Sails: from experimental to numerical
Sail performance analysis requires calculation tools of different levels of sophistication as the design process progresses. There are many methods available to estimate the characteristics of sails: experimental and numerical ones. One experimental method and two numerical ones are detailed in this paper to study the aerodynamics of sailing of a catboat (main sail only dinghy). 

This paper has been adapted from the MSc thesis of Damien Lafforgue, presented at the University of Southampton for the grade of MSc in Maritime Engineering Sciences in September 2007. The aim of this thesis was to compare the different methods available to study the aerodynamics of sailing applied to the case of a catboat designed by the Groupe FinotConq.

Damien Lafforgue
After his engineering school Damien specialized in hydrodynamics and aerodynamics at the University of Southampton with MSc in maritime engineering sciences. Fond of regattas and sailing in general, he is achieving the fitting out of a Passoa 47. 


Contact : 
tel: +33 (0) 6 63 37 05 30 email: damien.lafforgue@gadz.org 
The 3D sail can be assumed to be made up of slowly varying 2D profiles. For a specified air flow direction the representative 2D profiles are investigated in isolation as a first approximation.
The geometry of sail profiles is defined with different parameters. The intrados is the internal face of the sail, it is the face "seen by the wind", and it is generally a concave surface. The extrados is the other face of the sail, and it is generally a convex surface.
Each sail profile is then defined by its chord (c), its camber (f), the position of the camber (m), the leading and the trailing edge slope.
When the sail is in shape, the air particles are deviated from their original straight tack. On the extrados, the air particles are confined and thus they accelerate, on the intrados, the air particles are expanded and thus they decelerate. This simple phenomenon can be modelled by the superposition of a strait flow and a rotating flow (vortex) around the sail: it is the concept of ‘‘circulation’’ (Cf. Figure below).
Ideally in low speed flow the airflow follows the shape of the mast, but when the speed increases, in normal sailing conditions the air particles can no longer follow the mast, because of their inertia.
This in turn creates some recirculation zones along the mast. When the angle of attack or the camber of the sail becomes too large, the same phenomenon occurs at the trailing edge and the air particles are unable to follow the sail shape.
When the angle of attack or the camber increases, the mast and the training edge bubbles merge into a big bubble over the whole extrados and the sail loses most of its power. These bubbles increase the drag and reduce the lift of the profile. The drag and the lift are the two components of the resultant force in a system of coordinates aligned with the wind direction.
In a system of coordinates aligned with the boat track, the resultant force is decomposed into driving force and side force. The lift plays the main role of the driving force in closehauled condition, and in reaching conditions, the drag plays the main role in the driving force.
Then, the aim of a sail trimmer is to increase the lift and decrease the drag in closehauled condition (and inversely in reaching) to maximise the driving force.
The flow around 2D sail sections corresponds to the case of an infinite mast height (with an infinite luff) in a steady flow, where the wind blows horizontally with the same strength and direction at each altitude. These ideal 2D considerations give an approximation of the flow around theoretical sail and therefore must be refined to more precisely model the complex features of 3D flow.
In three dimensions, two additional vortices appear at the sail foot and head: the air particles slide from the high pressures of the intrados to the low pressures of the extrados. The sliding of particles generates these tip vortices. As we can see in the Figure bolow, the magnitude of the vortices is lower for high aspect ratio sails.
The aspect ratio is a nondimensional quantity defined by the ratio of the sail span² to the sail surface area:
The tip vortices create the main part of the sail drag: the induced drag. The induced drag depends on the sail lift according to the following relation:
If the sail creates no lift, then there will be no pressure differences between the intrados and the extrados, and then there will be no tip vortices, and finally no induced drag.
