2.1. Set-up

Flow, pressure and force measurements were carried out to assess the influence of three non-dimensional parameters controlling the tip leakage flow and associated noise sources: the geometric angle of attack $\alpha$ , the ratio of the maximum aerofoil thickness to gap size $\tau _{\textit{max}}/e$ and the ratio of gap size to boundary-layer thickness $e/\delta$ . To achieve this, configurations with aerofoils of different maximum thicknesses, gap sizes and angles of attack were analysed.

The set-up follows a similar design developed by Grilliat et al. (Reference Grilliat, Jacob, Camussi and Caputi-Gennaro2007). It consists of an interchangeable aerofoil, with a chord length of $c=200$ mm, positioned in the potential core of an open-jet wind tunnel between two Plexiglas plates, which are fixed at the top and the bottom of the nozzle exit. This set-up also allows for adjustments in the gap size $e$ between the bottom wall and the aerofoil tip, as well as the geometric angle of attack $\alpha$ , defined as the angle between the aerofoil chord line and the inflow direction. The coordinate system used hereafter is based on the inflow direction: the $x$ axis is aligned with it, pointing from the aerofoil leading edge to the trailing edge; the $y$ axis is in the cross-stream direction, oriented from the pressure to the suction side; and the $z$ axis is oriented upwards from the bottom wall. The inflow velocity $U_0$ was set to $40$ m s^–1, yielding a chord-based Reynolds number of ${Re}_c = 5.6 \times 10^5$ and a Mach number of $M = 0.12$ . This choice was supported by previous results showing similar flow behaviour across the range $20$ – $70$ m s^–1.

Table 1 lists all the configurations analysed, obtained by combining six different NACA ‘XXXX’ aerofoils with various gap sizes $e$ (in mm) to achieve a specific thickness-to-gap ratio $\tau_{max}/e$ . The NACA four-digit series defines the aerofoil profile as follows: the first digit represents the maximum camber as a percentage of the chord, the second digit indicates the position of maximum camber in tenths of the chord and the last two digits specify the maximum thickness of the aerofoil as a percentage of the chord (Anderson Reference Anderson2001). A wide range of geometric angles of attack, $\alpha =5^\circ , 7^\circ ,\ 10^\circ ,\ 12^\circ ,\ 15^\circ ,\ 17^\circ ,\ 20^\circ$ , was considered. Analysing the effect of the boundary layer has been challenging due to the limitations of the current set-up. However, to address this, a duct of $2$ m in length was added between the nozzle and the aerofoil, which increased the thickness of the turbulent boundary layer from $5$ to approximately $20$ mm, as shown in figure 3. This dimension was measured using a hot-wire at the location of the aerofoil leading edge, which had been removed.

Table 1. List of all of the configurations analysed.

Figure 3. Streamwise velocity profile measured at the leading edge of the aerofoil for $U_0=40$ m s^–1.

The experiments were conducted at the Institute of Sound and Vibration Research’s open-jet wind tunnel facility within an anechoic chamber measuring $8$ m $\times$ $8$ m $\times$ $8$ m. The chamber’s walls are acoustically treated with glass wool wedges, whose cut-off frequency is about $80$ Hz. The nozzle has dimensions of $150$ mm $\times$ $450$ mm and a contraction ratio of $25:1$ . To ensure that the acoustic signature of the tip leakage noise can be clearly captured, the background noise remains at least $10$ dB below the aerofoil self-noise, preventing contamination from facility noise, and the free-stream turbulence intensity is kept below $0.4\,\%$ in the potential core to minimise aerofoil–turbulence interaction noise. Further details of the facility are provided by Chong, Joseph & Davies (Reference Chong, Joseph and Davies2009).

2.2. Type of measurements

2.2.1. Velocity

Figure 4. Schematic representations of the regions assessed using the hot-wire: (a) within the gap and (b) in the suction-side region.

Flow measurements were conducted to characterise the tip leakage flow within the gap and the TLV developing in the suction-side region of the aerofoil. The velocity field was assessed in a midgap plane and at different chordwise sections in the suction side using the hot-wire anemometry technique, as shown in figures 4(a) and 4(b). In both cases, the single-wire probe was oriented parallel to the $x$ axis to measure the resultant velocity in the $x{-}z$ plane, i.e. $\sqrt {u^2+w^2}$ . Hot-wire anemometry has known limitations in regions of separated flow. We specifically targeted the low-velocity region within the gap, which we previously found to be associated with vortex-shedding instability responsible for the first tip noise source (Saraceno et al. Reference Saraceno, Chaitanya, Jaiswal, Palleja-Cabre, Long and Ganapathisubramani2024b ). In the suction-side region, the measurements aimed to localise the TLV trajectory, as Jacob et al. (Reference Jacob, Jondeau, Li and Boudet2016b ) observed maximum axial velocities in the vortex core, further supporting the usefulness of the chosen probe orientation.

A single hot-wire probe (Dantec type 55P11) was operated using a standard constant-temperature anemometry system (AA Lab System) operated at an over-hear ratio of $1.5$ . It was calibrated in the free stream using a standard Pitot tube, with a fourth-order polynomial applied as a calibration function. The probe acquired data at each grid point for $10$ s at a sampling frequency of $50$ kHz, ensuring convergence of the mean statistics.

