## Velocity Prediction Program Description

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
$F_L=C_L \: 0.5 \rho V^2 A$
where $C_L$ is the lift coefficient, $\rho$ 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
$C_D=C_{D,0}+\frac{C_L^2}{\pi\:e\:AR}$
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. $C_{D0}$ 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…

## How does a sail generate lift?

Despite the vast quantities of literature in the aeronautical field, and the original work of Arvel Gentry in the 1970’s, there is still a misunderstanding among sailing enthusiasts when it comes to the mechanism by which a sail generates lift. Most sailors refer to Bernoulli’s equation which states that the sum of the flow, kinetic, and potential energies is constant along a streamline.  Then, as the flow accelerates around the leeward side of the sail, the pressure decreases and lift is generated.  However, Bernoulli’s equation alone is not sufficient to explain lift on the sail. The real question is, what is the physical mechanism for the higher velocity on the suction side of the sail?  This is where misunderstandings may still persist.

For example, and quite surprisingly, the following statement was included in material handed out during a day-long seminar on spinnaker handling I attended about 6 years ago: “Because the wing is curved on the top, the air that follows this route has a longer way to go; therefore it must travel faster to meet the air on the bottom.”  A sketch was included that showed that the path along the windward side of a sail is shorter that that over the leeward side.  Clearly, this is not the case as the sail material is so thin that it represents not much more than a discontinuity in the flow field. Hence the distance from the leading to trailing edge (luff to leech) over the windward and leeward sides of the sail is nearly identical.  Nevertheless, a sail generates lift.

Another explanation for the higher velocities on the leeward side of the sail is simply that this is the solution to the governing equations of conservation of mass and momentum, which for a Newtonian fluid are known as the Navier-Stokes equations.  However, most do not find that to be a satisfying explanation.  However, if we make some simplifying assumptions, the problem becomes much more tractable.  In particular, let us assume that the effects of viscosity are limited to a very thin region along the sail, known as a boundary layer.  Outside this region viscous forces are quite small and hence the effects of viscosity can be neglected.  Given this simplification, the nonlinear Navier-Stokes equations can be replaced by what is commonly called the potential flow equation in fluid dynamics literature.  The major advantage here is that the potential flow equation is linear, and hence solutions can be superimposed.  That is, if phi1 is a solution to the problem, and phi2 is also a solution, then phi3=phi1+phi2 is also a solution.

How does this apply to our sail problem?  Let’s consider a horizontal section out of the sail such that we have an airfoil of zero thickness, with or without camber, set at a small angle to the approach flow.  It turns out that there are an infinite number of solutions for the potential flow about this airfoil.  However, observation indicates that for an airfoil at a small angle of incidence, the flow leaves a cusped trailing edge tangentially with the velocity of the fluid leaving the upper surface matching the velocity leaving the lower surface.  If the velocities are equal, then Bernoulli’s equation reveals that the pressures are also equal.  This observation, when enforced as part of our potential flow solution, is known as the Kutta condition.  What the Kutta condition does is add sufficient “circulation” about the airfoil such that the flow leaves tangentially from the trailing edge.  Consequently, we can consider the potential flow about the airfoil to consist of two parts that may be summed, a flow without circulation (that provides no lift) and a circulatory flow (that provides lift).

This idea of circulation about an airfoil can be further understood by considering an airfoil, initially stationary, subsequently translated at a constant speed.  As the airfoil motion is initiated, a boundary layer begins to form.  This boundary layer attempts to curl around the sharp trailing edge of the airfoil.  However, this fluid movement is not sustainable, and the airfoil ends up peeling off, or shedding, a circulatory “starting” vortex from the trailing edge.  However, initially there existed no circulation about the airfoil.  As a consequence of a theorem by Kelvin, the total circulation in an inviscid region surrounding the airfoil is preserved.  The circulation was zero before motion began, and hence the shedding of a starting vortex requires the development of circulation equal in magnitude but opposite in sign.  This is the circulatory flow about the airfoil, which, if the flow were left-to-right, would be in a clockwise direction.  Again, the addition of this circulatory field to the zero circulation potential flow solution provides a result which satisfies the Kutta condition and moves the rear stagnation point that occurs on an airfoil with zero circulation to the trailing edge.

A couple of codes are available to help with these concepts.  First, a Potential Flow Sail Analysis code is available that solves for the two-dimensional flow about a main/jib combination, or a mainsail alone.  One may plot velocity vectors of the complete flow, the zero circulation flow, and the circulatory flow.  To begin with, I recommend looking at the mainsail alone, in the form of a flat plate at small angle-of-attack.  Observe the location of the stagnation points near the leading and trailing edges.  Then plot the circulatory velocity vectors that provide sufficient circulation to move the stagnation point such that the flow leaves parallel to the trailing edge (not a stagnation point since the upper and lower surface flows are parallel to each other).

To observe the circulatory flow in an experimental setting, I recommend that students view the “Bathtub” video.”  This video shows the formation of a starting vortex, and the circulatory flow about the airfoil.  The final scene in the video shows the two counter-rotating vortices; one as the starting vortex, and another resulting from the circulatory motion about the airfoil (once the airfoil is lifted from the water).

## Update Potential Flow Sail Analysis Code

An error in the circulation velocity vector plot was fixed.  Added the option to plot flow about sail(s) with zero circulation.  This permits one to observe both the zero circulation flow vectors (no lift) and the circulation flow (sufficient to satisfy the Kutta condition).  Added together they comprise the potential flow solution one is interested in. Download potential flow sail analysis code.

## Zero Gaussian Curvature Sail Analysis Code

Zero Gaussian Curvature Sail Analysis

The Windows-based code displays Tight Luff, Tight Leech, and Perfect Blend sail shapes as described in the text “Physics of Sailing” by John Kimball.  The sails have zero Gaussian curvature, which means they are “developable” surfaces.  Section lift coefficients from foot to head are computed using a two-dimensional, higher-order panel method.

## Sailing Velocity Prediction Program

A basic velocity prediction program (VPP) has been released.  The code was developed as a teaching tool to help beginning sailors understand the influence of a number of parameters on boat speed and performance. The code is compiled for the Windows operating system and available under the Computer Codes link.  For further information the user should refer to the Help menu within the code.

## Potential Flow Solver for Sail Analysis

The sail analysis code runs under Windows and is available on the Computer Codes page. The code solves the two-dimensional potential flow equation using a vortex panel method in which the vortex strength varies linearly along the panel and is continuous from one panel to the next. In addition, the code permits the specification of either rigid or flexible sails.  The intended use of the code is to help users understand the fluid flow interaction between a mainsail and a jib. Note that since this is a potential flow solution, drag cannot be predicted.