The drag of a sail (or any appendices: daggerboard, rudder, fin, etc) can be split into:
· The form drag, always present, whatever the angle of attack of the sail. This drag depends only on the shape of the sail.
· The induced drag, depends on the sail aspect ratio, and it is proportional to the lift² for low angles of attack.
· The separation drag, is due to the flow separation. This drag appears for large angles of attack, when influenced by the trailing edge vortices.
Moreover, the sliding of the air particles from high pressures (intrados) to low pressures (extrados) at the tips of the sail creates some crossed flows, which tend to twist the streamlines after the leech.
In true sailing conditions, when the wind blows above the sea, a speed gradient develops because of the air viscosity. Indeed, by continuity, the particles of air in contact with the particles of water acquire the same velocity (at the sea surface level). In the lower part, the atmospheric boundary layer is the part of the atmosphere where the wind speed varies with altitude. In the upper part, the wind speed has a constant velocity. The atmospheric boundary layer thickness depends mainly on the wind strength, and the sea state. More generally, a boundary layer expands each time a fluid moves above a body (or another fluid) of different velocity.
The difference between the true wind speed vector and the boat speed vector is what we call the apparent wind speed vector: .
Because of the velocity gradient (seen above), the velocity and the angle of incidence of the apparent wind vary with altitude, this is the apparent wind twist.
To study the aerodynamics of a sail, one can build a model (sail(s), mast(s), hull(s)) and mount it on a turntable in a wind tunnel. A turntable is a structure composed of force blocs (dynamometers) which measure the forces and moments in the 3 directions x, y, and z. A wind tunnel can be simplified as a fan, which blows some wind in a tunnel section (where the model is mounted).
The dimensions of the tunnel limit the size of the model. For example, in the low speed section of the University of Southampton (3.7m width by 4.6m height), the length of the hull is limited to 1.4m and the mast to 2.2m. The model must be relatively distant from the ceiling and the walls to avoid the interaction with the sail wake.
Comment: There are only few wind tunnels, which take into account the apparent wind gradient and twist (Auckland and Valence by example).
The main sail is trimmed via a hoist that links the traveller and the boom. For each angle of attack studied, the trimmer tries to find the optimum combination of traveller and mainsheet, which will give the optimum angle of boom and sail twist. The main sheet and the traveller are trimmer with small winches.
The wind tunnel results are generally analysed with diagrams of lift versus drag for different apparent wind angles. These graphs are called sail polar diagrams. With this tool, many sails can be compared, and the optimum navigation point can be defined.
For example on this polar diagram (From The AeroHydrodynamics of Sailing, International Marine Publishing Company, ISBN : 0229986528 by C. A. Marchaj), two characteristic points are to be taken into account.
Point 1 is the optimum in term of ratio . It is the point where the sail is the most "efficient".
When the incidence angle increases, the lift increases to point 2 where the driving force is maximum.
When one moves from point 1 to point 2, the side force increases and so does the drag force (principally the induced drag)
Which point is then the optimum for this apparent wind angle: β ?
First, point 2 (biggest driving force) seems to produce the maximum boat speed. It is not necessarily true, because at point 2, the side force (and then the drag force) is more important than at point 1. The optimum then lies on the blue curve between point 1 and 2.
The optimum point is close to point 1 in closehauled conditions and close to point 2 in reaching conditions where the point 2 is closed to the maximum lift force.
Nowadays, sail makers tend to adopt the square head sail shape (IMOCA, ORMA, Class America, Mini 6.5, etc.).
There are many pros and cons concerning these two types of sails:
· The surface area of a square head sail will be bigger than a roached sail for the same mast length (luff), but the aspect ratio will be smaller,
· The square head allows a better control of the main sail twist, and selfregulates the sail shape during gusts,
· The head of a square head sail is more tolerant for small angles of attack, and then produces less induced drag,
