3.1.1. Total circulation and vortex structure

To compute ${\it\Gamma}_{\mathit{total}}$ , the domain is restricted to a square region aligned with the plate chord and therefore the data grid (figure 2 a); for all ${\it\omega}_{z}$ fields presented, data below and ahead of the plate are not captured due to a laser shadow and the camera lines-of-sight being blocked by the plate. Measured from the plate suction side, the total circulation region is $1c$ above the plate for $\AR =2$ and $1.5c$ for $\AR =4$ , and extends the same distances, respectively, from the leading edge aft; the $\AR =4$ region is greater due to its larger outboard flow. These domains capture all of the positive ${\it\omega}_{z}$ in the leading-edge region flow, plus any ${\it\omega}_{z}>0$ near the plate, above the threshold described next, but exclude that sometimes residing in the trailing-edge flow which would incorrectly contribute to ${\it\Gamma}_{\mathit{total}}$ as defined here.

Figure 2.

The ${\it\Gamma}_{\mathit{total}}$ processing methods. (a) Domain for calculating ${\it\Gamma}_{\mathit{total}}$ for $\AR =2$ , in plate-aligned ( $x^{\prime }$ , $y^{\prime }$ ) coordinates; data below and ahead of the plate are absent due to a laser shadow and line-of-sight blocking. (b) Effect of vorticity threshold, ${\it\omega}_{z,th}$ , on ${\it\Gamma}_{\mathit{total}}$ ( $\AR =2$ ); for each ${\it\phi}$ , the four curves indicate ${\it\omega}_{z,th}=0$ , $0.5\overline{{\it\omega}_{z}}$ , $1.0\overline{{\it\omega}_{z}}$ and $1.5\overline{{\it\omega}_{z}}$ , respectively.

A spanwise vorticity threshold, ${\it\omega}_{z,th}$ , is incorporated to remove any background vorticity below the uncertainly level. It is employed in obtaining both ${\it\Gamma}_{\mathit{total}}$ and ${\it\Gamma}_{\mathit{LEV}}$ . For use only to examine the sensitivity of the circulation to this threshold, similar to DeVoria & Ringuette (Reference DeVoria and Ringuette2012) a (normalized) reference vorticity value is chosen as the spatial and temporal mean of the positive ${\it\omega}_{z}$ residing within the square regions just described, referred to as $\overline{{\it\omega}_{z}}$ , using all rotation angles ${\it\phi}$ and all spanwise data planes ( $z$ positions). The values of $\overline{{\it\omega}_{z}}$ , normalized using $b$ , are 2.76 and 3.84 for $\AR =2$ and 4. Carr et al. (Reference Carr, Chen and Ringuette2013a ) showed that the maximum uncertainty in the non-dimensional vorticity for these data sets is ${\sim}$ 0.04 using $c$ for normalization, which becomes 0.08 and 0.16 for $\AR =2$ and 4, respectively, when normalizing via $b$ ; thus, $\overline{{\it\omega}_{z}}$ is much larger than the uncertainty.

To illustrate the effect of varying ${\it\omega}_{z,th}$ , figure 2(b) shows ${\it\Gamma}_{\mathit{total}}^{\ddagger }={\it\Gamma}_{\mathit{total}}/bU_{\mathit{tip}}$ (as discussed below) plotted against the normalized spanwise position, $z/b$ , for $\AR =2$ . The coloured curves indicate different values of ${\it\phi}$ , and for each ${\it\phi}$ four curves are shown, ranging from ${\it\omega}_{z,th}=0$ to $1.5\overline{{\it\omega}_{z}}$ in increments of $0.5\overline{{\it\omega}_{z}}$ . The highest values of ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ correspond to ${\it\omega}_{z,th}=0$ , and ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ decreases only slightly with each relatively large increase in ${\it\omega}_{z,th}$ . For the entire ${\it\omega}_{z,th}=0{-}1.5\overline{{\it\omega}_{z}}$ range, there is a ${\sim}$ 20 % decrease in ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ near 20 % span, where the circulation values are small, which reduces to ${\sim}$ 10 % in the peak regions near the tip. The value of ${\it\omega}_{z,th}$ has a negligible effect on the observed trends, and ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ is relatively insensitive to it since from ${\it\omega}_{z,th}=0$ to $0.5\overline{{\it\omega}_{z}}$ the ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ decreases only ${\sim}$ 7 % for 20 % span and ${\sim}$ 2 % in the tip region. A matching numerical value of ${\it\omega}_{z,th}=0.2$ is chosen for each $\AR$ , which is slightly above the span-normalized uncertainties reported earlier.

We first explore various normalizations of the circulation to better isolate $\AR$ effects, including those which produce the best agreement in the spatial and temporal trends. Figure 3 shows normalized curves of ${\it\Gamma}_{\mathit{total}}$ versus $z/b$ ; plotting the ${\it\Gamma}_{\mathit{total}}$ versus $z/c$ for $\AR =2$ and 4 shows poorer agreement in all cases. For each plot in figure 3, multiple curves are shown which indicate the circulation at different times. One method for comparing the flow evolution with time for each $\AR$ is to match the chord lengths travelled by the tip, $(s/c)_{\mathit{tip}}$ , which also matches the chords traversed at a given $z/b$ . Note that, neglecting the small wing root offset, $(s/c)_{\mathit{tip}}=\AR {\it\phi}$ . Therefore, when matching $(s/c)_{\mathit{tip}}$ for each $\AR$ , the value of ${\it\phi}$ for $\AR =4$ is half that for $\AR =2$ .

Figure 3.

Plots of ${\it\Gamma}_{\mathit{total}}$ using various normalizations, versus normalized spanwise location $z/b$ : (a)  ${\it\Gamma}_{\mathit{total}}/cU_{\mathit{tip}}$ for matching $(s/c)_{\mathit{tip}}$ ; (b)  ${\it\Gamma}_{\mathit{total}}/bU_{\mathit{tip}}$ for matching $(s/c)_{\mathit{tip}}$ ; (c)  ${\it\Gamma}_{\mathit{total}}/bU_{\mathit{tip}}$ for matching ${\it\phi}$ ; (d)  ${\it\Gamma}_{\mathit{total}}/bU_{\mathit{local}}$ for matching ${\it\phi}$ . Solid lines: $\AR =2$ ; dashed: $\AR =4$ . The legend for (a) applies to (b), the legend for (c) applies to (d).

