3.1.1. Acoustic measurement

Figure 4 shows some typical sound maps of the flat plate model and the background. The sound map has already been integrated within a 1/3 octave range to reduce statistical fluctuation. It is clear that, above 2000 Hz, the major noise sources from the flat plate model are located along the trailing edge. In terms of the background noise, at low frequencies, the major noise source is from the nozzle (on the left) and the jet collector (on the right); at higher frequencies (4000 Hz and 8000 Hz), the noise emitted from the trailing edges of the endplates is visible. However, it is clear that in the integration region, the level of trailing edge noise source is much larger than the background noise (by at least 10 dB), indicating that the beamforming method is effective in isolating the trailing edge noise from the background noise.

Figure 4. The 1/3 octave sound maps of the flat plate model (left column) and the background (right column) at three different frequencies at $U_0 = 20$ m s$^{-1}$. The solid rectangle in the middle represents the flat plate model, and the dotted-line rectangle represents the source integration region. The left grey rectangle and the two grey lines denote the wind tunnel nozzle and the endplates, respectively. The dynamic range of each plot is 12 dB.

The comparison between the flat plate trailing edge noise and the background noise is shown in figure 5. The reason that the background noise at several kHz is out of range is that, after diagonal removal, the conventional beamforming method can give a negative source distribution at some non-source regions, and the negative source intensity was set to zero. But even though the background noise within the integration region may be underestimated at these frequencies, it is still evident that the signal to noise ratio exceeds 10 dB. Therefore, the effect of background noise can be neglected except at low frequencies. The theoretical prediction based on the flat plate assumption by Amiet (Reference Amiet1978) and the empirical boundary layer statistics (Goody Reference Goody2004) is also plotted as a reference in figure 5. The fluctuation in the predicted spectrum at $90^\circ$ is due to the finite chord length of the model (Amiet Reference Amiet1978) and is independent of the flow speed. Since the diameter of the microphone array is not negligible with respect to the observation distance, it is necessary to take the average between the observation angles. The weighted average considering the distribution of the microphone observation angles is used here. This takes into account the effect that more microphones are located near the $90^\circ$ observation angle. This is compared with the simple averaging method, which assumes a uniform microphone distribution within the observation angle limits. After the averaging process, the fluctuation in the prediction is largely suppressed, and the theoretical prediction is in reasonable agreement with the measured spectrum. However, there is an underprediction at mid-to-high frequencies, which is likely caused by the inaccuracy of Goody's wall pressure spectrum for this case. A better prediction using the near wake turbulent statistics as input will be presented in § 4.2.

Figure 5. The flat plate trailing edge noise, the background noise and the theoretical predictions at two different free-stream velocities. The red dashed curve represents the predicted sound level at a $90^\circ$ observation angle. The thin dashed blue line and solid blue line represent the simple average and weighted average of the predicted noise within the observation angle of the microphone array, respectively.

The trailing edge noise of the baseline tripped flat plate model was measured twice with a six-month interval. Between these two measurements, there were many other experiments conducted in the wind tunnel, and the set-up of the wind tunnel changed continuously. Nevertheless, the average difference in terms of the power spectral density (PSD) in the range 200–10 000 Hz was 0.37 dB at 20 m s$^{-1}$ and 0.02 dB at 30 m s$^{-1}$, respectively. Therefore, the repeatability of the spectral measurement is estimated to be within 0.4 dB.

3.1.2. Near-wake flow measurement

Figure 6 shows the distribution of the mean streamwise velocity $U$ and the root mean square fluctuating component $u_{rms}$ at the near wake of the flat plate model ($x = 0.7$ mm) at $U_0 = 20$ m s$^{-1}$. It can be used to reveal the boundary layer properties in the upstream vicinity of the trailing edge. First, the flow is fully attached at the trailing edge, indicating that the smooth geometrical transition between the flat part and the trailing edge part was effective. Secondly, the symmetrical shape of the mean velocity distribution confirms that the install angle was $0^\circ$.

Figure 6. The distribution of (a) the mean velocity $U$ and (b) the velocity fluctuation $u_{rms}$ in the near wake of the flat plate model. Here, $U_0 = 20$ m s$^{-1}$.

Assuming that the near-wake velocity distribution is the same as the boundary layer velocity distribution just ahead of the trailing edge, we can estimate the boundary layer statistics, including the boundary layer thickness $\delta$, displacement thickness $\delta ^*$ and the momentum thickness $\delta _\theta$, as shown in table 2. The values in the table are the averages of the boundary layers on two sides. The empirical value (Schlichting & Gersten Reference Schlichting and Gersten2016) and the prediction by the program XFOIL (Drela Reference Drela1989) are also shown as a comparison. The measured $\delta ^*$ and $\delta _\theta$ are close to the XFOIL prediction, but higher than the empirical value. This is due to the slight adverse pressure gradient upstream of the trailing edge. The discrepancy between the measured and predicted $\delta$ will be explained in § 4.2.

Table 2. The boundary layer thickness $\delta$, displacement thickness $\delta ^*$ and the momentum thickness $\delta _\theta$ at the trailing edge. The near-wake velocity distribution was assumed to be the same as the boundary layer velocity distribution. Here, $U_0 = 20$ m s$^{-1}$.

The spectral information of the turbulent wake of the flat plate is shown in figure 7. The broadband feature of the spectrum also confirms a successful transition by the trip strip. The majority of the turbulent kinetic energy is concentrated within the boundary layer and at low frequencies. In order to characterize the overall turbulent energy distribution within the whole boundary layer, integration of the turbulent spectrum was conducted in the chord-normal direction ($y$-direction) and is shown in figure 8

(3.1)\begin{equation} {PSD_{u, overall}} = \int_{{-}H/2}^{H/2} {PSD_{u}} \, \mathrm{d} y , \end{equation}

where H is the length of integration and $H/2>\delta$ is required to capture the entire boundary layer. In the overall spectrum, there is an energy-containing range at low frequency and an inertial subrange following the $-$5/3 law between 2000 and 10000 Hz.

Figure 7. The spectrum of $u$ in the near wake of the flat plate. Here, $U_0 = 20$ m s$^{-1}$.

Figure 8. The integrated velocity spectrum of $u$ in the near wake of the flat plate along the transverse direction. Here, $U_0 = 20$ m s$^{-1}$.

