.
A number of porous leading edge treatments with and without centre plate were studied computationally with a lattice-Boltzmann method in a rod-aerofoil configuration by Teruna et al. [18]. The study also proposed and evaluated a 35 serrated porous leading edge to combine the benefits of porosity and serrations, which showed improved acoustic and aerodynamic performance relative to regular serrations. More recently, the concept of poro-serrations was further investigated to reduce rotor-stator interaction noise in a computational study of a full-scale aircraft model by [19]. It was found that applying the treatment 40 in the outer span of the OGV is most beneficial and can yield up to 1.5 dB of overall power level with a performance penalty below 1.5 %.
Although it is clear that leading edge porosity is a promising technology for the reduction of turbulence interaction noise, the physical mechanisms of noise reduction are not fully understood. Suggested mechanisms [11] include 45 hydrodynamic absorption of the impinging turbulence by the porous surface, viscous dissipation in the pores, and an increased effective aerofoil thickness due to thicker boundary layers. The first hypothesis was studied by Zamponi et al. [17] on a porous NACA-0024 aerofoil. Their measured data showed a weaker distortion of the turbulent flow in the vicinity of the leading edge stagnation 50 region due to the surface permeability, which was linked to a reduction in radiated noise at low frequencies. The computational work by Teruna et al. [18] also indicates that leading edge porosity reduces the strength of the sources at the 3   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 J o u r n a l P r e -p r o o f Journal Pre-proof leading edge by allowing the incoming turbulence to permeate into the porous medium.

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The noise reduction mechanisms of porous leading edges were also investigated by [16] for fully and partially porous flat plates. That is, extending over the full chord of the flat plate or over an extent l from the leading edge respectively. From here on, porosity is defined as the percentage of open area. In the case of a fully porous plate the noise reduction spectra were found to contain a 60 number of peaks of maximum noise reduction at the non-dimensional frequencies of f c 0 /U c = n, where c 0 is the chord. In the case of a flat plate in which porosity of extent l is introduced from the leading, however, peaks in the spectra were observed at f l/U c = (n − 1/2). Different noise reduction mechanisms were proposed to explain these two different configurations. The first for fully 65 porous aerofoils was attributed to a reduced radiation efficiency based on the assumption that the aerofoil response (pressure jump) across the porous section propagates at the flow convection speed U c and not the acoustic speed, as is the case for a rigid aerofoil. The second mechanism was attributed to destructive interference between compact sources located at the leading edge and at the 70 end of the porous section separated by l. Also, included in [16] were preliminary results for the case when the porous section was located downstream of the leading edge, thereby reducing adverse effects on aerodynamic performance.
Noise reductions of up to 6dB were observed by introducing porosity 32% of the chord length downstream of the leading edge. A significant feature of the 75 noise reduction spectra for this case was the presence of strong peaks occurring at frequencies of f l 0 /U c = n, where l 0 is the distance from the leading edge to the porous section. No satisfactory mechanism was provided to explain this phenomenon. The use of downstream porosity was also recently investigated by Ocker et al. [1] on 3D-metal-printed aerofoils and axial fans who showed that 80 the aerodynamic performance penalty is progressively reduced when installing the porous region further downstream from the leading edge, i.e. increasing l 0 . The use of downstream porosity was also investigated theoretically in [20].
In [20] the acoustic wave equation was solved with appropriate boundary con-4 J o u r n a l P r e -p r o o f Journal Pre-proof ditions using the Wiener-Hopf method to predict the far field noise in which 85 the problem was approximated by two flat plates separated by a known distance in a tandem configuration subject to an incoming harmonic vortical gust.
The problem therefore effectively assumes 100% porosity in the section between the plates, which we shall show in Section 4 below, exhibits almost identical behaviour to the case of non-zero porosity. Their predictions were compared 90 against our experimental results and found to give broad qualitative agreement, providing an approximate envelope for the measured noise reduction spectra.
However, peaks in the noise reduction spectra were absent from their predictions suggesting that some of the key physics were absent from this purely acoustical solution.

