Abstract
A study of a boat's motion is carried out in order to analyze the aerodynamic properties of the optimal sail for obtaining the maximum velocity when sailing to windward. The mechanics study shows the optimal C _{L} and C _{D} for a given sail and how the shape of the aerodynamic polar of the sail should be. A parametrical analysis of the aerodynamics of a sail is then carried out varying the maximum camber, position of the maximum camber in the chord direction and position of the maximum camber in the mast direction. The parametric analysis is done numerically with a vortex lattice method (VLM) and experimentally in a wind tunnel. The results show that the influence of the relevant parameters studied can be reduced to the variation of two parameters, A and B, defining the polar of the sail, C _{D} = B + A ^{2} C _{L} ^{2}; and the influence of parameters A and B on the maximum VMG obtainable are calculated.
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Abbreviations
 A :

coefficient of the polar of the sail defined as \( C_{{\text{D}}} = B + A^{2} C_{{\text{L}}}^{2} \)
 b :

sail span
 B :

coefficient of the polar of the sail defined as \( C_{{\text{D}}} = B + A^{2} C_{{\text{L}}}^{2} \)
 C :

profile chord
 C _{D} :

drag coefficient
 C _{Hx} :

hydrodynamic force coefficient in the direction normal to the yacht speed
 C _{Hy} :

hydrodynamic force coefficient in the direction of the yacht speed
 C _{L} :

lift coefficient
 C _{Lopt} :

optimal sailing lift coefficient
 C _{M} :

heeling moment coefficient made dimensionless with the apparent wind velocity
 \( C_{{\text{M}}}^{*} \) :

heeling moment coefficient made dimensionless with the true wind velocity
 C _{r} :

chord of the root section of the sail
 C _{t} :

chord of the tip section of the sail
 D:

drag
 \( \vec F_{\rm A} \) :

aerodynamic force
 \( \vec F_{{\text{\rm H}}} \) :

hydrodynamic force
 Fn :

Froud number, \( F_{n} = \frac{{\left {\vec V_{s} } \right}}{{\sqrt {gD_{{\text{h}}} } }} ,\) where g is the gravity and D _{h} is the characteristic length scale of hydraulic depth
 K :

\( \frac{{\rho _{{{\text{H}}_{ 2} {\text{O}}}} S_{{\text{H}}} }}{{\rho _{{\text{a}}} S_{{\text{A}}} }} \)
 L:

lift
 m:

maximum camber of the profile expressed as a percentage of its chord length
 m _{τ} :

slope of the twist profile
 Rn _{A} :

aerodynamic Reynolds number, \( R_{{n{\text{A}}}} = \rho _{a} \frac{{\left {V_{{\text{A}}} } \right\bar C_{{\text{A}}} }}{{\mu _{{\text{A}}} }} \), being \( \bar C_{{\text{A}}} \) the media chord of the sail and μ _{A} the dynamic viscosity of the water
 Rn _{H} :

hydrodynamic Reynolds number, \( R_{{{\it{n}\text{H}}}} = \rho _{{{\text{H}}_{ 2} {\text{O}}}} \frac{{V_{{\text{s}}} L_{{{\text{hull}}}} }}{{\mu _{{{\text{H}}_{ 2} {\text{O}}}} }}, \) being L _{hull} the length of the hull and \( \mu _{{{\text{H}}_{ 2} {\text{O}}}} \) the dynamic viscosity of the air
 S _{A} :

characteristic area of the sail
 S _{H} :

characteristic area of the hull
 \( \overrightarrow {V_{{\text{A}}} } \) :

apparent wind speed
 VMG:

Velocity Made Good
 V _{mg} :

velocity made good
 \( \overrightarrow {V_{{\text{S}}} } \) :

yacht speed
 \( \overrightarrow {V_{{\text{T}}} } \) :

wind speed
 ρ _{a} :

air density
 \( \rho _{{{\text{H}}_{ 2} {\text{O}}}} \) :

water density
 α :

angle of attack of the sail
 λ :

drift angle
 η _{A} :

aerodynamic efficiency of the sail
 η _{H} :

hydrodynamic efficiency of the hull
 η _{m} :

location of the profile with maximum camber along the mast direction as a percentage of the sail span
 ξ _{m} :

location of the maximum camber along the chord of the profile expressed as a percentage of the chord
 μ :

taper ratio
 τ :

twist angle of each profile
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Acknowledgments
This work has been sponsored by the Spanish Consejo Superior de Deportes (CSD) and by the Universidad Politécnica de Madrid (UPM) and is part of a more general endeavour for the study of sails design.
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LaverónSimavilla, A., Lapuerta, V., Franchini, S. et al. Sail optimization for upwind sailing: application in a Tornado, the Olympic class catamaran. J Mar Sci Technol 13, 190–206 (2008). https://doi.org/10.1007/s0077300800074
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DOI: https://doi.org/10.1007/s0077300800074