2.1. Wind-tunnel model

Experiments were conducted at the low-speed Small Low Turbulence facility, an open return type wind tunnel located at the Delft University of Technology. The tunnel was operated in an open-jet configuration, with a 900 mm $\times$ 600 mm rectangular nozzle, and a turbulence intensity of approximately $0.1 \,\%$ at the conditions tested. The tractor–propeller configuration is simulated using an off-the-shelf (OTS) propeller and a flat plate with an elliptic leading edge (LE), both mounted onto a custom aluminium beam structure. Figure 1 presents a simplified schematic of the relative positioning of the propeller, plate and measurement systems. The propeller sits to the side of the plate, with the entire set-up placed semi-immersed in the open jet of the wind tunnel. The coordinate system was selected to be right handed, with the respective positive directions indicated by the axes in figure 1. The origins of the coordinate system for $x$ , $y$ and $z$ are the start of the plate LE, flat section of the plate upper surface and propeller axis, respectively.

Figure 1. Experimental propeller–plate configuration semi-immersed in the open-jet flow. The shaded grey region represents part of the wind-tunnel nozzle exit area while the shaded green region indicates the plate-on particle image velocimetry (PIV) field(s) of view (FOV). (a) Front view. (b) Top view. The magenta and cyan dots indicate the origins of the $x, y, z$ and $x_p, y_p, z_p$ (§ 3) coordinate systems respectively. (c) Side view of semi-elliptic LE and PIV measurement domain.

The semi-immersed nature of the set-up was selected to allow for more accessible measurements of the small-scale BL interactions, through scaling up of the propeller and plate. This was supported by the region of interest for these interactions lying in the vicinity of the propeller-blade tip, away from the open-jet shear layer and propeller hub flow. Section 3 details the state of the slipstream in the vicinity of the blade tip, to further support this configuration. Propeller–wing integration effects such as the nacelle–BL interaction were then also neglected. Additionally, the open-jet configuration enables near-complete optical access with the PIV measurement system.

2.1.2. Propeller geometry and operating conditions

The propeller used was 3-bladed and 68.58 cm (27 inches) in diameter, with a fixed 34.08 cm (12 inches) pitch. It was manufactured from carbon fibre and is available OTS from Mejzlik Propellers™. A custom electronic speed controller cabinet with a 15 kW power supply unit was used to power the drivetrain system to which the propeller is attached. The propeller drivetrain is mounted onto an aluminium X-95 beam set, which is placed such that the propeller disk plane lies 150 mm downstream of the nozzle exit. The propeller rotates in the counter-clockwise direction when viewed from upstream and is positioned in the $y-z$ plane for two main purposes: first, to maximise the percentage of its blade immersed in the core flow of the open jet. Second, to maximise the time spent by the immersed blade in core flow within the angular sector above the plate. This allows for the unsteady loading of the blade to adjust to the desired operating condition as it slices into the open jet. In this configuration, approximately 25 % of the propeller disk area is exposed to the wind-tunnel jet, indicated nominally by the shaded grey area in figure 1(a,b).

The propeller was set to operate at a non-installed thrust coefficient of $T_c = 1$ ( $T_c = 2T/\rho U_{\infty }^2 D^2$ , where $T$ is thrust, $\rho$ is the air density), assuming a fully symmetric condition with uniform inflow over the entire propeller disk. This was nominally selected to simulate the high-thrust condition in which modern turboprop aircraft operate during take-off. As no instrumentation was used to measure the forces on the propeller, the conditions for $T_c = 1$ were estimated from performance data charts provided by the manufacturer i.e. a propeller rotational speed of $n = 48.6$ Hz, corresponding to $J = 0.3$ at a uniform speed of $U_{\infty }$ = 10 m s $^{-1}$ . While $J = 0.3$ is lower than what is typically expected for the simulated scenario, this was limited by the operating range of the propeller, with a maximum propulsive $J \sim 0.55$ . Within this operating range, the spatial separation between tip vortices ( $\sim U_\infty /nB$ ) does not increase much for a large decrease in the thrust coefficient. Thus, $J=0.3$ was selected primarily for the blade loading condition, over tip-vortex spacing, for the current study. It should also be noted that $T_c = 1$ for the 3-bladed propeller likely results in a higher loading per blade than what would be typical for a multi-bladed aircraft propeller operating at the same thrust coefficient. A consequence of this is stronger tip vorticity in the slipstream.