Normalizing circulation first by the chord length and reference velocity, $U_{\mathit{tip}}$ , gives ${\it\Gamma}_{\mathit{total}}^{\dagger }={\it\Gamma}_{\mathit{total}}/(cU_{\mathit{tip}})$ . Figure 3(a) shows ${\it\Gamma}_{\mathit{total}}^{\dagger }$ versus $z/b$ , with curves for $(s/c)_{\mathit{tip}}=0.4{-}4.0$ . For $(s/c)_{\mathit{tip}}=0.4$ , the $\AR =2$ circulation is larger. This may be due to the fact that the rotational acceleration is accomplished over matching ${\it\phi}=10^{\circ }$ and not $(s/c)_{\mathit{tip}}$ . At $(s/c)_{\mathit{tip}}=0.4$ the $\AR =2$ plate has travelled ${\it\phi}=12^{\circ }$ and acceleration is completed, whereas the $\AR =4$ plate is at ${\it\phi}=6^{\circ }$ and still accelerating. After the startup, for $(s/c)_{\mathit{tip}}=0.9$ and 2.2 the curves of ${\it\Gamma}_{\mathit{total}}^{\dagger }$ for $\AR =2$ and 4 show reasonably good agreement. However, as $(s/c)_{\mathit{tip}}$ increases further, the differences between the ${\it\Gamma}_{\mathit{total}}^{\dagger }$ become substantial outboard of $z/b\approx 0.4$ .

Next, following Harbig et al. (Reference Harbig, Sheridan and Thompson2013) the circulation is normalized by the plate span, giving ${\it\Gamma}_{\mathit{total}}^{\ddagger }={\it\Gamma}_{\mathit{total}}/(bU_{\mathit{tip}})$ ; alternatively, this can be written as ${\it\Gamma}_{\mathit{total}}^{\ddagger }={\it\Gamma}_{\mathit{total}}^{\dagger }/\AR$ . Figure 3(b) plots ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ versus $(s/c)_{\mathit{tip}}$ . With this normalization, the circulation curves for each $\AR$ are in comparatively worse agreement for all $(s/c)_{\mathit{tip}}$ , with the $\AR =2$ circulation being consistently higher, versus employing ${\it\Gamma}_{\mathit{total}}^{\dagger }$ . Another option is to compare the different $\AR$ data at matching ${\it\phi}$ , analogous to matching $(s/b)_{\mathit{tip}}$ , which Carr et al. (Reference Carr, Chen and Ringuette2013a ) indicated to be best for comparison of the vortex structures for the overall ${\it\phi}=120^{\circ }$ motion. In figure 3(c), ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ is plotted with colored curves signifying matching ${\it\phi}$ for $\AR =2$ and 4; the ${\it\phi}$ range is 12–108°. Note that the green curve for $\AR =2$ is for ${\it\phi}=24^{\circ }$ in figure 3(a) and ${\it\phi}=30^{\circ }$ in figure 3(c). By matching ${\it\phi}$ , the $\AR =2$ and 4 curves of ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ are much closer in magnitude for higher rotation angles, ${\it\phi}\geqslant 60^{\circ }$ , compared to matching $(s/c)_{\mathit{tip}}$ for ${\it\Gamma}_{\mathit{total}}^{\dagger }$ or ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ .

Overall, the $\AR$ differences are somewhat lessened if $b$ is used as the length scale in normalizing ${\it\Gamma}_{\mathit{total}}$ , consistent with Harbig et al. (Reference Harbig, Sheridan and Thompson2013), and if the results are compared at matching ${\it\phi}$ (or $(s/b)_{\mathit{tip}}$ ), particularly for larger ${\it\phi}$ . However, although these normalizations may reduce $\AR$ variations and therefore provide an appropriate framework for comparisons, there remain some significant differences with $\AR$ as discussed below.

The total circulation behaviour is described here. Figure 3(c) shows that for all values of ${\it\phi}$ and both $\AR\text{s}$ , there is a general increase in ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ with span after ${\sim}$ 20 % span, followed by a maximum within $0.7<z/b<1$ . A sharp drop to zero is found past the tip where reorientation of some of the LEV spanwise vorticity to streamwise vorticity, ${\it\omega}_{x}$ , occurs during aft tilt and lift-off (described later), and also ${\it\omega}_{x}$ from the TV is dominant.

Figure 3(c) also shows a clear variation with $\AR$ , namely the $\AR =2$ ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ is higher for nearly all data points across the span for each ${\it\phi}$ ; again, this difference is relatively larger for lower ${\it\phi}$ . Harbig et al. (Reference Harbig, Sheridan and Thompson2013) calculated the sum of the circulations of the dual-LEVs they found (§ 1), analogous to the present total circulation, for their $\mathit{Re}_{\mathit{span}}=3833$ case at ${\it\phi}=270^{\circ }$ and for $\AR =2.91{-}7.28$ Drosophila planforms. They normalized it using $b$ and the velocity at the radius of gyration, and showed a decrease in circulation with increasing $\AR$ across the span; therefore the current results are consistent with these simulations. Further, the present data show a weakening dependence of ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ on ${\it\phi}$ with larger values for ${\it\phi}>30^{\circ }$ , for both $\AR\text{s}$ . In the tip region, the differences between ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ for $\AR =2$ and 4 are relatively larger, particularly for ${\it\phi}=108^{\circ }$ . This can be explained by examining the three-dimensional flow structure shown by an isosurface of ${\mathcal{Q}}$ , which is provided in figure 4 for $\AR =2$ and 4 from ${\it\phi}=30^{\circ }$ to 90°.

