Some Mathematics of Sailing: Margot Gerritsen

Some Mathematics of Sailing�� Margot Gerritsen�� � Stanford Yacht Research �

� Institute for Computational & Mathematical Engineering

The famous America's Cup

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!

!

ACWS 2011-2012 Results

Team New Zealand �AC72 Unveiled

72 ft length at waterline!86 ft length overall!131 ft heigh mast!

13,000 lb!

11 person crew!

Let’s go back to 2000

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!

!!

Complex Horizontal Forces (Hull/Rig)

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!

Forces on rig and hull !

Side view! Front view!

Lowest vertical stays !20 tons!!Diagonal stays ! !5 tons!!Compression mast !45 tons!!

Velocity Prediction Program (VPP)

Optimize Sails for Different Wind Conditions

Yacht!

Apparent!True!

Upwind ! !Reaching ! Fully downwind!

Creating driving force

Lift = drive + heel!!

Apparent!

Drag = drive!

2D flow already challenging

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!

Wind tunnels in sail design University of Auckland !Twisted Flow Wind Tunnel!

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)!

Apparent Wind Angle 25°!

SST model!

3D Modeling Still Just a Flirt

3D Lamination (3DL)!

Sandwich of polyester film / kevlar yarns / polyester film!

From M Sawley (2002)!

Winglets used to reduce tip vortices, also provide lift when heeled!

!�  Design shape flying shape !

�  Square-rigged!

�  Twist onset flow small!

�  Extensive experimental data!

!280 ft

160 ft

≈

65 ft

Maltese Falcon ideal test case

AWA 30!

Drive/heel leads to more open arrangement

driv

e!dr

ive/

heel!

over-trimmed!leeward telltale lifting!

well-trimmed!telltales streaming back!

drive!

drive/heel!