3.2.1. Noise measurement

Figure 9 shows the narrow band trailing noise spectra corresponding to different velvety coatings and smooth coatings. The 1/3 octave spectra, shown in figure 10, have less fluctuation compared with the narrow-band spectra. The effect of velvety coatings on the trailing edge noise is largely frequency dependent. In general, for each velvety coating at a certain flow speed, there is a cross-over frequency $f_{c}$, below which the coating increases the trailing edge noise, while above $f_{c}$, the trailing edge noise is reduced. The noise increase in the low-frequency range has a hump-like shape, and the noise reduction at high frequencies is broadband. Due to the upper limit of the frequency response of the microphones, the effect of the velvety coating at frequencies above 10 000 Hz was not determined. From the trend of the spectrum, it is conjectured that the noise increase by the coating occurs at some frequency beyond 10 000 Hz. The hair dimension affects both the cross-over frequency and the magnitude of noise increase/reduction. For example, compared with other velvety coatings, the coating H1.5 gives a lower $f_{c}$, and a higher noise reduction for most of the frequencies above $f_{c}$. However, it also gives a higher noise increase in the low-frequency range.

Figure 9. The noise spectra in the trailing edge region for flat plates covered with different coatings.

Figure 10. The 1/3 octave spectra in the trailing edge region for flat plates covered with different coatings.

As a comparison, smooth coatings lead to additional tonal noise. The trailing edge thickness-based Strouhal number $St_h = f h/U_0$ for the tonal peak is around 0.15, which is similar to the results for the blunt trailing edge vortex shedding noise in the study by Brooks et al. (Reference Brooks, Pope and Marcolini1989), where $St_h$ lies between 0.12 and 0.2 for an aerofoil with plate extensions. The noise level at frequencies below the vortex shedding frequency is not affected by the presence of a smooth coating. This indicates that the mechanism for the low-frequency hump caused by the velvety coating is not due to vortex shedding. More details on the mechanism will be given in § 4. Smooth coatings also lead to a broadband noise reduction at high frequency. However, the magnitude is not as high as that of the velvety coating of the same thickness. For example, at 20 m s$^{-1}$, the maximum noise reduction by the H1.5 velvety coating is 18 dB, while it is 9 dB for the S1.5 coating.

Figure 11 shows the normalized 1/3 octave spectra at free-stream speeds between 16 and 30 m s$^{-1}$. The frequency is normalized to the chord-based Strouhal number $St = f c/U_0$. A reasonable collapse is achieved with a velocity scaling of $U_0^5$ except for the tonal peaks, which is consistent with the classical trailing edge noise theory (Ffowcs Williams & Hall Reference Ffowcs Williams and Hall1970). Figure 12 shows the change of the sound spectrum caused by the coating compared with the baseline flat plate case. A positive ${\rm \Delta} PSD$ represents a noise increase, and vice versa. All measurement results with a free-stream velocity between 16 m s$^{-1}$ and 30 m s$^{-1}$ are plotted. To reduce the fluctuation in the spectrum, a robust locally weighted regression (RLOWESS) filter (Cleveland Reference Cleveland1979) with a window size of 30 was applied to smooth the data. After data smoothing, the change of trailing edge noise ${\rm \Delta} PSD$ caused by the coating is found to be a quantity achieving good data collapse, especially when the frequency is close to $f_{c}$ or beyond $f_{c}$. This data collapse indicates the underlying mechanism for noise reduction remains the same within the measured flow speed range. The cross-over Strouhal number $St_{c}$ corresponding to $f_{c}$ appears to be nearly invariant within the measured speed range; $St_{c}$ is negatively related to the length of the velvety coating. For the velvety coatings, the magnitude and central frequency of the low-frequency hump do not collapse well with Strouhal number. As flow speed increases, the magnitude of the hump increases, while the central frequency of the hump decreases. This indicates that the disturbing effect of the velvety coating is larger at high speed, while the corresponding length scale is smaller. For the smooth coating, the central Strouhal number of the vortex shedding tone is nearly constant, indicating a speed-independent length scale. The magnitude of the tone increases with the incoming flow speed.

Figure 11. The scaled 1/3 octave spectra as a function of chord-based Strouhal number $St$ for flat plates covered with different coatings. The sound level is scaled with $U_0^5$.

Figure 12. The relationship between ${{\rm \Delta} PSD}$ and the chord based Strouhal number $St$, with data smoothed by a RLOWESS algorithm. The corresponding $U_0$ is between 16 and 30 m s$^{-1}$. A positive ${{\rm \Delta} PSD}$ represents a noise increase.

The sound maps are then used to reveal the location of the major noise sources. First, the origin of the low-frequency hump is investigated. Since the array resolution is better at a higher frequency, the sound maps of the baseline configuration and the H0.3 configuration are compared at 2000 Hz and 30 m s$^{-1}$. Although this frequency is not at the centre of the low-frequency hump, at this condition, ${{\rm \Delta} SPL}$ still exceeds 7 dB. Figure 13 shows that the major noise sources for the coated model are distributed along the trailing edge, and the magnitude is increased compared with the reference case. This indicates that the noise source for the low-frequency hump is at the trailing edge, even if the trailing edge is covered by the velvety coating.

Figure 13. The 1/3 octave sound maps at 2000 Hz for (a) the flat plate configuration and (b) the H0.3 configuration. Here, $U_0= 30$ m s$^{-1}$.

Next, the high-frequency noise reduction by the velvety coating is studied. The sound maps of the baseline configuration, H0.3 configuration, H1.5 configuration and S1.5 configuration are compared at 5000 Hz at 20 m s$^{-1}$ in figure 14. The corresponding ${{\rm \Delta} SPL}$ are $-9$ dB (H0.3), $-17$ dB (H1.5), $-9$ dB (S1.5), respectively. These coatings all reduce the noise source level at the central trailing edge position. It is interesting to notice that, for the H0.3 and S1.5 configurations, the dominant noise source is still distributed along the trailing edge, while for the H1.5 case, the major noise sources are at the wind tunnel nozzle and the plate–endplate junction. A direct comparison between the H1.5 and S1.5 configurations shows a superior noise reduction potential by the velvety coating, at the same coating thickness.

Figure 14. The 1/3 octave sound maps at 5000 Hz for (a) the flat plate configuration, (b) the H0.3 configuration, (c) the H1.5 configuration and (d) the S1.5 configuration. Here, $U_0= 20$ m s$^{-1}$.

