# Leading-Edge Vortex Characteristics of Low-Aspect-Ratio Sweptback Plates at Low Reynolds Number
## 3 Results and Discussion  
We first investigated effect of AR to understand basic aerodynamic characteristics at the targeted Reynolds number of 2.85 \(\times\) 10\({}^{4}\). Figure 3 shows lift and drag coefficients \(\mathrm{C_{L}}\) and \(\mathrm{C_{D}}\) for the three different ARs at \(\beta\) = 0 deg. One straight line on the left side indicates a theoretical value of a two-dimensional inviscid flow, i.e., \(\mathrm{C_{L\alpha}}\) = 2\(\pi\). The two dashed lines following the slopes of the \(\mathrm{C_{L}}\)-\(\alpha\) curves are the results of the lifting-line theory for a finite wing shown in Equation (4), where \(\mathrm{a_{0}}\) and \(\tau\) denote the \(\mathrm{C_{L\alpha}}\) and shape factor, respectively (refer to [26] for detail). It was found that \(\tau\approx\) 0.5, which is pretty larger than the typical range of 0.05 to 0.25, showed a sufficient agreement with the \(\mathrm{C_{L\alpha}}\) in this study. This is one typical characteristic for low Reynolds number flow [27]. \[\mathrm{C_{L\alpha}}=\frac{a_{0}}{1+\frac{a_{0}}{\pi\mathrm{AR}}(1+\tau)} \tag{4}\]  
Overall characteristics in the \(\mathrm{C_{L}}\)-\(\alpha\) curves, such as linear increments in the pre-stall region and gradual reductions after the stall angles, are completely in line with the previous study on LAR wings at low Reynolds number [5]. No stall peaks in the \(\mathrm{C_{L}}\)-\(\alpha\) curves also imply a laminar-dominant flow, which was insufficient to overcome the adverse pressure  
Figure 3: \(\mathrm{C_{L}}\)-\(\alpha\) and \(\mathrm{C_{D}}\)-\(\alpha\) curves for the three AR cases (\(\beta\) = 0 deg). Figure 2: Ensemble average results at aspect ratio (AR) = 4.50 near the stall angle of attack. on the suction side of the model plates (refer to [27] for the disappearance of the stall peak as Re decreases). Decreasing AR relieved the C\({}_{\text{L}\alpha}\) in the pre-stall region (arrow (1) in Figure 3), further suggesting a gradual growth in an encroachment of the TiVs into the inboard region as the previous studies have shown [1; 3]. At higher \(\alpha\), decreasing AR slightly increased the C\({}_{\text{L}}\) (arrow (2) in Figure 3; 15 < \(\alpha\) < 25 deg). This is in line with one recent study [6], which revealed that TiVs of a lower AR wing suppress the vortex shedding and assist in establishing the Kutta condition around the midspan, thereby enhancing the lift. This also explains the slight bump in the C\({}_{\text{D}}\) at AR = 2.03 (\(\alpha\)\(\approx\) 20 deg), because such TiV effect would practically augment the net force acting on the surface. No boost or bump in the C\({}_{\text{D}}\) at AR = 4.5 further suggests that such an aerodynamic benefit from the TiV may be only effective at AR < 4.5. Figure 4a-c show the C\({}_{\text{L}}\)-\(\alpha\) and C\({}_{\text{D}}\)-\(\alpha\) curves with respect to \(\beta\). The dashed straight lines in each graph are the results of the lifting-line theory with \(\tau\) = 0.5. In all cases, increasing \(\beta\) delayed growth of C\({}_{\text{L}}\) and C\({}_{\text{D}}\) (arrows (1) and (2) in Figure 4a). This is a typical feature of a sweptback plate as an increase in \(\beta\) gradually reduces the inflow velocity perpendicular to the wingspan as U\({}_{\text{inflow}}\) = U\({}_{\infty}\)cos\(\beta\) (as discussed later). The stall angles were gradually delayed (arrow (3) in Figure 4), also in line with previous studies on a sweptback angle (refer to [2] as an example). We noticed that the C\({}_{\text{L}}\) at AR = 2.03 shows the prominent peaks at each stall \(\alpha\) (Figure 4a). This feature, usually appears at a sufficiently high Reynolds number [27], is only found at this AR at \(\beta\)\(\geq\) 24 deg. The maximum lift coefficient C\({}_{\text{L,max}}\) depending on \(\beta\) was also noticeable. At AR = 2.03, the C\({}_{\text{L,max}}\) was obtained at \(\beta\) = 0 deg already (arrow (3) in Figure 4a). This implies that supplying a spanwise flow and developing an LEV by using \(\beta\) may not be effective to enhance a lift force, at least for this AR.