2. Experimental arrangement

The data analysed here were acquired experimentally in a room measuring $6.5\times 8.0\times 6.5$ rotor diameters; a schematic of this is shown in figure 1. The set-up comprised a 1.0 m diameter ( $D=2R$ ) single-bladed rotor (untwisted NACA 0012 airfoil with a constant chord of $c=52~\text{mm}$ and square tip) attached at the hub by a flap hinge. The rotor was balanced with a counterweight (statically up to 99 %) and was operated at ${\it\Omega}=1500$ revolutions per minute. This corresponds to a tip chord Reynolds number ( $\mathit{Re}_{c}$ ) of 218 000 and a tip Mach number of 0.23. The blade was set to a collective pitch angle of $7.3^{\circ }$ , thus resulting in a blade loading ( $C_{T}/{\it\sigma}$ ) of 0.066, as measured by a fixed-frame load cell.

Figure 1. Schematic of the experimental set-up and coordinate transformation.

Velocity measurements were performed along a two-dimensional slice of the rotor wake (along a plane normal to the vortex filament axis) by phase-aligning the rotor with a PIV system using a $1/\text{rev}$ optical switch. The orientation of the PIV system (laser and camera) relative to the measurement window is shown in figure 1. The laser sheet was oriented such that it was aligned along the quarter-chord of the blade at 0° vortex age. The focal distance of the PIV camera was chosen to capture only one vortex (including provisions for vortex wander) so as not to compromise on spatial resolution. At each vortex age, the camera was repositioned by way of a two-degree-of-freedom traverse so that the new vortex location was centred on the camera image. Measurement accuracies are estimated to be within $0.3^{\circ }$ in azimuth given the inter-frame timing ( $36~{\rm\mu}\text{s}$ ) between successive laser pulses (laser sheet thickness of 2 mm) and the rotation speed of the rotor. For the current set-up, misalignment of the measurement plane (in $z^{\prime }$ direction) relative to the calibration plane is of the order of the laser sheet thickness. Based on the object distance ( $z_{o}=905~\text{mm}$ ) and following § 2.2 in Discetti & Adrian (Reference Discetti and Adrian2012), the resultant magnification error was found to be less than 0.2 %.

Seeding was provided by a PIVTEC 14 cascadable Laskin nozzle olive oil seeder, which produced $1~{\rm\mu}\text{m}$ diameter particles. These small particles minimize particle tracking errors that are inherent to vortex-dominated flows (Leishman Reference Leishman1996). A check for peak locking errors was performed to ensure that particle image displacements were unbiased by the integer pixel dimension. Additional details concerning these measurements are described by Mula et al. (Reference Mula, Stephenson, Tinney and Sirohi2013).

Because of the relatively low blade loading on the rotor, the helical pitch ( $\hat{p}=H/(2{\rm\pi}R_{c})=0.046$ , where $H$ is the vertical displacement of the helix over one rotor revolution and $R_{c}=[1+(H/(2{\rm\pi}R))^{2}]R$ is the radius of curvature) and vortex filament torsion ( $\hat{{\it\tau}}=P/R=0.046$ using $P=H/(2{\rm\pi})$ as the reduced pitch) were also small so that the measurement plane was nearly orthogonal to the vortex axis (Leishman, Baker & Coyne Reference Leishman, Baker and Coyne1996; Han, Leishman & Coyne Reference Han, Leishman and Coyne1997; Mula et al. Reference Mula, Stephenson, Tinney and Sirohi2013). The resultant angle of inclination between the vortex axis and a line normal to the measurement plane was estimated to be $2.66^{\circ }$ . A total of 350 statistically independent PIV snapshots were acquired at each of the 10 vortex ages studied, ${\it\psi}=\{45^{\circ },90^{\circ },135^{\circ },180^{\circ },225^{\circ },270^{\circ },315^{\circ },405^{\circ },495^{\circ },585^{\circ }\}$ , using a field of view of $88~\text{mm}\times 66~\text{mm}$ . Images were processed using DaVIS v.7.2 with an initial window size of $64~\text{pixel}\times 64~\text{pixel}$ that iteratively reduced to a final window size of $16~\text{pixel}\times 16~\text{pixel}$ with 50 % overlap. The resulting measurement resolution ( $L_{m}$ ) was found to be 0.88 mm ( $\text{grid resolution}=L_{m}/2$ ). Spurious vectors were filtered by first establishing a threshold signal-to-noise ratio (set to 1.5), followed by the removal of groups (of less than five vectors) that then ended with a four-pass regional median filter (Westerweel Reference Westerweel1994; Nogueira, Lecuona & Rodriguez Reference Nogueira, Lecuona and Rodriguez1997). This often occurred in the inner core of the vortex due to the poor seeding levels that arise from centrifugal forces (see figure 2 in Mula et al. Reference Mula, Stephenson, Tinney and Sirohi2013). Missing vectors were then interpolated using a nearest-neighbour fit. Figure 2(a,b) demonstrates a sample raw PIV vector map (subsampled and filtering enabled), and the resultant vector map (after interpolation and smoothing) that was used for subsequent analysis. The post-processed vector map is shown to fit the overall flow structure quite well.

