# The Aerodynamics of New Design Soccer Balls Using a Three-Dimensional Printer
## 3 Results and Discussion
### Correlation Analyses  
There are several ways to characterize the roughness of the balls manufactured for this study. Dimple count could be used, but ball B, with its smooth panels, has seams that influence the ball's aerodynamic properties. Seam length could be used, but that parameter will not account for ball H's deeper seams. Dimple depth alone neither considers how many dimples are on a ball's surface nor the shape of those dimples.  
To account for the various ways a ball's surface may be roughened, surface roughness is measured here by forming a ratio of the new surface area created by the dimples and seams to the surface area of the remaining original surface. Because dimples and seams are carved into the ball's surface, the sum of the carved dimple and seam area exceeds the difference of a smooth ball's surface area and the remaining smooth surface area. The ratios formed here will thus not all have the same denominator.  
The starting surface area is that of smooth ball A, which has a surface area of \(A_{surface}=\pi\,(220\,\text{mm})^{2}\simeq 152,053\,\text{mm}^{2}\). The areas of all the dimples and seams were determined using the same 3D CAD software that created those surface features for the manufactured balls. Table 3 lists the software-measured areas, rounded to the nearest mm\({}^{2}\), of all the dimples on a ball, \(A_{dimples}\), of all the seams on a ball, \(A_{seams}\), and of all the remaining smooth portions on a ball, \(A_{smooth}\). Also listed in Table 3 are the ratios of total dimple surface area to total smooth surface area, \(\eta_{d}=A_{dimples}/A_{smooth}\); and total surface area of dimples and seams to total smooth surface area, \(\eta_{ds}=\left(A_{dimples}+A_{seams}\right)/A_{smooth}\).  
The critical Reynolds number is more strongly correlated with \(\eta_{ds}\) than with \(\eta_{d}\). The parameter \(\eta_{ds}\) is clearly more representative of a ball's surface roughness because both dimples and seams are accounted for. Note that \(\eta_{d}=0\) for both balls A and B because neither ball was manufactured with dimples. But ball B's seams are responsible for a rougher surface than smooth ball A and, as Table 2 shows, a smaller critical Reynolds number than that of ball A.  
A three-parameter fit was determined for the functional dependence of \(\text{Re}_{c}\) on \(\eta_{ds}\). The fitting equation is  
\[\text{Re}_{c}\times 10^{-5}=\alpha\,\exp\Bigl{(}-\beta\,\eta_{ds}^{N}\Bigr{)} \tag{6}\]  
with fitting parameters \(\alpha=4.06\pm 0.15\), \(\beta=0.81\pm 0.06\), and \(N=0.70\pm 0.01\). The errors in the fitting parameters are standard errors, i.e., standard deviations, in a nonlinear model fit. Figure 6 shows the data with Equation (6). All three parameters have a \(p\) value of \(p<10^{-4}\) and the quality of the fit is such that \(R^{2}=0.995\).  
The parameter \(\eta_{ds}=(A_{dimples}+A_{seams})/A_{smooth}\), which accounts for the roughness of a ball's surface, is thus an excellent parameter to use as a predictor for one of the manufactured ball's critical Reynolds number. To test the accuracy of Equation (6), a method will need to be devised to accurately determine \(\eta_{ds}\) for a real soccer ball.  
One study [3] showed that the critical Reynolds number was more strongly correlated with seam width than with seam depth. That work's conclusion is consistent with what has been shown here, namely that the critical Reynolds number depends strongly on the fraction of surface area that comprises rough features, such as seams and dimples, to the remaining smooth surface area. Increasing seam width does more to add roughness and remove smoothness than simply increasing the depth of a seam.  
Figure 6: Comparison between data and fitting equation, given by Equation (6). Letters next to points correspond to the ball labels given in Figure 1.