3.1. Oil flow visualisations

The first set of experiments was the OFV to identify qualitatively the surface topology associated with various flow structures present in the flow field prior to performing quantitative flow field measurements. First, the effect of the angle of attack and Reynolds number on the sweptback untapered model with ( $\varLambda _{\textit{LE}}, \varLambda _{\textit{TE}})=(30^\circ , 30^\circ$ ) was examined. Oil flow visualisations were performed at three angles of attack and three Reynolds numbers as shown in figure 5. The model tip is on the left side of the image and the root (mounted to the wall) is to the right. The free-stream flow direction is from top to bottom. This OFV represents an approximation of the skin friction lines on the surface, where regions with a high concentration of oil are associated with low shear stress and regions with a low oil accumulation represent high shear. The suction side of the model is shown, and the OFV indicates the signature of flow structures that are present in the mean flow.

Figure 5.

Oil flow visualisation over the sweptback, untapered model with ( $\lambda , \varLambda _{\textit{LE}} = 1, 30^\circ$ ) at different angles of attack and Reynolds numbers: (a) $\textit{Re}_c = 1.65\times 10^5$ , (b) $\textit{Re}_c = 2.48\times 10^5$ , and (c) $\textit{Re}_c = 3.3\times 10^5$ , at three angles of attack of $\alpha = 12^\circ , 18^\circ , 22^\circ$ (left, middle and right columns, respectively). Flow is from top to bottom.

For the three Reynolds numbers tested, the behaviour is similar for $\alpha = 12^\circ$ . At this angle of attack (left-most column of figure 5), there is a narrow region along the leading edge that extends to approximately $x/c=0.1$ , which indicates the presence of a leading edge separation bubble, similar to the discussion presented in Poll (Reference Poll1986). Downstream of this region the surface topology exhibits shear lines that are forward and attached to the surface of the model up to approximately mid-chord. Farther downstream, the oil lines veer towards the tip due to the outboard spanwise velocity component associated with the wing sweep. This trend appears to be consistent along the span, and the oil lines are fairly uniform throughout. As pointed out by flow visualisations of Lambourne & Pusey (Reference Lambourne and Pusey1955) and Poll (Reference Poll1986), attached flow for a swept wing always has some spanwise component of velocity even at low incidence ( $\alpha \approx 5^\circ$ ).

At the intermediate angle of $\alpha = 18^\circ$ (middle column of figure 5), there are some differences across Reynolds numbers. At $\textit{Re}_c = 1.65\times 10^5$ (figure 5 a), a surface spiral forms near the midspan, in the fashion of the ram’s horn Black (Reference Black1952), spiral vortex Poll (Reference Poll1986) or partial span vortex sheet flow fields Lambourne & Pusey (Reference Lambourne and Pusey1955). Directly downstream of this spiral, the shear lines are seen to diverge, where some get pulled into the spiral and some extend downstream and towards the tip. This behaviour is labelled with blue arrows in figure 5(a). At the higher Reynolds numbers $\textit{Re}_c = 2.48\times 10^5$ (figure 5 b) and $\textit{Re}_c = 3.3\times 10^5$ (figure 5 c), no surface spiral is present. Instead, the flow remains attached near the root, while the oil is drawn towards the tip from the spanwise flow caused by the sweptback planform. Near the trailing edge, a small arch-shaped region of shear line deflection is seen from near the midspan. This is due to the adverse pressure gradient at this high angle of attack. A similar phenomenon was reported by Lambourne & Pusey (Reference Lambourne and Pusey1955), who described this as a ‘thickening’ of the wake.

When the angle of attack is increased to $\alpha = 22^\circ$ (right column of figure 5), the surface topology exhibits a concentrated spiral in the oil in the shape of a ram’s horn vortex, as described by Black (Reference Black1956), for all three Reynolds numbers. This departure from the ordered oil lines at the lower angles of attack shows that a critical incidence for the formation of a partial span spiral vortex has been exceeded. Indeed, the imprint of the leading edge separation bubble is bisected by the swirling focus as the bubble bursts at this spanwise location. Across the majority of the suction surface, the oil follows the trend of this focus point’s rotation. The exceptions to this trend can be seen in two locations. Large amounts of oil are seen to agglomerate along the tip of the model and in an arc-shaped curve linking the corner of the tip and the leading edge to the core of the ram’s horn vortex. Downstream of the arc-shaped stagnation curve, or as named by Delery (Reference Delery2001) a separator line, the flow is reversed in the streamwise direction as the swirl vortex evolves downstream towards the tip vortex. Upstream of the stagnation line, the spanwise flow direction is reversed, i.e. the flow is towards the model root as it moves towards the low-pressure core of the surface-normal vortex. Furthermore, for the two higher Reynolds numbers, the stagnation curve is further upstream compared with the lower Reynolds number case, $\textit{Re}_c = 1.65\times 10^5$ .

Next, the effect of sweep and taper on the surface topology was explored using OFV on five swept and tapered planforms and is presented in figure 6 at $\textit{Re}_{\bar {c}}=247\,500$ and several angles of attack. For the unswept untapered model ( $\varLambda _{\textit{LE}}, \varLambda _{\textit{TE}} = 0^\circ , 0^\circ$ ), the separation begins at around $\alpha = 18^\circ$ , where reverse flow is seen at the midspan with two counter-rotating surface foci (marked with the blue arrows at $\alpha = 22^\circ$ ), which is indicative of a ‘mushroom-shaped’ stall-cell-type separation. The surface topology on the sweptback untapered model ( $\varLambda _{\textit{LE}}, \varLambda _{\textit{TE}} = 30^\circ , 30^\circ$ ) first exhibits a region of reversed flow at a similar angle of attack as the unswept model. However, in this case the flow field leaves the imprint of a single surface focus rotating in the same direction as the inboard focus in the pair seen on the unswept model. This single focus indicates the ram’s horn-type separation as identified by Black (Reference Black1956).

Figure 6.

