Some Mathematics of Sailing: Margot Gerritsen
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Stanford Asso. Prof. Margot Gerritsen's presentation on how mathematics relates to the America's Cup races.
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Some Mathematics of Sailing: Margot Gerritsen
- 1. Some Mathematics of Sailing Margot Gerritsen Stanford Yacht Research Institute for Computational & Mathematical Engineering
- 2. The famous America's Cup
- 3. 161 Year Old Tradition 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 Alinghi skipper! 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 World Series 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! !
- 9. Team New Zealand AC72 Unveiled 72 ft length at waterline! 86 ft length overall! 131 ft heigh mast! 13,000 lb! 11 person crew!
- 10. Let’s go back to 2000
- 11. Typically *very* close races 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 Maximize Performance? 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 Design Rule 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 limits max 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 rig and hull ! Side view! Front view!
- 17. Lowest vertical stays !20 tons! ! Diagonal stays ! !5 tons! ! Compression mast !45 tons! !
- 18. Velocity Prediction Program (VPP)
- 19. Optimize Sails for Different Wind Conditions Upwind ! !Reaching ! Fully downwind! Yacht! True! Apparent!
- 20. Creating driving force Apparent! Lift = drive + heel! Drag = drive! !
- 21. 2D flow already challenging
- 22. 3D adds a Literal 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 in sail design University of Auckland ! Twisted Flow Wind Tunnel!
- 24. CFD only option for 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 Flow Features > ideal (22º) Ideal angle (18º) > ideal (26º)! Choice turbulence models non-trivial when separated!
- 26. Downwind CFD in Early 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 Angle 25°! SST model!
- 28. 3D Modeling Still Just a Flirt
- 29. 3D Lamination (3DL)! Sandwich of polyester film / kevlar yarns / polyester film!
- 30. Winglets used to reduce tip vortices, also provide lift when heeled! From M Sawley (2002)!
- 32. Maltese Falcon ideal test case ! Design shape ≈ flying shape ! 160 Square-rigged! ft Twist onset flow small! Extensive experimental data! ! 280 ft 65 ft
- 33. Drive/heel leads to more open arrangement over-trimmed! leeward telltale lifting! drive! drive! drive/heel! drive/heel! AWA 30! well-trimmed! telltales streaming back!