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Sports Ball Aerodynamics

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Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 506))

Abstract

Aerodynamics plays a prominent role in defining the flight of a ball that is struck or thrown through the air in almost all ball sports. The main interest is in the fact that the ball can often deviate from its initial straight path, resulting in a curved, or sometimes an unpredictable, flight path. It is particularly fascinating that not all the parameters that affect the flight of a ball are always under human influence. Lateral deflection in flight, commonly known as swing, swerve or curve, is well recognized in cricket, tennis, soccer, volleyball and baseball. In most of these sports, the lateral deflection is produced by spinning the ball about an axis perpendicular to the line of flight, which gives rise to what is commonly known as the Magnus effect, named after the German chemist/physicist, Gustav Magnus.

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Bibliography

  • E. Achenbach, Experiments on the flow past spheres at very high Reynolds number, Journal of Fluid Mechanics, 54, pages 565–575, 1972.

    Article  Google Scholar 

  • E. Achenbach, Vortex shedding from spheres, Journal of Fluid Mechanics, 62, pages 209–221, 1974a.

    Article  Google Scholar 

  • E. Achenbach, The effects of surface roughness and tunnel blockage on the flow past spheres. Journal of Fluid Mechanics, 65, pages 113–125, 1974b.

    Article  Google Scholar 

  • F. Alam, A. Subic & S. Watkins, An experimental study on the aerodynamic drag of a series of tennis balls, In Proceedings of the International Congress on Sports Dynamics: sports Dynamics-Discovery and Application, pages 295–300, 1–3 September, Melbourne, Australia, 2003.

    Google Scholar 

  • F. Alam, A. Subic & S. Watkins, Effects of spin on aerodynamic properties of tennis balls, In The Engineering of Sport 5 (Hubbard M., Mehta, R.D. and Pallis, J.M., eds). Proceedings of the 5th International Sports Engineering Association Conference, Davis, California, USA, Vol 1, pages 83–89, 2004.

    Google Scholar 

  • F. Alam, A. Subic & S. Watkins, An experimental study of spin effects on tennis ball aerodynamic properties, In The Impact of Technology on Sport (Subic, A. and Ujihashi, S., eds.), pages 240–245, Tokyo, Japan, 2005a.

    Google Scholar 

  • F. Alam, A. Subic & S. Watkins, Measurement of aerodynamic drag forces of a rugby ball and Australian rules football, In The Impact of Technology on Sport (Subic, A. and Ujihashi, S., eds.), pages 276–279, Tokyo, Japan, 2005b.

    Google Scholar 

  • F. Alam, W. Chee, A. Subic & S. Watkins, A comparison of aerodynamic drag of a rugby ball using EFD and CFD, In The Engineering of Sport 6: Developments for Sports (Moritz, E. and Haake, S. eds.). Proceedings of the 6th International Sports Engineering Association Conference, Munich, Germany. Vol 2, pages 145–150, 2006.

    Google Scholar 

  • F. Alam, W. Tio, A. Subic & J. Naser, Effects of spin on tennis ball aerodynamics: an experimental and computational study, In Proceedings of the 16 th Australasian Fluid Mechanics Conference, pages 324–327, 3–7 December, Gold Coast, Australia, 2007.

    Google Scholar 

  • L.W. Alaways, Aerodynamics of the curve-ball: an investigation of the effects of angular velocity on baseball trajectories. Ph.D. dissertation, University of California, Davis, USA, 1998.

    Google Scholar 

  • L.W. Alaways & M. Hubbard, Experimental determination of baseball spin and lift, Journal of Sports Sciences, 19, pages 349–358, 2001.

    Article  Google Scholar 

  • W.F. Allman, Pitching rainbows, Science, 83, 3 (8), pages 32–39, 1982.

    Google Scholar 

  • W.F. Allman, Flight of the knuckler, Science, 83, 4 (5), pages 92–93, 1983.

    Google Scholar 

  • K. Aoki, Y. Kinoshita, E. Hirota, J. Nagase, & Y. Nakayama, The surface structure and aerodynamics of baseballs, In The Engineering of sport 4 (Ujihashi, S. and Haake, S., eds.) Proceedings of the 4th International Sports Engineering Association Conference, Kyoto, Japan, pages 283–289, 2002.

