# Aerodynamic Effects of Ceiling and Ground Vicinity on Flapping Wings
## 3 Results and Discussion
### The Reasons for the Combined Ceiling and Ground Effect at \(Re=10\)  
We summarize the reasons for the combined ceiling and ground effect as two types. One is called the \"downwash-reducing effect\", where the wall changes the aerodynamic forces by altering the strength of the downwash associated with the vortex wake; the other is called the \"narrow-channel effect\", where the fluid dynamics are changed by the channel formed between the wall and the flapping wing, with or without the presence of the vortex wake (downwash). \begin{table}
\begin{tabular}{c c c c c} \hline \hline _D/c_ & _CG(\%)_ & _CO + GO(\%)_ & _CO(\%)_ & _GO(\%)_ \\ \hline
1.0 & 217.35 & 190.82 & 86.37 & 104.08 \\ \hline
1.5 & 111.22 & 113.27 & 63.27 & 50.00 \\ \hline
2.0 & 81.63 & 69.39 & 45.92 & 23.47 \\ \hline
2.5 & 68.37 & 41.84 & 33.67 & 8.16 \\ \hline
3.0 & 57.14 & 30.61 & 25.51 & 5.10 \\ \hline
4.0 & 41.84 & 19.39 & 16.33 & 3.06 \\ \hline \(\infty\) & 0.00 & 0.00 & 0.00 & 0.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The relative increment of cycled-averaged lift coefficient relative to the \(D/c=\infty\) case (expressed as \((\overline{C_{L}}\ -\ \overline{C_{L\infty}})/\overline{C_{L\infty}}\)) at \(Re\ =\ 10\). _CG_, _CO + GO_, _CO_ and _GO_ represent the combined ceiling and ground case, the sum of the ceiling-only case and the ground-only case, the ceiling-only case and the ground-only case, respectively. Figure 6: The relative increment of cycled-averaged lift coefficient relative to the \(D/c=\infty\) case (expressed as \((\overline{C_{L}}\ -\ \overline{C_{L\infty}})/\overline{C_{L\infty}}\)) for the combined ceiling and ground case (referred to as _CG_), the ceiling-only case (referred to as _CO_), and the ground-only case (referred to as _GO_) and the sum of the ceiling-only case and the ground-only case (referred to as _CO_ + _GO_) versus \((D/c)\) at \(Re\ =\ 10\). Let us first see the narrow-channel effect in detail when the ceiling and ground exist together. Figure 7 compares the first stroke's force curves for \(D/c=1.0\), \(1.5\), \(2.0\), \(2.5\), \(3.0\), \(4.0\) and \(\infty\). The force enhancement shown in this figure is completely due to the narrow-channel effect, as the wake has not fully developed and there is no chance for the ground and the ceiling to affect the downwash. Compared to \(D/c=\infty\), the transient lift coefficient (\(C_{L}\)) curve at \(D/c=1.0\) still exhibits the largest systematic increase for nearly the whole half-flapping cycle, followed sequentially by \(D/c=1.5\), \(2.0\), \(2.5\). However, the transient lift-force coefficient curves at \(D/c=3.0\) and \(4.0\) are almost identical to that of \(D/c=\infty\). Moreover, the cycle-averaged lift coefficients (\(\overline{C_{L}}\)) for the first downstroke at \(D/c=3.0\) and \(4.0\) are increased by \(7.14\%\) and \(3.47\%\), respectively, compared to \(D/c=\infty\). These results indicate that the narrow-channel effect of the combined ceiling and ground effect will not occur when \(D/c\) is \(4.0\) or larger. Figure 8 plots the surface pressure (\(C_{p}\)) distributions in the middle of the first downstroke (\(\tau=0.25\)) when there is no vortex wake at \(D/c=1.0\), \(4.0\) and \(\infty\). Observing Figure 8, it is found that the \(D/c=1.0\) case exhibits both the largest negative pressure zone area on the upper surface and positive pressure zone area on the lower surface among the three cases. For the \(D/c=4.0\) case, the pressure on both sides is approximately the same as \(D/c=\infty\), which is consistent with approximately the same aerodynamic forces shown in Figure 7. Again, this confirms that the narrow-channel effect is insignificant at \(D/c=4.0\). Figure 8: Surface pressure contour plots on (**a**) the upper surface and (**b**) the lower surface for \(D/c=1.0\), \(4.0\) and \(\infty\) in the middle of the first downstroke (\(\tau=0.25\)) at \(Re=10\). Figure 7: The transient aerodynamic lift coefficient (\(C_{L}\)) distributions for the first downstroke at systematic wall clearances (\(D/c\)) and \(Re=10\).