## 3 Results and Discussion
### Fluid Force Experiment  
Figure 8 shows the \(C_{D}\) diagram for each ball. The drag coefficients of all the balls are large in the laminar flow region where the Reynolds number is low, but in the turbulent area where the Reynolds number is high, a drag crisis occurs, and the drag coefficient becomes small. In the recently released Merlin, the Reynolds number at which the drag crisis occurs is smaller than in other balls. The drag coefficients in the laminar and turbulent regions were relatively more extensive than those of the other balls. As shown in Figure 8, Merlin was characterized by a more considerable panel groove length and volume than the other balls. This larger total volume of the grooves has affected the drag crisis and drag coefficient. The changes in the drag coefficient of the other balls were consistent with the trends shown in previous studies [8; 9; 12; 21; 22; 23]. Thus, the drag coefficient dropped rapidly from about 0.5 to 0.2 in the Reynolds numbers range from 1.0 \(\times\) 10\({}^{5}\) to 2.4 \(\times\) 10\({}^{5}\). This drag  
Figure 6: 2D-PIV (particle image velocimetry) measurement. Figure 7: Specifications of ball groove. crisis corresponds to a change in the aspect of the flow around the ball from laminar to turbulent, but the Reynolds number at which the drag crisis occurs varies significantly from ball to ball. Therefore, we focused on the first and the last Reynolds numbers of the drag crisis to characterize these phenomena. The first and last Reynolds numbers of the drag crisis are smallest at \(1.0\times 10^{5}\) and \(1.2\times 10^{5}\) for Evopower and largest at \(1.8\times 10^{5}\) and \(2.4\times 10^{5}\) for Jabulani. The Reynolds number between the start and endpoints of the drag crisis, the critical region's width, is about \(0.3\) for Evopower and about \(0.6\) for Jabulani, which is also a significant difference between the balls. Figures 9 and 10 show the relationship between the volume of the grooves and the Reynolds number to clarify the dependence of the start and end of the drag crisis and the width of the critical region on ball panel geometry. As shown in the figure, there is a strong correlation between the grooves' volume and the Reynolds number of the critical region. The correlation coefficients are \(-0.63\) for the starting Reynolds number and \(-0.83\) for the ending Reynolds number. Figure 11 shows the relationship between the groove volume and the width of the critical region. Overall, the width of the critical region became smaller as the groove volume was increased. However, when Evopower was excluded, there was no clear correlation between the groove volume and the width of the critical region. Thus, the panel grooves significantly impact how the drag crisis occurs, i.e., the transition from laminar to turbulent flow. In order to elucidate the details of this mechanism, we continued to visualize the separation points by oil film experiments and 2D-PIV measurements to investigate how the transition from laminar to turbulent flow is induced. The relationship between groove volume and the drag coefficient in subcritical and supercritical regions is shown in Figures 12 and 13. There is a weak correlation between the groove volume and drag coefficient of \(0.40\) and \(-0.20\) in each velocity range, respectively. In the subcritical region, where the flow is laminar, the drag coefficient increases with the number of grooves, i.e., the rougher the surface, the higher the drag coefficient. In the turbulent, supercritical region, the drag coefficient decreased as the volume of the grooves increased. This may be due to the degree of accelerated turbulence, but the trend was not very clear. Figure 8: The drag coefficient (\(C_{D}\)) diagram of each ball. Figure 11: The relationship between the width of the critical region and total groove volume. Figure 10: The relationship between the Ending Reynolds number (\(Re\)) of drag crisis and total groove. volume.