Figure 4.

Top, or ( $x$ , $z$ )-plane, view of an iso-surface of non-dimensional ${\mathcal{Q}}=12$ at (a)  ${\it\phi}=30^{\circ }$ , (b)  ${\it\phi}=48^{\circ }$ , (c)  ${\it\phi}=60^{\circ }$ and (d)  ${\it\phi}=90^{\circ }$ ; $\AR =2$ (left column) and 4 (right column). The red arrow indicates LEV enlargement prior to breakdown.

Considering ${\it\phi}=30^{\circ }$ , the isosurfaces show some similarities between the $\AR =2$ and 4 flow structures, including outboard LEV lift-off into an arch-like structure, and a ‘stable’ LEV inboard; the latter persists for all ${\it\phi}$ . These features are in good agreement with the $\AR =2$ and 4 simulations of Garmann & Visbal (Reference Garmann and Visbal2014), the $\AR =4$ computation of Jardin et al. (Reference Jardin, Farcy and David2012) and the $\AR =1{-}3$ volumetric data of Ozen & Rockwell (Reference Ozen and Rockwell2012, Reference Ozen and Rockwell2013). As the first LEV lifts off, another forms ahead of it which connects to the TV for both $\AR\text{s}$ ( ${\it\phi}=30^{\circ }$ ). It is the contributions from these further, outboard LEVs for both $\AR\text{s}$ which cause the larger increase in ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ for $0.6<z/b<0.8$ and ${\it\phi}\geqslant 30^{\circ }$ (figure 3 c), although breakdown then acts to reduce the $\AR =4$ outboard circulation as described below.

Despite the flow similarities just mentioned, there is a substantial $\AR$ variation at ${\it\phi}=30^{\circ }$ . Namely, at a given $z/b$ the $\AR =4$ LEV structures are located farther aft from the leading edge than for the shorter-span $\AR =2$ plate. For example, a part of the $\AR =4$ lifted-off LEV has already moved past the trailing-edge location; also, the lifted-off LEV for $\AR =4$ appears larger. The $\AR =4$ aft LEV tilt at ${\it\phi}=30^{\circ }$ is thus more substantial at a given $z/b$ than for $\AR =2$ , as quantified in figure 9 (solid magenta line). As discussed in § 3.1.3, this leads to a reduction in spanwise circulation, and is likely the main reason for the smaller ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ for $\AR =4$ . In addition, the iso- ${\mathcal{Q}}$ surface at ${\it\phi}=30^{\circ }$ shows that the portion of the TV aft of the trailing edge for $\AR =4$ is positioned more inboard than the TV structure residing over the tip edge itself; for $\AR =2$ , these two TV segments are well-aligned, although similar behaviour occurs at higher ${\it\phi}$ .

As ${\it\phi}$ increases, flow structure variations with $\AR$ persist. For $\AR =4$ , the second outboard LEV lifts off to form an arch which connects to the TV ( ${\it\phi}=48^{\circ }$ ), and a further LEV forms ahead of this. By ${\it\phi}=60^{\circ }$ , the outboard $\AR =4$ flow shows three smaller LEVs ahead of the initial one, which has by this time moved aft of the trailing edge. Concurrently, the initial, main TV also travels off the plate and exists primarily behind the trailing edge; in addition, a smaller streamwise (tip-oriented) structure is visible just outside of the tip edge as well as aft of the outboard plate corner, connected to the smaller LEVs. For $\AR =4$ the outboard LEV becomes larger ( ${\it\phi}=60^{\circ }$ , red arrow), which is a sign of breakdown (Delery Reference Delery1994; Lucca-Negro & O’Doherty Reference Lucca-Negro and O’Doherty2001) and a similar LEV size increase was found by Lu & Shen (Reference Lu and Shen2008) and Garmann & Visbal (Reference Garmann and Visbal2014); also smaller vortical structures are visible around it. By ${\it\phi}=90^{\circ }$ , for $\AR =4$ the outboard breakdown has progressed substantially, qualitatively indicated by small, unorganized structures. This breakdown is responsible for the reduction in outboard ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ from ${\it\phi}=84^{\circ }$ to 108° (figure 3 c). The breakdown adversely affects the mean-flow convergence from phase averaging (Carr et al. Reference Carr, Chen and Ringuette2013a ), leading to the reduced coherence observed in the iso- ${\mathcal{Q}}$ surfaces and the outboard, high- ${\it\phi}$ vorticity fields (figures 7 and 8).

Conversely, for $\AR =2$ the initial, outboard LEV does not move past the trailing edge after lift-off, but by ${\it\phi}=60^{\circ }$ it has tilted aft and merged with the TV, and both are close to the plate. An inset giving the flow at ${\it\phi}=54^{\circ }$ shows an increase in the size of this aft-tilted LEV (red arrow), which as for $\AR =4$ indicates a breakdown event; this flow structure becomes more complex at ${\it\phi}=60^{\circ }$ . By ${\it\phi}=90^{\circ }$ , the outboard $\AR =2$ LEV/TV flow shows prominent breakdown with a leading-edge shear layer ahead of it that is deflected away from the plate; the $\AR =4$ case exhibits a similar leading-edge shear layer (see also figures 7 and 8, 70 % span). The present breakdown behaviour is similar in terms of spatial location and timing to that of the $\AR =2$ and 4 rotating-wing dye flow visualizations of Carr & Ringuette (Reference Carr and Ringuette2014). Despite the breakdown, the $\AR =2$ outboard flow remains more coherent and resides over plate, compared with $\AR =4$ , yielding the higher ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ .

A local minimum for ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ is found near or just outboard of 90 % span for $30^{\circ }\leqslant {\it\phi}\leqslant 60^{\circ }$ for $\AR =2$ and 4 (e.g. figure 3 c, black arrow); at higher ${\it\phi}$ the ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ reduction for $\AR =4$ is from breakdown. The iso- ${\mathcal{Q}}$ surfaces show that this coincides with the outboard ‘foot’ of the lifted-off, arch-like LEV (figure 4). This ‘foot’ tilts upward so some of its originally spanwise vorticity is converted into $x$ - and $y$ -direction vorticity, causing lower ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ . This also results in the ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ maximum moving somewhat inboard with higher ${\it\phi}$ .

