# Surface Patterns for Drag Modification in Volleyballs
## 3 Results and Discussion  
The first question is why aerodynamic characteristics along transverse and diagonal directions are different in conventional volleyballs with six panels. Traditionally, volleyballs are assembled from a total of six panels with each panel formed using three rectangular panel sections for a total of 18 sections and these volleyballs have been used in international competitions over a long period of time, as demonstrated in Figure 1. Conventional volleyballs assembled with six panels inevitably have two different panel directions in the production process. The transverse panel direction is parallel to the panel sections [Figure 1a] and the diagonal panel direction is perpendicular to the point where the three panels meet [Figure 1b]. Assembling volleyballs is like folding a cube with six sides of a square [Figure 1c]. The presence of two panel directions is supposed to be important for the anisotropic aerodynamic performance of volleyballs. The aerodynamic characteristics of volleyballs are demonstrated in Figure 2. The drag coefficients \(C_{d}\) with respect to the Reynolds number \(Re\) were measured in the wind tunnel experiments. The \(C_{d}\)-\(Re\) curves show that the drag crisis is dependent on the panel direction and the surface patterning. The different aerodynamic characteristics along the panel direction are verified for conventional balls: the drag crisis behaves differently between the transverse and the diagonal directions [Figure 3a,b]. The prime cause of the aerodynamic anisotropy is the difference in the panel direction. In contrast, the surface-modified balls have less difference in the drag crisis regardless of the panel direction [Figure 3c,d], which is due to the isotropic surface patterns with hexagons or dimples. Figure 1: Surface patterns of volleyballs with (**a**) transverse and (**b**) diagonal directions on panels. (**c**) In conventional volleyballs, the existence of two panel directions is a result of conventional assembly with six panels. (**d**) Surface patterns for conventional balls (Conv-1 and Conv-2) and modified balls with hexagon and dimple patterns. Figure 3: Flight characteristics of volleyballs. (**a**) An illustration for flight trajectory. (**b**) Lateral (L) and horizontal (H) flight distances were measured from the landing position experiment using a hitting robot. (**c**) All data of lateral and horizontal distances are summarized. Each panel with equidistant intervals is depicted at the same range of \(\pm\)2 m (lateral) and from 10 to 20 m (horizontal). Blue and red represents transverse and diagonal directions, respectively. Big circles and squares are the average values with error bars taken from the standard deviations. Figure 2: Drag crisis of volleyballs. Circles and squares represent transverse and diagonal directions, respectively, with different surface patterns: (**a**) Conv-1, (**b**) Conv-2, (**c**) Hexagon and (**d**) Dimple. Inset shows the wind tunnel experiment. To check out the experimental uncertainty, three data were included for each case. Discrepancy in drag crisis appears in conventional balls, while similarity in drag crisis in balls with isotropic patterns. The critical Reynolds numbers \(Re_{c}=dU_{c}/\
u\) for the onset of the drag crisis are measured in Figure 2, where \(U_{c}\) is the critical ball velocity and \(\
u\) is the kinematic viscosity of the air (\(\
u=1.5\times 10^{-5}\) m/s\({}^{2}\) at 20 \({}^{\circ}\)C [1; 15]). Figure 3 shows the flight characteristics of each ball taken from the lateral and horizontal landing positions. The longest flight distance (16.6\(\sim\)17.2 m on average) is found in the Hexagon ball. Interestingly, the shortest flight distance (12.7\(\sim\)14.1 m on average) is observed in the Dimple ball. The aerodynamic anisotropy of volleyballs originates from the panel direction through the volleyball production process.