· The square head gives a better aerodynamic efficiency in the upper part where the wind is stronger (velocity gradient), the square head does not increase the lift but reduces the drag,
· It is impossible to have a standing backstay with a square head sail, then one must use some running backstays which complicate the manoeuvres,
· For the same surface area, a square head sail will have a smaller mast than a roached sail, therefore the centre of gravity of the rigging, and the centre of effort of the sail will be lower, which increases the lateral stability of the boat,
· A square head sail is generally more difficult to hoist and to flake, because of the top battens,
· ...
The sail's polar diagrams are a powerful tool to compare sails (alone), nevertheless, it is necessary to appreciate the whole boat behaviour, from the aerodynamic of the sails, as well as from the hydrodynamics of the hull. The VPP (velocity prediction program) are codes that take into account the performances of sail(s) (sail polar diagram) and hull(s) (resistance versus speed curve), and the boat stability, to simulate the behaviour of the boat for different true wind angle, and true wind speed. The following diagram is a typical boat polar diagram produced by a VPP (the scale of the grid pattern is in knots)
The experimental tests are relatively expensive and time consuming. The numerical approaches tend to replace the experimental tests because they are more profitable in terms of cost and time length. The results of numerical simulations become more and more accurate but hardly as precise as the experimental ones. Some aspects of flows remain relatively difficult to model, such as turbulence, recirculation, boundary layer thickness, etc. The Computational Fluid Dynamics (CFD) is a very powerful tool, which needs a lot of experience. The CFD codes can be divided into two main groups: the RANS codes and the panel codes.
The NavierStokes equations govern the movements of fluid particles.
· The first is the continuity equation, it ensures that for each element of volume considered, there will be as many particles going in as particles goinig out (in steady state).
· The three next equations (momentum equations) ensure that: (Newton's second law) in the three directions of space for the fluid volume considered.
· The last equation ensures that the quantity of energy going in and out of the volume considered is equal (in steady state). It is the same principle as for the continuity equation.
It is relatively difficult to solve the NavierStokes equations directly (DNS: direct numerical simulation) because this method implies a very fine definition of the volumes (fine meshing) and then longer calculations. The actual computing tools are not powerful enough to solve these problems directly in a reasonable amount of time. The RANS method (ReynoldsAveraged NavierStokes) has been developed to solve the NavierStokes equations more rapidly, in defining each variable of the fluid by its average and fluctuating part.
The ReynoldsAveraged NavierStokes equations are detailed in the APPENDIX. This method solves fluid problems more rapidly but induces more variables than equations. The system of equations formed is then "unclosed" and a turbulence modelling is necessary to "close" the system, and then solve the equations. There are many types of turbulence modelling, adapted to different fluid mechanics problems.
First, a control volume must be defined around the body to be studied, and then this volume is discretised into cells. In this example, the control volume has the same cross section as the wind tunnel; generally, the control volumes are larger to ensure that the walls will not have any influences on the calculations.
The RANS equations are solved for each cell, thus the size of the cells must be adapted to the phenomenon to be observed. If the velocity (or pressure) gradients are low in the zone to be observed, then a coarse mesh will be sufficient, but at the boundary layer for example, and where the velocity (or pressure) gradients are more significant, the mesh must be particularly refined.
The following Figures present the velocity contour around three profiles of different camber position & magnitude, for angles of incidence between 2.5° and 12.5°. These Figures have been obtained with a RANS method (ANSYS CFX 11.0). The blue, green and red zones represent the low speed, far field speed and high speed respectively.
α 
f/c=1/27; m/c=1/2 
f/c=1/7; m/c=1/2 
f/c=1/7; m/c=1/3 
2.5 



