## 3 Results and Discussion
### Changes in the in Lift and Side Forces over Time  
Figure 6 plots the changes in standard deviations for the side and lift forces on each soccer ball for a 10 s duration at a constant wind velocity of 25 m/s. T1_A indicates the changes in the standard deviations of the side and lift forces on the Type 1 soccer ball positioned at orientation A for 10 s, and T1_B indicates the same at orientation B. Similar changes were observed for the side and lift forces on the Type 2 soccer ball (T2_A, T2_B) (10 s, wind velocity of 25 m/s). When T1 and T2 are compared, it is possible to see that the drag coefficient of T1 is more direction-dependent, but no significant difference was observed when the lift and side forces were changed. Non-stationary forces often exhibit irregular fluctuations; furthermore, the fluctuations in the lift and side forces acting on the ball varied depending on the type of soccer ball as shown in Figure 6. Specifically, we found that the irregular fluctuations for aerodynamic force differed more between different orientations of the same ball than between different ball types (Table 1). The variations in the lift and side forces were approximately 15% smaller (\(p<0.1\)) for the T2 ball with grooves compared to the T1 ball, which suggests that introducing a shallow groove structure on the surface of a soccer ball is an effective way to minimize the irregular vertical and horizontal force fluctuations and hence can  
Figure 5: Drag coefficient of the soccer balls. be expected to produce a more stable flight trajectory. Specifically, the results of Figure 6 indicate that the forces standard deviation (SD) vary more widely when the orientation of the same ball is changed than when different ball types are used (Table 1). Previous research has shown that adding a bumpy dimple pattern to the soccer ball's surface causes the air flow separation line to move backwards, thereby reducing the force (air resistance) exerted on the ball [17]. It has also been reported that the aerodynamics of a soccer ball are significantly affected by the characteristics of its seams, in particular the length, width, and depth of the seam [1]. With respect to the length of the seam in particular, it has been reported that the longer the seam, the smaller the critical Reynolds number (\(Re_{c}\)) [1,2]. However, in this study, the value of \(Re_{c}\) for the T1 soccer ball (with a seam length of 2300 mm) was \(\sim\)\(2.4\times 10^{5}\) (\(Cd\approx 0.16\)) for T1_A and \(\sim\)\(2.4\times 10^{5}\) (\(Cd\approx 0.16\)) for T1_B. For the T2 soccer ball with grooves (each 450 mm long) at four locations, with a seam length of 2300 mm and a total groove length of 1800 mm, the value of \(Re_{c}\) was \(\sim\)\(2.6\times 10^{5}\) (\(Cd\approx 0.15\)) for T2_A and \(\sim\)\(2.5\times 10^{5}\) (\(Cd\approx 0.15\)) for T2_B. The fact that the two balls exhibit little difference in the value of \(Re_{c}\), counter to the result from previous research [1,2] on the relationship between the length of the seam and the critical Reynolds number (\(Re_{c}\)). This is considered to be due to the fact that the depth (2 mm) and width (5 mm) of the groove introduced in this study are shallower and narrower than the depth (3.8 mm) and width (10 mm) of the seam. In addition, previous research on golf balls has shown that the form of the dimples on the surface of the ball has a large impact on the ball's aerodynamics, and that the ball's wind velocity range characteristics depend on the size of the dimples [18]. We speculate that the deep seams and shallow grooves on the surface of the soccer ball in this study act like dimples of different sizes on a golf ball. Furthermore, the fact that the shallow, narrow groove reduces the air resistance in the high-velocity segment suggests that the groove structure has a significant impact on the air flow around the ball in the supercritical region.