2.2.2. Pressure

Steady and unsteady pressure measurements were conducted in both near-field and far-field regions to evaluate the pressure distribution along the aerofoil and the far-field noise spectra. The pressure distribution $C_p$ over the aerofoil was evaluated by connecting surface pinholes to a pressure scanner with thin capillary tubes. These pressure taps are located at midspan and $1$ mm above the aerofoil tip on both aerofoil sides, at $12$ different locations along the chord. The formula used to evaluate $C_p$ is as follows:

(2.1) \begin{equation} C_p(x)=\frac {p(x)-p_\infty }{0.5 \rho _\infty U_0^2}, \end{equation}

where $p_\infty$ is the pressure measured without flow, $\rho _\infty$ is the density and $U_0$ is the inflow velocity. The lift coefficient $C_l$ is obtained by numerically integrating the pressure distributions along the chord:

(2.2) \begin{equation} C_l=\int ^c_0{[C_{p_{p}}(x)-C_{p_{s}}(x)}] {{\rm d}(x)}, \end{equation}

where $C_{p_{p}}(x)$ and $C_{p_{s}}(x)$ represent the pressure distribution along the pressure and suction sides of the aerofoil, respectively.

Far-field noise measurements were performed using four half-inch condenser microphones (B&K type $4189$ ) located at a radial distance of $1.5$ m from the aerofoil midchord. Two microphones were placed on the suction side and two on the pressure side at polar angles of $\theta =\pm 90^\circ$ and $\theta =\pm 40^\circ$ relative to the inflow direction. However, only data from the pressure side microphone at $\theta =-90^\circ$ , where noise levels were highest, are reported. Far-field pressure signals were acquired for $10$ s at a sampling frequency of $50$ kHz. The noise spectra are calculated with Welch’s method, using a window size of $1024$ data points with no overlap. This corresponds to a frequency resolution of $48.83$ Hz and a bandwidth–time product of $488.3$ , which is sufficient to ensure a $9\,\%$ uncertainty with a $95\,\%$ confidence level (Glegg & Devenport Reference Glegg and Devenport2017).

2.2.3. Aerodynamic forces

A load cell was mounted to the aerofoil to measure the forces acting on it. Specifically, an F/T Sensor Mini40 is mounted between two steel brackets. Four adjustable bolts regulate the gap size and set the position of the upper bracket. The lower bracket is attached to the aerofoil but can move freely. This set-up ensures that the lower bracket transfers these forces to the load cell for measurement as the aerofoil moves under aerodynamic forces.

The forces of interest are lift $L$ and drag $D$ , which are non-dimensionalised as follows:

(2.3) \begin{align} C_L&=\frac {L}{0.5 \rho _\infty U_0^2 S}, \end{align}

(2.4) \begin{align} C_D&=\frac {D}{0.5 \rho _\infty U_0^2 S}, \end{align}

Figure 5. (a) Pressure distributions measured at midspan for $\alpha =15^\circ$ in the current work, compared with other results in the literature (Jacob et al. Reference Jacob, Grilliat, Camussi and Gennaro2010; Koch, Sanjosé & Moreau Reference Koch, Sanjosé and Moreau2021), and the numerical prediction from XFOIL for $\alpha _{\textit{eff}}=5^\circ$ . (b) Variation of the ratio $\alpha _{\textit{eff}}/\alpha$ with the geometric angle of attack $\alpha$ .

where $S$ is the relevant surface area, calculated as the product of the aerofoil chord and the span length, which varies with the gap size. The load cell measures two forces: one perpendicular to the aerofoil chord and the other along it, $F_1$ and $F_2$ , respectively. The lift and drag are then calculated from $F_1$ and $F_2$ as

(2.5) \begin{align} L&=F_1 \cos (\alpha _{\textit{eff}}) - F_2 \sin (\alpha _{\textit{eff}}), \end{align}

(2.6) \begin{align} D&=F_1 \sin (\alpha _{\textit{eff}}) + F_2 \cos (\alpha _{\textit{eff}}), \end{align}

where $\alpha _{\textit{eff}}$ is the effective angle of attack of the aerofoil, which is lower than the geometric angle of attack $\alpha$ due to the wind tunnel deflection (Glegg & Devenport Reference Glegg and Devenport2017). The jet can be deflected due to the presence of an aerofoil, leading to a reduction in the effective angle experienced by the aerofoil. This effective angle can be estimated by matching the experimental pressure distribution $C_p$ , measured along the midspan where the tip flow effects are negligible (as shown in § 3), with the numerical predictions obtained using XFOIL (Drela Reference Drela1989). Figure 5(a) compares the experimental $C_p$ at $\alpha =15^\circ$ with the best matching numerical prediction obtained for $\alpha _{\textit{eff}}=5^\circ$ , along with additional measurements from the literature (Jacob et al. Reference Jacob, Grilliat, Camussi and Gennaro2010; Koch et al. Reference Koch, Sanjosé and Moreau2021). The ratio $\alpha _{\textit{eff}}/\alpha$ , shown in figure 5(b), exhibits a sharp increase between $\alpha = 2^\circ$ and $5^\circ$ , and then increases slightly for higher angles, assuming values of around $1/3$ .

3.1. Tip leakage vortex detachment on tip-separated flow

Figure 6. (a) Pressure distributions $C_p$ along the aerofoil measured at the tip for two configurations with gap sizes $e=0$ and $10$ mm, and at midspan for $e=10$ mm. The angle of attack is set to $\alpha =15^\circ$ . (b) Pressure distribution $C_p$ measured at the tip for three configurations with $\alpha =10^\circ ,\ 15^\circ$ and $20^\circ$ . The gap size is set to $e=10$ mm, corresponding to $\tau _{\textit{max}}/e=2$ .