Given the increase of ${\it\Gamma}_{\mathit{total}}^{\ddagger }$ with $z/b$ away from the tip, using instead the maximum plate azimuthal velocity at each (local) spanwise position, $U_{\mathit{local}}$ , to normalize it yields more comparable values across the span (figure 3 d); we define it as ${\it\Gamma}_{\mathit{total}}^{\ast }={\it\Gamma}_{\mathit{total}}/(bU_{\mathit{local}})$ . Circulation normalization using $U_{\mathit{local}}$ was also employed by Carr, DeVoria & Ringuette (Reference Carr, DeVoria and Ringuette2013b ) and Wojcik & Buchholz (Reference Wojcik and Buchholz2014a ). For ${\it\phi}=12^{\circ }$ , when the influence of the plate ends is small, the curves of ${\it\Gamma}_{\mathit{total}}^{\ast }$ are nearly flat between the root and tip. This indicates that the strength of the LEV scales proportionally to the local plate velocity. For higher ${\it\phi}$ , ${\it\Gamma}_{\mathit{total}}^{\ast }$ grows relatively larger toward the root and outboard of $z/b\approx 0.6$ except very near the tip, and the $U_{\mathit{local}}$ normalization is insufficient to remove spanwise variations in these regions. Outboard, this is due to the highly three-dimensional flow structure, including LEV lift-off, shown earlier. The values of ${\it\Gamma}_{\mathit{total}}^{\ast }$ near the root are somewhat amplified because of the division by very small $U_{\mathit{local}}$ . Between 20 % and 60 % span the variations in ${\it\Gamma}_{\mathit{total}}^{\ast }$ are significantly reduced, compared with using $U_{\mathit{tip}}$ . The 20–60 % spanwise range overlaps the locations where Carr et al. (Reference Carr, Chen and Ringuette2013a ) identified the LEV to be ‘stable’, i.e. for up to ${\sim}$ 40–50 % span for $\AR =4$ and up to ${\sim}$ 60 % for $\AR =2$ (also shown here in figure 4). Therefore, in the ‘stable’ LEV the circulation scales with the local azimuthal velocity; this holds even at the highest ${\it\phi}$ acquired prior to deceleration, 108°. Although this normalization is less successful near the plate ends, it remains useful for examining temporal trends and producing spanwise averages.

As shown above, the overall evolution of ${\it\Gamma}_{\mathit{total}}$ with time is most similar for $\AR =2$ and 4 at matching ${\it\phi}$ . Figure 5(a) plots ${\it\Gamma}_{\mathit{total}}^{\ast }$ against ${\it\phi}$ with multiple curves from 20 % to 80 % span in 10 % increments for $\AR =2$ and 4. For each $\AR$ the circulation has a higher initial slope, then beyond ${\it\phi}\approx 42^{\circ }$ the slope gradually reduces but remains generally positive, albeit with some fluctuations and differences at each spanwise position. Reflecting the results of figure 3(d), for each $\AR$ the individual-span curves have similar magnitudes for inboard locations, but increase in magnitude for 60–80 % span toward the tip region.

Figure 5.

Plots of ${\it\Gamma}_{\mathit{total}}$ versus  ${\it\phi}$ or chords travelled: (a)  ${\it\Gamma}_{\mathit{total}}/bU_{\mathit{local}}$ versus  ${\it\phi}$ at different spanwise locations; (b) spanwise-averaged ${\it\Gamma}_{\mathit{total}}/bU_{\mathit{local}}$ over 20–80 % span versus  ${\it\phi}$ ; (c) ${\it\Gamma}_{\mathit{total}}/bU_{\mathit{local}}$ versus  $(s/c)_{\mathit{local}}$ for $\AR =2$ ; (d) same, but for $\AR =4$ . Solid lines: $\AR =2$ ; dashed: $\AR =4$ .

Figure 5(b) shows the spanwise average of ${\it\Gamma}_{\mathit{total}}^{\ast }$ from 20 % to 80 % span, away from the plate ends, which yields clearer trends. The $\AR =2$ magnitudes are again larger, but what is most striking is the similarity in the slopes of the curves after ${\it\phi}=24^{\circ }$ , until deceleration. This feature is discussed again in the context of the lift coefficient in § 3.4.

We also examine the evolution of ${\it\Gamma}_{\mathit{total}}^{\ast }$ with local chord lengths travelled, $(s/c)_{\mathit{local}}$ , at each $z/b$ (figure 5 c,d for $\AR =2$ and 4, respectively); this was also done by Poelma et al. (Reference Poelma, Dickson and Dickinson2006) for the LEV. At a given $z/b$ , the $\AR =2$ plate travels approximately half the distance covered by $\AR =4$ for matching ${\it\phi}$ . For $\AR =2$ , up to $(s/c)_{\mathit{local}}\approx 1$ the ${\it\Gamma}_{\mathit{total}}^{\ast }$ curves from 30 % to 80 % span collapse reasonably well, then the slope of each decreases. The individual-span curves then diverge at slightly different values of $(s/c)_{\mathit{local}}$ . The initial collapse of the ${\it\Gamma}_{\mathit{total}}^{\ast }$ curves indicates a growth based on $(s/c)_{\mathit{local}}$ . This explains why matching ${\it\phi}$ is not as successful in characterizing temporal trends with $\AR$ for smaller ${\it\phi}$ . After $(s/c)_{\mathit{tip}}\approx 1$ , events such as outboard LEV lift-off and more complex tip-region flow occur which change the growth rate. Apart from the 20 % span values, the initial collapse of the $\AR =4$ curves occurs over a larger $(s/c)_{\mathit{local}}\approx 2$ , indicating a longer distance travelled for features such as plate end effects to manifest, which makes sense due to the higher $\AR$ . Following this, the individual curves diverge similar to $\AR =2$ . This initial ${\it\Gamma}_{\mathit{total}}$ collapse is consistent with that of the LEV circulation shown by Poelma et al. (Reference Poelma, Dickson and Dickinson2006) for a single $\AR$ . Jardin et al. (Reference Jardin, Farcy and David2012) found a linear increase in non-dimensional LEV circulation with non-dimensional distance travelled for $\AR =4$ at a given spanwise location, but that the coefficient varied with span; here we do not find a linear increase overall, however, it is characterized by chords travelled up to $(s/c)_{\mathit{local}}\approx 2$ .