3.2.2. Near-wake flow measurement

The mechanism of the noise spectrum modification was investigated through the measurement of the near-wake flow. The distributions of mean velocity $U$ and fluctuating velocity $u_{rms}$ are plotted in figure 15, and the spectral information is plotted in figure 16. It is clear that the velvety coatings broaden the boundary layer compared with the baseline configuration. The thicker velvety coatings have a larger effect on boundary layer broadening. In contrast, smooth coatings do not significantly increase the boundary layer thickness. This indicates the broadening effect by the velvety coatings is probably linked to their surface structure. For thicker coatings, there exists a region near $y=0$ where both mean velocity and fluctuating velocity are small. This is related to the bluntness caused by these coatings. For smooth coatings, we can see a large velocity gradient near the offset wall, similar to the baseline case. For the cases of velvety coatings, the near-wall velocity gradient is much reduced, which is consistent with the observation by Nishimura et al. (Reference Nishimura, Kudo and Nishioka1999). Similarly, in another study on porous aerofoils, Geyer & Sarradj (Reference Geyer and Sarradj2014) reported that, compared with non-porous aerofoils, porous aerofoils lead to a thicker boundary layer (up to three times the original boundary layer thickness) and smaller near-wall velocity gradient. In terms of the fluctuating velocity component, the velvety coatings widen the regions with high velocity fluctuations, and thicker velvety coatings have a larger effect. In contrast, the smooth coatings do not widen the region with high fluctuating velocity. Instead, the major effect is to offset the turbulent boundary layer outwards by approximately the thickness of the coating.

Figure 15. The distribution of (a) the mean streamwise velocity $U$ and (b) the root-mean-square (r.m.s.) fluctuating velocity $u_{rms}$. Here, $U_0= 20$ m s$^{-1}$.

Figure 16. The spectral density of the fluctuating velocity component $u$ at different chord-normal locations $y$. Here, $U_0= 20$ m s$^{-1}$.

The spatial distributions of the streamwise velocity fluctuation spectrum of several configurations are plotted in figure 16. It is clear that all spectra have a broadband component, which confirms that the flow at the trailing edge is turbulent. The turbulent energy is dominated by low-frequency components and is concentrated within the boundary layer. For the H0.3 and H1.5 configurations, there are three major differences from the clean configuration. First, the width of the region with high velocity spectrum is increased. Second, there is a wider region of low velocity spectrum around $y = 0$. It is clear that the high-frequency velocity fluctuation is reduced in this central zone, compared with the clean configuration. The increased width of this region can be attributed to the increased trailing edge bluntness caused by the coating. Third, at the outer region of the boundary layer, a broadband hump is visible. The central frequency for the hump is 1000 Hz for the H0.3 configuration and 500 Hz for the H1.5 configuration. These two frequencies match with the broadband hump in the corresponding noise spectra. For the smooth coating S1.5, the broadband spectrum is similar to the baseline configuration, except that the turbulent structure is approximately offset outward by the thickness of the coating. A significant tonal peak at 1 kHz and its harmonics are visible, suggesting vortex shedding behind the blunt trailing edge. However, vortex shedding is not present for all the velvety coating configurations. The possible mechanism is that the velvety structures can destroy the spanwise coherence of the boundary layer structures and suppress vortex shedding, as is the case with a porous surface (Ali, Azarpeyvand & da Silva Reference Ali, Azarpeyvand and da Silva2018).

The overall velocity spectra ${PSD_{u, overall}}$ under different configurations are plotted in figure 17. The calculation method was introduced in § 3.1.2. The velvety coatings consistently elevate the integrated turbulent spectrum within the whole frequency range, and the thicker coatings lead to a larger increase. The increase is more prominent in the low-frequency range. No tonal peak related to vortex shedding is detected for the velvet configurations. In contrast, the smooth coatings do not significantly change the broadband level of ${PSD_{u,overall}}$, but only add several tonal peaks to the spectra. These spectral peaks are directly related to vortex shedding behind the blunt trailing edge. Therefore, the increase of the velocity spectra by the velvety coatings is likely attributed to their extra surface roughness. It is counter-intuitive that, although the velvety coatings significantly elevate the overall high-frequency turbulent energy, they can still reduce more high-frequency noise compared with the smooth coating of the same thickness. This will be discussed in detail in § 4.2.

Figure 17. The integrated velocity spectrum of $u$, defined in (3.1), under different configurations. Here, $U_0= 20$ m s$^{-1}$.

4.1.1. Theoretical formulation

The basic formulation in Howe's model is about the scattering of a vortex past through a blunt trailing edge, as shown in figure 19. A rectangular-shaped trailing edge is used to match with the experimental set-up. The thickness $h$ is equal to twice the coating thickness. Since sharp separation occurs at the blunt trailing edge, it is assumed that the trajectory of the vortex is fully separated, i.e. parallel to the undisturbed mean flow direction. The magnitude of the vortex is assumed to be invariant along its trajectory under the frozen turbulence assumption (Taylor Reference Taylor1938). The flow elsewhere is assumed to be potential.

Figure 19. The formulation of the scattering problem of a single vortex past a rectangular trailing edge. The region outside the rectangular aerofoil in the $z$-plane is mapped to the upper half-plane in the $\zeta$-plane.

The two-dimensional (2-D) potential flow assumption enables the analysis using conformal mapping. The region outside the rectangular aerofoil in the $z$-plane ($z = x + \textrm {i} \,y$) is mapped to the upper half-plane in the $\zeta$-plane ($\zeta = \eta + \textrm {i} \,\xi$). Note that the symbol $z$ here is a complex number and does not represent the spanwise location. This is achieved by the following transformation:

(4.1)\begin{equation} \frac{z}{h} = f(\zeta)={-}\frac{1}{\rm \pi}\{\zeta \sqrt{\zeta+1}\sqrt{\zeta-1}-\ln (\zeta +\sqrt{\zeta+1} \sqrt{\zeta-1})\}-\frac{\textrm{i}}{2}. \end{equation}

Vortex sound is generated only if the vortex moves across the streamlines of ideal incompressible flow around the trailing edge (Howe Reference Howe2003), as illustrated in figure 20. After being transformed into the $\zeta$-plane, these streamlines correspond to a uniform flow along the $+\eta$ direction. For an infinitely thin trailing edge, the velocity potential is

