Leading-edge vortices over swept-back wings with varying sweep geometries

Micro air vehicles are used in a myriad of applications, such as transportation and surveying. Their performance can be improved through the study of wing designs and lift generation techniques including leading-edge vortices (LEVs). Observation of natural fliers, e.g. birds and bats, has shown that LEVs are a major contributor to lift during flapping flight, and the common swift (Apus apus) has been observed to generate LEVs during gliding flight. We hypothesize that nonlinear swept-back wings generate a vortex in the leading-edge region, which can augment the lift in a similar manner to linear swept-back wings (i.e. delta wing) during gliding flight. Particle image velocimetry experiments were performed in a water flume to compare flow over two wing geometries: one with a nonlinear sweep (swift-like wing) and one with a linear sweep (delta wing). Experiments were performed at three spanwise planes and three angles of attack at a chord-based Reynolds number of 26 000. Streamlines, vorticity, swirling strength, and Q-criterion were used to identify LEVs. The results show similar LEV characteristics for delta and swift-like wing geometries. These similarities suggest that sweep geometries other than a linear sweep (i.e. delta wing) are capable of creating LEVs during gliding flight.


Introduction
With an increased use of micro air vehicles (MAVs) new aerodynamic challenges arise [1]. The tasks of MAVs are many and include activities that require long-distance flight, large payloads and tight manoeuvres. Aerodynamic forces play a major role for a given task with respect to flight performance. The goal is to increase aerodynamic efficiency, which typically involves maximizing lift while minimizing drag. One approach to increasing lift through wing design involves the generation of a & 2019 The Authors. Published by the Royal Society under the terms of the Creative attack angles, a ¼ 0 , 20 , 30 , which were set using a digital level with an accuracy + 0.18 and re-checked/measured following each experiment to ensure that the wing angle did not change during data collection. Flow measurements were performed in the streamwise and wall-normal plane at three different spanwise positions over the wing, z

Results
The flow over the wings in all configurations is inherently turbulent and these analyses of the resulting LEVs represent ensemble averages of the vortex characteristics. There are many different methods to determine whether or not an area of interest is considered a vortex; thus, four main criteria were used to assess whether or not a particular velocity field had an LEV. We examined the streamlines of the mean velocities for areas of circulation, and identified locations of high vorticity, swirling strength, and positive Q-criterion. Each of these are evaluated on coherency and location with respect to the wing. Because vortex identification criteria are debatable, somewhat subjective, and prone to error, the use of multiple criteria helps to avoid associated biases. For example, the use of vorticity alone can be misleading because any shear in the flow also produces large vorticity. Also, the application of Q-criterion and swirling strength based on two-dimensional measurements of a three-dimensional flow can yield misleading results. Most differences between vortex identification methods applied in two-dimensional boundary layer flow are associated with the sensitivity of the methods to small-scale vortices [30]. Lastly, it should be noted that the measurements only provide a projection of any LEV onto the measurement plane and the magnitude of the projection varies with the wing geometry along the wingspan making it challenging to compare between planes (along the span). Results of using the various analysis techniques are described below, where collectively they suggest that an LEV was present for both the delta and swift-like wings for attack angles of a ¼ 208 and a ¼ 308.

Streamlines
We evaluate flow separation over the upper wing surface by observing streamlines of the fluid flow in a stationary frame of reference. The streamlines incident at the leading edge of the wing are examined for reattachment, and when there is no reattachment of these streamlines we presume that flow separation occurred. A streamline in the two-dimensional plane is defined by dy=dx ¼ v=u. Streamlines of the flow are computed for the mean velocity maps. Figure 3 depicts streamlines over both the delta and swift wings at the quarter plane. The streamlines depict flow reattachment patterns occurring at angles of attack varying from 08 to 308. The red lines highlight streamlines that intersect with the leading edge of the wing. At a ¼ 08 no separation is observed. Flow reattachment is observed for both wing geometries at a ¼ 208 with a degradation occurring for both at a ¼ 308. This similarity coupled with the similar formation of an area of circulation at the leading edge is indicative of a comparable vortex system forming.

