Magnus effect on a rotating soccer ball

^{Flight of the famous free kick taken by Roberto Carlos in the 1997 Soccer Photograph: Tony Marshall/Empics Sport, Image source Comsol and www.theguardian.com}

Author: Midhun Murali Mohan

Introduction 
The flight of a ball is vital across various sports, including soccer, golf, baseball, cricket, tennis, and volleyball. Research into the aerodynamics of sports balls traces back to 1672 when Newton observed deviations in tennis ball flight. Important fluid mechanics principles encompass boundary layer flow, turbulence, rough surface effects, the Magnus Effect, and wake characteristics. Mastery of these concepts’ benefits equipment designers, players, coaches, and governing bodies. 

In soccer, knowing how the ball moves through the air helps us understand its range, trajectories, and deflections, with or without spin. The Magnus Effect, which makes spinning balls curve, is greatly influenced by the ball’s surface texture. Thanks to advancements in computational power, Computational Fluid Dynamics (CFD) has become an essential tool for analysing the aerodynamics of sports balls. CFD technology allows us to better understand and compare airflow around various sports balls, influencing how sports equipment is designed and performs. 

The Magnus Effect 
During matches, spinning sports balls alter their trajectory. Side spin induces lateral movement, top spin causes downward trajectory, and backspin results in upward lift. When spinning, factors like lift coefficient (F_L) and side force coefficient (F_S) alongside drag coefficient (F_D) are relevant due to the Magnus Effect. This effect is depicted in Fig. 1 for a ball with side spin moving from right to left (as seen from above). In the absence of spin, streamlines would symmetrically flow around the ball. However, spinning rotates the air around it, increasing velocity at point A and decreasing it at point B. This asymmetry leads to delayed separation at the top and early separation at the bottom, creating an uneven wake. The pressure difference between points A and B generates a sideways Magnus Force (F_M) acting on the ball. 

Rotating soccer ball Experimental Investigations 
The Magnus effect exhibited on a prototype soccer ball, rotating perpendicular to the flow direction across various Reynolds number ranges, can be explored through aerodynamic force measurements and a survey of the flow field. Experiments were conducted by Thorsten Kray et al [2] using a rear sting support system, wherein the soccer ball was divided into two halves, each powered by an internal motor. Even when stationary, the force coefficients varied within the range observed in real soccer balls alongside boundary layer separation points. As depicted in Fig. 2, the prototype soccer ball, having a diameter of 226 mm, was affixed to an ‘L’-shaped sting featuring a NACA 0015 wing vertical section. Propelled by an internal DC motor, the ball halves could rotate at speeds of up to 540 rpm. This aluminium alloy ball, resembling the design of the 2006 World Cup Teamgeist ball, was adorned with 14 textured panels featuring triangle-based pyramidal peaks and valleys. 

Forces were measured using a six-component wind tunnel balance, recording drag force F_D, side force F_S, and Magnus force F_M as lift in the z-direction, with positive force corresponding to negative z-direction lift.  

Numerical study of the erratic motion of soccer balls 
Using CFD analysis, researchers S. Barber et al [3]. effectively compared the aerodynamics of different soccer balls. By examining various scanned and modified soccer balls, they discovered that the drag coefficient (F_D) remained consistent across different designs. However, the lift coefficient (F_L) and side force coefficient (F_S) varied significantly depending on the ball’s orientation and type. This variation was attributed to the asymmetrical geometry affecting the separation around seams. When these variations were approximated and inputted into trajectory simulation programs, the balls displayed distinct behaviours and erratic trajectories at spin rates below 2 revolutions per second. The study concluded that the surface geometry of a soccer ball significantly impacts its trajectory, with consistently performing balls achieving an optimal combination of amplitude and frequency of varying force coefficients relative to the applied spin. 

The famous free kick by Roberto Carlos in the 1997 Soccer Confederations Cup was scientifically explained by Asai et al [4]. The sideways deviation of the ball, rotating around an axis perpendicular to its flight direction from its initial straight path, is attributed to the ordinary Magnus effect. Additionally, the negative Magnus effect, which involves the reversal of the side force within a specific range of Reynolds numbers and spin parameters, also plays a role.

Watch the famous free kick !!!

Conclusion 
The aerodynamics of soccer balls is incredibly intriguing, not only because of the variety of kick types and trajectories but also due to recent advancements in manufacturing techniques. These innovations have enabled greater flexibility in designing and structuring the ball’s surface. The most reliable soccer balls strike an ideal balance between maximum force coefficient and frequency relative to the spin applied. Enhancing a ball’s consistency with flow visualizations and computational fluid dynamics (CFD) seems entirely possible. The future of soccer ball design is set to be both thrilling and controversial. 

References:  

1. S. Barber, and M.J. Carr´e. “Soccer Ball Aerodynamics Book Chapter Computational Fluid Dynamics for Sport Simulation ” Springer-Verlag Berlin Heidelberg 2009. 
2. Thorsten Kray, Jörg Franke and Wolfram Frank. ” Magnus effect on a rotating soccer ball at high Reynolds numbers.” Retrieved September 30, 2001. 
3. S. Barber, Chin, S.B, Carré, M.J.” Sports ball aerodynamics: a numerical study of the erratic motion of soccer balls”. Comput. Fluids 38 (6), 1091–1100 2009. 
4. Asai, T, Seo, K Kobayashi, O, Sakashita, R.” Fundamental aerodynamics of the soccer ball. Sports Engineering 10 (2)”, 101–109 2007. 
5. Bush, J.W.M., “The aerodynamics of the beautiful game, 2013. In Sports Physics, Ed. C. Clanet, Les Editions de l’Ecole Polytechnique”, pp. 171-192 2013.