The rotational speed of the propeller was measured with a differential rotary encoder (1024 CPR) mounted onto the drivetrain shaft. The encoder reading was used to control propeller rotational frequency with a custom-made control loop and features a 1-per-revolution trigger signal that was used to synchronise the phase-locked PIV measurements. A summary of the operating conditions for the propeller–plate set-up is presented in table 1.

Table 1. Operating conditions for the propeller-plate wind-tunnel set-up

2.2. Measurement techniques

2.2.2. Particle image velocimetry

Low-speed stereoscopic-PIV was used to quantify the interaction between the propeller tip vortices and plate BL, through velocity field measurements near the LE region of the flat plate. Figure 1(a) depicts the positioning of the two-camera and laser system, which provided a field of view (FOV) of 90 mm $\times$ 60 mm for each position of the stereo-image plane. The camera system was manually translated downstream to a second overlapping FOV of the same dimensions, for a second set of independent velocity field acquisitions. This allowed for a streamwise extension of the FOV to 140 mm $\times$ 60 mm, while maintaining resolution in the wall-normal direction. The two cameras and laser were additionally each mounted onto their own Zaber X-LRQ traverse, with movement of all three synchronised through external controllers. This was to enable precise translation of the PIV set-up in the out-of-plane direction. Independent acquisitions were conducted for each FOV at 16 spanwise locations separated by 4 mm, with a total spanwise measurement extent of 60 mm as shown in figure 1(a,b). The selected arrangement of the PIV set-up is used to obtain a scan of the slipstream in the region encompassing the tip vortices, allowing for a pseudo-volumetric reconstruction of the flow topology.

Table 2. Particle image velocimetry illumination and imaging

The fluid was seeded using a SAFEX Twin Fog DP generator, with a mixture of diethylene-glycol and water, and illuminated using a 200 mJ Quantel Evergreen low-speed laser. An optical lens arrangement consisting of two spherical lenses and one cylindrical lens of focal lengths $f = -50, 100$ and $-100$ respectively was selected to shape the beam into a 1.5 mm thick laser sheet. The laser pulse separation time was tuned for a minimum particle displacement of 5–6 pixels in low-speed regions. The two cameras used for acquisitions were identical, each a 16-bit LaVision Imager sCMOS with a 2560 $\times$ 2160 px $^2$ sensor and a pixel size of 6.5 μm. A Nikkor 105 mm lens with a Scheimpflug adaptor was mounted onto each camera to allow for the required stereoscopic FOV, while a numerical aperture (f-number) of 4 was set to obtain a particle image diameter of 2–3 pixels and a depth of field of 5 mm. Particle image pairs were acquired at a frequency of 15 Hz and processed using the LaVision DaVis 10.2 software. Stereoscopic cross-correlation was performed with 48 $\times$ 48 px $^2$ and 16 $\times$ 16 px $^2$ as the window sizes for the initial and final passes with 50 % overlap, resulting in a final spatial resolution of 0.32 mm. Spurious vectors were removed after each computation stage through rejecting vectors with a stereoscopic reconstruction error of less than 0.5 px, and through the use of a three-pass regional median filter (Westerweel Reference Westerweel1994). This resulted in an uncertainty of less than 0.8 % in the averaged velocity fields.

Uncorrelated acquisitions of 1000 image pairs were done for the propeller-off and propeller-on cases, triggered using a LaVision programmable timing unit (PTU). In addition to this, phase-locked measurements were performed for the propeller-on case by synchronising the PTU trigger to the 1P signal obtained from the propeller encoder. In the current work we consider three phases of $\phi = 0^\circ$ , $\phi = 40^\circ$ and $\phi = 80^\circ$ . For each phase a total of 400 image pairs were acquired for construction of the phase-averaged velocity statistics. Table 2 summarises the specifications of the PIV measurement system.

