# The Aerodynamics of New Design Soccer Balls Using a Three-Dimensional Printer
## 3 Results and Discussion
### Wind Tunnel Data  
Figure 5 shows drag coefficients for the eight manufactured balls as functions of the Reynolds number. To provide a comparison with an actual soccer ball, drag coefficient data from a previous investigation [6] for Telstar 18, which was used during the 2018 World Cup in Russia, are included in Figure 5. Points in Figure 5 are average values of 1000 force  
measurements at the various wind tunnel speeds. Error bars show one standard deviation on each side of the average. Because no wind tunnel is capable of creating a perfect flow of air in which every small cross section of area within the wind stream has the same time-independent flow velocity, force measurements vary slightly from one trial to the next. What the error bars in Figure 5 thus provide are rough estimates for sizes of variations in drag coefficient caused by inhomogeneities in wind tunnel flow. Horizontal error bars would be much smaller than the vertical ones because determining air-flow speed is much more accurate than deducing a drag coefficient from averaged force data. Drag coefficient data in Figure 5 follow the now-familiar trend [28; 30] that making a ball's surface rougher leads to a smaller critical Reynolds number and a larger drag coefficient at that critical Reynolds number. Table 2 lists the critical Reynolds numbers and corresponding drag coefficients for the eight manufactured balls and the Telstar 18. The critical Reynolds number and corresponding drag coefficient for ball A match quite well with what [PERSON] reported in his classic paper dealing with aerodynamics of smooth spheres [34], even though the minimum in ball A's \(C_{D}\) is admittedly at the third largest measured speed. Ball B's minimum \(C_{D}\) is easily identified, but Figure 5 shows that \begin{table}
\begin{tabular}{l l l} \hline \hline
**Ball** & \(\text{Re}_{e}\times\text{10}^{-5}\) & \(\text{C}_{D}\) (\(\text{Re}_{e}\)) \\ \hline A & 3.97 & 0.085 \\ B & 3.83 & 0.098 \\ C & 1.99 & 0.185 \\ D & 1.42 & 0.191 \\ E & 1.14 & 0.222 \\ F & 2.41 & 0.100 \\ G & 2.84 & 0.108 \\ H & 1.14 & 0.211 \\ Telstar 18 & 2.39 & 0.150 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Critical Reynolds number and corresponding drag coefficient for each of the manufactured balls and for Telstar 18. Figure 5: Drag coefficients of manufactured balls versus Reynolds number. Also shown for comparison are drag data for an actual soccer ball, the Telstar 18 [6]. there is not as much curvature at the location of the minimum as there is for the other printed balls. Ball B is modeled after a traditional 32-panel soccer ball. Numerous companies produce many types of 32-panel soccer balls with various ways of connecting the seams. Compared to the Molten Vantaggio [35], a 32-panel soccer ball with thermally bonded panel edges, ball B's critical Reynolds number is approximately 13% larger and its corresponding drag coefficient is about 41% smaller. The manufactured 32-panel ball is smoother than the Vantaggio soccer ball, with ball B's seams not as wide and deep as those between Vantaggio's thermally bonded panels. That explains why ball B has a larger critical Reynolds number compared to Vantaggio's. The Telstar 18 drag coefficient data fall somewhere between balls C and F for \(\text{Re}<\text{Re}_{c}\), but look to be closer to ball F, and perhaps ball G, for \(\text{Re}>\text{Re}_{c}\). Recall that balls C, F, and G all have conical dimples on their surfaces. The aerodynamics of a modern soccer ball may then be similar to a 3D-printed 32-panel ball with conical dimples. Such a proposition's validity will need more study to demonstrate. Because the focus of this work is on the connection between a ball's surface roughness and the critical Reynolds number, wind tunnel data for the side and lift forces will not be shown.