Motivation and scope of the paper
This paper is a detailed experimental investigation into the use of downstream porosity for the reduction of aerofoil -turbulence interaction noise. This paper was motivated, not only by its noise reduction effectiveness and improved aerodynamic performance [1], but because it involves new physical principles 100 not previously reported in the literature. Preliminary results relating to the use of downstream porosity were presented by the authors in the conference paper [21].
Noise reduction measurements were acquired in two different open jet wind tunnels to demonstrate the robustness and consistency of the noise reduction 105 principles for aerofoils with downstream porosity. The sensitivity of the noise reductions to variations in porosity, length of the porous section, and its distance from the leading edge, were conducted on a flat plate. Porous flat plates have been demonstrated to be representative of the behaviour of thin aerofoils [16]. One of the key results of the current paper is that the noise reduction 110 performance of a flat plate with downstream porosity is shown to have identical behaviour to that of two flat plates in a tandem configuration, i.e., where the porosity is effectively 100%. The measured data is compared to a simple 5   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 analytical model in an attempt to understand the underlying noise reduction mechanisms. The flat plate results and predictions are later compared to data 115 obtained for a thin aerofoil of 5% thickness typical of OGVs.

Porous and tandem flat plates
For economy and ease of manufacture, measurements were made of the far field noise due to a flat plate with various arrangements of downstream porosi-120 ties, situated within a turbulent flow. The baseline and porous flat plates had a mean chord (c 0 ) of 125 mm and a span of 450 mm. They were constructed by joining together two 1 mm thick metallic sheets to allow porous flat plate inserts 2 mm thick to be inserted between them. The step arising from these inserts into the two flat plates were grounded to ensure a smooth transition between the 125 inserts and the flat plates. All corners were rounded and the trailing edges were sharpened to eliminate vortex shedding noise. The slotted flat plate is sketched in Fig. 1. Further details of a similar flat plate construction can be found in [22]. A parametric study was performed at the ISVR with one or more rows of circular holes downstream of the leading edge. The parameters investigated 130 were the hole diameter (D), the spacing between holes (T ), the distance from the leading edge to the (first) row of holes (l 0 ), and the length of the porous section (l). A diagram of the porous flat plate is depicted in Fig. 2a, where s is the span and c 0 is the total chord of the assembled slotted flat plate with the porous insert. Note that the diagram is not to scale and it represents a top view 135 of Fig. 1. Cases with rectangular slots instead of circular holes were also tested but not included here since they provided almost identical noise reductions to the circular holes at the same porosity. J o u r n a l P r e -p r o o f

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Noise measurements were also made for two rigid flat plates in a tandem configuration and separated by a variable air gap to provide the limiting case of 100% porosity (see Fig. 2b).

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In addition to the flat plate study, the effectiveness of downstream porosity was also investigated on a NACA4505 aerofoil of 150 mm chord to verify whether the findings obtained from flat plates also apply to more realistic aerofoil ge- A summary of the test cases is provided in Table 2.   The inflow turbulence was generated by using a bi-planar rectangular grid [24] of 630 x 690 mm 2 made of wooden bars of 12 mm width separated by 34 mm.  the pressure spectra over the polar array of microphones using the procedure described in [22]. The data used all along the manuscript was obtained in the ISVR open-jet test facility except for the source localisation results of Section 6,185 which were obtained at the BTU Cottbus open-jet test facility described below.
However, the precise opposite behaviour is observed, and the two edges must radiate in anti-phase to produce these peaks in the noise reduction spectra.

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The PWL spectra for three values of l 0 /c 0 are shown in Fig. 6 in the presence of grid-generated turbulence, labelled in the figure as Turbulence Interaction Noise (TIN). Also shown are the Self Noise spectra, labelled SN, produced in the absence of the grid to allow the relative contributions between interaction noise and self-noise to be determined. Note that the term Self Noise is used to 260 refer to all noise sources present with clean inflow, such as trailing edge noise and roughness noise.
Peak noise reductions in total noise can be clearly observed in Fig. 6 (thin solid curves) at the frequencies close to f l 0 /U c = n, as reported in [16], which are characterised by 'deeps' in the spectrum at those frequencies. Noise reductions 265 become progressively weaker as the row of holes is located further downstream of the leading edge from Fig. 6a to Fig. 6c. However, the reductions in total noise between the porous case (thin solid curve) and the baseline (thick solid curve) are limited over some frequencies by additional self-noise induced by the porous section (dashed curve), which is discussed next.