The pressure distributions $C_p$ along the aerofoil measured close to the tip for two gap sizes $e=0$ and $10$ mm, and same angle of attack $\alpha =15^\circ$ , are shown in figure 6(a), together with that evaluated at midspan for $e=10$ mm. In the presence of the gap, the pressure distribution near the tip changes significantly compared with that at midspan, which corresponds to the loading at the tip without clearance in agreement with previous studies (Jacob et al. Reference Jacob, Grilliat, Camussi and Gennaro2010; Koch et al. Reference Koch, Sanjosé and Moreau2021). This indicates that the gap mainly influences the pressure distribution near the tip edge. The loading on the pressure side tends to be lower in the presence of a gap, particularly upstream of the midchord. On the suction side, the pressure decreases downstream of the leading edge reaching a peak around the $60\,\%$ of the chord, followed by a subsequent drop. The magnitude of the peak and its location along the chord vary with changes in the angle of attack, as shown in figure 6(b). Specifically, when the angle of attach decreases from $20^\circ$ to $10^\circ$ , the magnitude of the peak on the suction side diminishes, and the drop shifts downstream, from midchord to $75\,\%$ of the chord.

Figure 7. Mean velocity fields of the resultant velocity in the $x{-}z$ plane measured for three configurations with $\alpha =10^\circ$ (a), $15^\circ$ (b) and $20^\circ$ (c) in different chordwise sections: at midchord, at $75\,\%$ of the chord and at $2$ mm downstream the trailing edge. The gap size is set to $e=10$ mm, corresponding to $\tau _{\textit{max}}/e=2$ .

Previous studies linked the loading on the aerofoil suction side to the TLV (Graham Reference Graham1986; Kang & Hirsch Reference Kang and Hirsch1993), which remains near the aerofoil surface. The pressure peak, followed by the drop, occurs when the vortex moves away from the surface (Intaratep Reference Intaratep2006). According to figure 6(b), this vortex detachment shifts progressively upstream with the increase of the angle of attack. This behaviour is qualitatively shown in figure 7, which displays the resultant velocity contained in the $x{-}z$ plane measured by the hot-wire probe in different chordwise sections, precisely at $x/c=50\,\%$ , $75\,\%$ and at $2$ mm downstream of the trailing edge, for the three configurations with $\alpha =10^\circ ,\ 15^\circ$ and $20^\circ$ . The resultant velocity reaches maximum values in the vortex core. The location of this maximum velocity indeed matches with the centre of the TLV identified by Jacob et al. (Reference Jacob, Jondeau and Li2016a ) at the same chordwise section located downstream of the trailing edge, for the configuration with $\alpha =15^\circ$ shown in figure 7(b). For this configuration, the trajectory of the TLV can be qualitatively inferred from the three sections, showing that it is almost aligned with the inflow direction, consistent with other works (Jacob et al. Reference Jacob, Jondeau, Li and Boudet2016b ; Koch et al. Reference Koch, Sanjosé and Moreau2021). The vortex core remains close to the aerofoil suction side up to the midchord, and then it starts to move away, as expected from its pressure distribution () in figure 6(b). For $\alpha =10^\circ$ in figure 7(a), the vortex core remains close to the aerofoil suction side up to $75\,\%$ of the chord, whereas for $\alpha =20^\circ$ in figure 7(c), the core begins to move away from the surface around the midchord. In summary, the relationship between the TLV and the pressure distributions on the aerofoil suction side is clear, with the location of vortex detachment shifting as the angle of attack changes. Furthermore, the vortex is located near the aerofoil tip, so it does not influence the pressure at the midspan. As a result, the pressure distribution at the midspan remains similar to that of the configuration without a gap, as seen in figure 6(a).

Figure 8. Mean velocity fields of the resultant velocity in the $x{-}z$ plane measured within the gap for three configurations with $\alpha =10^\circ$ (a), $15^\circ$ (b) and $20^\circ$ (c). The gap size is set to $e=10$ mm, corresponding to $\tau _{\textit{max}}/e=2$ . The colour map is chosen to maintain consistent colour representation of velocity values with figure 7.

Hot-wire measurements were used to trace the vortex core in the suction-side area of the aerofoil and evaluate the flow within the gap. The tip flow exhibits low velocity values at midchord for the configuration with $\alpha =20^\circ$ in figure 7(c), and at $75\,\%$ of the chord for $\alpha =15^\circ$ and $20^\circ$ , in figures 7(b) and 7(c). Figure 8 displays this characteristic more clearly, showing the flow fields in the midgap plane for these three configurations with $\alpha =10^\circ ,\ 15^\circ$ and $20^\circ$ . The low-velocity regions, found for each case in figure 8, arise downstream of the pressure peaks identified in the pressure distributions in figure 6(b). The low-velocity region shifts upstream as the angle of attack increases, following the same shifting trend as the pressure peaks. Given the connection between the pressure peak and low-velocity region, as well as between the peak and vortex detachment, it can be suggested that the TLV plays a role in the development of the low-velocity region.

Saraceno et al. (Reference Saraceno, Chaitanya, Ganapathisubramani, Alguacil, Moreau and Sanjose2024a ) used large-eddy simulations to investigate two chordwise sections of the configuration with $\alpha =15^\circ$ , at midchord and at $75\,\%$ of the chord, of the configuration with $\alpha =15^\circ$ . At midchord, the flow was found to separate at the tip but to reattach before leaving the gap. At $75\,\%$ of the chord, the tip flow remained separated until the exit of the gap and low axial velocity was measured. This low-velocity region corresponds to those observed in figures 7 and 8. Figure 7 also indicates that the extended separated flow occurs when the TLV is away from the aerofoil suction side. A similar trend was observed by Liu et al. (Reference Liu, Wang, Tan, Tucker and Moller2024), who found an extended separation developed within the gap downstream of the TLV detachment from the aerofoil surface, while upstream, the flow remained steady and attached. This trend suggests that when the TLV is close to the aerofoil surface, its low pressure can influence the separated flow to reattach to the aerofoil tip. However, as the TLV moves away, this effect vanishes, and the tip flow stays separated until it exits the gap. Figure 8 also shows that the velocity values within the low-velocity region decrease as the angle of attack increases. This behaviour appears to be correlated with the intensity of the pressure peaks: as the pressure peak magnitude increases, the vortex is characterised by lower pressure, which may lead to a stronger separation upon its detachment.

