3.1. Mean flow and turbulence statistics

We first focus on the visualization and analysis of the tip vortices generated by the propeller's blades that interact with the wall boundary layer. The wall-normal-streamwise plane, in figure 2, captures the distinct trace of the helicoidal vortices produced by the two-blade propeller. Phase-averaged field analysis at $\phi =0^\circ$ reveals that the decay of tip vortices is faster when the blades are mounted closer to the wall, as depicted in figure 2(a). The manifestation of tip vortices is profoundly affected by the wall-bounded turbulence within the boundary layer, and this influence varies with the tip clearance ($\epsilon$). Interestingly, at $\varepsilon /\delta _0=0.1$ and at a distance of $3.5\delta _0$ behind the propeller, the tip vortices undergo complete disruption (figure 2b), leading to the breaking of the helicoidal tip blade vortex. Our observations are in line with the work of de Vries et al. (Reference de Vries, van Arnhem, Avallone, Ragni, Vos, Eitelberg and Veldhuis2021) who demonstrate a similar vorticity interaction, describing the reasons for the significant downward displacement of the tip vortices. The viscous shear generated by this interaction is responsible for the varying forces applied to the tip vortices and their disruption. Conversely, when the tip clearance is comparably larger (i.e. $\varepsilon /\delta _0=0.5$), the decay is more gradual (figure 2d) and the interaction with the wall vorticity is notably absent.

Figure 2.

Phase-locked mean flow field and vorticity at $\phi =0^\circ$: (a,c) dimensionless streamwise velocity component; (b,d) dimensionless spanwise vorticity component; (a,b) $\varepsilon =0.1\delta _0$; (c,d) $\varepsilon =0.5\delta _0$; the region upstream of the blade is being masked due to the presence of its shadow during PIV measurements, which decreases the signal-to-noise ratio.

The propeller, acting as an energy source, imparts acceleration to the incoming boundary layer, reaching velocities up to 1.8 times that of the free stream (see figure 3a,c). Analysis of the mean flow behind the blades reveals a distinctive contraction in the slipstream tube generated by the propeller rotation, inducing local APGs and deceleration of the wall-bounded flow. The assumption of the pressure gradients as a consequence of the slipstream contraction behind the propeller is demonstrated experimentally by both Marcus et al. (Reference Marcus, de Vries, Kulkarni and Veldhuis2018) and de Vries et al. (Reference de Vries, van Arnhem, Avallone, Ragni, Vos, Eitelberg and Veldhuis2021), who measured a rise in the wall pressure coefficient immediately downstream of the propeller using an array of static pressure ports. Notably, there is no discernible evidence of flow separation in the mean flow. The creation of shear behind the blade tips is attributed to the tip vortices (shown in figure 2), which travel along the edge of the wall-bounded stream tube. Examining the boundary-layer profiles in figures 3(b) and 3(d), it becomes evident that in the presence of rotating blades, the flow behaves akin to being in an APG condition, with a reduction of wall-normal extent in the log region and an enhanced outer wake region. The values of the latter for both the tip clearances are shown in figure 4(b). The reduced extension of the logarithmic region in canonical boundary layers with APG was previously investigated by Monty, Harun & Marusic (Reference Monty, Harun and Marusic2011) who also showed that the wake region of APG boundary layers begins earlier, at $y^+=150$ ($Re_{\tau }=2860, \beta =1.9$). The authors suggest that the reduced size of the log layer extension depends on both the $Re$ and the strength of the pressure gradient.

Figure 3.

Axial mean flow and boundary-layer profiles: (a,c) dimensionless streamwise velocity component, (b,d) boundary-layer profiles in wall units, (a,b) $\varepsilon =0.1\delta _0$, (c,d) $\varepsilon =0.5\delta _0$; red dashed line, log-law equation with values of k and B from Marusic et al. (Reference Marusic, Monty, Hultmark and Smits2013); circles, velocities in wall units; vertical dashed lines, streamwise positions of the blades’ tip in wall units; blue gradients from lighter to darker, $x/\delta _0=0.5$, 1, 1.5, 2.5, 3.5, 4.5; red circles, flat plate at $Re_{\theta }=12\,200$ ($Re_{\tau }=4400$). Due to the resolution of the PIV system, the mean velocity points shown in (b,d) below $y^+=30$ are not valid data.

Figure 4.

Skin friction coefficient and wake parameter at different streamwise positions: (a) $C_f = 2(u_\tau /U_\infty )^2$, (b) $\varPi$; black dot-dashed line, $C_f$ function of $Re_x$ for a smooth flat plate (Schlichting & Gersten Reference Schlichting and Gersten2016); red dashed line, position of the two-blade propeller; red circle, flat plate boundary layer ($C_f=0.0026$ and $\varPi =0.33$) at $Re_{\theta }=12\,200$ ($Re_{\tau }=4400$); crosses, $\varepsilon =0.1\delta _0$; circles, $\varepsilon =0.5\delta _0$; black, values of $C_f$ and $\varPi$ upstream of the propeller, at $x/\delta _0=-0.5$, $-1$; blue gradients from lighter to darker, $x/\delta _0=0.5$, 1, 1.5, 2.5, 3.5, 4.5.