Surface OFV for the $\textit{sAR} = 2$ cantilevered models (the wall is on the right-hand side in each image) at $\textit{Re}_{\bar {c}} = 247\,500$ and $\alpha = 14^\circ , 16^\circ , 18^\circ , 20^\circ$ , $22^\circ$ and $\lambda =$ 1, 0.27. Blue up-pointing arrows indicate reversed flow and black blue outlined arrows denote the direction of the shear lines. Flow is from top to bottom.

The surface topology for the tapered model, $\lambda =0.27$ , with an unswept leading edge and a forward swept trailing edge $(\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,-30^\circ )$ exhibits the formation of reversed flow near the midspan at around $\alpha = 16^\circ$ . This reversed flow region grows with the angle of attack, leading to the formation of a single surface spiral near the wingtip, indicating the formation of an inverted ram’s horn vortex similar to that documented on a forward swept wing by Breitsamter & Laschka (Reference Breitsamter and Laschka2001). For the sweptback tapered models with $(\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,0^\circ )$ and $(\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(35.8^\circ ,8.2^\circ )$ , a small reversed flow region forms near the wing tip at the lowest angle of attack tested, leading to the formation of a single inboard surface spiral. As the angle of attack increases, the reversed flow region grows due to an increasing adverse pressure gradient, thus shifting the surface spiral towards the root. This is the same behaviour of the ‘spiral vortex’ that Poll (Reference Poll1986) observed with surface visualisations over sweptback wings.

3.2. Time-averaged volumetric flow fields

The effects of planform over swept and tapered wings on the flow field over a sweptback untapered model $(\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,30^\circ )$ is presented first as a baseline, against which the more complex swept and tapered models will be compared. Some of the pertinent flow features in the flow volume above the suction surface are visualised in figure 7 at $\alpha =22^\circ$ . Specifically, streamlines are computed near the surface of the model and coloured in white, the 3-D contour of $\bar {U}=0$ representing the reversed flow region is coloured in peach and select 3-D streamlines are calculated and are coloured in green. The goal with these flow field representations is to show the large-scale structures present in the flow field without cluttering the image. The OFV study revealed that the structure of the separation for this model took the form of a ram’s horn vortex. The visualisation of this structure at $\alpha = 22^\circ$ is made even clearer as the 3-D streamlines give a clear sense of rotation for the trajectories spiralling about the ram’s horn core. As a result, inboard of the vortex core the flow is attached, whereas the flow is separated outboard of the core where the separation region increases towards the tip.

Figure 7.

Flow field representation over the $30^\circ$ sweptback untapered model, including near surface streamlines (white lines), reverse flow region (isosurface of $\bar {U}=0$ ) and 3-D streamlines (coloured in green). Here $\alpha = 22^\circ$ and $\textit{Re}_{\bar {c}} = 247\,500$ .

The location of the reverse flow region has a strong influence on the location of maximum flow field unsteadiness. To explore this on the sweptback untapered model, the components of the normal Reynolds stresses $\overline {u'u'}/U_\infty ^2$ , $\overline {v'v'}/U_\infty ^2$ and $\overline {w'w'}/U_\infty ^2$ are shown in figure 8. At this angle of attack ( $\alpha = 22^\circ$ ) where separation is present, all components of the normal Reynolds stress exhibit the highest magnitudes downstream of or around the reverse flow region, which is represented by the peach isosurfaces of $\bar {U}=0$ . Moreover, all three normal Reynolds stresses exhibit a 3-D distribution along the span. Furthermore, the streamwise Reynolds stress is dominant over the model, due to the leading edge separated shear layer, whereas $\overline {v'v'}/U_\infty ^2$ and $\overline {w'w'}/U_\infty ^2$ dominate in the wake, due to the vortex shedding into the wake. This will be discussed further in § 3.3.

Figure 8.

Isosurfaces of the three components of the normal Reynolds stresses over the sweptback untapered model. Results are shown for (a) $\overline {u'u'}/U_\infty ^2$ , (b) $\overline {v'v'}/U_\infty ^2$ and (c) $\overline {w'w'}/U_\infty ^2$ . Here $\alpha = 22^\circ$ and $\textit{Re}_{\bar {c}} = 247\,500$ .

Note that the work of Burtsev et al. (Reference Burtsev, He, Zhang, Theofilis, Taira and Amitay2022) using TriGlobal modal linear stability analysis on unswept and sweptback models having semi-aspect ratios of 2 and 4 at Reynolds number of 400 and an angle of attack of $\alpha =22^\circ$ showed interesting similarities to the present work. They showed that multiple amplified modes exist along the span, where the spanwise location of the most amplified mode at a sweep angle of $30^\circ$ corresponds to the spanwise location where the highest unsteadiness found in the present work is at a three orders of magnitude higher Reynolds number.

Next, the effect of the taper ratio and sweep angle is explored. Some of the pertinent flow features in the flow volumes above the suction surface of each swept and tapered model are visualised in figure 9. Specifically, streamlines are computed near the surface of each model and coloured in white, the 3-D contour of $\bar {U}=0$ , representing the reversed flow region, is coloured in peach, and select 3-D streamlines are calculated and are coloured in green. The goal with these flow field representations is to show the large-scale structures present in the flow field and the corresponding separated flow regions.

Figure 9.

Flow field representations over the models, including near surface streamlines (white lines), reverse flow region (isosurface of $\bar {U}=0$ in peach) and 3-D streamlines (coloured in green). Results are shown for (a) ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,0^\circ$ ), (b) ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,0^\circ )$ , (c) $(\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(35.8^\circ ,8.2^\circ )$ , (d) ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,30^\circ )$ and (e) ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,-30^\circ )$ . Here $\alpha = 22^\circ$ and $\textit{Re}_{\bar {c}} = 247\,500$ .