Figure 9. Lift coefficient $C_l$ plotted against the non-dimensionalised gap size $\tau _{\textit{max}}/e$ , obtained for different angles of attack. The markers refer to the angles of attack: plus, $5^\circ$ ; circle, $7^\circ$ ; diamond, $10^\circ$ ; cross, $12^\circ$ ; star, $15^\circ$ ; square, $17^\circ$ .

Figure 9 shows the lift coefficient $C_l$ as a function of the angle of attack $\alpha$ for different gap sizes $e$ . The gap size has been non-dimensionalised using the maximum thickness of the $5510$ aerofoil, i.e. $\tau _{\textit{max}}=20$ mm. The lift coefficient remains nearly constant as the gap size varies and for a fixed angle of attack. This is unexpected given that the pressure distributions with and without the gap differ significantly, as shown in figure 6(a). Based on this figure, it can be hypothesised that, to keep the lift coefficient constant, the loading on the pressure side is reduced in the presence of the gap to counterbalance the loading on the suction side caused by the TLV. However, the effect of the gap size on the lift coefficient becomes evident for large gaps ( $\tau _{\textit{max}}/e\gt 2$ ) and small angles of attack ( $\alpha \lt 12^\circ$ ), resulting in a significant reduction in lift.

3.2. First tip noise source

Figure 10. Tip leakage noise obtained by subtracting the sound pressure levels measured with and without the gap, for the three configurations with $\alpha =10^\circ$ , $15^\circ$ and $20^\circ$ and gap size $e=10$ mm, corresponding to $\tau _{\textit{max}}/e=2$ .

The focus on the low-velocity region is due to its association with a vortex-shedding-type phenomenon, which was identified as responsible for one of the tip leakage noise sources (Saraceno et al. Reference Saraceno, Chaitanya, Ganapathisubramani, Alguacil, Moreau and Sanjose2024a , Reference Saraceno, Chaitanya, Jaiswal, Palleja-Cabre, Long and Ganapathisubramanib ). This noise source was linked to the spectral hump found in figure 10, within the non-dimensional frequency range $St_c=fc/U_0=2{-}5.5$ . The curves plotted in this figure are obtained by subtracting the far-field spectra obtained with and without the gap to highlight the noise increase due to the tip leakage noise. This subtraction partially removes the contribution of other noise sources, such as background noise and leading- and trailing-edge noise, since these are expected to be similar with and without the gap. As the angle of attack decreases from $20^\circ$ to $10^\circ$ , the spectral hump narrows and its peak shifts towards higher frequencies. This behaviour was associated with the location where the low-velocity region starts to appear. As a result of the previous considerations, the first tip noise source depends on the TLV detachment location.

Figure 11. Overall sound pressure levels evaluated in the frequency range of the first tip noise source against the maximum difference between $C_{p_{p}}$ and $C_{p_{s}}$ , for different configurations characterised by $\tau _{\textit{max}}/e=1,\ 2$ and $3$ (a), and $\tau _{\textit{max}}/e=4,\ 5$ and $6$ (b). Tip leakage noise for different configurations with $\alpha =10^\circ$ (c) and $15^\circ$ (d). The markers refer to the angles of attack: circle, $7^\circ$ ; diamond, $10^\circ$ ; cross, $12^\circ$ ; asterisk, $15^\circ$ ; square, $17^\circ$ ; star, $20^\circ$ .

The peaks on the pressure distributions along the suction side have been linked to the flow separation: a higher peak indicates stronger separation, which may lead to more intense vortex-shedding phenomena responsible for the first tip noise source, explaining the louder spectral peak observed in figure 10 as the angle of attack increases. This relationship between pressure peak and noise strength is further shown in figure 11(a), where the spectral curves, integrated within the frequency range of the first noise source, are plotted as a function of the maximum pressure peak of the pressure distribution along the aerofoil suction side, $C_{p_{s}}$ , on the abscissa. Specifically, the abscissa has been ‘normalised’ by considering the maximum difference between $C_{p_{p}}$ and $C_{p_{s}}$ , to account for the effect of the TLV on the pressure distribution of aerofoil pressure side $C_{p_{p}}$ , as suggested previously. The blue data points in figure 11(a) correspond to the configurations with $e=10$ mm, i.e. $\tau _{\textit{max}}/e=2$ , for different angles of attack. The strength of the first noise source exhibits a sudden increase when passing from $\alpha =7^\circ$ to $10^\circ$ , followed by a rise that reaches its maximum at $\alpha =20^\circ$ . Beyond this point, stall occurs at $\alpha =30^\circ$ , as reported by Saraceno et al. (Reference Saraceno, Chaitanya, Jaiswal, Palleja-Cabre, Long and Ganapathisubramani2024b ). The other cases shown in figure 11(a) with $\tau _{\textit{max}}/e=1$ and $3$ display the same trend, albeit with lower maximum intensity. On the contrary, the points for the cases with $4\leqslant \tau _{\textit{max}}/e \leqslant 6$ mainly cluster below the dotted line in figure 11(b).