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A distinct feature in the self-noise spectra of porous plates in Fig. 6 (dashed curve) is that it exhibits a number of peaks also at the frequencies close to f l 0 /U c = n, at which maximum noise reductions in total noise (thin solid curve) 13   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 have been observed. This phenomenon is particularly pronounced when the row of holes is located closest to the leading edge (Fig. 6a) and negligible when is 275 located well downstream (Fig. 6c), and it is therefore likely to be related to the higher levels of unsteady lift in this region, although more detailed analysis is required to confirm this hypothesis.
However, the mechanisms proposed in [16,20,21] individually cannot explain the shape of the noise reduction spectra, which comprises a number of distinct peaks at f l d /U c = n, superimposed on a broad 'envelope' that decays with increasing frequency. The peaks are particularly pronounced in the case 400 of the tandem flat plates in Fig. 5b where noise reductions of up to 10dB are observed. In this case, the air gap cannot support a pressure jump and hence the cut-off mechanism cannot exist. Therefore, a possible mechanism that can explain the peaks at f l d /U c = n is the presence of highly coherent interference between the radiation from the two leading edges. However, to explain destruc-405 20   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  must also be present. We now propose a new mechanism in order to explain the cause of this additional phase inversion that does not appear to have been recognised in the literature. 410 We hypothesise that the additional phase shift necessary to explain the noise reduction peaks at f l d /U c = n is caused by secondary vorticity generated near the first leading edge interacting with edges further downstream. The phase inversion arises because its sense of rotation is opposite to that of the initial vortex. The existence and significance of secondary vorticity in aerofoil noise 415 generation were first recognised by [32] in a fundamental numerical study of leading edge noise due to flat plates. In [32] it was shown that secondary vorticity is induced as a result of nonlinear interactions between the aerofoil and the impinging vortex. This secondary vortex then convects towards the trailing edge and was a significant source of trailing edge noise.

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Recent, detailed analysis has suggested that all noise reduction mechanisms proposed in [16,20,21] must be simultaneously present to explain the characteristics of the noise reduction spectra described above. In this section, we further develop the simple analytic model proposed in [16] that combines all aforementioned noise reduction mechanisms, namely (i) edge-to-edge interfer-425 ence, (ii) phase inversion due to secondary vorticity, (iii) cut-off effects across the porous section and (iv) reduced effective chord. Predictions from the model will be compared against experimental data.
Consider a flat plate aligned with the mean flow along the y 1 direction, as sketched in Fig. 11. The figure shows three distinct sections. The first section 430 is rigid and extends from the leading edge at y 1 = 0 to the start of the porous section at y 1 = l 0 . The second is a porous section of length l that extends between l 0 < y 1 < l d . The third section is a rigid flat plate with leading edge at y 1 = l d and extends to infinity in the streamwise direction since the trailing edge is not believed to contribute significantly to the noise reduction mechanism. 435 We now elucidate the main assumptions of the simple analytic model based 21   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 on the proposed noise reduction mechanisms, whose main components are listed below and sketched in Fig. 11: 1. The impinging initial vortex, assumed here to rotate in the clockwise sense, approaches the first leading edge (y 1 = 0) at the free stream velocity U , 440 causing a downwash velocity (Fig. 11a). The initial vortex interacts with the upstream leading edge to generate a localised compact pressure jump at y 1 = 0 equal to ∆p 0 δ(y 1 ), where δ is the Dirac delta function, as shown in Fig. 11d. Interaction of the leading edge with the initial vortex induces a pressure jump that propagates along the plate to y 1 = l 0 , which behaves 445 similar to a trailing edge. Radiation from this upstream section therefore occurs with an effective chord equal to l 0 , which is therefore generally weaker than the baseline airfoil of larger chord.