(4.2)\begin{equation} \varphi^*(x,y) = \sqrt{r} \sin (\theta/2), \end{equation}

where $(x, y ) = (r \cos \theta , r \sin \theta )$, $r$ is the distance between the observer and the trailing edge and $\theta$ is the observation angle with the downstream direction set as zero. For a rectangular-shaped trailing edge with a thickness of $h$, the streamlines near the blunt edge is different from that of the sharp case. However, the velocity potential approaches $\varphi ^*(x,y)$ far from the trailing edge. Denoting the velocity potential around the blunt edge by $\varPhi ^*(x,y)$, then

(4.3)\begin{equation} \varPhi^*(x,y) \to \varphi^*(x,y) \text{ as } \sqrt{x^2+y^2} \to \infty . \end{equation}

Here, $\varPhi ^*$ can then be represented as a function of $\zeta$ by

(4.4)\begin{equation} \varPhi^* \equiv \varPhi^*(z) ={-} \mu\, Re \,\zeta, \quad \mu > 0 , \end{equation}

where $\mu$ is a constant coefficient to ensure the condition in (4.3). The main result from Howe's model is that, for the same incoming turbulent wall pressure fluctuation that is convected by a speed $U_c$, the effect of trailing edge shape on the far-field acoustic pressure spectrum can be represented using a factor $|\mu \mathcal {I}(\omega /U_c)|^2$, where

(4.5) \begin{equation} \mathcal{I}(k_1) = \mathcal{I}\left(\frac{\omega}{U_c}\right) = \begin{cases} \displaystyle \left(\textrm{e}^{{-}k_1 h /2} \int_{-\infty}^{\infty} \textrm{e}^{-\textrm{i} k_1 z}\, \mathrm{d} \zeta\right)^*, & \text{if } k_1 \geq 0,\\ -(\mathcal{I}({-}k_1))^*, & \text{if } k_1 < 0, \end{cases} \end{equation}

where $\omega$ is the angular frequency, $k_1$ is the streamwise hydrodynamic wavenumber and the asterisk denotes the complex conjugate. When the frequency of interest is very low, or when the trailing edge is very thin, the domain can be viewed as a half-plane such that

(4.6)\begin{equation} \mu\mathcal{I}\left(\frac{\omega}{U_{c}}\right) \sim \mu\mathcal{I}_0 \left(\frac{\omega}{U_{c}}\right) = \mathrm{e}^{-\mathrm{i} {\rm \pi}/ 4} \sqrt{\frac{{\rm \pi} U_{c}}{\omega}}, \quad \frac{\omega h}{U_{c}} \ll 1 . \end{equation}

Figure 20. The streamlines (thin grey lines) of an ideal incompressible flow around a trailing edge (thick black lines): (a) rectangular trailing edge with a thickness $h$; (b) infinitely thin trailing edge.

Therefore, we can represent the effect of reduced scattering efficiency of a finite-thickness trailing edge compared with the ideal zero-thickness trailing edge by

(4.7)\begin{equation} {{\rm \Delta} SPL_{th}} = 20 \log \left| \frac{\mathcal{I}(\omega/U_c)}{\mathcal{I}_0(\omega/U_c)} \right| . \end{equation}

The above analysis assumes a semi-infinite aerofoil and thus does not account for the effect of leading edge back scattering on trailing edge noise (Roger & Moreau Reference Roger and Moreau2005). However, it can be argued that ${{\rm \Delta} SPL_{th}}$ remains the same in the case of a finite chord, since the leading edge has no effect on the hydrodynamic field at the trailing edge, and the back-scattered sound intensity is proportional to the sound generated directly at the trailing edge. For completeness, the detailed derivation of the results can be seen in Appendix A.

4.1.2. Comparison with experimental results

It is straightforward to use numerical integration to evaluate the effect of ${{\rm \Delta} SPL_{th}}$, as shown in figure 21. The convection velocity $U_c$ is set to be 0.7 times the free-stream velocity $U_0$ in the calculation (Roger & Moreau Reference Roger and Moreau2004). For a trailing edge thickness-based Strouhal number $St_h <10^{-2}$, the thickness has almost no effect on the scattering efficiency, and the trailing edge can be regarded as sharp. As $St_h$ increases, the reduction of ${{\rm \Delta} SPL_{th}}$ first comes from the reduction of pressure fluctuation at the lower side of the aerofoil by a factor of $\exp (-k h)$. At $St_h>1$, the contribution from the lower side of the aerofoil to trailing edge noise can be neglected, and the noise is only caused by the scattering of boundary layer turbulence by a right-angled wedge.

Figure 21. The prediction of the change of turbulent boundary layer-trailing edge noise due to the bluntness effect, ${{\rm \Delta} SPL_{th}}$, (a) as a function of trailing edge thickness-based Strouhal number $St_{h}$; and (b) as a function of frequency at $U_0 = 20$ m s$^{-1}$. The measurement results are shown for comparison for the latter case.

The measured noise reduction level is also plotted for comparison. The noise increase in the experiment below the critical frequency is caused by other mechanisms, as discussed in § 3.2.1. The prediction is consistent with the observation that the addition of smooth coating only slightly modifies the low-frequency noise (below vortex shedding frequencies), and the trend in the mid- to high-frequency range is qualitatively consistent with the experiment. However, it does not predict a correct ${{\rm \Delta} SPL_{th}}$ value for both velvety coatings and smooth coatings at high frequencies, suggesting that there could be additional reasons for the large modification of the noise spectrum apart from the bluntness effect. In the above analysis to calculate ${{\rm \Delta} SPL_{th}}$, we made the assumption that the spectrum of the fluctuating wall pressure is not altered. This is not necessarily true in the experiment, and the effect of coatings on the wall pressure spectrum will then be discussed in § 4.2.