3.1.2. LEV circulation and flow fields

To obtain ${\it\Gamma}_{\mathit{LEV}}$ , following DeVoria & Ringuette (Reference DeVoria and Ringuette2012) the centre of the first LEV formed is identified and tracked using the maximum positive ${\it\omega}_{z}$ . The search for subsequent LEV centres is restricted to a window ( $9\times 9$ grid points for $\AR =2$ , $7\times 7$ for $\AR =4$ ) surrounding the previous ${\it\omega}_{z}$ maximum, which aids in avoiding other vortices or any stronger ${\it\omega}_{z}$ in the leading-edge shear layer; this plants a ‘seed’ point in the first LEV at each time step. Starting with this point, a clustering algorithm connects adjacent points which make up the LEV, if the in-plane ${\mathcal{Q}}$ , i.e.  ${\mathcal{Q}}_{2D}$ , is greater than 0; this indicates a local rotation rate greater than the strain rate (Chakraborty, Balachandar & Adrian Reference Chakraborty, Balachandar and Adrian2005). The three-dimensional ${\mathcal{Q}}$ is not used since it is influenced by out-of-plane flow, and a threshold of 0 is chosen to obtain the maximum possible identified vortex points. The actual LEV centre location is then determined from the ${\it\omega}_{z}$ -weighted centroid of each cluster, after the following procedure is performed.

The ${\mathcal{Q}}_{2D}$ is often too restrictive and does not identify data in the outer portion of the LEV having comparable ${\it\omega}_{z}$ , which yields lower ${\it\Gamma}_{\mathit{LEV}}$ . We correct for this, as done by DeVoria & Ringuette (Reference DeVoria and Ringuette2012), by searching for any further grid points having ${\it\omega}_{z}>{\it\omega}_{z,th}$ that are no greater than $2\sqrt{2}$ grid spacings from at least one point in the original cluster, and then appending them into the (augmented) set. Figure 6(a) shows a ${\it\omega}_{z}$ field for $\AR =2$ with a black contour indicating the originally identified cluster and a magenta contour showing the augmented cluster. Similar to ${\it\Gamma}_{\mathit{total}}$ , the ${\it\Gamma}_{\mathit{LEV}}^{\ast }$ exhibits insensitivity to the ${\mathcal{Q}}_{2D}$ threshold used, in part due to the cluster augmentation, with reductions of typically less than 6 % when comparing results using ${\mathcal{Q}}_{2D}>0$ versus (non-dimensional) ${\mathcal{Q}}_{2D}>5$ .

Figure 6.

LEV circulation: (a) LEV identified using ${\mathcal{Q}}_{2D}>0$ only (black contour) and additional clustering (magenta contour); (b) spanwise-averaged ${\it\Gamma}_{\mathit{LEV}}^{\ast }$ over ‘stable’ LEV (20–60 % span for $\AR =2$ , 20–40 % span for $\AR =4$ ); (c)  ${\it\Gamma}_{\mathit{LEV}}^{\ast }$ versus ${\it\phi}$ , $\AR =2$ ; (d)  ${\it\Gamma}_{\mathit{LEV}}^{\ast }$ versus ${\it\phi}$ , $\AR =4$ ; solid lines: $\AR =2$ ; dashed: $\AR =4$ .

To examine the temporal trends of ${\it\Gamma}_{\mathit{LEV}}$ and compare $\AR =2$ and 4, we plot ${\it\Gamma}_{\mathit{LEV}}^{\ast }$ versus ${\it\phi}$ (figure 6). Figure 6(b) gives black lines showing the spanwise average of ${\it\Gamma}_{\mathit{LEV}}^{\ast }$ across the perpetually ‘stable’ LEV, from 20–60 % span and 20–40 % span for $\AR =2$ and 4, respectively. Reflecting the behaviour of ${\it\Gamma}_{\mathit{total}}^{\ast }$ , the span-averaged ${\it\Gamma}_{\mathit{LEV}}^{\ast }$ for $\AR =2$ is generally larger. For each $\AR$ , these mean curves have their highest growth rates for ${\it\phi}=24{-}36^{\circ }$ , after which the $\AR =2$ circulation essentially levels off or saturates (apart from some fluctuations) while that for $\AR =4$ increases at a slow rate and peaks before deceleration; fluctuations in the $\AR =4$ span-averaged curve are due to values at 40 % span as explained below. The initial greater increase of ${\it\Gamma}_{\mathit{LEV}}$ followed by a leveling off or slower growth is consistent with the rotating $\AR =2.3$ plate results of Kim & Gharib (Reference Kim and Gharib2010) for trapezoidal velocity programs. Poelma et al. (Reference Poelma, Dickson and Dickinson2006) also showed ${\it\Gamma}_{\mathit{LEV}}$ levelling off, or slightly increasing or decreasing, depending on the spanwise location.

Figure 6(c,d) show curves of ${\it\Gamma}_{\mathit{LEV}}^{\ast }$ at 20 % and 40 % span, within the ‘stable’ LEV, and outboard at 70 % span, for $\AR =2$ and 4, respectively. In figures 7 and 8, corresponding ${\it\omega}_{z}$ fields are shown for ${\it\phi}=30^{\circ }{-}90^{\circ }$ . In these figures, the solid black contour outlines the identified LEV, but in some instances a value of ${\mathcal{Q}}_{2D}=0.1$ is given (dash-dot black line) to indicate problems with identification of the first LEV caused by breakdown.