A secondary vortex is induced in response to the impinging vortex due to
the non-linear mechanisms proposed in [32] (Fig. 11b). The secondary 450 vorticity is assumed to remain unchanged as it convects downstream at the convection speed U c . The sense of rotation of the secondary vortex will be opposite to the main vortex and will therefore introduce a phase inversion of 180 • upon interaction with edge discontinuities relative to the impinging vortex (Fig. 11c). The initial vortex has now become bisected 455 by the flat plate, as shown in Fig. 11b. 3. The secondary vortex interacts with the porous section to generate a pressure jump that propagates at the convection speed U c across the porous section l 0 < y 1 < l d , which at a single frequency is of the form ∆p(y 1 )e −iωy1/Uc , as indicated in Fig. 11d. Since in subsonic flows the 460 propagation speed is lower than the speed of sound a, radiation from this section is considerably less efficient than for a rigid aerofoil, in which the ∆p propagates at the supersonic speed a + U c , and is therefore essentially 'cut-off'. It is well established in classical radiation theory that an essential requirement for efficient radiation is that the phase speed of a 465 22   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  The mean square pressure p 2 due to a flat plate with downstream porosity is therefore assumed to be the sum of the radiation p 2 c=l0 due to the upstream section with effective chord l 0 and the radiation p 2 Int due to the interference 480 mechanisms described above, where α is a non-dimensional factor that controls the balance between the radiation due to the upstream section and due to interference mechanisms, which will be adjusted to give best fit to the measured data.

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The radiation p 2 c=l0 due to the upstream section, with chord l 0 , is predicted using the classical theory for interaction noise due to Amiet [34] with prescribed inflow mean square velocity and turbulence integral length-scale taken from measured values.
We note that the tandem flat plate configuration is a special case of the We first consider the form of the solution for the limiting case when the extent of downstream porosity is very small and hence f l/U c < 1. This case corresponds to the single row of holes studied experimentally, which we assume to be located at a distance l 0 downstream of the flat plate leading edge. In this case only the compact sources at the leading edge ∆p 0 and downstream edge of the holes ∆p l0 are considered (l d ≈ l 0 ) and hence ∆p(y 1 ) = 0. For generality, we assume that the two compacts source strength differ by a factor K and hence ∆p l0 = K∆p 0 , which upon substitution into Eq. (2) and performing the   25   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 integration yields, The corresponding mean square pressure due to a single row of holes is therefore given by, At moderate Mach numbers M maximum noise reductions are therefore predicted at the peak frequencies f l 0 /U c ≈ n, n = 1, 2, 3, etc. Note that this prediction differs from that due to edge-to-edge interference presented in [16] for leading edge porosity, which predicts that f n l 0 /U ≈ n−1/2 where no additional phase inversion was accounted for. we further assume that the pressure jump over the porous section per unit length is equal to that at the two leading edges ∆p 0 and ∆p ld . After integration, the resultant acoustic pressure is of the form, The corresponding mean square pressure due to the presence of the three sources assumed in the model is of the form,

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Journal Pre-proof where sinc(X) = sin(X)/X. The first term is identical to Eq. (4) which accounts for edge-to-edge interference with equal source strengths (K = 1). The second term accounts for the contribution due to radiation from the region of extended 530 porosity, referred to in [16] as 'cut-off radiation'. The remaining four terms account for the various interactions between the three sources.
At the peak frequencies close to f l d /U c = n the term p 2 Int due to edgeto-edge interference is weakest, the radiated mean square pressure is therefore largely determined by the radiation due to the upstream chord section. From 535 Eq. 1, therefore, and hence the radiation from the upstream section provides the envelope of noise reductions.
To assess the validity of the simple model we now compare in Fig. 12 predictions of the sound power against the measured noise reduction spectra for 540 the case of multiple rows of holes and a tandem configuration. Sound power predictions are obtained by integrating Eq. 1 over the polar angles of the microphone array, with p 2 Int given by Eq. 7. Note that the factor α was adjusted to provide best fit to the measured sound power spectrum in each case. Also shown in Fig. 12 is the spectrum of sound power reductions due solely to the 545 radiation from the upstream section of chord l 0 , p 2 c=l0 . The simple model can be observed to provide acceptable qualitative agreement to the behaviour of the measured sound power reduction spectrum, with both the interference peaks and general spectral shape being closely predicted.
We note from Fig. 12 that the noise reduction performance at low frequencies, f l d /U c < 1, is determined by the interference sources, which are now in-555 creasingly in-phase in this low frequency limit. In this limit, the radiation from the upstream section is relatively small, tending to 14dB reduction in noise as

