# Leading-Edge Vortex Characteristics of Low-Aspect-Ratio Sweptback Plates at Low Reynolds Number
## 3 Results and Discussion
In other words, wings/plates with an AR of approximately two or below may be not able to produce an additional lift force supported by an LEV. At AR = 3.13 and 4.50, on the other hand, the C\({}_{\text{L,max}}\) was gradually improved with increasing \(\beta\) (arrow (4) in Figure 4b). The maximum value of C\({}_{\text{L,max}}\) appeared at \(\beta\approx 30\) deg for both cases. The rate of growth in C\({}_{\text{L,max}}\) was more conspicuous at AR = 3.13, implying that C\({}_{\text{L,max}}\) does not have simple monotonic relation with AR. This is similar to previous studies on a revolving/flapping wing (refer to [11] as an example). Figure 5 represents slopes of the C\({}_{\text{L}}\)-\(\alpha\) curves in the pre-stall region, i.e., C\({}_{\text{L}\alpha}\). These were extracted at \(\alpha<9\) deg where the curves have a linear slope. The Helmbold equation, which is known to provide a closer fit at AR < 4 [26], are also added in Figure 5. The Helmbold equation is,  
\[C_{\text{L}\alpha}=a_{0}\frac{\text{AR}}{2+\sqrt{4+\text{AR}^{2}}} \tag{5}\]  
\(\tau=0.50\) is quite larger than the typical range as mentioned before, but the C\({}_{\text{L}\alpha}\) of the lifting-line theory is still higher than the measurement. The Helmbold equation gave a better fit than that of the theory with \(\tau=0.25\), but also overestimates the C\({}_{\text{L}\alpha}\). As [PERSON] and [PERSON] [28] pointed out, these would be stemmed from the lower \(a_{0}\) at the low Reynolds number, which had been assumed as \(2\pi\). The higher AR resulted in higher C\({}_{\text{L}\alpha}\) at each \(\beta\). Increasing \(\beta\) resulted in a gradual reduction in C\({}_{\text{L}\alpha}\). These were steadily converged to C\({}_{\text{L}\alpha}\approx 1.0\) at \(\beta=54\) deg. Figure 6a represents the maximum lift coefficient C\({}_{\text{L,max}}\) with respect to \(\beta\). In addition, the chordwise component of C\({}_{\text{L,max}}\) at AR = 2.03, i.e., C\({}_{\text{L,max}\oplus\beta=0}\text{cos}^{2}\beta\) is given as the black dashed line (as discussed later). The C\({}_{\text{L,max}}\) at AR = 2.03 remained nearly unchanged at \(\beta<36\) deg. At AR = 3.13 and AR = 4.50, on the other hand, increasing \(\beta\) resulted in a gradual increment in the C\({}_{\text{L,max}}\). This effect continued until the \(\beta\) reached 42 deg. The model plate of AR = 4.50 showed a higher rate of increment in the C\({}_{\text{L,max}}\), but the largest C\({}_{\text{L,max}}\) appeared at AR = 3.13 with \(\beta=36\) deg. The AR = 3.13 also showed superior lift production at \(\beta>36\) deg. In order to decompose an effect of the spanwise flow and an LEV contribution from the measurement, we assumed that the C\({}_{\text{L,max}}\) at \(\beta=0\) deg can be the maximum value without support of the LEV, because there is no spanwise flow at \(\beta=0\) deg. The portion of this value in the other \(\beta\) cases becomes then C\({}_{\text{L,max}\oplus\beta=0}\text{cos}^{2}\beta\) as shown in the black dashed line in Figure 6a, because a chordwise component of a freestream is U\({}_{\infty}\text{cos}\beta\). This approach, dividing a lift force into two components, obviously cannot decompose effect of TiVs, wake, or their interaction in practice. Other aerodynamic effects at \(\beta=0\) deg, such as the boost and bump of the C\({}_{\text{L}}\) as shown in Figure 3, also could be blended in this approach. Figure 5: The slope of the C\({}_{\text{L}}\)–\(\alpha\) curves in the pre-stall region. Figure 6b shows the LEV contribution \(\rm C_{L,LEV}\), which is the remaining part of the \(\rm C_{L,max}\) after extracting the \(\rm C_{L,max}\)\(\oplus\)\(\rm\beta\)=\(\rm cos^{2}\beta\). At \(\rm\beta\)\(<\) 30 deg, the \(\rm C_{L,LEV}\) showed a rapid increment with increasing \(\rm\beta\).