By David Vacanti

In the design of a sailboat keel, it is not possible simply to write down all the optimum proportions and draw the keel accordingly. A designer must cope with several limitations, and he or she is not always free to use those optimum values. The greatest concerns, in addition to generating side force, are (1) to provide sufficient volume for ballast, (2) to locate properly the center of lateral resistance to balance the helm, and (3) to position the ballast in the keel to achieve level trim fore and aft. Also, the area of the keel must be just large enough to balance sail forces and must provide low ballast placement for good stability. Often, rating rules, cruising requirements, or moorage depth limitations limit maximum draft of the keel.

In this article, while operating with these limitations, we shall see how various keel planform factors affect overall keel and boat performance. We will be using data compiled from a computer program written with the aid of a government aerodynamic design document that is called Datcom. I'll spare you the details of this program, but those who wish to tinker with numbers can refer to the list of equations in Table 1, which are accurate for many keel design applications. I used a computer program with more complex equations to make it as accurate as possible.

The computer program is currently used in Robert Perry's design office, so I'll use a few of his designs to illustrate various keel features. Table 2 shows the calculations made for Meridian, a 70-foot ultralight-displacement boat (ULDB). The calculations let us observe the forces on the keel and its efficiency at various angles of attack, or leeway. The side force, or lift, is measured in pounds and represents the hydrodynamic power of the keel to resist the sail forces at a given speed V and leeway angle ct (alpha). Drag forces, also computed in pounds, can be thought of as pulling from the keel's rear toward the stem.

Just as forward thrust and side force from the sails depend on wind speed, the keel's angle of leeway depends on the speed of the boat. For example, let's assume that the 70-footer is going 10 knots, as shown in Table 2. Now ignore the hull's own ability to resist side forces and concentrate only on the keel. If the wind in the sails is producing 3,387 pounds of side force, the keel must be at an angle of attack that generates an equal and opposite force of 3,387 pounds. The lift-force column in Table 2 shows us that at 4.0 degrees of leeway the keel develops the required 3,387 pounds of lift.

On a boat that sails slower for the same 3,387 pounds of wind, or sail side force, the lift produced by the keel would be much lower, since the lift force depends on the square of the boat speed V. If the speed were 9.5 knots, the lift force at 4 degrees leeway would be 3,063 pounds using the lift force equation, Table 1, and numbers from Table 2, 2.85 X 0.196 X 60.77 X 9.5 X 9.5 = 3,063 pounds. That amounts to a loss of 324 pounds of lift, nearly a 10 percent reduction. A little calculation tells us that the boat would have to sail at 4.42 degrees of leeway in order to balance the wind forces.

If, on the other hand, the keel was more efficiently designed (that is, without some of the limitations I mentioned at the outset), it might produce more lift at 4.0 degrees of leeway and sufficient lift at a lower angle of leeway, allowing the boat to point higher.

Assuming that we could make changes in our 70-footer's keel, how could we tell at a glance that the changes would improve its efficiency? This is where the lift-to-drag ratio, or L/D, comes into play. L/D numbers tell us how many pounds of lift force the keel generates for every pound of drag it produces.

Notice in Table 2 that Meridian's keel's L/D changes as a function of leeway angle. This illustrates how every keel achieves its highest efficiency at a particular leeway angle and does less well at other angles. The trick is to design a keel, within a given set of size limitations that will produce its highest L/D, or efficiency, over the range of leeway angles at which the boat will sail.

We have laid an effective groundwork to evaluate efficiency. The next step is to show how the various parameters of a keel affect the outcome. There are five major parameters that affect not only efficiency but other characteristics as well, such as stall.

They are:

The first three parameters are called planform characteristics, as they relate primarily to the looks of the keel as it is viewed in side plan or profile-that is, broadside and standing vertically. While we will consider these planform characteristics one at a time, bear in mind that they influence one another in actual design. The last two parameters are foil characteristics; they will be covered in a second article.

Before we dive into the charts and graphs that follow, there remain a few more definitions to be made clear. First, let's look at aspect ratio. In our computer program the effective aerodynamic aspect ratio is calculated by dividing twice the foil's span by the mean chord (the average of the keel's root. and tip chords). The formula 2W in Table 1 takes into consideration the "mirroring" effect of the water surface on the keel's performance. For everyday comparisons, though, geometric aspect ratio can be simply measured as keel height divided by the mean chord, without using the factor of 2. In the strict scientific sense the effective aspect ratio is defined as the aspect ratio multiplied by the efficiency factor e. This little e is a very tough value to calculate; it involves all the other planform characteristics. You may set e equal to 0.75 as an average value. You will see it used in some complex calculations later on.

Taper ratio is defined as the tip chord divided by the root chord, or ct/ cr. It is referred to as the variable X.

Leading-edge angle (ALE) is Simply the angle the forward edge of the keel makes with a vertical line. Designers may prefer to take the. sweep angle from one-quarter of the way aft on the keel's chord (AC/4), but Datcom uses the leading edge, so we shall, too.

NACA foil section comes from the National Advisory Committee for Aeronautics (NACA). In the 1930s NACA was established to further the study of aircraft technology. During its tenure extensive studies of airfoil shapes were compiled and published in the well-known Theory of Wing Sections (Ira Abbott and Albert von Doenhoff, Dover Books, New York). NACA was disbanded about 1960 and replaced by NASA, the National Aeronautics and Space Administration. NASA, unlike its predecessor, has not compiled extensive files of airfoil shapes that are also readily available to the public, but several do exist in special reports. Foils used for sailboat keels often come from NACA 63, 64, 65, or 00 series.

