# Investigation of Reverse Swing and Magnus Effect on a Cricket Ball Using Particle Image Velocimetry
## 2 Literature Review  
### Flow Separation on Spheres and Cricket Balls  
[PERSON] published the results from a definitive set of experiments showing how the Reynolds number affects the location of boundary layer separation on hydraulically-smooth spheres [6]. These locations were derived from skin-friction measurements; the results are presented in Figure 2, denoted by the open circles. Below a critical Reynolds number of \(\mathrm{Re}=2\times 10^{5}\) (i.e., sub-critical flow), [PERSON] showed that the boundary layer is laminar and the adverse pressure gradient on the sphere causes it to separate at around \(\theta_{\rm L}\)\(\sim\)80\({}^{\circ}\). Within the critical flow regime (\(2\times 10^{5}<\mathrm{Re}<4\times 10^{5}\)), the boundary layer transitions to turbulence and the separation angle is delayed to \(\theta_{\rm T}\)\(\sim\)120\({}^{\circ}\). As the Reynolds number increases beyond critical flow to a supercritical flow regime, the point of boundary layer separation gradually retreats upstream. It was later shown by T on a smooth sphere, that the increase in separation angle at the critical Reynolds number is associated with an LSB--see inset in Figure 2[7]. An LSB forms as the laminar boundary layer separates from the sphere (\(\theta_{\rm L}\)) before transitioning to turbulence and reattaching downstream (\(\theta_{\rm R}\)); the turbulent boundary layer separates yet further downstream (\(\theta_{\rm T}\)). [PERSON] measured \(92^{\circ}<\theta_{\rm L}<110^{\circ}\), \(107^{\circ}<\theta_{\rm R}<127^{\circ}\), and \(123^{\circ}<\theta_{\rm T}<147^{\circ}\) (depending on Re). Importantly, \(\theta_{\rm T}\) is larger in the presence of an LSB during the critical flow regime, than in the absence of the LSB during the supercritical flow regime. [PERSON] later investigated the effect of surface roughness on the separation angles of spheres [8]. He found that the value of Re in the critical regime reduces with increasing roughness, as the roughness destabilizes the boundary layer and causes it to transition to turbulence earlier. The size of the critical region also increases, described by the range of Re across which the flow regime changes. These results are also shown in Figure 2 by the open triangle and square symbols. In the scenario of a cricket ball with an angled seam, the two hemispheres will feature a different roughness, the separation angles will be different, and a lateral aerodynamic force will be caused by the asymmetric distribution of pressure. Figure 2 demonstrates this asymmetry for CS and RS in the context of a cricket match. Consider a swing bowler delivering a new ball at 90 mph (Re-\(2\times 10^{5}\)). The relative size of the primary seam (hence primary, dominating roughness) on the SS is approximately equivalent to \(k/d=1250\times 10^{-5}\) (denoted by the squares). For the sake of demonstration, it is assumed that the surface of the NSS for a new ball is hydraulically-smooth. At these conditions, \(\theta_{\rm L}=82^{\circ}\) on the NSS (in the sub-critical regime) and \(\theta_{\rm T}=100^{\circ}\) on the SS (in the super-critical regime), resulting in CS. As the ball ages during the course of play, the NSS roughens slightly to \(k/d=250\times 10^{-5}\) (despite careful polishing of the ball by the fielding side). Assuming the bowler delivers the ball at the same speed (and therefore, same Re), now \(\theta_{\rm T}=120^{\circ}\) on the NSS (in the critical regime) and \(\theta_{\rm T}=100^{\circ}\) on the SS (in the super-critical regime), resulting in RS. The red region demonstrates where the separation angle on the NSS is less than the SS, corresponding to CS, and the solid blue region corresponds to RS for an aged ball. At larger Re, the hatched blue region corresponds to RS for a new ball. [PERSON]'s results show that RS for a new cricket ball would occur if it were bowled at an impossible speed of \(>\)75 m/s (Re \(>3\times 10^{5}\)); here the surface on the NSS is assumed to be hydraulically-smooth and the ball to be a perfect sphere.