To explore this further, the friction velocity $u_{\tau }$ and the wake strength $\varPi$ are determined from the mean velocity profile using the method described by Rodríguez-López, Bruce & Buxton (Reference Rodríguez-López, Bruce and Buxton2015). The assumption validating the use of the aforementioned method relies on the existence of a limited portion of the boundary layer that adheres to the log-law function. In the presented tip vortices/TBL interaction, the boundary layer is disrupted by the movement of the tip vortices at each rotational phase of the blades. However, statistically speaking, by considering the ensemble of velocity fields at all phases $\phi$, a portion of the log layer remains intact. This is demonstrated in figures 3(b) and 3(d), where the extent of the log layer exists and particularly depends on $x/\delta _0$ and tip clearance. The procedure finds the optimum solution $\{u_{\tau }, \kappa, \delta, \varPi \}$ that minimizes the error between the measured mean velocity profile and the analytical functions, i.e.

(3.1)\begin{equation} U^+_{canonical} =\begin{cases} U_{musker} + U_{bump} + \dfrac{2\varPi}{\kappa} W, & 0 \leqslant y \leqslant \delta, \\ U_{\infty}, & \delta \leqslant y \leqslant \infty, \end{cases} \end{equation}

with

(3.2)\begin{equation} U_{musker} = U_{musker}(y^+, \kappa, s) \end{equation}

obtained from

(3.3)\begin{gather} \frac{{\rm d}U_{musker}}{{{\rm d}y}^+} = \frac{\dfrac{(y^+)^2}{k} + \dfrac{1}{s}}{(y^+)^3 + \dfrac{(y^+)^2}{k} + \dfrac{1}{s}} , \end{gather}

(3.4)\begin{gather}U_{bump} = U_{bump}(y^+) = \exp[-\log^2(y^+{/}30)]/2.85 , \end{gather}

(3.5)\begin{gather}W = W(y/\delta) , \end{gather}

(3.6)\begin{gather}B = B(\kappa) , \end{gather}

(3.7)\begin{gather}s = s(\kappa). \end{gather}

Equation (3.1) represents the analytical form used to describe the canonical flow from the inner layer to the log-law region. It also depends on $s$, which is the constant relating the eddy viscosity in the inner and overlap regions. In fact, $U_{musk}$ takes into account a continuous change in the eddy viscosity from the wall to the overlap region, adapting the linear to the log region (Musker Reference Musker1979). Contrary, $U_{bump}$ does not depend on s and includes an overshoot over the log law onto the buffer layer, which is generally used for a better description of mean velocity profiles at high Re (Rodríguez-López et al. Reference Rodríguez-López, Bruce and Buxton2015). Equation (3.5) is the exponential wake function described in Chauhan, Monkewitz & Nagib (Reference Chauhan, Monkewitz and Nagib2009). During the optimization routine, the values of the von Kármán constants $\kappa$ and $B$ are kept fixed as in Chauhan et al. (Reference Chauhan, Monkewitz and Nagib2009), at values respectively of 0.39 and 4.2 as our aim is to explore the effects of the propeller wake on the boundary-layer state. This assumes that all velocity profiles will have a log region with differences exhibited only in the wall-normal extent compared with a canonical TBL. This choice allows for evaluating how propeller/boundary-layer interactions affect the outer region, and thus, the wake structure. Having said that, we also allowed $\kappa$ and $B$ as part of the optimisation and this did not show sufficient change in these values (relative to uncertainty). This further reinforces our approach of fixing these values and examining the differences in the resulting values of other quantities.

When the blade tips rotate within the log region of the TBL ($\varepsilon =0.1\delta _0$, see figure 3b), the most significant deviation from the canonical case occurs at an axial distance of $x/\delta _0=0.5$ from the propeller. Subsequently, as the axial distance increases, the wake amplification diminishes and the log region gradually extends, signifying a partial recovery of the boundary layer towards its canonical state. Conversely, at a tip clearance of $0.5\delta _0$ (see figure 3d), the attenuation of the wake of the TBL profiles is less pronounced with $x/\delta _0$. All profiles behind the rotating blades collapse onto a single curve, indicating that a greater distance from the propeller is required to observe the damping effects of the wall on the tip vortices.

Upstream of the propeller, the flow adhering to the wall undergoes acceleration, resulting in a notable rise in skin friction (figure 4a). Immediately behind the propeller, within the zone marked by the contraction of the slipstream, local APGs emerge at the wall, causing a localized reduction in skin friction ($C_f$) to its minimum (see figure 4a). This phenomenon is consistent for both tip clearances, with a more pronounced decrease observed when $\varepsilon =0.1\delta _0$. The proximity of the propeller to the wall intensifies the streamline contraction within the logarithmic region of the TBL (see figure 3a), concurrently amplifying the magnitude of the APG (Marcus et al. Reference Marcus, de Vries, Kulkarni and Veldhuis2018). In practical scenarios, the presence of a flap system at the airfoil's trailing edge can further amplify these APGs, potentially leading to flow separation, especially during short takeoffs (de Vries et al. Reference de Vries, van Arnhem, Avallone, Ragni, Vos, Eitelberg and Veldhuis2021). As the flow progresses downstream (towards $x/\delta _0=4.5$), a noteworthy trend emerges, wherein the skin friction coefficient ($C_f$ in figure 4a) experiences an increase. Values up to 1.3 times and 2 times larger than those observed in a canonical case are reached, respectively for tip clearances of $\varepsilon /\delta _0=0.5$ and 0.1. This suggests that the APG effects diminish with downstream distance and that the flow starts to feel the new ‘free stream’ where the flow is accelerated leading to higher skin friction.