The clustering below the dotted line, along with the sudden drop in the noise strength at small angle of attack ( $\alpha \lt 10^\circ$ ), is associated with the absence of the first noise source. The dotted line is defined based on the observation that the case with $\tau _{\textit{max}}/e=4$ and $\alpha =15^\circ$ was found to exhibit neither the first noise source nor extended separated flow (Saraceno et al. Reference Saraceno, Chaitanya, Jaiswal, Palleja-Cabre, Long and Ganapathisubramani2024b ). Figure 11(d) shows the tip leakage noise obtained for this case (), along with those obtained for $1\leqslant \tau _{\textit{max}}/e \leqslant 6$ . The spectral humps remain negligible, below $5$ dB, for $\tau _{\textit{max}}/e\geqslant 4$ . Other data points are found along the dotted line for $\alpha =10^\circ$ in figures 11(a) and 11(b), and their tip noise spectra are shown in figure 11(c). The case with $\tau _{\textit{max}}/e=2$ () is characterised by a shallow spectral peak and low-velocity region, as seen in figure 8(a). This hump decreases as $\tau _{\textit{max}}/e$ increases, approaching $5$ dB for $\tau _{\textit{max}}/e=4$ , and eventually becomes negligible, below $5$ dB, for higher values. In summary, the first noise source does not develop for $\tau _{\textit{max}}/e\geqslant 4$ and $\alpha \lt 10^\circ$ . The reason why the data for $\tau _{\textit{max}}/e\geqslant 4$ collapse below the dotted line will be shown to be linked to the ratio between the aerofoil thickness and the gap size, despite these configurations having their tips immersed within the boundary layer. The absence of the first noise source for $\alpha \lt 10^\circ$ is attributed to the lack of extended separated flow, which may result from the vortex detachment occurring even further downstream of $75\,\%$ of the chord. Additionally, it should be considered that the tip flow changes as the angle of attack decreases, with a much weaker cross-stream flow forming even further downstream of $75\,\%$ , as observed by Jacob et al. (Reference Jacob, Grilliat, Camussi and Gennaro2010) for the configuration with $\alpha =5^\circ$ and $\tau _{\textit{max}}/e=2$ .

3.3. Second tip noise source

Figure 12. Overall sound pressure levels evaluated in the frequency range of the second noise source against the lift coefficient $C_l$ measured at the tip for different configurations characterised by $\tau _{\textit{max}}/e=1,\ 2$ and $3$ (a), and $\tau _{\textit{max}}/e=4,\ 5$ and $6$ (b). The markers refer to the angles of attack: plus, $5^\circ$ ; circle, $7^\circ$ ; diamond, $10^\circ$ ; cross, $12^\circ$ ; asterisk, $15^\circ$ ; square, $17^\circ$ ; star, $20^\circ$ .

The second tip noise source was associated with a shear-layer roll-up (Saraceno et al. Reference Saraceno, Chaitanya, Ganapathisubramani, Alguacil, Moreau and Sanjose2024a , Reference Saraceno, Chaitanya, Jaiswal, Palleja-Cabre, Long and Ganapathisubramanib ), which develops within the gap due to the separation of the flow at the pressure side tip edge. Figure 10 identifies the noise increase due to this fluid-dynamic instability within $St_c=5.5{-}13$ . As the angle of attack increases, the levels rise reaching the greatest values for the configuration with $\alpha =20^\circ$ . The integration of the curves in figure 10 over the corresponding frequency range of the second noise source results in the scatter plot of figures 12(a) and 12(b), where the abscissa represents lift coefficient $C_l$ , obtained by integrating the pressure distributions along the chord. These figures have been enhanced with additional data points obtained by varying the angle of attack, between $5^\circ$ and $20^\circ$ , and gap size, in the range $1\leqslant \tau _{\textit{max}}/e \leqslant 6$ . Specifically, the cases with $\tau _{\textit{max}}/e\leqslant 3$ , also characterised by the aerofoil tip outside the boundary layer $e/\delta \gt 1$ , are shown in figure 12(a), while the others are presented in figure 12(b). Figure 12(a) shows a proportionality between the strength of the second tip noise source and the lift coefficient. In figure 12(b), for $e/ \delta \lt 1$ , the data appear slightly more dispersed but still show an increase of the noise source with the lift coefficient. A detailed analysis of these data, including the effects of the casing boundary layer, is conducted in the next sections.

The proportionality, shown in figure 12(a), between the overall loading acting on the aerofoil tip and the strength of the second tip noise source is not straightforward. This noise source was linked to the shear-layer roll-up, which develops from the separation of the flow entering the gap at the pressure side edge. The roll-up is due to the Kelvin–Helmholtz instability, which arises from a velocity difference across the interface of two parallel fluid streams (Kundu, Cohen & Dowling Reference Kundu, Cohen and Dowling2016). In the tip flow scenario, these streams are represented by the separated flow near the aerofoil tip and a high-speed jet flow close to the bottom wall, as depicted in the zoomed-in aerofoil cross-section in figure 2 and observed by Saraceno et al. (Reference Saraceno, Chaitanya, Jaiswal, Palleja-Cabre, Long and Ganapathisubramani2024b ) and Storer & Cumpsty (Reference Storer and Cumpsty1991). Since the separated flow near the aerofoil has negligible velocity, the roll-up mechanism is primarily influenced by the flow velocity near the wall, $V_L$ . A way to estimate of this velocity, normal to the chamber line, was provided by Rains (Reference Rains1954), using the following expression: $V_L/U_0=\sqrt {C_{p_{p, m}} - C_{p_{s}}}$ , where $C_{p_{p, m}}$ is the pressure coefficient measured at midspan on the aerofoil pressure side and $C_{p_{s}}$ is that measured at the tip on the suction side. This formula was derived from the ideal model, proposed by Rains (Reference Rains1954), which applies the Bernoulli equation under the assumption of an ideal, unattached flow entering the gap normal to the chamber line. Consequently, as noted by Storer & Cumpsty (Reference Storer and Cumpsty1991), this expression provides a good estimate of the flow velocity when applied to sections downstream of the minimum pressure on the suction side, where the flow tends to be unattached.