4.1. Triple decomposition of velocity fields

Statistical analysis of the interacting flow field is carried out through two types of averaging, which yield either time-averaged ( $\overline {u_i}$ ) or phase-locked ( $\overline {u_i}^{\text{PL}}$ ) velocity quantities, through the following operations, respectively:

(4.1) \begin{align} \overline {u_i}(x,y,z) = \frac {1}{T} \int _0^T u_i(x,y,z,t)\,dt, \end{align}

(4.2) \begin{align} \overline {u_i}^{\text{PL}}(x,y,z,t) = \frac {1}{N} \sum _{n=0}^{N} u_i(x,y,z,t + nT), \end{align}

with $i = 1,2,3$ corresponding to the axial, wall-normal and spanwise coordinate directions, respectively. Using the above definitions for these operations, we employ a triple decomposition of the flow field (Ambrogi et al. Reference Ambrogi, Piomelli and Rival2022; Hussain & Reynolds Reference Hussain and Reynolds1970) where the instantaneous flow field is separated into time-averaged, periodic and stochastic components as

(4.3) \begin{align} u_i(x,y,z,t) = \overline {u_i}(x,y,z) + \widetilde {u}_i(x,y,z,t) + u_i'(x,y,z,t). \end{align}

The various components of the triple decomposition can be connected to each other through the relations

(4.4) \begin{align} \overline {u_i}^{\text{PL}} & = \overline {u_i} + \widetilde {u}_i, \end{align}

(4.5) \begin{align} u_i' &= u_i - \overline {u_i}^{\text{PL}}. \end{align}

It follows from (4.4) that the periodic velocity fields $\widetilde {u}_i$ can be obtained by subtracting the time-averaged velocity fields from the phase-locked velocities, with the components of the triple-decomposed $u$ -velocity presented in Figure 3. Comparison between Figures 3(b) and 3(c) shows that the large-scale periodic perturbation is of overall higher amplitude than the turbulent fluctuations, with the spatial standard deviation of the former being approximately 3 times larger than the latter. This implies that the TBL response is primarily driven by the tip-vortex-induced velocities, while the turbulence field plays a smaller, secondary role. Thus, our discussion of unsteady flow features in this study will focus on the phase-locked velocity fields, which is presented later in § 5.

Figure 3. Contours of the triple-decomposed $u$ -velocity components at $z/R = 1$ , $\phi = 0^\circ$ . (a) Time averaged ( $\overline {u}/U_\infty$ ), (b) periodic ( $\widetilde {u}/U_\infty$ ) and (c) stochastic ( $u'/U_\infty$ ).

4.2. Time-averaged velocity fields

The interaction between a propeller slipstream and downstream surface can be described in a steady sense through time averaging of the flow field. We discuss the pertinent features of this time-averaged interaction here to establish the mean flow topology of the slipstream within the measurement domain as a baseline for later analyses. Figure 4 presents slices of the flow field along the three orthogonal planes within the measurement domain. The propeller acts as an actuator, adding momentum to the flow within its slipstream. This is evidently seen in figure 4(a), a wall-parallel slice at 15 mm ( $y/R = 0.044$ , $y/\delta _0 = 3.41$ ) off the surface, where fluid is approximately 1.7 times the free-stream velocity. The interior of the slipstream can thus be identified as this region of increased dynamic pressure, with an accompanying increase in local $Re$ . The exact slipstream edge, however, is ambiguously defined, as the velocity gradient across it is continuous. We identify a band of maximum fluctuation through a time-averaged fluctuation kinetic energy

(4.6) \begin{align} \overline {k} = \frac {1}{2}\overline {(\widetilde {u}_{i} + u'_{i})(\widetilde {u}_{i} + u'_{i})} . \end{align}

The slipstream edge is then defined as lying within the $\overline {k}/U_\infty ^2 = 0.065$ isoline band, with the two isolines in figure 4(a) demarcating the outboard and inboard bounds of this band. The choice of $\overline {k}/U_\infty ^2 = 0.065$ is through Gaussian fitting of the spanwise profiles of $\overline {k}$ , which leads to the two isolines lying approximately one standard deviation away from the $z\hbox{-}$ location of each spanwise $\overline {k}$ -profile peak. As a result, the selected band is representative of the largest fluctuations in the domain. The use of flow field fluctuation in the context of identifying the propeller slipstream boundary is also seen in Avallone et al. (Reference Avallone, Casalino and Ragni2018) and Sinnige et al. (Reference Sinnige, De Vries, Corte, Avallone, Ragni, Eitelberg and Veldhuis2018 a), although the fluctuating quantity used is the unsteady pressure coefficient.

