# Aerodynamic Effects of Ceiling and Ground Vicinity on Flapping Wings
## 3 Results and Discussion
### Combined Ceiling and Ground Effect at \(Re=100\)  
In the study of the ground effect in Ref. [24], the reasons for \(Re=10\) and \(Re=100\) are different. Thus, it is interesting to see the combined ceiling and ground effect on the aerodynamics of a flapping wing at a higher \(Re\). Using the same parameters and kinematics as those at \(Re=10\), we employed the equivalent code to simulate the \(Re=100\) case. Figure 17 shows the transient aerodynamic lift and drag coefficient (\(C_{L}\) and \(C_{D}\)) distributions for varying wall clearances. Compared to \(D/c=\infty\), the transient lift and drag coefficients at \(D/c=1.0\) exhibit a large systematic increase for nearly the whole flapping cycle. The \(D/c=1.0\) case is followed sequentially by the \(D/c=1.5\) case. However, the transient lift and drag coefficients decrease rapidly to the lowest values for nearly the whole flapping cycle from \(D/c=1.5\) to \(D/c=2.5\). From \(D/c=2.5\) to \(D/c=4.0\), the transient lift and drag coefficients rise again and almost overlap with the \(D/c=\infty\) case. Consequently, it is shown in Figure 18 that the cycle-averaged lift and drag coefficients (\(\overline{C_{L}}\) and \(\overline{C_{D}}\)) firstly decrease to the minimum at about \(D/c=2.5\), and then recover as \(D/c\) increases further. Thus, the force behavior at \(Re=100\) exhibits a non-monotonous trend of 'three force regimes' as the wall clearance \(D/c\) changes. This behavior is different from the monotonous trend at \(Re=10\). Figure 16: Side 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 red line means the position of the ceiling and/or the ground. The non-transparent wing in the first row indicates the position of the wing at the corresponding time. Figure 17: The transient aerodynamic (**a**) lift and (**b**) drag coefficient (\(C_{L}\) and \(C_{D}\)) distributions for one flapping cycle at systematic wall clearances (\(D/c\)) and \(Re~{}=~{}100\). To know whether or not there exists any aerodynamic interaction (coupling) effect between the ground-only effect and ceiling-only effect at \(Re=100\), we also made more computations for the flapping wing when only the ceiling or only ground exists. Figure 19 shows the force enhancement relative to the no-wall case (expressed as \(\left(\overline{C_{L}}\ -\ \overline{C_{Lo}}\right)/\overline{C_{Lo}}\)) under 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 \(GO\)), and the sum of the ceiling-only case and the ground-only case (referred to as \(CO\ +\ GO\)). The ceiling-only case shows that the aerodynamic force increases monotonically as the wall clearance decreases. However, the ground-only case and the combined ceiling and ground case both show that the aerodynamic forces exhibit a non-monotonous trend of 'three force regimes' as the wall clearance \(D/c\) decreases. By comparing the results of the \(CG\) with those of the \(CO\ +\ GO\), it can be observed that the changes in the aerodynamic force caused by the combined ceiling and ground effect are approximately equal to the sum of the ceiling-only effect and the ground-only effect. Table 2 gives the exact values from Figure 19. It is shown that the difference between the combined ceiling and ground effect, and the sum of the ceiling-only effect and the ground-only effect, are always smaller than 10% (the maximum difference is 8.54% at \(D/c=1.0\)). These results show that the coupling effect of the ceiling and the ground is small at \(Re=100\).