Figure 13. Tip flow velocity $V_L$ plotted against the lift coefficient $C_l$ , obtained for $\tau _{\textit{max}}/e=1,\ 2$ and $3$ . The markers refer to the angles of attack: plus, $5^\circ$ ; circle, $7^\circ$ ; diamond, $10^\circ$ ; cross, $12^\circ$ ; asterisk, $15^\circ$ ; square, $17^\circ$ ; star, $20^\circ$ .

In this study, the flow velocity $V_L$ is evaluated by considering the pressure coefficients measured at the location of the minimum pressure peak and at the aerofoil tip of both sides, using this formula $V_L/U_0=\sqrt {C_{p_{p}} - C_{p_{s}}}$ , where $C_{p_{p}}$ is the pressure coefficient measured at the pressure side tip. The scatter plot of figure 13 displays $V_L$ plotted against the lift coefficient $C_l$ , obtained by varying the angle of attack for all the configurations with the aerofoil tip outside the boundary layer to satisfy the assumption of ideal flow, i.e. $e/\delta \gt 1$ corresponding to $1\leqslant \tau _{\textit{max}}/e \leqslant 3$ . The data show a linear relationship between $V_L$ and $C_l$ , which is evident for $\alpha \geqslant 10^\circ$ , but not for lower angles of attack, $\alpha \lt 10^\circ$ . This discrepancy may be due to the method of evaluating $V_L$ , which relies on pressure coefficients measured at the same chordwise location on both aerofoil sides, potentially causing inaccuracies in the velocity estimation at low angles of attack. At low angles of attack, the pressure tap locations almost align with the cross-stream direction, providing an estimate of the cross-stream flow, while the actual tip flow has both stream and cross-stream components, $V_{L,x}$ and $V_{L,y}$ , as shown in figure 2. Additionally, at low angle of attack, the cross-stream flow is weaker, as observed by Jacob et al. (Reference Jacob, Grilliat, Camussi and Gennaro2010). As the angle of attack increases, the tip flow changes with a higher cross-stream component and the alignment of the pressure taps with the actual tip flow direction improves, leading to a more accurate estimation of $V_L$ .

This section illustrates a linear relationship between the lift coefficient and the tip flow velocity, without claiming an exact value of $V_L$ , due to the several assumptions inherent in the Rains model. This relationship helps to explain the connection between the lift coefficient and the strength of the second noise source observed in figure 12. As the lift coefficient increases, the greater pressure difference between the suction and pressure side results in a faster flow close to the bottom wall, enhancing the shear within the gap. This results in stronger shear-layer vortices, which in turn induce larger unsteady aerodynamic forces and, consequently, a more intense tip noise source. In § 4, a power relationship between lift coefficient and strength of the second noise source is presented.

4.1. First tip noise source

Figure 14. (a) Tip leakage noise and (b) pressure distributions $C_p$ along the aerofoil measured at the tip for the two configurations with $\tau _{\textit{max}}/e=2$ , $4$ and angle of attack set to $\alpha =12^\circ$ , $15^\circ$ .

Two different NACA aerofoils, $5505$ and $5510$ with $\tau _{\textit{max}}=10$ and $20$ mm, at different angles of attack $\alpha =12^\circ$ and $15^\circ$ , were analysed in detail. Both featured a gap size of $e=5$ mm, corresponding to $e/\delta =1$ . Figure 14 displays the tip leakage noise and the pressure distribution $C_p$ along the chord obtained for these two configurations with $\tau _{\textit{max}}/e=2$ and $4$ . Although the pressure distributions are nearly identical, only the configuration with $\tau _{\textit{max}}/e=2$ exhibits a distinct spectral hump in the frequency range of the first noise source, $St_c=2{-}5.5$ , with a peak of $11$ dB. This noise source suggests the development of a low-velocity region, expected to form at the location where the vortex detaches. According to the pressure distributions in figure 14(b), the vortex detaches from the aerofoil surface around $25\,\%$ of the chord, corresponding to the location of the maximum aerofoil thickness. This detachment is confirmed in figure 15, which displays the resultant velocity in the $x{-}z$ plane measured at the midchord and $75\,\%$ of the chord. In both configurations, the trajectory of the vortex core, identified by maximum velocity values, is located further from the aerofoil surface in those sections downstream of $25\,\%$ of the chord. At $25\,\%$ of the chord, the difference between the two configurations is the thickness-to-gap ratio: $2$ for the $5505$ aerofoil and $4$ for the $5510$ profile. The thickness-to-gap ratio determines whether the tip flow remains separated or reattaches to the aerofoil: for $\tau _{\textit{max}}/e=2$ , the flow remains separated leading to the formation of the first noise source; for $\tau _{\textit{max}}/e=4$ , the flow reattaches preventing the formation of the low-velocity region. This aligns with the findings of Saraceno et al. (Reference Saraceno, Chaitanya, Jaiswal, Palleja-Cabre, Long and Ganapathisubramani2024b ), which showed the absence of the low-velocity region within the gap for the case with $\tau _{\textit{max}}/e=4$ .

Figure 15. Mean velocity fields of the resultant velocity in the $x{-}z$ plane measured for two configurations with $\tau _{\textit{max}}/e=4$ , $\alpha =15^\circ$ (a) and $\tau _{\textit{max}}/e=2$ , $\alpha =12^\circ$ (b) in different chordwise sections, midchord and $75\,\%$ of the chord. Different aerofoils characterise these configurations, NACA $5510$ and $5505$ .

Figure 16. Overall sound pressure levels evaluated in the frequency range of the first tip noise source against the maximum difference between $C_{p_{p}}$ and $C_{p_{s}}$ , for different configurations characterised by $\tau _{\textit{max}}/e=2$ (a), $3$ (b) and $4$ (c). The markers refer to the aerofoils: plus, $5505$ , circle, $5507$ , asterisk, $5510$ ; cross, $5512$ ; square, $5515$ . The colours refer to the angles of attack: $7^\circ$ (), $10^\circ$ (), $12^\circ$ (), $15^\circ$ (), $17^\circ$ (), $20^\circ$ ().