4.2.1. Theoretical formulation

In the following, for notational convenience, we use $\boldsymbol {x} = (x_1, x_2, x_3)$ to represent the spatial coordinate, where the subscripts 1, 2 and 3 represent the streamwise, wall-normal and spanwise directions, respectively. Blake (Reference Blake1986) derived the turbulent wall pressure fluctuation $p'$ as a solution of Poisson's equation, with the source term $\mathcal {S}$

(4.8)\begin{equation} \frac{\partial^{2} p^{\prime}}{\partial x_{i} \partial x_{i}} ={-}\mathcal{S}={-}\mathcal{S}^{M S}-\mathcal{S}^{T T} , \end{equation}

where

(4.9)$$\begin{gather} \mathcal{S}^{M S}= 2 \rho \frac{\partial U_{i}}{\partial x_{j}} \frac{\partial u_{j}^{\prime}}{\partial x_{i}} , \end{gather}$$

(4.10)$$\begin{gather}\mathcal{S}^{T T}= \rho \frac{\partial u_{i}^{\prime}}{\partial x_{j}} \frac{\partial u_{j}^{\prime}}{\partial x_{i}}-\rho\left\langle\frac{\partial u_{i}^{\prime}}{\partial x_{j}} \frac{\partial u_{j}^{\prime}}{\partial x_{i}}\right\rangle , \end{gather}$$

where $U_{i}$ is the mean velocity components, $u_{i}$ represents the fluctuating velocity components, $\mathcal {S}^{M S}$ is the source term corresponding to the mean shear–turbulence interaction and $\mathcal {S}^{T T}$ is the source term corresponding to the turbulence–turbulence interaction. In strong shear flows such as boundary layer flow, the second term is negligible. In addition, for the mean boundary layer flow, $\partial U_{1}/\partial x_{2}$ is the leading-order term such that the source tensor $\mathcal {S}^{M S}$ only has one non-zero term

(4.11)\begin{equation} \mathcal{S}^{M S}=2 \rho \frac{\partial U_{1}}{\partial x_{2}} \frac{\partial u_{2}^{\prime}}{\partial x_{1}} . \end{equation}

The wavenumber–frequency wall pressure spectrum as a solution of (4.8) is

(4.12)\begin{align} \varPhi_{p}(k_{1}, k_{3}, \omega) &= 4 \rho^{2} \frac{k_{1}^{2}}{k^{2}} \int_{0}^{\infty} L_{2}(x_{2})\left(\frac{\partial U_{1}(x_{2})}{\partial x_{2}}\right)^{2} \overline{u_{2}^{2}}(x_{2}) \nonumber\\ &\quad \times \phi_{22}(k_{1}, k_{3},x_2) \phi_{m}(\omega-U_{c}(x_{2}) k_{1})\, \textrm{e}^{{-}2 |k| x_{2}}\, \mathrm{d} x_{2} , \end{align}

where $\rho$ is the fluid density, $L_2(x_2)$ is the wall-normal integral length scale of the turbulence, $\overline {u_{2}^{2}}(x_2)$ is the mean square of the wall-normal velocity fluctuation, $k_1$ and $k_3$ are the streamwise and spanwise hydrodynamic wavenumbers, $k = \sqrt {k_1^2+k_3^2}$, $\phi _{22}(k_1,k_3)$ is the dimensionless energy density corresponding to $\overline {u_{2}^{2}}(x_2)$ and $\phi _{m}(\omega -U_{c}(x_{2}) k_{1})$ is the moving axis spectrum. A constant convective velocity $U_c(x_2) = 0.7 U_0$ is used in this study. The upper limit of the integral can be shrunk to the turbulent boundary layer thickness $\delta$, as the integrand is negligible outside the boundary layer. Under the frozen turbulence assumption, the moving axis spectrum contains a Dirac delta function (denoted as $\delta _D$ to avoid confusion) at the convective ridge,

(4.13)\begin{equation} \phi_{m}(\omega-U_{c}(x_{2}) k_{1})=\delta_D(\omega- U_{c}(x_{2}) k_{1}). \end{equation}

For an observer located at the mid-span plane, only the component $\varPhi _{p}(k_{1}, k_{3}=0, \omega )$ matters. Therefore, the surface pressure spectrum can be simplified as

(4.14)\begin{align} S_{q q}(\omega)&= \frac{\rm \pi}{U_{c}} \frac{1}{L_{3}(\omega)} \varPhi_{p}\left(k_{1}=\frac{\omega}{U_{c}}, k_{3}=0, \omega\right) \nonumber\\ &= \frac{4 {\rm \pi}\rho^{2}}{L_{3}(\omega)} \int_{0}^{\delta} \frac{L_{2}(x_{2})}{U_{c}}\left(\frac{\partial U_{1}}{\partial x_{2}}\right)^{2} \overline{u_{2}^{2}}(x_{2}) \phi_{22}\left(\frac{\omega}{U_{c}}, 0, x_2\right) \,\textrm{e}^{{-}2|k| x_{2}}\, \mathrm{d} x_{2} , \end{align}

where $L_3(\omega )$ is the spanwise correlation length scale, which can be estimated as $L_3(\omega ) = 1.6 U_c /\omega$, according to Brooks & Hodgson (Reference Brooks and Hodgson1981). The non-dimensional energy density is estimated using a von Kármán energy density spectrum adjusted by anisotropic stretching factors $\beta _i \equiv \overline {u_{i}^{2}}/ \overline {u_{1}^{2}}$ (Bertagnolio, Fischer & Zhu Reference Bertagnolio, Fischer and Zhu2014)

(4.15)\begin{equation} \phi_{22}(k_{1}, k_{3}, x_2)=\frac{4}{9 {\rm \pi}} \frac{\beta_{1} \beta_{3}}{k_{e}^{2}} \frac{( \beta_1 k_1/k_e)^{2}+(\beta_3 k_3/k_e)^{2}}{[1+(\beta_1 k_1/k_e)^{2}+(\beta_3 k_3/k_e)^{2}]^{7 / 3}} , \end{equation}

where $\beta _1 = 1$, $\beta _2 = 1/2$ and $\beta _3 = 3/4$. Here, $k_e$ is the streamwise wavenumber of the energy bearing eddies

(4.16)\begin{equation} k_{e}(x_{2})=\frac{\sqrt{\rm \pi}}{L_{1}(x_{2})} \frac{\varGamma (5/6)}{\varGamma(1/3)} , \end{equation}

where $L_1$ is the longitudinal length scale, and $\varGamma$ is the gamma function; $L_1 = 2 L_2$ under the isotropic assumption. Lastly, the vertical integral length scale $L_2(x_2)$ can be derived from the mixing length scale (Kamruzzaman et al. Reference Kamruzzaman, Herrig, Lutz, Würz, Krämer and Wagner2011)

(4.17)\begin{equation} L_{2}(x_{2})=\frac{l_{mix}(x_{2})}{\kappa} , \end{equation}

where

(4.18)\begin{equation} l_{{mix}}(x_{2})=\frac{0.085 \delta \tanh (\kappa x_2 / (0.085 \delta))}{\sqrt{1+B(x_2/\delta)^{6}}} , \end{equation}

where $\kappa = 0.38$ is the Kármán constant and $B=5$ (Stalnov et al. Reference Stalnov, Chaitanya and Joseph2016).