    Google Scholar 

  • K. Aoki, M. Nonaka, & T. Goto, Effect of dimple structure on the flying characteristics and flow patterns of a golf ball, In The Engineering of Sport 5 (Hubbard M., Mehta, R.D. and Pallis, J.M., eds.), Proceedings of the 5th International Sports Engineering Association Conference, Davis, California, USA. Vol 1, pages 49–55, 2004.

    Google Scholar 

  • S. Aoyama, A modern method for the measurement of aerodynamic lift and drag on golf balls, In Science and Golf (Cochran, A.J., ed.), pages 199–204, E. & F.N. Spon, London, UK, 1990.

    Google Scholar 

  • T. Asai, T. Akatsuka & S.J. Haake, The physics of football, Physics World, 11–6, pages 25–27, 1998.

    Google Scholar 

  • T. Asai, K. Seo, O. Kobayashi, & R. Sakashita, Flow visualization on a real flight non-spinning and spinning soccer ball, In The Engineering of Sport 6: Developments for Sports (Moritz, E. and Haake, S., eds.), Proceedings of the 6th International Sports Engineering Association Conference, Munich, Germany. Vol 1, pages 327–332, 2006.

    Google Scholar 

  • T. Asai, K. Seo, O. Kobayashi, & R. Sakashita, Fundamental aerodynamics of soccer ball, Sports Engineering, 10, No. 2, pages 101–109, 2007.

    Google Scholar 

  • T. Bahill & D.G. Baldwin, The rising fastball and other perceptual illusions of batters, Biomedical Engineering Principles in Sports (Hung, G.K. and Pallis, J.M., eds.), Kluwer Academic, New York, USA, pages 257–287, 2004.

    Google Scholar 

  • S. Barber, S.J. Haake, & M.J. Carré, Using CFD to understand the effects of seam geometry on soccer ball aerodynamics, In The Engineering of Sport 6: Developments for Sports (Moritz, E. and Haake, S., eds.), Proceedings of the 6th International Sports Engineering Association Conference, Munich, Germany. Vol 2, pages 127–132, 2006.

    Google Scholar 

  • N.G. Barton, On the swing of a cricket ball in flight, Proceedings of the Royal Society London A, 379, pages 109–131, 1982.

    Article  Google Scholar 

  • P.W. Bearman, & J.K. Harvey, Golf ball aerodynamics, Aeronautical Quarterly, 27, pages 112–122, 1976.

    Google Scholar 

  • D. Beasley, & T. Camp, Effects of dimple design on the aerodynamic performance of a golf ball, In Science and Golf IV (Thain, E., ed.), pages 328–340, 2002, Routledge, London and NY.

    Google Scholar 

  • J.H. Bell & R.D. Mehta, Contraction design for small low-speed wind tunnels, NASA-CR 177488, 1988.

    Google Scholar 

  • K. Bentley, P. Varty, M. Proudlove, & R.D. Mehta, An experimental study of cricket ball swing, Imperial College Aero Technical Note 82–106, 1982.

    Google Scholar 

  • L.J. Briggs, Effect of spin and speed on the lateral deflection of a baseball, and the Magnus effect for smooth spheres, American Journal of Physics, 27, pages 589–96, 1959.

    Article  MATH  Google Scholar 

  • A.M. Binnie, The effect of humidity on the swing of cricket balls, International Journal of Mechanical Sciences, 18, pages 497–9, 1976.

    Article  Google Scholar 

  • W. Bown, & R.D. Mehta, The seamy side of swing bowling, New Scientist, 139, No. 1887, pages 21–24, 1993.

    Google Scholar 

  • L.O. Bowen, Torque and force measurements on a cricket ball and the influence of atmospheric conditions, Transactions of Mechanical Engineering, IE Australian, ME20, No. 1, pages 15–20, 1997.