The two cases analysed in this section were selected to show the influence of the thickness-to-gap ratio on the development of the first tip noise source, as they are characterised by similar pressure distribution along the chord, resulting in the same vortex detachment position. Figures 11(a) and 11(b) have been expanded by considering different aerofoils, obtaining the three plots shown in figure 16. In figure 16(a), the data characterised by $\tau _{\textit{max}}/e=2$ follow a proportional trend above the dotted line, but appear sparse below it, as expected, since the first noise source tends to disappear for small angles of attack ( $\alpha \lt 10^\circ$ ). A similar trend is observed for the cases with $\tau _{\textit{max}}/e=3$ , shown in figure 16(b), with lower values of the maximum strength of the noise source. Almost all data points are found below the dotted line for the cases with $\tau _{\textit{max}}/e=4$ in figure 16(c). Therefore, we can conclude that the first tip noise source can develop when $\tau _{\textit{max}}/e\lt 4$ and depending on the location of the vortex, a function of the angle of attack. Unfortunately, $\tau _{\textit{max}}/e\geqslant 4$ cannot be considered as a condition that prevents the formation of the first noise source. Furthermore, the tip flow behaviour appears independent of the boundary layer as figures 16(b) and 16(c) contain cases with and without the tip immersed within the boundary layer. However, a more detailed analysis of the boundary layer is presented in § 5.

Figure 17. (a) Tip leakage noise measured for configurations with $\tau _{\textit{max}}/e=4$ . (b) Mean velocity field of the resultant velocity in the $x{-}z$ plane measured within the gap for the configuration with $\tau _{\textit{max}}/e=4$ , NACA $5515$ aerofoil and $\alpha =17^\circ$ .

Figure 17(a) displays the tip leakage noise obtained for three different configurations with NACA $5512$ and $5515$ , where the gap size was chosen to maintain $\tau _{\textit{max}}/e=4$ , the identified threshold for the development of the first noise source. At an angle of attack of $\alpha =15^\circ$ , the grey spectral humps nearly collapse within $St_c=2{-}5.5$ , reaching a negligible peak of about $5$ dB. However, a slight increase in the angle of attack to $17^\circ$ results in the characteristic spectral hump (). For this configuration with $\tau _{\textit{max}}/e=4$ and $\alpha =17^\circ$ , the low-velocity region is found at around $60\,\%$ of the chord, as shown in figure 17(b), which illustrates the velocity field within the gap. At this position, far from the location with maximum thickness localised at $30\,\%$ of the chord, the ratio between the local thickness and the gap size assumes values less than $4$ , for which the tip flow remains separated until the gap exit. In summary, it is not possible to determine in advance whether the first noise source will develop solely based on the ratio $\tau _{\textit{max}}/e\geqslant 4$ . It is necessary to evaluate the local thickness-to-gap ratio at the location where the tip vortex detaches.

4.2. Second tip noise source

Figure 18. Overall sound pressure levels evaluated in the frequency range of the second noise source against the lift coefficient $C_l$ measured for different configurations with $\tau _{\textit{max}}/e=[1,6]$ . The markers refer to the NACA aerofoils: plus, $5505$ ; circle, $5507$ ; asterisk, $5510$ ; cross, $5512$ ; square, $5515$ ; diamond, $5520$ . Muted colours refer to the configurations with the aerofoil tip immersed in the wall boundary layer with $e/\delta \lt 1$ . Specifically, $5505$ and $5507$ with $e/\delta =0.66$ and $0.92$ (c); $5505$ and $5507$ with $e/\delta =0.5$ and $0.7$ (d); $5505$ , $5507$ , $5510$ and $5512$ with $e/\delta =0.4$ , $0.56$ , $0.8$ and $0.96$ (e); $5505$ , $5507$ , $5510$ and $5512$ with $e/\delta =0.33$ , $0.46$ , $0.66$ and $0.8$ (f).

The scatter plots of figures 12 (a) and 12(b) suggested a proportionality between the lift coefficient and the strength of the second tip noise source. Specifically, these data points were obtained using the $5510$ aerofoil, where the gap size was changed. Figure 17(a) also illustrates the noise increase due to this noise source, within $St_c=5.5{-}13$ , for the configurations with two different aerofoils and two angles of attack, $5512$ – $5515$ and $\alpha =15^\circ$ – $17^\circ$ . Notably, for a fixed $\tau _{\textit{max}}/e$ , the noise source strength is primarily governed by the angle of attack: for the same angle of attack, the two grey lines, corresponding to cases with different aerofoils, collapse well, while as the angle of attack increases the noise levels increase.