Figure 5 compares the streamwise and wall-normal turbulent fluctuations at different axial distances from the propeller to the flat plate reference case for two different tip clearances. It appears that both streamwise and wall-normal fluctuations do not conform to local scaling (with $u_\tau$) as the data do not exhibit collapse (for both values of $\epsilon$). This suggests that non-local flow parameters are required to capture the trends in turbulence statistics. This is analogous to history effects of pressure gradients where local $u_\tau$ does not necessarily collapse the data since the outer-layer large scales and the smaller scales nearer to the wall experience different effects. This needs further exploration in future work. Furthermore, the peak values and their wall-normal positions appear to depend on the axial distance ($x/\delta _0$) from the propeller position. With the scaling in wall units, as the streamwise distance ($x/\delta _0$) increases, the peaks of $u$ and $v$ fluctuations diminish (see figure 5). The same occurs particularly for the wall-normal statistics when scaling the data with the outer scale (see figure 6b,d). The dependence of the peak of the streamwise velocity fluctuations on $x/\delta _0$ is less evident if compared with the results in wall units (see figure 5a,c). When the tip clearance is small ($\varepsilon =0.1\delta _0$), specifically at $x/\delta _0=0.5$ behind the propeller, the ${\rm r.m.s.}(u)^+$ profile manifests a distinctive double-peak shape (see figure 5a). The same pattern is shown with the outer scaling of the flow statistics (see figure 6a,c). This is a consequence of the large scales generated by the rotation of the propeller at the height of the blade's tip. A similar trend is depicted in the work of Harun et al. (Reference Harun, Monty, Mathis and Marusic2013), where the impact of a mild APG on the large scales of a TBL was investigated. Subsequently, with an increase in $x/\delta _0$, these profiles transit to a single peak, which then progressively attenuates.

Figure 5.

Root mean square of both axial and wall-normal fluctuations in wall units. Comparison at two different tip clearances: (a,b) $\varepsilon =0.1\delta _0$, (c,d) $\varepsilon =0.5\delta _0$, (a,c) r.m.s. of $u$ in wall units, (b,d) r.m.s. of $v$ in wall units; circles, velocities in wall units; vertical dashed lines, streamwise positions of the blades’ tip in wall units; blue gradients from lighter to darker, $x/\delta _0=0.5$, 1, 1.5, 2.5, 3.5, 4.5; black circles, flat plate at $Re_{\theta }=12\,200$ ($Re_{\tau }=4400$).

Figure 6.

Root mean square of both axial and wall-normal fluctuations in outer units. Comparison at two different tip clearances: (a,b) $\varepsilon =0.1\delta _0$, (c,d) $\varepsilon =0.5\delta _0$, (a,c) r.m.s. of $u$, (b,d) r.m.s. of $v$; circles, velocities; vertical dashed line, streamwise positions of the blades’ tip; blue gradients from lighter to darker, $x/\delta _0=0.5$, 1, 1.5, 2.5, 3.5, 4.5; black circles, flat plate at $Re_{\theta }=12\,200$ ($Re_{\tau }=4400$).

The key disparity in assessing the effects of different tip clearances lies in the manifestation of the turbulence intensities. As shown by You et al. (Reference You, Wang, Moin and Mittal2007), the streamlines of the tip-gap flow experience a sudden change of directions along the span of the blade's tip, which generates spanwise derivatives of the mean $U$ and $V$. When the propeller is mounted closer to the wall, the derivatives ${\partial U}/{\partial z}$ and ${\partial V}/{\partial z}$ increase in magnitude and so does the turbulent mixing. For $\varepsilon =0.1\delta _0$, the increase in turbulence intensities spans the entire log-law region and wake, commencing from $y^+=100$. In contrast, for the alternative tip clearance case ($\varepsilon =0.5\delta _0$), the effect is observed as a transient impulse of turbulent energy solely in the wake (see figure 5c,d). Importantly, the mean profile (and the log region) remains unaffected in this scenario. Crespo, Hernandez & Frandsen (Reference Crespo, Hernandez and Frandsen1999) reviewed different wake modelling methods for wind turbine farms design and defined the aforementioned localized peak of turbulence intensity seen at $\varepsilon =0.5\delta _0$ as ‘added turbulence’, i.e. the TKE added by the propeller locally to the periodic and random fluctuations at the blade tip's location.

3.2. Two-dimensional coherent structures

The spatial two-point correlation analysis of streamwise fluctuations (see figure 7) offers valuable insights into the coherent structures arising from the interaction between the propeller and the wall boundary layer. The two-point correlation coefficient between two velocity fluctuations $R_{ij}$ is defined as in Ganapathisubramani et al. (Reference Ganapathisubramani, Hutchins, Hambleton, Longmire and Marusic2005), i.e.

(3.8)\begin{equation} R_{ij}(x,y) = \frac{\overline{u_i(x,y)u_j(x+\Delta x,y+\Delta y)}}{\sigma_i \sigma_j}, \end{equation}

where the overline notation is an ensemble average over multiple time realizations at all the angular positions of the blade, $\sigma$ represents the standard deviation of the physical quantity and $(\Delta x, \Delta y)$ are the distances from the reference point $(x,y)$. Equation (3.8) is therefore a covariance of two fluctuating velocity components evaluated at two different points normalized in a way that $R_{ii}(0,0) = 1$. Conducted at six reference points spanning the entire axial distance within the field of view and fixed at both $y=0.1\delta _0$ for $\varepsilon =0.1\delta _0$ (figure 7a,c,e,g,i,k) and $\varepsilon =0.5\delta _0$ (figure 7b,d,f,h,j,l), and $y=0.5\delta _0$ for $\varepsilon =0.1\delta _0$ (figure 8a,c,e,g,i,k) and $\varepsilon =0.5\delta _0$ (figure 8b,d,f,h,j,l), this statistical tool reveals distinctive observations for different tip clearances.

