The purpose of a velocity prediction program is to predict boat speed for given set of conditions which include the wind speed and the direction of boat movement with respect to the wind direction. For upwind sailing, this direction will be slightly to leeward of the boat centerline by what is termed the leeway (or yaw) angle. The leeway angle is necessary to provide lift on the keel.
The physics of this velocity prediction are governed by Newton’s second law. That is, the sum of the forces acting on the boat is equal to the mass of the boat times its acceleration. If we limit ourselves to the case of constant speed and direction, then the equation reduces to the sum of the forces equals zero. This is a vector equation and we choose to balance the components parallel and normal to the direction of motion. In other words, the aerodynamic driving force on the sails is balanced by the hydrodynamic drag from the underbody and keel, and the aerodynamic heeling force is balanced by the lift force of the keel. A more extensive analysis would also include righting moments of the keel and sail plan, but for our purposes we will assume the boat is sailed upright.
Applying Newton’s second law we end up with two equations (resulting from the force balance parallel and normal to the direction of motion) and two unknowns, which are the speed and leeway angle. However, given that the system of equations is nonlinear, no closed form solutions are available and a numerical iterative approach must be taken. The remainder of this post looks at the assumptions and solution procedures involved in my simplified VPP available for download.
In essence, the accuracy of the prediction depends on how accurately the forces acting on the sail and keel may be predicted. However, before we get to that a quick discussion of the force components acting on the sail is necessary. In particular, the traditional means of decomposing the force on a wing is to look at components parallel and perpendicular to the freestream. The component parallel to the freestream is the drag, and the perpendicular component is the lift. These vectors are shown in the VPP in red. However, for our purposes it is preferable to decompose the force vector into components in the direction of movement through the water, and perpendicular to that direction. These vectors are shown in the VPP in green. The component in the direction of motion is the driving component, whereas the component perpendicular to this represents what would lead to a heeling moment. These components are then parallel to the drag and lift components, respectively, acting on the keel. This facilitates the balance of force calculations that result in the boat speed and leeway angles.
Unfortunately, the forces acting on the sail are difficult to compute. The most accurate method currently used involves the solution to the three-dimensional Reynolds-averaged Navier-Stokes equations. Much too complex for our purposes. What is done in the VPP is much simpler. The user specifies the sail lift slope, sail area, sail aspect ratio, Oswald efficiency factor, and zero lift parasitic drag coefficient. The sail lift slope is used to compute the lift coefficient. In particular, the lift coefficient is equal to the lift slope times the angle of attack. Given the lift coefficient, the actual lift force may then computed from the relation
where is the lift coefficient, is the air density, V is the apparent wind speed, and A is the sail area.
Given the lift coefficient, the drag coefficient is then computed from the relation
where e is the Oswald efficiency factor which for an airplane usually varies between 0.7 and 0.85. AR is the aspect ratio which, for a triangular shape sail, is defined as twice the luff length divided by the foot length. represents a “parasitic” drag coefficient at zero lift.
The force vector on the keel is decomposed into lift and drag components which are perpendicular and parallel to the direction of motion, respectively. The computation of lift and drag coefficients follows the approach taken for the sail.
The unknowns are the leeway angle and the boat speed. Increasing the leeway angle increases both the lift and drag on the keel, as does increasing the boat speed. At some combination of boat speed and leeway angle, the drag on the keel balances with the driving force on the sail, and the lift on the keel balances with the side force on the sail. At this point, we have a solution. Readjusting the wind speed and/or direction, and the main sheet angle alters this force balance resulting in a new boat speed and new leeway angle.
The VPP outputs a number of results just below the graphics plot showing the force decomposition on the sail, and actual and apparent wind directions. (Note vectors for the keel are not shown, but are equal in magnitude and opposite in direction to the green colored vectors representing driving heeling (or side) forces.) For reasonable results the sail lift coefficient should be about one. Since the VPP does not take wave drag into account, the boat speeds are likely to be high. The program will also let the user know if the main sheet angle settings are likely to result in stall or luffing. Since the equations are nonlinear and the solution follows an iterative procedure, it is possible that the numerical algorithm may diverge. This occurrence is noted on the vector plot should it occur.
The program was designed to let users experiment with several basic parameters to see their influence on boat performance. Two of the most important parameters are the sail and keel lift slopes. The higher the value, the better the performance, but it is up to the designer and sailor to maximize these values. Let me know if you have any questions…