# 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 \(Re=10\)  
From Figure 6, it is shown that at \(D/c=4.0\), there is a coupling effect when the ceiling and ground exist together, and the force enhancement is much larger than (almost twice) the sum of the ceiling-only effect and ground-only effect. Since the narrow-channel effect disappears at \(D/c=4.0\), we speculate that this coupling effect is achieved through the vortex-wake manipulation. Figure 14a,b exhibit different views of the vortex structures (iso-Q surfaces--green), and the downwash (iso-vertical velocity surface--yellow) at mid-downstroke (\(\tau=0.25\)). It includes 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 \(GO\)) at \(D/c=4.0\) and infinity case (\(D/c=\infty\) case). Observing Figure 14a,b, the vortex structures are almost the same for the ground-only case at \(D/c=4.0\) and the infinity case. There is an accumulated and stable vortex wake ahead of the flapping wing as marked in Figure 14b, which causes the  
Figure 13: (**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 \(D/c=1.0\), 4.0 and \(\infty\) at \(Re=10\). The red line in (**a**) gives the slice position. The red arrow in (**a**) indicates the stroke direction. largest downwash strength. This similar vortex wake, plus the similar aerodynamic forces in Figure 6 and Table 1, indicates that the ground-only effect disappears at \(D/c=4.0\) or larger. However, for the ceiling-only case at \(D/c=4.0\), the vortex wake left by the last flapping cycle is reduced in size and is much more scattered and weaker, resulting in a weaker downwash. Hence, the forces of the flapping wing are increased for the ceiling-only case at this wall clearance, as shown in Figure 6 and Table 1. In the case of the combined ceiling and ground, the vortex wake of the preceding stroke is further reduced in size, and is the weakest among the four cases, resulting in the weakest downwash. Figure 14c gives the spanwise vorticity contour of a spanwise slice, the position of which is shown in Figure 14a, along with an inset showing the velocity vectors ahead of the wing. It further shows that the vortex wake and associated downwash for the \(GO\) case at \(D/c=4.0\) is similar to that of the \(\infty\) case, but stronger than that of the \(CO\) case. For the \(CG\) case, the vortex wake and the downwash are the weakest. In addition, when observing the \(GO\) case, the \(CO\) case, and the \(CG\) case in order, the shape of the vortex wake changes from aligning more horizontally to aligning more vertically. This shape change makes the vortex wake contact the ground in the \(CG\) case, which does not happen in the \(GO\) case. This phenomenon will be further discussed in the following paragraphs. From the analysis of the fluid structures in Figure 14, it is seen that at \(D/c=4.0\), the ground-only effect disappears and the ceiling-only effect reduces the strength of the vortex wake to some extent. However, when the ceiling and ground coexist, the coupling effect on the vortex wake happens, with much quicker vortex-wake dissipation and a more substantial downwash-reducing effect. To explore how this coupling effect on the vortex wake happens, Figures 15 and 16 plot the evolution of the vortex wake over time from mid-downstroke (\(\tau=2/8\)) to the successive mid-upstroke (\(\tau=6/8\)) for three cases above at \(D/c=4.0\) (also referred to as \(CG\), \(CO\) and \(GO\)). Again, the bigger of the volume of the iso-Q surface, the stronger the vortex wake. Based on the findings of Ref.