some mathematics of sailing: margot gerritsen
DESCRIPTION
Stanford Asso. Prof. Margot Gerritsen's presentation on how mathematics relates to the America's Cup races.TRANSCRIPT
Some Mathematics of Sailing�� Margot Gerritsen�� � Stanford Yacht Research �
� Institute for Computational & Mathematical Engineering
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. !!
!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!
!! !!
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!!!!!
America’s Cup World Series AC45
Wingsail catamaran by!Mike Drummond/Oracle !honeycombcore/carbonfiber!
44 ft; 2850 lb; 70 ft heigh!
!
!
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!
!
Team New Zealand �AC72 Unveiled
72 ft length at waterline!86 ft length overall!131 ft heigh mast!
13,000 lb!
11 person crew!
Typically *very* close races
Time!Min:sec!
Delta in !seconds!
Start! 0!
1! 26:11! 12!
2! 24:14! -34!
3! 26:37! -26!
4! 22:43! -14!
5! 27:33! -26!
finish! 25:43! 7!
Three upwind, three downwind legs!
Each 3 nautical miles!
!
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!!!
!
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!
!!
heel angle
buoyancy force
aerodynamic side foce
hydrodynamic side force
crew weight
weight
Heeling force limits max driving force
Vertical plane
Stability through crew,!lead ballast in lower!part keel and/or bulb!!In IACC, bulb is 20!tons out of total of 24!
Optimize Sails for Different Wind Conditions
Yacht!
Apparent!True!
Upwind ! !Reaching ! Fully downwind!
3D adds a Literal Twist
UT
US UA
Twisted onset flow!
Sails twisted!(to operate close to ideal angle)!
!
!
Angle and strength apparent wind varies with height!
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)!
Resolve Turbulent Flow Features
Ideal angle (18º) > ideal (22º) > ideal (26º)!
Choice turbulence models non-trivial when separated!
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)!
!� Design shape flying shape !
� Square-rigged!
� Twist onset flow small!
� Extensive experimental data!
!280 ft
160 ft
≈
65 ft
Maltese Falcon ideal test case