Figure 2. (a) Sample original vector map (subsampled) at ${\it\psi}=45^{\circ }$ with (b) corresponding post-processed vector map. The vortex centre and core boundary are also identified. (c) Standard deviation of vortex wander along the principal major (♢, ——) and minor (▵, $\cdots \cdots$ ) axes.

3. Vortex characteristics

Vortex wander is an inactive motion of the vortex that produces artificially high-velocity fluctuations in the vortex core (Devenport et al. Reference Devenport, Rife, Liapis and Follin1996). Corrections for this were implemented here using an integral-based approach (the ${\it\Gamma}_{1}$ method) that allowed for the vortex centres to be identified; the details of this are described elsewhere (Graftieaux, Michard & Grosjean Reference Graftieaux, Michard and Grosjean2001; Mula et al. Reference Mula, Stephenson, Tinney and Sirohi2013). Previous studies using multi-bladed rotors have shown the vortex wandering motion to be anisotropic (Kindler, Mulleners & Richard Reference Kindler, Mulleners and Richard2010; Mula et al. Reference Mula, Stephenson, Tinney and Sirohi2013). Similar results are observed here, where figure 2(c) shows the standard deviation of wander along the principal major and minor axes across all vortex ages. A third-order least-squares fit has also been added (identified by the solid and dashed lines) to help with the interpretation. This wandering motion is shown to grow linearly along the principal minor axis. Wander along the major axis grows faster than that of the minor axis at earlier vortex ages. However, the wandering motion (along the major axis) does not appear to show a continuous growth, as it deviates from the least-squares fit in the vicinity of the first blade passage (360°). It is postulated that this discontinuity is the consequence of long-wave instabilities with non-integer wavenumbers (Bhagwat & Leishman Reference Bhagwat and Leishman2000). The growth rate of these long-wave instabilities has been shown to manifest a local minimum near the first blade passage, which is evident in figure 2(c).

The average spatial topography of the mean vorticity and turbulent kinetic energy per unit mass ( $\text{TKE}=0.5\langle u_{1}^{2}+u_{2}^{2}\rangle$ with $u_{i=1,2}$ being the in-plane fluctuating velocity components), for vortices captured at ${\it\psi}=180^{\circ }$ and 495°, are shown in figure 3(a–d) after wander correction. A noticeable asymmetry is observed, which persisted across all other vortex ages (not shown). Core boundaries are identified by the locations of peak swirl velocity and so it is evident in figure 3(c,d) that TKE levels peak inside the vortex core (Devenport et al. Reference Devenport, Rife, Liapis and Follin1996; Han et al. Reference Han, Leishman and Coyne1997; Ramasamy et al. Reference Ramasamy, Johnson and Leishman2009b ). These same features are observed in figure 4(a,b), which shows slices across the average swirl velocity and TKE at ${\it\psi}=180^{\circ }$ and 495°, respectively. It is worth mentioning that the total resolved turbulent kinetic energy, based on the in-plane velocity, varies between 2 % and 8 % of the total resolved kinetic energy associated with the vortex over the range of vortex ages studied.

Figure 3. (a,b) Mean axial vorticity, ${\it\omega}_{z}/{\it\Omega}\times (-0.1)$ at (a)  ${\it\psi}=180^{\circ }$ and (b)  ${\it\psi}=495^{\circ }$ . (c,d) Plots of $(\text{TKE}/V_{t}^{2})\times 10^{3}$ at (c)  ${\it\psi}=180^{\circ }$ and (d)  ${\it\psi}=495^{\circ }$ ; $V_{t}$ is the blade-tip velocity.