Figure 4. Time-averaged $u$ -velocity slices of the flow field. Cyan, magenta lines indicate the respective positions of each slice plane. (a) Shows $y/R = 0.044$ . Thick black lines are the $\overline {k}/U_\infty ^2=0.065$ band, thin black lines are in-plane streamlines. (b) Shows $z/R = 0.965$ . Dashed line is $\overline {\delta _{99}}$ with estimation uncertainty indicated by the grey shaded region. (c) Shows $x/R = 0.5$ . Black lines are $\overline {k}/U_\infty ^2=0.065$ band with in-plane velocity vectors.

The slipstream over the surface experiences positive spanwise deflection, as indicated by the mean streamlines and orientation of the edge isolines in figure 4. This is due to two primary mechanisms: an image vortex effect associated with the presence of the solid surface, and spanwise surface pressure gradients induced by the propeller–plate interaction. Both have been well established in previous works (Johnston & Sullivan Reference Johnston and Sullivan1993; Muscari et al. Reference Muscari, Dubbioso and Di Mascio2017; Sinnige Reference Sinnige2018). Additionally, the location of the slipstream edge in the domain is a function of the interaction at the plate LE (Felli Reference Felli2021) and the spacing between the propeller and plate, lying slightly inboard of the propeller-blade tip here. Figure 4b presents a streamwise–wall-normal slice of the slipstream at $z/R = 0.965$ . The relative orientation of the slice to the spanwise shearing effect results in the axial velocity gradient observed. It should be noted that, due to the 3-D nature of the flow field and arbitrary choice of coordinate system, this gradient is not indicative of streamwise fluid acceleration.

The mean swirl induced by the propeller within the slipstream is minimal in the measurement domain, as seen in a plane at $x/R = 0.5$ relative to the plate LE (figure 4 c). This could be attributed to vanishing inductions due to the propeller close to the blade tip in combination with the low advance ratio. It is apparent that the orientation of the vectors in figure 4(c) is in contrast to figure 2(b). This is once more due to the effect of the plate on the slipstream, and the resulting spanwise deflection discussed earlier. A mean vertical deflection of fluid is also observed on the outboard edge of the slipstream, far above the wall. Given the helical nature of the discrete tip-vortex system and the positioning of the measurements on the downgoing side of the propeller blades, this aligns with the expected mean induced velocities in this region. Additionally, there is a local variation in BL thickness in the spanwise direction around the slipstream edge, which is further discussed in the following section.

4.3. Boundary-layer modulation

The discrete vortical system comprising the propeller slipstream induces unsteady pressure fluctuations over the plate surface. In a time-averaged sense, this changes the steady loading experienced by the developing BL. Figure 5 presents the averaged static pressure coefficient ( $c_p = 2(p - p_\infty )/\rho _\infty U_\infty ^2$ ) distribution measured from the two arrays of pressure taps, indicative of the change in loading between the propeller-off and propeller-on cases. The reference pressure $p_\infty$ here is the ambient pressure outside of the open-jet flow. In the absence of a slipstream, the TBL flow continues to accelerate beyond the trip location to attain a mild suction peak at $x/R \sim 0.28$ , before starting to relax into a zero-pressure-gradient flow. The TBL flow in the PIV measurement domain is hence exposed to a very low adverse pressure gradient (APG). With the propeller on, the solid $c_p$ curves in figure 5 exhibit higher $c_p$ values overall. This due to the slipstream-induced change in loading experienced by the plate (Veldhuis Reference Veldhuis2005). The section immersed in the slipstream at $r/R = 0.7$ ( $\square$ ) experiences a swirl-induced negative angle of attack, with the $c_p$ distribution exhibiting a strong FPG up to $x/R = 0.4$ . It should also be noted that there is increased total pressure within the slipstream which provides a positive offset to the $c_p$ distribution measured here. The FPG at $z/R = 1.3$ ( $\circ$ ), external to the slipstream, is much milder in comparison. This is due to lower-amplitude secondary inductions caused by the spanwise loading gradient at the slipstream edge (Sinnige Reference Sinnige2018; Veldhuis Reference Veldhuis2005).

