# Leading-Edge Vortex Characteristics of Low-Aspect-Ratio Sweptback Plates at Low Reynolds Number
## 3 Results and Discussion
This clearly indicates that there might be identical characteristics in between the two LEVs, even they are created and stabilized by the two distinctive sources, i.e., a part of freestream in a sweptback wing versus a radial pressure gradient in a revolving/flapping wing. As another way of decomposing the effect of spanwise flow, we replaced the reference velocity \(\text{U}_{\infty}\) with the chordwise component of the freestream, i.e., \(\text{U}_{\infty}\text{cos}\beta\). The related lift coefficient \(\text{C}_{\text{L,cw}}\) is then,  
\[\text{C}_{\text{L,cw}}=\frac{2L}{S\rho U_{\infty}^{2}\cos^{2}\beta}=\frac{ \text{C}_{\text{L}}}{\cos^{2}\beta} \tag{6}\]  
In this concept, it can be seen that the steady spanwise flow of \(\text{U}_{\infty}\text{sin}\beta\) is added to the model plates that were placed in a freestream of \(\text{U}_{\infty}\text{cos}\beta\). Note that in this approach, the \(\text{C}_{\text{L,cw}}\) does not directly reflect the level of the lift force; the \(\text{C}_{\text{L,cw}}\) at higher \(\beta\) would result in a smaller lift. Figure 7 shows the \(\text{C}_{\text{L,cw}}\) for the three AR cases. We noticed that the slopes of \(\text{C}_{\text{L,cw}}\)-\(\alpha\) curves in a pre-stall region eventually show consistency with the other \(\beta\) cases at each AR. This indicates that the reduction in \(\text{C}_{\text{L,cw}}\) with increasing \(\beta\) (Figure 4) was mainly stemmed from the reduction in chordwise inflow speed. Increasing \(\beta\) enhanced both the maximum \(\text{C}_{\text{L,cw}}\) and stall \(\alpha\), implying that the spanwise flow stabilized the LEV and flow over the model plates at the higher \(\alpha\), as a spanwise flow on a revolving/flapping wing stabilizes an LEV. The identical slopes at each AR further suggest satisfaction of the Kutta condition, as an LEV on a flapping wing does [16]. Figure 8 shows the maximum \(\text{C}_{\text{L,cw}}\) with respect to \(\beta\), which is in rough proportion to \(\beta\). It was found that the \(\text{C}_{\text{L,cw}}\) at AR = 3.13 eventually reached the superior lift force at \(\beta\) > 36 deg. This, once again, suggests that an AR of approximately three would be most effective for the lift enhancement with a sufficient sweptback angle. Surface oil-flow visualization is one adequate technique to observe an attached flow and/or vortex including an LEV [25]. Figure 9a-c show surface oil-flow patterns and streamlines on the suction side of the AR = 3.13 plate for three selected \(\beta\) cases of 0, 24, and 36 deg. Let us see the results at \(\beta\) = 0 deg first (Figure 9a). At \(\alpha\) = 2 deg, the flow was separated at the leading edge and reattached near the quarter-chord line. Part of the reattached flow went against the freestream and produced the line of separation beyond the line of reattachment, creating a leading-edge bubble (LEB) in between them. Laminar separation bubble seemed not to appear because the plate does not have streamlined curvature around the leading edge and, thus, cannot provide a favorable pressure gradient region for a laminar attached flow [30]. At \(\alpha\) = 5 deg, the two lines were stepped back,and TiV traces became prominent. At \(\alpha=10\) deg (the stall angle), the TiV was extent reached \(\sim\)28% from each tip, occupying \(\sim\)56% of the entire span. A line of reattachment was also clearly found at \(\beta=24\) deg (Figure 9b). At \(\alpha=2\) deg, the flow over the LEB smoothly flowed out to the trailing edge. The attached flow running in parallel with the freestream here indicated little effect from the LEB. The LEV traces were found at \(\alpha=10\) deg. The flow ran toward the leading edge from the line of reattachment, and separated before reaching the leading edge, creating the line of separation. At higher \(\alpha\), the LEV trace was gradually moved to the left side with increasing \(\alpha\); a substantial unsteady wake region remained on the right side.