Figure 4. Swirl velocity, $(V_{{\it\Theta}}/V_{t})\times 10$ (▿), and $(\text{TKE}/V_{t}^{2})\times 10^{3}$  (

) along the vortex slice labelled by ${\it\xi}$ in figure 3(c,d) at (a)  ${\it\psi}=180^{\circ }$ and (b)  ${\it\psi}=495^{\circ }$ . (c) Comparison of the core radius with the previous studies listed in table 1.

As for the core radius, close inspection of figure 4(c) shows a monotonic increase from 10 % to 15 % of the blade chord up until the first rotor revolution (this is after wander correction). A comparison of our findings with those reported in the open literature is also provided; see table 1 for a listing of the rotor operating conditions associated with these studies. At the earliest vortex age studied here ( ${\it\Psi}=45^{\circ }$ ), $L_{m}/r_{c}=0.17$ ( ${<}0.2$ ), which satisfies the measurement resolution requirements suggested by Martin et al. (Reference Martin, Leishman, Pugliese and Anderson2000) for determining core radius. In general, the trends in figure 4(c) are consistent with the open literature. That is, up until the first blade passage (360°), diffusion causes the core radius to increase with increasing vortex age. Aside from the studies of Thompson, Komerath & Gray (Reference Thompson, Komerath and Gray1988) and Richard et al. (Reference Richard, Van der Wall, Raffel and Thimm2008), the core radius is shown to decrease after the first blade passage due to vortex stretching induced by the oncoming blade. The corresponding vortex Reynolds number of our study was found to range between $8.3\times 10^{4}$ and $8.7\times 10^{4}$ over the vortex ages studied.

Table 1. Overview of experimental conditions reported by others.

Table 2. Estimates of the binormal induced velocity measured from the current experiment compared with the theoretical predictions using (3.2).

Since the torsion of the helix is very small ( $\hat{{\it\tau}}=0.046$ ), its influence is significant on the binormal induced velocity (of the vortex), which is primarily responsible for the displacement of the vortex filament in a fluid (Ricca Reference Ricca1994). Here, the binormal velocity is computed as the net in-plane velocity of the vortex,

(3.1) $$\begin{eqnarray}\boldsymbol{v}_{b}=\frac{1}{A}\iint \boldsymbol{U}(x^{\prime },y^{\prime })\,\text{d}x^{\prime }\,\text{d}y^{\prime },\quad A=\iint \text{d}x^{\prime }\,\text{d}y^{\prime },\end{eqnarray}$$

where $\boldsymbol{U}(x^{\prime },y^{\prime })$ is the mean (in-plane) velocity field within the vortex, $A$ is the area of the vortex, and the limits of integration are confined to ${\sim}2r_{c}$ (from the vortex centre). The dimensionless binormal velocity is then estimated using $\hat{v}_{b}=|\boldsymbol{v}_{b}|4{\rm\pi}R_{c}/{\it\Gamma}_{v}$ and is provided in table 2 at various vortex ages throughout the measurement envelope using ${\it\Gamma}_{v}$ as the circulation strength of the vortex. A comparison of our estimates with the theoretical predictions is also provided based on the following analytical expression from Ricca (Reference Ricca1994),

(3.2) $$\begin{eqnarray}\hat{v}_{b}=\log \frac{R_{c}}{r_{c}}+C_{MS},\end{eqnarray}$$

where $C_{MS}$ depends on the geometry of the helix and is determined using the closed analytical expression

(3.3) $$\begin{eqnarray}C_{MS}=-{\textstyle \frac{1}{4}}+\hat{p}^{-1}+\log \hat{p}+1-{\textstyle \frac{1}{2}}\hat{p}+[{\textstyle \frac{3}{8}}{\it\zeta}(3)-{\textstyle \frac{1}{2}}]\hat{p}^{2}-{\textstyle \frac{5}{8}}\hat{p}^{3}+O(\hat{p}^{4})\end{eqnarray}$$

derived by Boersma & Wood (Reference Boersma and Wood1999) for thin helical filaments of small pitch. These theoretical estimates are shown to compare favourably with our laboratory measurements.

