# Aerodynamic Effects of Ceiling and Ground Vicinity on Flapping Wings
## 3 Results and Discussion
### The Reasons for the Combined Ceiling and Ground Effect at \(Re=10\)
Figure 9a,b show the streamlines relative to the wings, along with the incoming flow velocity (\(u\)) contours on the \(2/3 RR\) spanwise slice in the middle of the first downstroke (\(\tau=0.25\)), for the \(D/c=1.0\) and \(\infty\) cases (\(D/c=4.0\) is almost the same as \(D/c=\infty\), so is not shown here). Judging from the density of the streamlines and the velocity contours, especially in the squared area, the velocity of the incoming flow of \(D/c=1.0\) is larger than that of \(D/c=\infty\). This is because a narrow channel was formed between the leading edge of the wing and the ceiling. Based on the directions of the velocity vector, the effective angle of attack (\(\alpha_{e}\)) of the wing was also calculated and is shown in Figure 9c,d. Comparing the \(\alpha_{e}\) contours for the two cases, it is seen that the effective angle of attack of the \(D/c=1.0\) is larger than that of \(D/c=\infty\). This can easily be seen by comparing the surrounded area of the same contour line with the value of \(60^{\circ}\) in Figure 9c,d. With a larger incoming velocity and effective angle of attack, the wing would produce a larger LEV. This is validated by the fact that the area of negative spanwise vorticity (\(\omega_{z}\)) at \(D/c=1.0\) is larger than that of \(D/c=\infty\), as shown in Figure 9e,f. This helps to explain the larger negative pressure on the upper wing surface in Figure 8. The corresponding circulation of the LEV calculated in Figure 10 also shows the consistency, with the circulation along the wingspan for \(D/c=1.0\) being larger than that of \(D/c=\infty\). Here, the increase in the incoming flow velocity and angle of attack in Figure 9 may also be regarded as the result of the existence of an image LEV, as explained in Ref. [31], which studied the ceiling-only effect. The classical image approach was also employed in Ref. [17] to study the ground effect for a fixed-wing. This classical method is based on the linear potential theory. Thus, it should be applied cautiously in the analysis of a wall effect for unsteady motion at a high angle of attack. In addition, the wake vortex, which is hard to convect downstream in hovering flight, is one of the main factors affecting the aerodynamics of a flapping wing. The application of the image method to a flapping wing will be more complex than to a fixed wing, because it is necessary to simulate the image's leading-edge vortex and all the shedding wake vortices at the same time. Figure 9: The streamlines and velocity (\(u\)) contours relative to the wings (**a**,**b**), the effective angle of attack (\(\alpha_{e}\)) contours (**c**,**d**), and the spanwise vorticity (\(\omega_{z}\)) contours (**e**,**f**) in the middle of the first downstroke (\(\tau~{}=~{}0.25\)) at \(Re~{}=~{}10\). The first row is for \(D/c=1.0\), and second row is for \(D/c=\infty\). To explain why there is larger positive pressure on the lower wing surface for the combined ceiling and ground case, the velocity vectors in the inertial frame around the wing section, plus the pressure contours, are plotted in Figure 11. It is found that due to the restriction of the ground at \(D/c=1.0\), a narrow channel is formed between the trailing edge of the wing and the ground, and the velocity vectors are'squeezed' to align more horizontally than that at \(D/c=4.0\) and \(D/c=\infty\). This leads to a larger pressure on the lower surface in Figure 8, which is also evident in the pressure contour distribution and the high-pressure area encircled by the single contour line with a value of 2.0 in Figure 11b. As the clearance increases to \(D/c=4.0\), the air squeezing effect is very weak, the vectors are almost the same between the \(D/c=4.0\) and \(\infty\), and finally, the pressure distributions of these two cases are almost the same. From the above analysis of the first stroke when there is no vortex wake, the increase in aerodynamics is only due to the narrow-channel effect.