Figure 7.

Spanwise vorticity fields at specified spanwise positions and ${\it\phi}$ , $\AR =2$ . Contour lines range from $-44$ to $-4$ and 4 to 44 in steps of 5. Solid black line, identified LEV; dashed black line, ${\mathcal{Q}}_{2D}=0.1$ : (a) 20 % span, ${\it\phi}=30^{\circ }$ ; (b) 40 % span, ${\it\phi}=30^{\circ }$ ; (c) 70 % span, ${\it\phi}=30^{\circ }$ ; (d) 20 % span, ${\it\phi}=48^{\circ }$ ; (e) 40 % span, ${\it\phi}=48^{\circ }$ ; (f) 70 % span, ${\it\phi}=48^{\circ }$ ; (g) 20 % span, ${\it\phi}=60^{\circ }$ ; (h) 40 % span, ${\it\phi}=60^{\circ }$ ; (i) 70 % span, ${\it\phi}=60^{\circ }$ ; (j) 20 % span, ${\it\phi}=90^{\circ }$ ; (k) 40 % span, ${\it\phi}=90^{\circ }$ ; (l) 70 % span, ${\it\phi}=90^{\circ }$ .

Figure 8.

Spanwise vorticity fields at specified spanwise positions and ${\it\phi}$ , $\AR =4$ . Same contour levels as figure 7: (a) 20 % span, ${\it\phi}=30^{\circ }$ ; (b) 40 % span, ${\it\phi}=30^{\circ }$ ; (c) 70 % span, ${\it\phi}=30^{\circ }$ ; (d) 20 % span, ${\it\phi}=48^{\circ }$ ; (e) 40 % span, ${\it\phi}=48^{\circ }$ ; (f) 70 % span, ${\it\phi}=48^{\circ }$ ; (g) 20 % span, ${\it\phi}=60^{\circ }$ ; (h) 40 % span, ${\it\phi}=60^{\circ }$ ; (i) 70 % span, ${\it\phi}=60^{\circ }$ ; (j) 20 % span, ${\it\phi}=90^{\circ }$ ; (k) 40 % span, ${\it\phi}=90^{\circ }$ ; (l) 70 % span, ${\it\phi}=90^{\circ }$ .

At 20 % span the ${\it\Gamma}_{\mathit{LEV}}^{\ast }$ curves for each $\AR$ reflect the temporal trends of the span average, although the magnitudes for $\AR =2$ and 4 are more comparable. Inspection of the ${\it\omega}_{z}$ fields indicates that the LEV remains closely attached to the plate at 20 % span, and that there are small changes in the LEV vorticity distribution and its identification with time that lead to the fluctuations in ${\it\Gamma}_{\mathit{LEV}}^{\ast }$ for both $\AR$ .

For $\AR =2$ , ${\it\Gamma}_{\mathit{LEV}}^{\ast }$ at 40 % span is higher than at 20 %, but with a similar time evolution. Correspondingly, the LEV is larger, it does not evolve substantially although its size fluctuates, and it remains close to the plate (figure 7). However, the $\AR =4$ case shows substantial variations between ${\it\phi}=36^{\circ }$ and 90°. The vorticity fields in figure 8 reveal the cause, namely after ${\it\phi}=30^{\circ }$ the leading-edge vorticity is typically organized in one of two ways: (i) one large LEV as well as further positive ${\it\omega}_{z}$ ahead of this, which is the LEV shear layer ( ${\it\phi}=48^{\circ }$ ); (ii) two LEVs of either approximately equal size ( ${\it\phi}=60^{\circ }$ ) or with a larger initial one and smaller second one ahead of it ( ${\it\phi}=42^{\circ }$ and 72°, not shown), both smaller than the single LEV of case (i); the tracking algorithm identifies the first LEV formed. The ${\it\Gamma}_{\mathit{LEV}}^{\ast }$ is therefore larger for case (i) and yields a local minimum for case (ii) (figure 6 d). The iso- ${\mathcal{Q}}$ surfaces (figure 4) indicate a ‘split’ of the $\AR =4$ LEV around 40 % span, whose spanwise position fluctuates, as previously shown by Carr et al. (Reference Carr, Chen and Ringuette2013a ). For $\AR =2$ , inspection of the ${\it\omega}_{z}$ fields and iso- ${\mathcal{Q}}$ surfaces indicates an LEV ‘split’ location of ${\sim}$ 60 % span. As mentioned above, Harbig et al. (Reference Harbig, Sheridan and Thompson2013) also showed ‘split’ LEVs for all $\AR\text{s}$ tested for $Re_{\mathit{span}}\approx 1000$ and higher. However, unlike Harbig et al. (Reference Harbig, Sheridan and Thompson2013) the present ‘split’ locations are not the same for both $\AR\text{s}$ despite the matching $\mathit{Re}_{\mathit{span}}$ . This may be a function of the slightly higher $\mathit{Re}_{\mathit{span}}$ here, the wing geometry (rectangular versus fruit fly), or the S-DPIV resolution.

The ${\it\Gamma}_{\mathit{LEV}}^{\ast }$ at 70 % span is larger for $\AR =2$ compared with 4, consistent with the outboard trends of ${\it\Gamma}_{\mathit{total}}^{\ast }$ . For $\AR =2$ , ${\it\Gamma}_{\mathit{LEV}}^{\ast }$ for 70 % span peaks at ${\it\phi}=36^{\circ }$ then decreases, while for $\AR =4$ it peaks at ${\it\phi}=42^{\circ }$ . The curves at 70 % span for each $\AR$ do not extend over the full ${\it\phi}$ range, because after they stop the first LEV can no longer be clearly identified using ${\mathcal{Q}}_{2D}$ . This is illustrated by plotting dash-dot black contours of ${\mathcal{Q}}_{2D}=0.1$ for $\AR =2$ at ${\it\phi}=90^{\circ }$ (figure 7 l) and for $\AR =4$ at ${\it\phi}=60^{\circ }$ , 90° (figure 8 i,l). There are multiple and more complex vortex structures than at earlier ${\it\phi}$ (and compared with smaller spanwise locations), and the identification and tracking algorithm becomes unreliable. This is due to factors such as LEV breakdown, and an LEV–TV interaction for $\AR =2$ (figure 4).