Figure 7.

Two-point correlation of streamwise fluctuations at several axial reference points $y_{Ref}/\delta _0=0.1$: (a,c,e,g,i,k) $\varepsilon =0.1\delta _0$, (b,d,f,h,j,l) $\varepsilon =0.5\delta _0$, (a,b) $x/\delta _0=0.5$, (c,d) $x/\delta _0=1$, (e,f) $x/\delta _0=1.5$, (g,h) $x/\delta _0=2.5$, (i,j) $x/\delta _0=3.5$, (k,l) $x/\delta _0=4.5$, (a–l) $y/\delta _0=0.1$; $\Delta x/\delta _0=0$, x coordinate of the reference point.

Figure 8.

Two-point correlation of streamwise fluctuations at several axial reference points with $y_{Ref}/\delta _0=0.5$: (a,c,e,g,i,k) $\varepsilon =0.1\delta _0$, (b,d,f,h,j,l) $\varepsilon =0.5\delta _0$, (a,b), $x/\delta _0=0.5$, (c,d) $x/\delta _0=1$, (e,f) $x/\delta _0=1.5$, (g,h) $x/\delta _0=2.5$, (i,j) $x/\delta _0=3.5$, (k,l) $x/\delta _0=4.5$, (a–l) $y/\delta _0=0.1$; $\Delta x/\delta _0=0$, x coordinate of the reference point.

At $\varepsilon =0.5\delta _0$, the coherent structure, captured at each different reference point, closely resembles those obtained in a flat plate boundary layer without pressure gradients. Their size and tilt angle are found to be respectively $(0.8 \pm 0.1)\delta _0$ and $(11.5 \pm 1)^\circ$, which closely match the values for the ZPG boundary layer reported in the work of Volino (Reference Volino2020) (for a ZPG, ${\rm size}=0.6\delta _0\unicode{x2013}1.0\delta _0$, ${\rm angles}=10^\circ \unicode{x2013}12^\circ$). The size of the coherent structures is measured as twice the maximum distance between the reference point of the two-point correlation and the isoline $R_{uu}=0.5$. In Volino (Reference Volino2020) it is shown that when the boundary layer is subjected to APGs, the streamwise length of the coherent structures shrink to $0.6\delta _0$ at angles up to $15^\circ$. Therefore, it can be inferred that in figures 7(b), 7(d), 7(f), 7(h), 7(j) and 7(l) there is little or insignificant influence of the rotating blades on the developing boundary layer in terms of velocity fluctuations, as their structure resembles that of a ZPG. Importantly, both the shape and intensity of coherency remain unaltered with the distance between the reference point and the propeller tip. By shifting the reference point further from the wall to $y/\delta = 0.5$, the shape of the correlation coefficient distribution (centred at $y_{Ref}$) changes, aligning the structures with the wall-normal direction (see figures 8 and 10). In figures 8(b), 8(d), 8(f), 8(h), 8(j) and 8(l), the size of the coherent structure, depicted by $R_{uu}$ and centred at the reference point, diminishes to $(0.4 \pm 0.1)\delta _0$. Notably, unlike in figures 7(b), 7(d), 7(f), 7(h), 7(j) and 7(l), where a single structure (as seen in canonical TBLs) is shown, several portions in the outer layer are correlated in the region $y > 0.3\delta _0$. Conversely, when the blades are in closer proximity to the wall ($\varepsilon =0.1\delta _0$), the two-point correlations behind the propeller unveil the generation of correlated and anti-correlated structures stacked in pairs (see figure 7a), which is consistent with the increased turbulence levels throughout the log region (see figure 5a,b). The vertical alignment of these structure pairs tilts with downstream distance ($x/\delta _0$), and beyond $2.5\delta _0$ from the propeller (see figure 7g), the coherence of these structures is disrupted. As the reference point moves downstream, the intensity of the two-point correlation structures diminishes, fading away at $4.5\delta _0$ from the tip of the blades. For longer distances, a single, smaller structure remains, weakly correlated to the outer region of the boundary layer. Interestingly, with a reference point at $y=0.5\delta _0$, a single large structure is observed, spanning the range $1.2 > y/\delta _0 > 0.2$, regardless of the axial distance chosen as a reference for the two-point correlation (see figure 8a,c,e,g,i,k). This implies that the outer region is more influenced by the wake of the propeller and less by the wall-bounded tip vortices.

Similar outcomes are observed in the two-point correlation analysis of wall-normal fluctuations (see figure 9), with the primary difference being the stretched wall-normal orientation of $R_{vv}$ contours. This occurs only when the tip clearance is $\varepsilon =0.1\delta _0$, in the left column of figure 9, indicating that the wall/propeller interaction significantly influences the coherence of velocity fluctuations in the wall-normal direction throughout the developing boundary layer. For the larger tip clearance ($\varepsilon =0.5\delta _0$), the contours once again resemble that of a canonical boundary layer that does not seem to be influenced by the presence of the propeller. With the reference point at $y=0.5\delta _0$, the correlation patterns change slightly. At $\varepsilon =0.5\delta _0$ (see figure 10b,d,f,h,j,l), more structures are correlated in vertically oriented pockets of size $1\delta _0$, confined to the region $y > 0.3\delta _0$, and thus, do not affect the log layer of the TBL. In contrast, at $\varepsilon =0.1\delta _0$ (see figure 10a,c,e), the wall-normal oriented structures, centred at the reference point, are correlated with the wall-bounded structures behind the propeller over an extension of $\Delta x=2\delta _0$. In this case, the flow in the outer region impacts the wall-normal turbulence intensity more significantly compared with $\varepsilon =0.5\delta _0$.