The OFV showed that sweeping back the model would remove the outboard focus of the stall cell pattern (as seen on the unswept model) and gradually concentrate the inboard spiral. In the SPIV results in figure 9, the reversed flow region exhibits a similar trend to the change in sweep angle. For the unswept model (figure 9 a), the reversed flow region has a slight spanwise inboard preference (downstream limit of the reversed flow region is inboard of the midspan), which is due to the inboard induced velocity by the tip vortex. However, the overall structure of the downstream edge of the reverse flow region is U-shaped, characteristic of an arch vortex. Note that there is some discrepancy between the OFV and SPIV data, especially for the unswept case, ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,0^\circ )$ . For this case, the OFV exhibits two foci whereas the SPIV data shows only one. This is due to limited optical access in the SPIV experiment close to the model root where data was not collected.

For the cases with the leading edge sweptback, quarter-chord sweptback and sweptback models (figure 9 b–d, respectively), the downstream limit of the reversed flow region moves further outboard with an increase of the effective sweep angle. As a result, the green 3-D streamlines begin to reveal a different structure of the arch vortex. For the leading edge sweptback case (figure 9 b), the arch vortex is still clearly seen in the 3-D streamlines, though it is angled such that the outboard focus does not form on the suction surface, but rather downstream of the trailing edge near the tip vortex. Increasing the effective sweep angle from the leading edge sweptback case to the quarter-chord sweptback case (figure 9 c) results in a disruption of the arch vortex as some of the green 3-D streamlines begin to break out of the arch and follow the path expected of the spiral vortex over a sweptback wing. Finally, increasing the effective sweep angle of the quarter-chord swept case by moving to the sweptback untapered case (figure 9 d) demolishes the arch vortex and replaces it with the spiral vortex. This suggests that there is a critical sweep angle where the formation of an arch vortex over a stalled wing is no longer supported, and instead a partial span spiral vortex will form.

Sweeping the trailing edge forwards (figure 9 e) distorts the arch vortex, present on the unswept untapered model, by moving the outboard focus upstream and the inboard focus downstream. The inboard velocity due to the forward sweep is parallel to the direction that the inverted ram’s horn vortex evolves. As shown by Iannelli, Donelli & De Gregorio (Reference Iannelli, Donelli and De Gregorio2010), axial flow added through the core can stabilise a vortex, so it could be that the spanwise inboard flow from the forward sweep is promoting the formation of the inverted ram’s horn-type vortex instead of an arch vortex. A very large reversed flow region is the result, and the green 3-D streamlines through the reverse flow region indicate that the vortex emanates from the tip towards the root and bends in the streamwise direction into the wake, similar to the inverted ram’s horn vortex visualised qualitatively by Breitsamter & Laschka (Reference Breitsamter and Laschka2001).

Sweep angle and taper ratio have distinct effects on the location of the reversed flow on the suction surface of the wing due to the formation of the inverted ram’s horn vortex at effective forward sweep angles, which shifts the reversed flow region towards the wing root, or the formation of a classical ram’s horn vortex that effects backwards sweep angles and shifts the reverse flow region towards the wing tip. To quantify this effect, the area of the reversed flow region, $A_{\textit{RF}}$ , was calculated at each spanwise plane along the span. The reverse flow area was calculated around the convex hull enclosing points in the plane with $\bar {U}\leqslant 0$ . An example of such a region is displayed in figure 10(a) as the orange contour. The normalised lengths $x/\bar {c}$ and $y/\bar {c}$ are used to calculate the reverse flow area, so $A_{\textit{RF}}$ is dimensionless. The distribution of $A_{\textit{RF}}$ along the normalised span for each model is plotted in figure 10(b). In general, as the leading edge is swept further back, the peak reverse flow area shifts towards the wing tip due to the outboard velocity component, and when the trailing edge is swept forward, the peak shifts towards the wing root due to the inboard velocity component. The sweptback untapered model has the furthest outboard peak of the reverse flow area, while the tapered model with the forward swept trailing edge has the furthest inboard peak at $z/\bar {c}=0.4$ . The unswept untapered model has reversed flow peaking at $z/\bar {c} = 0.6$ . The tapered models have an increase in reverse flow area that is linked to increasing chord size towards the root. The larger inboard chord substantially increases the adverse pressure gradient that the flow needs to overcome to reattach. This explains why the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,-30^\circ )$ model has the highest peak of $A_{\textit{RF}}$ of any of the models studied here: it has the large inboard chord and the forward swept trailing edge that shifts the reverse flow region inboard.

Figure 10.

(a) An example $\bar {U}=0$ contour for calculation of the normalised reverse flow area, $A_{\textit{RF}}$ , and (b) the variation of $A_{\textit{RF}}$ along the models’ normalised span. Here $\alpha = 22^\circ$ and $\textit{Re}_{\bar {c}} = 247\,500$ .

It is noteworthy that similar large-scale vortical structures were shown by our research collaborators in a laminar flow at a Reynolds number of 400 using direct numerical simulation on sweptback and swept forward wings (Zhang et al. Reference Zhang, Hayostek, Amitay, Burtsev, Theofilis and Taira2020 and Zhang & Taira Reference Zhang and Taira2022, respectively). They showed the evolution of the vortical structures as a function of semi-aspect ratio, sweep angle and angles of attack. In their numerical simulations, for a given aspect ratio, they showed that, for the sweptback wings, as the sweep angle increases, the unsteadiness of vortex shedding decreases and the large-scale structures move towards the wing tip. However, for the swept forward cases, the large-scale vortices move towards the wing’s root with increasing forward sweep angle. For both sweptback and swept forward wings, they mapped out these flows over the $\varLambda -\alpha$ space for different aspect ratios.

