# Aerodynamic Effects of Ceiling and Ground Vicinity on Flapping Wings
## 3 Results and Discussion
### The Reasons for the Combined Ceiling and Ground Effect at \(Re=10\)
The channel between the wing  
Figure 11: (**a**) The velocity vectors; (**b**) the pressure contours on the slice at 2/3 of wingspan in the middle of the first downstroke (\(\tau~{}=~{}0.25\)) for \(D/c=1.0\), 4.0 and \(\infty\) at \(Re~{}=~{}10\). Figure 10: The change curves of LEV circulation along the wingspan in the middle of the first downstroke (\(\tau~{}=~{}0.25\)) for \(D/c=1.0\), 4.0 and \(\infty\) at \(Re~{}=~{}10\). and the ceiling increases the relative oncoming flow velocity near the wing's leading edge, and also increases the effective angle of attack. Meanwhile, the narrow channel between the wing and the ground squeezes the fluid and increases the pressure on the lower wing surface. Referring to Figure 4 again, which shows the force curves after the periodicity has been established, it is noted that the lift coefficient (\(C_{L}\)) at \(D/c=4.0\) becomes larger than that of \(D/c=\infty\). Since the narrow-channel effect is not functioning at \(D/c=4.0\), the force enhancement at this wall clearance, after the vortex wake is fully developed, can be attributed to the restriction of the wall on the development of the vortex wake and the associated downwash (discussed below). Note that all the results in the remaining section discuss the fourth flapping cycle. Figure 12 shows the downwash velocity (denoted by \(w\)) contours when the wing moves from left to right in the top view at \(D/c=1.0\), \(D/c=4.0\) and \(\infty\). The dashed boxes are shown to exhibit the regions encountered by the wing at mid-stroke. It is seen that the downwash strength increases as the wall clearance increases. When the wing flaps through this downwash region, smaller downwash enlarges the effective angle of attack, and leads to larger forces. Figure 13 shows the fluid features around the wing for \(D/c=1.0\), \(4.0\) and \(\infty\) when the wing is in the mid-downstroke (\(\tau=0.25\)). Figure 13a,b exhibit different views of the vortical structures (iso-Q surfaces--green), and the downwash (iso-vertical velocity surface--yellow). The size of the iso-vertical velocity surface on the wing's head indicates the strength of the downwash. For \(D/c=\infty\), because of the high viscosity at such low \(Re\) (\(Re=10\)), the trailing-edge vortex cannot separate from the trailing edge of the wing. Rather, it forms an accumulated and thick vorticity layer that remains attached to the wing during a single stroke (marked in Figure 13a,b). When the wing moves back, the thick vorticity layer (or the vortex wake) causes a large downwash at the head of the wing. At \(D/c=4.0\), due to the existence of the ceiling and ground, the vortex wake of the previous stroke dissipates quickly. This can be seen by comparing the size of the iso-Q surface of the vortex wake of \(D/c=4.0\) with that of \(D/c=\infty\) in Figure 13a. Consequently, the downwash caused by the vortex wake is lower near the wing's head. At \(D/c=1.0\), the vortex wake from the preceding stroke already disappears, so that the downwash is negligible. Figure 13c gives the spanwise vorticity contour in a spanwise slice, the position of which is shown in Figure 13a. The boxed region in Figure 13c is exaggerated as an inset to show the velocity vectors. It is seen that at low wall clearance, i.e., at \(D/c=1.0\), there is no vortex wake remaining near the wing head, and the velocity vectors are nearly  
Figure 6:0 The vertical velocity (\(w\)) contours on the stroke plane (top view) at the beginning of a stroke for (**a**) \(D/c=1.0\); (**b**) \(D/c=4.0\); and (**c**) \(D/c=\infty\) at \(Re=10\). The dashed boxes indicate the regions encountered by the wing at mid-stroke. The wing’s stroke direction is given by the red arrow. horizontal compared to the infinity case (\(D/c=\infty\) case). When the wall clearance increases to \(D/c=4.0\), the strength of vortex wake becomes larger and the velocity vectors begin to deflect downwards, meaning that the downwash strength is larger.