Figure 5. (a) Pressure coefficient distributions for the propeller-on (solid black lines) and off (dotted black lines) cases, with uncertainty represented by the grey shaded regions. Leading-edge geometry is shown as the solid grey line. Shaded green area indicates the PIV measurement domain for the plate-on measurements.

Figure 6. (a) Relative change in TBL thickness with $\overline {U_e}$ vectors (grey). Solid black lines are the $\overline {k}/U_\infty ^2 = 0.065$ band, dashed lines are where BL velocity profiles are shown, $z/R = 1.00$ (magenta), $z/R = 0.95$ (black), $z/R = 0.92$ (cyan). (b) Streamwise ( $\circ$ ) and cross-flow ( $\square$ , scaled by a factor of 5 for visualisation) velocity profiles.

A relative change in the time-averaged TBL thickness $\overline {\delta _{99}}$ between the propeller-off and propeller-on cases is used as a global estimator to quantify BL modulation caused by the slipstream, within the measurement domain. This is computed using the criterion described in § 2.1.1 for each spanwise PIV measurement plane in the domain and presented in Figure 6(a). There is observed to be a local thickening of the TBL in a band roughly centred around the slipstream edge, within an amplitude range of $1.2 {-} 1.5$ times the propeller-off $\overline {\delta _{99}}$ . On the contrary, the increase in local $Re$ within the slipstream due to increased dynamic pressure as well as the strengthened FPGs on the downgoing blade side would indicate an overall decrease in BL thickness. The observed change in $\overline {\delta _{99}}$ is thus attributed to the unsteady passage of the propeller tip vortices within the $\overline {k}/U_\infty ^2 = 0.065$ band. Here, the maximum variation in relative $\overline {\delta _{99}}$ change is observed to be on the outboard side of this band, where the unsteady vortical inductions oppose the free stream. Boundary-layer thickening is also noted to occur on the inboard side of the slipstream edge, where the vortical inductions accelerate the free-stream fluid. The relative $\overline {\delta _{99}}$ increase within the slipstream decays at more inboard sections with BL thinning observed at the farthest inboard point in figure 6(a), as would be expected. The origins of the local time-averaged thickening of the BL around the $\overline {k}/U_\infty ^2 = 0.065$ isolines in figure 6(a) are further discussed in § 5 from the unsteady point of view.

The TBL also exhibits strong three-dimensionality in the mean flow, owing to the inherently 3-D nature of the external flow imposed by the slipstream. This can be seen in figure 6(a) where the vectors represent the local in-plane external velocity at the edge of the BL. The positive deflection of the slipstream induces a mean spanwise flow into the BL throughout the measurement domain. The full effect of this can be seen by decomposing the 3-D BL profiles into streamwise and cross-flow velocity profiles (figure 6 b) based on the local $\overline {U_e}$ vectors. There is a non-zero component of cross-flow for all extracted profiles, largest at the most inboard location within the domain. This corresponds to the increased shearing effect of the slipstream at more inboard locations, as can be seen with the vectors in figure 6(a).

5.1. Three-dimensional vortical structures

Coherent vortical structures are visualised using the normalised $Q$ -criterion (Hunt et al. Reference Hunt, Wray and Moin1988; Mula et al. Reference Mula, Stephenson, Tinney and Sirohi2013), where

(5.1) \begin{align} Q = -\frac {1}{2} \frac {\partial \overline {u}^{\textrm {PL}}_i}{\partial x_j} \frac {\partial \overline {u}^{\textrm {PL}}_j}{\partial x_i}, \end{align}

isolating rotationally dominant regions of fluid. As shown in figure 7, isosurfaces of the $Q$ -criterion at a fixed rotation phase $\phi = 80^\circ$ reveal a distinct, well-organised tip-vortex system located in the $z/R = 0.9 {-} 1$ vicinity. This system comprises a double-vortex structure, with the two component features labelled the primary and secondary vortices for the remainder of this discussion. As the TBL flow is resolved up to approximately 0.5 $\overline {\delta _{99}}$ in the vicinity of this vortex system, only the outer-region TBL interaction is captured.