Complicating the definitions of the lift coefficient CL (and also of the drag coefficient CD mentioned below) are multiple definitions. In the case of the lift coefficient CL, we must distinguish between the airfoil lift coefficient and the keel lift coefficient. The airfoil lift coefficient is a value measured by NACA in a wind tunnel in which both ends of the wing being tested were sealed by the wind tunnel walls. This is the theoretical maximum value that is rarely, if ever, achieved in practice. This value is used to help calculate the actual keel lift coefficient, which has only one end sealed by the hull of the boat and the other end open.

The drag coefficient CD is made up of two parts: the friction drag and the induced, or vortex, drag. The value measured by NACA and shown in its tables and graphs is the friction drag only. No induced drag is measured in the wind tunnel since the tunnel walls seal the ends of the wing section, thereby preventing any fluid from escaping around the wing tip. Induced drag is present on a keel because water escapes from the leeward (high-pressure) side to the windward (low-pressure) side of the keel at the open tip. The overall drag coefficient is computed in Table 1, with friction equal to Cd. and the remaining terms equal to the induced drag.

It is not possible simply to look up the drag and lift coefficients of a given keel configuration in Theory of Wing Sections. The values listed there pertain to the airfoil shape themselves and to no keel shape in particular. The lift and drag coefficients for every keel result from the combination of the 3 planform characteristics outlined above, with the chosen foil shape, and may be calculated with the equations in Table 1.

Having struggled at length with the definitions of all the keel parameters, we are at last free to explore the effects of the planform characteristics on the lift and drag of the keel. Aspect ratio is perhaps talked about the most of all of the planform characteristics. It provides a means of telling the designer how many times taller a keel is than it is wide. Generally speaking, a high-aspect-ratio (tall and thin) keel is more efficient than a low aspect-ratio (short and squat) keel.

Notice in Figure 1 that the keel lift-curve slope (lift coefficient per degree of leeway) increases along the vertical axis with increasing aspect ratio. The lift-curve slope is reduced significantly, however, when the leading edge of a high-aspect-ratio keel is swept back. Note that a keel of aspect ratio 2.5 loses more than 5.0 percent of its lift when swept back 30 degrees, but when a keel with an aspect ratio of 0.5 is swept even 60 degrees, there is virtually no loss of lift. High-aspect ratio keels are less forgiving of design compromise and of sailing inefficiency.

In Table 3 we see the effects of sweep angle on three recent sailboat designs. The 70-foot ULDB in the first column has the highest aspect ratio keel. The design sweep angle is 30 degrees, but had it not been swept, the L/D ratio would have increased 0.2. L/ft2 would have increased 0.8 pounds, and the leeway would be reduced 0.28 degrees, as seen in Delta Alpha..

The Fryer 42 keel in the second column is of moderate aspect ratio and shows improvement with reduced sweep angle as well. The Newton 52, however, has a low-aspect-ratio keel, and its efficiency is actually reduced by lower sweep angle, as it drops from an L/D of 12.1 to one of 11.8 when sweep is eliminated.

These results support the data presented by Pierre De Saix in SAIL several years ago ("Yacht Keels, an Experimental Study," May 1974). Tank tests showed that highest efficiency and lowest drag were achieved when low-aspect-ratio keels were highly swept back and high-aspect ratio keels were left unswept. A designer may still choose a sweep angle because he believes it will shed kelp or to achieve a particular location for the center of gravity, however.

Taper ratio plays an important role in keel efficiency. It is possible to choose the ratio of keel tip length to keel root length so that the minimum possible induced, or vortex, drag is produced for that keel design. This minimum induced drag condition is theoretically reached under the following conditions. When Aerodynamicists analyze a fin keel, they calculate the lift produced by small sections of the keel distributed along the keel span. The lift force produced by each section of the keel can be plotted on a chart like the one in Figure 2. If the plotted points fall along a curve in the shape of an ellipse, the keel is said to be elliptically loaded and will have a minimum induced drag for that design. There are some "elliptical" keels currently in vogue, but the adjective frequently refers to the keel's physical profile, not the loading pattern.

In order to minimize the induced drag, a designer can refer to a curve like the one shown in Figure 3. This curve tells him that there is a taper ratio that corresponds to the sweep angle he has chosen that will yield elliptical loading, or minimum induced drag. It's important to remember that the minimum drag achieved by this method may be greater or less than the drag obtained for another keel of the same area or aspect ratio but of different planform or NACA section. The curve will only insure that for a given keel configuration the minimum drag is being produced. If you are familiar with the 12Meter Australia II's forward-swept keel, you will notice from Figure 3 that forward sweep requires an inverted taper keel for minimum drag. For example, a keel with 30 degrees of forward sweep requires a taper ratio of 1.4, yielding a keel whose tip is 40 percent longer than the root.

So far we have indicated that high aspect-ratio keels are very desirable from the standpoint of efficiency, but as in so many areas, too much of a good thing can have poor results. Figure 4 shows that as aspect ratio increases, the leeway angle at which a keel or rudder stalls decreases. When a keel stalls, water will separate from some portion of the keel's area, resulting in a loss of lift and a large increase in drag. Typically, though, the angle where stall occurs is much larger than the normal range of sailing leeway angles.

However, Figure 4 also shows that as a keel is increasingly swept, either forward or aft, stall will occur at constantly smaller angles of leeway. Theory indicates that in general forward-swept keels will stall or develop turbulence near the root first, and aft swept keels will stall near the tip.