# Aerodynamic Effects of Ceiling and Ground Vicinity on Flapping Wings
## 3 Results and Discussion  
### Combined Ceiling and Ground Effect at Re = 10  
From previous ground-only effect studies, it is known that the force trends are different between \(Re\) = 10 and \(Re\) = 100 or higher, and the underlying fluid mechanisms are also different due to the different viscosity effects. Therefore, the combined ceiling and ground effect on aerodynamic forces were first investigated at a very low Reynolds number (\(Re\) = 10). In the simulations, the combined ceiling and ground effect were examined at \(D/c\) = 1.0, 1.5, 2.0, 2.5, 3.0, 4.0 and \(\infty\). In the simulations, the transient forces achieved periodicity after three flapping cycles. Thus, the fourth cycle was chosen for analysis and the cycle-averaged lift and drag coefficient (\(\overline{C_{L}}\) and \(\overline{C_{D}}\)) were calculated on the fourth cycle. Figure 4 shows the transient aerodynamic lift and drag coefficient (\(C_{L}\) and \(C_{D}\)) distributions for varying wall clearances. According to Figure 4, after the periodicity is reached, the transient forces distributions are symmetrical between the upstroke and the downstroke because of the symmetrical wing kinematics. Compared to \(D/c\) = \(\infty\), the transient lift and drag coefficients at \(D/c\) = 1.0 exhibit the largest systematic increase for nearly the whole flapping cycle, followed sequentially by \(D/c\) = 1.5, 2.0, 2.5, 3.0 and 4.0. Then, it is natural that the \(\overline{C_{L}}\) and \(\overline{C_{D}}\) decrease monotonically as \(D/c\) increases as shown in Figure 5. At \(D/c\) = 1.0, the \(\overline{C_{L}}\) and \(\overline{C_{D}}\) increase by 217.35% and 43.89%, respectively; this causes a 120.55% increment in \(\overline{C_{L}}/\overline{C_{D}}\), indicating that the wing can obtain a significant aerodynamic benefit when flapping in the vicinity of the ceiling and ground. At \(D/c\) = 2.0, the \(\overline{C_{L}}\) and \(\overline{C_{D}}\) increase by 81.63% and 11.11%, respectively. When \(D/c\) increases to 4.0, the \(\overline{C_{L}}\) can still increase by 41.84%, but the increase in \(\overline{C_{D}}\) is only 5.00%. The lift-drag ratio \((\overline{C_{L}}/\overline{C_{D}})\) in Figure 5 also shows the monotonic force trend. These monotonous trends with the changing wall clearance \(D/c\) are similar to those in ceiling-only effect in the Ref. [31] and ground-only effect in Ref. [24] at such low \(Re\). To know whether or not there exists any aerodynamic interaction (coupling) effect between the ground-only effect and ceiling-only effect, we made more computations for the flapping wing when only a ceiling or only ground exists. The forces of three cases, namely the ground-only case (referred to as \(GO\)), ceiling-only case (referred to as \(CO\)), and combined ceiling and ground case (referred to as \(CG\)), were compared at each of the wall clearance to explore the coupling effect. Figure 6 shows the cycle-averaged lift force enhancement relative to the no-wall case (expressed as \(\left(\overline{C_{L}}~{}-~{}\overline{C_{Lo}}\right)/\overline{C_{Lo}}\)) for the above three cases. The sum of the ceiling-only case and the ground-only case (referred to as \(CO~{}+~{}GO\)) is also shown in Figure 6. Table 1 shows the exact values from Figure 6.  
From Figure 6, the \(GO\) case, the \(CO\) case, and the \(CG\) case show that the aerodynamic force increases monotonically with decreasing wall clearance. However, the force enhancement in the \(CG\) case is much larger than that in the \(GO\) case and the \(CO\) case when \(D/c\) is smaller than or equal to 4.0. From Table 1, it is seen that the increase in the aerodynamic forces for the \(CG\) case at \(D/c\) = 1.0 is approximately equal to the sum of the separate effect of the ceiling and ground. This is also true at \(D/c\) = 1.5 and 2.0.