Figure 7. Isosurfaces of vortex core regions for $QR^2/U_\infty ^2 = 24$ (dark grey), $1200$ (black) from phase-locked velocity fields at $\phi = 80^\circ$ . White vectors represent local vorticity vectors on the $QR^2/U_\infty ^2 = 24$ isosurface. Surface slices are $\overline {u}^{\textrm {PL}}/U_\infty$ and black lines are streamlines, both at $y/\delta _0 = 0.72$ . Magenta and cyan lines respectively indicate the outboard (I) and inboard (II) $z$ -locations discussed in § 5.2. (a) Inboard side, downstream view. (b) Outboard side, upstream view.

The primary vortex structure exhibits a 3-D character largely focused in the wall-normal $y$ -direction, associated with the impinging propeller tip vortex. The spatial organisation of the $QR^2/U_\infty ^2 = 1200$ isosurface with respect to the wall-normal component of the $QR^2/U_\infty ^2 = 24$ surface indicates that both structures comprise the primary vortex, with the former representing a strongly rotational core region of the tip vortex. Away from the wall, the primary vortex displays a negative inclination in the spanwise ( $z$ ) direction and a positive axial ( $x$ ) tilt, consistent with a helical organisation of discrete vorticity in the propeller slipstream. In the near-wall region, the axial inclination of the vortex increases in the positive $x$ -direction and the inner core exhibits contraction. This shift in orientation near the wall can stem from two sources: first, the interaction between the impinging vorticity and the LE of the plate. Second, viscous shear of the BL flow exerted on the tip-vortex core. With regard to the former, Felli (Reference Felli2021) and Thom (Reference Thom2011) have indicated that the ratio of LE-to-vortex core size can strongly influence the extent of vortex wrapping around the plate LE, with the ratio here estimated to be near unity. This would correlate to the mild axial deformation of the tip vortex observed in figure 7, although the full development from the point of impingement is not resolved here.

The secondary vortex structure is observed to emerge from within the TBL and wraps around the primary vortex as it progresses downstream (figure 8). This secondary structure is characterised by its proximity to the wall and its eventual convergence with the primary vortex. Near the wall, the secondary vortex exhibits predominantly axial and spanwise vorticity components, consistent with its origin in the BL. As it convects upward and outward, the structure becomes more fully three-dimensional, indicating entrainment and roll-up by and into the primary vortex. This process is elucidated by the surface vorticity vectors in figure 7. This wrapping and merging process appears to intensify over the passage of the tip vortex through the measurement domain, with the secondary vortex growing and progressing away from the wall at more downstream locations (figure 8).

Figure 8. Isosurfaces of vortex core regions at phases $\phi = 0^\circ , 40^\circ , 80^\circ$ . See figure 7 caption for colours. Surface slices are wall-normal velocity $\overline {v}^{\textrm {PL}}/U_{\infty }$ and black lines are streamlines, both at $y/\delta _0 = 0.72$ . Left column: outboard side, downstream view. Right column: inboard side, upstream view.

We postulate that the genesis of the secondary vortex is driven by the forcing introduced by the primary vortex on the BL. A velocity deficit region of approximately 60 % the free-stream velocity is observed on the outboard side of the vortex (region I in figure 7), aligned with the direction of induced velocities due to the primary wall-normal vorticity in this region. Conversely, the inboard side of the vortex (region II in figure 7) corresponds to a region of accelerated flow, with the flow locally reaching 1.8 $U_\infty$ . These regions of locally decelerated and accelerated flow are associated with strong local velocity gradients, causing a local amplification of BL vorticity. Additionally, the BL undergoes spanwise deflection in the vicinity of the primary vortex, illustrated by the streamlines in figures 7 and 8. The resulting spanwise gradients, in combination with the amplified BL vorticity, facilitate the development of the secondary vortex structure which couples with the primary vortex to feed back into the overall 3-D interaction. This viscous–inviscid coupling mechanism has been similarly detailed by Didden & Ho (Reference Didden and Ho1985) for 2-D vortex–BL interaction. The relative contributions of each vortex structure to the BL response are difficult to isolate, due to their overlapping and mutual inductions. A deeper analysis of the dynamics of this 3-D interaction is carried out in the following section.