    Google Scholar 

  • K. Bray, & D.G. Kerwin, Modeling the long throw in soccer using aerodynamic drag and lift, In The Engineering of Sport 5 (Hubbard M., Mehta, R.D. and Pallis, J.M., eds.) Proceedings of the 5th International Sports Engineering Association Conference, Davis, California, USA. Vol 1, pages 56–62.

    Google Scholar 

  • F.N.M. Brown, See the wind blow, Aeronautical Engineering Department Report, University of Notre Dame, South Bend, Indiana, USA, 1971.

    Google Scholar 

  • T.M.C. Brown, & A.J. Cooke, Aeromechanical and aerodynamic behavior of tennis balls, In Tennis Science and Technology (Haake, S.J. and Coe, A., eds.) Proceedings of the 1st International Conference on Tennis Science and Technology. Blackwell Science, Oxford, UK, pages 145–153, 2000.

    Google Scholar 

  • T.W. Cairns, Modeling the Lift and Drag Forces on a Volleyball, In The Engineering of Sport 5 (Hubbard M., Mehta, R.D. and Pallis, J.M., eds.) Proceedings of the 5th International Sports Engineering Association Conference, Davis, California, USA. Vol 1, pages 97–103, 2004.

    Google Scholar 

  • M.J. Carré, T. Asai, T. Akatsuka, & S.J. Haake, The curve kick of a football II: flight through the air, Sports Engineering, 5, No. 4, pages 193–200, 2002.

    Article  Google Scholar 

  • M.J. Carré, S.R. Goodwill, S.J. Haake, R.K. Hanna, & J. Wilms, Understanding the aerodynamics of a spinning soccer ball, In The Engineering of Sport 5 (Hubbard M., Mehta, R.D. and Pallis, J.M., eds.) Proceedings of the 5th International Sports Engineering Association Conference, Davis, California, USA. Vol 1, pages 70–76, 2004.

    Google Scholar 

  • M. Cavendish, Balls in flight, Science Now, 1, pages 10–13, 1982.

    Google Scholar 

  • S.G. Chadwick, & S.J. Haake, The drag coeffcient of tennis balls, In The Engineering of Sport. Research, Development and Innovation (Subic, A.J. and Haake, S.J., eds.) Proceedings of the 3rd International Conference on the Engineering of Sport. Blackwell Science, Oxford, UK, pages 169–176, 2000a.

    Google Scholar 

  • S.G. Chadwick, & S.J. Haake, Methods to determine the aerodynamic forces acting on tennis balls in flight, In Tennis Science and Technology (Haake, S.J. and Coe, A., eds.) Proceedings of the 1st International Conference on Tennis Science and Technology. Blackwell Science, Oxford, UK, pages 127–134, 2000b.

    Google Scholar 

  • S.G. Chadwick, The aerodynamics of tennis balls, Ph.D. thesis, University of Sheffield, UK, 2003.

    Google Scholar 

  • A. Chase, A slice of golf, Science 81, 2(6), pages 90–91, 1981.

    Google Scholar 

  • T. Chikaraishi, Y. Alaki, K. Maehara, H. Shimosaka, & F. Fokazawa, A new method on measurement of trajectories of a golf ball, In Science and Golf (Cochran, A.J. ed.), pages 193–198, E. & F.N. Spon, London, UK, 1990.

    Google Scholar 

  • A. Cochran, & J. Stobbs, Search for the Perfect Swing, pages 161–162, Lippincott: Philadelphia/New York, USA, 1968.

    Google Scholar 

  • A.J. Cooke, An overview of tennis ball aerodynamics, Sports Engineering, 3, No. 2, pages 123–129, 2000.

    Article  Google Scholar 

  • J.C. Cooke, The boundary layer and seam bowling, The Mathematical Gazette, 39, pages 196–199, 1955.

    Article  Google Scholar 

  • J.M. Davies, The aerodynamics of golf balls, Journal of Applied Physics, 20, pages 821–828, 1949.

    Article  Google Scholar 

  • C. Frohlich, Aerodynamic drag crisis and its possible effect on the flight of baseballs, American Journal of Physics, 52 (4), pages 325–334, 1984.

