# Aerodynamic Effects of Ceiling and Ground Vicinity on Flapping Wings
## 3 Results and Discussion
### The Reasons for the Coupling Effect of the Ceiling and Ground at \(Re=10\)
[43], a vortex ring needs energy input to keep  
Figure 14: (**a**) Top view; (**b**) oblique view of the iso-Q surface (green) and iso-vertical velocity surface (yellow) showing the strength of downwash; and (**c**) spanwise vorticity distributions on a slice at \(\tau~{}=~{}0.25\) for the combined ceiling and ground case (referred to as \(CG\)), the ceiling-only case (referred to as \(CO\)), the ground-only case (referred to as \(GO\)) at \(D/c=4.0\) and \(D/c=\infty\) case at \(Re~{}=~{}10\). The red arrow and red line in (**a**) give the stroke direction and the slice position, respectively. its form and intensity. The self-induced downwash associated with the vortex ring of the flapping wing can also be regarded as the energy source. The fluid's viscosity and the no-slip boundary on the ceiling could decrease the self-induced downwash above and within the TEV layer. Thus, the input energy of the vortex ring is reduced. Considering that the ground-only effect disappears at \(D/c=4.0\), it is reasonable that the vortex wake of the ceiling-only case would dissipate more rapidly than that of the ground-only case. This can be seen in the top view of the time evolution of the vortex structure in Figure 15b,c. In addition, it shows that the dissipation of the vortex wake for the combined ceiling and ground case is faster than that of the ceiling-only case. The side view of the time evolution of the vortex structures in Figure 16 may explain the reasons for this coupling effect. It can be seen that accompanying the dissipation of the vortex wake, its shape also changes from aligning more horizontally to aligning more vertically from the \(GO\) case to the \(CO\) case, then to the \(CG\) case (see Figure 16, \(\tau=4/8\)). This observation is consistent with those in Figure 14c. This elongating in the vertical direction makes the vortex wake nearer to the ground, as shown in Figures 14c and 16a. In turn, the ground, which has no chance to restrict the self-induced downwash below and within the TEV layer at \(D/c=4.0\) in the ground-only case, now has the chance to affect the vortex wake and restrict its downwash. It is believed that these are the reasons for the coupling effect between the ceiling and the ground. Accompanying the further downwash-reducing effect from the ground, the vortex wake loses more energy to keep its status and dissipate much more quickly. This can be seen from the last column for \(\tau=6/8\) in Figures 15 and 16, where most of the vortex wake has dissipated for the combined ceiling and ground case. However, for the ground-only case, the vortex wake still keeps its shape and intensity, while the size of the vortex wake for the ceiling-only case is between that of the two of the other cases. Figure 15: Top view of the iso-Q surface (green) with vortex wake for (**a**) the combined ceiling and ground case (referred to as \(CG\)); (**b**) the ceiling-only case (referred to as \(CO\)); and (**c**) the ground-only case (referred to as \(GO\)) at \(D/c=4.0\) for different times at \(Re~{}=~{}10\). The non-transparent wing in the first row indicates the position of the wing at the corresponding time.