Board vs. Wind Speed
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Board Speed vs. Wind Velocity
Contributed by Dr.Peter I Somlo
A theoretical solution of the numerous parametric equations which determine speed as a function of course and wind velocity is of great interest to the technical windsurfing enthusiast. Dr. Peter Somlo has developed such a solution utilizing the aeodynamic and fluid dynamic relatonships for lift and drag of the sail, fin and board over a range of courses to the true wind and at two wind velocities. Peter's solution is presented below for true winds of 12 and 15 knots and a sail size of 7.0 sqm.
The calculation was performed for the board-speed (vector) as follows:
1. Set up the equations for sail-lift, sail & board-drag as functions of air temperature, air pressure, sail size, wind-velocity and board-velocity. Here we must remember that sail-lift and sail-drag depend on the velocity of the apparent wind (which we don't know yet because it depends on the board-speed) and the board-drag depends on the board-speed (not known).
2. Set up an equation for the apparent wind, which is a vectorial subtraction of the wind vector minus the board-speed vector. (The faster the board, the more the apparent wind comes from the front.) To make the vectorial calculations simple, all vectors were represented as complex numbers (which have magnitudes and angles).
3. Assume a sheeting-in angle which will be used to modify the angle of the sail-lift.
4. Assume that the sail-lift is at right angles to the sail direction.
5. Break this sail-lift force into two components: in the direction of the board and right angles to it.
6. Chose a direction of board-movement (relative to the true wind direction).
7. Realising that when moving at a steady speed, the forward-force must equal the backward drag (otherwise the board would be speeding up or slowing down), solve the equation(s) for the board-velocity vector - for the condition of the net forward-force to be zero - yielding the board-velocity vector, and so all the forces will become known.
a. Although the computation was carried out in SI units, speeds were converted to knots (from m/s) and forces were converted in results to kilogram-force (from N) - more familiar units.
b. An artificial function was devised to increase the hull-drag smoothly by a factor of 3 at the planing transition as the board-speed drops from 12 to 10 knots . (The choice of 11 knots was the result of a calculated hull-speed-limit of a commonly used board.) The factor of 3 came from estimating the reduction of the wetted surface when planing properly - or not.
c. The 'Forces' diagram shows the total sail-lift. The fin-lift is the component which is at right angles to the board, and the forward-force is the component in the forward direction. We see that most of the sail-lift is transferred to the fin (which is preventing the board going sideways), and the forward-force is a fraction of the sail lift only. For a 7m^2 sail, reaching in a 15 knot wind, the sail-lift is about 68 kgf (so you can hang most of your weight on the sail), but the force that makes you go forward is only about 11 kgf. The curves are not quite applicable for sailing downwind (square running at 180 degrees) because the transfer of lifting force to drag force only at square running is unknown, i.e. there is no sail-lift, only drag.
d. The Cartesian graph for board-speed shows two wind-speeds 12 and 15 knots, for a 7m^2 sail. Note the 'kinks' in the curves indicating the transition to planing above 12 knots.
e. The polar plot is the curve re-plotted for the 15 knots case, with laser-gun measurements by Ken Winner, technical editor of American Windsurfer magazine superimposed. It can be seen that the curves are very similar, differing mainly in scale (Ken suggested to reduce the drag coefficients.) However, at close-hauled sailing there may be another reason for the difference. If a competitor is asked to see how fast can he/she sail up-wind, they will bear off first to get on the plane, and then sail upwind. The computer program does not accomodate this behaviour. In the 10 to 12 knots regime the results are multi-valued, and one can chose which root to find: the planing one or the non-planing one. Below 11 knots the non-planing one was chosen.
For more info contact:
Dr.Peter I Somlo
Microwave Consultant | email: firstname.lastname@example.org