# 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 \(r=0.75R\)  
Figure 10 shows the time courses of lift coefficients of the five wings, and the amount of changes in lift coefficients of the four corrugated wings, compared to those of the S wing, respectively. Figure 11 shows the time courses of drag coefficient and the amount of changes in the drag coefficients of corrugated wings, compared to those of the S wing. Table 2 shows the amount of changes in stroke and cycle average of lift and drag coefficients and gliding ratios of the four wings, compared to those of the S wing. Figure 10b shows that during the translational stage of downstroke (\(0.1T<t/T<0.4T\)), the lift coefficient of the LE3 wing is reduced significantly. Meanwhile, that of the Z wing is increased in the period of \(t=(0.2\) to \(0.4)T\). As a result, the average lift coefficient of the LE3 wing during downstroke is reduced the most, by 7%, while that of the Z wing is increased by 2.1%, as shown in Table 2. The amount of change in downstroke average lift coefficient of the LE2 and LE3 wings is (1.3 and \(-1.4\))%, respectively, compared to that of the S wing, as shown in Table 2.  
Figure 9: Flow fields, swirling strength, and pressure acting on the surfaces of the five wings at \(r=0.375R\). Figure 11: Comparison of drag coefficient: (**a**) the time courses of drag coefficients (\(\rm C_{D}\)) and (**b**) the changes in the drag coefficients (\(\rm C_{D}\) change) of the Z, LE1, LE2, and LE3 wings, compared to those of the S wing, at \(r=0.75R\). \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline  & & \(\mathbf{Z}\) & & & & & & & & & & & & & \\  & & \(\mathbf{C_{L}}\) & \(\mathbf{C_{D}}\) & \(\mathbf{C_{L}/C_{D}}\) & \(\mathbf{C_{L}}\) & \(\mathbf{C_{D}}\) & \(\mathbf{C_{L}}\) & \(\mathbf{C_{D}}\) & \(\mathbf{C_{L}/C_{D}}\) & \(\mathbf{C_{L}}\) & \(\mathbf{C_{D}}\) & \(\mathbf{C_{L}/C_{D}}\) \\ \hline Downstroke (\%) & 2.1 & 0.8 & 1.3 & 1.3 & \(-\)0.7 & 2.1 & \(-\)1.4 & \(-\)0.9 & \(-\)0.5 & \(-\)7.0 & \(-\)4.0 & \(-\)3.1 \\ Upstroke (\%) & \(-\)12 & \(-\)7.6 & \(-\)4.7 & \(-\)2.3 & 0.6 & \(-\)2.8 & \(-\)5.0 & \(-\)0.5 & \(-\)4.5 & \(-\)5.3 & \(-\)3.9 & \(-\)1.4 \\
1 cycle (\%) & \(-\)4.5 & \(-\)3.2 & \(-\)1.4 & \(-\)0.4 & \(-\)0.1 & \(-\)0.3 & \(-\)3.1 & \(-\)0.7 & \(-\)2.4 & \(-\)6.2 & \(-\)3.9 & \(-\)2.4 \\ \hline \end{tabular}
\end{table} 
Table 2: Changes in the average lift coefficients (\(\rm C_{L}\)), drag coefficients (\(\rm C_{D}\)), and gliding ratios (\(\rm C_{L}\)/\(\rm C_{D}\)) of the Z, LE1, LE2, and LE3 wings, compared to those of the S wing, at \(r=0.75R\). Figure 10: Comparison of lift coefficient: (**a**) the time courses of lift coefficients (\(\rm C_{L}\)) and (**b**) the changes in the lift coefficients (\(\rm C_{L}\) change) of the Z, LE1, LE2, and LE3 wings, compared to those of the S wing, at \(r=0.75R\). For the upstroke, the lift coefficients of the three leading-edge corrugated wings are reduced significantly at \(t=0.6T\) but are maintained in the period of \(t=(0.7\) to \(1)T\). The lift coefficient of the Z wing is reduced significantly during almost the entire upstroke period. Due to the decrease, the upstroke average lift coefficients of the four corrugated wings are decreased, as shown in Table 2. The decrease in the upstroke average lift coefficient of the Z wing is the most, by 12%. The decreases in the upstroke average lift coefficient of the LE1, LE2, and LE3 wings are (2.3, 5, and 5.3)%, respectively. Over a flapping cycle, the cycle average lift coefficients of the LE1 wing are reduced the least, by 0.4%. For the Z and LE2 wings, the reductions are (4.5 and 3.1)%, respectively. That of the LE3 is reduced the most, by 6.2%, as shown in Table 2.  
Table 2 shows that in a flapping cycle, the drag coefficients of the Z and LE3 wings are reduced by (3.2 and 3.9)%, respectively. The drag coefficient of the LE2 wing is slightly decreased, by 0.7%, compared to the drag coefficient of the S wing. The reduction in drag coefficient of the LE1 wing is negligible, at 0.1%.