4. POD of a spiralling vortex

Lumley’s (Reference Lumley, Yaglom and Tatarsky1967) POD is used here to identify the most energetic features of the turbulence fluctuations that reside within the blade-tip vortex. The technique has been rigorously used in both the experimental and numerical disciplines, with an extensive description of its mathematical intricacies having been provided by Berkooz, Holmes & Lumley (Reference Berkooz, Holmes and Lumley1993). In the current study, we follow the approach outlined by Glauser & George (Reference Glauser, George, Comte-Bellot and Mathieu1987), Citriniti & George (Reference Citriniti and George2000) and Tinney, Glauser & Ukeiley (Reference Tinney, Glauser and Ukeiley2008), whereby the vortex (and its surrounding fluid) is first decomposed in azimuth using Fourier decomposition followed by a radial decomposition using POD. This is applied to only the fluctuating part of the signal and after correcting for wander; the effect of vortex wandering correction on the results presented here is discussed in the Appendix A. While it is customary for one to use the snapshot POD (Sirovich Reference Sirovich1987) approach with PIV data, (given the tradeoff between spatial resolution and the number of realizations), the spatial modes described by harmonic analysis are more easily interpreted.

The process begins by first transforming the raw PIV images from Cartesian coordinates to cylindrical coordinates (i.e.  $x^{\prime },y^{\prime },z^{\prime }~\rightarrow ~r,{\it\theta},z^{\prime }$ ). At each vortex age, the azimuthal grid resolution, ${\it\delta}{\it\theta}$ , is determined such that it equals the resolution of the original grid at $r=r_{c}$ , ${\it\delta}{\it\theta}=\tan ^{-1}(L_{m}/(2r_{c}))$ . For example, at ${\it\psi}=135^{\circ }$ , ${\it\delta}{\it\theta}$ is equal to 4° (though it was found to vary only from 3° to 5° over the entire measurement envelope). The fluctuating part of the in-plane velocity field ( $u_{i=1,2}$ , where $\tilde{u} _{i}=U_{i}+u_{i}$ , and $1=r$ and $2={\it\theta}$ ) is then transformed along the azimuthal coordinate to obtain the Fourier-azimuthal modes for each of the 350 PIV snapshots at a given vortex age as follows:

(4.1) $$\begin{eqnarray}\hat{u_{i}}(r,{\it\psi},t;m)=\frac{1}{2{\rm\pi}}\int _{-{\rm\pi}}^{{\rm\pi}}u_{i}(r,{\it\theta},{\it\psi},t)\text{e}^{-\text{i}m{\it\theta}}\,\text{d}{\it\theta},\end{eqnarray}$$

from which a two-point tensor is then formed,

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D609}_{ij}(r,r^{\prime },{\it\psi};m)=\langle \hat{u_{i}}(r,{\it\psi},t;m)\,\hat{u_{j}}^{\ast }(r^{\prime },{\it\psi},t;m)\rangle .\end{eqnarray}$$

Here $\langle ~\rangle$ is used to denote ensemble averaging. In (4.1), the same azimuthal starting position is employed in order to preserve the asymmetries that are shown to reside in the TKE profiles in figures 3 and 4. Symmetry considerations for statistically axisymmetric flows without swirl are provided in appendix A of Jung, Gamard & George (Reference Jung, Gamard and George2004). Here we assume these symmetries cannot be applied. An integral eigenvalue problem is then formed for each vortex age,

(4.3) $$\begin{eqnarray}\int _{R}^{}\unicode[STIX]{x1D609}_{ij}(r,r^{\prime },{\it\psi};m)\,{\it\Phi}_{j}^{(n)}(r^{\prime },{\it\psi};m)r^{\prime }\,\text{d}r^{\prime }={\it\Lambda}^{n}({\it\psi};m)\,{\it\Phi}_{i}^{(n)}(r,{\it\psi};m),\end{eqnarray}$$

and is solved to produce an ordered sequence of eigenvalues ( ${\it\lambda}^{n}\geqslant {\it\lambda}^{n+1}$ ) with eigenfunctions ${\it\Phi}_{i}^{n}(r,{\it\psi};m)$ . This vector decomposition ensures that the eigenfunctions, corresponding to the in-plane velocity components, are coupled. In order to construct the spatial modes that characterize the velocity field, the POD eigenfunctions ${\it\Phi}_{i}^{n}$ and Fourier eigenfunctions $\text{e}^{\text{i}m{\it\theta}}$ are combined as follows:

(4.4) $$\begin{eqnarray}\mathfrak{U}_{i}^{(m,n)}(r,{\it\theta},{\it\psi})=\text{e}^{\text{i}m{\it\theta}}{\it\Phi}_{i}^{(n)}(r,{\it\psi};m).\end{eqnarray}$$

When compared with the spatial modes obtained using the snapshot POD method (which also does not force axisymmetry), the general features of the first few most energetic modes were essentially the same (Mula & Tinney Reference Mula and Tinney2014).