The sweep angle and taper ratio also affect the development of the tip vortex as presented in figure 11. The tip vortex is identified with the blue isosurface of the normalised streamwise vorticity, and the reversed flow region is identified with the white isosurface of $\overline {U}=0$ . Note that the vorticity isosurface was manually clipped over the suction side of the model to isolate the tip vortex. As shown in figures 9 and 10, the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,30^\circ )$ and ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,0^\circ )$ models experience a shift of the reversed flow region towards the wing tip due to the outboard velocity component. However, for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,-30^\circ )$ model, the reversed flow region shifts markedly towards the root. For the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,30^\circ )$ model, the reversed flow region exhibits a triangular shape when viewed from above, as opposed to the more horseshoe-shaped reverse flow regions of the sweptback and tapered models. For the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,30^\circ )$ model, the inboard point of the triangle corresponds to the single surface focus identified with the OFV and SPIV, while for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,-30^\circ )$ model near the leading edge, the outboard point corresponds to the (oppositely rotating) surface focus. For both models, the reversed flow region grows moving away from this point: towards the wing tip for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,30^\circ )$ model and towards the wing root for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,-30^\circ )$ model. As the reversed flow grows towards the wing tip for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,30^\circ )$ model, destructive interactions between the reversed flow and the tip vortex serve as an outboard barrier for the growth of the reversed flow region resulting in a weak tip vortex. On the other hand, the reverse flow region for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,-30^\circ )$ model grows away from the wing tip, resulting in a long-lasting tip vortex where its trajectory is angled inboard (i.e. towards the root) by about $10^\circ$ . The trajectory of the tip vortex for the cases where the leading edge is sweptback is relatively parallel to the free stream and the strength of the vortex is reduced compared with the tip vortex of the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,-30^\circ )$ model due to the outboard spanwise velocity. The ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,0^\circ )$ model has a larger tip vortex than the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(35.8^\circ ,8.2^\circ )$ model, as the unswept trailing edge results in a slight shift of the reversed flow region towards the root and, thus, less interference with the tip vortex. These results are similar to the findings of Zhang et al. (Reference Zhang, Hayostek, Amitay, Burtsev, Theofilis and Taira2020), Zhang & Taira (Reference Zhang and Taira2022) and Burtsev et al. (Reference Burtsev, He, Zhang, Theofilis, Taira and Amitay2022) who performed direct numerical simulation of similar geometries as in the present paper but with a Reynolds number of 400. Burtsev et al. (Reference Burtsev, He, Zhang, Theofilis, Taira and Amitay2022) explored the effect of the sweep angle (no taper) on the instantaneous vortical structures. For the unswept wing with $\textit{sAR} = 2$ , a single row of braid-like vortices was formed, where its time-average flow field resembles the flow field in the present paper. Moreover, horn-like vortices were formed at $\varLambda =30^\circ$ , with the flow over most of the wing being steady. They also performed TriGlobal analysis and found that for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,0^\circ )$ model, two instability modes are present, i.e. one at the midspan and one closer to the root, whereas for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,30^\circ )$ model, a single amplified mode is present and is located closer to the wing tip, which corresponds to the location where the normal Reynolds stresses exhibit the highest magnitude in the present work, as shown in figure 8. In a subsequent paper, Ribeiro et al. (Reference Ribeiro, Neal, Burtsev, Amitay, Theofilis and Taira2023) explored numerically the effect of taper ratio at a mean chord Reynolds number of 600. They showed that, for tapered wings with sweptback leading edges, unsteadiness emerges near the wing tip. On the other hand, wings with swept forward trailing edges are shown to concentrate wake shedding structures near the wing root.

Figure 11.

Tip vortex visualised with isosurface of streamwise vorticity, $\omega _x \bar {c}/U_\infty ^2 = -0.5$ , and the reversed flow region visualised with an isosurface of $\overline {U}=0$ for swept and tapered models. Here $\alpha =22^\circ$ and $\textit{Re}_{\bar {c}} = 247\,500$ .

To further explore the effect of the sweep angle and taper ratio, the Reynolds shear stresses $\overline {u'v'}/U_\infty ^2$ , $\overline {u'w'}/U_\infty ^2$ and $\overline {v'w'}/U_\infty ^2$ are shown for the swept and tapered models in figures 12(a–b), 12(c–d) and 12(e–f), respectively. The high-magnitude concentrations of positive $\overline {u'v'}/U_\infty ^2$ is above the reversed flow region, while the high-magnitude negative $\overline {u'v'}/U_\infty ^2$ lies below the reversed flow region. In both cases, the regions of $\overline {u'v'}/U_\infty ^2$ are most prominent at the spanwise location of the peak $A_{\textit{RF}}$ . For example, the $(\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,-30^\circ )$ model (figure 12 a) has an $A_{\textit{RF}}$ peak near the root, and the $\overline {u'v'}/U_\infty ^2$ contours are most prominent near the root for this model. As shown by Neal & Amitay (Reference Neal and Amitay2023) for an unswept $\textit{sAR} = 4$ wing, the high $\overline {u'v'}/U_\infty ^2$ regions are correlated to momentum transfer across the shear layers associated with the shedding of the leading edge and trailing edge vortices into the wake. Here, for the more complex swept and tapered planforms, it is not expected that the shedding into the wake will share such quasi-two-dimensionality. Still, the mechanism for $\overline {u'v'}/U_\infty ^2$ production here still seems to be the interaction with the reversed flow region and the high momentum free stream, more so at the downstream limit of the reversed flow region where the effect of sweep angle is less pronounced.

Figure 12.

Isosurfaces of Reynolds shear stresses: (a,b) $\overline {u'v'}/U_\infty ^2$ , (c,d) $\overline {u'w'}/U_\infty ^2$ and (e, f) $\overline {v'w'}/U_\infty ^2$ , for the models with (a,c,e) ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,-30^\circ$ ) and (b,d, f) ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,0^\circ$ ). Here $\alpha =22^\circ$ and $\textit{Re}_{\bar {c}} = 247\,500$ .

The Reynolds shear stress $\overline {u'w'}/U_\infty ^2$ (figure 12 c–d) has the lowest magnitude of all three Reynolds stresses, and it exhibits a similar distribution as $\overline {u'v'}/U_\infty ^2$ , except for the absence of concentration near the wing tip for the $(\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,-30^\circ )$ model.