The $\AR =2~{\it\omega}_{z}$ fields in figure 7 at 20 % span additionally show that the TEV (largest region of ${\it\omega}_{z}<0$ ) travels aft with ${\it\phi}$ as expected. Discrete vortices having ${\it\omega}_{z}<0$ are visible in the separated trailing-edge flow at ${\it\phi}=60^{\circ }$ and 90°, interspersed with ${\it\omega}_{z}>0$ . At 40 % span the TEV is larger, and the discrete trailing-edge flow vortices are stronger. The $\AR =4$ vorticity fields (figure 8) at 20 % and 40 % span show similarities to the $\AR =2$ flow. However, the TEV moves farther from the plate at the same ${\it\phi}$ due to the greater distance travelled for $\AR =4$ . Also, note that the $\AR =4$ LEVs are larger, consistent with the $\AR$ trend of Harbig et al. (Reference Harbig, Sheridan and Thompson2013).

At 70 % span the $\AR =2$ LEV is much larger than inboard (figure 7). From ${\it\phi}=30^{\circ }{-}48^{\circ }$ , the LEV grows perpendicularly to the plate and secondary, opposite-sign (negative) vorticity is visible ahead of its lower, upstream portion, between it and the leading-edge shear layer. The opposite-sign vorticity is created by the LEV flow over the plate surface. For ${\it\phi}\geqslant 48^{\circ }$ , the leading-edge separated shear layer ahead of the LEV is deflected away from the plate, as also found by Ozen & Rockwell (Reference Ozen and Rockwell2012) for $\AR =2$ and larger ${\it\phi}$ . At ${\it\phi}=60^{\circ }$ the negative vorticity is lifted farther from the plate by the LEV velocity (see also Harbig et al. Reference Harbig, Sheridan and Thompson2013; Garmann & Visbal Reference Garmann and Visbal2014; Wojcik & Buchholz Reference Wojcik and Buchholz2014a ), and some of this vorticity has wrapped around the LEV itself (left portion). This overall behaviour is associated with the arch-like LEV observed in the iso- ${\mathcal{Q}}$ surfaces (figure 4). By ${\it\phi}=90^{\circ }$ , the LEV appears as a region of ${\it\omega}_{z}>0$ wrapped around a patch of ${\it\omega}_{z}<0$ . This complexity is due to the LEV interacting with the TV (figure 4) and breakdown. Despite the high ${\it\phi}$ , the LEV is still relatively close to the plate. The 70 % span primary TEV is larger than before, with the smaller discrete vortices wrapped around it and merging with it.

Substantial $\AR$ differences are found at 70 % span. At ${\it\phi}=30^{\circ }$ the $\AR =4$ LEV extends over about two-thirds of the plate chord, which is approximately the maximum size of the $\AR =2$ LEV. Next at ${\it\phi}=48^{\circ }$ , the $\AR =4$ LEV is lifted higher off the plate compared with $\AR =2$ , and is nearly the size of the chord (see also figure 4); in addition, secondary ${\it\omega}_{z}<0$ exists between the plate and LEV. At ${\it\phi}=60^{\circ }$ the LEV structure is much more complex, with smaller vortices as identified by ${\mathcal{Q}}_{2D}$ (figure 8 i, also figure 4). These are the multiple LEVs described earlier, and the less-coherent flow structure again indicates breakdown. At ${\it\phi}=90^{\circ }$ , the $\AR =4$ flow is transitional with continued breakdown effects, and the coherent leading-edge shear layer is deflected away from the plate as for $\AR =2$ . To gain further insight into the larger $\AR =2~{\it\Gamma}_{\mathit{LEV}}^{\ast }$ at 70 % compared with $\AR =4$ , prior to breakdown, we next examine LEV tilt and vorticity flux.

3.1.3. LEV tilt, axial flow and vorticity flux

Here we analyse the LEV tilting, axial flow and vorticity flux. We visualize any aft LEV tilting with time by plotting $x$ -direction trajectories of the LEV centre from 20 % to 80 % span, in steps of 10 %, in figure 9. The trajectories are connected, coloured lines for each ${\it\phi}$ value, and outboard LEVs are only identified up to a certain ${\it\phi}$ . A top or ( $x$ , $z$ )-plane view of the plate is shown and a grey region gives the projected plate area; the span scale is $z/b$ so that the plate spans have the same length. For each $\AR$ the LEV centre locations begin parallel to the leading edge as expected, then by ${\it\phi}=36^{\circ }$ there is substantial aft tilt and the LEV centre also shifts rearward as the vortex grows. Where the LEV is ‘stable’ for each $\AR$ , the tilt and trajectory variation with ${\it\phi}$ are smaller; the outboard tilt is much larger. The $\AR =4$ outboard LEV centre moves aft of the trailing edge at 70 % span for ${\it\phi}>36^{\circ }$ , and at 80 % span and ${\it\phi}=48^{\circ }$ it is ${\sim}0.2c$ past the trailing edge. Conversely, for $\AR =2$ at 80 % span and ${\it\phi}=60^{\circ }$ (highest angle with an identified LEV), the LEV centre position does not move past the trailing edge, indicating less tilt.

Figure 9.

LEV centre $x$ -direction trajectories versus ${\it\phi}$ , top-view of plate: (a)  $\AR =2$ ; (b)  $\AR =4$ . Data for ${\it\phi}=6{-}120^{\circ }$ , incrementing by ${\rm\Delta}{\it\phi}=6^{\circ }$ , are given by the colour progression: red, green, blue, yellow, magenta, cyan, black, which repeats but line styles also progress as solid, dashed, dash-dot, ending with cyan and dash-dot for ${\it\phi}=120^{\circ }$ . As described in the text, an outboard LEV is not found for all ${\it\phi}$ . LE: leading edge.