    Article  Google Scholar 

  • S.R. Goodwill, & S.J. Haake, Aerodynamics of tennis balls — effect of wear, In The Engineering of Sport 5 (Hubbard M., Mehta, R.D. and Pallis, J.M., eds.) Proceedings of the 5th International Sports Engineering Association Conference, Davis, California, USA. Vol 1, pages 35–41, 2004.

    Google Scholar 

  • S.R. Goodwill, S.B. Chin, & S.J. Haake, Aerodynamics of spinning and non-spinning tennis balls, Journal of Wind Engineering and Industrial Aerodynamics, 92, pages 935–958, 2004.

    Article  Google Scholar 

  • C. Grant, A. Anderson, & J.M. Anderson, Cricket ball swing — the Cooke-Lyttleton theory revisited, In The Engineering of Sport — Design and Development. (Haake, S.J., ed.). Blackwell Publishing, Oxford, UK, pages 371–378, 1998.

    Google Scholar 

  • Guinness, Guinness World Records, Millennium Edition. Guinness World Records Ltd, Bantam, London, UK, 2000.

    Google Scholar 

  • S.J. Haake, S.G. Chadwick, R.J. Dignall, S. Goodwill. & P. Rose, Engineering tennis — slowing the game down, Sports Engineering, 3, No. 2, pages 131–143, 2000.

    Article  Google Scholar 

  • S.J. Haake, S.R. Goodwill, & M. J. Carré, A new measure of roughness for defining the aerodynamic performance of sports balls, Journal of Mechanical Engineering Science, Part C, 221, pages 789–806, 2007.

    Article  Google Scholar 

  • J.H. Horlock, The swing of a cricket ball, ASME Symposium on the Mechanics of Sport, 1973

    Google Scholar 

  • A. Imbrosciano, The swing of a cricket ball., Project Report, Newcastle College of Advanced Education Newcastle, Australia, 1981.

    Google Scholar 

  • R.A. Lyttleton, The swing of a cricket ball, Discovery, 18, pages 186–191, 1957.

    Google Scholar 

  • J. Maccoll, Aerodynamics of a spinning sphere, Journal of the Royal Aeronautical Society, 32, page 777, 1928.

    Google Scholar 

  • R.D. Mehta, & D.H. Wood, Aerodynamics of the cricket ball, New Scientist, 87, No. 1213, pages 442–447, 1980.

    Google Scholar 

  • R.D. Mehta, K. Bentley, M. Proudlove, & P. Varty, Factors affecting cricket ball swing, Nature, 303, pages 787–88, 1983.

    Article  Google Scholar 

  • R.D. Mehta, Aerodynamics of sports balls, Annual Review of Fluid Mechanics, 17, pages 151–189, 1985.

    Article  Google Scholar 

  • R.D. Mehta, Cricket ball aerodynamics: myth versus science, In The Engineering of Sport. Research, Development and Innovation. (Subic, A.J. and Haake, S.J., eds.) Blackwell Science, London, pages 153–167, 2000.

    Google Scholar 

  • R.D. Mehta & J.M. Pallis, Sports ball aerodynamics: effects of velocity, spin and surface roughness, In Materials and Science in Sports. (Froes, F.H. and Haake, S.J., eds.), pages 185–197, The Minerals, Metals and Materials Society [TMS], Warrendale, USA, 2001a.

    Google Scholar 

  • R.D. Mehta & J.M. Pallis, The aerodynamics of a tennis ball, Sports Engineering, 4, No. 4, pages 1–13, 2001b.

    Article  Google Scholar 

  • R.D. Mehta & J.M. Pallis, Tennis ball aerodynamics and dynamics, In Biomedical Engineering Principles in Sports (Hung, G.K. and Pallis, J.M., eds.), pages 99–124, Kluwer Academic/Plenum Publishers, Norwell, MA, USA, 2004.

    Google Scholar 

  • R.D. Mehta, An overview of cricket ball swing, Sports Engineering, 8, No. 4, pages 181–192, 2005.

    Article  Google Scholar 

  • R.D. Mehta, Swinging it three ways, The Wisden Cricketer, 3, No. 7, pages 50–53, 2006a.