5.2. Dynamics of boundary-layer response

The BL response to the tip-vortex structures reveals two distinct regimes of interaction, corresponding to the outboard (region I, figure 7) and inboard (region II, figure 7) regions relative to the primary vortex core. Similar near-wall flow topology is observed at varying spanwise distance from the tip-vortex cores, within each of these respective regions. This dual-regime behaviour underscores the inherently asymmetric, 3-D nature of the BL modulation induced by the 3-D vortex system.

In the outboard region the TBL exhibits localised axial velocity deficits (figure 7 b), induced by the largely wall-normal primary vortex structures. This implies local deceleration of fluid into this region followed by acceleration as fluid is convected out of it. Figure 9 presents phase-locked, axial-wall-normal slices of the $u$ -velocity field at two spanwise locations: one in the vicinity of the tip-vortex core and the other at a farther outboard location. Away from the vortex cores (left column in figure 9), axial gradients in the velocity field are less pronounced. Here, the deficit regions in the BL are in phase with the external induced deficits and BL vorticity is observed to be attached to the surface. It should be noted that, since the slipstream undergoes spanwise deflection, the streamwise–wall-normal slice planes shown in figure 9 also progressively slice into the slipstream along the $x$ -direction. This coincides with time-averaged thickening of the BL within the slice.

Figure 9. Slices of $\overline {u}^{\textrm {PL}}/U_\infty$ on the outboard side of the tip vortices (region I in figure 7). Left column: $z/R=1.02$ , right column: $z/R=0.99$ . Black solid and dashed–dotted lines are isolines of negative and positive $z$ -vorticity ( $\overline {\omega _z}^{\textrm {PL}}R/U_\infty$ ), respectively. Thick dashed line is $\overline {\delta _{99}}_{J=0.3}$ for each respective slice.

Closer to the vortex cores (right column in figure 9) the BL is seen to respond more intensely, with stronger velocity deficit regions and a roll-up of BL vorticity. This appears to be strongly correlated to the presence of spanwise vorticity from the secondary vortex structure discussed in § 5.1, which wraps around the primary vortex. The flow topology here bears strong resemblance to the classical case of 2-D rectilinear spanwise vortices interacting with a BL (Cassel & Conlisk Reference Cassel and Conlisk2014; Luton et al. Reference Luton, Ragab and Telionis1995; Serra et al. Reference Serra, Crouzat, Simon, Vétel and Haller2020), where the presence of the vortex induces a local unsteady roll-up of the BL flow. Slices of the phase-locked $v$ -velocity in figure 8 show that there is a focused wall-normal transport of fluid away from the wall, on the outboard side of the tip-vortex cores. This correlates to the more pronounced velocity deficit observed on the outboard side, as low-momentum fluid within the BL is transported away from the wall. While positive $\overline {v}^{\textrm {PL}}$ is also observed on the inboard side of the tip vortex, this is less than half as intense as that in the outboard region and not as localised. This reinforces the interpretation that the BL in the outboard regime is more dominantly forced by the $\overline {\omega _x}^{\textrm {PL}}$ and $\overline {\omega _z}^{\textrm {PL}}$ components of the secondary vortex, in the vicinity of the primary vortex core.

Despite the intense modulation and the presence of a pronounced velocity deficit, reverse flow is not detected within the near-wall region. This, however, is consistent with findings from the literature on unsteady and 3-D BL separation, which show that the classical indicators of separation may not be appropriate in this scenario (Doligalski et al. Reference Doligalski, Smith and Walker1994; Sears & Telionis Reference Sears and Telionis1975). It is also evident from the right column of figure 9 that the flow is topologically indicative of separating flow forced by the presence of a convecting vortex.