Substantial aft LEV tilt will reduce the spanwise LEV circulation, due to the conversion of some of the ${\it\omega}_{z}$ to streamwise vorticity ${\it\omega}_{x}$ . Kim & Gharib (Reference Kim and Gharib2010) and Ozen & Rockwell (Reference Ozen and Rockwell2012) also reported significant rearward LEV tilt and streamwise vorticity for their $\AR \approx 2$ wings. The LEV tilting over the entire span generally increases up to ${\it\phi}\approx 36^{\circ }$ , after which its evolution slows. This attainment of maximum tilt coincides with the reduction in slope of both ${\it\Gamma}_{\mathit{LEV}}^{\ast }$ and ${\it\Gamma}_{\mathit{total}}^{\ast }$ . However, after this the decrease in spanwise circulation due to tilting will not change significantly. An exception is the 70–80 % region for $\AR =4$ . The continued aft tilting here yields the lower ${\it\Gamma}_{\mathit{LEV}}^{\ast }$ at 70 % span compared with $\AR =2$ (figure 6 d).

Next we analyse the volume flow rate and vorticity flux along the LEV. We account for the LEV tilting by calculating an axial unit vector, ${\bf\beta}$ , which is parallel to the LEV centre axis. This is done using the information from figure 9, for each spanwise position and ${\it\phi}$ , again only when an LEV can be identified. For example, for a given ${\it\phi}$ , the ${\bf\beta}$ at 20 % span is taken to be the unit vector whose direction is parallel to the line connecting the LEV centre at 20 % span to that at 30 % span, with positive defined as outboard. Therefore, for a given spanwise location ${\bf\beta}$ is computed using the LEV ( $x$ , $y$ , $z$ ) centre position there and at 10 % span outboard. The ${\bf\beta}$ are not calculated at 80 % span, because no 90 % span LEV data are considered. The LEV axial velocity is defined at each grid point as the scalar product of the velocity vector and the axial unit vector, $\boldsymbol{u}_{axial}=(\boldsymbol{u}\boldsymbol{\cdot }{\bf\beta}){\bf\beta}$ . We define the non-dimensional vorticity flux as $(\boldsymbol{u}\boldsymbol{\cdot }{\bf\beta})|{\bf\omega}|=u_{axial}|{\bf\omega}|$ .

We first examine the non-dimensional volume flow rate, which is the area integral of the non-dimensional $u_{axial}$ over the non-dimensional LEV area; this is done for each identified LEV at the selected spanwise positions ( $z/b$ ) and for each ${\it\phi}$ . Note that Carr et al. (Reference Carr, Chen and Ringuette2013a ) showed both positive (outboard) and negative (inboard) spanwise velocity in the vicinity of the LEV for $\AR =2$ and 4, which was also found by e.g. Lu & Shen (Reference Lu and Shen2008), Jardin et al. (Reference Jardin, Farcy and David2012), Phillips & Knowles (Reference Phillips and Knowles2013) and Garmann & Visbal (Reference Garmann and Visbal2014). To gain insight into the individual contributions of the positive and negative $u_{axial}$ , we calculate the non-dimensional volume flow rate conditionally on its positive and negative components; this is given in figure 10(a,b) for $\AR =2$ and 4, respectively. The overall trend for each $\AR$ is that the positive non-dimensional LEV volume flow rates generally have greater magnitudes than the negative components. In addition, in some instances the magnitudes increase with greater spanwise position, particularly from 20 % to 50 % span and for the positive components. Further, the volume flow rate magnitudes for $\AR =2$ are typically larger than for $\AR =4$ . For both $\AR\text{s}$ , the volume flow rates at 50 % span (and 60 % for $\AR =2$ ) exhibit substantial fluctuations for ${\it\phi}>48^{\circ }$ , due to breakdown.

Figure 10.

Non-dimensional volume flow rate and average vorticity flux for the LEV, for 20–70 % span, $\AR =2$  (a,c) and $\AR =4$  (b,d). For the non-dimensional volume flow rate, solid lines indicate the calculation done conditionally using only positive volume flow rate regions, while the dashed lines similarly give it for negative regions.

The average non-dimensional LEV vorticity flux at different spanwise locations is shown for $\AR =2$ and 4 in figure 10(c,d), respectively. It is calculated as the area integral of the non-dimensional vorticity flux, divided by the LEV area. The positive and negative components of the average vorticity flux (not shown) have trends similar to the volume flow rate for each $\AR$ . As figure 10(c,d) demonstrate, the average vorticity flux is generally positive (outboard) for both $\AR\text{s}$ , except for a small number of fluctuations below 0. Unlike the volume flow rate, the vorticity flux values are comparable for each $\AR$ at a given spanwise position. For $\AR \approx 2$ , Kim & Gharib (Reference Kim and Gharib2010) also found primarily positive LEV vorticity flux (in this case, the area integral of the flux), while Ozen & Rockwell (Reference Ozen and Rockwell2012) showed positive flux when the LEV is concentrated and near the plate, but negative flux when the LEV is lifted away from the plate and grows longer aft. For ${\it\phi}\geqslant 36^{\circ }$ and between 30 % and 50 % span for both $\AR\text{s}$ , in some instances the vorticity flux increases with span, which Kim & Gharib (Reference Kim and Gharib2010) state indicates transport outboard toward the tip. This will also act to reduce ${\it\Gamma}_{\mathit{LEV}}^{\ast }$ .

In the LEV flow fields we observe patches of both positive and negative $u_{axial}$ , as found by others, however the spatial resolution of the present data sets does not allow for these to be fully resolved. We believe this is the explanation for the fluctuations observed in the curves of the non-dimensional volume flow rate and average vorticity flux; however, again for 50 % and 60 % span and ${\it\phi}>48^{\circ }$ , the larger fluctuations are due to breakdown.