Having discretized (4.3), the total number of POD modes is governed by the product between the number of points measured ( $N$ ) and the number of in-plane components $({\it\eta})$ used to construct $\unicode[STIX]{x1D609}_{ij}$ . The radial extent is confined to ${\sim}2r_{c}$ at each ${\it\psi}$ . So, given the grid resolution employed here ( $L_{m}/2$ ; $L_{m}/r_{c}<0.2$ ), $N>20$ , at least 40 POD modes are generated at each vortex age. Further, the total resolved energy (Citriniti & George Reference Citriniti and George2000; Tinney et al. Reference Tinney, Glauser and Ukeiley2008) of the flow, ${\it\Pi}({\it\psi})$ , and the normalized eigenspectra, ${\it\beta}^{n}({\it\psi};m)$ , are obtained as

(4.5a,b ) $$\begin{eqnarray}{\it\Pi}({\it\psi})=\mathop{\sum }_{n}^{}\mathop{\sum }_{m}^{}{\it\Lambda}^{n}({\it\psi};m),\quad {\it\beta}^{n}({\it\psi};m)=\frac{{\it\Lambda}^{n}({\it\psi};m)}{{\it\Pi}({\it\psi})}.\end{eqnarray}$$

While (4.4) allows one to view the spatial topography of the modes associated with the average turbulence statistics, an instantaneous low-dimensional representation of the fluctuating velocity field can be obtained from the following established expressions (Lumley Reference Lumley, Yaglom and Tatarsky1967),

(4.6) $$\begin{eqnarray}\hat{\mathfrak{u}}_{i}(r,{\it\psi},t;m)=\mathop{\sum }_{n=1}^{k}\boldsymbol{a}^{(n)}({\it\psi},t;m)\,{\it\Phi}_{i}^{(n)}(r,{\it\psi};m),\end{eqnarray}$$

using uncorrelated and time-varying coefficients,

(4.7) $$\begin{eqnarray}\boldsymbol{a}^{(n)}({\it\psi},t;m)=\int _{R}\hat{u} _{i}(r,{\it\psi},t;m)\,{\it\Phi}_{i}^{(n)\ast }(r,{\it\psi};m)r\,\text{d}r.\end{eqnarray}$$

The mean-square energies of these coefficients are the eigenvalues themselves, ${\it\lambda}^{(n)}=\langle \boldsymbol{a}^{(n)}\boldsymbol{a}^{(n)}\rangle$ , while for $k={\it\eta}N$ , $\hat{\mathfrak{u}}_{i}=\hat{u} _{i}$ . A low-dimensional reconstruction of the fluctuating velocity is then obtained using the following inverse transformation:

(4.8) $$\begin{eqnarray}\mathfrak{u}_{i}(r,{\it\theta},{\it\psi},t)=\mathop{\sum }_{m}\hat{\mathfrak{u}}_{i}(r,{\it\psi},t;m)\text{e}^{\text{i}m{\it\theta}}.\end{eqnarray}$$

4.1. Low-dimensional axial vorticity

Given the kind of flow being studied here and the arrangement of our measurement system, it is only natural that we seek to construct a low-dimensional representation of the axial vorticity. This is calculated using the standard expression

(4.9) $$\begin{eqnarray}{{\it\varpi}_{z}}^{(m,n)}=\frac{1}{r}\frac{\partial (r{\mathfrak{U}_{2}}^{(m,n)})}{\partial r}-\frac{1}{r}\frac{\partial {\mathfrak{U}_{1}}^{(m,n)}}{\partial {\it\theta}},\end{eqnarray}$$

with gradients in $r$ being determined using a second-order central difference scheme. Contrarily, gradients in ${\it\theta}$ are calculated analytically by the nature of Fourier functions,

(4.10) $$\begin{eqnarray}\frac{\partial \mathfrak{U}_{1}^{(m,n)}}{\partial {\it\theta}}=\text{i}m\,\text{e}^{\text{i}m{\it\theta}}{\it\Phi}_{1}^{(n)}(r,{\it\psi};m).\end{eqnarray}$$

Likewise, the axial vorticity of the low-dimensional instantaneous fluctuating velocity can be calculated by simply replacing ${\mathfrak{U}_{i}}^{(m,n)}$ in (4.9) with $\mathfrak{u}_{i}$ from (4.8).