    Google Scholar 

  • R.D. Mehta, The unpredictable flight of the World Cup ball explained, Sports Traders Magazine, Vol. 27, No. 4, page 38, August 2006b.

    Google Scholar 

  • R.D. Mehta, Cricket ball tampering, New Scientist, Issue 2569, page 23, September 16, 2006c.

    Article  Google Scholar 

  • R.D. Mehta, Swing is not Colour Deep, Cricinfo Magazine, 1, 12, page 20, December, 2006d.

    Google Scholar 

  • R.D. Mehta, The art and science of ball tampering, Cricket International Quarterly Magazine, 1, No. 1, pages 11–13, March, 2007a.

    Google Scholar 

  • R.D. Mehta, Does a white cricket ball swing more than a red one?, Cricket International Quarterly Magazine, 1, No. 1, pages 42–43, March, 2007b.

    Google Scholar 

  • A.M. Nathan, J. Hopkins, L. Chong, & H. Kaczmarski, The effect of spin on the flight of a baseball, In The Engineering of Sport 6: Developments for Sports (Moritz, E. and Haake, S. eds.) Proceedings of the 6th International Sports Engineering Association Conference, Munich, Germany. Vol 1, pages 23–28, 2006.

    Google Scholar 

  • A.M. Nathan, The effect of spin on the flight of a baseball, American Journal of Physics, 76, Part 2, pages 119–124, 2008.

    Article  Google Scholar 

  • I. Newton, New theory of light and colours, Philosophical Transactions of the Royal Society London, 1, pages 678–688, 1672.

    Google Scholar 

  • D. Oslear, & J. Bannister, Tampering with Cricket, Collins Willow (Harper Collins) Publishers, London, UK, 1996.

    Google Scholar 

  • J.M. Pallis, & R.D. Mehta, Tennis science collaboration between NASA and Cislunar Aerospace, In Tennis Science and Technology (Haake, S.J. and Coe, A., eds.) Proceedings of the 1st International Conference on Tennis Science and Technology. Blackwell Science, Oxford, UK, pages 135–144, 2000.

    Google Scholar 

  • J.M. Pallis, & R.D. Mehta, Balls and ballistics, In Materials in Sports Equipment (Jenkins, M., ed.), pages 100–125, Woodhead Publishing Limited, Cambridge, England, UK, 2003.

    Google Scholar 

  • J.M.T. Penrose, D.R. Hose, & E.A. Trowbridge, Cricket ball swing: a preliminary analysis using computational fluid dynamics, In The Engineering of Sport. (Haake, S.J., ed.) A.A. Balkema, Rotterdam, Holland, pages 11–19, 1996.

    Google Scholar 

  • Lord Rayleigh, On the irregular flight of a tennis ball, Messenger of Mathematics, 7, pages 14–16, 1877.

    Google Scholar 

  • W.J. Rae & R.J. Streit, Wind-tunnel measurements of the aerodynamic loads on an American football, Sports Engineering, 5, No. 3, pages 165–172, 2002.

    Article  Google Scholar 

  • T. Sajima, T. Yamaguchi, M. Yabu, & M. Tsunoda, The aerodynamic influence of dimple design on flying golf ball, In The Engineering of Sport 6: Developments for Sports (Moritz, E. and Haake, S., eds.) Proceedings of the 6th International Sports Engineering Association Conference, Munich, Germany. Vol 1, pages 143–148, 2006.

    Google Scholar 

  • G.S: Sawicki, M. Hubbard, & W.J. Stronge, How to hit home runs: optimum baseball bat swing parameters for maximum range trajectories, American Journal of Physics, 71 (11), pages 1152–1162, 2003.

    Article  Google Scholar 

  • A.T. Sayers, & A. Hill, Aerodynamics of a cricket ball, Journal of Wind Engineering and Industrial Aerodynamics, 79, pages 169–182, 1999.

    Article  Google Scholar 

  • K. Sherwin, & J.L. Sproston, Aerodynamics of a cricket ball, International Journal of Mechanical Engineering Education, 10, pages 71–79, 1982.