5 



7.5 



10 



12.5 



On the above Figures, the wind blows horizontally and the profiles rotate around the mast.
These Figures show that the flow detaches more rapidly for low cambered profiles. The flow is also more disturbed when the camber is positioned forward, the profile with an important camber positioned in the middle of the chord seems to be the most efficient.
The flow can also be visualised with streamlines; it is the tack followed by the air particles.
Thanks to the RANS method, one can obtain the velocity distribution above the sail surface, and verify that the flow is mainly horizontal, except at the foot and the head of the sail. One can also observe the zones of recirculation on the extrados, on the following Figure, the recirculation zone extends particularly, because the sail trimming was not optimised.
Thanks to the pressure contour on intrados and extrados, one can see that the suction on the extrados is more significant than the overpressure on the intrados; the sail is then mainly sucked on the leeside more than pushed on the weather side.
With RANS codes, one can plot the 3D streamlines around the sail. The following Figures show the magnitude of the vortices at the head and the foot of the sail. It also shows the twisting of the streamlines at the middle of the leech.
Nevertheless, the lift and drag forces integrated in the sail area remain the main interest of the aerodynamician. The values of these forces are calculated with RANS codes.
The potential flow method is a simplification of NavierStokes equations. With this method, the fluid is assumed inviscid, incompressible, and the flow is assumed irrotational. In fact, the viscous effects take place mainly in the sail boundary layer, which is a small layer a few millimetres thick around the sail. For low wind speed (less than velocity of sound), the air is assumed to be incompressible.
With the potential flow method, the wall of the sail is not directly modelled. The shape of the flow around the wall is created in superposing a straight flow with some local flows. These local flows are modelled by singularities: source, sink, vortex, or doublet. These singularities are illustrated on the following Figure for the case of 2D flows.
· The source is a point from which fluid issues uniformly in all directions. Superposing a straight flow and a source gives an upwind stagnation point, and the flow follows a shape of nose cone from this stagnation point.
· The sink (negative source) is a point towards which fluid converge uniformly in all directions. Superposing a straight flow and a sink gives a downwind stagnation point.
· The doublet is obtained in superposing a source and a sink slightly aft of the source.
Superposing a straight flow with a doublet gives an upwind and a downwind stagnation point, the flow follows a disc shape.
· A vortex is a rotating flow around a point (whose speed is proportional to the inverse of the distance to this point), superposing a straight flow and a vortex gives a flow around a rotating disc. This flow creates a lifting force (perpendicular to the flow) because the particles are accelerated on one side and decelerated on the other side, it is the phenomenon observed around a sail.
To model the flow around the sail, the sail area and the wake must be discretised. Singularities are then distributed on these surfaces.
The values of forces and pressure on each facet but also the total forces values can be obtained with panel codes.
Finally, which of these three methods is the most appropriate to estimate sail's performances?
Wind tunnel testing gives the most accurate results, but the experiments are long to settle and the use of a wind tunnel and the model manufacture are relatively expensive.
The panel codes (potential flow) are relatively rapid but less accurate than wind tunnel because of the assumptions made on the fluid and the flow.
The RANS codes are generally more accurate than the potential flow codes, but much longer.
For the two numerical methods mentioned, the computing time depends essentially on the complexity of the problem to solve and the number of cells or facets used. These three methods are synthesized on the following diagram:
The last breakthroughs in numerical modelling and computational time make the numerical method nearly as accurate as experiments; nevertheless, every method must be validated with an experiment before any industrial use. The numerical methods remain a very good comparison tool, and are intensively used in the early stages of design where many sails have to be compared in a limited amount of time. In the last stages of sail design, the few remaining sails are studied more precisely in wind tunnel.
APPENDIX ReynoldsAveraged NavierStokes Equations
Continuity Equation
, this equations ensure mass conservation in space and time.
x momentum Equation
y momentum Equation
z momentum Equation
In these equations u=U+u', v=V+v', and w=W+w'.
Scalar Equation:
Here Γ is the circulation. The scalar S represents: the kinetic energy k, the dissipation rate ε or the enthalpy h, with s=S+s'. The additional terms, S_{u}, S_{v}, S_{w} and S_{s} in the momentum equations and in the scalar equation represent the transfer of momentum due to turbulent fluctuation.
Nomenclature
Latin letters
f 
Sail camber 
m 
Sail camber position 
P  Pressure 
s 
Scalar for the NavierStokes equations 
s' 
Fluctuating part of scalar s 
S 
Time averaged part of scalar s 
S_{s} 
Source term in the scalar equation 
S_{u} 
Source term in the xmomentum equation 
S_{v} 
Source term in the ymomentum equation 
S_{w} 
Source term in the zmomentum equation 
t 
Time (s) 
u 
x component of velocity (m/s) 
u' 
Fluctuating part of velocity u (m/s) 
U 
Time average part of velocity u (m/s) 
v 
x component of velocity (m/s) 
v' 
Fluctuating part of velocity v (m/s) 
V 
Time average part of velocity v (m/s) 
V 
Velocity vector (m/s) 
w 
x component of velocity (m/s) 
w' 
Fluctuating part of velocity w (m/s) 
W 
Time average part of velocity w (m/s) 
x 
Coordinate in the stream wise direction (m) 
y 
Coordinate in lateral direction (m) 
z 
Coordinate in vertical direction (m) 
Greek letters
α 
Angle of attack of wind to sail section profile (deg) 
β 
Apparent wind angle (deg) 
Γ 
Circulation 
ε 
Turbulent dissipation rate (m^{2}/s^{3}) 
λ 
Leeway angle (deg) 
μ_{D} 
Dynamic viscosity (N s/m^{2}) 
μ_{T} 
Turbulent viscosity (N s/m^{2}) 
ν 
Kinematic viscosity (m^{2}/s) 
ρ 
Density of fluid (kg/m^{3}) 
ω 
Turbulence frequency, ε/k 
Glossary
CAD 
Computer Aided Design 
CFD 
Computational Fluid Dynamic 
FEA 
Finite Element Analysis 
FSI 
Fluid/Structure Interaction 
IMOCA 
International 60 feet Monohull Class Association 
ORMA 
Ocean Racing Multihull Association 
RANS 
ReynoldsAveraged NavierStokes 