Figure 9.

Two-point correlation of wall-normal fluctuations at several axial reference points with $y_{Ref}/\delta _0=0.1$: (a,c,e,g,i,k) $\varepsilon =0.1\delta _0$, (b,d,f,h,j,l) $\varepsilon =0.5\delta _0$, (a,b) $x/\delta _0=0.5$, (c,d) $x/\delta _0=1$, (e,f) $x/\delta _0=1.5$, (g,h) $x/\delta _0=2.5$, (i,j) $x/\delta _0=3.5$, (k,l) $x/\delta _0=4.5$, (a–l) $y/\delta _0=0.1$; $\Delta x/\delta _0=0$, x coordinate of the reference point.

Figure 10.

Two-point correlation of wall-normal fluctuations at several axial reference points with $y_{Ref}/\delta _0=0.5$: (a,c,e,g,i,k) $\varepsilon =0.1\delta _0$, (b,d,f,h,j,l) $\varepsilon =0.5\delta _0$, (a,b) $x/\delta _0=0.5$, (c,d) $x/\delta _0=1$, (e,f) $x/\delta _0=1.5$, (g,h) $x/\delta _0=2.5$, (i,j) $x/\delta _0=3.5$, (k,l) $x/\delta _0=4.5$, (a–l) $y/\delta _0=0.1$; $\Delta x/\delta _0=0$, x coordinate of the reference point.

Given that the structure of the fluctuations either conform to that of the boundary layer or that of the propeller wake, it is important to understand if the periodic velocity component from the propeller is a significant contributor to these fluctuations. This is explored in the next section.

3.3. Mechanics of the periodic motion

The triple decomposition methodology (Reynolds & Hussain Reference Reynolds and Hussain1972; Lignarolo et al. Reference Lignarolo, Ragni, Scarano, Ferreira and Van Bussel2015) serves as a valuable tool to disentangle the statistics of the fluctuations contributed solely by the periodic motions, generated through the rotation of the blades. The tool is capable of shedding light on the influence of the periodic motions on the background random fluctuations within the boundary layer. In general, according to the triple decomposition, a fluctuating velocity component can be divided into

(3.9)\begin{equation} u_i = \overline{u_i} + \tilde{u}_{i} + u_{s,i} , \end{equation}

where $\overline {u_i}$ is the time-average contribution, $\tilde {u}_{i}$ is the periodic motion, $u_{s,i}$ is the background turbulent fluctuation of the signal and $i$ indicate the coordinate directions, with $i=1, 2$ corresponding to axial and wall-normal directions, respectively. The organized motion ($\tilde {u}_{i}$) is obtained by subtracting the time-average value ($\overline {u_i}$) from the phase average ($\langle u \rangle$), i.e. the conditional average over the points that have the same phase with respect to an oscillator, which in this case is the propeller:

(3.10)\begin{equation} \tilde{u}_{i} = \langle u_i \rangle - (\overline{u_i}) . \end{equation}

Effectively, the phase averaging of the velocity components allows rejecting the background turbulence nature of the field and extracting only the organized periodic motions from the signal (Reynolds & Hussain Reference Reynolds and Hussain1972).

Figure 11 provides a comprehensive analysis of the impact of the phase of the rotating propeller on the triple decomposed fluctuations. The investigation spans from $0^\circ$, where the blades are perpendicular to the wall, to $50^\circ$ with an increment of $10^\circ$.

Figure 11.

Phase-locked profiles behind the propeller ($x=0.5\delta _0$) of both organized periodic and random fluctuations at two different tip clearances: (a,c,e,g) $\varepsilon =0.1\delta _0$, (b,d,f,h) $\varepsilon =0.5\delta _0$, (a,b) $\tilde {u}^+$, (c,d) $\tilde {v}^+$, (e,f) r.m.s. of $u_s^+$, (g,h) r.m.s. of $v_s^+$, (a–h) $x/\delta _0=0.5$; red dashed line, blades tip position in wall units at $\phi =50^\circ$; red gradients from darker to lighter, $\phi =0^\circ$, $10^\circ$, $20^\circ$, $30^\circ$, $40^\circ$, $50^\circ$; the velocity error estimation in wall units is $\varDelta ^+ = 0.25$.

At $\varepsilon =0.1\delta _0$ (see figure 11a,c), the amplitude of the periodic motions is significant, affecting an extensive portion of the boundary layer. As the blade tip moves further away from the wall, the amplitude of $\tilde {u}$ and $\tilde {v}$ decreases. During the rotation from $\phi =0^\circ$, the blades draw the flow toward the wall-normal direction, diminishing the periodic fluctuations, which are particularly peaking at $\phi =10^\circ$ for $\tilde {u}$ and $\phi =0^\circ$ for $\tilde {v}$. Differently, at $\varepsilon =0.5\delta _0$ (see figure 11b,d), the trend is inverted. The amplitude of the periodic structures amplifies as the blade tip moves away from the wall (as $\phi$ increases from $0^\circ$ to $50^\circ$). Therefore, an initial observation can be made: with the rotating blades closer to the wall ($\varepsilon =0.1\delta _0$), the amplitude of the periodic motion ($\tilde {u}$ and $\tilde {v}$) is lowered as the angle of rotation $\phi$ increases, especially in the region of maximum slipstream contraction ($x=0.5\delta _0$). Further downstream (see figure 12a–d) for $x=2.5\delta _0$), the periodic features of the flow gets attenuated, when compared with figures 11(a)–11(d).

