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Some Mathematics of Sailing: Margot Gerritsen
The document discusses the history and mathematics of sailing, particularly focusing on the America's Cup, a prestigious sailing competition established in 1851. It outlines key events, victories, and challenges from various teams through the years, emphasizing engineering complexities and design rules involved in yacht racing. Additionally, it highlights how mathematics and computer simulation tools are utilized to optimize yacht performance under varying wind and water conditions.
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Some Mathematics of Sailing: Margot Gerritsen
- 1. Some Mathematics ofSailing Margot Gerritsen Stanford Yacht Research Institute for Computational & Mathematical Engineering
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- 3. 161 Year OldTradition 1851 ! ! Royal Yacht Squadron gives cup for Isle of Wight race! ! ! Won by schooner America, donated to New York Yacht Club!! ! ! ! ! ! Deed of Gift: cup for perpetual international competition! till 1970 !Twenty successful defenses with always just one challenger! ! 1970 ! !Challenger selection series starts – the Louis Vuitton Cup! ! 1983 ! !Royal Perth Yacht Club steals trophy from NYYC (finally!)! ! 1987 ! !Dennis Connor (USA) takes revenge in Stars & Stripes in Perth! ! 1988 ! ! Surprise challenge by New Zealand (KZ1 90 ft "big boat")! ! ! Connor won with 45 ft cat. ! !
- 4. ! ! ! !! 1992 ! !US successfully defends Cup in San Diego! ! 1995 ! !Team New Zealand sails to victory in San Diego! ! 2000 ! !Team New Zealand beats Italian challenger Prada! ! 2003 ! !Swiss challenger Alinghi beats Team New Zealand in Auckland! ! 2007 ! !Alinghi beats challenger Team New Zealand in Valencia! ! 2008-9! ! Team Oracle wins court case for rightful AC challenge! ! ! (Battle of the Billionaires Ellison and Bertarelli)! ! ! ! 2010 ! !Oracle beats Alinghi in Valencia!
- 5. 115ft long! ! Swiss Alinghiskipper! Brad Butterworth! ! USA Oracle skipper! Jimmy Spithill! Captain Russell Coutts! ! ! Oracle easily beats Alinghi with metallic wing rather than traditional sail! ! ! ! !
- 6. America’s Cup WorldSeries AC45 Wingsail catamaran by! Mike Drummond/Oracle ! honeycombcore/carbonfiber! 44 ft; 2850 lb; 70 ft heigh! ! !
- 7. 2013 America’s Cup(number 34) America’s Cup World Series ! Regular circuit of regattas with fleet and match racing in each event! 2011-2012 !completed! 2012-2013 !San Francisco, Aug 21-26 and Oct 2-7, 2012! ! ! !Venice, April 16-21, 2013! ! ! !Naples, May 14-19, 2013! ! ! !! ! Louis Vuitton Cup !San Francisco, July 4 – Sept 1, 2013! ! America’s Cup Finals !San Francisco, Sept 7 – 22, 2013! !
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- 9. Team New Zealand AC72Unveiled 72 ft length at waterline! 86 ft length overall! 131 ft heigh mast! 13,000 lb! 11 person crew!
- 10. Let’s go backto 2000
- 11. Typically *very* closeraces Time! Delta in ! Min:sec! seconds! Start! 0! 1! 26:11! 12! 2! 24:14! -34! 3! 26:37! -26! 4! 22:43! -14! Three upwind, three downwind legs! 5! 27:33! -26! Each 3 nautical miles! ! finish! 25:43! 7!
- 12. How to MaximizePerformance? Can Mathematics Help? Balance forces and moments and maintain stability! Make sure structural design limits are met! Respect any race and design rules ! Take into account uncertainty in wind, waves and crew handling! ! !! A *very* complex engineering problem!!! !
- 13. The Main DesignRule in 2000/2003 ( length + 1.25 x (sail area)1/2 – 9.8 x (displacement)1/3 ) / 0.679 = 24! The international America’s Cup Class Rule! ! Originally hoped to introduce difference designs! ! ! But, two typical designs:! ! Length overall !82 ft ! !79.5 ft! Beam ! !14.1 ! !13.3 ft! Sail area! !3600 sq ft !3600 sq ft! ! !
- 14. Complex Horizontal Forces(Hull/Rig)
- 15. Heeling force limitsmax driving force Vertical plane aerodynamic side foce heel angle crew weight Stability through crew,! lead ballast in lower! part keel and/or bulb! ! In IACC, bulb is 20! hydrodynamic buoyancy force weight tons out of total of 24! side force
- 16. Forces on rigand hull ! Side view! Front view!
- 17. Lowest vertical stays !20 tons! ! Diagonal stays ! !5 tons! ! Compression mast !45 tons! !
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- 19. Optimize Sails forDifferent Wind Conditions Upwind ! !Reaching ! Fully downwind! Yacht! True! Apparent!
- 20. Creating driving force Apparent! Lift = drive + heel! Drag = drive! !
- 21. 2D flow alreadychallenging
- 22. 3D adds aLiteral Twist Angle and strength apparent wind varies with height! UT Twisted onset flow! US UA Sails twisted! (to operate close to ideal angle)! ! !
- 23. Wind tunnels insail design University of Auckland ! Twisted Flow Wind Tunnel!
- 24. CFD only optionfor upwind analysis Flat sail (small camber)! Possibly TE separation, LE bubble (re-attached)! ! For AC upwind main+genoa! • Re O(106), L/D approximately 8-9! • Pressure drag approximately 10% of sail drag! • Induced drag up to 15% of total drag (high roach)!
- 25. Resolve Turbulent FlowFeatures > ideal (22º) Ideal angle (18º) > ideal (26º)! Choice turbulence models non-trivial when separated!
- 26. Downwind CFD inEarly Stage Wind tunnel tests lead to 15% efficiency gain ! What improvement can we get with CFD?! ! AC genneker operates at maximum lift (high-lift foil)!
- 27. Apparent Wind Angle25°! SST model!
- 28. 3D Modeling StillJust a Flirt
- 29. 3D Lamination (3DL)! Sandwichof polyester film / kevlar yarns / polyester film!
- 30. Winglets used toreduce tip vortices, also provide lift when heeled! From M Sawley (2002)!
- 32. Maltese Falcon idealtest case ! Design shape ≈ flying shape ! 160 Square-rigged! ft Twist onset flow small! Extensive experimental data! ! 280 ft 65 ft
- 33. Drive/heel leads tomore open arrangement over-trimmed! leeward telltale lifting! drive! drive! drive/heel! drive/heel! AWA 30! well-trimmed! telltales streaming back!