# Surface Patterns for Drag Modification in Volleyballs
## 3 Results and Discussion
Importantly, the aerodynamic anisotropy can be eliminated by the isotropic surface modification, for instance, by adopting the isotropic hexagonal patterns and the dimples. For the unmodified (Adidas and Mizuno) volleyballs, the aerodynamics is anisotropic, originating from the panel direction: the drag crisis behaves separately with the transverse and diagonal panel directions. As a result, their flight distances are relatively short. By adopting the surface modification with the isotropic patterns such as hexagonal patterns (Molten) and dimples (Mikasa), the modified volleyballs can have the improved aerodynamics. Their aerodynamic coefficients with respect to the Reynolds number behave almost identically regardless of the panel direction, their aerodynamic forces are stabilized and their flight distances can be controlled to have the longest distance (Molten) or the shortest distance (Mikasa). The flight distance modification by surface pattern may be explained by the drag coefficients. The Dimple ball, which is currently used in international competitions (e.g., Olympics), has the shorter flight distance compared to the Hexagon ball, which exhibit the longest flight distance. At the ball velocity \(U=15\) m/s (\(Re\)\(\sim\)2 \(\times 10^{5}\)) in the hitting robot tests, the drag coefficients for the Hexagon ball, estimated as \(C_{d}\approx\) 0.26\(\sim\)0.34 from Figure 2, are lower than those for the Dimple ball as 0.45\(\sim\)0.50. The conventional balls show the intermediate \(C_{d}\) quantities as 0.25\(\sim\)0.48 for the Conv-1 and 0.21\(\sim\)0.43 for the Conv-2 balls. Interestingly, the hexagonal pattern in the Hexagon ball significantly reduces the critical \(Re\) for the drag crisis, while the dimple pattern in the Dimple ball enhances the critical \(Re\) for the drag crisis. Similar results were found in the dimpled soccer balls [27] and the dimpled golf balls [28]. We suppose that the drag enhancement by dimples is associated with the boundary-layer instability. This study suggests that the critical \(Re\) number for the drag crisis can be reduced or enhanced with the surface design in a volleyball. The relatively far or short flight of a ball can be controlled by the surface design of a ball. The lower the drag coefficient, the farther the flight of the ball. The design of the surface pattern can contribute to the drag reduction or the drag enhancement [29]. The surface-modified balls exhibit the quite isotropic aerodynamics and the well-controlled flight distances. In ball sports, the well-controlled flight trajectory of a ball would be useful to lure opponents and score [15]. For instance, float serves are important skills for ace players and they utilize the knuckle effect of a ball, allowing a ball to fall suddenly or to have irregular flight paths using no- or slow-rotating while flying [25]. Therefore, controlling the flight characteristics of a ball would be important to ball designers and players. This work shows an answer for the hypothesis: from a comprehensive study on the aerodynamic singularity in volleyballs, the present work reveals how the isotropic surface modification can eliminate the aerodynamic anisotropy and reduce or enhance the drag coefficient. As described in Figure 4, the results of lift and side forces show that the isotropy of the aerodynamics would be relevant to the isotropy of the surface pattern. The previous work is limited to identify difference in the critical Reynolds number or the drag coefficient between unmodified and modified volleyballs [19]. Unmodified volleyballs assembled with traditional 18 sections and modified volleyballs with isotropic hexagonal patterns or isotropic dimples represent all feasible designs of volleyballs that are actually used in volleyball games. The main result would be generalizable beyond the specific brands of volleyballs. Recently, significant changes were tried in design of the volleyball surface.