Figure 12.

Phase-locked profiles behind the propeller ($x=2.5 \delta _0$) of both organized periodic and random fluctuations at two different tip clearances: (a,c,e,g) $\varepsilon =0.1\delta _0$, (b,d,f,h) $\varepsilon =0.5\delta _0$, (a,b) $\tilde {u}^+$, (c,d) $\tilde {v}^+$, (e,f) r.m.s. of $u_s^+$, (g,h) r.m.s. of $v_s^+$, (a–h) $x/\delta _0=2.5$; red dashed line, blades tip position in wall units at $\phi =50^\circ$; red gradients from darker to lighter, $\phi =0^\circ$, $10^\circ$, $20^\circ$, $30^\circ$, $40^\circ$, $50^\circ$; the velocity error estimation in wall units is $\varDelta ^+ = 0.25$.

The periodic structures induced by the rotating blades exhibit a phase-driven movement, and their interaction with the mean flow contributes to the generation of background random fluctuations. At $\varepsilon =0.1\delta _0$ (see figure 11e), the amplitude of the root-mean-square (r.m.s.) profiles of $u_{s}^+$ decreases with increasing angular position of the propeller. Conversely, the r.m.s. peaks of the wall-normal fluctuation $v_{s}^+$ profiles increase with the blade phase angle (see figure 11g). This suggests that at the smaller investigated tip clearance, the presence of the wall attenuates both the periodic and random fluctuations, with an exception for the wall-normal variance ${\rm r.m.s.}(v_s)^+$.

Further downstream from the propeller, at $x=2.5\delta _0$ (see figure 12e,g), both the streamwise and wall-normal stochastic fluctuations of the flow, specifically at $\epsilon = 0.1\delta _0$, are not dependent on the angular rotation of the blades, whose amplitudes are much lower compared with $x=0.5\delta _0$ in figures 11(e) and 11(g). No anomalies are seen in this region due to the lack of slipstream/boundary-layer interaction. At $\varepsilon =0.5\delta _0$ (see figures 11f,h and 12f,h), the r.m.s. profiles of $u_{s}^+$ and $v_{s}^+$ behave in the same fashion with the angular phase, showing peak values at both $\phi =50^\circ$. Comparing these results with the statistics for the tip clearance case $\varepsilon =0.1\delta _0$, the portion of boundary layer affected by the interaction by the tip vortices is lower, extending from $y^+=2000$ to $y^+=6000$, ensuring that the log region of the boundary layer is not affected (contrary to the case $\varepsilon =0.1\delta _0$ where the tip-vortex influence extends down to $y^+=100$).

3.4. Phase-locked turbulent structures

The divergent trends observed in the angular phase of background fluctuations between the streamwise and wall-normal components at $\varepsilon /\delta _0 = 0.1$ (figure 11e,g) suggest a distinct phenomenon, which becomes apparent as the blade tip diverges from the wall. These findings necessitate a further look at the turbulent flow structure at specific locations to explore the features that may be responsible for these statistics. The location of the slipstream contraction is a major factor that affects the observations in the previous section and, therefore, we examine two-point correlations of the random fluctuations at specific phases angles (of the propeller) in the vicinity of the slipstream.

Figure 13 provides an examination of coherent structures captured at different angular positions using two-point correlations, with reference points located at $x/\delta _0=0.5$. The wall-normal reference is taken to be at the location of the slipstream (i.e. $y_{ref} = \epsilon$). The analysis focuses on fluctuations in the wall-normal direction only as these showed opposing trends to the streamwise fluctuations in the previous section. The analysis spans angular positions from $0^\circ$ (blade perpendicular to the wall) to $50^\circ$, with the effects beyond this range remaining unknown.

Figure 13.

Two-point correlation of wall-normal fluctuations at phase-locked positions evaluated for two different tip clearances: (a,c,e,g,i,k) $\varepsilon /\delta _0=0.1$, (b,d,f,h,j,l) $\varepsilon /\delta _0=0.5$. For all subfigures, (a–l) $x/\delta _0=0.5$: (a,b) $\phi =0^\circ$, (c,d) $\phi =10^\circ$, (e,f) $\phi =20^\circ$, (g,h) $\phi =30^\circ$, (i,j) $\phi =40^\circ$, (k,l) $\phi =50^\circ$; negative values of $R_{vv}$ filtered in (d,f,h,j,l) due to a lack of anti-correlated structures; the region upstream of the blade is being masked due to the presence of its shadow during PIV measurements, which decreases the signal-to-noise ratio; first column on the left, schematics of the phase-locked propeller positions, in the plane perpendicular to the flow (y–z).