Spanwise vorticity
Given that the measurements were performed using two-dimensional PIV, the spanwise vorticity component, v z ¼ @v=@x À @u=@y, is estimated from the instantaneous velocity map series. The spanwise vorticity series is ensemble averaged over all maps providing a single map of v z , where the overbar denotes an ensemble mean. Figure 4 compiles the mean spanwise vorticity distribution for both the delta and swift wing at the quarter plane using the same range of angles of attack as in figure 3. One can observe analogous vorticity contours forming over each wing geometry. The vorticity regions show similar characteristic shapes and magnitudes, and vary in a similar way with changes in the angle of attack. At a ¼ 08, mean vorticity is negligible relative to the cases with higher angles of attack. At a ¼ 208 and a ¼ 308, a concentrated region of spanwise vorticity can be observed for both wing configurations in the leading-edge region-the same region where fluid circulation was observed in figure 3. The coincidence of these regions provides more evidence of an LEV.

Vortex identification
In addition to the streamlines and vorticity occurring simultaneously, swirling strength and Q-criterion also appear in the same region for both the delta ( figure 5) and swift (figure 6) wing geometries. Both

Vertical momentum and circulation
Here we examine whether the observed circulation patterns can be associated with increased lift. In this case, with the wing upside down, vertical momentum associated with lift would be in the negative y-direction (i.e. the direction normal to the surface). To unambiguously identify negative vertical velocity, we consider vertical displacements (from the PIV measurements) of at least 1.5 pixels in the downward direction as indicative of negative velocities (i.e. v , 20.03 m s 21 ). At each position along the chord, the mean of these negative vertical velocities over the region beneath the wing was computed and squared to estimate the vertical momentum distribution over the chord per unit span (rv 2 ), where r ¼ 1000 kg m 23 , and is shown in figure 7 for various planes and angles of attack. This distribution integrated over the chord is proportional to the vertical forces per unit span acting on the wing and is used to evaluate the similarity of the LEV as a mechanism contributing to (total) lift generation for the delta and swift wings. In figure 7, the wing leading and trailing edges are shown as thin vertical lines. For both the delta and swift-like wings, there is no apparent vertical momentum at 08 angle of attack, while at higher attack angles (a ¼ 208 and a ¼ 308), we observe vertical momentum near the leading edge that overlaps with the LEV regions previously identified. Similar trends between the two geometries are also observed as the angle of attack and plane change, such as peak vertical momentum location relative to the wing and characteristic widening, which is approximately linear with the increase in angle of attack. Clearly, both wing geometries produce comparable vertical momentum profiles; thus, one can suggest that this lift-generating mechanism is also analogous. The circulation of the flow associated with the LEV is estimated as G ¼ Ð Ð R v z dA, where the region, R, covered by the LEV is defined as the area below the wing that yields swirling strengths greater than 2 s 21 . The circulation is another contributor to the overall lift. The circulations for each spanwise plane and angle of attack for both wings are shown in table 1. The circulation peaks at the mid-span of the wing for both sweep geometries at a ¼ 208, and also for the delta wing at a ¼ 308. The circulations also increase with the angle of attack suggesting more lift is needed at larger angles. The trends of circulation for the delta and swift-like wing geometries were similar supporting    wing chord was also examined as an indicator of transport, and circulation was estimated for each wing geometry at each spanwise location and angle of attack. No wing geometry exhibited an LEV at a ¼ 08. LEVs are observed for both the delta and the swift at a ¼ 208 and 308.

Conclusion
Overall, there appears to be a high level of similarity in the identified LEVs for delta and swift wings. They both show similar development of the LEV along the spanwise direction from the root towards the tip at several attack angles. They also show similarity in the movement of the vortex system centre towards the trailing edge at increasing distances from the root along the wing span, and similar distribution of vertical momentum along the chord. Because an LEV is present with both sweep geometries, there is support for the idea that there may be several leading-edge geometries for swept wings that aid in the creation of an LEV, which enhances lift, as has been shown for the delta wing. Other variables such as trailing-edge geometries, not varied in this study, may also affect the formation of LEVs, as studied by Muir