# Effect of Wing Corrugation on the Aerodynamic Efficiency of Two-Dimensional Flapping Wings
## 3 Effect of Wing Corrugation on the Two-Dimensional Flapping Wing
### Effect of Corrugation at \(r=0.375R\)  
Figures 7a and 8a show the time courses of lift and drag coefficients for the five flapping wings at spanwise location of \(r=0.375R\), respectively. Figures 7b and 8b compare the amount of changes in lift and drag coefficients of the four corrugated wings, respectively, compared to those of the S wing in a flapping cycle. Table 1 summarizes the amount of changes in stroke and cycle averages of lift and drag coefficients and gliding ratios (C\({}_{\text{L}}\)/C\({}_{\text{D}}\)) of the four wings compared to those of the S wing. The gliding ratios (C\({}_{\text{L}}\)/C\({}_{\text{D}}\)) are the ratios between the average lift coefficients and average drag coefficients in each stroke and in one cycle. Figure 7b shows that the lift coefficients of the three leading-edge corrugated wings are increased in a flapping cycle, while the lift coefficient of the Z wing is decreased, compared to that of the S wing. The lift coefficients of the LE1 and LE2 wings during the translational stage of downstroke (\(0.1T<t/T<0.4T\)) are slightly increased and almost the same during the translational stage of the upstroke (\(0.6T<t/T<0.9T\)). Meanwhile, the lift coefficient of the LE3 wing during the translational stage of downstroke is almost the same, compared to that of the S wing. However, during the translational stage of the upstroke, the lift coefficient of the LE3 wing is increased. In downstroke, the lift coefficient of the Z wing is close to that of the S wing, but in upstroke, it is significantly lower compared to that of the S wing. In the rotation stage (\(t/T=(0.5-0.6)\)), the lift coefficients of the three leading-edge corrugated wings are increased significantly, compared to that of the S wing. Among them, the increase in lift coefficient of the LE3 is the largest over this period, and the increase in lift coefficient of the LE1 wing, which has the lowest corrugation, is the smallest, as shown in Figure 7b. Over the entire flapping cycle, the cycle average lift coefficients of the LE1, LE2, and LE3 wings are (1.5, 2.1, and 3.6)% higher than that of the S wing, respectively. In contrast, the average lift coefficient of the Z wing is 2.6% lower than that of the S wing, as shown in Table 1.  
For drag coefficient, it is evident in Figure 8b that the drag coefficients of the four corrugated wings are mostly decreased over a flapping cycle, compared to those of the S wing. Due to the decrease, the cycle average drag coefficients of the four corrugated wings are decreased, as shown in Table 1. Among the four corrugated wings, as shown in Table 1, the decrease in the drag coefficient of the LE3 wing is the largest (\(-4.1\%\)), while that of the LE1 wing is the smallest (\(-1.1\%\)). As a result, the gliding ratio of the LE3 wing is the highest, at around 7.9% higher than that of the S wing. Meanwhile, the increases in the gliding ratios of the LE1 and LE2 are (2.7 and 4.3)%, respectively, compared to that of the S wing. Only the gliding ratio of the Z wing is reduced for 0.9%. Figure 8: Comparison of drag coefficient: (**a**) time courses of drag coefficient (\(\rm C_{D}\)) and (**b**) the changes in the drag coefficients (\(\rm C_{D}\) change) of the LE1, LE2, and LE3 wings, compared to those of the S wing, at \(r=0.375R\). Figure 7: Comparison of lift coefficient: (**a**) time courses of lift coefficient (\(\rm C_{L}\)) and (**b**) the changes in the lift coefficients (\(\rm C_{L}\) change) of the LE1, LE2, and LE3 wings, compared to those of the S wing, at \(r=0.375R\). Figure 9 shows the flow fields and swirling strength, which is a vortex indicator for the strength of swirling motion, around the wing at \(t=(0.5,\,0.6,\) and \(1.0)T\), where the changes in lift coefficients of the corrugated wings are large. In general, the flow fields and swirling strength wings in the trailing edge of the five wings are similar. At \(t=(0.5\) and \(1.0)T\), which is the rotation state, the position of the wings is almost vertical, and the leading-edge vortex (LEV) disappears.