For $\varepsilon /\delta _0=0.5$ (see figure 13b,d,f,h,j,l), when the blade is perpendicular to the wall, an alternate arrangement of positive and negative correlation is observed behind the blade tip (figure 13b). The magnitude of correlation diminishes as the blade rotates away from the wall, suggesting that the tangential sweeping of the blades is not intense enough to impact the phase-locked ‘random’ vortices created by the interaction between the boundary layer and the propeller. As the blades rotate, they generate a swirling motion in their vicinity, adding a tangential component to the flow. Closer to a wall, along the height of the blade, spanwise derivatives of U and V are produced, becoming stronger near the wall (You et al. Reference You, Wang, Moin and Mittal2007). These stronger flow derivatives, which vary over time with the passage of the blades, enhance turbulent activity. Consequently, local random fluctuations depend on the phase position of the blade and the spanwise derivatives of the flow. Therefore, a decreased correlation $R_{ii}$ indicates that flow structures in that region fluctuate coherently with a lower intensity, as a result of the smaller magnitudes of the spanwise derivatives generated by the blades’ tangential motion. Conversely, for $\varepsilon /\delta _0=0.1$ (see figure 13a,c,e,g,i,k), when the blades are mounted closer to the wall, a different scenario is observed. As the blades rotate, the rotational effect amplifies the interaction between the propeller and the boundary layer. Coherent vortices behind the propeller exhibit strong correlation with absolute values that get close to unity. This is particularly evident in the train of vertical structures created by the propeller's rotation. These vertical and narrow regions in the flow field fluctuate with the same intensity in the wall-normal direction, becoming more pronounced as the blades move further from the wall.

In figure 14 the same correlation analysis is shown only for $\varepsilon /\delta _0=0.1$, focusing on reference points at two streamwise positions: $x/\delta _0=1$ (see figure 14a,c,e,g,i,k) and $x/\delta _0=2.5$ (see figure 14b,d,f,h,j,l). This reveals that as the reference point moves downstream, the intense correlation values of the turbulent structures exhibit a modest increase with $x/\delta _0$ when compared with the $x/\delta _0=0.5$ case. It can be inferred that the presence of more intense coherent structures during the blades rotation is associated with a narrower contraction of the stream tube (at approximately $x/\delta _0=0.5$) and, consequently, with stronger APGs generated by the propeller itself. This suggests that the periodic motions generated by propeller rotation also generate corresponding turbulent fluctuations that remain in phase. Therefore, there should be a mechanism of energy transfer between the periodic motions to the random ones and this is explored further in the next section.

Figure 14.

Two-point correlation of wall-normal fluctuations at phase-locked positions evaluated for $\varepsilon /\delta _0=0.1$: (a,c,e,g,i,k) $x/\delta _0=1$, (b,d,f,h,j,l) $x/\delta _0=2.5$, (a,b) $\phi =0^\circ$, (c,d) $\phi =10^\circ$, (e,f) $\phi =20^\circ$, (g,h) $\phi =30^\circ$, (i,j) $\phi =40^\circ$, (k,l) $\phi =50^\circ$; the region upstream of the blade is being masked due to the presence of its shadow during PIV measurements, which decreases the signal-to-noise ratio; first column on the left, schematics of the phase-locked propeller positions, in the plane perpendicular to the flow (y–z).

3.5. Viscous dissipation and energy transfer

The triple decomposition method also allows further investigation of how the processes of energy transfer among mean flow, wave-induced and fluctuating fields take place. The travelling waves generated by the periodic motion of the propeller distort the surrounding turbulent field and interact distinctively with the boundary layer at different tip clearances. As a consequence, the altered energetic balance between dissipation into thermal energy, production and diffusion of turbulent kinetic energy can be assessed. The total time-averaged kinetic energy per unit mass at a specific point is

(3.11) \begin{equation} \tfrac{1}{2} (\overline{u_i u_i}) = \tfrac{1}{2} \left( \bar{u_i}\, \bar{u_i} + \overline{\tilde{u}_{i} \tilde{u}_{i}} + \overline{u_{s,i} u_{s,i}} \right). \end{equation}

The formulation for the second component of the kinetic energy on the right-hand side of (3.11) is obtained by first doing a phase averaging and then a time averaging of the momentum equation of the periodic motion multiplied by $\tilde {u}_{\imath }$ (Reynolds & Hussain Reference Reynolds and Hussain1972):

(3.12) \begin{align} \frac{\bar{D}}{D t}\left(\frac{1}{2} \overline{\tilde{u}_{\imath} \tilde{u}_{\imath}}\right) &={-}\frac{\partial}{\partial x_j}\left[\overline{\tilde{u}_{\jmath}\left(\tilde{p}+\frac{1}{2} \tilde{u}_{\imath} \tilde{u}_{\jmath}\right)}\right] \nonumber\\ & \quad + \left(-\overline{\tilde{u}_{\imath} \tilde{u}_{\jmath}}\right) \frac{\partial \bar{u}_{\imath}}{\partial x_j} - \overline{\left(-\left\langle u_{s, \imath} u_{s, j} \right\rangle\right) \frac{\partial \tilde{u}_{\imath}}{\partial x_j}} \nonumber\\ & \quad -\frac{\partial}{\partial x_j}\overline{\left[\tilde{u}_i\left\langle u_{s, i} u_{s, j}\right\rangle\right]} + \frac{1}{R e} \frac{\partial}{\partial x_j}\left[\overline{\tilde{u}_i \left(\frac{\partial \tilde{u}_t}{\partial x_j}+\frac{\partial \tilde{u}_j}{\partial x_i}\right)}\right] \nonumber\\ & \quad -\frac{1}{2 \,{Re}} \overline{\left(\frac{\partial \tilde{u}_i}{\partial x_j}+\frac{\partial \tilde{u}_j}{\partial x_i}\right)\left(\frac{\partial \tilde{u}_i}{\partial x_j}+\frac{\partial \tilde{u}_j}{\partial x_i}\right)} . \end{align}