Discussion and limitations of the model
The analytic model for predicting the noise reduction performance of the 560 porous flat plates based on simple source mechanisms has been shown to provide acceptable qualitative agreement with the experimental noise reduction spectra.
We now interrogate the solution to identify the dominant terms and hence the principal noise reduction mechanisms.
Equation (7) for the contribution to the radiation due to interference com-565 prises three components. The first two terms account for the radiation due to edge-to-edge interference and to 'cut-off' behaviour, respectively. The remaining terms account for the radiation due to the interaction between the different 28 J o u r n a l P r e -p r o o f Journal Pre-proof sources.
We observe from Fig. 12 that, consistent with measurement, the greatest 570 noise reductions occur at the peak frequencies of f l d /U c ≈ n, strongly implying the dominance of interference between two edges separated by l d with an additional phase shift of 180 • included.
Inspection of Eq. (7) indicates that, apart from the first term due to edgeto-edge interference, the remaining terms include factors containing the non-575 dimensional frequencies of (U c /ωl) 2 and U c /ωl. These terms arise from the radiation due to the porous section and are most significant at low frequencies and responsible for the weaker noise reductions in the low frequency limit.
At the higher frequencies, these terms become progressively smaller with increasing non-dimensional frequency. As a consequence of the (U c /ωl) frequency 580 dependence, noise reductions at the first peak frequency steadily improve with increasing l, as observed in the experimental noise reduction spectra in Fig. 8.
It is noted that the intrinsic simplifying assumptions considered in the model, such as the assumption of compact sources of equal source strength, are aimed at predicting the general shape of the noise reduction spectra due to the intro-585 duction of downstream porosity. However, the model cannot predict absolute levels of noise reduction and hence the inclusion of an arbitrary factor α to control the balance of the two reduction mechanisms p 2 Int and p 2 c=l0 described above. The current model is therefore limited in this aspect and would require additional information on the spectral and spatial distribution of the pressure 590 jump over the rigid and porous sections. The assumption of completely undisturbed convection of the secondary vorticity over the plate is also known to be a major simplification since it has been reported in [32] that the vortex energy content is transferred from larger to smaller scales as the vorticity convects over the flat plate. Improvements to the current model and testing of the underlying 595 assumptions will be undertaken in the future by extending the computational work of [32] for cases with downstream porosity and evaluating the pressure jump over the flat plate. The current model however represents a starting point for more complex analytical studies and provides insight into the understanding 29   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  The introduction of downstream porosity can be clearly seen to increase the thickness of the boundary layer on the suction side of the aerofoil. This behaviour is most likely due to flow feeding the boundary layer through the holes 640 driven by the pressure difference across the aerofoil. Self noise therefore increases 31   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  We hypothesise that larger turbulent structures created within the thicker boundary layer scattered from the trailing edge are responsible for the increase in self-noise at low frequencies. This is confirmed from measurements of the hot wire turbulence velocity frequency spectrum (not shown here). Self-noise 650 therefore increases with increasing AoA but there remain good levels of noise reduction overall of up to 2.8dB for the cases investigated as summarised in Table 3.   2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 J o u r n a l P r e -p r o o f Journal Pre-proof  A principal finding of this paper is that the noise reduction spectra for a flat plate with downstream porosity and a thin aerofoil are almost identical in 665 shape to that of two flat plates in a tandem configuration. It has been shown experimentally that in both cases the noise reduction spectra collapse when plotted against non-dimensional frequency f l d /U c , where l d is the distance between the leading edge and the downstream edge and U c is the convection velocity. Narrowband peaks of noise reduction have been identified at frequencies  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  proposing a new noise reduction mechanism not previously discussed in the literature in which subsequent interactions downstream of the first leading edge are due to secondary vorticity driven by the initial vortex. These narrowband peaks are observed to be superimposed on a broad 'envelope', whose spectral 675 shape has been shown to be closely related to the noise reduction due to a shorter chord equal to that of the upstream section of length l 0 .
Based on these proposed mechanisms and previous analytic work on porous aerofoils, an analytical model has been proposed in an attempt to explain the general characteristics of the noise reduction spectra. The model appears to 680 capture the general behaviour of the measured noise reduction spectra, including the peaks and the spectral envelope subject to the appropriate choice of a single empirical constant α.
Further work is required to more fully understand and establish the underpinning mechanisms of noise reduction in aerofoils with downstream porosity.

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Nevertheless, it is clear that this approach is potentially very useful for reducing broadband interaction noise and for achieving a good compromise between aeroacoustic benefits without significant degradation in aerodynamic performance.