This dependency is further explored in figure 18, where the strength of the second tip noise source is plotted against the lift coefficient for a variety of aerofoils, ranging from the thinnest $5505$ to the thickest $5520$ . Each scatter plot corresponds to a specific $\tau _{\textit{max}}/e$ ratio, with the gap size adjusted accordingly. For each plot, the data points are distributed linearly, increasing with the lift coefficient. An empirical power relationship can be defined as $\overline {p'^2} \propto {C_l}^K$ , where $\overline {p'^2}$ is the mean square of the fluctuating pressure linked to the second tip noise source, $C_l$ is the lift coefficient measured at the tip and $K$ is the slope of the least-squares line superimposed on the scatter plots. The slope $K$ remains almost constant for $\tau _{\textit{max}}/e=1$ , $2$ and $3$ . For $\tau _{\textit{max}}/e\geqslant 4$ , the dispersion of the data increases with a consequent decrease in the slope. This effect can be due to the immersion of the aerofoil tip within the boundary layer, rather than a change in the tip flow behaviour associated with the $\tau /e$ ratio. By removing all the configurations characterised by $e/\delta \lt 1$ (muted colours in the figure), the dashed least-square lines present a slope $K$ that aligns closely with that obtained for $\tau _{\textit{max}}/e\lt 4$ , suggesting a significant role of the boundary layer. Due to the linear relationship between $C_l$ and $V_L$ shown in figure 13, the power law can be expressed as $\overline {p'^2} \propto {V_L}^{4.5}$ , which is close to the $V_{\infty }^5$ scaling law characteristic of a dipole source close to a sharp edge (Howe Reference Howe2002). This suggests that the observed slope may be a consequence of the dipole nature of the second tip noise source. The power relationship is then finalised as $\overline {p'^2} = a {C_l}^{4.5}$ , where the coefficient $a$ depends on the thickness-to-gap ratio, decreasing as $\tau _{\textit{max}}/e$ increases. Ideally, when the thickness-to-gap ratio tends to infinity (with the gap size approaching zero), the tip flow does not enter the gap, thereby preventing the development of the second noise source.

The second tip noise source develops whenever the tip flow separates at the pressure side edge singularity, regardless of whether the tip flow reattaches to the aerofoil or remains separated until exiting the gap. This behaviour is reflected in the power relationship between the strength of the noise source and the lift coefficient, where the slope remains independent of the thickness-to-gap ratio. The dipole nature of the source dictates the slope value. However, as the thickness-to-gap ratio increases and the aerofoil tip is more immersed in the boundary layer, the noise strength decreases.

The power relationship between the intensity of the second tip noise source and the lift coefficient is useful during the design process, as it allows for a qualitative prior estimation of the noise intensity based on the $\tau _{\textit{max}}/e$ ratio and the lift coefficient $C_l$ . The lift coefficient can be obtained by integrating the pressure coefficient $C_p$ at either the midspan or the aerofoil tip. In particular, knowing $C_p$ at the tip enables the prediction of the TLV detachment location, which, together with the $\tau _{\textit{max}}/e$ ratio, provides insight into the development of the first tip noise source. Future studies with different chamber lines are needed to extend the correlations established in this work.

4.3. Aerodynamic implications

Figure 19. Variation of the lift and drag coefficients, $C_L$ and $C_D$ , with the non-dimensional gap size $\tau _{\textit{max}}/e$ , considering NACA $5510$ aerofoil with a geometric angle of attack $\alpha =15^\circ$ .

This section evaluates the overall aerodynamic performance of the configurations obtained by varying the angle of attack and gap size. Figure 19 displays the lift and drag coefficients, $C_L$ and $C_D$ , as a function of the non-dimensional gap size $\tau _{\textit{max}}/e$ for the configuration with NACA $5510$ aerofoil and geometric angle of attack $\alpha =15^\circ$ . In the evaluation of these force coefficients, the effective angle of attack $\alpha _{\textit{eff}}$ , which results to be approximately one-third of $\alpha$ , was considered to account for wind tunnel deflection (Glegg & Devenport Reference Glegg and Devenport2017). It can be seen that $C_L$ decreases as the gap increases, i.e. $\tau _{\textit{max}}/e$ decreases, consistent with the findings of Higgens et al. (Reference Higgens, Joseph, Lidtke and Turnock2019) and the considerations of Denton (Reference Denton1993) regarding the drop in lift due to the reduction in the aerofoil spanwise extent. Indeed, as the gap between the aerofoil tip and the casing wall increases, the effective span of the aerofoil decreases. The gap also causes a rise in drag, with a significant increase when $e$ increases from $e=5$ to $10$ mm, i.e. $\tau _{\textit{max}}/e=4$ to $2$ . As observed in the previous section, the first noise source develops when $\tau _{\textit{max}}/e\lt 4$ , corresponding to a tip flow that does not reattach to the aerofoil surface and remains separated until exiting the gap. Bindon (Reference Bindon1989) found that most of the performance loss occurs when the separation bubble is ejected into the gap, corresponding to the tip flow remaining separated. Therefore, this extended separation may contribute to both increased drag and generation of the first tip noise source.

Figure 20. Variation of the lift and drag coefficients, $C_L$ (a) and $C_D$ (b), with the effective angle of attack $\alpha _{\textit{eff}}$ , considering NACA $5510$ aerofoil with $\tau _{\textit{max}}/e=2$ and $4$ .

The development of the first noise source was found to depend on the thickness-to-gap ratio and the geometric angle of attack, as shown in figures 11 and 17. This noise source was not observed when $\tau _{\textit{max}}/e\lt 4$ and $\alpha \lt 10^\circ$ , or when $\tau _{\textit{max}}/e\geqslant 4$ regardless of $\alpha$ . This behaviour helps explain the trend of the drag coefficients plotted against $\alpha _{\textit{eff}}$ in figure 20(b), measured for two gap sizes $e=5$ and $10$ mm, i.e. $\tau _{\textit{max}}/e=4$ to $2$ . The drag coefficients remain similar at small angles of attack but begin to diverge at a certain $\alpha _{\textit{eff}}$ , which corresponds to $\alpha =10^\circ$ , the point at which the first noise source develops for $\tau _{\textit{max}}/e=2$ . At higher angles of attack, the drag produced in the case with $\tau _{\textit{max}}/e=2$ is higher than that of the small gap. The lift forces, shown in figure 20(a), follow a linear trend with the effective angle of attack, with slightly higher values for the smaller gap, as expected from figure 19.

In summary, this section highlights that drag losses are connected to the tip flow behaviour and, similarly to the first noise source, depend on the angle of attack and the thickness-to-gap ratio. This suggests a relationship between the first noise source and drag losses, indicating that optimising one can improve both aerodynamic performance and noise reduction in turbomachinery design.