# Aerodynamic Effects of Ceiling and Ground Vicinity on Flapping Wings
## 4 Conclusions  
The combined aerodynamic effects of the ceiling and the ground on normal hovering flapping wings were studied using the computational fluid dynamics method at various wall clearances and at two Reynolds numbers (\(Re=10\) and \(100\)). The underlying mechanisms of the combined ceiling and ground effect and the coupling effect at \(Re=10\) are explained.  
The combined effect of the ceiling and the ground changes the aerodynamic forces through two effects, namely the narrow-channel effect and the downwash-reducing effect. The narrow-channel effect enhances the suction force on the upper wing surface by increasing the incoming flow velocity and the effective angle of attack. It also produces more significant positive pressure on the lower wing surface by squeezing fluid. The narrow-channel effect decreases monotonically with increasing wall clearance at both \(Re=10\) and \(100\). However, at a higher \(Re\), it disappears at a smaller wall clearance.  
The downwash-reducing effect decreases the strength of the downwash associated with the vortex wake, and consequently, increases the aerodynamic forces. At a low \(Re\) (\(Re=10\)), the ceiling and ground decrease the downwash strength monotonically with decreasing wall clearance, by helping the vortex wake to shed from the wing, and meanwhile, dissipate quickly. However, at \(Re=100\), the downwash strength is changed non-monotonically by the interaction between the vortex wake and the walls. As the wall clearance increases, the distance between the vortex wake and the wing first decreases and then increases, leading to a non-monotonic change in the downwash intensity caused by the vortex wake on the wing. This non-monotonic downwash-reducing effect and the narrow-channel effect eventually lead to the three force regimes (force enhancement, force reduction, and force recovery) at \(Re=100\).  
At \(Re=10\), there is a coupling effect, where the force enhancement caused by the combined ceiling and ground effect is much larger than (almost twice) the sum of the ceiling-only effect and the ground-only effect. The underlying fluid physics are as follows. When the wall clearance is large enough, the ground-only effect on the vortex wake disappears first. However, at the same wall clearance, the ceiling-only effect still functions. It changes the vortex-wake shape from aligning more horizontally to aligning more vertically, thus bringing the shed vortex wake close enough to the ground, and then the ground's downwash-reducing effect comes into play. Finally, this coupling effect causes the vortex wake for the combined case to dissipate much more quickly than in the ceiling-only case and the ground-only case. Unlike at \(Re=10\), the coupling effect at \(Re=100\) is small.  
Conceptualization, X.M. and G.C.; methodology, X.M.; Z.C. and A.G.; formal analysis, X.M.; Y.H.; Z.C. and G.C.; data curation, Y.H.; writing--original draft preparation, X.M.; Y.H. and G.C.; writing--review and editing, X.M.; A.G. and G.C.; visualization, Y.H. and Z.C.; supervision, G.C.; funding acquisition, X.M. and G.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by grants from the National Natural Science Foundation of China (Grant Nos. 11872293 and 12172276) and the Natural Science Foundation of Shaanxi Province (Grant No. 2022 JC-03).  
Not applicable.  
Please contact the corresponding author for any additional data or models used in this paper.  
The authors declare no conflict of interest.