In contrast to the outboard region, the inboard BL interaction first features strong local acceleration of the fluid near the tip-vortex core (figure 10, left column). This acceleration is driven by the induced velocity field of the primary vortex structure, which exerts a suction-like effect on the inboard side due to its orientation and proximity to the wall. Downstream of the local acceleration zone the flow rapidly decelerates, recovering to an average lower velocity between the tip vortices. This transition between rapid acceleration to deceleration leads to a BL evolution reminiscent of a turbulent separation bubble. The flow does not exhibit clear signs of full detachment within the measured domain, although the $u$ -velocity field reveals a strong velocity deficit in the deceleration section. These features are consistent with those in unsteady TBLs subjected to rapid pressure-gradient changes (Ambrogi et al. Reference Ambrogi, Piomelli and Rival2022, Reference Ambrogi, Piomelli and Rival2023; Parthasarathy & Saxton-Fox Reference Parthasarathy and Saxton-Fox2023). In the vicinity of the primary vortex cores, the TBL additionally exhibits a strong amplification in mean $\overline {\omega _z}^{\textrm {PL}}$ close to the wall. Contours of positive $\overline {\omega _z}^{\textrm {PL}}$ also reveal a more complex forcing structure associated with the tip vortices as compared with the outboard side, with $\overline {\omega _z}^{\textrm {PL}}$ isolines indicating a merging of the primary and secondary vortices. However, the amplification of vorticity in the BL is quite large in comparison with the magnitude of $\overline {\omega _z}^{\textrm {PL}}$ in the forcing structure. This is not the case on the outboard side, where the forcing structure and BL response possess $\overline {\omega _z}^{\textrm {PL}}$ of a similar order of magnitude. This would seem to imply that the TBL response on the inboard is driven more by the strong wall-parallel induced velocities associated with the $\omega _y$ -component of the primary vortex.

Figure 10. Slices of $\overline {u}^{\textrm {PL}}/U_\infty$ on the inboard side of the tip vortices (region II in figure 7). Left column: $z/R=0.95$ , right column: $z/R=0.92$ . Black solid and dashed–dotted lines are isolines of negative and positive $z$ -vorticity ( $\overline {\omega _z}^{\textrm {PL}}R/U_\infty$ ), respectively. Thick dashed line is $\overline {\delta _{99}}_{J=0.3}$ for each respective slice.

At a more inboard location (figure 10, right column), the scale of the velocity deficit region is diminished. Despite this contraction in wall-normal extent, the streamwise extent of the velocity deficit region increases. This can be attributed to the lower amplitude of the axial velocity gradients further into the slipstream, as induced velocities due to the primary structure becomes less intense and are not as localised. However, amplification of $\overline {\omega _y}^{\textrm {PL}}$ in the TBL remains significant, although this concentration of vorticity shifts closer to the wall. The structure of the flow in this region indicates that the dominant forcing is still by the primary vortex, with the resulting pressure field a result of 3-D vortical induction. Wall-normal $\overline {\omega _y}^{\textrm {PL}}$ is likely the principal contributor to the observed inboard BL response, although the 3-D spatial orientation of the primary vortex makes it difficult to fully decouple the induction effects of the $\overline {\omega _z}^{\textrm {PL}}$ and $\overline {\omega _y}^{\textrm {PL}}$ components. Additionally, no discernible influence from the blade-wake vorticity is evident within the slipstream, suggesting that the TBL response is predominantly governed by the influence of the tip vortices and not the propeller-blade wake for the selected configuration.

Although the inboard and outboard BL interactions exhibit distinct topological and dynamical characteristics, they are fundamentally connected through the strongly 3-D nature of the vortex–wall interaction, as postulated in the preceding section. In the inboard region, the primary vortex induces pronounced local shear and a velocity deficit in its vicinity, giving rise to strong amplification of mean vorticity within the TBL. Spanwise induced velocities associated with the organisation of the primary vortex facilitate the transport of this vorticity from the inboard region toward the outboard region. On the outboard side, this transported vorticity interacts with the local decelerated environment, resulting in further amplification and vertical displacement of BL vorticity away from the wall (figure 8) through the unsteady separation process. The lifted vorticity layer connects through a vortex line to the inboard region, forming a coherent, wrapped structure around the core of the primary vortex. This structure constitutes the secondary vortex, which is most prominent in the outboard region and plays an active role in shaping the local BL response. By modulating momentum transport and shear, it reinforces the spanwise asymmetry and feeds back predominantly into the outboard BL separation dynamics. This coherent tip-vortex–BL vorticity system evolves in the wall-normal direction as it convects downstream, previously depicted in figure 8. The unsteady velocity deficits induced by the primary and secondary vortices leave a time-averaged imprint on the BL, observable in figure 6a as a relative (with respect to the propeller-off scenario) local increase in $\overline {\delta _{99}}$ on the inboard and outboard sides of the $\overline {k}/U_\infty ^2 = 0.065$ isoline. This relative thickening increases downstream, correlating with the unsteady evolution of the tip-vortex–BL structures.