In summary, (4.14) provides the required input for Amiet's model (Amiet Reference Amiet1976). Here, $S_{q q}(\omega )$ is a double-sided spectrum and should be doubled if the single-sided spectrum is required. The velocity gradient $\partial U_1/\partial x_2$ is drawn from the hot-wire measurement in the near wake, at a distance very close to the trailing edge. The fluctuating velocity $\overline {u_{2}^{2}}(x_2)$ is indirectly obtained from the measurement of $\overline {u_{1}^{2}}(x_2)$ and the stretching factor $\beta _2$. In this study, an X-type hot-wire probe was not used since the spatial resolution in the $x_2$ direction is much impaired compared with the single wire probe. The resolution of the X-probe is of the order of the length of the hot-wire, while the resolution of the single wire probe is of the order of the diameter of the hot-wire.

The boundary layer thickness $\delta$ is a crucial parameter for this prediction model as it determines the length scales $L_1$ and $L_2$. In the literature, $\delta$ is mostly determined as the location where the mean flow velocity is equal to 99 % of the outer velocity. In experiments, however, this approach may lead to large experimental uncertainties, as $\partial U_1/\partial x_2$ is small near the edge of the boundary layer. Since the main function of the boundary layer thickness in the TNO-Blake model is to acquire the turbulence length scale, it is reasonable to determine the boundary layer thickness according to the distribution of the r.m.s. velocity, i.e. $(u_1)_{rms}$. It was found that, in the measurement by Stalnov et al. (Reference Stalnov, Chaitanya and Joseph2016), at 2.5 % chord length upstream of the trailing edge,

(4.19)\begin{equation} (u_1)_{rms}/U_0 = 0.0137\text{ at } x_2 = \delta . \end{equation}

In their study, the relation in (4.19) achieves better data collapse between various Reynolds numbers, compared with the approach using mean velocity distribution. In the current study, the velocity and turbulence distributions are measured at 0.5 % chord length downstream of the trailing edge. However, it can be argued that, at this streamwise location, the turbulent velocity fluctuation at the outer wake region should resemble that of the boundary layer at the close proximity of the trailing edge. Hence, we use the same value in (4.19) to define the boundary layer thickness at the trailing edge. Figure 22 shows that the current approach is more robust in various flow conditions, and can correctly reveal the region of high turbulence distribution. In addition, for the flat plate, the calculated boundary layer thickness using the new method is 4.65 mm, which is closer to the XFOIL prediction (4.48 mm), compared with the definition based on 99 % of the outer velocity (8.35 mm).

Figure 22. The boundary layer edge location defined using 99 % free-stream velocity rule (dashed line) and by (4.19) (solid line). The velocity spectrum is plotted to show the spatial distribution of the turbulence. Here, $U_0= 20$ m s$^{-1}$.

4.2.2. Comparison of noise computations with measurements

The far-field noise prediction using Blake's model is first checked with the baseline flat plate configuration, as shown in figure 23. The effect of the finite thickness of $h = 0.2$ mm is also taken into account, using the method shown in § 4.1. It is found that the bluntness correction only leads to a marginal modification of the prediction at this condition, as the trailing edge of the baseline model is relatively sharp. In general, the prediction using Blake's model is better compared with that using Goody's empirical wall pressure spectrum.

Figure 23. The comparison between prediction and measurement of the baseline flat plate trailing edge noise. The ‘current model’ is based on the Amiet theory (Amiet Reference Amiet1976) and the wall pressure wavenumber–frequency spectrum obtained from (4.12).

For coated configurations, the $x_2$ coordinate is offset by the thickness of the coating. The comparison between the predicted sound spectra and the measured sound spectra is presented in figure 24. The first model, which combines Blake's model for wall pressure spectrum estimation and Amiet's model for scattering, uses a sharp trailing edge assumption in Amiet's original model, and the second model, i.e. the current model, further takes the bluntness effect into account. It is clear that the current model can predict both the magnitude and the slope of the trailing edge noise spectra of velvet-coated configurations very well. Interestingly, for the ‘H1.0, $[-120\ -65]$’ configuration, the prediction captures the considerable low-frequency noise increase and only mild high-frequency noise reduction. The good agreement suggests that the related physics of the effect of velvety coating on trailing edge noise is well captured by the model. The major mechanism is that the wall pressure spectrum is modified due to the influence of velvety structure on the boundary layer turbulence. The secondary mechanism is the reduction of trailing edge scattering through the bluntness effect, which is accounted for using Howe's theory, as revisited in § 4.1. The peak value of the low-frequency hump is underpredicted, possibly due to the fact that the empirical relations for the boundary layer turbulence in the model are not suitable for the large coherent turbulent structures above velvety coatings (Finnigan, Shaw & Patton Reference Finnigan, Shaw and Patton2009). For the smooth coatings, the combined model can well predict the noise level below the vortex shedding frequency. The missing of vortex shedding tone in the prediction is reasonable as the effect is not considered in the model. At frequencies higher than the vortex shedding frequency, the combined model well predicts the slope of the spectrum. However, the noise level is consistently over-predicted, which is possibly due to the effect of strong vortex shedding on the movement trajectory of small eddies and thus the scattering process. Further investigation is required to improve the prediction model accuracy at these conditions.

Figure 24. The comparison between predictions and the measured trailing edge noise of coated configurations. The ‘current model’ is based on the Amiet theory (Amiet Reference Amiet1976) and the wall pressure wavenumber–frequency spectrum obtained from (4.12). Here, $U_0 = 20$ m s$^{-1}$; (a) H0.3, (b) H1.0, (c) H1.0, $[-120\ -65]$, (d) H1.0, $[-55\ 0]$, (e) H1.5, (f) S0.3, (g) S1.0 and (h) S1.5.