The Reynolds shear stress $\overline {v'w'}/U_\infty ^2$ is shown in figure 12(e–f). Due to the spanwise component of velocity from the sweep angle and taper ratio, this shear stress component is an order of magnitude higher than $\overline {u'v'}/U_\infty ^2$ and $\overline {u'w'}/U_\infty ^2$ . The tapered models have a high-magnitude concentration of negative $\overline {v'w'}/U_\infty ^2$ along the leading edge. This leading edge region grows tip-to-root for the $(\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,-30^\circ )$ model (figure 12 e) and root-to-tip for the sweptback model (figure 12 f). The models exhibit a streamwise concentration of negative $\overline {v'w'}/U_\infty ^2$ directly outboard of the tip vortex. For the $(\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,-30^\circ )$ model, a large region of high-magnitude positive $\overline {v'w'}$ grows inboard of the tip vortex. For the $(\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,0^\circ )$ model, a smaller positive $\overline {v'w'}/U_\infty ^2$ region forms inboard of the tip vortex, correlated to weakening and shortening of the tip vortex.

3.3. Time-resolved planar velocity fields

In order to explore the spectral content of the 3-D flow field, a spectral analysis on the vertical velocity component was performed from the TR-SPIV data. Power spectral density (PSD) contours were calculated using Welch’s method with window size $N/10$ and overlapping windows at $95\,\%$ of the window size. The PSD was calculated at a series of $y/\bar {c}$ locations to create a contour plot as shown in figure 13.

Figure 13.

Colour contours of time-averaged normalised streamwise velocity (left), and the vertical variance of power spectra (right) of the vertical velocity component directly downstream from the trailing edge (marked by the vertical black line) for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,0^\circ$ ) model. Results are shown for (a) $z/\bar {c}=0.3$ , (b) $z/\bar {c}=0.6$ , (c) $z/\bar {c}=1$ and (d) $z/\bar {c}=1.3$ . Here $\alpha = 22^\circ$ and $\textit{Re}_{\bar {c}}=247\,500$ .

Note that since the flow field among the different models is very different, and it also varies along the span, the ranges of $x$ and $y$ coordinates in figures 13–20 are not the same among the various cases. Moreover, since the magnitude of the power in the power spectra is very different for each frequency, as well as along the span, we used different ranges of the colour bars to show the location and extent of each mode.

The left subfigures in figure 13 show the mean flow field with colour contours of normalised streamwise velocity superimposed with in-plane streamlines, while the right subfigures show the PSD contour plots generated at the $x/\bar {c}$ location indicated by the black vertical line in the corresponding mean flow plots. The frequencies are represented with Strouhal number, where $St = f \bar {c} sin(\alpha )/U_{\infty }$ .

The power spectral densities at four spanwise locations (at a single streamwise location for each plane) are shown in figure 13 for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,0^\circ$ ) model. Near the root and near the tip ( $z/\bar {c}=0.3$ and $z/\bar {c}=1.3$ , figures 13 a and 13 d, respectively) the flow is mostly attached, with only a small reversed flow region near the trailing edge. As shown above, this planform exhibits the formation of the stall cell at $\alpha =22^\circ$ , where the flow is reversed near the midspan and attached near the spanwise limits with two counter-rotating surface spirals. At these spanwise locations, the PSD exhibits peaks near the vertical locations of velocity deficit due to the reversed flow. At these two spanwise locations the PSD peaks are at $St \leqslant 0.1$ . Note that Burtsev et al. (Reference Burtsev, He, Zhang, Theofilis, Taira and Amitay2022) found that for the unswept model with the same $\textit{sAR}$ and angle of attack, but at $\textit{Re}_{\bar {c}}=400$ , the two amplified modes are at $St \approx 0.1$ , similar to the present finding at a much higher Reynolds number. Near the midspan at $z/\bar {c}=0.6$ and $z/\bar {c}=1$ (figures 13 b and 13 c, respectively), a band of frequencies have high power for $St \leqslant 0.2$ . This indicates that the regions near the foci exhibit fluctuations at lower frequencies, most likely wandering of the stall cell on the suction surface, as also shown by Manolesos & Voutsinas (Reference Manolesos and Voutsinas2014). The regions near the midspan at $z/\bar {c}=0.6$ and $z/\bar {c}=1$ have a much larger reversed flow region associated with vortex shedding from the leading edge. As such, the PSD peaks at higher frequencies and with larger power than at the further inboard and outboard planes.

To explore the most dominant frequencies in each spanwise plane for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,0^\circ$ ) model, the PSD was calculated on the vertical velocity component at every grid point within the TR-SPIV data using Welch’s method. Then, the power at each frequency was averaged along the $y$ axis. The resulting power as a function of $x/\bar {c}$ and Strouhal number is plotted for each spanwise plane for the unswept, untapered wing in the left column of figure 14. The $y$ -averaged power spectra exhibit clear concentrations at specific frequencies that grow in the streamwise direction. Note that the spanwise slices were selected at locations where the dominant frequencies can be easily distinguished. Moreover, the spanwise locations are different among the various cases due to the different flow field over each model. For example, at $z/\bar {c}=0.3$ the power spectra exhibits peaks at $St = 0.04$ and $St = 0.08$ (figure 14 a). To understand where in the spanwise plane these frequencies have high power, the power for each Strouhal number is plotted in the middle and right subfigure of figure 14(a). This can be thought of as a band-pass filter keeping a single frequency range, then plotting the power of this frequency as a function of $x/\bar {c}$ and $y/\bar {c}$ . These low frequency modes are dominant near the suction surface close to the trailing edge. Similar low frequency modes at $St = 0.04$ and $St = 0.09$ are visible at $z/\bar {c}=1.3$ (figure 14 d). These low frequencies are present near the surface spirals of the stall cell, as also shown by Manolesos & Voutsinas (Reference Manolesos and Voutsinas2014). Near the midspan, the power is concentrated at frequencies near the expected shedding frequency of $St = 0.15$ at $z/\bar {c}=0.6$ (figure 14 b) and $St = 0.13$ at $z/\bar {c}=1$ (figure 14 c). These frequency modes increase in power deeper into the wake.

Figure 14.

