# Aerodynamic Effects of Ceiling and Ground Vicinity on Flapping Wings
## 2 Materials and Methods  
### Wing Motion  
As shown in Figure 1a, two three-dimensional models of the insect wing were made to flap horizontally in the middle of two horizontal surfaces: a ceiling above and a ground below the wings. For better visualization, the insect's body is represented in Figure 1a, but was not considered in actual computations. The wing's planform shape resembled a hoverfly wing, as shown in Figure 1b. The wing's aspect ratio \(R/c\) was 3.75 (\(R\) and \(c\) being the wing length and mean chord length, respectively) [33], and \(R\) was equal to 13.20 mm for the actual hoverfly wing. The radius of gyration of the wing (\(r_{2}\)) was 0.56\(R\). We modeled wing as a flat plate with rounded edges, where its thickness was 3% of \(c_{l}\) (local chord length). In Figure 1c, \(D_{c}\) and \(D_{g}\) are used to denote the distance between the wing base and the wall (ceiling and ground, respectively) and named as the wall clearances. In this paper, \(D_{c}\) equals \(D_{g}\), denoted by \(D\).  
The kinematics of the wings implemented in our study is referred to as \"normal hovering\" which is a simplified model of the actual insect wing kinematics with a horizontal stroke plane. The spanwise axis of the wing, which connects the wing base to the wing tip, rotates about the wing base in the stroke plane reciprocally, and at the same time, the wing flips around the spanwise axis (see Figure 2a). Following  
Figure 1: (**a**) Schematic motion of the wing between the ceiling and the ground, a wing model is shown as an insect; (**b**) wing’s planform shape; and (**c**) side view of a wing section’s motion. The black dot indicates the wing base, and \(D_{c}\) and \(D_{g}\) mean the ceiling clearance and the ground clearance, respectively. In this paper, \(D_{c}=D_{g}=D\). The position of the section on the wing is shown in (**a**).  
methods, the forward and backward motions are called the downstroke and the upstroke, respectively. In Figure 2a, two independent angles, \(\phi(t)\) and \(\psi(t)\), are defined to describe the wing motion relative to the wing base. \(\phi(t)\) means the stroke position angle, and \(\psi(t)\) means the rotational angle. In the inertial frame \(OXYZ\), the origin \(O\) coincides with the wing base and the \(XOY\) plane overlaps with the stroke plane. \(\phi(t)\) is the angle between the spanwise axis and Y-axis (Figure 2a). Time variation of \(\phi(t)\) is given by:  
\[\phi(t)\ =\ 0.5\,\Phi\cos(2\pi nt) \tag{1}\]  
where \(\Phi\) is called the stroke amplitude, and \(n\) represents the wingbeat frequency and equals about 150 Hz.  
The rotation angle \(\psi(t)\) is the angle between the wing surface plane and the stroke plane. The wing moves with a constant angle \(\psi\) during the mid-portion of each half stroke while it flips over at stroke reversals. \(\alpha\) denotes the mid-stroke angles of incidence. In the downstroke, \(\alpha\ =\ \psi\), while in the upstroke, \(\alpha\ =\ 180^{\circ}\ -\ \psi\). The function of \(\psi(t)\) is described in our previous work [31; 32].  
Figure 2b demonstrates the time course curves of the two angles (\(\psi(t)\) and \(\phi(t)\)) over a stroke period. The dimensionless time \(\tau\) is used, which equals \(t/T\ -\ (n\ -\ 1)\) at the \(n\)th beat to implement the period normalization, and facilitates the representation of the different stages of any beat (\(T\) is the wingbeat period, equal to \(1/n\)). Thus, \(\tau\) ranges from 0 to 1 for any complete flapping cycle. The first half is the downstroke (\(0<\tau<0.5\)) and the second half is the upstroke (\(0.5<\tau<1\)). As per the data measured [33; 34; 35], \(\Phi\) and \(\alpha\) are fixed to be 110 and 40 degrees, respectively. The wing rotation interval for a stroke's reversal is set as 30% of \(T\).