4.2.3. Analysis of source terms

It is now worthwhile checking which terms in (4.14) cause a major modification of the wall pressure spectrum. The approach used here is similar to the noise source characterization by Lee (Reference Lee2019). The term outside the integral is independent of the turbulent distribution. The first step is to non-dimensionalize equation (4.14), using $\delta$ as the length scale and $U_0$ as the velocity scale. We use the symbol with $\widetilde {(\cdot )}$ to represent non-dimensional values and functions. Then (4.14) can be written as

(4.20)\begin{align} S_{q q}(\tilde{\omega}) =\frac{4 {\rm \pi}\rho^{2} \beta_2 \tilde{\omega}}{1.6 (U_c/U_0)^2} U_0^3 \delta \int_{0}^{1} \tilde{L}_2(\tilde{x}_{2})\left(\frac{\partial \tilde{U}_{1}}{\partial \tilde{x}_{2}}\right)^{2} \overline{\tilde{u}_{1}^{2}}(\tilde{x}_{2}) \tilde{\phi}_{22}\left(\frac{\tilde{\omega}}{U_{c}/U_0}, 0 , \tilde{x}_2\right)\, \textrm{e}^{{-}2|\tilde{k}_1| \tilde{x}_{2}}\, \mathrm{d} \tilde{x}_{2} . \end{align}

Thus, the overall scaling factor of trailing edge noise is $U_0^3 \delta$. In this sense, increasing the boundary layer thickness will increase the overall noise level. Denoting the integral as $\tilde {A}(\tilde {\omega })$, then

(4.21)\begin{equation} S_{q q}(\tilde{\omega}) = \frac{4 {\rm \pi}\rho^{2} \beta_2 \tilde{\omega}}{1.6 (U_c/U_0)^2} U_0^3 \delta \cdot \tilde{A}(\tilde{\omega}) . \end{equation}

The integrand can be decomposed into three terms, i.e.

(4.22)$$\begin{gather} \tilde{A}(\tilde{\omega}) = \int_{0}^{1} \tilde{Q}_1(\tilde{x}_2) \tilde{Q}_2(\tilde{x}_2,\tilde{\omega}) \tilde{Q}_3(\tilde{x}_2,\tilde{\omega})\, \mathrm{d} \tilde{x}_2 , \end{gather}$$

(4.23)$$\begin{gather}\tilde{Q}_1(\tilde{x}_2) = \tilde{L}_2(\tilde{x}_{2})\left(\frac{\partial \tilde{U}_{1}}{\partial \tilde{x}_{2}}\right)^{2} \overline{\tilde{u}_{1}^{2}}(\tilde{x}_{2}) , \end{gather}$$

(4.24)$$\begin{gather}\tilde{Q}_2(\tilde{x}_2,\tilde{\omega}) = \tilde{\phi}_{22}\left(\frac{\tilde{\omega}}{U_{c}/U_0}, 0 , \tilde{x}_2\right) , \end{gather}$$

(4.25)$$\begin{gather}\tilde{Q}_3(\tilde{x}_2,\tilde{\omega}) = \textrm{e}^{{-}2|\tilde{k}_1| \tilde{x}_{2}} , \end{gather}$$

where $\tilde {\omega } = \omega \delta /U_0 = 2 {\rm \pi}f \delta /U_0$ is the non-dimensional frequency, $\tilde {x}_2 = x_2/\delta$ is the non-dimensional distance and $\tilde {k}_1 = k_1 \delta = \tilde {\omega } /(U_c/U_0)$ is the non-dimensional wavenumber. The value of $\tilde {x}_2$ stays between 0 and 1 in the boundary layer. The range of $\tilde {\omega }$ depends on the boundary layer thickness, range of interest in frequency and flow speed. The conversion between $\tilde {\omega }$ and frequency is shown in figure 25, with the boundary layer thickness listed in table 3.

Figure 25. The conversion between the frequency and the non-dimensional frequency $\omega$ for three configurations. Here, $U_0 = 20$ m s$^{-1}$.

Table 3. Comparison of boundary thickness $\delta$ (defined by (4.19)), displacement thickness $\delta ^*$ and momentum thickness $\delta _\theta$ for different test configurations. Here, $U_0 = 20$ m s$^{-1}$.

The first term $\tilde {Q}_1$ is only a function of the wall-normal distance $\tilde {x}_2$, and characterizes the spatial distribution of source strength. The distribution of $\tilde {Q}_1$ for different configurations is plotted in figure 26(a). The magnitudes are shown in dB scale, with 1 set as the reference value for the physical quantities. Since the flow is symmetric in the experiment, we only plot the distribution at $x_2>0$. Note that $\tilde {Q}_1$ is negligible near the edge of the boundary layer. This justifies the shrinkage of the integration upper limit in (4.14). It can be seen that the velvety coating can reduce the peak value of $\tilde {Q}_1$ near the wall, but increases $\tilde {Q}_1$ consistently in the mid-to-outer region of the boundary layer. The smooth coatings, on the other hand, increase the peak value of $\tilde {Q}_1$, while they do not modify the outer part of the $\tilde {Q}_1$ distribution.

Figure 26. The distribution of different non-dimensional quantities along $\tilde{x}_{2}$. (a) Non-dimensional source term $\tilde{Q}_{1}$. (b) Non-dimensional vertical integral length $\tilde{L}_{2}$. (c) The square of the non-dimensional mean shear $\left(\frac{\mathrm{d}\tilde{U}_1}{\mathrm{d}\tilde{x}_2}\right)^{2}$. (d) Non-dimensional mean square streamwise velocity fluctuation $\overline{\tilde{u}_{1}^{2}}$.

The contributing factors are clear under a further decomposition of $\tilde {Q}_1$, as shown in figure 26(b–d). Note that $\tilde {L}_2(\tilde {x}_2)$ is the same between different configurations. For the flat plate, for the smooth coating configurations as well as the ‘H1.0, $[-120\ -65]$’ configuration, there is a peak in the velocity gradient term $(\partial \tilde {U}_1/\partial \tilde {x}_2)^2$ near the wall. This location also corresponds to a high turbulence level. This peak of the velocity gradient term is much reduced if the trailing edge is covered by velvety coatings. It can be seen that the H1.5 configuration gives the lowest peak value in $(\partial \tilde {U}_1/\partial \tilde {x}_2)^2$, as a result of the smooth transition at the surface of the velvety coating. However, the velvety coatings, no matter what their locations are, lead to a relatively large velocity gradient in the mid-to-outer part of the boundary layer, compared with the flat plate and smooth coating configurations. In addition, the velvety coatings also produce a larger turbulence intensity for the majority of boundary layer locations, compared with other settings. These two factors explain why the distribution of $\tilde {Q}_1$ is larger at the mid-to-outer region of the boundary layer for the velvet coating configurations.