    Google Scholar 

  • A.J. Smits, & D.R. Smith, A new aerodynamic model of a golf ball in flight, In Science and Golf II, (Cochran, A.J., ed.), pages 341–347, E. & F.N. Spon, London, UK, 1994.

    Google Scholar 

  • A.J. Smits, A Physical Introduction to Fluid Mechanics, John Wiley & Sons, New York, NY, USA, 2000.

    Google Scholar 

  • A.J. Smits, & S. Ogg, Golf ball aerodynamics, In The Engineering of Sport 5 (Hubbard M., Mehta, R.D. and Pallis, J.M., eds.) Proceedings of the 5th International Sports Engineering Association Conference, Davis, California, USA. Vol 1, pages 3–12, 2004a.

    Google Scholar 

  • A.J. Smits, & S. Ogg, Aerodynamics of the golf ball, In Biomedical Engineering Principles in Sports (Hung, G.K. and Pallis, J.M., eds.), pages 1–27, Kluwer Academic/Plenum Publishers, Norwell, MA, USA, 2004b.

    Google Scholar 

  • J.P. Spampinato, N. Felten, P. Ostafichuk, & L. Brownlie, A test method for measuring forces on a full-scale spinning soccer ball in a wind tunnel., In The Engineering of Sport 5 (Hubbard M., Mehta, R.D. and Pallis, J.M., eds.) Proceedings of the 5th International Sports Engineering Association Conference, Davis, California, USA. Vol 1, pages 111–117, 2004.

    Google Scholar 

  • A. Stepanek, The aerodynamics of tennis balls — the topspin lob, American Journal of Physics, 56, pages 138–141, 1988.

    Article  Google Scholar 

  • P.G. Tait, Some points in the physics of golf. Part I, Nature, 42, pages 420–423, 1890.

    Article  Google Scholar 

  • P.G. Tait, Some points in the physics of golf. Part II, Nature, 44, pages 497–498, 1891.

    Article  Google Scholar 

  • P.G. Tait, Some points in the physics of golf, Part III, Nature, 48, pages 202–205, 1890b.

    Article  Google Scholar 

  • S. Taneda, Visual-observations of the flow past a sphere at Reynolds numbers between 104 and 106, Journal of Fluid Mechanics, 85, pages 187–192, 1978.

    Article  Google Scholar 

  • M. Van Dyke, An Album of Fluid Motion, The Parabolic Press, Stanford, California, USA, 1982.

    Google Scholar 

  • R.G. Watts & C. E. Sawyer, Aerodynamics of a knuckleball, American Journal of Physics, 43, pages 960–963, 1975.

    Article  Google Scholar 

  • R.G. Watts, & E. Sawyer, The lateral force on a spinning sphere: aerodynamics of a curveball, American Journal of Physics, 55, pages 40–44, 1987.

    Article  Google Scholar 

  • R.G. Watts, & A.T. Bahill, Keep your eye on the ball: Curve balls, Knuckleballs, and Fallacies of Baseball, W. H. Freeman: New York, NY, USA, 2000.

    Google Scholar 

  • Q. Wei, R. Lin, & Z. Liu, Vortex-induced dynamic loads on a non-spinning volleyball, Fluid Dynamics Research, 3, pages 231–237, 1988.

    Article  Google Scholar 

  • H. Werlé, Transition and separation — visualizations in the ONERA water tunnel, In Recherche Aérospace, 1980–5, pages 35–49, 1980.

    Google Scholar 

  • B. Wilkins, The Bowlers Art, A&C Black Publishers Ltd., London, UK, 1991.

    Google Scholar 

  • M.V. Zagarola, B. Lieberman, & A.J. Smits, An indoor testing range to measure the aerodynamic performance of golf balls, In Science and Golf II (Cochran, A.J. and Farally, M.R., eds.), pages 348–354, E&F.N. Spon, London, UK, 1994.

    Google Scholar 

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Mehta, R.D. (2008). Sports Ball Aerodynamics. In: Nørstrud, H. (eds) Sport Aerodynamics. CISM International Centre for Mechanical Sciences, vol 506. Springer, Vienna. https://doi.org/10.1007/978-3-211-89297-8_12

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