Power spectra calculated from the vertical velocity $\hat {v}$ averaged across $y/\bar {c}$ direction (left column), and power spectra associated with specific Strouhal numbers as a function of $x/\bar {c}$ and $y/\bar {c}$ (right two columns) for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,0^\circ$ ) model. Results are shown for (a) $z/\bar {c}=0.3$ , (b) $z/\bar {c}=0.6$ , (c) $z/\bar {c}=1$ and (d) $z/\bar {c}=1.3$ . Here $\alpha = 22^\circ$ and $\textit{Re}_{\bar {c}}=247\,500$ .

The normalised normal Reynolds stresses, quantifying the velocity fluctuations, are plotted for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,0^\circ$ ) model in figure 15. As expected, the streamwise Reynolds stress, $\overline {u'u'}$ , has high magnitude in the separated shear layer, which is associated with the vortex shedding (Kelvin Helmholtz vortices). Note that the sampling frequency was not high enough to temporally resolve these shear layer instabilities in the spectra, but their effect can still be quantified in the variance of the velocity history. The vertical Reynolds stress, $\overline {v'v'}$ , has the highest magnitude near the midspan at $z/\bar {c}=1$ (figure 15 b) due to the large reversed flow region and the associated shedding of vortices into the wake. The spanwise Reynolds stress, $\overline {w'w'}$ , has high magnitudes in the wake and in the separated shear layers, which are most dominant at $z/\bar {c}=1$ (figure 15 b).

Figure 15.

Colour contours of normal Reynolds stress components for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,0^\circ$ ) wing. Results are shown for (a) $z/\bar {c}=0.6$ , (b) $z/\bar {c}=1$ and (c) $z/\bar {c}=1.3$ . Here $\alpha = 22^\circ$ and $\textit{Re}_{\bar {c}}=247\,500$ . Flow is from left to right.

The power spectral densities for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,30^\circ$ ) wing are shown in figure 16. Near the root ( $z/\bar {c}=0.5$ and $z/\bar {c}=0.8$ figures 16 a and 16 b, respectively) the flow is mostly attached, with only a small reversed flow region near the leading edge at $z/\bar {c}=0.8$ . The flow field over this model exhibits the ram’s horn vortex at this angle of attack, where this spanwise location is near the associated surface focus. At all spanwise locations, the PSD has peaks near the vertical locations of the reversed or otherwise slowed flow. At $z/\bar {c}=0.5$ (figure 16 a), since the flow is attached, there is no pronounced peak and a low-level power is present across the entire range of frequencies at the vertical locations of the trailing edge. At $z/\bar {c}=0.8$ (figure 16 b) the PSD exhibits a higher magnitude for $St \leqslant 0.4$ , still the power is low. At the outboard planes at $z/\bar {c}=1.2$ and $z/\bar {c}=1.8$ (figures 16 c and 16 d, respectively) the flow is massively separated with the ram’s horn pattern, which exhibits some discrete frequencies with higher power. At $z/\bar {c}=1.2$ the highest peak is at $St = 0.125$ and at $z/\bar {c}=1.8$ a band of frequencies has high power from $St \leqslant 0.125$ . This indicates that periodic vortex shedding occurs at these spanwise planes.

Figure 16.

Colour contours of time-averaged normalised streamwise velocity (left), and the vertical variance of power spectra (right) for the vertical velocity component directly downstream from the trailing edge for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,30^\circ$ ) model. Results are shown for (a) $z/\bar {c}=0.5$ , (b) $z/\bar {c}=0.8$ , (c) $z/\bar {c}=1.2$ , (d) $z/\bar {c}=1.8$ . Here $\alpha = 22^\circ$ and $\textit{Re}_{\bar {c}}=247\,500$ .

The power spectra for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,30^\circ$ ) model, averaged in the $y$ direction, as a function of $x/\bar {c}$ is plotted for each spanwise plane in the left column of figure 17. At $z/\bar {c}=0.5$ (figure 17 a) the integrated power spectrum exhibits two dominant frequencies at $St = 0.1$ and $St = 0.15$ . Both modes reside directly downstream of the leading edge. Note that these powers are very low compared with the other spanwise locations, as the flow is mostly attached at this spanwise plane with small velocity fluctuations. At $z/\bar {c}=0.8$ (figure 17 b) the power spectra exhibits some higher frequency modes at $St = 0.11$ and $St = 0.27$ . These modes are present near the spiral, which is the cross-section of the ram’s horn vortex (see the time-averaged flow in figure 16). Further outboard at $z/\bar {c}=1.2$ (figure 17 c), two modes with higher power can be seen at $St = 0.05$ and $St = 0.13$ . As can be seen from the middle and right subfigures of figure 17(c), these modes both reside in the wake and are of the expected vortex shedding frequency. However, the magnitude of the mode at $St = 0.05$ is much smaller than at $St = 0.13$ . Moreover, there are two distinctive regions of high fluctuation magnitudes at $St = 0.05$ , which are located at the two shear layers one from the leading edge and the other one from the trailing edge. Close to the wing tip at $z/\bar {c}=1.8$ (figure 17 d), modes at $St = 0.08$ and $St = 0.12$ are present. The $St = 0.12$ mode is linked to the vortex shedding in the wake.

Figure 17.

Power spectra calculated from the vertical velocity $\hat {v}$ averaged across $y/\bar {c}$ direction (left column), and power associated with specific Strouhal numbers as a function of $x/\bar {c}$ and $y/\bar {c}$ (right two columns) for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,30^\circ$ ) model. Results are shown for (a) $z/\bar {c}=0.5$ , (b) $z/\bar {c}=0.8$ , (c) $z/\bar {c}=1.2$ and (d) $z/\bar {c}=1.8$ . Here $\alpha = 22^\circ$ and $\textit{Re}_{\bar {c}}=247\,500$ .