The second term $\tilde {Q}_2$ is the non-dimensional velocity spectrum that determines the dominant source frequency. Figure 27 shows the distribution of $\tilde {Q}_2$ with respect to the non-dimensional location $\tilde {x}_2$ and non-dimensional frequency $\tilde {\omega }$. This distribution is invariant between different configurations. The peak occurs at approximately $\tilde {x}_2 = 0.4$ and $\tilde {\omega } = 1$. When $\tilde {\omega } \gg 1$, we have

(4.26)\begin{equation} \tilde{Q}_2 \sim \tilde{\omega} ^{{-}8/3} . \end{equation}

Since we have an additional $\tilde {\omega }$ term in (4.20), we have

(4.27)\begin{equation} S_{qq}(\tilde{\omega}) \sim \tilde{\omega} ^{{-}5/3}, \text{ at } \tilde{\omega} \gg 1 , \end{equation}

which represents the inertial sub-range of the Kolmogorov turbulence spectrum, as the model is based on the von Kármán energy density spectrum.

Figure 27. The distribution of $\tilde {Q}_2$ as a function of $\tilde {\omega }$ and $\tilde {x}_2$. The step size of the contour plot is 2 dB.

The third term $\tilde {Q}_3$ is an exponential decay term that reduces the influence region for high-frequency eddies and thus lowers the peak acoustic frequency. This distribution is also invariant between different configurations. Figure 28 shows the distribution of $\tilde {Q}_3$ with respect to $\tilde {x}_2$ and $\tilde {\omega }$. The interpretation of this function is straightforward.

Figure 28. The distribution of $\tilde {Q}_3$ as a function of $\tilde {\omega }$ and $\tilde {x}_2$. The step size of the contour plot is 2 dB.

The combined effect of function $\tilde {Q}_2 \cdot \tilde {Q}_3$ is shown in figure 29. We can regard this function as a window function that is added to the source distribution $\tilde {Q}_1$, at different frequencies. Compared with $\tilde {Q}_2$, the peak of $\tilde {Q}_2 \cdot \tilde {Q}_3$ is slightly shifted to the smaller $\tilde {\omega }$ and $\tilde {x}_2$ values. For $\tilde {\omega } \lessapprox 1$ , all the boundary layer turbulence contributes to the wall pressure spectrum while, for $\tilde {\omega }\gtrapprox 1$, the contribution is only from the inner part of the boundary layer. If the source term $\tilde {Q}_1$ is mainly distributed near the wall, we expect that the high-frequency roll-off rate will be slower than that when the source term $\tilde {Q}_1$ is more spread out within the boundary layer.

Figure 29. The distribution of $\tilde {Q}_2 \cdot \tilde {Q}_3$ as a function of $\tilde {\omega }$ and $\tilde {x}_2$. The step size of the contour plot is 2 dB.

Now we can study the overall effect of $\tilde {Q} = \tilde {Q}_1 \cdot \tilde {Q}_2 \cdot \tilde {Q}_3$. To be concise, we only show $\tilde {Q}$ of representative configurations in figure 30. For the flat plate case, the peak is located at a higher frequency and close to the wall, compared the distribution of $\tilde {Q}_2 \cdot \tilde {Q}_3$, since the source term $\tilde {Q}_1$ is concentrated close to the wall. The velvety coatings strongly increase the intensity of $\tilde {Q}$ in the low-frequency range, which can be attributed to the extended region of high $\tilde {Q}_1$ value within the boundary layer. In the high-frequency range, however, the velvety coatings that cover the trailing edge reduce the intensity of $\tilde {Q}$, due to the reduced $\tilde {Q}_1$ term near the wall, which is directly related to the reduced non-dimensional velocity gradient $\partial \tilde {U}_1/\partial \tilde {x}_2$ and turbulence intensity $\overline {\tilde {u}_1^2}$ in the near-wall region. In comparison, for the S0.3 and S1.5 configurations, the $\tilde {Q}$ distribution is similar to that of the flat plate, except in the region close to the wall, where $\tilde {Q}$ is elevated by the presence of strong shear flows.

Figure 30. The distribution of $\tilde {Q}$ as a function of $\tilde {\omega }$ and $\tilde {x}_2$, for the baseline configuration and different coated configurations. The step size of the contour plot is 2 dB.

From the spatial distribution of $\tilde {Q}$, we also expect a large difference in the high-frequency roll-off rate of the $S_{qq}$ among different configurations. For the H1.5 case, as the major source region is distributed near $\tilde {x}_2 \approx 0.3$, after integration over $\tilde {x}_2$, the high-frequency roll-off is affected by both $\tilde {Q}_2$ and $\tilde {Q}_3$. In contrast, for the flat plate, $\tilde {Q}$ is concentrated near the wall such that the decay from the exponential decay factor $\tilde {Q}_3$ is less prominent, and the high-frequency roll-off rate is mainly due to the term $\tilde {Q}_2$, except at very high frequencies. The non-dimensional integral $\tilde {A}(\tilde {\omega })$ introduced in (4.21) is shown in figure 31. The slope of the spectrum at the intersection between the velvet-coated cases and the flat plate case ($\tilde {\omega } \approx 5$) is clearly different.

Figure 31. The non-dimensional integral $\tilde {A}(\tilde {\omega })$ in (4.21).

In (4.20), the non-dimensional surface pressure spectrum is also proportional to the boundary layer thickness $\delta$. The dimensional surface pressure spectrum is proportional to $\delta ^2$ since there is an $\tilde {\omega }$ term in front of the integral. In Amiet's model, with the flow speed and frequency specified, the far-field noise spectrum is directly proportional to the dimensional surface pressure spectrum, thus $\delta ^2$. In the experiment, the boundary layer thickness is at maximum doubled by the attachment of the H1.5 velvety coating. This would result in a 6 dB increase of far-field noise spectrum according to this scaling, and would not change the slope of the noise spectrum. Therefore, the modification of the non-dimensional velocity distribution and turbulence distribution within the boundary layer is regarded as the major cause for the large modification of the wall pressure spectrum and the associated trailing edge noise. This can explain the good collapse of ${\rm \Delta} PSD$ against Strouhal number in the acoustic measurements in figure 12. The slope in figure 12 near ${\rm \Delta} PSD =0$ is well predicted for the cases of velvety coatings.

Lastly, in the bird flyover noise measurement conducted by Sarradj et al. (Reference Sarradj, Fritzsche and Geyer2011), the authors found that the roll-off of in the 1/3 octave-band spectrum is 15 dB decade$^{-1}$ for the Barn Owl, but only 10 dB decade$^{-1}$ for other birds. Our analysis indicates that this is achievable by moving the boundary layer source term $\tilde {Q}_1$ away from the wall. This might be a potential explanation for the experimental finding by Sarradj et al. (Reference Sarradj, Fritzsche and Geyer2011).