# 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 = 100  
To explain the reasons for the non-monotonous aerodynamic force behavior for the combined ceiling and ground case at \(Re=100\), the vortex structures and downwash were also studied. Figure 20 shows the downwash contours when the wing moves from left to right in the top view at \(D/c=1.0\), \(D/c=2.5\) and \(\infty\). It is seen that the downwash strength firstly increases as the wall clearance (\(D/c\)) changes from 1.0 to 2.5, and then decreases as the wall clearance changes from 2.5 to \(\infty\). When the wing flaps through this downwash region, a smaller downwash enlarges the effective angle of attack, and leads to larger forces. Moreover, the non-monotonous downwash strength causes the non-monotonous aerodynamic force behavior. Figure 21a,b are different views of the vortex structures (iso-Q surfaces) at mid-downstroke (\(\tau=0.25\)) for \(D/c=1.0\), \(2.5\) and \(\infty\). The corresponding iso-vertical velocity surfaces (yellow) on the wing's head are also given. Because of the relatively low viscosity at \(Re=100\), the trailing-edge vortex easily separates from the trailing edge of the wing. Consequently, the LEV, root vortex (RV), TEV, and tip vortex (TV) form the vortex ring for all the three wall clearances. Additionally, the blue dash-dot line and red dashed line indicate the vortex ring formed in this stroke and the vortex wake of the preceding stroke, respectively. For the \(D/c=1.0\) case, the wall clearance is relatively small; thus, the wake is quickly dissipated as it is stretched away from the wing by the walls. Finally, the lowest interaction occurs between the wing and the wake. As the wall clearance increases to \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{D/c}\) & \(\mathbf{CG(\%)}\) & \(\mathbf{CO+GO(\%)}\) & \(\mathbf{CO(\%)}\) & \(\mathbf{GO(\%)}\) \\ \hline
1.0 & 53.66 & 45.12 & 25.61 & 19.51 \\ \hline
1.5 & 4.88 & 10.37 & 12.20 & \(-1.83\) \\ \hline
2.0 & \(-7.93\) & \(-1.83\) & 6.71 & \(-8.54\) \\ \hline
2.5 & \(-10.98\) & \(-6.71\) & 3.66 & \(-10.37\) \\ \hline
3.0 & \(-3.05\) & \(-6.10\) & 1.83 & \(-7.93\) \\ \hline
4.0 & 1.83 & 0.61 & 0.61 & 0.00 \\ \hline \(\infty\) & 0.00 & 0.00 & 0.00 & 0.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: 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 \(Re\ =\ 100\). \(CG\), \(CO\ +\ GO\), \(CO\) 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 20: The vertical velocity (\(w\)) contours on the stroke plane (top view) with vortex wake at the beginning of a stroke for (**a**) \(D/c=1.0\); (**b**) \(D/c=2.5\); and (**c**) \(D/c=\infty\) at \(Re\ =\ 100\). The dashed boxes indicate the regions encountered by the wing at mid-stroke. The wing’s stroke direction is given by the red arrow. a medium clearance (\(D/c=2.5\)), the vortex wake becomes less stretched and dissipates more slowly; hence, the downwash area is mainly on the wing's path. Thus, it causes more interaction of the wing wake and the forces decrease. However, from \(D/c=2.5\) to \(D/c=\infty\), due to the low viscosity at \(Re=100\) and without the bottom wall's restriction, the vortex wake can advect downward and out of the wing's path (see Figure 21b, \(D/c=\infty\)). Consequently, the downwash caused by the vortex wake on the wing declines, and the forces recover. The evolution of the vortex wake and the associated downwash is similar to that of the ground-only effect in Ref. [24]. Figure 21c shows the spanwise vorticity distribution of a slice pointed in Figure 21a. From \(D/c=1.0\) to \(2.5\), the vortex wake changes from being further away to being close in the horizontal direction, and in the meantime, becomes stronger. As \(D/c\) further increases to \(\infty\), the wake can convect downwards.