The normal Reynolds stresses for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,30^\circ$ ) model are shown in figure 18. At the spanwise location near the root ( $z/\bar {c}=0.5$ , figure 18 a) the flow is attached, and each Reynolds stress component has a low magnitude except for $\overline {u'u'}$ right near the wing surface close to the leading edge, suggesting the presence of a very small separation bubble. At $z/\bar {c}=0.8$ (figure 18 b), which is the spanwise location where the focus of the ram’s horn vortex is located, $\overline {u'u'}$ is elevated in the separated shear layer feeding the focus, whereas $\overline {v'v'}$ and $\overline {w'w'}$ are only slightly elevated due to wandering of the ram’s horn vortex core. At $z/\bar {c}=1.2$ (figure 18 c) the reverse flow region is very large, leading to concentrations of $\overline {u'u'}$ in the leading and trailing edge shear layers. Moreover, there is a high magnitude of $\overline {v'v'}$ from the vortex shedding in the wake. Similarly, the flow is massively separated at $z/\bar {c}=1.8$ (figure 18 d); thus, the wake has a high magnitude of $\overline {v'v'}$ , though it is not as high as at $z/\bar {c}=1.2$ in figure 18(c). Moreover, $\overline {w'w'}$ (figure 18 d) has a higher magnitude than at the other spanwise planes due to interactions between the ram’s horn vortex associated with outboard flow and the wing tip vortex inducing inboard flow.

Figure 18.

Colour contours of normal Reynolds stress components for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,30^\circ$ ) wing. Results are shown for (a) $z/\bar {c}=0.5$ , (b) $z/\bar {c}=0.8$ , (c) $z/\bar {c}=1.2$ and (d) $z/\bar {c}=1.8$ . Here $\alpha = 22^\circ$ and $\textit{Re}_{\bar {c}}=247\,500$ . Flow is from left to right.

Figure 19 shows colour contours of the time-averaged streamwise velocity (left column) and the power spectral densities on the right column at several spanwise locations for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(35.8^\circ ,8.2^\circ$ ) wing. For this specific model, the xlocations are different for the different zslices due to the significant variation of the extent of separation along the span. The time-averaged flow fields show that the flow is partially separated at $z/\bar {c}=0.5$ near the leading edge. The flow field over this model has the formation of a surface spiral inboard of this spanwise location, which is partly arch-type separation and partly ram’s horn separation. At $z/\bar {c}=0.5$ (figure 19 a), the flow reattaches downstream of the ram’s horn spiral indicating a long separation bubble discussed by Black (Reference Black1952). Here, a band of frequencies has slightly elevated power at the range of expected shedding frequencies from $0.1\lt St\lt 0.2$ . The flow is massively separated from $z/\bar {c}=0.8$ to $z/\bar {c}=1.1$ (figures 19 b and 19 c, respectively). At both spanwise locations, a discrete frequency peak is seen at $St = 0.12$ at the vertical location corresponding to the focus of the separation vortex. At $z/\bar {c}=1.1$ , the power for this frequency is spread across all of the $y$ range. This is due to the presence of alternate shedding of vortices from the leading edge and trailing edge. At $z/\bar {c}=1.8$ (figure 19 d), the PSD exhibits lower-level peaks at frequencies at $St \leqslant 0.4$ .

Figure 19.

Colour contours of time-averaged normalised streamwise velocity (left), and the vertical variance of power spectra (right) of the vertical velocity component downstream from the trailing edge for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(35.8^\circ ,8.2^\circ$ ) model. Results are shown for (a) $z/\bar {c}=0.5$ , (b) $z/\bar {c}=0.9$ , (c) $z/\bar {c}=1.1$ and (d) $z/\bar {c}=1.8$ . Here $\alpha = 22^\circ$ and $\textit{Re}_{\bar {c}}=247\,500$ . Flow is from left to right.

Next, the spectral content for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,-30^\circ$ ) model is discussed. Figure 20 shows colour contours of the time-averaged streamwise velocity (left column) and the power spectral densities on the right column at several spanwise locations. All the spanwise planes exhibit massive separation that decreases in severity approaching the wing tip. This is due to the inverted ram’s horn-type separation that is present for this model. At ( $z/\bar {c}, x/\bar {c})=(0.5, 0.95)$ and ( $z/\bar {c}, x/\bar {c})=(0.8, 1.1)$ (figures 20 a and 20 b, respectively), the PSD exhibits peaks at multiple discrete frequencies for $St \leqslant 0.2$ . Near the midspan at ( $z/\bar {c}, x/\bar {c})=(1.1, 1.35)$ (figure 20 c), an even larger band of distinct frequency peaks are present for $St \leqslant 0.3$ . At all three spanwise planes, the high power is at a vertical location corresponding to the focus of the separated flow over the wing. Near the tip at ( $z/\bar {c}, x/\bar {c})=(1.8, 1.6)$ (figure 20 d) the flow is also separated, though there is minimal extension of the reverse flow beyond the trailing edge, and the PSD peaks at a very low frequency, which corresponds to the wandering of the focus of the inverted ram’s horn vortex.

Figure 20.

Colour contours of time-averaged normalised streamwise velocity (left), and the vertical variance of power spectra (right) for the vertical velocity component directly downstream from the trailing edge for the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,-30^\circ$ ) model. Results are shown for (a) $z/\bar {c}=0.5$ , (b) $z/\bar {c}=0.8$ , (c) $z/\bar {c}=1.1$ , (d) $z/\bar {c}=1.8$ . Here $\alpha = 22^\circ$ and $\textit{Re}_{\bar {c}}=247\,500$ . Flow is from left to right.

The TR-SPIV measurements have indicated a complex flow field with interactions of multiple 3-D modes for each of the swept and tapered models. For all the models, the spanwise locations with the largest extent of reversed flow had high power modes oscillating near the expected shedding frequency, while at the spanwise locations near the surface foci associated with the global separation patterns had fluctuations at an order of magnitude lower frequency associated with the wandering of these surface foci on the models. A similar low frequency was also observed by Manolesos & Voutsinas (Reference Manolesos and Voutsinas2014) for stall cells on unswept wings. Note that bispectral analysis was performed (not shown here), which showed nonlinearly coupled waves and spontaneously excited waves that vary along the span and varied between different planforms. This will be shown and discussed in detail in a separate paper that will focus on the nonlinear interaction among different modes.