The right-hand side of (3.12) delineates the main components governing the net change rate of the kinetic energy content (left-hand side of (3.12)) of the organized periodic motions. The second and third terms on the right-hand side of (3.12) are the two most dominant entities affecting the energy balance. The second term, $-\overline {(\tilde {u}_{i} \tilde {u}_{j})} (\partial (\overline {u_i})/\partial x_j)$, represents the turbulent production due to the action of the mean flow on the Reynolds stresses of the periodic motion generated by the rotation of the blades. Thus, it represents the amount of energy drained from the mean flow to produce turbulence in the periodic organized field. The third term, $-(-\overline {\langle u_{s,i} u_{s,j} \rangle (\partial \tilde {u_i}/\partial x_j)})$, is a drain of turbulent energy due to the interaction between the wave field and the Reynolds stresses of the background turbulence. The statistical components of these two terms are computed as described in Lignarolo et al. (Reference Lignarolo, Ragni, Scarano, Ferreira and Van Bussel2015), considering solely the fluctuations in the near wall rotation of the blades, at $\Delta \phi =(0\unicode{x2013}50)^\circ$ . Another important term is the last component of the right-hand side of (3.12), which is the viscous dissipation of the periodic motion. The mechanics behind the interaction propeller/wall-bounded flow can therefore be depicted as follows.

1. (i) The rotation of the blades injects periodically organized structures that alter the boundary layer by taking away energy from the mean flow to support its organized wave movement (Reynolds & Hussain Reference Reynolds and Hussain1972).

2. (ii) The presence of the wall disrupts the periodic motions, which then break down and amplify the Reynolds stresses of the turbulent background field.

The mechanisms of vortex disruption are provided also in the work of Lignarolo et al. (Reference Lignarolo, Ragni, Scarano, Ferreira and Van Bussel2015) for the case of a propeller rotating into a free stream. The authors show that the mechanism triggering the breakup of helicoidal tip vortices and the amplification of turbulent mixing is due to the mutual interaction among vortices rather the presence of wall vorticity, as in our case. Their work illustrates that the pairing instability of the vortices in the free stream occurs at a streamwise distance of $1.5D$ from the blades. However, in our case, we demonstrate that the presence of the boundary layer is the dominant factor that influences the tip vortices displacement, reducing the distance at which the structures lose their coherence and periodicity to ${\leqslant }0.9D$ (see figure 2). This mechanism is further examined using triple decomposition to particularly emphasize the effects of different propeller tip clearances on energy exchanges.

In the wake of the propeller, particularly at $x/\delta _0=0.5$, a detailed analysis of the turbulent production and energy transfer reveals intricate dynamics within the boundary layer (see figure 15). The turbulent production associated with the energy transfer between decomposed fields is particularly relevant for $y^+$ values higher than $100$ and for small tip clearances, as shown in figures 15(a) and 15(c). For $\varepsilon /\delta _0=0.1$ and at $x/\delta _0=0.5$, in the log layer up to $y^+=800$, energy is extracted from the mean flow, contributing to the turbulent production of periodic motions (see figure 15a). Simultaneously, the wake region experiences negative production, indicating an energy transfer in the opposite direction. This interplay results in the energizing of the turbulent mean flow by the periodic motion generated by the propeller. As the downstream distance increases ($x/\delta _0$), the turbulent production becomes negative throughout the log region, gradually attenuating the amplitude of this energy transfer until null values are reached. Conversely, the turbulent production within the background fluctuating field is predominantly positive from $y^+=100$ to $y^+=1000$ (see figure 15c). An exception occurs at $x/\delta _0=0.5$, where there is a relevant negative peak inside the log law, indicating energy transfer from the background field to the periodic motion due to turbulent production from the shear of periodic waves. For $\varepsilon /\delta _0=0.5$ (see figure 15b,d), the energy exchange for turbulent production within the two different fields is negligible compared with $\varepsilon /\delta _0=0.1$ (see figure 15a,c). With increased tip clearance, reduced interaction among the triple decomposition fields is observed, and turbulent productions and energy dissipation by viscosity predominantly occur in the far wake for $y^+$ values exceeding $2500$ (see figure 15f). In the case of $\varepsilon /\delta _0=0.1$, significant viscous dissipation is observed (see figure 15e), particularly at the streamwise position of $x/\delta _0=0.5$, where slipstream contraction occurs. This region experiences amplified turbulent production in the periodic field, indicative of intense shear.

Figure 15.

Dimensionless turbulent production in the organized periodic motion, in the background turbulence and energy dissipated by viscosity for two different tip clearances: (a,c,e) $\varepsilon =0.1\delta _0$, (b,d,f) $\varepsilon =0.5\delta _0$. (a,b) Turbulent production in the periodic motion due to shear flow of the time average field. (c,d) Turbulent production of the background field due to shear flows of the periodic motion. (e,f) Dimensionless viscous dissipation. The blue gradients from lighter to darker represent $x/\delta _0=0.5$, $1$, $1.5$, $2.5$, $3.5$, $4.5$.

Considering the findings presented earlier, a correlation becomes apparent between local APGs at $\varepsilon /\delta _0=0.1$ (induced by the strong slipstream contraction) and turbulent production along with viscous dissipation. A scenario quite different from the case of a propeller rotating in a free stream, where Lignarolo et al. (Reference Lignarolo, Ragni, Scarano, Ferreira and Van Bussel2015) demonstrate that there is no turbulent production or kinetic energy mixing in the near wake (known as the tip-vortex shield effect).