The Reynolds stress figures present the relative magnitude of the velocity fluctuations along the span as a result of the various large-scale flow structures that arise at high angles of attack for the different wing planforms. The band-pass filtered PSD plots for dominant frequencies in the $\hat {v}$ spectrum can be thought of as the subset of the $\overline {v'v'}$ that is due to a specific frequency. As such, it is important to recognise that the power of specific frequencies varies along the wing span, and many frequencies are present. While these measurements can elucidate the local fluctuations and their spanwise variance, global flow measurements (or aerodynamic load measurements) are required to find the most dominant global frequencies for each wing. The next section will discuss the effects of these flow structures on the global aerodynamic loads for each wing and the dominant frequencies.

3.4. Global forces and moments

The aerodynamic forces and moments for the unswept, swept and swept and tapered wings at $\textit{Re}_{\bar {c}}= 247\,500$ are presented in figure 21. For the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,0^\circ$ ) wing, the lift coefficient increases with the angle of attack until $\alpha = 17^\circ$ , followed by a sharp drop in the lift coefficient due to the formation of the stall cell as shown in figure 6. After the stall angle of attack, the wing experiences the largest decrement of lift of all models tested. The ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,30^\circ$ ) wing has, as expected, the shallowest lift slope of all the models, though it has the highest stall angle of attack at $\alpha = 19^\circ$ . The ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,0^\circ$ ) and ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(35.8^\circ ,8.2^\circ$ ) models have a stall angle of around $\alpha = 13^\circ$ , but the lift continues to increase as the angle of attack increases past the stall angle. The ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,-30^\circ$ ) wing stalls at $\alpha = 15^\circ$ . The pre-stall lift slopes are compared with the equation that was originally developed by Helmbold (Reference Helmbold1942) and extended by Weissinger (Reference Weissinger1942) to include the sweep angle (note that it does not include the taper ratio):

(3.1) \begin{equation} {C_L{_\alpha }} = \frac {C_l{_\alpha } \textit{cos}\varLambda }{\sqrt {1+[(C_l{_\alpha }\textit{cos}\varLambda )/(\pi \textit{AR})]^2}+(C_l{_\alpha }\textit{cos}\varLambda /\pi \textit{AR})} .\end{equation}

Figure 21.

Variation of the aerodynamic loads with the angle of attack for several wings at $\textit{Re}_{\bar {c}} = 247\,500$ . (a) Lift coefficient, (b) drag coefficient, (c) pitching moment coefficient, and (d) lift-to-drag ratio. The solid lines were calculated using (3.1) from Weissinger (Reference Weissinger1942).

Here, the lift slope for an infinite aspect ratio thin aerofoil is assumed to be $C_{l_\alpha }=2\pi$ . Most of the models show good agreement with the lift slope corresponding to $\textit{sAR} = 2$ , as predicted by the Weissinger equation (Weissinger Reference Weissinger1942).

Each of the models have a similar drag behaviour at low angles of attack, though the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,0^\circ$ ) wing clearly has higher drag values than the rest of the models, as expected. For the ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,0^\circ$ ), ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,30^\circ$ ) and ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,-30^\circ$ ) wings, the stall angle is accompanied by a large jump in drag of about $\Delta C_D = 0.2$ . The models with ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,30^\circ$ ) and ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(35.8^\circ ,8.2^\circ$ ) have a much smaller and gradual jump in drag at the stall angle, and remain with lower drag values than all of the other models for the post-stall angles of attack. These stalling qualities of the sweptback and tapered wings are exemplified by the high $C_L/C_D$ (figure 21 d) that these wings exhibit compared with the other wings. All of the tapered wings outperform the $C_L/C_D$ of the untapered wings. This is due to the reduced lift-induced drag from the reduced tip vortex from the smaller tip chord that the tapered wings have compared with the untapered wings.

The pitching moment coefficient, $C_M$ , for all the wings is presented in figure 21(c). The unstable pitch break is an undesirable stalling characteristic that can be dangerous, and the effect of planform can dictate whether the wing experiences this phenomenon. The ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,0^\circ$ ) wing experiences a zero crossing of the $C_M$ post-stall, indicating an unstable pitch break. The ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(30^\circ ,30^\circ$ ) wing also experiences a less stable pitch break than the tapered planforms, with the post-stall $C_M$ values approaching a zero crossing. None of the tapered wings experience unstable pitch break, which is due to their decreased tip chord. With the inclusion of taper, the wings experience largely attached flow near the root or near the tip (for the sweptback or swept forward wings, respectively) at post-stall angles of attack. The ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,-30^\circ$ ) wing experiences the smallest $\Delta C_M$ post-stall of all the tapered wings. This is because the loss of lift at the reversed flow region, which contributes to the pitch-up moment, occurs at different regions along the span for the sweptback and tapered wings versus the swept forward tapered wing.

The unsteady load measurements are leveraged to examine the global frequency content experienced by the wings. Spectrograms of the lift force are generated for each model and are shown in figure 22 as a function of the angle of attack. Frequency is expressed by the Strouhal number and is listed on the left $y$ axis, while the variance of the lift coefficient $\sigma _{C_L}$ is overlaid with the white line as a function of the angle of attack on the right $y$ axis. All of the wings experience a sharp increase in $\sigma _{C_L}$ at stall as the massive separation begins and vortex shedding induces large fluctuations in the lift coefficient. For each wing, these fluctuations seem to be centred around a narrow band of frequencies corresponding to the shedding frequency, while the sweptback tapered wings also exhibit the harmonic frequency. The ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,0^\circ$ ) and ( $\varLambda _{\textit{LE}},\varLambda _{\textit{TE}})=(0^\circ ,-30^\circ$ ) wings (figures 22 a and 22 e, respectively) have the steepest slope of $\sigma _{C_L}$ , while the sweptback wings (figure 22 b–d) increase in $\sigma _{C_L}$ much more gradually as a function of $\alpha$ .

Figure 22.

Spectrograms of the lift coefficient as a function of the angle of attack at $\textit{Re}_{\bar {c}} = 247\,500$ . The white line traces the variance of the lift coefficient as a function of the angle of attack (right axis).